CONTENTS EDITORIAL TEAM EUROPEAN MATHEMATICAL SOCIETY

EDITOR-IN-CHIEF ROBIN WILSON Department of Pure Mathematics The Open University Milton Keynes MK7 6AA, UK e-mail: [email protected] ASSOCIATE EDITORS STEEN MARKVORSEN Department of Mathematics Technical University of Denmark Building 303 NEWSLETTER No. 38 DK-2800 Kgs. Lyngby, Denmark e-mail: [email protected] December 2000 KRZYSZTOF CIESIELSKI Mathematics Institute Jagiellonian University EMS News: Reymonta 4 Agenda, Editorial, Edinburgh Summer School, London meeting ...... 3 30-059 Kraków, Poland e-mail: [email protected] KATHLEEN QUINN Joint AMS-Scandinavia Meeting ...... 11 The Open University [address as above] e-mail: [email protected] The World Mathematical Year in Europe ...... 12 SPECIALIST EDITORS INTERVIEWS The Pre-history of the EMS ...... 14 Steen Markvorsen [address as above] SOCIETIES Krzysztof Ciesielski [address as above] Interview with Sir Roger Penrose ...... 17 EDUCATION Vinicio Villani Interview with Vadim G. Vizing ...... 22 Dipartimento di Matematica Via Bounarotti, 2 56127 Pisa, Italy 2000 Anniversaries: John Napier (1550-1617) ...... 24 e-mail: [email protected] MATHEMATICAL PROBLEMS Societies: L’Unione Matematica Italiana ...... 26 Paul Jainta Werkvolkstr. 10 D-91126 Schwabach, Germany The Price Spiral of Mathematics Journals ...... 29 e-mail: [email protected] ANNIVERSARIES Digital Models and Computer Assisted Proofs ...... 30 June Barrow-Green and Jeremy Gray Open University [address as above] e-mail: [email protected] Forthcoming Conferences ...... 31 and [email protected] and CONFERENCES Recent Books ...... 35 Kathleen Quinn [address as above] RECENT BOOKS Designed and printed by Armstrong Press Limited Ivan Netuka and Vladimir Sou³ek Crosshouse Road, Southampton, Hampshire SO14 5GZ, UK Mathematical Institute telephone: (+44) 23 8033 3132 fax: (+44) 23 8033 3134 Charles University Published by European Mathematical Society Sokolovská 83 18600 Prague, Czech Republic ISSN 1027 - 488X e-mail: [email protected] and [email protected] NOTICE FOR MATHEMATICAL SOCIETIES ADVERTISING OFFICER Labels for the next issue will be prepared during the second half of February 2001. Vivette Girault Please send your updated lists before then to Ms Tuulikki Mäkeläinen, Department of Laboratoire d’Analyse Numérique Mathematics, P.O. Box 4, FIN-00014 University of Helsinki, Finland; e-mail: [email protected] Boite Courrier 187, Université Pierre et Marie Curie, 4 Place Jussieu INSTITUTIONAL SUBSCRIPTIONS FOR THE EMS NEWSLETTER 75252 Paris Cedex 05, France Institutes and libraries can order the EMS Newsletter by mail from the EMS Secretariat, e-mail: [email protected] Department of Mathematics, P. O. Box 4, FI-00014 University of Helsinki, Finland, or by e- OPEN UNIVERSITY mail: ([email protected]). Please include the name and full address (with postal code), tele- PRODUCTION TEAM phone and fax number (with country code) and e-mail address. The annual subscription fee Liz Scarna, Barbara Maenhaut (including mailing) is 60 euros; an invoice will be sent with a sample copy of the Newsletter.

EMS December 2000 1 EMS NEWS EMS Committee EMS Agenda EXECUTIVE COMMITTEE PRESIDENT (1999–2002) 2001 Prof. ROLF JELTSCH Seminar for Applied Mathematics 15 February ETH, CH-8092 Zürich, Switzerland Deadline for submission of material for the March issue of the EMS Newsletter e-mail: [email protected] Contact: Robin Wilson, e-mail: [email protected] VICE-PRESIDENTS Prof. ANDRZEJ PELCZAR (1997–2000) 10-11 March Institute of Mathematics Jagellonian University Executive Committee Meeting in Kaiserslautern (Germany) at the invitation Raymonta 4 of the Fraunhofer-Institut für Techno- und Wirtschafts Mathematik PL-30-059 Krakow, Poland e-mail: [email protected] 4-6 May Prof. LUC LEMAIRE (1999–2002) EMS Workshop, Applied Mathematics in Europe, Berlingen, Switzerland Department of Mathematics Université Libre de Bruxelles Contact: R. Jeltsch, e-mail: [email protected] C.P. 218 – Campus Plaine Bld du Triomphe 11-12 May B-1050 Bruxelles, Belgium EMS working group on Reference Levels in Mathematics: e-mail: [email protected] Conference on Mathematics at Age 16 in Europe (venue to be announced) SECRETARY (1999–2002) Contact: V. Villani or A. Bodin, Prof. DAVID BRANNAN Department of Pure Mathematics e-mail: [email protected] or [email protected] The Open University Walton Hall 15 May Milton Keynes MK7 6AA, UK Deadline for submission of material for the June issue of the EMS Newsletter e-mail: [email protected] Contact: Robin Wilson, e-mail: [email protected] TREASURER (1999–2002) Prof. OLLI MARTIO Department of Mathematics 9-25 July P.O. Box 4 EMS Summer School at St Petersburg (Russia) FIN-00014 University of Helsinki Asymptotic combinatorics with applications to mathematical physics Finland Organiser: Anatoly Vershik, e-mail: [email protected] e-mail: [email protected] ORDINARY MEMBERS Prof. BODIL BRANNER (1997–2000) 19-31 August Department of Mathematics EMS Summer School at Prague (Czech Republic) Technical University of Denmark Simulation of fluids, and structures interactions Building 303 Organiser: Miloslav Feistauer, e-mail: [email protected] DK-2800 Kgs. Lyngby, Denmark e-mail: [email protected] 24-30 August Prof. DOINA CIORANESCU (1999–2002) Laboratoire d’Analyse Numérique EMS lectures at the University of Malta, in association with the 10th Université Paris VI International Meeting of European Women in Mathematics 4 Place Jussieu Lecturer: Michèle Vergne (Ecole Polytechnique, Palaiseau, France) 75252 Paris Cedex 05, France Title: Convex polytopes e-mail: [email protected] Contact: Dr. Tsou Sheung Tsun, e-mail: [email protected] Prof. RENZO PICCININI (1999–2002) Dipartimento di Matematica e Applicazioni These lectures will also be given at University of Rome, jointly arranged by Università di Milano-Bicocca ‘Tor Vergata’ and ‘Roma Tre’, at dates to be announced. Via Bicocca degli Arcimboldi, 8 Contact: Maria Welleda Baldoni, e-mail: [email protected] 20126 Milano, Italy e-mail: [email protected] 1-2 September Prof. MARTA SANZ-SOLÉ (1997–2000) EMS Executive meeting in Berlin (Germany) Facultat de Matematiques Universitat de Barcelona Gran Via 585 3-6 September E-08007 Barcelona, Spain 1st EMS-SIAM conference, Berlin (Germany) e-mail: [email protected] Organiser: Peter Deuflhard, e-mail: [email protected] Prof. ANATOLY VERSHIK (1997–2000) P.O.M.I., Fontanka 27 30 September 191011 St Petersburg, Russia e-mail: [email protected] Deadline for proposals for 2002 EMS Lectures. EMS SECRETARIAT Contact: David Brannan, e-mail: [email protected] Ms. T. MÄKELÄINEN Department of Mathematics 30 September P.O. Box 4 Deadline for proposals for 2003 EMS Summer Schools FIN-00014 University of Helsinki Finland Contact: Renzo Piccinini, e-mail: [email protected] tel: (+358)-9-1912-2883 fax: (+358)-9-1912-3213 2002 telex: 124690 1-2 June e-mail: [email protected] EMS Council Meeting in Oslo (Norway) website: http://www.emis.de 2 EMS December 2000 EDITORIAL EditorialEditorial Anatoly Vershik (St Petersburg)

Member of EMS Executive Committee (1997-2000)

President of St Petersburg Mathematical Society

The EMS and cooperation in mathematics the internet and electronic media which both positive and negative aspects to the Cooperation in science, and in mathemat- step-by-step are changing the type of coop- internet. To some extent, cooperation and ics in particular, can be discussed from sev- eration between individual mathematicians exchange of information using the internet eral points of view – first as exchange of and has forced a change in the daily behav- deprive us of more vivid forms of commu- scientific information, then as cooperation iour of scientists. nication. But at the same time we have (in in joint research projects or other kinds of Let me give an example. As everybody our country) no other way of finding need- scientific activity (organisation of confer- knows, Russia has fewer computers than ed information, especially because in ences, schools, etc.), and finally as social the West and access to the internet is not recent years there has been a decrease in cooperation between different communi- the subscriptions on our journals, in buy- ties. All three types of cooperation are very ing mathematical books, and so on. Under important for Eastern European countries. these circumstances it is very important for Below I discuss them and give my view on us to have access to MatSciNet, the role of the EMS in encouraging them. Zentralblatt, and other such systems. I have spent many years organising the St Ultimately we will need to use the internet Petersburg Mathematical Society, as a instead of printed matter, because it will be member of the committee (1970-78), as impossible to maintain enough subscrip- Vice-president (1978-97) and as President tions, even for the main journals and (1997-now), so I can present some conclu- books, in our libraries. sions about the role of local societies in the mathematical life in Russia, looking in par- The EMS as a mediator between East ticular at the activities of the Moscow and and West St Petersburg Mathematical Societies. The creation of the EMS in the early 1990s Mathematical work is mainly very indi- has had several consequences, especially vidual (in contrast to the experimental and for Eastern Europe after the collapse of the technical sciences), so the tendency to sep- Soviet block. The original goal was to aration is rather strong. The principal role establish an organisation that could unify of the local mathematical societies in cooperation between mathematical com- Russia is to provide regular meetings with munities in the different European coun- interesting talks, discussions, information, common. Nevertheless, these new forms of tries. etc. In mathematics departments in the communication between people have I think that it is wrong to compare the West (for example, in the US), part of this raised all discussions on the transmission EMS with the American Mathematical role is played by regular colloquiums, of scientific and administrative informa- Society, because their functions and roles because most mathematicians are tion to a completely new level. In our are very different. First of all, unlike the employed in the universities. In contrast, mathematical society, almost all of the 350 European situation, the US has no local for many reasons, professional mathemati- members use the internet, and so all infor- (state) mathematical societies. In contrast, cians in Russia were (and are) dispersed mation about meetings, special and regu- Russia has no national mathematical soci- over various institutes and centres, some- lar lectures, new books, jobs, prizes, prob- ety (although we tried to organise one in times without any mathematical environ- lems, and even discussions on special areas the 1980s!), but we have about ten local ment. So the meetings of local mathemati- (such as education) can be propagated by ones (Moscow, St Petersburg, Kazan’, cal societies play the role of scientific cen- e-mail or the web. The role of local mathe- Voronez, Niznii Novgorod, Ural, tres, and provide almost the only opportu- matical societies now becomes a little dif- Novosibirsk, etc.). Also each new state, nity for mutual discussions on mathematics ferent from before. The societies now have such as the Baltic states Ukraina, and the place for exchanging information. new duties – to present information about Belorussia, Armenia and Georgia, has at Another role of mathematical societies is mathematical life to individual members, least one mathematical society. Some of publishing activity: some of our societies to help them avoid long trips on the inter- these are now very active, while others have have their own journals or proceedings. To net in order to find links, web-pages, etc., many difficulties but try to keep going. my mind this is important, but less so than and to support fast contact with other soci- Cooperation between former Soviet the reason above, because there are many eties. Union states, as well as cooperation with journals and many possibilities for publica- As an aside, I believe that the EMS still other countries, now helps with these prob- tion. A much more important function of does not fully use the possibilities of the lems. I believe that one of the essential local societies is to represent its communi- internet. Doing so could help to solve some roles of the EMS is to assist former Iron ties in other scientific organisations and problems that have appeared with individ- Curtain mathematical communities to international societies. ual members, or to discuss other urgent become incorporated in the European and The last has a direct connection with the questions. One could even vote on issues World mathematical communities. role of the EMS and other international via the internet. The EMS’s web-page is This is not only a question of financial organisations, but before I discuss this con- still too short. help – indeed, I don’t even think that it’s nection I want to emphasise the new role of As with all kinds of progress, there are the main thing. A more delicate problem EMS December 2000 3 EDITORIAL is to mediate with European organisations visas for visits to other countries. I under- absence of such special programmes for and other communities. In order to do stand that this is a question for bureaucra- short visits by young mathematicians is a this, it is important to understand better cy at the very highest level, but my impres- major deficiency of our interrelations. We the situation in the scientific life of Eastern sion is that the scientific community can at must try to correct this – for example, we Europe. I will mention at least two serious least raise the question. The procedures recently held a special conference for current problems: scientific cooperation for obtaining visas to many countries is young mathematicians from Moscow, St and the survival of mathematical schools. very complicated and humiliating, and Petersburg and Stockholm on dynamical It is useful to recall how the activities of remains one of the worst Soviet legacies. systems and combinatorics. It would be most mathematicians were previously sup- Invitations from universities to respectable good if such meetings could become a fre- pressed by official institutions (the prob- scientists (including young ones) must be quent occurrence in Europe. lems of having a job, defending a thesis, given preference and must be freed from In July 2001 we will hold the first travelling abroad, having contact with such procedures. European summer school in Russia, which Western colleagues, and so on), especially But the main problem is still the prob- should provide opportunities for contacts for some categories of mathematicians. lem of how to prevent decay in our mathe- between young mathematicians (see EMS Even admittance to the main mathematical matics. The traditions of the Russian math- Newsletter 37 or the website: http://www. departments was forbidden to many peo- ematical schools are distinguished and dif- dmi.ras.ru/EIMI/2001/emschool/index.html). ple before the 1990s, and few Soviet math- ferent from the West. It is completely At the round table during the Barcelona ematicians were able to participate at wrong to say, as I have heard many times Congress I suggested the establishment of mathematical congresses and conferences: (especially from some former Russians), 30-40 stipends (from UNESCO, the EC, even Fields Medallists and invited speakers that there are now no serious mathemati- UNTAS, etc.) to be awarded to the best were denied permission to go to the ICMs! cians in Russia – we have many outstand- Russian (and other Eastern European) The situation has now changed drastical- ing mathematicians and most seminars graduate students, so that they can spend ly. At the Zurich, Berlin and Barcelona and schools are still active. time in their countries and can devote Congresses there were hundreds of partic- But what is true is that we are in a criti- themselves to mathematics for 2-3 years, ipants from Russia. But we are now faced cal situation, and the essential question is without having to search for a job. In our with new problems. On the one hand there about young mathematicians. Russia had, dramatic situation this gives a chance for are now no serious obstacles to going and still has, an excellent mathematical our mathematics to survive during a diffi- abroad and having contact with colleagues education in the elementary and high cult period. from the West; indeed, many mathemati- schools, and particularly in the special In conclusion, I wish to say that the role cians from Russia and the former Soviet mathematical schools. So we still have of EMS and EC should be more construc- Union, as well as many emigrants of the enough young and talented people who tive in all these aspects. My impression is 1970 and 1980s, now have permanent or want to study mathematics. But the miser- that recently we have concentrated too temporary positions in the West, and this able stipends awarded to students (under- much on technical questions, such as links must simplify and intensify contact and graduate and graduate) as well as some liv- between EMS and other organisations, cooperation. On the other hand the prob- ing difficulties have forced most students institutes, and so on. It is more important lem is how to make this cooperation more who have already finished university either to work with local societies to encourage efficient and, most importantly, how to to drop mathematics for other things contacts and interrelations by organising preserve mathematics in the Eastern coun- (computing or business) or to go abroad. appropriate European conferences. tries. My colleagues and I have received many It is also very important to pay much Moreover, there are now many special letters from the West requesting us to send more attention to the organisation of the grants for Eastern Europe, such as those our former students to other countries for four-yearly European Mathematical organised by the Sorosz Foundation, the graduate school or postdoctoral positions. Congresses. In future, they must be more AMS and Promatematika (France). They Indeed, the students from Russia have a balanced – both geographically, and by were rather small, but well organised. high reputation. subject area – and more original, and must There are also a few local grants in In a sense, the brain drain is a natural provide a real forum for all European Western countries such as Germany and thing. But we must pay attention to the fact mathematicians. It is important to have a Holland which are given to mathemati- that the collapse of Russian mathematics better financial base for the EMS, and I cians from both East and West – a great would be catastrophic for world mathemat- think there are possibilities for this. There and disinterested form of support that has ics. In order to prevent this disaster we is a similar question about the EMS council provided a good illustration of the solidar- need to keep at least some young mathe- meetings which take place each two years. ity of mathematicians. At the same time maticians in our community. If most of They must be more widely based and less there have been many complaints about them leave just after finishing at university, technical; it is better for the EMS Executive the INTAS-system from Brussels; for it is bad for both sides. It is clear why it is Committee to discuss and solve such tech- example, one of the INTAS grants finished bad for the Russian mathematical schools, nical problems previously. two years ago but participants from but it is now clear – and we have some sta- Moscow, St Petersburg and Niznii tistics – that in general they will not stay in Novgorod did not obtain their salaries and mathematics in the West either. They nobody from Brussels answered their e- arrive without sufficient grounding from EMS Committee for mail messages. I think that one of the roles their Russian mathematical school, so they Women and Mathematics of EMS and its East European Committee need to start their education from the is to help with this. It is important to beginning. At the same time difficulties Correction understand that it is still very difficult for arising from their first period abroad The September issue of the EMS Eastern Europeans to communicate with forces many of them to go to computer Newsletter contained a short account of bureaucrats from the EC. centres or banks. the EMS Committee for Women and I believe that there must be more joint There are many solutions to this para- Mathematics. Unfortunately, the research teams in various areas, as well as dox. First, it is possible to establish com- author’s name and e-mail address more visits to Russia from the West; there mon graduate schools – say, between a uni- given there were incorrect, for which are now fewer visits than during the years versity in Russia and another in Germany, the Editor apologises. The contact of stagnation. But during the last decade with two advisors whose areas are close to details of the author and Committee the mathematical community in Russia each other, so that a student can share Chair are: Emilia Mezzetti, and the Eastern European countries has his/her time between the two countries. Dipartimento di Scienze Matematiche, undoubtedly started to return step-by-step Alternatively, one could establish a few Università di Trieste, Via Valerio 12/1, to World community, although they have special sufficiently high stipends for our 34127 Trieste, Italy; e-mail: mezzette@ made only the first few steps. graduate students enabling them to make univ.trieste.it Another very serious problem relates to short visits to a western university. The 4 EMS December 2000 EMS NEWS NewNew MembersMembers ofof thethe ExecutiveExecutive CommitteeCommittee

At the Council Meeting in Barcelona, Victor Buchstaber and Mina Teicher were elected to the Executive Committee, and Marta Sanz- Solé was re-elected. A mini-biography of Marta Sanz-Solé appeared in EMS Newsletter 32; biographies and statemnts of the others appear below. Thanks were given to Andrzej Pelczar and Anatoly Vershik who leave the Committee after several years of service.

Victor M. Buchstaber (e-mail: [email protected]) graduated from the Moscow State University (MSU) in 1969 and went on to postgraduate study there, with advisors Sergei P. Novikov and Dmitri B. Fukhs. He received a Ph.D. in 1970 and a Dr.Sc. in 1984. He has been Research Leader of the Topology Division of the Steklov Mathematical Institute of the Russian Academy of Science, Professor in Higher Geometry and Topology at the MSU, and Head of the Mathematical Modelling Division at the National Scientific and Research Institute for Physical, Technical and Radio-technical Measurements. He has been on the Council of the Moscow Mathematical Society, Deputy Editor- in-Chief of Uspekhi Mat. Nauk, and Head of the Expert Committee in Mathematics in the Russian Foundation for Basic Research.

Statement: I represent the Moscow Mathematical Society (MMS), one of the oldest in Europe (1864). I believe that the EMS must play an important role in developing and deepening the relations between its corporate members, leaning on the best achievements of the national mathematical societies. The eminent achievements of the MMS over the past 60 years have undergone a period of rapid fruitful development with a world-wide reputa- tion. Thinking over the experience of the past, the following approaches to organising the life of a mathematical society seem the most significant: – strong relations between mathematical schools working in different directions, mainte- nance of generation succession in mathematics, and involvement of new young talent; – stimulating interest in modern achievements of mathematics, while nourishing a love for, and respect towards, its classical results. The importance of this approach can be demon- strated by the bright and deep applications of classical Abelian function theory and alge- braic geometry to the top modern problems of mathematical physics; – raising an interest in the sciences related to mathematics, considering them both as spheres of application and as important motive forces and grounds for further development. As a convincing example, ideas from physics, especially quantum field theory, have affected the modern state of mathematics. I see my participation in the work of the EMS Executive Committee, in connection with pro- moting and putting into life these approaches.

Mina Teicher (e-mail: teicher:macs.biu.ac.il) is Chair of the Mathematics and Computer Science Department and Director of the Emmy Noether Research Institute for Mathematics (Minerva Center) at Bar-Ilan University in Ramat- Gan, Israel. She received her Ph.D. from Tel-Aviv University for a thesis entitled ‘Factorization of birational morphisms between 4-folds’. Since then, her research interests have developed into geometry and topology, group theory, artificial vision, and mathematical models in brain research. She has travelled widely, spending the year 1981-82 at the Institute for Advanced Study, Princeton, and paying short-term visits to countries ranging from China, Japan and Tibet, to India and South Africa.

Statement: The EMS should acquire the financial means to enable it to support a broad spectrum of activities on a large scale, and should work to enhance governmental and public attitudes towards mathematics. To achieve this we should: – establish The Society of Friends of the EMS to promote donations, public awareness, government contacts, etc.; – strengthen the scientific relationship with European-based industries and encourage their financial investment in basic research via the EMS (and individual institutions); – convince Ph.D. students to join the EMS; – work, through governmental and other means, to influence the EU to develop new pro- grammes better suited to mathematics and to non-governmental organisations like the EMS. Concerning Mathematics Education, we should work towards: – a unified curriculum based on the current advanced high-school European pro- grammes; – programmes for the identification and education of especially talented high school stu- dents (where they do not already exist), especially in underprivileged regions. EMS December 2000 5 EMS NEWS Executive Committee meeting London, 11-12 November 2000 Tuulikki Mäkeläinen and David Brannan

Present: Rolf Jeltsch (President, in the of the President, Bodil Branner and Bernd bers and others invited to EMS meetings. Chair), David Brannan (Secretary), Olli Wegner had attended a meeting in Lecce, The Treasurer reported that there are Martio (Treasurer), Bodil Branner, Doina Italy, on Information Science and now few complications with collecting cor- Cioranescu, Luc Lemaire, Andrzej Pelczar, Libraries in Mathematics; Luc Lemaire porate member fees, and that the Society

Renzo Piccinini, Marta Sanz-Solé and had attended a celebration of the French now has a well-functioning system for Anatoly Vershik; (by invitation) Victor EMS Prizewinners in Paris; and Jean- invoicing for advertisements in the Buchstaber, Tuulikki Mäkeläinen, David Pierre Bourguignon had represented the Newsletter. Salinger, Mina Teicher and Robin Wilson; EMS at the RPA event on 11 November in and (by invitation to a portion of the meet- Portugal. Membership ing) Chris Lance, Ari Laptev, Anders It was agreed to accept an offer from EMS membership currently stands at around Lindquist, Ulf Persson and Bernd Wegner. Oxford University Press (OUP), who pub- 2000. It was agreed that membership dri- Apologies were received from Carles lish the journal Interfaces and free bound- ves are needed in various countries. Casacuberta. aries, of a discounted price of US$65 Several possible new Corporate Member The President thanked the London instead of US$90, to members of the EMS; applications seemed to be coming into the Mathematical Society for its hospitality. the arrangements will be publicised in the pipeline. The benefits of the EMS having a Newsletter and in EMIS. A section in EMIS contact person with each corporate mem- Officers’ reports will shortly be started outlining EMS mem- ber were also emphasised. The President reported that Volker bers’ membership benefits, such as the The reciprocity agreement with the Mehrmann had moved to the Technical OUP and International Press discount Australian Mathematical Society had now University of Berlin, allowing closer coop- offers. been signed, and it was hoped soon to eration with Bernd Wegner. He had visit- The Secretary reminded Committee agree a reciprocity agreement with the ed Nigeria’s first national mathematical members of the Executive Committee (EC) Canadian Mathematical Society. The possibil- centre in July; IMPA in Rio de Janeiro for that the EMS covers the expenses of EC ity of the EMS forming reciprocity mem- the Latin American Congress; the members and others invited to EMS meet- bership agreements with further societies University of California at Los Angeles for ings, for expenses incurred in connection was also discussed and approved. The the opening of its Institute of Pure and with the meeting if they cannot be covered EMS routinely exchanges the EMS Applied Mathematics; and Budapest for from other sources, but that the local host Newsletter with those of its reciprocity soci- the Rényi Institute celebrations. On behalf society pays all local expenses of EC mem- eties. 6 EMS December 2000 EMS NEWS Prizes for 2004 and later; they were asked to report to the March meeting of the Executive Committee.

Stop Press: It has just been announced that the European Congress in 2004 will be held in Stockholm (Sweden).

Council Meeting in Barcelona on 7-8 July 2000 There was a discussion of various possible changes for the following Council meeting in 2002. Among the topics were: – should elections be held for individual members’ delegates in any case? – should a Committee member (or its Chair) present the report of each EMS Committee, in order to stimulate a dis- cussion on topics of interest to dele- gates? – how could EMS activate people between EMS President Rolf Jeltsch (left) and London Mathematical Society President Martin Taylor meetings? – should delegates be encouraged to start Matters agreed by electronic voting – having an accent on young people; a discussion? since the previous EC Meeting – the differences between an ECM and an – projects where EMS is a partner, like Laurent Guillopé was elected as Chair of ICM, including mini-symposia, round LIMES and EULER (see below), should the Data Base Committee for the period tables, etc.; it was thought that local be presented; in Oslo a presentation on 2001-2004; Christian Houzel was elected organisers should be encouraged to the proposed EMS publishing house was as the EMS representative on the Abel think widely as to the actual format of suggested. bicentennial conference programme committee; the whole event; – highlights of the past two years should the reciprocity agreement with the – the possibility of involving the various be put forward; Australian Mathematical Society was European Union ‘networks’. – of the two days of the Council, perhaps accepted; Bernd Wegner was elected as The Executive Committee wanted one day could be for business matters Chair for the Electronic publishing committee Scientific Committees to choose a wide range and the second day for discussions, or a for the period 2001-2004; and the meeting of topics for speakers; to interest as many seminar for planning the future. Parallel processing and applied mathematics people as possible across both pure and The French delegation to the Barcelona 2001 (PPAM 01) was accepted as an EMS- applied mathematics; and to discuss Council meeting had expressed the wish SIAM satellite meeting. whether the lectures should be shorter for EMS to have more interaction with cor- than in the past, in order to accommodate porate members, and the Italian European Congresses of Mathematicians more lectures and to avoid listeners losing Mathematical Union had also expressed a (ECM) interest after a while. wish for more frequent exchange of infor- A lengthy discussion was held of a possible The Committee agreed that there is a mation. It was agreed to discuss these mat- site for 4ecm, The Fourth European Congress clear need for establishing rules for the ters at Kaiserslautern in spring 2001. of Mathematicians, in summer 2004. It was EMS Prizes for 2004 and later; for the tim- The next Council meeting will be held on hoped to be able to finalise the site selec- ing of the Prize Committee’s activities; for Saturday-Sunday 1-2 June 2002 in Oslo, tion by the end of 2000. It was noted that the working of the Prize Committee and Norway, with the first session starting at 10 the dates of the meeting must be carefully selection of candidates, including the age a.m. on 1 June. coordinated with those of ICME (The 10th limit (currently 32), gender balance, geo- International Congress of Mathematics graphical distribution and definition of Changes of EMS Statutes Education), which will be held in ‘European’ in this context; for the balance The Committee discussed various items of Denmark, in the week of 4-11 July 2004. between pure and applied mathematics; the EMS Statutes and EMS By-laws that There was a brief discussion of the com- and for the call for nominations and seemed to require change, noting that any position and operation of the Scientific timetables. The identity of the Chairman change in the Statutes need the approval Committee, and of how different aspects of the Prizes Committee will be known of the Finnish authorities, but that changes could be taken properly into considera- publicly from the start, and an open invita- in the By-laws do not require such tion. The Executive Committee felt it tion for nominations for prizes will be pub- approval. important that ECMs attract all active licised. Among the topics were: the possibility of mathematicians in Europe, not just the top The Committee expressed its thanks to allowing mathematics departments to mathematicians and new mathematicians. the local organisers of 3ecm, especially become EMS members; the notion of a Various ideas to improve the working of Marta Sanz-Solé, for an efficient and President-elect and a Past President; the the various committees set up for the friendly organisation of the 3ecm in need for gender and geographic and pure- Congress by the EMS were discussed, and Barcelona in July 2000. The first volume of applied balance in the EMS; the idea of the importance of close collaboration the Proceedings will include the plenary Officers having 2-year terms, not 4-year between local mathematicians and the var- lectures, section lectures, mini-symposia terms; whether the President needed to be ious committees was emphasised. and presentations of the prizewinners. The a Council delegate; how to expel EMS Among the topics raised in the interest- second volume will consist of material from members who do not pay their dues; allow- ing discussion of a site for 4ecm were: the round tables, including contributions ing reciprocity membership; the possibility – why the ECM should be held in a partic- from panellists and discussions. The of joining EMS via the EMS-Zentralblatt ular location; Proceedings are planned to come out in scheme; and omitting Articles 5.10 and – what was the purpose of the ECMs?; the first part of 2001. 5.8. – possible benefits to the local mathemat- David Brannan and Mina Teicher were Andrzej Pelczar, David Brannan, Olli ics community; appointed as an ad hoc Committee to pre- Martio and Mina Teicher were elected to – the possibility of many satellite meet- pare a set of rules and a schedule for the an ad hoc committee to formulate the ings; operation of the Prize Committee for EMS changes needed to the EMS Statutes.

EMS December 2000 7 EMS NEWS EMS Projects tance of the Zentralblatt/MATH database as logues using a common metadata profile The Committee decided to organise a a Large European Infrastructure. method for providing a homogeneous meeting of its member societies (especially The LIMES project [Large Infrastructure access to heterogeneous resources. The those with a strong interest in applied in Mathematics – Enhanced Services: for project has developed a metadata maker mathematics), applied mathematics soci- details, see EMS Newsletter 37 or the web- with a de-duplication facility, and has test- eties outside EMS, European Union math- site www.emis.de/projects/LIMES] started ed a beta version. The project had received ematics networks, and some influential officially in April 2000, and a meeting had very high marks from its reviewers. individual European applied mathemati- been held to divide the tasks: data The partners in the Euler Project were: The cians in spring 2001 to increase the visibil- improvement, input structure and national State Library of Lower Saxony and the ity and acceptance of EMS among the applied access nodes. The EMS is a supervising University Library of Göttingen; the J. mathematics community, to involve them in body for the project. The director of the Hadamard Library, University of Orsay; shaping future EMS policy, and to help project is Michael Jost; Bernd Wegner and the Centrum voor Wiskunde en make them feel at home within the EMS. Rolf Jeltsch are the Scientific Directors. Informatica library, Amsterdam; the The EMS had received encouragement The partners are: FIZ Karlsruhe University of Florence; the library of the from several sources for the creation of a (Zentralblatt-MATH, Berlin) (Coordinator); Institut de Recherche Mathématique publishing house and preparations had pro- Cellule de Coordination Documentaire Avancée, University of Strasbourg; ceeded both by e-mail and at meetings in Nationale pour les Mathématiques; NetLab, the Research and Development Zurich and London. The Committee Eidetica; Coordinamento SIBA, Università Department at Lund University Library; decided that a foundation should be creat- degli Studi di Lecce; Danmarks Tekniske MathDoc Cell, Grenoble; FIZ Karlsruhe; ed to be the legal owner of the publishing Videncenter & Bibliotek; Universidade de Zentralblatt für Mathematik; EMS; and the house, called the European Mathematical Santiago de Compostela; Hellenic Department of Mathematics of the Foundation, ‘EMF’, with its seat in Mathematical Society; Technische Technical University of Berlin. Switzerland. Swiss law places no restric- Universität Berlin; and the European The EULER Project had formally termi- tions on the nationalities of the persons Mathematical Society. There would be a nated in September 2000. The Executive involved; in Switzerland a foundation can workshop in December for the partners Committee felt that there is a need for a have tax-free status; it will be a non-profit and editorial units, and a later meeting in product like EULER; that EULER pro- organisation; and the Statutes must be Copenhagen would be held in April 2001. vides good tools; that effort is needed for accepted by the Swiss authorities. The The 2000 Mathematics Subject further development – e.g., to provide Publishing House will be a legal body sep- Classification is a joint project of Zentralblatt searchable data and to become more user arate from the EMS. The EMS logo will be and Mathematical Reviews. It was agreed friendly; and it decided that the EMS used for the EMF, but inserting the abbre- that the EMS should be one owner of the should join the consortium to continue viation EMF instead of EMS. copyright to the classification, the other work on the EULER Project – as a ‘spon- The Committee decided to commit being the American Mathematical Society. soring partner’, rather than a source of 10000 euros to be the founding capital of The reason for ownership of the copyright manpower or finance. the European Mathematical Foundation. having to be made clear was in order to It was reported that the EU-funded The tasks of the EMF will be to establish avoid abuse of the classification, not in any Reference Levels Project will have a final and run the publishing house; any surplus way to limit its free usage. meeting in in May 2001 (a report of the could be used to support the work of the The President reported that he had working group was nearly ready), and that EMS. signed the papers for EMS involvement in the contract on TOME [Test of Rolf Jeltsch and Jean-Pierre Zentralblatt for Didactics of Mathematics. Mathematics for Everybody] would be Bourguignon had attended the meeting of Bernd Wegner made a brief presenta- signed on 16 November 2000. the Zentralblatt Consultative Committee in tion to the Executive Committee of the Berlin at the end of October. It was noted EULER Project (European Libraries and EMS Committees that management of the subscribers’ list Electronic Resources in Mathematical It was decided that the Electronic Publishing has now been moved to the editorial office, Sciences – for details, see the website Committee will be chaired by Bernd Wegner and that the price of Zentralblatt is below www.emis.de/projects/EULER) and gave a in 2001-2004; and that the other members the price of Mathematical Reviews. The demonstration of EULER during the lunch of the committee will be: Slawomir Cynk, Jahrbuch project has now a coverage of break. Its current server is in Göttingen. Its Laura Fainsilber, Aviezri Fraenkel, Eva 70%, of which 40% has been edited by purpose is to provide access via a search Bayer-Fluckiger, Laurent Guillopé, Hvedri experts so far. engine (‘EULER’) to access various web Inassaridze, Michael Jost, Jerry L. Kazdan, It was agreed to send a paper to EU com- resources, including OPAC, databases, Volker Mehrmann, Peter Michor, Andrew missioner M. Busquin describing the impor- preprints, e-journals, and WWW cata- Odlyzko, Colin Rourke, Laurent Siebenmann, Jan Slovak and David Wilkins. The committee will be responsible for all aspects of electronic publication, and will develop a new remit. The composition of the Education Committee will be discussed at the March meeting of the Executive Committee. It was reported that Laurent Guillopé had accepted to serve as chair to the Database Committee, with the term 2001- 2004. The Executive Committee agreed that the other members of the Database Committee should be: Francisco Marcell’an, Alberto Marini, Steen Markvorsen, Peter Michor, Marek Niezgodka and Bernd Wegner. It was agreed to invite ERCOM members [Committee on European Research Centres of Mathematics] to write on their web home pages that they are members of the EMS, to use the EMS logo there too, and to keep the EMS fully informed of dis- Bernd Wegner addresses the Committee on the Euler project cussions between themselves and the EU. 8 EMS December 2000 EMS NEWS The purpose of ERCOM is to enable mem- and Coding) were accepted. ber institutions to discuss matters of mutu- The composition of the al interest, to enable them to approach Summer Schools Committee was funding bodies (such as the European recalled as: R. Piccinini Union) with a wider scientific and geo- (Chair 2000-2003), C. Broto, graphic base than individual institutions C. Casacuberta, D. Cioran- can, to facilitate the exchange of informa- escu and R. Fritsch; M. tion, etc. Teicher was added to this It was agreed to add Georg Bock, Tsou list. Sheung Tsun and Doina Cioranescu to the membership of the Committee on Developing EMS Lectures Countries. It was agreed to invite It was agreed to add George Jaiani and Michèle Vergne to give the Victor Buchstaber to the membership of EMS Lectures in 2001, possi- the Committee on Support for Eastern bly in Rome and Malta. European Mathematicians (CSEEM). The The lecture notes of Nigel annual budget of the CSEEM was 10000 Cutland (1997 EMS Lecturer) euros, and they had supported around on Loeb Measures in Practice: thirty mathematicians in 2000. It was felt Recent Advances will appear that there was a continuing need to soon as Lecture Notes in improve contact between the committee Mathematics 1751, published and mathematicians in Russia – but the by Springer-Verlag. total resource was of necessity limited. The idea that EMS member societies should Relations with various help to disseminate information on the institutions and organisa- Committee’s activities was welcomed. tions The Committee received a report on a It was reported that the meeting for Large Infrastructures, held in Venice office of UNESCO has Strasbourg, and noted that mathematics is awarded the EMS a grant of included in six different programmes, US$25000 to support various going across several DGs. It considered a EMS activities in 2000. set of possible future projects with the Plans are in hand for the European Union, including discussion of joint EMS-SIAM Conference to the Sixth Framework Programme. be held in Berlin on 2-6 It was agreed that a draft of the Executive September 2001; full details Title page of Nigel Cutland’s book (see EMS Lectures, above) Committee agenda should be circulated will appear in the EMS beforehand to committee chairs, with an Newsletter. There will be a reduction in the ings. The article that David Salinger had invitation to them to suggest agenda items conference fee for the meeting for EMS written for the LMS newsletter should be and supply discussion papers – possibly members. The International Conference sent to all EMS corporate members, for around three weeks ahead of an EC meet- on Stochastic Programming in Berlin on them to adapt to local situations. ing; and that Committee Chairs should be 25-31 August 2001 (with about 150 partic- At the GAMM annual meeting in Zurich added to the Newsletter mailing list if they ipants) was awarded the status of a satellite on 12-15 February 2001, EMS will share a are not already EMS individual members. meeting of the conference. booth with Zentralblatt. The Committee received a report on the Diderot Mathematical Forums (DMF) activities of the Banach International Center EMS Newsletter The Committee held a general discussion from its representatives on the Center’s The contents of the Newsletter were of its Diderot Mathematical Forum pro- Scientific Committee: F. Hirzebruch; M. applauded by the Committee. gramme, including items such as whether Sanz-Solé; D. Wallace (1998-2001). There had been repeated requests to they actually worked well, whether it would The Society had been informed of the have the Newsletters on EMIS. The articles be easier to set them up with only two intention of establishing an Institute for of the 1999 issues of the Newsletter would simultaneous sites rather than three, the Scientific Information at the University of shortly be sent as text files to EMIS. critical dependence on local organisers, Osnabrück, devoted to mathematics-related The Committee noted that the question and the need for the dates/locations of activities, especially in support of the raised at the Barcelona Council meeting DMFs to be advertised well and ahead of MPRESS project. The institute is to have about the uneven distribution of the book time. Plans for the Fifth Diderot both institutional and individual members. reviews of different publishers was being Mathematical Forum, probably on It was decided that the EMS should be studied by the Editor-in-Chief. Telecommunications, were moving ahead. involved in the plans for the Institute, and that Rolf Jeltsch should attend the found- Future meetings of the Executive Summer Schools ing meeting of the Institute on 30 Committee The EMS has two summer schools planned November 2000, if possible. The following outline schedule was for 2001. The Prague Summer School has Rolf Jeltsch was appointed as the EMS approved: received funding from the European representative to ICIAM [International 10-11 March 2001: ITWN Kaiserslautern Science Foundation. The St Petersburg Council of Industrial and Applied 1-2 September 2001: Berlin, prior to the Summer School (which will be held at the Mathematics] during 2000-2003. EMS-SIAM Conference. Euler Institute) has received support from The Publicity Officer reported on the suc- the US National Science Foundation, the cess of the EMS booth at 3ecm in American Mathematical Society, and Barcelona, commenting that the practice CNRS (France). AMS cooperation in gain- of sharing a booth with Zentralblatt should Reciprocity arrangement ing support swiftly and smoothly from NSF be repeated; the Executive Committee Following the reciprocity arrangement for the summer school had been much thanked Tuulikki Makelainen and Mrs signed in July with the American appreciated. The EMS had given a 5000 Martio for their invaluable efforts in Mathematical Society (see EMS euro guarantee to the St Petersburg staffing the EMS booth in Barcelona. It was Newsletter 37, page 8), a further reci- Summer School. agreed to produce a number of high-qual- procity arrangement was signed in Applications for Summer Schools in ity posters with the EMS logo, for decora- Shanghai on 21 October 2000 with the 2002 in Brasov and in Israel (on Geometry tion of EMS booths at various future meet- Australian Mathematical Society. EMS December 2000 9 EMS NEWS EMSEMS SummerSummer SchoolSchool inin EdinburEdinburghgh Erkki Somersalo

The European Mathematical Society conductivity of a body by injecting electric for the choice of the summer school topic Summer School on New Geometric and currents into the body and measuring the was to bridge the gap between the realms Analytic Methods in Inverse Problems was held voltages at the surface. In 1980, Alberto of pure and applied mathematics. It is vital in Edinburgh, Scotland, from 24 July to 2 Calderón published a groundbreaking for the high quality of mathematical August. It was combined with a meeting on article in which he formulated the mathe- research in Europe that the young genera- Recent Developments in the Wave Field and matical problem, and since then, consider- tion of applied mathematicians have the Diffuse Tomographic Inverse Problems, from 3-5 August. Both meetings were supported by the European Commission and the London Mathematical Society. The Organising Committee consisted of Professors Yaroslav Kurylev (Lough- borough University, UK), Brian Sleeman (University of Leeds, UK) and Erkki Somersalo (Helsinki University of Technology, Finland). The conference was organised in collaboration with ICMS at Heriot-Watt University.

Why geometry and analysis? Inverse problems constitute an active and increasing field of applied mathematics. Roughly speaking, in inverse problems the aim is to retrieve information of inaccessi- ble quantities based on indirect observa- tions. A typical inverse problem is an inverse boundary value problem of a par- tial differential equation, where the objec- tive is to reconstruct the unknown coeffi- cient functions of the equation in a Heriot-Watt University lecture theatre domain, based on a knowledge of the boundary values of its solutions. able progress has been achieved in the most advanced tools at their disposal; it is Application areas of such problems include mathematical research of this problem. equally important for pure mathematicians medical imaging, geophysical sounding Despite the efforts, several aspects of this to have some insight into the possibilities and remote sensing. problem are still open, in particular when in applied areas of their research. The whole area of inverse problems is the conductivity is allowed to be anisotrop- far too wide to be covered in any single ic (direction dependent). It turns out that Participants summer school or meeting. In the present the anisotropic problem can essentially be The summer school lecturers were one, the focus was on modern geometric rephrased in terms of differential geome- Professors Victor Isakov (University of and analytic methods applied to inverse try: can one reconstruct the Riemannian Kansas, USA), Dmitrii Burago (Penn State, problems. The role of differential geome- metric of a manifold from the knowledge USA), Vladimir Sharafutdinov (Novosi- try in inverse problems is becoming of the Cauchy data of the Laplace-Beltrami birsk, Russia), Lassi Päivärinta (University increasingly significant as more complex operator? This rephrasing, of course, of Oulu, Finland), Anders Melin systems are studied. brings the well-developed machinery of (University of Lund, Sweden), Gunther To get an idea, one can consider the Riemannian geometry to our disposal. Uhlmann (University of Washington, inverse conductivity problem. In physical Currently, new ideas and techniques are USA), Alexander Kachalov (Steklov terms, the goal is to reconstruct the electric sought in the direction of differential Institute, St. Petersburg, Russia) and Dr geometry to treat this and other anisotrop- Matti Lassas (University of Helsinki, ic inverse problems. Finland). The participants were mostly The need for new ideas coming from graduate students and young researchers harmonic analysis and control theory is from the EU area and associated countries. also recognised among the inverse prob- The summer school and the conference lems community. Almost every boundary also attracted a number of first-rate measurement carries inherently a bound- researchers on inverse problems from all ary control problem: inversion techniques over the world, the total number of partic- often rely on ideas such as focusing of ipants amounting to over 70. The idea waves or, more generally, sounding by behind arranging the summer school waves of prescribed form. The need to together with a conference was to give the recover discontinuities and other singular- students and young researchers a view of ities of the coefficient functions requires how the newly acquired ideas work in cur- techniques for treating partial differential rent mathematical research. The written equations with non-smooth coefficients. material of the summer school, as well as a These are only a few of the problems of selection of the invited talks, will be pub- interest in this field. The emphasis of the lished as lecture notes. EMS Summer School programme was on harmonic analysis and control theory of Erkki Somersalo teaches at the Institute of partial differential equations. Mathematics, Helsinki University of Erkki Somersalo and Jari Kaipio One of the most important motivations Technology. 10 EMS December 2000 NEWS Joint 11th Update AMS-Scandinavia ECMI Conference on meeting (ECMI2000) Vincenzo Capasso, President of ECMI 3ecm June 2000 ECMI2000, the 11th Conference of ECMI (The European Consortium for Hans J. Munkholm Mathematics in Industry) was held in Barcelona Palermo (Torre Normanna) from 26 to 30 Chair of the Organising September 2000. There were about 270 The Proceedings of 3ecm were distributed to participants from 24 countries, with more participants in preliminary form as a com- Committee than 200 speakers, including nine plenary pact disc, and will be published in two vol- lectures, 37 mini-symposia (many organ- umes by Birkhäuser in its Progress in The first joint meeting of the American ised in collaboration with industry) and Mathematics series. Articles are also accessi- Mathematical Society and the about 50 contributed talks. About 10% of ble as ‘pdf’ files on the 3ecm website Mathematical Societies of Denmark, the participants were industrial delegates. (www.iec.es/3ecm). There will also be a Finland, Iceland, Norway and Sweden took Everybody felt that an ECMI conference book of Proceedings of the Round Tables, edit- place in Odense, Denmark, from 13-16 provides an excellent forum for discussing ed jointly by the Catalan Mathematical June 2000. It was simultaneously the twen- the role of mathematics in (and for) indus- Society and CIMNE. The videos exhibited ty-third in a sequence of Scandinavian try at a European level. during the Congress are distributed by Congresses of Mathematicians which start- As usual, the conference was organised Springer-Verlag in their VideoMath series, ed in Stockholm in 1909. around industrial themes, as they are the and are also available in DVD format. Plenary lectures were delivered by ECMI’s Special Interest Groups. The focus The earlier list of Invited Lecturers omit- Tobias Colding (New York), Nigel J. topics of the conference were micro-elec- ted Nicolas Burq: Lower bounds for shape res- Hitchin (Oxford), Johan Håstad tronics, glass, polymers, composite materi- onance width of Schrödinger operators. (Stockholm), Elliott Lieb (Princeton), als, fuel pipelines, finance, biomedical, The full list of participants at the mini- Pertti Mattila (Jyväskylä), Curtis McMullen ecosystems, multi-body dynamics, informa- symposium on Mathematical Finance: Theory (Harvard), Alexei Rudakov (Trondheim), tion and communication technologies, and Practice (Chair: Hélyette Geman) was: Karen K. Uhlenbeck (Austin) and Dan- automatic differentiation and sensitivity M. A. H. Dempster, Stanley Pliska, Dilip analysis, scientific computing and visuali- Madan, Ernst Eberlein, Tomas Bjork, Ton sation. Most of the mini-symposia showed Vorst, Ezra Nahum and Rainer Schobel. unique examples of scientific coordination The talk by Charles H. Bennett in the and collaboration at a European level for mini-symposium on Quantum Computing high quality research within the ECMI was held by video-conference. The talk by Special Interest Groups. Selected papers Hans Föllmer had the title Probabilistic will appear in the newly established ECMI aspects of financial risk. subseries of Springer volumes on The Chair of the round table How to Mathematics in Industry. increase public awareness of mathematics was At the conference all appeared happy and Vagn Lundsgaard Hansen; sadly, Felipe enthusiastic; encouraged by the excellent Mellizo died the week before the Congress. environmental setting, clusters of people Jean-Pierre Bourguignon was a panellist at could often be seen discussing future col- the last round table, not Rolf Jeltsch. laboration. Particularly relevant was the Rafael de la Llave should be deleted from large participation of young scientists, the list of panellists of the round table most of whom had been involved in the Building networks of cooperation in mathemat- ECMI’s educational activities – in particu- ics. He was in fact the Chair of the round lar, the modelling weeks that have greatly table The impact of new technologies on math- contributed to establishing long-lasting ematical research, in which Bruno bridges between European students. Most Buchberger was a panellist. of those who had participated in these modelling weeks were among the speakers and mini-symposium organisers. Highlights of the conference included the Journal of the ‘Alan Tayler lecture’ delivered by Helmut Neunzert, the ‘Wacker prize lecture’ deliv- European Virgil Voiculescu (Berkeley). In addition ered by Carl F. Stein, a student from there were more than 100 lectures in Gotheborg, and the ECMI honorary mem- Mathematical twelve special sessions. bership offered to Carlo Cercignani. Titles and abstracts for (almost) all the A delicious 9-course social dinner was Society lectures are available through the confer- organised in a princely palace in Palermo, ence home page: http://www.imada.ou.dk/$\ after an excursion to the marvellous cathe- Volume 3, Number 1 of JEMS contained: sim$hjm/AMS.Scand.2000.html dral of Monreale, where a fusion of D. Mucci, A characterization of graphs which There were 269 registered participants, Byzantine, Arab and Norman architecture can be approximated in area by smooth graphs roughly 25% of whom came from the shows how Sicily was one of the fundamen- G. Bouchitté and G. Buttazzo, United States. Another 25% were Danes, tal melting pots of modern Europe. Characterization of optimal shapes and masses while the rest of Scandinavia accounted for The next ECMI conference will take place through Monge- Kantorovich equation 19%. The remaining 31% represented in 2002 in Latvia; information about it will M. Harris and A. J. Scholl, A note on trilin- twenty different countries, with France and be available on the ECMI web page: ear forms for reducible representations and Germany as the major contributors. http://www.ecmi.dk. Beilinson’s conjectures

EMS December 2000 11 WMY NEWS TheThe WWorldorld MathematicalMathematical YYearear inin EurEuropeope Vagn Lundsgaard Hansen (Chair of the EMS committee for the WMY) with the assistance of Ronnie Brown and Mireille Chaleyat-Maurel

The World Mathematical Year is coming to an idea of bringing Europeans and Arabs Sweden and the UK. Mathematical lec- end and it is time to look back and ask ourselves: together in the old city with the famous tures for the general public have been pre- what did we accomplish? what did we learn? Alhambra, castle of the Moorish kings, to sented in all countries, and there have how should we proceed? The celebration of the discuss the historical perspectives of both been many mathematical articles in news- mathematical year has taken place all around cultures to our present mathematical papers and magazines. knowledge. It was a magnificent confer- It has been interesting to observe the ence, and included a visit to the Alhambra, strongly varying degree to which it has guided by the Spanish mathematician been possible to catch the interest of radio Rafael Pérez-Gómez who in the mid-1980s and television in various European coun- established that all the seventeen planar tries. In most countries it has been very dif- crystallographic groups are represented in ficult indeed – in fact close to impossible – the fascinating tilings at the Alhambra. with a few notable exceptions, like France. It might be useful, although difficult, to A short list of accomplishments study at the European level whether there In almost all European countries at least is a connection between the general status one poster has been produced, motivated of mathematics in society and schools in by the WMY, giving suitable links to places the various countries and the willingness to where further information about the year can be found. In several countries a series of posters were produced, usually based on ideas submitted for the poster competi- tion, arranged by the EMS and collected in the ems-gallery at http://www.mat.dtu.dk/ ems-gallery/. Series of posters have been WMY2000 stamp produced in Belgium, Portugal, Spain, from Monaco Italy, France, Germany, the UK, and other countries, while sets of postcards based on the globe, for the language of mathematics is ideas from the EMS poster competition common to all peoples, and mathematics is inde- were produced in Denmark, France and pendent of nations, religions and races. For Germany. Stamps related to WMY 2000 good measure, it should therefore be said that were issued in the following European present mathematics in the media. How when I focus this article mainly on what has countries: Belgium, Croatia, the Czech can we otherwise explain the rather large happened in Europe, my intention is not to Republic, Hungary, Italy, Luxembourg, variations? neglect the rest of the world but only to find a Monaco, Slovakia, Spain and Sweden. way of selecting events where European mathe- Mathematical exhibitions and workshops Two projects funded by the European maticians have had the most direct opportunity have been presented in Denmark, Finland, Commission to influence what has taken place during the France, Germany, Italy, Portugal, Spain, To help in raising public awareness of sci- year. ence and technology in Europe, the European Commission has funded a pro- Conferences dedicated to the WMY posal in mathematics. The contract has During the year 2000, several internation- three partners: the EMS, represented by its al conferences have been dedicated to the WMY-committee, with a coordinating role, WMY. In most cases, the conferences and actual deliveries to be produced by a would have taken place independently of team based in Paris, and a UK-team based this special occasion. Nevertheless, they in Bangor. The proposal consisted of two have helped to make WMY2000 visible to projects for presentation in connection mathematicians and in many cases they with the European Science and contributed to making mathematicians Technology Week (ESTW) from 6 to 12 aware of the need for communicating November 2000. More information can be mathematics to the public, by arranging found at http:/www.cpm.sees.bangor.ac.uk/ discussions on this topic during the confer- rpamath/. ences. In particular, 3ecm, the Third As a result of the first project, twelve European Congress of Mathematics, which posters were presented during the ESTW took place in Barcelona, Spain, from 10 to in a series with the title Les mathematiques du 14 July, contained a well attended Round quotidien (Mathematics in daily life), at Table on Raising Public Awareness of locations in France, the UK and Denmark, Mathematics (RPA). with Paris as the central city. The produc- A very special conference was arranged tion of such posters has been of consider- in Granada, Spain, from 3 to 7 July, as a able interest to schools, where teachers satellite conference to 3ecm. This confer- Robin Wilson as Edmond Halley at a find it increasingly difficult to motivate ence was one of the main projects of the WMY2000 lecture to schoolchildren at and inspire pupils. EMS-committee of the WMY and had the London’s Royal Institution. One result of the second project has 12 EMS December 2000 WMY NEWS

EMS Vice-President Luc Lemaire with European Commissioner Philippe Busquin and a WMY2000 poster that appeared in fifty Brussels under- ground stations, in the context of the RPA programme funded by the European Commission and co-organised by the EMS. The picture was taken at a science festival at the Université Libre de Bruxelles. been the production by the Bangor team of Mathematics, which mainly presents Hansen_Vagn_lundsgaard/rpa.html a booklet on Presenting Mathematics to the Mathematics and Knots, and Symbolic A major concern for the new EMS Public, which was distributed to all partici- Sculptures. This was launched officially at Committee is to ensure a continuing pres- pants of the 3ecm with a CD-Rom of John meetings in Obidos (Portugal) on 11 ence for mathematics at future ESTWs. Robinson’s Symbolic Sculptures, donated by November, and at Bangor (UK) on 16 This needs further support for the meet- Edition Limitée. The major work has been November. The above-mentioned posters ings involved in the planning and prepara- the production by the Bangor team of a were also on exhibit at these occasions, tion of proposals to the European CD-Rom Raising Public Awareness of together with posters in the London Commission. The experience gained from Underground produced by the Isaac the first contract with the European Newton Institute, Cambridge. Commission has been very valuable. The RPA-committee will be eagerly looking for What did we learn? good ideas that can form the basis for First of all, many more professional math- future proposals. ematicians have now gained first-hand One of the first tasks of the RPA-com- experience that presenting mathematics to mittee will be to arrange a competition for the public is non-trivial and difficult. The the best newspaper article on mathematics. EMS Committee of the WMY has been The competition will be announced at the deeply impressed by the dedicated work beginning of 2001, allowing authors of done by many individuals in several articles published in national newspapers European countries, who have taken time during WMY 2000 to submit the best of out of their usual positions to work for the their work for the competition. WMY project. It is important that we all The RPA-committee of the EMS will also work together in such endeavours and help investigate the possibilities for establishing with encouragement and by supplying a web-site containing eps-files of posters good ideas. But everyone should be aware (graphics only, but with open fields where of the fact that realising a good idea usual- texts in various languages can be added), ly needs hard work. Among the benefits of possibly in connection with the ems- such efforts is that it enriches, and puts gallery. From the experiences gained in into perspective, your future work, not connection with the production of posters only as a mathematical educator but prob- during the WMY, it is clear that such an ably also as a research mathematician. archive will be of considerable value to future producers of mathematical posters. How do we proceed? It is also planned to create a web-site The main issue of the WMY has been to containing a collection of short mathemat- raise the public awareness of mathematics. ical web-stories directed towards secondary This important task cannot be restricted to school pupils and the general public. one particular year; it is an on-going process which will take time and needs Altogether, the EMS Committee of the WMY constant attention. To continue the work finds that the year 2000 has been a terrific year begun by its committee of the WMY, the for mathematics. Some of the results obtained Executive Committee of the EMS has may not seem impressive right at this moment, appointed a new committee with the but as with many things related to mathematics, acronym RPA; further information can be the products have lasting value and will have Two of John Robinsons mathematical sculptures found at http://www.mat.dtu.dk/persons/ impact in years to come. EMS December 2000 13 FEATURE on a greatly reduced scale, the project was so large that it was separated from the EMC to become an independent project, and a legal entity called the European Mathematical Trust was founded for its upkeep. TheThe PPrre-historye-history This Euromath project proceeded with substantial financing from the European Community, but not at the pace that had been expected. An Euromath Centre was ofof thethe EMSEMS founded, by a Danish subvention, in Copenhagen. After many delays the pro- ject produced its first (and only) product, Aatos Lahtinen the Euromath editor. A widespread criti- cism was that this editor did not contain any essentially new features. It did not achieve any substantial success, and little by little the project was closed. As far as I There is nothing more untrustworthy than an ment to form an informal body called the understand, the parent European eyewitness, except another eyewitness. European Mathematical Council (EMC), Mathematical Trust does not exist any (Sir Walter Raleigh, 1552-1618) under the chairmanship of Professor more. Sir Walter Raleigh was one of the Michael Atiyah. The idea of Euromath was attractive and favourites of Elizabeth I. During the reign After Helsinki this informal body met in worth trying, but it was before its time. In of James I he fell from grace and was final- Oberwolfach (1980), in Warsaw (1982 and fact, in spite of the internet and the great ly imprisoned in the Tower of London. progress with EMIS, we are still far away To pass the time Raleigh started to write a from the goal. history of the world. There is a story that one day while writing he heard noises of Liblice tumult from the courtyard. Two other I attended the meeting of the EMC in prisoners had started a violent fight which November 1986 at Liblice, near Prague. caused others to join in. Later the same In addition to such normal items as day Raleigh tried to find out what had ‘Survey of journal prices’ and ‘Report of happened and interviewed several fellow the databank committee (Euromath)’, it prisoners who had been present. Everyone contained ‘Future activities and structure had a different opinion on the course of of the EMC’. In introducing this item the events. Raleigh, frustrated, tore his man- Chairman, Sir Michael Atiyah, pointed out uscript to pieces. (He changed his mind that the present informal Council was not afterwards and started again!) intended to be a permanent structure; for I have been asked to tell some of my instance, the financial contributions to the reminiscences on the pre-history of our EMC were on a voluntary basis and there- Society. While making an honest try I am fore small and erratic. Many Societies do aware that I am only one of the fellow pris- not have their own means, and so have to oners of Sir Walter Raleigh. justify the use of their funds to their Government; this makes the formation of Prelude a more formal association desirable. Such The earliest attempt to form a European a body should also include the Moscow Society for mathematicians that I am Mathematical Society. This formal body aware of took place in the 1978 ICM in should have links with the International Helsinki. A meeting was organised to con- Mathematical Union (IMU), but not be sider the foundation of a Federation of dependent on it. European Mathematical Societies. The 1983) and again in Oberwolfach (1984). There followed a lively discussion in a possibility of founding a European Society Representatives of some twenty societies positive atmosphere. It was commented with individual membership instead was participated at these meetings. that physicists and biochemists already also discussed, and it was admitted that Mathematicians from the USSR and DDR have European organisations and that one the latter would give mathematicians a were present as observers. True, the ECM would similarly benefit mathematicians. more immediate feeling of being members provided a forum for the exchange of The new association must be so flexible of the European Mathematical information, but because of the informali- that the participation of all European Community. However, a federation was ty, common action was scarce. The EMC Countries would be possible. The preferred because it would involve consid- carried out biannual surveys of prices of European Mathematical Trust would be erable less organisation and expense. European mathematical journals, in the entirely separate, but its relation to the One purpose of such a federation would hope that these could be used to reduce, main body should be clarified. A form of be to provide a forum for exchange of or at least maintain, prices of commercial- federation would allow the formation of information and for common action – in ly published journals. The Council, how- subgroups performing independent activ- particular, the following were mentioned: ever, had no effective means towards this ities. An alternative to this Federation of the coordination, planning and publicity aim. European Mathematical Societies could be of conferences; fellowships and exchange The most imposing action initiated by a European Mathematical Society with visits; the sponsoring of research centres; the EMC was beyond doubt the Euromath individual members. the foundation of a Federation Newsletter; project. It was a very ambitious idea for a It was decided that this formalisation of the fostering of cooperation between gigantic mathematical database, together structure needed a more profound investi- mathematical societies in matters of with a sophisticated system of data trans- gation before any decisions could be European interest. fer, storing and editing. The database taken. A small committee was appointed During the discussions it turned out would contain all the mathematical knowl- to consider the subject, under the chair- that, despite the positive attitude towards edge in Europe, from preprints to mono- manship of Sir Michael Atiyah. A draft of such an attempt, it was not possible to graphs and reviews. In the 1980s this the committee’s conclusions should be realise it. The only outcome was an agree- dream was too huge to be realised – even sent to the ECM member societies as soon 14 EMS December 2000 FEATURE as possible. A discussion of the commit- addition to the normal two days, the been thought of beforehand. tee’s proposals would take place at the Chairman Michael Atiyah had reserved an In this open situation I tried to forward next meeting of the ECM in 1988. extra day for the meeting. His insight was the process by suggesting to Michael right as we used all three days, from morn- Atiyah that Finland might be willing to Suggestions of the committee ing to evening and even more. The focus host this new association. I assured him The considerations of the committee were at the meeting was on the future structure that Finland would fulfil the three condi- sent out in May 1988. It had decided that of the EMC. tions of the committee, by giving a short the best way to proceed was to formulate The question was considered seriously. description on the process needed in certain principles for discussion at the After a lively debate there was general Finland with a rough timetable and esti- next ECM meeting. agreement that the time was now ripe for mation of costs. Fortunately I was aware of The unanimous view of the committee revival of the idea, from ten years previ- the legal requirements in Finland and I was that the association would be a ously, on closer cooperation between could certify that the legalisation would be Federation of National Mathematical mathematicians in Europe, based on a a fairly simple, and not expensive, proce- Societies. However, the membership legally accepted structure. There was, dure. should be defined more flexibly, because however, no mutual agreement on the Sir Michael Atiyah apparently saw possi- some countries have several mathematical form of the cooperation. In particular, the bilities in this proposal because he pre- societies while others have none. Also, it French mathematicians opposed a federa- sented the case to the council, where it was should be possible for institutional mem- tion of national societies. They wanted to received with interest. After a short dis- bers to join. The membership fee would follow the structure of the American cussion the Council decided to accept the suggestion and to place the seat of the future association in Helsinki. This agree- ment somehow opened the deadlock we were in and progress was also made in other directions. As a compromise it was decided that the future association should have both corporate and individual mem- bers. The way the power of decision would be distributed between these two cate- gories was resolved only in principle; the details would be settled at the foundation meeting. The question about the name raised a long debate, too. Finally the Council accepted the name European Mathematical Society. The Council authorised me to take care of the formal registration of the Society in Finland. In practice, this contained also the formulation of a draft of the Statutes of the new Society, according to the guide- lines presented and to the requirements of Finnish law. The next Council meeting in 1990 in Poland would make the final deci- sion on the founding of the Society, on its Statutes and on the principles of the actions the Society would undertake.

Sir Michael Atiyah signing the official charter of the Foundation in M¹dralin, 1990. Others from The response in Finland left to right: Fritz Hirzebruch, Lászlo Marki, Aatos Lahtinen, Jean-Marc Deshouillers, Andrzej My suggestion that the legal office of the Pelczar and Chris Lance. (Photo Courtesy of Prof. Ivan Ivansic.) Society should be in Finland was at my own initiative; the Finnish mathematical be related to the size and resources of the Mathematical Society and found a community had not discussed the question societies, but each society would still have European Mathematical Society of indi- of the location at all. I therefore returned only one vote at the Council, which was the vidual members. to Helsinki with some anxiety. It was, supreme authority. There was a long and animated debate however unnecessary. The Finnish The Federation would be formally regis- on the pros and cons of these two alterna- Mathematical Society approved my action tered in some country, but the committee tives. It became clear that the supporters and promised its support for the project. made no suggestion as to the location. of each model were not giving up but were The Chancellor of the University of They only pointed out three important digging in. Thus it was not possible to Helsinki, Olli Lehto, promised the sup- factors: come to a unanimous resolution along port of the University. All the officials I – the legal arrangement for charities in these lines, while a non-unanimous deci- contacted at the Ministry of Education, the country; sion was out of the question. The only way Justice and Home Secretary took a very – the stability and convertibility of its cur- left was to find some kind of a compro- positive attitude and saw no difficulties in rency; mise. the foundation of the Society as a Finnish – the initial availability of mathematicians The first agreement was reached on the Learned Society. I also learned in the in that country prepared to assist in the question on the legal form of the new asso- process that the Government was prepar- establishment of the Federation. ciation. The EMC decided to give it a ing a Bill for a new law on associations. legal status by registering it under the law This would make the registration of the For the problem of paying membership of some European country. The next European Mathematical Society even easi- fees with non-convertible currencies, the question was which country, and this was er. committee suggested setting up a special tied up with the debate on the form of the Thus by October 1988 I could already East European Secretariat. association. Each of the mathematically write a short report to Michael Atiyah, eminent countries France, Germany and confirming that Finnish mathematicians Oberwolfach Great Britain was firmly behind its own were ready to take on the task and carry it The next meeting of the EMC took place concept and none of them was suggested. to a conclusion. The foundation process in Oberwolfach, 15-17 October 1988. In Possible alternatives had apparently not could be initiated. EMS December 2000 15 FEATURE Preparing the Statutes ing to accept that the President would be porarily the fee of any member. It was Preparing the Statutes appeared to be an the chairman of both the Executive decided that the more serious problem on iterative process in the sum of two sub- Committee and the Council. the upper limit should be left to the foun- spaces. One was the mathematical com- I made the alterations that the lawyer dation meeting. munity and the other the civil servants demanded and so by November 1989 we who were taking care of the legal registra- had the fourth draft of the Statutes which M¹dralin tion. The iteration was complicated by the would be acceptable to the Ministry but The foundation meeting of the European multi-dimensionality of both subspaces. which were in some places more cumber- Mathematical Society, which started as the In fact, it soon turned out to be impossible some than the third draft. At the same last meeting of the European to prove that the iteration would con- time we raised the 25% upper limit on the Mathematical Council, was held in verge. other Council delegates to 40%. In the Madralin, near Warsaw, on 27-29 October The procedure started with the first By-laws we raised the upper limit of dele- 1990. The meeting of the EMC was draft composed by Atiyah in December gates representing individual members to chaired, as always, by Michael Atiyah. 1988, with the Statutes of European 5, and we also added a statement that the There was a lively discussion on the pur- Physical Society as a model. The most upper limit should be reconsidered when pose and modes of action of the forthcom- essential points were: the Society had 2000 individual members. ing Society. Concerning the Statutes there – the members of the Society could be The Eastern European Secretariat was were several comments most of which were individuals and organisations other removed from the Statues and the By-laws, of a technical nature. As expected, the than mathematical societies; because Professor Schwabik from Prague only principal question was raised by the – the supreme authority of the Society was considered that the rapid changes in French mathematicians. They repeated the Council; Eastern Europe had made it unnecessary. what they had already said in Kyoto – – each member society had one, two or This fourth draft was sent once again to namely, that they would accept only a soci- three Council delegates; the Societies. Not unexpectedly, many ety of individual members and not one of – the number of delegates of other organ- questioned the sense in having an Societies. In particular, they wanted the isations and individual members was Assembly that does not meet. After a dis- upper limit on delegates of individual restricted to at most 25% of the total cussion with Atiyah I approached the members to be removed. number of Council delegates; the draft Ministry of Justice once more. By coinci- In the discussion they did not get sup- did not specify how to elect the Council. dence there was a different official in port. However, the unanimous accep- charge. He took our criticism seriously tance of the Statutes was considered This draft was sent for comments to math- and admitted that we could as well have a important, and therefore a small ad hoc ematical societies. Simultaneously I tried direct postal vote of the Council without a committee of Michael Atiyah, Jean-Pierre to specify the requirements of the Finnish fictitious Assembly. Bourguignon, Fritz Hirzebruch, Lászlo law. A problem was that the registration I was then quite happy to produce in Marki and myself was set up to try to nego- would be under the future law, and no June 1990 the fifth draft, in which the tiate a compromise. We convened at lawyer was yet willing to make interpreta- Assembly was deleted. Also the number of lunchtime. After some attempts it was tions. delegates of individual members in agreed to propose a package of four reso- Anyway, in June 1989 we had the second Council was redefined by the formula lutions: draft of the Statutes ready. There was a min{(n–1)/300)+1, 2C/5}, where C is the – the member societies should encourage small meeting in Oxford in July, with M. total number of Council delegates. I their members to become individual Atiyah, J-M. Deshouilles, W. Schwarz, J. believed that the convergence of the members of the EMS and should be Valenca, J. Wright and myself. Based on Statues had essentially taken place, and committed to collect individual mem- the received comments and legal require- that only minor adjustments were needed. bership fees for the EMS; ments, we iterated it to the third draft I was wrong. – the statutes of EMS will be reconsidered which was accepted by all present. In par- when the individual membership of the ticular, the 25% upper limit on the other Kyoto EMS has reached 4000; Council delegates remained intact. We As a final check before the foundation – in the formula defining the number of also produced the first draft by-laws. meeting in October 1990, a meeting was delegates for individual members the There it was stated that the Council is arranged in August in conjunction with denominator 300 should be replaced by elected by postal vote where (for example) the ICM 1990 in Kyoto. It seemed appro- 100; the number of delegates representing priate that the process that had started at – in the same formula the upper limit individual members was restricted to at ICM 1978 would essentially also end at an 2C/5 should be retained. most four. There was also a mention of ICM. Michael Atiyah was not present in the Special Secretariat in Prague (by the Kyoto, and therefore I chaired the meet- The council accepted the first three reso- offer of Professor Kufner). The essential ing. There was general contentment at lutions unanimously and the fourth by 28 factors seemed to us to be fairly complete. the deletion of the Assembly, and it votes to 12. However, when the French Armed with a Finnish translation of the seemed that the Mathematical Societies were still unhappy with the upper limit, it third draft Statutes, I approached the were ready to accept this version as final. was decided to replace it by 2C/3. department of the Ministry of Justice that Unexpectedly the French delegates After some minor items the Chairman took care of the registration of associa- announced that they were not content and proposed that the Council should formal- tions, and asked their opinion. They were that they would not recommend accep- ly establish the European Mathematical helpful, but not happy with our draft. The tance of the Statutes. They opposed the Society with its seat in Helsinki, which was lawyer in charge complained that our style upper limit of Council delegates elected agreed. Professor Atiyah was accepted as and formulations differed so much from by individual members and said that these the first individual member of the Society. the Finnish praxis that it was difficult to delegates must be able to have a majority The official charter for the foundation say whether the Statutes were acceptable in the Council. They also raised the point (written in Finnish, of course) was signed, or not. The most serious point was the that individual members from ‘poor’ a toast was raised, and we had the lawyer’s demand that the Council must be countries may not be able to pay even a European Mathematical Society. elected by a General Assembly of all mem- modest fee. The first point came as a sur- bers. I pointed out the difficulties of prise to me because I had understood that What happened after that is another story. arranging such an assembly in a also the French Mathematical Society had A part of this story, told by David Wallace, European-scale society, but it resulted accepted the upper limit. can be read at the address http://turn.to/ only to a poor compromise where the In due course I consulted Atiyah. We EMSHISTORY99. It cannot be more than Assembly may elect the Council by a postal decided to treat the latter problem by a part, because the story of our Society vote without actually meeting. Another adding a new By-law giving the Executive continues and continues forever, I hope. difficulty was that the lawyer was not will- Committee the authority to waive tem- Vivat, crescat, floreat! 16 EMS December 2000 INTERVIEW InterviewInterview withwith SirSir RRogeroger PPenrenroseose part 1 Interviewer: Oscar Garcia-Prada

Roger Penrose was formerly Rouse Ball conclusion he came to was that the prob- parents were rather annoyed when I got Professor at Oxford University, and currently lem was much more complicated than any- home; my medical career had disappeared holds the position of Gresham Professor of body had thought before, which is proba- in one stroke. Geometry in London. bly the right answer. This interview is in two parts – the second Where did you go to university? part will appear in the March 2001 issue. This was before going to Canada? I went to University College London for Yes. Then we went over first to the US my undergraduate degree. My father was When did you first get interested in mathe- when it started to become clear there was professor there and so I could go there matics? going to be a war. He had this opportuni- without paying any fees. My older brother From quite an early age – I remember ty to work overseas and he took it. had also been there as an undergraduate making various polyhedra when I was and he then went to Cambridge to do a about 10, so I was certainly interested in And when did you return to England? Ph.D. in physics. I went to Cambridge mathematics then – probably earlier, but it Just after the War, in 1945. I went to afterwards to do my Ph.D. in mathematics. became more serious around the age of 10.

Are there other mathematicians in your family? Yes, my father was a scientist – he became a professor of human genetics, but he had broad interests and was interested in math- ematics – not on a professional level, but with abilities and genuine interests in mathematics, especially geometrical things. I also have an older brother Oliver who became a professor of mathematics. He was very precocious – he was two years older than I was, but four years ahead in school. He knew a lot about mathematics at a young age and took a great interest in both mathematics and physics; he did a degree in physics later on. My mother also had an interest in geometry; she was med- ically trained as my father was.

Did you have good teachers at school? I did have at least one teacher who was quite inspiring. I found his classes inter- esting, although maybe not terribly excit- ing.

Where did you go to school? I was at school in Canada between the ages of 8 and 13. I don’t know that I got much mathematics interest from there. Then I was back in England at the age of 14.

But you were born in England? Yes. We went over to the US just before University College School in London I was mainly a pure mathematician in the War. My father had a job in London where I became more and more interested those days. I’d specialised in geometry (Ontario) at the Ontario hospital, where he in mathematics, but I didn’t think of it as a and went to Cambridge to do research in later became the Director of Psychiatric career. I was always the one who was sup- algebraic geometry, where I worked under Research. He was interested in mental posed to become a doctor, but I remember William Hodge. disease and its inheritance, the sort of an occasion when we had to decide which A contemporary who was also starting at thing that he became particularly expert subjects to do in the two final years. Each the same time was Michael Atiyah, who on later. So the question of inheritance of us would go up to see the headmaster, later won the Fields Medal and became versus environmental influence were of one after the other, and he said ‘Well, what President of the Royal Society, Master of great interest to him. subjects do you want to do when you spe- Trinity College, Cambridge, and the first In fact I was born in Colchester in Essex cialise next year’. I said ‘I’d like to do biol- director of the Isaac Newton Institute. – it’s an old Roman town, possibly the old- ogy, chemistry and mathematics’ and he When you first become a research student est town in England. My father took on a said ‘No, that’s impossible – you can’t do you’ve no idea who the other people are. project called the Colchester survey, which biology and mathematics at the same time, It took me a while to realise there was had to do with trying to decide whether we just don’t have that option’. Since I had something special about him. So it was a environmental or inherited qualities were no desire to lose my mathematics, I said bit intimidating, I remember, at the begin- more important in mental disease. The ‘Mathematics, physics and chemistry’. My ning. EMS December 2000 17 INTERVIEW I worked with Hodge for only one year, orems about, and what do they say about material, especially if it started with a mass because he decided that the kind of prob- space-time? of, say, ten times the mass of the sun. lems I was interested in were not in his line Well, singularities are regions of space- So what happens to it? Round about of interest. I then worked under John time where the laws of physics break down. 1939, Robert Oppenheimer and various Todd for two years, but during that period The main singularity one hears about is students of his – in particular, Hartland I became more and more interested in the big bang, which represents the origin of Snyder – produced a model of the collapse physics, largely because of my friendship the universe. Now cosmological models of a body. As an idealisation, they consid- with Dennis Sciama who rather took me were introduced in accordance with the ered a body made of pressureless material, Einstein equations of general relativity, which was assumed to be exactly spherical- which describe curvature of space-time in ly symmetrical – and they showed that it terms of the matter content. The equa- will collapse down to produce what we now tions determine the time-evolution of the call a black hole. universe. You apply these equations to a A black hole is basically what happens very uniform universe, which is what peo- when a body is concentrated to such a ple did originally, assuming that the uni- small size for such a large mass that the verse is homogenous and isotropic, in escape velocity is the velocity of light, or accordance with the standard models that exceeds the velocity of light, the escape are used to describe cosmology on a large velocity being that speed at which an object scale. If you extrapolate Einstein’s equa- thrown from the surface of the body tions backwards you find that at the very escapes to infinity and doesn’t ever fall beginning was this moment where the den- back again. It’s about 25000 miles an hour sity became infinite and all matter was con- for an object on the surface of the earth. Roger Penrose and Dennis Sciana at the 1962 centrated in a single place. The big bang But if you concentrate the earth so much, Relativity Conference in Warsaw represents the explosion of matter away or take a larger body with a mass of, say, from this – in fact, the whole of space-time twice the mass of the sun and concentrate under his wing. He was a good friend of originated in this single event. it down, it reaches the region in which the my brother’s, and I think I made some- Some people used to worry about this, as velocity will reach the speed of light when thing of an impression on him when I vis- I did, because it represents a limit to what it’s just a few miles across. And then it ited Cambridge and asked him some ques- we can understand in terms of known becomes a black hole once the escape tions about the steady-state universe which physical laws. The same situation arose velocity exceeds the velocity of light, so I don’t think he’d quite thought about. So later when people started to worry about that nothing can escape, not even light. he thought it was worth cultivating my what happens to a star which is too massive This is exactly what happened in the interest in physics. to hold itself apart and singularities arise. model that Oppenheimer and Snyder put Back in the 1930s Chandresekar showed forward in 1939. But it didn’t catch on. So was he one of the most influential people that a white dwarf star, which is a really Nobody paid any attention to it, least of all you came across? concentrated body, can have the mass of Einstein, as far as one knows. I think the He was very influential on me. He taught the sun, or a bit more. We know that such view of many people was that if you remove me a great deal of physics, and the excite- objects exist – the companion of Sirius is the assumption of spherical symmetry then ment of doing physics came through; he the most famous one – but if such a body the exact model that Oppenheimer and was that kind of person, who conveyed the has more than about one-and-a-half times Snyder had suggested would not be appro- excitement of what was currently going on the mass of the sun then, as Chandresekar priate, and who knows what would hap- in physics – it was partly Dennis Sciama showed, it cannot hold itself apart as a pen? Maybe it would not concentrate into and partly lectures that I attended ‘on the white dwarf and will continue to collapse; a tiny thing in the centre, but would just side’ when I was in my first year. nothing can stop it. A white dwarf is basi- swirl around in some very complicated I remember going to three courses, none cally held apart by what’s called electron motion and come spewing out again – I of which had anything to do with the degeneracy pressure – this means that the think this was the kind of view some peo- research I was supposed to be doing. One electrons satisfy an exclusion principle ple had. And you wonder about whether was a course by Hermann Bondi on gener- which tells you that two electrons cannot be assuming that there’s no pressure is the al relativity which was fascinating; Bondi in the same state, and this implies that fundamental assumption, because matter had a wonderful lecturing style which when they get concentrated they hold the does have pressure when it gets concen- made the subject come alive. Another was star apart. So it’s this exclusion principle trated. a course by Paul Dirac on quantum in effect that stops a white dwarf star from This was revived in the early 1960s when mechanics, which was beautiful in a com- collapsing. the first quasars were discovered. These pletely different way; it was just such a per- However, what Chandra showed is that extremely bright shining objects seemed to fect collection of lectures and I really gravity will overcome this if the star is too be so tiny, yet so massive that one would found them extremely inspiring. And the massive, and that its electron degeneracy have to worry about whether an object had third course, which later on became very pressure cannot hold it apart. This prob- actually reached the kind of limits that I’ve influential although at the time I didn’t lem occurs again in what’s called neutron just been talking about, where you would- know it was going to, was a course on degeneracy pressure, which is again the n’t see it if it was really inside what’s called mathematical logic given by Steen. I exclusion principle but now applied to the event horizon, and where the escape learnt about Turing machines and about neutrons. What happens is that the elec- velocity exceeds the velocity of light, but if Gödel’s theorem, and I think I formulated trons get pushed into the protons and you you get close to that then very violent during that time the view I still hold – that have a star made of neutrons. Those neu- processes could take place which could there is something in mental phenomena, trons hold themselves apart by not being produce extraordinarily bright objects. something in our understanding of mathe- able to be in the same state. But again the When the first quasar was observed, people matics in particular, which you cannot Chandrasekar argument comes to bear on began to worry again about whether what encapsulate by any kind of computation. the neutron stars and you find that they we now call black holes might not really be That view has stuck with me since that also have a maximum mass which isn’t there out in the universe. period. believed to be much more than that of a So I began thinking about this problem white dwarf. So anything with, say, twice and the whole question of whether the You’ve worked in many areas, but let me the mass of the sun would seem to have no assumption of exact spherical symmetry start with your 1960s work on cosmology. resting place and would go on collapsing could be circumvented, using techniques With Stephen Hawking you discovered the unless it could throw off some of its mater- of a topological nature which I had started singularity theorems that won you both the ial. But it seems unlikely that in all cir- to develop for quite other reasons. What prestigious Wolf prize. What are these the- cumstances it would throw off enough people had normally done would be just to 18 EMS December 2000 INTERVIEW solve complicated equations, but that’s not would come about if material is dragged impressive recent evidence of material very good if you want to introduce irregu- into a tiny region and gets heated in the being swallowed by one without trace. larities and so on, because you simply can’t process of being dragged in; the material There’s also another potential possibility solve the equations. So I looked at this probably forms a disc, which is the normal of the direct observation of a black hole: from a completely different point of view, view people have. The material gets when I say ‘direct’, it’s more because the which was to look at general topological dragged off the companion star, the blue theory of black holes is so well developed issues: could you obtain a contradiction supergiant star, and it spirals into the hole, that one knows very closely what the geom- from the assumption that the collapse in the standard picture. It gets hotter and etry should be. There’s a geometry known takes place without any singularities? hotter until it reaches X-ray temperature, as a Kerr geometry which seems to be the Basically what I proved was a theorem which is the source of these X-rays, and unique endpoint of a collapsed object to which was published in 1965 in Physical that’s what’s seen. form a black hole, and this geometry has Review Letters, where I showed that if a col- Now it doesn’t tell you that this object is very interesting specific properties. Some lapse takes place until a certain condition actually a black hole, but the dynamics of of these could be tested to see whether holds (a qualitative condition which I the system are such that the invisible com- these concentrated objects that we know called the existence of a trapped surface), then ponent has to be much too massive to be are there really conform with the Kerr you would expect some type of singularity. either a white dwarf or a neutron star, geometry. That would add much more What it really showed is that the space-time because of the Chandrasekar argument, direct evidence for black holes, but it’s could not be continued, it must come to an and so on. So the evidence is indirect: something for the future. end somewhere – but it doesn’t say what what one knows is that there is a tiny high- the nature of that end is, it just says that ly concentrated object which seems to be What would be the most striking physical the space-time cannot be continued indef- dragging material into it, and from the implications of the singularities here? initely. neighbourhood of which one sees X-rays. What the singularities tell us is that the Also gamma ray sources seem to be black laws of classical general relativity are limit- Can you test this theory in our universe? hole systems, and there may now be many ed. I’ve always regarded this as a strength Well, the first question is: do black holes other examples, other double star systems in general relativity. It tells you where its exist? They are almost a theoretical conse- or black holes in galactic centres. Indeed, own limitations are. Some people thought quence of the kind of discussion I’ve just there is convincing evidence for a very con- it was a weakness of the theory because it referred to. Then Stephen Hawking came centrated dark object at the centre of our has these blemishes, but the fact that it in as a beginning graduate student work- own galaxy, of the order of something like really tells you where you need to bring in ing with Dennis Sciama, and he took off a million solar masses. other physics is a powerful ingredient in from where I’d started, introducing some It seems to be a standard phenomenon the theory. other results mainly to do with cosmology that galaxies may have these highly con- Now what we believe is that singularities rather than black holes. Later we put our centrated objects which we believe to be are regions where quantum theory and results together and showed that singular- black holes in their centres. Some galaxies general relativity come together, where ities arise in even more general situations than we had individually been able to han- dle before. Now there is a big assumption here to which we still don’t know the answer. It’s called cosmic censorship, a term I introduced to emphasise the nature of this hidden presumption, that is often tacitly made. Cosmic censorship asserts that ‘naked sin- gularities’ do not occur. We know from the singularity theorems that singularities of some kind occur at least under appropri- ate initial conditions that are not unrea- sonable – but we don’t know that those sin- gularities are necessarily hidden from external view. Are they clothed by what we call a horizon, so you can’t actually see them? With a black hole you have this horizon which shields that singularity from view from the outside. Now it’s conceiv- able that you could have these naked sin- gularities, but they’re normally considered to be more outrageous than black holes. The general consensus seems to be that may have large ones, and quasars are things are both small and massive at the they don’t happen, and that tends to be my believed now to be galaxies which have at same time. ‘Small’ is where quantum view also. If you assume that they don’t their centres objects that are much effects become important, and ‘massive’ is occur, then you must get black holes. So brighter than the entire galaxy, so all you where general relativity becomes impor- it’s a theoretical conclusion that if you have see is this central region which is extraor- tant. So when you get the two things hap- a collapse of a body which is beyond a cer- dinarily bright. It’s bright because it has pening together, which is what happens in tain size, then you get black holes. dragged material into it, and it gets extra- singularities, then the effects of both gen- Now one type of system that ordinarily hot and spews out in certain eral relativity and quantum mechanics astronomers have observed is where there directions at nearly the speed of light. You must be considered together. is a double star system, only one member see examples of things where jets come out Now this applies in the big bang and in of which is visible. The invisible compo- of centres of galaxies and things like this. the singularities in black holes, and it nent is taken to be a black hole – Cygnus X- But all this evidence is indirect. It’s not would also apply if the whole universe were 1 was the first convincing example. It’s an that one knows that black holes are there, ever to collapse, although that is just a con- X-ray source, and what is seen is a blue it’s just that the theory tells us that there glomeration of all the black holes into one supergiant star which is in orbit about ought to be black holes there and the the- big black hole. There’s one thing I find something; the ‘something’ is invisible ory fits in very well with the observations. particularly interesting, however, which is through a telescope, but seems to be the But most observations do not directly say the stark contrast between the big bang source of the X-rays. Now the X-rays that those are black holes, although there’s and the singularities in black holes. It’s a EMS December 2000 19 INTERVIEW bit ironic, because in the earlier stages of rules of quantum mechanics as they are What are twistors, and how are they more the black hole singularity discussions, their and try to apply them to some classical the- fundamental than a point in space-time? reasonableness was that we already know ory, but I prefer not to use that word. I say Well, you see, if I follow the complex analy- there’s a singularity in the big bang. It was that the theory we seek involves also a sis I can come back to this. First of all, argued that the singularities in black holes change in the very structure of quantum complex analysis is just mathematics, and are just the same as the big bang, but time mechanics. It’s not quantising something; it’s beautiful mathematics that’s tremen- is going the other way – so if you have one it’s bringing in a new theory that has stan- dously useful in many other areas of math- you should have the other. This was quite dard quantum theory as a limit. It also has ematics. But in quantum theory you see it a plausible kind of argument, but when we standard general relativity theory as at the root of the subject – for the first time look at these things in detail we see that another limit, but it would be a theory that one sees that it’s really there in nature, and the structures are completely different: the is different in character from both those that nature operates (at least in the small structure that the big bang had was very theories. scale) according to complex numbers. smooth and uniform, whereas the struc- Now the thing that struck me from quite ture we expect to find in singularities is Let me come to another aspect of your work. early on – it’s one of the earliest things I very complicated and chaotic – at a com- One of your greatest inventions is twistor did in relativity – is if you look out at the pletely different end of the spectrum. theory, which you introduced about 30 sky you see a sphere; but if you consider In fact, this is all tied up in a deep way years ago. What is twistor theory? two observers looking out at the same sky, with the second law of thermodynamics. This Well the main object of twistor theory is to one of whom is moving with a high speed law tells us that there is a time asymmetry find the appropriate union between gener- relative to the other, then they see a slight- in the way things actually behave. This is al relativity and quantum mechanics. I ly transformed sky relative to each other, normally traced far back in time to some suppose I had that view over thirty years and the transformation of that sky pre- very ordered structure in the very early ago (actually, 1963) before I talked about serves circles and takes angles to equal stages of the universe – and the more you this singularity issue and the asymmetry, angles. Now those people who know about trace it back, the more you find that this and so on. I’d already felt that one needs complex analysis know that this is the way ordered structure is indeed the big bang. a radically different way of looking at you look at the complex numbers: you So what is the nature of that ordered things, and twistor theory was originally have infinity as well, and they make a structure in the big bang, and what is its motivated by such considerations. Since sphere – the Riemann sphere – and the cause? Well, in relation to what I was just we can’t just ‘quantise’, we need other transformations that send that sphere to saying, it is quantum gravity. We believe guiding principles. itself, the complex analytic transforma- that this is where quantum theory and Let me mention two of them. One was tions, are precisely those that send circles gravitational theory come together. And non-locality, because one knows about to circles and preserve angles. I was com- what this tells us – and I’ve been saying this phenomena in which what happens at one pletely struck by this phenomenon, as it for quite a few years but few people seem end of a room seems to depend on what seems to me that what you’re doing when to pick up on this completely obvious point happens at the other end. These experi- you look at the sky is you’re seeing the – is that the singularity structure, which is ments were performed about twenty years Riemann sphere – they are the complex where we see general relativity and quan- ago by Alain Aspect in Paris – all right, numbers out there in the sky, and it tum mechanics coming together most bla- those experiments hadn’t been performed seemed to me that that’s the kind of math- tantly, is time-asymmetrical. So it tells me when I introduced twistor theory, but the ematical connection. It seemed to me such that the laws involved in quantum gravity, original ideas were there already – I mean a beautiful fact, and in a sense the trans- combining quantum theory with general the Einstein-Podolsky-Rosen phenomena, formations of relativity are all contained in relativity, must be time-asymmetrical, which tell you that quantum mechanics that fact. Surely that means something. whereas the laws we normally see in says that you have ‘entanglements’ – things We already know that complex numbers physics are time-symmetrical. at one end of the world seem to be entan- are fundamental to quantum theory, and It also tells us, it seems to me, that the gled with things at the other end. Now here we see complex numbers fundamen- laws of quantum mechanics are not just that’s only a vague motivation: it’s not real- tal to relativity when we look at it this way. concerned with applying quantum ly something that twistor theory even now My view was to say ‘all right, don’t think mechanics to general relativity – when I say has a great deal to say about, but it does say of the points you see when you look at the ‘just’, it’s a gross understatement because that somehow non-locality is important in sky; what you are doing is seeing light rays. nobody knows how to do that, but I think it our descriptions, and twistor theory (as it You and a star in the distance are connect- must be a union between these two theo- has developed) certainly has features of ed by a light ray, and the family of things ries, giving a new theory of a different non-locality, over and above those I was that you see as you look at the sky is the character. It’s not just quantum mechan- aware of when I started thinking about family of light rays through your eye at ics: quantum mechanics itself will have to these ideas. that moment. So the thing with the com- change its structure and it will have to Originally, rather than having points in plex structure is light ray space, telling you involve an asymmetry in time, but I have space-time as the fundamental objects, I that maybe you can see this link between reason to believe that this is all tied in with thought more in terms of entire light rays space-time structure and complex num- the measurement problem – the collapse as fundamental. The reason for thinking bers if you concentrate not on points but of the wave function, the curious features about light rays actually came from some- on light rays instead. that quantum theory has which make it in thing quite different, which I regard as So that was really the origin of twistor many respects a totally unsatisfying theory perhaps the most important motivation theory – well, that’s cheating slightly, but I from the point of view of a physical picture underlying twistor theory. In the union suppose one cheats when one gets used to or a philosophically satisfying view of the between quantum mechanics and general a certain idea, because although these phe- world. Quantum mechanics is very pecu- relativity, I feel strongly that complex nomena were known to me and I realised liar, because it involves incompatible pro- numbers and complex analytic structures their importance, it was something else cedures. My own view is that this is some- are fundamental for the way that the phys- that really steered me in the direction of thing that we will only understand when ical world behaves. I suppose that part of twistor theory. It’s a bit technical, but had we’ve brought Einstein’s general relativity my reason for this goes way back to my to do with complex numbers – all right, in with quantum mechanics and combined mathematical training. When I first learnt you see them in the sky, but you also see them into a single theory. about complex analysis at university in them in all sorts of other places – in solu- So my view on quantum gravity is quite London I was totally ‘gob-smacked’ – it just tions of Einstein’s equations, and so on – different from that of most people. What seemed to me an incredible subject; some they started to come up when people most people seem to say is ‘Oh, you’ve got of the simplest ideas in complex analysis, looked at specific solutions of Einstein’s to try and quantise general relativity, and such as if a function is smooth then it’s ana- equations. It turned out very often that quantise gravitation theory, and quantise lytic, are properties which I always thought you could express things very nicely if you space-time’: to ‘quantise’ means to take the were totally amazing. used complex numbers, and it suggested 20 EMS December 2000 INTERVIEW to me that somehow – I had this image like problem. I regard it as a very long round- prime interest, so it’s kind of ironic that an iceberg, you see – what you see is a lit- about route, but one needs first to under- here’s a theory that’s supposed to be tle bit at the top and there’s the rest of it stand how Einstein’s general relativity real- answering the problems of physics, and yet down underneath which is invisible. It’s ly fits in with twistor theory. Although con- it’s not caught on at all on the physics side. really a huge area where these complex siderable advances have been made, some numbers at the tip poke up through the dating back to twenty years ago, it’s still a You mentioned string theory. Are there con- water, while the rest of it is underneath. question mark. We don’t completely know nections between twistor theory and string So these solutions, where one sees the how to represent Einstein’s theory in rela- theory? complex numbers, seemed just the tip of tion to twistors; there are some very strong I think there probably are. It’s not some- an iceberg, and they were really under- indications that there’s a good connection thing that has been deeply explored, and neath governing the way that the first- between the two, but how one does it is still the groups of people who work on these hand structure works. It was a search to try not clear. subjects are more-or-less disjoint. There and find what that complex structure was, So my view is that the major problem in have been some attempts to bring the the- and it wasn’t until certain things that are twistor theory is to see how to incorporate ories together, but I think that the right not appropriate to describe here, but Einstein’s theory into the twistor frame- vehicle for doing so hasn’t come about yet. which relate to solutions of Maxwell’s work, and it’s still not complete. What we I wouldn’t be at all surprised to find that in equations and Einstein’s equations which seem to see is that in the process of incor- the future some more significant link show you that the space of light rays, porating Einstein’s theory into twistors, we between these two areas is found, but I although it’s not quite a complex space also have to incorporate ideas of quantum don’t see it right now. because it’s got the wrong number of mechanics. So my hope is that in bringing dimensions – but if you look at the right classical general relativity into the scope of These new theories involving p-branes seem structure you see it as part of another twistor theory, one will also see how quan- to be more suitable, somehow? structure, a slightly extended one with six tum theory must be made to combine with Well, there is a connection, but I don’t dimensions, and it produces complex general relativity, and in that combination know how significant it is. I was talking to objective space which is complex projective one will see how to deal with singularities, Ed Witten recently and he was telling 3-space. because that is the place where the combi- about the 5-branes they’re interested in. Now with hindsight I can describe these nation of the two theories comes in. Also, But that’s curious, because in work that things more satisfactorily. Let me put it there must be a time asymmetry in the way Michael Singer did some years ago with like this. When you think of a light ray, it comes together, and that will explain the Andrew Hodges and me, the suggestion that is an idealised photon idealised in a difference between the past and the future was made that what one should really be specific respect, you are just thinking of it singularities. But all these things are looking at is generalisations of strings. as a path through space-time. But you hopes – they’re not something I can do Whenever you see an ordinary string, you have to bear in mind that massless parti- now. should really think of it as a surface, cles (photons, in particular) also have spin because it’s a string in time. It’s one- (they spin about their direction of motion), Twistor theory has been tremendously suc- dimensional in time, so that gives you two and if you introduce the spin they also cessful in applications within mathematics, dimensions. These things are studied very have energy. The spin is a discrete para- but has it been helpful in understanding the much in connection with complex one- meter. It’s either left-handed or right- nature of the physical world? dimensional spaces (Riemann surfaces), so handed, but when the particle has spin, Not very much, I would say. It’s rather they are in some natural way associated introducing the energy (a continuous para- curious, but I would say that this is not with these Riemann surfaces. meter) gives one more degree of freedom. unique to twistor theory. One sees it in Now what we had in mind, which was So instead of having just five dimensions of other areas – like string theory, for much more in line with twistor theory, is to light rays, you find a six-dimensional space instance – where people start with great look at a complex three-dimensional ver- that is naturally the complex 3-space. So ambitions to solve the problems of physics, sion of this, which we called pretzel twistor you’ve got the whole thing, the right-hand- and instead come up with ideas that have spaces; they’re complex three-dimensional ed ones, the light rays and the left-handed had implications within mathematics; this spaces, so they are six real-dimensional, ones, and they all fit together to form a is certainly the case with twistor theory, its and if you can think of them as branes in space that’s called projective twistor space. applications and its interest. If you round- some sense, then they are 5-branes. Now And it seemed to me that once you take ed up all the people who claimed they is there a connection between those 5- this space as being more fundamental than worked on twistor theory, you’d find, I branes and the 5-branes of the string theo- space-time (the main reason being that it’s would think, that a vast majority of them ry? I just don’t know, and I haven’t complex), it ties in with other things that were mathematicians with no particular explored it. I didn’t mention it to Witten I’ve been interested in for years – the use interest in physics – they might be interest- when I talked to him, but there might be of spinors and how you treat general rela- ed in differential geometry, or integrable something to explore here. That’s just off tivity, things which I’d learnt in Bondi’s systems, or representation theory. Very the top of my head, I don’t know, but yes, and Dirac’s lectures. This notion of spinors, few of them would have physics as their it might be that there’s a connection there. as a way of treating general relativity, was something I found to be powerful, but it didn’t quite do what I wanted, which was to get rid of the points. That was what twistor theory achieved, and it’s still going on.

So how do twistors actually relate to these singularity theorems? Do they have any- thing to say about those theorems? The short answer to that question is no – or, not yet. The hope is that they will, but the subjects have been going off in two quite different directions. Twistor theory is motivated by trying to bring general rel- ativity and quantum mechanics together. If it’s successful in that direction, then it would have something to say about the sin- gularity problem, but at the moment it has very little direct bearing on the singularity EMS December 2000 21 INTERVIEW

InterInterviewview withwith VVadimadim G.G. VVizingizing interviewers: Gregory Gutin and Bjarne Toft

In 1964 the Russian mathematician Vadim G. Tomsk in 1954 and graduated from there Vizing published, in a Siberian journal., a in 1959. paper with the title ‘On an estimate of the chro- I was then sent to Moscow to the famous matic class of a p-graph’ (in Russian). Its main Steklow Institute to study for a Ph.D. The result is a theorem that today can be found in area of my research was function approxi- most textbooks on graph theory. Vizing is now mation, but I did not like it. I asked my one of the best-known names in modern graph supervisor for permission to do something theory. In 1976 he initiated the study of ‘list else, but was not allowed to change. So I colourings’, a topic that has received much did not finish my degree and returned to attention recently. Novosibirsk in 1962. From 1962 to 1968 I spent a happy peri- Vizing’s Theorem (1964). The edges of a od at the Mathematical Institute of the graph with maximum degree d can be Academy of Sciences in Academgorodoc, coloured in at most d + 1 colours so that no outside Novosibirsk. In 1966 I obtained a two edges with a common vertex are coloured Ph.D. I did not have a formal supervisor, the same. Moreover, the edges of a p-graph but A. A. Zykov helped me. with maximum degree d (where any two ver- Because of the very cold climate I want- tices are joined by at most p edges) can be ed to move back to Ukraine, but I could coloured in at most d + p colours so that no not get permission to live in Kiev. After two edges with a common vertex are coloured living in various provincial towns, I finally the same. ended up in Odessa, where I taught math- ematics at the Academy for Food In October 2000 Vizing visited the University of Technology from 1974. Southern Denmark in Odense, where the follow- ing interview was conducted by Gregory Gutin How was life in Academgorodoc in the (Royal Holloway College, University of 1960s? Vadim Vizing in 1975 London) and Bjarne Toft (University of It was nice and quiet, and the atmosphere in Novosibirsk in Metody Diskret. Analiz. It Southern Denmark). was good. Zykov let me present my results appeared in 1964 when I had also solved in his seminar, and he became my friend. the p-graph case. By this time the result Where did you grow up, and where did you And later the place attracted some very had already been mentioned in the West get your education? good students, like Oleg Borodin, when Zykov stated it in the proceedings of I was born on 25 March 1937, in Kiev in Alexander Kostochka and Leonid S. a meeting in Smolenice that was published Ukraine. After the war, when I was 10, my Melnikov. jointly by the Czechoslovak Academy of family was forced to move to the Sciences and Academic Press. Novosibirsk region of Siberia because my What made you choose mathematics in the mother was half-German. I started to first place? Did you expect that your result that would study mathematics at the University of Because I was not happy doing anything eventually find its way into almost all books else! on graph theory? No! And I did not consider that the topic How did you conceive the idea of your had reached its final form by 1964. For famous theorem? example, I looked for algorithms and had In Novosibirsk I started to work on a prac- many open problems. In 1968 I published tical problem that involved colouring the a paper on ‘Some unsolved problems in wires of a network. To solve the problem I graph theory’ (English translation in studied a theorem of Shannon from 1949 Russian Math. Surveys, 1968), summarising (that the edges of any p-graph can be these and many other problems. Some are coloured in 3d/2 colours). Through now classical and still unsolved, like the Shannon’s theorem I got interested in total graph colouring conjecture (posed more theoretical questions. independently by Behzad). Shannon’s Theorem is best possible for p-graphs in general, but I asked myself The Total Graph Colouring Conjecture what the situation would be for graphs (Vizing 1964, 1968, Behzad 1965). The without multiple edges. I improved vertices and edges of a graph with maximum Shannon’s bound stepwise. At one point I degree d can be coloured in at most d + 2 had something like 8d/7, but eventually I colours so that no two adjacent or incident proved the best possible result, d + 1. The elements are coloured the same. Moreover, next step was to consider p-graphs. the vertices and edges of a p-graph with I sent the graph result to the prestigious maximum degree d can be coloured in at journal Doklady, but they rejected it. The most d + p + 1 colours so that no two adja- referee said that it was just a special case of cent or incident elements are coloured the Shannon and not interesting. They did same. Vadim Vizing in 1965 not understand it. So I published it locally 22 EMS December 2000 INTERVIEW There is only little social protection. The bureaucratic system has survived, but now without control. This has led to open cor- ruption. However, the market economy is devel- oping. Consumer goods are easily avail- able if you have money. If you are healthy and energetic you can earn much more and live better than before. I like the gen- eral development. Of course there are many mistakes, but the direction is right.

How often have you travelled outside the former Soviet Union? Three times, all of them to Denmark. Before Perestroika I had many invitations, more than twenty, but I was never allowed to go, not even to other socialist countries. I tried twice, but was stopped. It was hopeless. Very few people from the Ukraine could go. From Novosibirsk it was perhaps easier, but from Ukraine almost impossible, especially if you were not from Kiev. You were looked upon with suspicion if you wanted to travel. When I received foreign letters they were opened by the KGB and afterwards sent to Cover page of Discret Analiz. First page of Vizing’s 1964 paper me privately without comment.

What makes a mathematical result out- to graph theory. During my stay I solved What was your relationship to the commu- standing? a problem with Melnikov that was later nist party? A mathematician should do research and published in the Journal of Graph Theory. I was asked to join, but I never wanted to find new results, and then time will decide do so for political and moral reasons. I what is important and what is not! What did you like least before Perestroika? did not want to lose the freedom I had. I At first we had a police state. Then it am glad that I did not join, even though What were the most interesting periods in became bureaucratic. It ended in econom- my life in some ways might have been eas- your scientific life? ic failure. ier as a member. Definitely my years in Novosibirsk, when I worked in graph theory. And now, being How has life changed in Ukraine after What are your research plans? able to do research again with time to Perestroika? To work on graph-theoretical questions. think about unsolved problems. The In general, the direction is positive, but But great discoveries are not planned. I INTAS grant from the European Union there are many negative aspects also. will work and see where I get! has helped. [The INTAS grant is a 3-year grant initiated by the Technical University of Ilmenau, with participation from Odense, Nottingham, Odessa and Novosibirsk.]

How? I have retired. My pension is around $70 per month. This corresponds almost to my earlier salary, for which I had to teach up to 20 hours per week. I earned some extra money by writing a mathematics book for those wanting to pass a universi- ty entrance exam. The INTAS grant now gives me $45 extra per month, and it makes it possible for me to travel and meet colleagues. Last year in August we had an interesting meeting in Novosibirsk.

Have you carried out research during your years in Odessa? In 1976 I stopped my graph theory research and moved to scheduling. I was writing a habilitation thesis and finished it in 1985. It did not work out, more for political and economical reasons than for scientific. It was partly my own fault. I could submit it now, but my interests have changed and I would rather use my time on something more useful. In 1995 I was invited to Odense for the first time. This motivated me to go back Vadim Vizing at Odense with Gregory Gutin and Bjarne Toft EMS December 2000 23 ANNIVERSERIES 20002000 AnniversariesAnniversaries John Napier (1550-1617) abdication of Mary not long afterwards, thynges which must shortly come to passe . . . and the coronation of her son James VI Happy is he that redith, and they that heare the John Fauvel which helped mark the Protestantisation of wordes of the prophesy . . . [Revelation Chapter Scotland. The next we hear of Napier him- 1, Tyndale translation] Scotland has produced many creative and self is in the early 1570s. His father remar- – and given that in the succeeding 1500 or influential mathematicians – one thinks of ried in 1570 (Napier’s own mother had so years some of the predicted events must James Gregorie, James Stirling, Colin died shortly after he went to St Andrews), have happened, then this gives clues about Maclaurin and their many successors – but and Napier himself married Elizabeth how to match up the language of predic- arguably the greatest and most original of Stirling in 1573, receiving the Merchiston tion with the historic record. So what all was the first Scottish mathematician of estate from his father as part of the wed- Napier was seeking to establish was a func- international renown, John Napier, who ding settlement. tion, if you like, between two continua: the was born 450 years ago. Napier was indeed There are five books in Napier’s textual historic time-line from the time of Christ the first Scottish mathematician that we corpus, which were all first printed in onwards, and the narrative time-line of St know about, and it is extraordinary that he Edinburgh: John’s vision as presented in the created mathematics of the highest quality – Napier’s first and indeed best selling Apocalypse which is being mapped onto it. from within a country with no other math- book in its day was A plaine discovery of the To evaluate the functional correlation, he ematicians, with no mathematical tradi- whole Revelation of St John, published in had to make considered judgements about tion, and plunged into religious, political 1593. This anti-papist tract made his repu- what trumpets are, what seals are, what and social feuding. As his descendant Mark tation as a leading theologian, and went candlesticks are, and so on, the conclusions Napier wrote in 1834: into numerous editions in many languages. of which he presented in a series of 36 As for Scotland, until Napier arose, it was – His next book, which did not appear for numbered propositions. Once the function only famed for mists that science could not pen- another twenty-one years, was on a quite is established, from the information about etrate, and for the Douglas wars, whose baro- different subject. Mirifici logarithmorum the past which you have, you are then in a nial leaders knew little of the denary system canonis descriptio, of 1614, ‘Descriptio’ for position to use the correlation to work out beyond their ten fingers. short, was the book that introduced loga- the things you don’t know – in particular, Born in 1550, Napier was the eldest son rithms to the world and established his the date of the last judgement. in a wealthy and well-connected family who reputation among mathematicians across I’ve described Napier’s procedure in his had been playing an increasingly impor- Europe. Plaine discovery in this functional way in tant part in Scottish court and civic life – His next book, in 1617, the year he died, order to point up the similarities with what over the hundred years leading up to his was called Rabdologiae. This was not he was later doing in constructing loga- birth. His parents Sir Archibald Napier about logarithms but about other devices rithms. Napier constructed logarithms and his wife Janet Bothwell were both and means of calculation. through considering two moving points, P barely 16 when their son was born, and – Two years after Napier’s death, in 1619, and L, say, moving along a finite and an from the start John Napier was living in an his son Robert brought out from his infinite line respectively, in such a way that atmosphere of political and religious dis- manuscripts a companion work, as it while L is moving at constant speed, in putation and intrigue: the Scottish were, to the 1614 Descriptio, called arithmetical progression, P is moving geo- Reformation was in full spate and Sir Mirifici logarithmorum canonis constructio, metrically, it’s slowing down, in the origi- Archibald was strongly on the Protestant ‘Constructio’ for short, which explained nal construction, its speed being propor- side, as his son was to be. This wasn’t mere- how logarithm tables were constructed. tional to the distance it still has to go. Then ly a theological but also a political-cum – Finally, 220 years later, another descen- he defined the logarithm of the distance P constitutional position, given the swirl of dant, Mark Napier, edited more of his had still to go as the distance the other intrigue surrounding the Catholic Queen papers under the name of De arte logisti- point L has travelled. The idea that multi- Mary, James V’s daughter, and her ca (1839). plication of terms in a geometric progres- Protestant-inclining son James VI (as he sion correspond to addition of terms in an became). [First editions of the Descriptio, Rabdologiae arithmetical progression had long been At the age of thirteen, young John was and De arte logistica, as well as early editions familiar, from Greek times if not earlier. sent to the University of St Andrews, where of the other two, were in the Turner The fresh insight that Napier brought was he lodged with the principal, John Collection at Keele University, UK, before to situate this in two continuous move- Rutherfurd, and where he tells us he devel- that university secretly sold off the collec- ments – he even uses the word ‘fluxion’ at oped his theological interests and strongly tion to a second-hand book dealer for a one point – so that he could make infer- anti-Papist views. There is no record of mess of pottage (see EMS Newsletter 31, ences about one from what happened on Napier graduating from St Andrews, and it pp.10-12, and 32, pp.14-15.)] the other. is supposed that he probably went to study Napier’s fame in his own day was as the So in very broad terms, both the Plaine abroad, as was fashionable among young author of A plaine discovery of the whole discovery and Napier’s construction of loga- Scots of his generation and class. He may Revelation of St John. This remarkable best- rithms involve functional relationships well have studied in Paris, where he would seller explains such pressing issues as just between two continua, using information have had an opportunity to develop his why the Pope is the Antichrist and how we from one to make deductions about the mathematical knowledge, and perhaps in know that judgement day will fall between other. It might be ill advised to push this Geneva too, where he could have learned 1688 and 1700. It is worth more attention parallel too far, but both are examples of Greek in a fiercely Protestant environ- from historians of mathematics than it has Napier’s overwhelming characteristic, his ment. received, if less for its conclusions then for lateral thought in the service of making His being out of the country during the the process by which he reaches and calculations easier. In some ways he was a latter 1560s meant that he missed the explains those conclusions. computationalist, a calculator, even more excitements at the Scottish court such as Given the assumption that the text of the strongly and more pervasively than he was the murder of Queen Mary’s secretary book of Revelation contains predictions the inventor of logarithms. David Rizzio, the murder of the Queen’s about the subsequent course of human his- Part of his subsequent success and fame husband Lord Darnley, the Queen’s mar- tory – which is not an unfair inference echoing down the ages is due to luck. It was riage to the Earl of Bothwell (the wedding from the opening words: amazing good fortune, which he could not ceremony being performed by Napier’s The revelacion of Jesus Christe, which God have anticipated, that logarithms turned uncle the Bishop of Orkney), the forced gave unto him, for to shewe unto his servauntes out in the course of the century after his 24 EMS December 2000 ANNIVERSERIES death to be not only a calculating device before that. Thus the concept was old but than multiplication in base 10, which of for astronomers and navigators – doubling its physical realisation was new, demon- course computers got around to realising the life of astronomers, as Laplace strating Napier’s lateral technological 350 years later. So in Napier’s procedure remarked – but also central to the devel- thinking. Rabdologiae explains how to con- decimal numbers are converted into bina- opment of mathematics itself. Already by struct the rods as well as how to use them ry, the operation is carried out (multiplica- the 1650s and 1660s it was becoming clear for multiplication and division, taking tion, division or whatever) and then the result is converted back into a decimal number. Notice two things. One is that this transformation of base is really quite radi- cal and innovative – no-one else had done this kind of thing before. The other thing you might notice is that the process of con- verting into different numbers, carrying out your operation and then coming back, is structurally the same as the logarithmic procedure; and indeed, one might argue, of his theological procedures. Why do we remember John Napier? His deep significance may be that, along with others of his time, Napier was a central fig- ure in the transformation of the mediaeval into the modern world-view, in a very spe- cific way arising from his deep concern for computation and calculating effectiveness. We know the immediate context of loga- rithms and why they were taken up so widely and so rapidly: the need for ways of doing mathematical calculations was becoming evident to the navigators and others who were beginning to lay the foun- dations for the British imperial adventure. For some years, a century or more, it was increasingly clear that European expan- sion, geographically, in military engineer- ing, in terms of trade and business prac- tices, was predicated upon better mathe- matical skills. Napier happened to be working at a time when the idea of quan- tification was settling deep into the mind- set of the movers and shakers of Renaissance Europe, and supplied a num- ber of justifications for considering that how you handle and compute with num- bers is a really important issue. In some ways there was nothing else like this con- ceptual revolution in the applicability of mathematics to the world until the statisti- cisation of inquiries in the 19th century. Bibliography that logarithms were much deeper mathe- square roots and cube roots, and doing the Baron, Margaret, ‘John Napier’, Dictionary of scientific matical objects than their initial motivation rule of three. These became very popular biography ix, Charles Scribner’s Sons, New York, 1974, might suggest, relating to the area mea- and there are still many sets of rods, gen- 609-613. sure of hyperbolas, and thus a vital tool for erally in wood or ivory, in our museums. Bryden, D. J., Napier’s bones: a history and instruction the integral calculus, as well as being The promptuary was a more complicated manual, Harriet Winter Publications, 1992. thought of as an infinite series, which and more powerful modification of the Fauvel, John, ‘Revisiting the history of logarithms’, opened up another great swathe of mathe- rods, enabling ready handling of much Learn from the masters! (ed. F. Swetz, J. Fauvel et al.), matical analysis. The fact that we still teach larger numbers. It uses flat cards rather Mathematical Association of America, 1995, 39-48. the logarithm function to young people than rods, but with rather similar markings Knott, Cargill Gilston (ed.), Napier tercentenary memorial who wouldn’t have a clue how to use loga- and factorings. It is sufficiently sophisticat- volume, Edinburgh, 1915. rithm tables or even what they are for, ed that it has been called ‘the first calculat- Macdonald, William Rae, A catalogue of the works of John indicates how Napier’s invention has tran- ing machine’, though it’s not quite a Napier, pp.101-169 of his translation of Napier’s scended its original use and purpose. machine as we usually understand the Constructio, Edinburgh, 1889. Napier’s genius was fundamentally that term, its operation depending on quite a Napier, John, Rabdology (tr. W. F. Richardson), MIT of an amazingly gifted and innovative cal- lot of manual manipulation. The only Press, 1990. culator. His all-pervading interest in calcu- known example of a promptuary from the Sherman, Francis, Flesh and bones: the life, passions and lating showed itself especially strongly in time of Napier is in the Archeological legacies of John Napier, Napier Polytechnic, 1989. his remarkable little book Rabdologiae Museum in Madrid, and was only recog- which appeared in 1617, the year of his nised for what it is in the last twenty years. John Fauvel is Senior Lecturer in Mathematics death. This described three inventions for The third of Napier’s calculational at the Open University, UK. This article is aiding calculations, the so-called Napier’s devices, his chessboard abacus, is the most based on a lecture to commemorate the 450th rods, the promptuary, and a chessboard abacus. innovative and of greatest conceptual anniversary of John Napier’s birth, given at Napier’s rods, or Napier’s bones, are a phys- interest, even though he described it as Napier University, Edinburgh, on 1 December ical realisation of an old method of multi- ‘more of a lark than a labour’. The funda- 2000. The event was organised jointly by plying numbers, known since the middle mental insight is that multiplication of Napier University and the International Centre ages in Europe and maybe in India long binary numbers is more straightforward for Mathematical Sciences, Edinburgh. EMS December 2000 25 SOCIETIES Currently the UMI organises a general meeting every four years, with a wide turn- out of mathematicians from Italy and over- LL’Unione’Unione MatematicaMatematica ItalianaItaliana seas, in one of the cities where there is a mathematics department. Under the Giuseppe Anichini influence of Pincherle, the Bollettino dell’Unione Matematica Italiana was found- ed: at the beginning there were two sec- The UMI (Italian Mathematical Union) (1891), the American Mathematical Society tions devoted to ‘Short communications’ was established in 1922. On 31 March of (1891) and, particularly, the International and ‘Abstracts of papers published in other that year, Salvatore Pincherle, an eminent Mathematical Union (1920). journals, Letters to the Society, News from mathematician from Bologna, circulated a Before long, distinguished members, the members, Book reviews, etc.’. In 1939 letter to all Italian mathematicians in among them Luigi Bianchi and Vito a special section devoted to the history of which a possible national mathematical Volterra, strongly supported Pincherle’s mathematics and to mathematical educa- society was proposed. In July a tentative initiative and they were presenting papers tion was introduced. issue of the future Bollettino was published. in the first issues of the formative journal. Since then the Bollettino has increased About 180 mathematicians supported the One of the major achievements of the substantially the number of papers of out- proposal, and in December the first meet- UMI during this period was the organisa- standing scientific quality. At present ing was held and the first of the Society’s tion of the meeting of the International there are two sections of the Bollettino by-laws received its approval. The UMI Mathematical Union (IMU) in Bologna in (Sections A and B) in which expository membership swelled to 400 by 1940 and 1928. Not only was the organisation car- papers and high-level scientific papers are currently stands at over 2700. The regis- ried out effectively, but a difficult political published. tered office remains in Bologna, in the question was successfully confronted. Recent years have seen a huge increase Department of Mathematics. Salvatore Pincherle, elected IMU in the diversity of activities in which the

Four presidents: (from left to right) A. Figà-Talamanca, C. Pucci, E. Magenes and C. Sbordone

The goals of the UMI were to communi- President in 1924, had successfully worked Unione is involved, from education, to pop- cate the nature of the mathematical sci- to get together all those with a keen inter- ularisation, to institutional policy. The ences and how mathematics contributes to est in mathematics, despite their nationali- UMI has encouraged, and will continue to society, and to promote the understanding ty, by overcoming a consequence of the do so, the participation of research mathe- of mathematics by publishing high-level First World War. A strong difference of maticians in the reform of mathematics papers, by encouraging the output of high- opinion existed between French mathe- education at university level and at any quality expositions for members and stu- maticians and the mathematicians of other high-school level. The UMI, in coopera- dents at all levels, and by organising meet- countries, mainly concerning the invitation tion with the MPI (Italian Department of ings in order to stimulate the production to the German delegation. In the event, Education) and the MURST (Italian and the exchange of ideas. around 840 mathematicians assembled in Department of University and Research), The formation of the UMI was inspired Bologna, and besides 336 people from supports efforts to review and reform the by the existence and the increasing growth Italy there were 76 mathematicians from undergraduate mathematics curriculum in of other mathematical societies, such as the Germany, 56 from France and 52 from the response to current changes in the world. Société Mathématique de France (1872), United States. The opening address was in Traditionally the UMI supports, as an the Deutsche Mathematik Vereinigung Latin. institutional task, the Committee for 26 EMS December 2000 SOCIETIES (approximately 4100 euros).

Publications Apart from the Bollettino, several other publications are produced by the UMI. Since 1974 the Notiziario dell’UMI has appeared. This presents news of interest, about meetings, prizes and awards, educa- tion, Ph.D. achievements, and so on. There are ten issues each year, with sup- plements on special occasions. The Unione Matematica also edits a series of Quaderni: a series of textbooks for young researchers aimed at arguments outside the usual mathematical path to the Ph.D. degree; a series of monographs covering a wide range of subjects in mathematics; a series of Opere dei Grandi Matematici includ- ing all (or a selection) of the papers of well- known Italian mathematicians. Among these have been Felice Casorati, Paolo Ruffini, Luigi Bianchi, Leonida Tonelli, Ulisse Dini, Giuseppe Peano, Gregorio Ricci-Curbastro, Vito Volterra, Ernesto Cesaro, Corrado Segre, Guido Fubini, Giuseppe Vitali, Renato Caccioppoli and Salvatore Pincherle.

The structure and the members The UMI has an Executive Committee of four elected members: the President, the Vice-President, the Treasurer and the Secretary. The Scientific Committee con- sists of 19 members: those of the Executive Committee and 15 other elected members; elections take place every four years. The Scientific Committee often nominates spe- cial committees for specific reasons (math- ematics education, Publications, the teach- ing of non-degree-level mathematics, etc.). Of the 2700 members, many are university researchers, while many others are school- teachers or belong to industries or to pub- Professor Carlo Pucci, Honorary President lic research centres. Finally we recall the Presidents of the Mathematics Education, the Committee ory of his son Giuseppe. This prize is Unione. for Research and the Education of awarded every two years in recognition of After Salvatore Pincherle, the founding Mathematics in Engineering, and the an outstanding contribution to mathemati- President, the Presidents of the UMI have Committee for the Mathematical Olympic cal research by an Italian mathematician of been Luigi Berzolari, Enrico Bompiani, games. no more than 33 years old. The prize Giovanni Sansone, Alessandro Terracini, amounts to three million Italian lira Giovanni Ricci, Guido Stampacchia, Prizes and awards (approximately 1500 euros). Enrico Magenes, Carlo Pucci, Vinicio Since the sixties the Unione has sought to The Premio Franco Tricerri was estab- Villani, Alessandro Figà Talamanca and promote and reward mathematical lished in 1995, using funds collected by Alberto Conte; the current President is achievements, mainly for young people, by colleagues, friends and students of Franco Carlo Sbordone. In addition, Enrico means of prizes and awards. Currently Tricerri, Professor of Geometry at the Bompiani (1952-75) and Carlo Pucci (from four prizes are awarded by the UMI. University of Firenze, who tragically died 1995) were appointed Honorary These prizes are awarded on the recom- in a plane accident in China in 1994. This Presidents by plenary meetings of the mendation of a Committee especially prize is awarded every two years in recog- members. appointed by the Ufficio di Presidenza nition of an outstanding contribution to The author is very indebted to the fol- (Officers of the Society) of the UMI. differential geometry by a graduate of not lowing papers for information: The Premio Renato Caccioppoli was estab- more than 3 years’ standing in mathemat- Carlo Pucci, L’Unione Matematica Italiana lished in 1960 with a donation by his fam- ics or physics. The prize amounts to two dal 1922 al 1944: documenti e riflessioni, ily in memory of Renato Caccioppoli, late million Italian lira (approximately 1000 Symposia Mathematica XXVII, Istituto Professor of Mathematical Analysis at the euros). Nazionale di Alta Matematica Francesco University of Napoli. This prize is award- The Premio Calogero Vinti was established Severi, Roma, 1992, 187 pages. ed every four years in recognition of an in 1998 with a donation by the family and Giovanni Sansone, Le attivit dell’Unione outstanding contribution to mathematical former students of Calogero Vinti, Matematica Italiana nel primo cinquanten- analysis by an Italian mathematician of no Professor of Mathematical Analysis at the nio della sua fondazione, Bollettino UMI, more than 38 years old. The prize University of Perugia. This prize is award- suppl. fasc. 2, Bologna, 1974, 8 pages. amounts to ten million Italian lira (approx- ed every four years in recognition of an imately 5200 euros). outstanding contribution to mathematical Giuseppe Anichini is Professor of Mathematical The Premio Giuseppe Bartolozzi was estab- analysis by an Italian mathematician of no Analysis in the Engineering Faculty of Firenze, lished in 1969 with a donation by Professor more than 40 years old. The prize Italy, and has been Secretary of the Unione Federico Bartolozzi and his family in mem- amounts to eight million Italian lira Matematica Italiana since 1988. EMS December 2000 27 EULER PROJECT contents as a freely accessible resource in the project. The EMS provides its The EULER project: Electronic Library of Mathematics as a resource, distributed through its EMIS sys- achievements and continuation tem of Internet servers; scientific co-ordi- nation of this library is currently organised Laurent Guillopé (Nantes) and Bernd Wegner (Berlin) with the Department of Mathematics of the Technical University of Berlin, the final In this article we report on the EULER good Internet service and a self-explaining partner of the EULER project. project, which has developed a web based structure. Users have one single entry search engine for distributed mathematical point to start their information search; this The EULER service sources. The main features of this EULER entry point contains browsing indices of Based on the structure described above, a prototype are uniform access to different authors and keywords, form-based search- subgroup of the current EULER partners sources, high precision of information, de- es for authors, titles and other relevant bib- decided to develop a service from the duplication facilities, user-friendliness and liographic information, and a selection of EULER prototype. It had been guaran- an open approach enabling participation different information sources. Good de- teed during the project work that the of additional resources. We describe the duplication facilities enable to display the EULER engine and other tools could be functionalities of the EULER engine availability of the same item at different installed at new sites, and thus the group report on the transition from the proto- sites within the same record. was able to go on with the current offer type developed in the project to a consor- from EULER. It is expected that all tium-based service in the internet. The partners of the EULER project resources made accessible during the pro- The currently accessible contents in the ject phase will remain open for the service, Aims and achievements EULER prototype are provided by the even if the corresponding partner cannot The aim of the EULER project was to pro- partners of the project. This group contribute further work to run the service. vide a system for strictly user-oriented, includes libraries from all over Europe, The current members of the group care integrated-network-based access to mathe- which represent several different types of about improvements to the EULER offer matical publications. The period for the libraries: the State Library of Lower and handle software problems potentially project terminated in September 2000. Saxony and University Library of coming up with new partners. In particu- The EULER system has been designed to Göttingen and the J. Hadamard library lar, the administration of access control offer a ‘one-stop shopping site’ for users (University of Orsay) represent libraries (for resources beyond free metadata) is a interested in Mathematics. An integration with a national responsibility for collecting challenge for the future. This will lead to of all types of relevant resources has been all publications in pure mathematics; the an essential improvement of the document taken into account: bibliographic databas- library of the Centrum voor Wiskunde en delivery facilities, and will motivate scien- es, library online public access catalogues, Informatica (Amsterdam) represents the tific publishers to support the system and electronic journals from academic publish- typical research library of a national make their contents accessible through ers, online archives of pre-prints and grey research centre; the University of Florence EULER. literature, and indexes of mathematical represents a typical university library with But also as a free search portal EULER Internet resources. They have been made its distributed department libraries; the will provide a very useful gateway to math- interoperable, using common Dublin Core library of the Institut de Recherche ematics. The present aim is to get more based metadata descriptions. A common Mathématique Avancée (University of libraries interested in participating in user interface, called the ‘EULER engine’, Strasbourg) represents a typical library of EULER. This means that they should assists the user in searching for relevant an important mathematical institute. make their catalogues accessible by provid- topics in different sources in a single effort. In addition, a partner specialised in dig- ing the metadata of their holdings. The As a matter of principle, the EULER sys- EULER engine will include their resource tem has been designed as an open, in the searches made by the user, and users scaleable and extensible information sys- will get a bigger choice of providers where tem. Mathematicians and librarians from they may ask for a copy of the documents mathematics in research, education and they are interested in. The de-duplication industry will be the main users and check will provide them with comprehen- providers of such an enterprise. sive lists of where to find the documents (a EULER is an EMS initiative and espe- book, journal article or other source), and cially focuses on real user needs. The pro- having a good coverage of the main ject has been funded by the European libraries in Europe, they will probably get a Union within the programme ‘Telematics reference for where they will be offered the for Libraries’. Standard, widely used and resource at very low cost. non-proprietary technologies such as Several additional libraries are already HTTP, SR/Z39.50 and Dublin Core (DC) interested in discussing participation in are used. Common resource descriptions the EULER service. A definite decision of document-like objects enable inter- will take some time and the preparation of operability of heterogeneous resources. Bernt Wegner with Einstein the metadata possibly even more, but most One of the main achievements of the pro- of these first contacts are very promising. ject is the development of a DC-based ital libraries and net-based information is The first bricks of a European catalogue of metadata structure that can be used as a represented by NetLab, the Research and mathematics resources in the libraries have common target into which the metadata of Development Department at Lund been installed, while others will be added. the given resources could be converted. University Library: they give a large set of Access to EULER is available through At distributed servers, multi-lingual classified internet resources, complement- EMIS, the Information Service of the EULER service interfaces are provided as ing a similar collection, the ‘Math Guides’, European Mathematical Society on the entry points to the EULER engine, offer- organised by the Göttingen Library. web (with 39 servers world-wide, see ing browsing, searching, some document MathDoc Cell (Grenoble), as a national www.emis.de). delivery and user support (help texts, tuto- centre for coordination and resource-shar- rial, etc.). The interface is based on com- ing of mathematics research libraries in Laurent Guillopé (e-mail: laurent.guillope mon user-friendly and widely used web France, contributes also in giving metada- @math.univ-nantes.fr) is at the Université de browsers (public domain or commercial), ta of its national indexes on preprints and Nantes; Bernt Wegner (e-mail: wegner@math. such as Netscape. The multi-lingual user thesis. Via the partner FIZ Karlsruhe, tu-berlin.de) is at the Technische Universität interface has the common features of every Zentralblatt MATH provides a part of its Berlin. 28 EMS December 2000 PRICE SPIRAL TheThe PPricerice SpiralSpiral ofof MathematicsMathematics JournalsJournals andand WhatWhat toto DoDo AboutAbout ItIt Ulf Rehmann (Bielefeld)

The material given here was presented by the more per year was during a time, when, in the lication: every mathematician should know author in a talk at the annual assembly of the western world, the average price inflation was more about journal prices. For that pur- chairmen of German Math departments usually below 2% or so. And this is true both for pose I will, in accordance with the AMS, (KMathF) in May 2000, and also in a talk at the price increase per volume and for the price annually update the price tables as soon as the DMV Jahrestagung in September 2000, increase per page! new data is available, and I hope this will both in Dresden; see the website http:// My conclusion is that mathematicians help others to make the right decisions www.mathematik.uni-bielefeld. are funny consumers: they buy the materi- concerning their local library budget. de/\char126rehmann/BIB/ al which they produce themselves from We also should take appropriate deci- In common with scientists of other disci- people – the commercial scientific publish- sions ourselves when acting as author, ref- plines, many of us mathematicians are, ers – who do nothing other than distribute eree or editor, asking ourselves: Why are concerned about the rapid price increase that material at prices that increase beyond we submitting to an expensive journal? for scientific journals. Recently a major any reasonable measure. Not only that: Why are we refereeing for it? And if you reduction in the library budget for the mathematicians work hard for the publish- are an editor, why are you not taking any Mathematical Library at Bielefeld ers, usually without pay, by acting as their measures to produce the journal by your- University forced me, as the person editors, collecting and refereeing the self? responsible for our departmental library, material written by their colleagues, and as Meanwhile, there are successful journals to take some measures to decide which authors, by perfectly typesetting their run by mathematicians themselves, such as journals we should cancel. Since many manuscripts, leaving almost nothing to do Geometry and Topology, The Electronic Journal departments are in a similar position, I for the publishers but count their profits. of Combinatorics, or DOCUMENTA MATH- think it may be useful to publicise the We have a really strange situation: it EMATICA, just to mention a few among information that I gathered for my depart- seems that serious people are willing to many. For example, I proved that, using ment. accept such price differences. For exam- the facilities of DOCUMENTA MATHE- As a first step I listed all the mathemati- ple, consider the following information MATICA it was possible to produce a seri- cal journals at Bielefeld, in a table on the (provided by the publishers themselves): ous work such as the ICM98 Proceedings in Web, including the publishers, the 1998 in 1999 ‘Inventiones Mathematicae’ published shorter time, with better quality, and for price, and also some information from the 2894 pages for US$2760, a price per page of much less money, than most of the earlier citation index ISI: http://www.isinet.com/, US$0.95, while ‘Annals of Mathematics’ pub- productions of ICM proceedings. To do such as the number of citations of that lished 2294 pages for US$220, a price per page such things is not hard nowadays, since journal and the average impact of each of US$0.10. many electronic tools are at our fingertips: article, insofar as this information was I chose these particular journals, since I I gave a public description of that produc- available to us. think that they have a similar reputation. tion process at the Berkeley Workshop on With a little perl script, I made this list But when I mention these figures to col- The Future of Mathematical Communication: more transparent by ordering it with leagues, many are surprised by the drastic 1999, including financial details and the respect to various data: by publisher, by price difference. This is not an isolated sit- technical tools used: see the website price, or by ISI number, so by a mouse uation: checking the tables will show you http://www.mathematik.uni-bielefeld. click, I could locate the most expensive several similar cases. de/\char126rehmann/EP/index.html. And I journals, those with the strongest impact A typical pattern might occur to you guess most of our colleagues working factor, and so on. I then made this list when you scan these tables. Journals that actively in the area of publication are will- public, not only to my department, but also are cheap are very often produced by ing to share their experiences and knowl- to some colleagues and librarians world- learned societies or by universities, while edge in order to support similar projects wide, on the website http://www.mathe- expensive journals are produced by private by others. matik.uni bielefeld.de/\char126rehmann/BIB/, publishers. (Using the word “produced” and received a good response. In particu- here is often an abuse, since I pointed out lar, I learned that at the same time the above that the production is essentially American Mathematical Society (AMS) had done by the mathematicians themselves, 2001 anniversaries collected data on about 250 journals, while the publisher just does the distribu- The following mathematicians have including their respective numbers of tion.) anniversaries during 2001. pages and prices for the years 1994-99: see Another fact might strike you. Whatever If you would like to write an anniver- the website . Since that table was not very you might think about citation indices and sary article about any of them, please transparent at a first glance I decided, with impact factors, at least they don’t seem to contact the Editor. permission from the AMS, to extract the provide any arguments for preferring Muhammad al-Tusi (b. 1201) high-priced journals above others. If they data in a similar way as I did with the Girolamo Cardano (b.1501) Bielefeld list, using some perl script to do suggest anything, it seems to be the oppo- computations of derived data, such as the site: if you click on the list ordered by Pierre de Fermat (b. 1601) price per page and the price increases over ‘impact’ (see the website http://www.mathe E. W. Tschirnhaus (b. 1651) the years, and also to sort the table accord- matik.uni- bielefeld.de/\char126 Mikhail Ostrogradsky and Julius ing to these data. rehmann/BIB/impact.html, you will find at Plücker (b. 1801) These tables contained some surprises. I the top many journals run by learned soci- Carl Jacobi (d. 1851) learned that these 250 journals published eties or universities and offered at moder- 323,786 pages of refereed mathematics in ate prices. Charles Hermite and Peter Guthrie 1999. I also learned to my great surprise This situation is no longer acceptable. Tait (d. 1901) that many journals had an average annual So what is to be done? It is certainly nec- Richard Brauer, P. S. Novikov, I. G. price increase of 15% or more during the essary for us all to become better acquaint- Petrovsky and A. Tarski (b. 1901) last five or six years. This inflation of 15% or ed with the facts concerning scientific pub- EMS December 2000 29 DIGITAL MODELS

DigitalDigital modelsmodels andand computercomputer-assisted-assisted prproofsoofs Michael Joswig and Konrad Polthier

The first collection of reviewed electronic thetic arguments. On the other hand, the mats we want to ensure that the server’s geometry models is available online at the inherent property of a proof is its verifia- information can be kept up to date on a new Internet server http://www.eg-models.de bility; that is, verifiable by someone who is technical level. Additional files in arbitrary [1]. This archive is open for any geometer sufficiently trained. But this very property formats are welcome for explanatory pur- to publish new geometric models, or to of a proof might be challenged in individ- poses. browse this site for material to be used in ual cases, where a computer is involved to The Electronic Geometry Models Server education and research. Access to the serv- solve a task too arduous or too tiring for opened in November 2000. er is free of charge. any human. We are not going to raise the The geometry models in this archive general question about the development of References cover a broad range of mathematical topics the mathematical culture, but we do 1. Electronic Geometry Models, http://www.eg- from geometry, topology and, to some believe that the installation of a server for models.de extent, numerics. Examples are geometric mathematical models can help to improve 2. Udo Hertrich-Jeromin, Isothermic cmc-1 surfaces, algebraic surfaces, topological the transparency of computer assisted Cylinder, Electronic Geometry Models, No. knots, simplicial complexes, vector fields, proofs. For instance, think of a proof that 2000.09.038, DarbouxSphere_Master.jvx. curves on surfaces, convex polytopes, and is established by a computer construction 3. Michael Joswig and Günter M. Ziegler, A in some cases, experimental data from of some complicated geometric shape. A neighborly cubical 4-polytope, Electronic finite element simulations. standardised description, independently Geometry Models, No. 2000.05.003, All models in this archive are reviewed checked by experts and available to every- C45_Master.poly. by an international team of editors. The one, would provide an criteria for acceptance follow the basic enormous potential for rules of mathematical journals and are validation. based on the formal correctness of the data Using the digital set, the technical quality, and the mathe- models, interested matical relevance. This strict reviewing mathematicians can process ensures that users of the EG- verify the claims on Models archive obtain reliable and endur- their own, using appro- ing geometry models. For example, the priate software of their availability of certified geometry models choice. Moreover, allows for the validation of numerical once there is a model experiments by third parties. All models available, it is possible are accompanied by a suitable mathemati- to perform one’s own cal description. The most important mod- computational experi- els will be reviewed by the Zentralblatt für ments on this data set. Mathematik. This could be a numer- We are advocating the construction and ical evaluation as well submission of digital geometric models as a search for another from various areas of mathematics. The property yet to be advantages of these digital models go analysed for this beyond those of the classical plaster shapes model. Darboux transform of a spherical discrete isothermic net [2]. and dynamic steel models of earlier days. Each model comes Given the data it is easy to verify that this describes an isother- At the end of the 19th century several with a detailed descrip- mic surface. Additionally, it can be checked that this surface has mathematicians felt the need to handle tion that identifies the discrete constant mean curvature. physically the geometric objects they author, explains the thought about. In particular, Felix Klein mathematical purpose, and Hermann Amandus Schwarz in and includes references Göttingen built many models of curves, to other sources of surfaces and mechanical devices for teach- information. Each ing and other educational purposes. model has a unique What are the main reasons for today’s identification number mathematicians to construct digital models for unambiguous cita- of geometric shapes and make them avail- tion. Each model is able via the EG-Models server? There are equipped with quali- obvious educational aspects, as for the his- fied metadata informa- torical models, and the means of interac- tion, and therefore, the tive visualisation are definitively useful for archive can be scientific purposes, too. searched via special- But the focus of this article is another, ized search engines somewhat different, view. Nowadays com- such as those from puter generated or assisted proofs enter EMIS and MathNet /. virtually all areas of mathematics, and still Each model itself is the majority of the mathematicians are represented by a mas- reluctant to accept the validity of such ter file from a fixed set results. On the one hand, it seems some- of file formats, includ- what strange to abstain completely from ing XML formats spec- Schlegel diagram of a cubical 4-polytope whose graph is iso- using tools such as the computer for doing ified by DTDs. By morphic to the graph of the 5-dimensional cube [3]. mathematics, disregarding, maybe, aes- restricting the data for- 30 EMS December 2000 CONFERENCES (Poland), E.I. Moiseev (Russia), A.M. Nakhushev (Russia), Yu.V. Obnosov (Russia), Ja.V. Radyno (Belarus), F. Rebbani (Algeria), FForthcomingorthcoming conferconferencesences M. Reissig (Germany), O.A. Repin (Russia), M. Stojanovich (Yugoslavia), J.J. Trujillo Compiled by Kathleen Quinn (Spain), N.A. Virchenko (Ukraine), Vu Kim Tuan (Kuwait), N.I. Yurchuk (Belarus), L.A. Yanovich (Belarus), P.P. Zabreiko (Belarus), Please e-mail announcements of European confer- Joel Lebowitz (New Brusnwick), Francois E.I. Zverovich (Belarus) ences, workshops and mathematical meetings of Ledrappier (Paris), Russ Lyons Languages: Russian, English interest to EMS members, to k.a.s.quinn@ (Bloomington), Gregory Margulis (New Call for papers: the deadline for one-page open.ac.uk. Announcements should be written in a Haven), Fabio Martinelli (Rome), Stansilav abstracts is 1 January; see Web site below style similar to those here, and sent as Microsoft Molchanov (Charlotte), Sergei Nechaev Programme committee: V.I. Korzyuk Word files or as text files (but not as TeX input (Paris), Amos Nevo (Haifa), Yuval Peres (Belarus), L.A. Aksent’ev (Russia), V.I. files). Space permitting, each announcement will (Jerusalem), Ben-Zion Rubshtein (Beer- Burenkov (UK), P. Butzer (Germany), R. appear in detail in the next issue of the Newsletter Sheva), Laurent Saloff-Coste (Ithaca), Oded Gorenflo (Germany), V.I. Gromak (Belarus), to go to press, and thereafter will be briefly noted in Schramm (Rehovot), Jeff Steif (Goeteborg), V.A. Kakichev (Russia), V.S. Kiryakova each new issue until the meeting takes place, with a Toshikazu Sunada (Sendai), Domokos Szasz (Bulgaria), G.S. Litvinchuk (Portugal), O.I. reference to the issue in which the detailed (Budapest), Balint Toth (Budapest), Anatoli Marichev (USA), S.A. Minyuk (Belarus), Yu.V. announcement appeared Vershik (St. Petersburg), George Willis Obnosov (Russia), Ya.V. Radyno (Belarus), (Newcastle, NSW) V.N. Rusak (Belarus), S. Rutkauskas January 2001 Programme: there will be two separate main (Lithuania), H.M. Srivastava (Canada), J.J. periods of activity in the first Trujillo (Spain), M.A. Sheshko (Poland), N.A. (February/March) and in the second Virchenko (Ukraine), L.A. Yanovich (Belarus), 8-18: ICMS Instructional Conference on (May/June/July) halves of the semester. The P.P. Zabreiko (Belarus), E.I. Zverovich Nonlinear Partial Differential Equations, first period will start with a two-week work- (Belarus). Organizing committee: Edinburgh, UK shop with the general theme Random Walks Academician I.V. Gaishun (Belarus), Information: and Statistical Physics, 19 February - 2 March Academician V.A. Il’in (Russia), A.V. Kozulin Web site: http://www.ma.hw.ac.uk/icms/currrent 2001. Towards the end of the second period (Belarus), A.A. Kilbas (Belarus), M.V. there will be another two-week workshop with Dubatovskaya (Belarus), S.V. Rogosin 28-3 February: 2001 XXI International the general theme Random Walks and Geometry, (Belarus), H. Begehr (Germany), H.-J. Seminar on Stability Problems for Stochastic 25 June - 6 July 2001 Glaeske (Germany), V.V. Models, Eger, Hungary Organising committee: Vadim A. Gorokhovik(Belarus), N.A. Izobov (Belarus), Information: Kaimanovich (Rennes, France), Klaus Schmidt N.K. Karapetyants (Russia), A. Kufner e-mail: [email protected] or (Vienna, Austria), Wolfgang Woess (Graz, (Czech), M. Lanza de Cristoforis (Italy), P.A. [email protected] Austria) Mandrik (Belarus), V.V. Mityushev (Poland), Web site: http://neumann.math.klte.hu/~stabil Site: Erwin Schrödinger Institute E.I. Moiseev (Russia), M. Saigo (Japan), S.G. http://bernoulli.mi.ras.ru Information: Samko (Portugal), A.A. Sen’ko (Belarus), N.I. [For details, see EMS Newsletter 36] e-mail: [email protected] Yurchuk (Belarus) Web site: http://www.esi.ac.at/Programs/rwalk Proceedings: to be published in Integral February 2001 2001.html Transform and Special Functions and in Proc. Inst. Math. (Minsk) 15-16: Workshop on Fractional Brownian Information: February – July: Random Walks special Motion: Stochastic Calculus and e-mail: [email protected] semester, Vienna, Austria Applications, Barcelona Web site: http://amade.virtualave.net Scope: the semester will be dedicated to vari- Speakers include: Coutin, Hu, Memin, ous problems connected with stochastic Mishura, Nualart, Oksendal, Qian, Russo, 19-23: New Trends in Potential Theory and processes on geometric and algebraic struc- Valkeila, Zaehle Applications, Bielefeld, Germany tures, with an emphasis on their interplay, Site: Facultat de Matemàtiques, Universitat de and also on their interaction with theoretical Barcelona 25-1 March: NATO Advance Research physics Information: Workshop: Application of Algebraic Topics: some of the focal points are: proba- Web site: http://orfeu.mat.ub.es/~gaesto/ Geometry to Coding Theory, Physics, and bility on groups; products of random matrices welcome.htm Computation, Eilat, Israel and simplicity of the Lyapunov spectrum; Information: boundary behaviour, harmonic functions and 15-19: Analytic Methods of Analysis and e-mail: [email protected] other potential theoretic aspects; Brownian Differential Equations (AMADE-2001), Web page: http://www.mat.uniroma2.it/ motion on manifolds; combinatorial and spec- Minsk, Belarus ~cilibert/workshop.html tral properties of random walks on graphs; Topics: integral transforms and special func- [For details, see EMS Newsletter 37] random walks and diffusion on fractals tions; differential equations and applications; Main speakers: Alano Ancona (Paris), Martine integral, difference, functional equations and March 2001 Babillot (Orléans), Martin Barlow fractional calculus; real and complex analysis (Vancouver), Itai Benjamini (Rehovot), Rob Main speakers: P. Adler (France), A.B. van den Berg (Amsterdam), Donald Antonevich (Belarus), A.E. Barabanov 18-24 Geometric Analysis and Index Theory Cartwright (Sydney), Davide Cassi (Parma), (Russia), H. Begehr (Germany), V.I. Burenkov Conference, Trieste, Italy Thierry Coulhon (Cergy), Bernard Derrida (UK), L. de Castro (Portugal), I.V. Gaishun Information: (Paris), Persi Diaconis (Stanford), Steven (Belarus), Yu.V. Gandel (Ukraine), H.-J. Web site: Evans (Berkeley), Alex Furman (Chicago), Glaeske (Germany), R. Gorenflo (Germany), http://www.sissa.it/~bruzzo/ncg2001/ncg2001. Hillel Furstenberg (Jerusalem), Rostislav V.V. Gorokhovik (Belarus), V.I. Gromak html Grigorchuk (Moscow), Geoffrey Grimmett (Belarus), V.A. Il’in (Russia), N.A. Izobov (Cambridge), Yves Guivarch (Rennes), David (Belarus), N.K. Karapetyanz (Russia), A. 26-29: Numerical Methods for Fluid Handelman (Ottawa), Pierre de la Harpe Karlovich (Mexico), A.A. Kilbas (Belarus), V. Dynamics, Oxford, UK (Genève), Frank den Hollander (Nijmegen), Kiryakova (Bulgaria), V.I. Korzyuk (Belarus), Aim: to bring together mathematicians, engi- Barry Hughes (Melbourne), Felix Izrailev M. Lanza de Cristoforis (Italy), A. Laurincikas neers and other scientists in the field of com- (Puebla), Michael Keane (Amsterdam), Yuri (Lithuania), F. Mainardi (Italy), L.G. putational fluid dynamics, to review recent Kifer (Jerusalem), Gregory Lawler (Durham), Mikhailov (Tadzhikistan), V.V. Mityushev advances in mathematical and computational EMS December 2000 31 CONFERENCES techniques for modelling fluid flows Jeu, Kenneth Falconer, Cameron Gordon, J.C. Wood (UK) Topics: all areas of CFD but with particular Alexander Ivanov, Mark Jerrum, Paul Martin, Programme: one-hour lectures and thirty- emphasis given to adaptivity, biomedical mod- Steffen König, Oleg Kozlovski, Ian Leary, Ran minute talks elling and innovative methods in CFD Levi, James McKee, Viacheslav Nikulin, Robin Call for papers: prospective speakers should Invited speakers: include: M.J. Baines Wilson contact M. Ville (e-mail below) (Reading), T.J. Barth (NASA Ames), J.-D. Registration: £30 before 26 February, £40 Programme committee: J. Eells (Cambridge); Benamou (INRIA-Rocquencourt), F. Brezzi afterwards L. Lemaire (Brussels); J.C. Wood (Leeds) (Pavia), S.M. Deshpande (IISC-Bangalore), C. Information: Department of Mathematics, Organising committee: M. Ville (Ecole Farmer (Geoquest), D. Kr”ner (Freiburg), R. University of Glasgow, Glasgow G12 8QW Polytechnique), E. Loubeau (Brest), S. LeVeque (Washington), D. Noble (Oxford), R. e-mail: Montaldo (Cagliari). Rannacher (Heidelberg), P.L. Roe (Michigan), Web site: http://www.maths.gla.ac.uk/bmc2001 Sponsors: CIRM, Ministries S.J. Sherwin (Imperial-London), E. Süli 15-21 : Spring School in Analysis, Paseky Proceedings: will be submitted to a publisher (Oxford), N.P. Weatherill (Swansea) nad Jizerou, Czech Republic Site: Centre International de Rencontres Programme: invited lectures, 20-minute con- Theme: Banach spaces Mathématiques, Luminy tributed talks and poster sessions. These will Main speakers: Joram Lindenstrauss (The Grants: for information on financial support, be selected mainly, but not exclusively, on the Hebrew University of Jerusalem), Israel contact M. Ville (e-mail below) basis of their likely contribution to the above Gideon Schechtman (The Weizmann Institute Deadlines: no deadline but limited number of themes of Science, Rehovot, Israel), Yoav Benyamini seats Organiser: this is the seventh international (The Technion, Haifa, Israel), Gilles Lancien Information: conference on CFD organised by the ICFD (Université de Franche-Comte, Besancon e-mail: [email protected] (Institute for Computational Fluid Dynamics), Cedex France), W. B. Johnson (not yet con- Web-site: http://beltrami.sc.unica.it/harmor/ a joint research organisation at the firmed, Texas A&M University, United States) Universities of Oxford and Reading Language: English June 2001 Organising committee: M.J. Baines Organising committee: Jaroslav Lukes, Jan (Reading), M.B. Giles (Oxford), M.T. Arthur Rychtar (Czech Republic) (DERA, Farnborough), M.J.P. Cullen Grants: probably support for a limited num- 4-9: Fractals in Graz 2001, Analysis- (ECMWF), M. Rabbitt (British Energy) ber of students Dynamics-Geometry-Stochastics, Graz, Prize: a feature of the meeting will be the Deadlines: for reduced fee, 15 January; for Austria third award of ‘The Bill Morton Prize’ for a support, 15 January [Loosely linked to a special semester on ran- paper on CFD by a young research worker. Information: dom walks at the Erwin Schrödinger Institute The Prize papers will be presented by the e-mail: [email protected] in Vienna. For further information see authors at a special session of the Conference Web site: http://www.karlin.mff.cuni.cz/ http://www.esi.ac.at/Programs/rwalk2001.html] and the prize will be presented at the katedry/kma/ss/apr01/ss.htm Theme: fractals Conference dinner Scope: analysis on fractals, fractals in dynam- Information: contact Mrs B. Byrne, Oxford May 2001 ics, geometry of fractals, stochastic processes University Computing Laboratory, Wolfson on fractals Building, Parks Road, Oxford OX1 3QD, UK, Aim: to bring together researchers from vari- tel: +44-1865-273883, fax: +44-1865-273839 27-2 June: Spring School in Analysis: ous mathematical areas who share a common e-mail: [email protected] Function Spaces and Interpolation, Paseky interest in fractal structures, with open-mind- Web site: http://web.comlab.ox.ac.uk/ oucl/ nad Jizerou, Czech Republic edness to interaction between different fields people/bette.byrne.html Theme: function spaces and interpolation inside and outside the fractal world. The sub- Topics: function spaces, interpolation, title of the conference gives the range of top- 26-29: Quantum Field Theory, rearrangement estimates, Sobolev inequalities, ics Noncommutative Geometry and Quantum K-divisibility, Calderon couples, extrapolation Main speakers: Martin Barlow (Canada), Probability Workshop, Trieste, Italy Main speakers: A. Cianchi (University of Thierry Coulhon (France), Kenneth Falconer Web site: http://www.sissa.it/~bruzzo/ncg2001/ Florence, Italy), M. Cwikel (Technion, Haifa, (UK), Hillel Furstenberg (Israel), Ben Hambly ncg2001.html Israel), M. Milman (Florida Atlantic (UK), Jun Kigami (Japan), Takashi Kumagai University, USA) (Japan), Michel Lapidus (USA), Andrzej April 2001 Language: English Lasota (Poland), Michel Mendès-France Organizing committee: Jaroslav Lukes, Lubos (France), Robert Strichartz (USA), Alexander Pick (Czech Republic) Teplyaev (Canada) 2-6: Lévy Processes and Stable Laws, Lecture notes: notes containing main talks to Language: English Coventry, UK be published Organising committee: Martin Barlow Information: Grants: probably support for a limited num- (Vancouver), Robert Strichartz (Ithaca), Peter Web site: http://science.ntu.ac.uk/msor/conf/ ber of students Grabner (Graz), Wolfgang Woess (Graz) Levy/ Deadlines: for reduced fee, 15 February; for Site: Technical University of Graz support, 15 February Information: 7-9: 16th British Topology Meeting, Information: e-mail: [email protected] Edinburgh, UK e-mail: [email protected] Web site: http://finanz.math.tu-graz.ac.at/ Information: Web site: http://www.karlin.mff.cuni.cz/ ~fractal/ Web site: katedry/kma/ss/jun01/ss.htm 9-12: 53rd British Mathematical 28-1 June: Harmonic morphisms and har- 18-22: Fourth European Conference on Colloquium, Glasgow, Scotland monic maps, Marseille, France Elliptic and Parabolic Problems: Theory, Sponsors: The Edinburgh Mathematical Aim: to gather specialists four years after the Rolduc, Netherlands Society, the Glasgow Mathematical Journal first international conference on harmonic [The former Pont-à-Mousson meeting is now Trust and the London Mathematical Society maps and harmonic morphisms, brest97 split into two conferences. This one is devoted Special sessions: partial differential equations Main speakers: (to be confirmed) P. Baird to more theoretical aspects; the other, with (Jean-Yves Chemin, Pierre Collet, Emmanuel (France), R. Bryant (USA), F.E. Burstall (UK), more emphasis on applications, takes place in Grenier, John Toland) and modular forms J. Eells (UK), B. Fuglede (Danemark), S. Gaeta, Italy, 24-28 September 2001] (Kevin Buzzard, Ernst-Ulrich Gekeler, Jacques Gudmundsson (Sweden), F. Helein (France), Topics: besides elliptic and parabolic issues, Tilouine) S. Ianus (Romania), D. Kotschick (Germany), topics include geometry, free boundary prob- Plenary speakers: Henri Berestycki (Paris), L. Lemaire (Belgium), P. Li (USA), M. lems, fluid mechanics, evolution problems in Michel Broué (Paris), Henri Darmon Micallef (UK), C. Negreiros (Brazil), Y. general, calculus of variations, homogeniza- (Montreal), Clifford Taubes (Harvard) Ohnita (Japan), Y.L. Ou (China), F. Pedit tion, control, modelling and numerical analy- Other speakers: Nikolaos Bournaveas, Rob de (USA), Z. Tang (China), B. Watson (Jamaica), sis 32 EMS December 2000 CONFERENCES

Invited speakers: include C. Bandle (Basel), July 2001 Language: English H. Beirão da Veiga (Pisa), X. Cabré Sponsors: RFBR, EMS, NATO, local funds (Barcelona), P. G. Ciarlet (Paris), M. Escobedo Information: (Bilbao), H. Farwig* (Darmstadt), M. Fila 1-6: Eighteenth British Combinatorial e-mail: [email protected] (Bratislava), D. Hilhorst* (Orsay), D. Conference, Brighton, United Kingdom Web site: .pdmi.ras.ru/EIMI/2001/emschool/ Kinderlehrer (CMU), Yan-Yan Li (Rutgers), Information: index.html F.H. Lin (New York), S. Luckhaus (Leipzig), e-mail: [email protected] H. Matano (Tokyo), U. Mosco (Rome), J.C.C. Web sites: http://www.maths.susx.ac.uk/Staff/ 15-20: Algorithms for Approximation IV Nitsche (Minneapolis), F. Otto (Bonn), M. JWPH/ International Symposium, Huddersfield, UK Padula (Ferrara), P. Pedregal* (Ciudad Real), http://hnadel.maps.susx.ac.uk/TAGG/Confs/ [in celebration of the 60th Birthdays of L.A. Peletier (Leiden), J.F. Rodrigues* BCC/index.html Claude Brezinski, Maurice Cox and John (Lisbon), C.J. van Duijn (Amsterdam); [For details, see EMS Newsletter 36] Mason] * organisers of thematic sessions Theme: approximation theory Organising committee: J. Bemelmans 4-6: MathFIT workshop: The Representation Aim: to provide an opportunity for exchange (Aachen), B. Brighi, A. Brillard (Mulhouse), and Management of Uncertainty in of ideas about current theoretical and practi- M. Chipot (Zurich), F. Conrad (Nancy), I. Geometric Computations, Sheffield, UK cal research on approximation Shafrir (Haifa) V. Valente (IAC, Rome), G. Information: Topics: radial basis functions, splines, rational Vergara-Caffarelli (Rome) Web site: http://www.shef.ac.uk/~geom2001/ approximation, computer-aided geometric Programme: in addition to the main lectures design, shape preserving methods, wavelets, parallel sessions of short communications will 5-7: British Congress of Mathematics support vector machines and neural networks, be organized. Education, Keele, UK non-linear approximation, spectral methods, Deadline: for submission of abstracts, 1 April Information: orthogonal polynomials, approximation on a Note: The division between theory and appli- Web site: http://www.bcme.org.uk sphere, special functions, applications cations will not be enforced, but a theoretical 8-13: Second Workshop on Algebraic Graph Main speakers: M. Buhmann (Germany), subject will certainly have a greater audience Theory, Edinburgh, Scotland M.G. Cox (UK), K. Driver (South Africa), M. in Rolduc, and an applied one a greater audi- Main topics: the new fullerenes, eigenspace Floater (Norway), T. Goodman (UK), W. Light ence in Gaeta techniques, generalisations from distance-reg- (UK), C.A. Micchelli (USA), L. Nielsen Information: ular graphs, topological considerations (Denmark), G. Plonka (Germany), T. Poggio e-mail: [email protected], Key speakers: N.L. Biggs (London School of (USA), L.L. Schumaker (USA), G.A. Watson [email protected] Economics), P.J. Cameron (Queen Mary & (UK) Web site: http://www.math.unizh.ch/ Westfield College), D. Cvetkovic (Belgrade), Special sessions: splines, wavelets, orthogonal rolducgaeta P.W. Fowler (Exeter), M.A. Fiol (Barcelona), polynomials and pade approximation, inte- W. Haemers (Tilburg), P. Hansen (Directeur grals and integral equations, the mathematics 19-22: Computational Intelligence, Methods du GERAD, Montreal), B. Mohar (Ljubljana), and statistics of metrology, mathematical and Applications (CIMA 2001), Bangor, UK B. Shader (Wyoming) modelling methods in medicine Information: Information: Proceedings: to be published as a special e-mail: [email protected]; e-mail: [email protected] issue of The Journal of Numerical Algorithms by [email protected]; Web site: http://www.ma.hw.ac.uk/icms/current/ Kluwer Academic/Plenum Publishers [email protected] graph/index.html Programme committee: C. Brezinski Web site: http://www.icsc.ab.ca/cima2001.htm (France), M. Buhmann (Germany), T. [For details, see EMS Newsletter 37] 9-22: European summer school: Asymptotic Goodman (UK), T. Lyche (Norway), L.L. combinatorics with application to mathemat- Schumaker (USA), G.A. Watson (UK). 25-29: Cmft2001, Computational Methods ical physics, St Petersburg, Russia Organising committee: I.J. Anderson (UK), and Function Theory, Aveiro, Portugal Aim: to observe the recent progress in the J.C. Mason (UK), D.A. Turner (UK), M.G. Theme: the various aspects of interaction of asymptotic theory of Young tableaux and ran- Cox (UK), A.B. Forbes (UK), J. Levesley (UK), function theory and scientific computation; dom matrices from the point of view of com- W. Light (UK) other areas from complex variables (including binatorics, representation theory and theory Sponsors: European Office Of Aerospace generalisations such as quaternions, etc.), of integrable systems. Systematic courses on Research And Development, London approximation theory and numerical analysis the subjects and current investigations will be Mathematical Society, Software Support For are also covered. presented Metrology (National Physical Laboratory, Aim: to assist in the creation and mainte- Scientific committee: E. Brezin (ENS, Department Of Trade And Industry) nance of contact between scientists from France), O. Bohigos (Orsay, France), P. Deift Site: School of Computing and Mathematics, diverse cultures; there is a strong effort to (U.Penn, USA), L. Faddeev (POMI, Russia), V. University of Huddersfield, Queensgate, encourage the participation of highly quali- Malyshev (INRIA, France), A. Vershik (POMI, Huddersfield, HD1 3DH, UK fied scientists who normally have only limited Russia, Chair) Grants: a limited amount of money will be opportunity to attend international confer- Main speakers: P. Biane (Paris), E. Brezin available as grants for bona fide research stu- ences (Paris), P. Deift (USA), K. Johansson dents and people from less advantaged coun- Organisers: H. Malonek (Aveiro, Portugal), (Stockholm), V. Kazakov (Paris), R. Kenyon tries N. Papamichael (Nicosia, Cyprus), St (Orsay), M. Kontsevich (France), A. Lascoux Deadlines: for abstracts, 31 December 2000 Ruscheweyh (Würzburg, Germany), E. B. Saff (France), A. Okoun’kov (USA), G. Ol’shansky (please e-mail the symposium committee at (Tampa, USA) (Moscow), L. Pastur (Paris), R. Speicher the address below if you require an extended Note: anyone interested in being invited (Heidelberg), R. Stanley (MIT), C. Tracy deadline); for registration, 15 June (late regis- should send the following details by ordinary (USA), H. Widom (USA) tration will be allowed, but will incur a ten mail or e-mail: name, affiliation, address, Topics: asymptotic combinatorics and its percent surcharge) phone/fax/e-mail, please send me the Second applications in the theory of integrable sys- Note: registration forms are available at the Announcement, I intend to submit a commu- tems, random matrices, free probability, quan- Web site below nication (yes or no) tum field theory, etc. Also those topics con- Information: Information: contact: H. R. Malonek, cerned with low-dimensional topology, QFT, e-mail: [email protected] Departamento de Matematica Universidade new approach in Riemann-Hilbert problem, Web-site: Http://Helios.Hud.Ac.Uk/A4a4/ de Aveiro, Portugal asymptotics of the orthogonal polynomials, tel./fax: +351-234-370359 /+351-234-382014 symmetric functions, representation theory 23-27: 20th IFIP TC 7 Conference on e-mail: [email protected] and random Young diagrams System Modelling and Optimization, Trier, Web site: http://event.ua.pt/cmft2001/ Local organising cmmittee: A. Vershik, Ju. Germany Neretin., K. Kokhas., E. Novikova Scope: IFIP TC7 promotes applications, the Site: International Euler Institute development of new techniques and theoreti- EMS December 2000 33 CONFERENCES cal research in all areas of system modelling generated modules, derived categories, con- September 2001 and optimisation. Each biennial conference nections to the commutative setting brings together TC7 working groups and a Programme: the meeting is in two parts: in wide scientific and technical community, who the first part the participants lecture on intro- 24-28: Fourth European Conference on share information through lectures and dis- ductory topics; the second part is a workshop Elliptic and Parabolic Problems: cussions where specialists in the area lecture on recent Applications, Gaeta, Italy Main speakers: A. Ben-Tal (Haifa), O.L. results [The former Pont-à-Mousson meeting is now Mangasarian (Madison), K.-H. Hoffmann Workshop specialists: Luchezar L. Avramov split into two conferences. This one is devoted (Bonn), J.-S. Pang (Baltimore), F. Jarre (USA), Edward L. Green (USA), Dieter to applications; the other, with more emphasis (Düsseldorf), R. Rackwitz (München), C.T. Happel (Germany), Birge Huisgen- on theory, takes place in Rolduc, Netherlands, Kelley (Raleigh), R. Schultz (Duisburg), K. Zimmermann (USA), Bernard Keller (France), 18-22 June 2001] Kunisch (Graz), P.L. Toint (Namur) Claus M. Ringel (Germany) Topics: besides elliptic and parabolic issues, Language: English Organisers: Peter Dräxler (draexler@mathe- topics include geometry, free boundary prob- Deadlines: for submission of extended matik.uni-bielefeld.de, Universität Bielefeld), lems, fluid mechanics, evolution problems in abstracts, 31 December 2000 Henning Krause ([email protected] general, calculus of variations, homogenisa- Organising committee: E. Sachs (Chair), bielefeld.de, Universität Bielefeld), Øyvind tion, control, modelling and numerical analy- Universitaet Trier FB IV - Mathematik Solberg ([email protected], NTNU, sis Information: Trondheim) Invited speakers include: H. Amann e-mail: [email protected] Sponsors: support is provided by the TMR (Zurich), C. Baiocchi (Rome), J. Ball (Oxford), Web site: http://ifip2001.uni-trier.de scheme of the EC; further support applied for A. Bermúdez (Santiago), M. Bertsch (Rome), Information: contact Øyvind Solberg, C.M. Brauner* (Bordeaux), A. Capuzzo- August 2001 ([email protected], NTNU, Trondheim) Dolcetta* (Rome), J. Escher (Hannover), E. Web sites: http://www.mathematik.uni- Fereisl (Prague), A. Friedman (Minneapolis), bielefeld.de/~sek/summerseries.html G. Geymonat (Montpellier), W. Hackbusch 5-18 2001 BALTICON 2001, BALTICON http://www.math.ntnu.no/~oyvinso/ (MIP), A. Henrot* (Nancy), M. Iannelli* 2001, Banach algebra theory in context, Nordfjordeid/ (Trento), M. Mimura (Hiroshima), P. Podio- Krogerup Hojskole, Humlebaek, Denmark Guidugli (Rome), J. Rubinstein (Haifa), E. [15th in a series of conferences and work- 24-30: 10th International Meeting of Sanchez-Palencia (Paris), S. Sauter* (Zurich), shops] European Women in Mathematics, Malta A. Sequeira (Lisbon) Topics: the emphasis will be on the connec- Programme: pure session on Cohomology theo- * organisers of thematic sessions tions between Banach algebra theory and ries, applied session on Mathematics applied to Organising committee: J. Bemelmans other areas of mathematics; for instance (list- finance, interdisciplinary session on The uses of (Aachen), B. Brighi, A. Brillard (Mulhouse), ed alphabetically), automatic continuity theo- geometry, social session on Mathematics outside M. Chipot (Zurich), F. Conrad (Nancy), I. ry, Banach spaces, homological algebra theo- the classroom: cultural differences Shafrir (Haifa) V. Valente (IAC, Rome), G. ry, locally compact groups and harmonic Information: contact Dr Tsou Sheung Tsun Vergara-Caffarelli (Rome) analysis, operator theory, spectral theory, (EWM01), Mathematical Institute, 24-29 St Programme: in addition to the main lectures topology Giles, Oxford OX1 3LB, United Kingdom, parallel sessions of short communications will Invited speakers include: G. Dales, J. Esterle, fax: +44-01865-273583 be organised. A.Ya. Helemskii, B.E. Johnson, C. Read and Web site: http://www.maths.ox.ac.uk/~ewm01/ Deadline: for submission of abstracts, 1 April G. Willis Note: The division between theory and appli- Local organising committee: Niels Grønbæk 27-31: Equadiff 10, Czechoslovak cations will not be enforced, but a theoretical and Kjeld Bagger Laursen, both University of International Conference on Differential subject will certainly have a greater audience Copenhagen Equations and their Applications, Prague, in Rolduc, and an applied one a greater audi- Sponsors: include the Mathematics Czech Republic ence in Gaeta Department of the University of Copenhagen, Honorary presidents: Ivo Babuska, Jaroslav Information: the Danish Science Research Council, Pomona Kurzweil e-mail: [email protected], College Topics: ordinary differential equations, par- [email protected] Call for papers: all interested are urged to tial differential equations, numerical methods Web site: http://www.math.unizh.ch/ sign up and to submit papers and application rolducgaeta Site: Krogerup Hojskole, approx. 25 miles Language: English north of Copenhagen Organising committee: Jiri Jarnik (Chair), October 2001 Deadlines: for abstracts, 15 February Bohdan Maslowski (Secretary), Jan Chleboun, Note: around 60 speakers and contributors Vladimir Dolezal, Eduard Feireisl, Miroslav are expected Krbec, Alexander Lomtatidze, Josef Malek, 16-22: Conference of the Austrian Information: Pavol Quittner, Milan Tvrdy, Jaromir Mathematical Society and the Deutsche e-mail: [email protected] Vosmansky Mathematiker Vereinigung, Vienna, Austria Web site: http://www.math.ku.dk/conf/balticon Advisory board: H. Amann (Switzerland), D. Plenary speakers: V. Capasso (Milano), 2001/ Arnold (USA), F. Brezzi (Italy), P. Brunovsky M.H.A. Davis (London), I. Ekeland (Paris), (Slovakia), F. Clarke (France), G. Da Prato W.T. Gowers (Cambridge), M. Kreck 5-18: Groups St Andrews 2001, Oxford, (Italy), N. Everitt (UK), B. Fiedler (Germany), (Heidelberg), N.J. Mauser (Vienna), V.L. England J. Hale (USA), W. Jaeger (Germany), I. Popov (Moskau), T. Ratiu (California), D. Information: Groups St Andrews 2001, Kiguradze (Georgia), P.L. Lions (France), J. Salamon (Zürich), G. Teschl (Vienna), J.-C. Mathematical Institute, North Haugh, St Mawhin (Belgium), P. Raviart (France), K. Yoccoz (Paris), D. Zagier (Bonn), G.M. Ziegler Andrews, Fife KY16 9SS, Scotland Schneider (Germany), N. Trudinger (Berlin) e-mail: [email protected] (Australia), A. Valli (Italy), W. Wendland Local organiser: Karl Sigmund (University of Web site: http://www.bath.ac.uk/~masgcs/ (Germany) Vienna) gps01/ Site: Charles University of Prague, Faculty of Site: Technical University [For details, see EMS Newsletter 36] Law Information: Karl Sigmund, University of Notes: 2nd announcement including all forms Vienna, Institute of Mathematics, 12-19: Summer School 2001: Homological is available at the Web site Strudlhofgasse 4, 1090 Vienna, conjectures for finite dimensional algebras, Deadlines: for registration, 31 May; for tel: +43 1 4277 506 02 or 506 12; fax: +43 1 Nordfjordeid, Norway abstracts, 31 March 4277 9506 Topics include: origin of conjectures, resolu- Information: e-mail [email protected] tions and syzygies, homologically finite subcat- e-mail: [email protected] Web site: http://www.mat.unive.ac.at/~oemg/ egories, some geometrical aspects, infinitely Web site: www.math.cas.cz/~equadiff/ Tagungen/2001/ 34 EMS December 2000 RECENT BOOKS body who wants to learn more about this part of combinatorics with quite a strong algebro-categorical aesthetic appeal. As a bonus, it has one of the never boring fore- RecentRecent booksbooks words by G.-C. Rota. (mkl) edited by Ivan Netuka and Vladimír Sou³ek P. Berthelot, D-modules arithmétiques II. Descente par Frobenius, Mémoires de la Books submitted for review should be sent to the of prefix rewriting to the subgroup prob- SMF, 81, Société Mathématique de France, following address: lem in combinatorial group theory, and a Paris, 2000, 136 pp., FRF 150, ISBN 2- Ivan Netuka, MÚUK, Sokolovská 83, 186 75 paper of Mislin and Tolleli is concerned 85629-086-8 Praha 8, Czech Republic. with periodic Farrell and Tate cohomolo- This is a continuation of the author’s pro- gies for hierarchically decomposable ject of developing foundations of crys- A. S. Asratian, T. M. J. Denley and R. groups. Finally, a paper of Nekrashevych talline/rigid cohomology with coefficients Häggkvist, Bipartite Graphs and their and Sushchansky studies automorphism in terms of ‘arithmetic D-modules’. The Applications, Cambridge Tracts in groups of spherically homogeneous root- basic objects of interest are schemes (resp. Mathematics 131, Cambridge University Press, ed trees. (ad) formal schemes) smooth over a given base Cambridge, 1998, 259 pp., £40, ISBN 0-521- Z/pnZ-scheme (resp. a p-adic formal 59345-X H. Bercovici and C. Foias (eds.), scheme). This book is devoted to the study of a par- Operator Theory and Interpolation, The volume is devoted to various func- ticular class of graphs. Yet the book International Workshop on Operator Theory tionality properties of arithmetic D-mod- demonstrates that this is a rich class that and Applications, IWOTA 96, Operator ules with respect to (a lift of) the Frobenius captures many important properties of Theory, Advances and Applications 115, morphism F. The main result (‘Frobenius graphs in general. Birkhäuser, 2000, 309 pp., ISBN 3-7643- descent’) is a far-reaching generalisation The book is divided into twelve chapters 6229-4 of the classical Cartier isomorphism. This which include metric properties (with an The papers in this volume were presented is used by the author to establish, among appendix: addressing schemes for com- at the International Workshop on other things, compatibility of F* with vari- puter networks), connectivity (with an Operator Theory and Applications ous cohomological operators. (jnek) appendix on the construction of linear (IWOTA), held at Indiana University, superconcentrators), and expanding prop- Bloomington, in June 1996. They repre- A. Böttcher and S. M. Grudsky, Toeplitz erties (with an appendix on expanders sent most of the areas that were discussed Matrices, Asymptotic Linear Algebra, and and sorters). Curiously, algorithms for the at the workshop, with some emphasis on Functional Analysis, Birkhäuser, Basel, minimum spanning tree problem are modern interpolation theory, a topic that 2000, 116 pp., DM58, ISBN 3-7643-6290-1 included in an appendix devoted also to has seen much progress in recent years. This text is a self-contained introduction the travelling salesman problem. Graph Much of the work in this volume is related to some problems for Toeplitz matrices on theory is maturing: one day every class of to Béla Sz.-Nagy’s results on interpolation the border between linear algebra and graphs will have a book. (jnes) and dilation theory. functional analysis. The text looks at The book may serve as an inspiration for Toeplitz matrices with rational symbols, M. Atkinson, N. Gilbert, J. Howie, S. further research, and can be recommend- and focuses attention on the asymptotic Linton and E. Robertson (eds.), ed to researchers and postgraduate stu- behaviour of the singular values; this Computational and Geometric Aspects of dents involved in these fields. (knaj) includes the behaviour of the norms, the Modern Algebra, London Mathematical norms of the inverses, and the condition Society Lecture Note Series 275, Cambridge F. Bergeron, G. Labelle and P. Leroux, numbers as special cases. The text illus- University Press, Cambridge, 2000, 279 pp., Combinatorial Species and Tree-like trates that the asymptotics of several linear £27.95, ISBN 0-521-78889-7 Structures, Encyclopedia of Mathematics and algebra characteristics depend in a fasci- This volume of proceedings contains 18 its Applications 67, Cambridge University nating way on functional analytic proper- papers, for which it is hard to find any uni- Press, Cambridge, 1998, 457 pp., £55, ISBN ties of infinite matrices. Many conver- fying description other than the title of the 0-521-57323-8 gence results can be comfortably obtained conference. There are papers on group This book gives, in English for the first by working with appropriate C*-algebras, presentations, term rewriting, string time, a thorough presentation of the com- while refinements of these results (for rewriting, cancellation diagrams with non- binatorial theory of species (which origi- example, estimates of the convergence positive curvature, a new proof of the cut- nated in the work of A. Joyal in 1980). speed) nevertheless require hard analysis. point conjecture for negatively curved The introductory Chapter 1 explains asso- This book is warmly recommended to groups, papers on discontinously cocom- ciated power series and the operations of beginners specialising in functional analy- pact actions by isometries, with computa- addition, multiplication, substitution and sis and algebra. (knaj) tions in word-hyperbolic groups and in differentiation of species. Chapter 2 intro- groups with exponent 6, and several fur- duces further operations, weighted A. Candel and L. Conlon, Foliations I, ther topics. species, virtual species (enabling species Graduate Studies in Mathematics 23, I will explicitly mention the papers that subtraction), and molecular and atomic American Mathematical Society, Providence, exceed 20 pages. Bartholdi and Grigor- species. Chapter 3 is devoted to combina- 2000, 402 pp., US$54, ISBN 0-8218-0809- chuk present a paper whose expository torial functional equations; among other 5 part associates a graded Lie algebra to a things, Lagrange inversion, iterative This is the first volume of a two-volume group G with a given N-series, discusses methods, and a useful overview of asymp- monograph on the qualitative theory of the questions of growth; the authors then totic analysis are presented. Chapter 4 foliations. It consists of three parts. The construct two examples of groups of inter- deals with unlabelled enumeration and first part The Foundations is designed as an mediate growth that can be used as asymmetric structures and gives proofs for introduction to the theory of foliated man- counter-examples to a conjecture on the substitution formulas for weighted species. ifolds for postgraduate students. The structure of just-infinite groups of finite Chapter 5 presents species on totally authors state that the readers of this part width. A paper by Huch and Rosebrock is ordered sets and combinatorial theory of are assumed to have a fairly good back- concerned with two mutually dual small differential equations. ground in manifold theory, but I think cancellation conditions that generalise The book contains more than 350 exer- that the authors do not require any extra- (C6), (C4) (T4), (C3) and (T6); they solve cises and an extensive bibliography. The ordinary knowledge. This first part can the conjugacy problem for the groups with exposition is illuminated by many dia- also be considered as a necessary prereq- a respective presentation. A paper by grams. In the appendix, numerous tables uisite for the next two parts. The second Madlener and Otto surveys the application are given. The book is essential for any- and third parts have the titles Codimension EMS December 2000 35 RECENT BOOKS One and Arbitrary Codimension. These are chronology of their lives. The mathemati- Azbelev, Stability and asymptotic behavior devoted to the study of the foliations of cal work of the Youngs can be convenient- of solutions of equations with aftereffect, codimension 1 and to the foliations of ly divided into three broad categories: the C. T. H. Baker and A. Tang, Generalized higher codimension, and this division is theory of real functions, Fourier analysis, Halanay inequalities for Volterra function- quite understandable, because the meth- and miscellaneous. The bibliography is al differential equations and discretized ods used in these two cases are quite dif- based entirely on that of I. Grattan- versions, P. Clément and G. da Prato, ferent. They can be compared with the Guinness in Historia Mathematica 2 (1975), Stochastic convolutions with kernels aris- theory of flows on surfaces, and the theory 43-58. The authors have grouped the arti- ing in some Volterra equations, H. Engler, of flows on manifolds of dimension >2. cles according to the year of their publica- An example of Lp-regularity for hyperbol- Already this first volume covers a lot of tion. The three books of the Youngs fol- ic integro-differential equations, V. material on foliated manifolds, and a great low the list of the articles. The two obitu- Lakshmikantham and A. S. Vatsala, The part deals with more general foliated aries, by G. H. Hardy (1877-1947) and M. present status of UAS for Volterra and spaces. L. Cartwright (1900-1998) respectively, delay equations, I. W. Sandberg, Myopic In a way, the book is built on examples. give a balanced account of the mathemati- maps and Volterra series approximation, This means that the authors first demon- cal work of the Youngs, as viewed by their and O. J. Staffans, State space theory for strate various phenomena on examples, almost-contemporaries. A brief overview abstract Volterra operators). and only when the reader understands of the totality of their mathematical work This is followed by 43 contributed them do they present any systematic theo- from a modern viewpoint is given in the papers addressing a great variety of prob- ry. The authors pay great attention to essay by Chatterji. lems. In particular, they deal with stabili- examples, and you can find a large num- This book should form an ideal resource ty theory, stochastic processes, classical ber of them in the book. We also find for mathematicians and specialists in the Volterra equations (also in connection with many exercises. This is very important, history of mathematics. (knaj) dynamical systems and blow-up type prob- especially in the book of this extent. They lems), numerical problems (with attention are well chosen, and will keep the interest B. Cipra, What’s Happening in the to finite-element method and generalisa- of a reader on a high level. The biography Mathematical Sciences: 1998-1999, tions of known discretisation methods for has 149 items, and goes up to 1999. The American Mathematical Society, Providence, ordinary differential equations), periodic book is surely not a short introduction into 1999, 126 pp., ISBN 0-8218-0766-8 solutions, control theory (especially opti- foliations or a concise survey of foliation The contents of the fourth volume in this mal control), infinite-dimensional systems, theory, but is a fundamental source for series is well expressed by the titles given integro-differential equations, approxima- everybody with a serious interest in folia- below. This lively presentation of an tion methods, abstract Volterra operators tions. (jiva) amazingly wide spectrum of happenings in and equations, applied problems in mathematics is impressive. I believe that physics and engineering, and other top- R. Cerf, Large Deviations for Three this should be presented to a wide audi- ics. Dimensional Supercritical Percolation, ence even outside mathematics, which This volume will be of interest for both Astérisque 267, Société Mathématique de could be fascinated by the ideas, concepts, pure and applied mathematicians, as well France, Paris, 2000, 177 pp., FRF 250, and beauty of the mathematical topics. as theoretically oriented engineers and ISBN 2-85629-091-4 The contents: A blue-letter day for comput- graduate students seeking a broad state- The aim of the work is to propose a er chess (the end of the long way to beat of-the-art insight into Volterra equations method for analysing phase separation Kasparov does not mean solving the com- and their applications. (trou) and coexistence for the three-dimensional binatorial games problem); A prime of chaos Bernoulli percolation model. The main (on quantum chaology and algebraic num- W. A. de Graf, Lie Algebras: Theory and results concern the large deviation princi- ber theory); Proof by example: a mathemati- Algorithms, North-Holland Mathematical ples and their application to the Wulff cian’s mathematician (on the impact of Paul Library 56, North-Holland, Amsterdam, 2000, crystal. The case of a single cluster, as well Erdõs); Computers take algebraic geometry 393 pp., US$118, ISBN 0-444-50116-9 as the whole configuration, are consid- back to its roots (algorithmic questions in The theory of Lie algebras has many ered. algebraic geometry); As easy as EQP (on explicit constructions and concrete algo- The book is divided into twelve chapters automatic theorem proving); Beetlemania: rithms (Levi decomposition, branching with the headings: Introduction, The large chaos in ecology (on experimental evidence rules, Hall-Shirshov and Grbner bases, deviation principles (LDP), Sketch of the for chaotic dynamics); From wired to weird etc.). This book contains a standard proofs, The model, Surface tension, The (on revolutionary quantum computing); course in Lie algebras and includes practi- surface tension, Coarse graining, The cen- Tales from the cryptosystem (computational cally all existing algorithms in this theory. tral lemma, Proof of the LDP for a single complexity and cryptographic systems); The approach simplifies proofs of some cluster, Collections of sets, The surface But is it math? (mathematics and art: important theorems and makes them energy of a Caccioppoli partion, Proof of Escher, etc.); Mathematical discovery by more transparent and clear. Moreover, the LDP for the whole configuration. Henri Poincaré (Henri Poincaré’s since current research on Lie algebras The large deviation principles are stated thoughts). (jslo) requires the use of computers, such an in Chapter 2, together with their applica- exposition facilitates understanding and tion to the Wulff crystal. Chapter 3 is an C. Corduneanu and I. W. Sandberg practical use of computational methods informal sketch of the proofs for the single (eds.), Volterra Equations and for solving concrete problems. The author cluster case. The notation and the model Applications, Stability and Control: Theory, is also the author of a sub-package Lie alge- are introduced in Chapter 4. Important Methods and Applications 10, Gordon and bras in the programme package GAP. This facts on the theory of Caccioppoli sets and Breach, Amsterdam, 2000, 496 pp., £ 75, has enabled him to create a book that will the Wulff Isoperimetric Theorem are ISBN 90-5699-171-X be useful for experts, as well as for inter- summed up in Chapter 6. (mhusk) This volume contains 52 papers out of ested researchers from other fields of more than 60 presentations of the sympo- mathematics and mathematical physics. S. D. Chatterji and H. Wefelscheid (eds.), sium held at University of Texas at (ae) Selected Papers. G. C. Young, W. H. Young, Arlington in 1996 to celebrate the 100th Presses Polytechniques et Universitaires anniversary of Vito Volterra’s (1860-1940) J. J. Duistermaat and J. A. C. Kolk, Lie Romandes, Lausanne, 2000, 484 pp., publications on integral equations. Groups, Universitext, Springer, Berlin, 2000, CHF149, ISBN 2-88074-445-8 It begins with nine invited papers 344 pp., DM79, ISBN 3-540-15293-8 In this volume the authors present a selec- addressing both history (M. Schetzen, This book is devoted to the theory of tion of 52 of the 215 published articles of Retrospective of Vito Volterra and his finite-dimensional Lie groups and their Grace Chisholm Young (1868-1944) and influence on nonlinear system theory, and representations, mainly from the differen- William Henry Young (1863-1941), a com- R. K. Miller, Volterra integral equations at tial geometry point of view. Lie algebras plete list of which appears next to a brief Wisconsin) and recent developments (N. are studied in the first chapter, together 36 EMS December 2000 RECENT BOOKS with their relations to Lie groups. The Berkeley, MacLaurin and d’Alembert, 2-85629-092-2 proof of Lie’s third fundamental theorem continues with a selection from Kant and In this work the authors consider weak on the existence of a simply connected Lie Lambert and valuable translations of the shocks for systems of conservation laws in group with a given Lie algebra is included. texts of Bernard Bolzano. It then contin- any space dimension. The main result is a Proper actions of groups on manifolds, the ues with excerpts or complete texts of construction on a space-time domain, corresponding stratification of manifold Gauss, Gregory, De Morgan, Hamilton, independent of the parameter ε, of fami- into orbit types and the related blowing- Boole, Sylvester, Cayley, Peirce, Baire, lies of weak solutions uε, discontinuous up process are the main topics of the sec- Hilbert, Brouwer, Zermelo and Hardy. along a smooth hypersurface Σε, with ond chapter. In the third chapter, the The book is in some sense complementary jumps of order ε. For a fixed ε, the prob- authors study compact Lie groups and to van Heijenoort’s source book in mathe- lem can be recast as a non-linear mixed algebras, their fundamental group, the matical logic ‘From Frege to Gdel’ (for hyperbolic problem with a free non-char- corresponding Weyl group and Stiefel dia- example, it contains no texts by Frege, acteristic boundary, which has been solved grams. Peano, Russell, or Weyl), and represents a by A. Majda. When ε tends to 0, the front Invariant densities and problems of traditional and widely accepted view on tends to be characteristic; this induces a invariant integration are discussed, the foundations of mathematics. This loss of stability and regularity. As a conse- together with the classical Weyl integra- point of view is expressed by the very last quence, the classical non-linear methods tion formula. The last chapter presents a text of this collection, ‘The architecture of based on Picard’s iterations and differenti- good overview of the representation theo- mathematics’ (Bourbaki, 1948). (jmlc) ations do not apply. To prove suitable a ry of compact Lie groups, including the priori estimates and construct the solutions Peter-Weyl theorem, induced representa- J. Faraut, S. Kaneyuki, A. Korányi, Qi- the authors use more sophisticated meth- tions, character formulas and real forms of keng Lu and G. Roos, Analysis and ods, such as the para-differential calculus complex representations. There is also a Geometry on Complex Homogeneous and Nash-Moser’s type iteration schemes. nice description of the right regular repre- Domains, Progress in Mathematics 185, These results have important applications sentation of Lie groups, the Borel-Weil Birkhäuser, Boston, 2000, 540 pp., DM138, to Euler’s equations of gas dynamics, both theorem and its applications. The book ISBN 0-8176-4138-6 and 3-7643-4138-6 to the full system and the isotropic system. can be recommended as a higher level The book is an introduction to several Weak solutions of both two systems are introduction to theory of (compact) Lie basic topics in complex analysis and geom- constructed and compared. groups and their representations. (jbu) etry at an advanced graduate level; a cer- The authors start the book with a nice tain amount of preliminary knowledge is introduction, giving a summary of existing Y. Eliashberg and L. Traynor (eds.), required. It is based on lectures delivered literature, pointing out the general Sympletic Geometry and Topology, at the CIAMPA Autumn School in Beijing scheme of proofs, indicating crucial points IAS/Park City Mathematics Series 7, American in 1997, and extended in several interest- and difficulties that must be overcome and Mathematical Society, Providence, 1999, 430 ing directions. It consists of five parts writ- briefly describing how this can be done. pp., US$69, ISBN 0-8218-0838-9 ten by different authors. The parts are (jkop) The seventh volume in this series is devot- more or less independent. ed to various aspects of symplectic topolo- The first part (by J. Faraut) deals with H. Gordon, Discrete Probability, gy and related topics. The individual the theory of function spaces on complex Undergraduate Texts in Mathematics, parts present the contents of the following semi-groups, and gives an overview of the Springer, New York, 1998, 266 pp., DM68, lectures: Introduction to symplectic topology by theory of Hilbert spaces of holomorphic ISBN 0-387-98227-2 Dusa McDuff, Holomorphic curves and functions on complex manifolds endowed This is an undergraduate text designed for dynamics in dimension three by Helmut with the action of a (real) Lie group. The an introductory course in probability theo- Hofer, An introduction to the Seiberg-Witten main problems discussed are the decom- ry. With a few exceptions, only elementary equations on symplectic manifolds by Clifford position of the Hilbert space into irre- mathematics is used throughout. The Taubes, Lectures on Floer homology by ducible invariant subspaces and a descrip- book starts with the definition of probabil- Dietmar Salamon, A tutorial on quantum tion of the reproducing kernel on it. The ity on discrete sample spaces. It proceeds cohomology by Alexander Givental, Euler second part (by S. Kaneyuki) on graded to discuss combinatorial probability (sam- characteristics and Lagrangian intersections by Lie algebras and related geometric struc- pling with and without replacement), Robert MacPherson, Hamiltonian group tures gives a nice survey of recent results independence of events and conditional actions and symplectic reduction by Lisa on semi-simple pseudo-Hermitian sym- probability, and random variables and Jeffrey, and Mechanics, dynamics, and sym- metric spaces and Siegel domains. The their mean and variance. Independence metry by Jerrold Marsden. third part (by A. Korányi) presents an of random variables is treated only briefly. The result is a lively exposition of recent introduction to the theory of holomorphic One section is devoted to the weak law of developments in this exciting branch of functions on Cartan domains. It is based large numbers. The Poisson distribution mathematics, often starting with quite ele- on the Harish-Chandra approach arising as a limit of a sum of independent mentary and introductory facts and reach- from the theory of semi-simple groups. Bernoulli variables, the Stirling formula, ing far beyond standard textbooks, up to The fourth part (by Q. Lu) is devoted to and the De Moivre-Laplace theorem (with- sketches of proofs of most recent deep the study of properties of Laplace- out proof) are all treated in one chapter. results. In particular, this volume will be Beltrami operator and various integral The rest of the book is devoted to moment useful reading for graduate students and transforms. The last part (by G. Ross) on generating functions, random walks and experts. (jslo) Jordan triple systems contains another discrete Markov chains. approach to study of geometry and analy- The strengths of the book are undoubt- W. Ewald, From Kant to Hilbert. A Source sis of Hermitian bounded symmetric edly its exercises (over 400 of them), all Book in the Foundations of Mathematics, I, domains. with numerical solutions, and the many II, Clarendon Press, Oxford, 2000, 648 and All contributions are written carefully interesting remarks on the history of prob- 690 pp., £50, ISBN 0-19-850537-X, 0-19- and systematically. Let me mention espe- ability theory and biographies of impor- 850535-3 and 0-19-850536-1 cially Parts 2 and 3 which bear a strong tant personalities that are scattered This is an excellent collection of carefully relation to the geometry and analysis of throughout the book. On the other hand, selected and edited classical texts on the invariant operators for special geometric important definitions and facts are some- foundations of mathematics. Each text is structures. (jbu) times hidden in the text. (mkul) preceded by an introduction and notes and a comprehensive bibliography is J. Francheteau and G. Métivier, Existence T. V. Gramchev and P. R. Popivanov, included at the end of each volume. Many de chocs faibles pour des systèmes quasi- Partial Differential Equations. texts appear in a reliable English transla- linéaires hyperboliques multidimension- Approximate Solutions in Scales of tion for the first time. nels, Astérisque 268, Société Mathématique de Functional Spaces, Mathematical Research The selection starts with the texts of France, Paris, 2000, 198 pp., FRF250, ISBN 108, Wiley-VCH, Berlin, 2000, 155 pp., EMS December 2000 37 RECENT BOOKS DM148, ISBN 3-527-40138-5 was published with a small print run in In this book the authors present in a uni- 1888, and has never been reissued in its H. Koch, Number Theory. Algebraic fied form the results of their research over entirety (only an extract was printed in Numbers and Functions, Graduate Studies the last two decades in micro-local analysis Peano’s Opera Scelte and Hubert Kennedy in Mathematics 24, American Mathematical of pseudo-differential operators. The included an English translation of the Society, Providence, 2000, 368 pp., US$59, reader is supposed to be familiar with Introduction and Chapter 1 in his Selected ISBN 0-8218-2054-0 basic facts from the theory of Gevrey class- Works of Giuseppe Peano). Now a complete According to the preface, it is the author’s es, pseudo-differential operators, micro- and reliable translation is available to a conviction ‘that an area of mathematics local analysis, Fourier integral operators wider audience. such as number theory that has developed and differential geometry. The authors The preliminary chapter of the book is over a long period of time can be proper- study micro-local properties (solvability, Peano’s first publication in mathematical ly studied and understood if one proceeds hypo-ellipticity) of pseudo-differential logic: he first develops a calculus of classes through this entire development in abbre- operators in Sobolev spaces and Gevrey and then a calculus of propositions, intro- viated form, much as an organism recapit- classes; construct approximate solutions ducing for the first time modern notation ulates its evolutionary path in abbreviated with non-classical phase functions and (such as the symbols ∪ and ∩). Of partic- form during its embryonic development. amplitudes; investigate linear differential ular interest is his treatment of From this I derived the concept of allow- operators with multiple characteristics and Grassmann’s regressive product. Chapter ing the reader to take part from chapter to quasi-homogeneous operators; and pre- IX represents one of the first attempts to chapter in the historical development of sent applications of Airy operators to axiomatise the idea of a linear vector number theory’. oblique derivative problems for hyperbol- space. Comments on two errors, discov- Leaving aside the allusion to a by-now- ic equations and applications of Gevrey ered and corrected by Honbo Li, are discredited biological principle, let us classes to dynamical systems (approximate included in a short Editorial Note. The examine the contents of the book in the normal forms for pairs of glancing hyper- book (unfortunately) contains no other light of the author’s intentions. Chapter 1 surfaces). The language of the book is comments on this classical text. (jmlc) consists of several topics in elementary very general and abstract. Unfortunately, number theory, such as Pythagorean the misprints make the technically A. Khrennikov, Interpretations of triples, the (incomplete) history of Zell’s demanding notions even more difficult to Probability, VSP BV, Utrecht, 1999, 228 pp., equation and Fermat’s last theorem, con- read. The book is suitable for experts in ISBN 90-6764-310-6 gruences, quadratic reciprocity law and the field. (efa) The book presents an interesting discus- the distribution of primes. Chapter 2 is sion on quantum mechanics from a proba- devoted to elementary theory of orders in A. Guichardet, Groupes quantiques, bility point of view. It is well known that number fields, including Dirichlet’s theo- Mathématiques, EDP Sciences, Les Ulis, 1995, the theory of quantum mechanics gives rem on units, finiteness of class number 149 pp., FRF 160, ISBN 2-7296-0564-9 and strange results in some specific situations: and Minkowski’s theorem in the geometry 2-271-05272-6 the Einstein-Podolsky-Rosen paradox and of numbers. Chapters 3 and 4 develop the This book gives an excellent introduction Bell’s inequalities seem to be the most theory of Dedekind rings and valuations. into the field of quantum groups. It is a popular of these. The author points out These are used in Chapter 5, which treats relatively subtle volume, but nevertheless that such difficulties could be caused by an function fields in one variable over perfect contains a lot of interesting material. The inconvenient measurement of random- constant fields, up to the Riemann-Roch text is written with the necessary mathe- ness, and instead of values in the ordinary theorem. Chapter 6 is on higher ramifica- matical rigour, which ensures that the interval [0,1], he proposes the space of p- tion groups (including Herbrand’s theo- book will be well received by mathemati- adic numbers as the most convenient rem) and their applications, such as the cians. On the other hand, its reading range for probability employed in quan- decomposition of prime ideals in cyclo- requires no extraordinary mathematical tum mechanics. tomic and Kummer extensions. Chapter preparation, and will be understandable to The book begins with a survey on the 7 begins with an introduction of adèles physicists. notion of probability. Kolmogorov’s mea- and idèles and reproduces Tate’s The book starts with a chapter present- sure-theoretical approach and von Mises’ approach to the functional equation of ing some prerequisites from algebra. It idea on collectives giving frequency prob- Hecke L-series, as well as F. K. Schmidt’s then continues with a chapter introducing ability theory and proportional approach proof of the functional equation of the the main concepts concentrated around to randomness are introduced and com- zeta-function of a function field. Analytic the notion of a Hopf algebra. We find pared. After that, the author proceeds to properties of Hecke L-functions are used here also the notion of compact quantum random principles in quantum mechanics. in Chapter 8 to prove various distribution group, in the sense of Woronowicz, and The Einstein-Podolsky-Rosen paradox is results for prime ideals that generalise relations to the Poisson structures. The formulated and compared with Bell’s Dirichlet’s theorem on primes in arith- third chapter deals with formal deforma- inequality for probabilities as well as for metic progressions. Chapter 9 is devoted tions of the objects introduced in the pre- covariances and with the idea on hidden to the arithmetic of quadratic fields, and vious chapter. From the fourth chapter variables. The next two sections are devot- treats the correspondence between classes the author passes to very concrete consid- ed to the necessary theory of p-adic num- of binary quadratic forms and ideal classes erations: namely, a chapter about the bers and their calculus. The book con- in quadratic fields, units and class number quantum group Uh sl(2,k), a chapter about cludes with a discussion on tests for ran- formulas. Finally, Chapter 10 gives a brief the quantum group Uh sl(n+1,k), and a domness for p-adic-valued probability. survey of class field theory. There are very interesting chapter about deforma- The book is intended as a deep presen- three appendices, on elements of divisibil- tions of homogeneous spaces. tation of the author’s idea that p-adic-val- ity (including the structure theory of The book reads very well. One reason ued probability is able to remove, and finitely generated modules over PID’s and for this is that it contains many interesting even to explain, such difficulties as the Euclidean domains), on traces, norms and examples, and hints for further studies are Einstein-Podolsky-Rosen Paradox and to discriminants, and on Fourier analysis on given. It can be strongly recommended. validate places where the theory produces locally compact abelian groups. (jiva) ‘negative’ probabilities. It will be valuable The book requires as a prerequisite a for theoretical physicists, especially those good knowledge of basic algebra and L. C. Kannenberg, Geometric Calculus. working in quantum mechanics and relat- Galois theory and is meant to be an intro- Giuseppe Peano, Birkhuser, Boston, 2000, ed fields. On the other hand, it has value ductory text aimed at Ph.D. students in 150 pp., DM138, ISBN 0-8176-4126-2 and for mathematicians dealing with probabil- number theory and related areas. 3-7643-4126-2 ity theory, since the book is an interesting The brief description of its contents The first edition of Peano’s important attempt to use special Banach space-val- shows that the book is concerned mainly work on geometrical calculus, ‘preceded ued probability for description of observed with the general theory of number fields by the first operations of deductive logic’, phenomena. (pl) and function fields. In fact, a significant 38 EMS December 2000 RECENT BOOKS part of the material goes beyond what one (Volume III). (trou) sion), showing the role played by mani- would expect from an introductory text. A folds in topology, geometry, complex disadvantage of this approach, however, is K. B. Laursen and M. M. Neumann, An analysis, algebra, algebraic geometry, clas- the absence of the full-flavoured ‘concrete Introduction to Local Spectral Theory, sical mechanics, general relativity and arithmetic’, regardless of its place in the London Mathematical Society Monographs quantum field theory. He then introduces historical development of algebraic num- New Series 20, Clarendon Press, Oxford, an important tool, simplicial complexes, ber theory. The most significant omis- 2000, 591 pp., £75, ISBN 0-19-852381-5 and presents triangulation theorems for sions include cubic and biquadratic reci- This monograph develops the local spec- manifolds of dimensions 1, 2 and 3. Using procity laws, genus theory of quadratic tral theory for bounded linear operators simplicial complexes, he describes 1-man- forms, Hilbert symbols, a more detailed on Banach spaces. Chapter 1 is devoted to ifolds and gives a complete classification of analysis of the class number formula and decomposable operators. The authors compact 2-manifolds. He introduces the examples of zeta-functions of function derive several basic characterisations of fundamental group, paying much atten- fields. For these reasons this book can be decomposability, explore the role of the tion to this notion. We find here various recommended to students of number the- local spectrum, and establish the impor- methods enabling us to compute the fun- ory for its rigour and emphasis on theory, tant connection with the theory of spectral damental group, the Seifert-Van Kampen but its study should be complemented by capacities. Chapter 2 centres around cer- theorem, and covering spaces. Of course, reading other, more ‘concrete’ texts, such tain characterisations and applications of the central objects of interest are the fun- as Borevich and Shafarevich or Ireland- Bishop’s property ß and the decomposi- damental groups of compact 2-manifolds. Rosen. (jnek) tion property δ for bounded linear opera- To make the theory of 2-manifolds rela- tors on an arbitrary complex Banach tively complete, the author introduces the I. Lasiecka and R. Triggiani, Control space; the main goal of this chapter is to notions of homology and cohomology. Theory for Partial Differential Equations: show that property ß describes precisely The book is very carefully written. In Continuous and Approximation Theories, the restrictions of decomposable operators the text we find many exercises, classified 1: Abstract Parabolic Systems, Encyclopedia to closed invariant subspaces, that proper- as simpler problems, which should not be of Mathematics and its Applications 74, ty δ characterises the quotients of decom- omitted because some are used later in the Cambridge University Press, Cambridge, 2000, posable operators by closed invariant sub- main text. At the end of each chapter are 644 pp., £75, ISBN 0-521-43408-4 spaces, and that there is a complete duali- problems that are classified as more diffi- This volume represents a comprehensive ty between the two properties. In Chapter cult. The author has written this course as and up-to-date treatment of quadratic 3, distinguished parts of the spectrum and a first course in topology, and as a prepa- optimal control theory for linear parabol- their relationships to local spectra are ration for more advanced courses on ic-like partial differential equations (PDEs) studied, and several important classes of topology and differential geometry. This over a finite or infinite time horizon and spectral subspaces are considered; particu- is the reason why he did not touch PL- related differential (integral) and algebra- lar emphasis is placed on the relations structures or differential structures on ic Riccati equations. A semigroup between the spectra and essential spectra manifolds. It can be used to in full or par- approach is systematically used. Besides of two operators that are connected with tially for various basic courses on topology. continuous problems, numerical approxi- each other through some intertwining It is especially good that the course is writ- mation theory is pursued. On an abstract condition. Chapter 4 collects essentially ten in a very clear and attractive way, and level, the controlled system is assumed to everything that is known about the spec- we can expect that it will attract the atten- have the form dy/dt = Ay + Bu, with A and tral theory of multipliers, and particularly tion of students. (jiva) B linear possibly unbounded operators, about convolution operators on group and the former generating a C0- (and even measure algebras. Chapter 5 illustrates M. Liebeck, A Concise Introduction to analytic) semigroup on a Hilbert space, the usefulness of local spectral theory in Pure Mathematics, Chapman & Hall/CRC, and with the control u being an L2-func- automatic continuity. Finally, Chapter 6 Boca Raton, 2000, 162 pp., £17.95, ISBN 1- tion in time. The quadratic cost function- contains a list of open problems. The 584888-193 al to be minimised then involves still an modest prerequisites from functional This well-written book is based on the observation operator R. analysis and operator theory that the author’s lectures ‘Foundations of Analysis’ The abstract theory for such problems authors require are collected in the at Imperial College for students in the first applicable to a broad class of PDEs is pre- Appendix. (dmed) term of their degree. It contains sections sented in Chapters 1 and 2 for the finite on number systems, combinatorics, geom- and infinite horizon cases. Chapter 3 pre- J. M. Lee, Introduction to Topological etry, and a basic introduction to analysis sents many PDE illustrations with Dirichlet Manifolds, Graduate Texts in Mathematics and set theory. or Neumann boundary control or point 202, Springer, New York, 2000, 385 pp., 138 The aim of this book is to fill the gap control. This includes the heat equation, fig., DM69, ISBN 0-387-95026-5 and 0- between high-school mathematics and the Kelvin-Voight, Kirchhoff, and Euler- 387-98759-2 mathematics taught at university. In par- Bernoulli equations, and thermo-elastic This is a first course on topology for post- ticular, the reader is shown what it means plates. Chapter 4 provides a detailed graduate students, written by an author to prove something rigorously. The book numerical approximation, including opti- who has evidently great experience in begins with topics often taught at high mal rates of convergence, detailed illustra- teaching this subject. In order to reduce school, and goes further; for example, it tion being then given in Chapter 5. the prerequisites to the minimum, the starts with a naive understanding of real Finally, Chapter 6 returns to an abstract author has included an appendix in which numbers and ends with least upper level, dealing with a min-max game theo- he reviews the necessary notions from set bounds and a proof of the existence of the ry over an infinite time interval. theory, the theory of metric spaces, and nth root. The author illustrates the theory This thorough very detailed exposition group theory. Later, we find a special with number of exercises. largely expands the lecture notes of these chapter ‘Some group theory’, whose aim is In order to keep the book easily read- experienced authors, published in 1991 by to have the necessary algebraic techniques able, a few mathematical proofs are inac- Springer-Verlag, and is primarily available when studying the fundamental curate; however, these omissions are not addressed to applied mathematicians and group. Moreover, he assumes no knowl- explicitly mentioned, which might confuse theoretical engineers interested in optimal edge even of general topology, and devel- a thorough reader. This book is easy to control, in particular, of linear distributed- ops the relevant part of this theory in full read for anyone with a high-school mathe- parameter systems, as well as to graduate detail. matics background. (sh) students in this area. This volume will be The main part of the book is concen- followed by optimal control theory for trated around the topology of 2-mani- V. G. Maz’ya and S. B. Poborchi, hyperbolic or Petrowski-type PDEs folds. The author has devoted the whole Differentiable Functions on Bad Domains, (Volume II) and for hyperbolic-like introductory chapter to motivating the World Scientific, Singapore, 1997, 481 pp., dynamics and coupled PDE systems notion of a manifold (of arbitrary dimen- £56, ISBN 981-02-2767-1 EMS December 2000 39 RECENT BOOKS Sobolev spaces of functions whose partial This textbook provides an elegant and References and Textbooks 6, Marcel Dekker, derivatives belong to Lp hold an excep- serious introduction to the basic concepts New York, 2000, 392 pp., US$165, ISBN 0- tional position among spaces of differen- and results of (elementary) algebraic 8247-0447-9 tiable functions. These spaces are well geometry. The computational and algo- Difference equations are a powerful tool adapted for solving boundary value prob- rithmic aspects provide the guidelines of for solving many problems arising in lems in the theory of partial differential the exposition, but the synthetic approach applications involving health-related equations. For these applications, it is is also presented. The result is a pleasant research. The authors start with a general important to know under what circum- combination of intuitive and technical introduction to difference equations and stances the inequalities and theorems on exposition of the material. develop the iterative solution for first- embedding, extension and traces hold. The main topics include: simple inter- order equations. The main method used The validity depends on the quality of polation and spline theory, conic sections, for solving difference equations is the domain. If the boundary is locally repre- an introduction to algebraic projective method of generating functions. General sented as an isometric image of a graph of geometry, the theory of algebraic curves properties of generating functions are a Lipschitz function, then the domain is (including resultants), the Maclaurin- described (scaling, the convolution princi- still relatively good, although it can have Bézout theorem, resolutions of singulari- ple, the use of partial fractions, coefficient ‘corners’. Most of the material of the book ties and the genus of curves, and the theo- collection) and the role of probability gen- is devoted to domains with non-Lipschitz ry of algebraic surfaces. Much space is erating functions is emphasised. Among singularities or even to general domains. devoted to applications and examples. the applications of difference equations we The introductory chapter contains a The book is designed as a text for a gen- find a model of unusual heart rhythm, the self-contained exposition to the general uine course on algebraic geometry and its random walk problem, a model of clinic theory of Sobolev spaces. Next, many applications, and selections for shorter visits, run theory, drought prediction and examples of wild domains are shown to courses are also possible. I believe that follow-up losses in clinical trials. The end demonstrate the failure of basic statements professionals seeking applied mathemat- of the book is devoted to applications of of the theory when the assumptions on ics, as well as students and researchers, will difference-differential equations in epi- domain are violated. The second part make good use of this text. (jslo) demiology that are derived from the deals with parameter-dependent domains. Chapman-Kolmogorov forward equations. Typically, a family of domains depending Y. Meyer and R. Coifman, Wavelets. The book is written in an understand- on a small positive parameter ε is consid- Calderón-Zygmund and Multilinear able style for students in biostatistics and ered. The family exhibits a certain degen- Operators, Cambridge Studies in Advanced for researchers in this field. It is surpris- eracy as ε tends to 0. The asymptotic Mathematics 48, Cambridge University Press, ing that such a fundamental concept as the behaviour of norms of extension operators Cambridge, 2000, 314 pp., £42,50, ISBN 0- characteristic equation is not introduced, and trace operators is then investigated. 521-42001-6 specialised to difference equations. In the third part, a domain with an inner This book is a translation of Ondelettes et Mathematically, the book should be read or outer cusp (peak) is mostly considered. opérateuers, Opérateuers de Calderón- with some care; for example, the inter- Here, the results that depend on the Zygmund, by Yves Meyer, and the volume change of derivative and infinite summa- domain shape include Friedrichs’ inequal- Opérateuers multilinéaires, by R. R. Coifman tion is frequently used, but not discussed. ity, Hardy’s inequality, estimates of the and Yves Meyer. The original numbering Nevertheless, the book describes some extension operator (also the weighted of the chapters and of the theorems has useful methods for solving difference case), trace theorems and embedding the- been retained. equations and can be recommended as a orems (the Sobolev inequality). Another In this volume the theory of paradiffer- source of interesting examples of applica- type of domains considered are domains ential operators and the Cauchy kernel on tions. (ja) between two graphs, of type {[x, y]|φ(x) < Lipschitz curves are discussed, with the y < ψ(x)}, which even for smooth graphs emphasis firmly on their connection with P. J. Nahin, Duelling Idiots and Other may have singularities at boundary points wavelet bases. Calderón-Zygmund opera- Probability Puzzlers, Princeton University belonging to the contact set {[x, y]| φ(x) = tors have a special relationship with Press, Princeton, 2000, 232 pp., £15.95, y = ψ(x)}. Some results on traces are con- wavelets and with classical pseudo-differ- ISBN 0-691-00979-1 sidered on arbitrary domains. ential operators, of which they are a There are many textbooks on probability Specialists in function spaces will remarkable generalisation. They form the theory, but unfortunately, books with already have this book, as well as others in subject of an independent theory which interesting problems and examples from the excellent series of books by V. G. the authors expand completely and probability theory are extremely rare. This Maz’ya. For the same reason, the book is autonomously in Chapters 7-11. is such an exception. The author is an widely known among experts in boundary Multilinear analysis is one of the routes experienced teacher. His collection of value problems for elliptic partial differen- into the non-linear problems studied in twenty-one puzzles (with solutions) is tial equations. Although such equations Chapters 12-16. designed for students who are eager to try are not explicitly studied in the book (with This route is possible only for those non- their skills on challenging problems. one exception), the theory developed linear problems with a holomorphic struc- The first problem, whose name forms there is needed for an analysis of such ture, enabling them to be decomposed the title of the book, can be formulated as problems. However, the book may be use- into a series of multilinear terms of follows. Two idiots A and B decide to duel, ful and interesting for mathematicians increasing complexity. The multilinear but they have only one six-shot revolver working in other related areas, such as the operators turn out to be the Calderón- and only one bullet in it. First A spins the rest of PDE theory, the calculus of varia- Zygmund operators, whose continuity is cylinder and shoots at B. If the gun does tions, numerical analysis and the theory of established using the earlier chapters. not fire, then B spins the cylinder and functions of several real variables. The Wavelets make a final appearance, as shoots at A. The process continues until ‘bad domains’ are not artificial products eigenfunctions of certain realisations of one fool shoots the other. What is the invented only for counter-examples, and paraproducts, in the final chapter, which is probability that A will win? How many the emphasis is put on simple shapes with devoted to J. M. Bony’s theory of paradif- trigger pulls will occur (on average) before cusps that occur in real life. The book is ferential operators. The bibliography lists somebody wins? Most of the problems are strongly recommended to researchers and 239 items. This book can be strongly rec- formulated in this style: for example, find advanced students. (jama) ommended to those wishing to learn about the distribution function and density of the mathematical foundations of operator the length of a walk through a square gar- R. J. Y. McLeod and M. L. Baart, theory and wavelets. (knaj) den if one enters it at a randomly chosen Geometry and Interpolation of Curves and point on the border and continues in a Surfaces, Cambridge University Press, L. A. Moyé and A. S. Kapadia, Difference random direction. A few problems are for- Cambridge, 1998, 414 pp., £50, ISBN 0-521- Equations with Public Health mulated mathematically: for example, 32153-0 Applications, Biostatistics: A Series of find the density of the random variable Z 40 EMS December 2000 RECENT BOOKS = XY when X and Y are independent vari- The text yields a detailed insight into con- nances and quasi-modes. (jnek) ables with rectangular distribution on (0, cepts involving type and co-type of Banach 1). The author proposes simulating the spaces, B-convexity, super-reflexivity, the M. Serfati (ed.), La recherche de la vérité, problems on a computer to check the the- vector-valued Fourier and Hilbert trans- L’écriture des mathématiques, ACL, Les éditions oretical results. The last part of the book forms, and the unconditionality property du Kangourou, Paris, 1999, 335 pp., ISBN contains MATLAB programs that serve for martingale differences. The list of sec- 2-87694-057-4 this purpose. To help understanding of tions includes ideal norms, operator This book presents 10 articles based on the principles of simulation, a short chap- ideals, Khintchine constants, Sidon con- the history of mathematics seminar at the ter on random numbers generators is stants, Riemann, Dirichlet and Parseval Université Paris VII. The texts are written included. This book can be recommended ideal norms, Gauss versus Rademacher, with a view to the historical analysis of as inspiration for teachers of introductory the Maurey-Pisier theorem, J-convexity, ideas and to epistemology and present courses on probability theory. (ja) unconditional norms, super weakly com- actual researches into the history of math- pact operators, and uniform convexity and ematics. D. Perrin, Géometrie algébrique, uniform smoothness. The text also The first contribution by the editor is Mathématiques, EDP Sciences, Les Ulis, 1995, includes challenging unsolved problems. devoted to the birth of the procedure for 301 pp., FRF 240, ISBN 2-7296-0563-0 and The book is accessible to graduate stu- solving cubic (algebraic) equations in 16th- 2-271-05271-8 dents and researchers interested in func- century Italy. The second article, written This is a well-composed first course on tional analysis and is understandable with by M. Waldschmidt, is a historical survey algebraic geometry, based on the author’s a basic knowledge of Banach space theory of the theory of transcendental numbers courses from 1991-94 at the Université together with a background from real up to 1900 (when Hilbert posed his 7th Paris Sud (Orsay). Covering the material analysis, probability and algebra. (jl) problem, whose solution was the source of should take approximately 50 hours, and a a new era for this theory); in particular, it quarter of this time should be devoted to B. P. Rynne and M. A. Youngson, Linear presents works of Liouville, Cantor, exercises. The methods are entirely alge- Functional Analysis, Springer Hermite, Lindemann and Weierstrass on braic, but the author requires from a read- Undergraduate Mathematics Series, Springer, this topic. In the third article, M. Barbut er only a fairly standard knowledge of London, 2000, 273 pp., DM59, ISBN 1- describes an approach for teaching proba- algebra. He very skilfully introduces the 85233-257-3 bility to beginners which is based on intu- really necessary ideas from commutative This book provides an introduction to the ition of the economical value: the expect- algebra, and has included an appendix ideas and methods of linear functional ed value of a (discrete) random variable is Mémento d’algébre, where one can find a analysis and is addressed mainly at under- axiomatically introduced and then the compact summary of the necessary defini- graduate students. The opening chapter Kolmogorov’s ‘axioms’ for probability tions and results with references. outlines the basic ideas from linear alge- measures are derived. R. Langevin sur- The text in fact represents an introduc- bra, metric spaces and Lebesgue measure veys the history of integral geometry and tion into contemporary algebraic geome- and integration that are required through- its interactions with probability theory, try. The principal notion is an algebraic out the book. Further chapters are devot- measure theory, Riemannian geometry variety, always endowed with the corre- ed to the fundamental properties of and topology, starting from 1777. sponding sheaf. The author’s explicitly normed linear and Hilbert spaces and to A philosophical article of M. Serfati, La stated idea behind the exposition is to linear transformations between these dialectique de l’indétermité, de Viète à Frege et start with problems that can be simply for- spaces (also the open mapping theorem Russell, is devoted to the representation of mulated, but whose solution is non-trivial. and its equivalent forms). Elementary ‘datum’ in symbolic mathematical writing. Concerning the important notion of a properties of linear operators on Hilbert In P. Cegielski’s contribution, the history scheme, in the text we meet only schemes spaces and of compact operators are of presentation, formalisation and of dimension 0, but, being aware of the included in the next chapters. The last axiomatisation of mathematics from importance of this notion, the author chapter is concerned with applications of ancient times is described, leading to the includes an appendix Les schémas. In the previous results to integral and differential Zermelo-Fraenkel set theory as a base of text are many exercises and problems, equations. More sophisticated theorems, mathematics. O. Hudry’s article deals with including those used at the examinations such as the Hahn-Banach theorem, the the four-colour problem, methods and organised by the author. In summary, the principle of uniform boundedness and the results of its study in historical order, and main feature of this book is a good choice notions as reflexivity, are not included. meditates over the ‘correctness’ of com- of topics and a very nice presentation of The text includes many exercises with puter-aided proofs. The eighth article is a them. (jiva) complete solutions. The book is under- philosophical survey on the conception of standable with only standard undergradu- scientific discovery, due to the scientist- A. Pietsch and J. Wenzel, Orthonormal ate linear algebra and real analysis. (jl) philosophers Poincaré and Einstein: the Systems and Banach Space Geometry, creative process is based on ‘free construc- Encyclopedia of Mathematics and its Séminaire Bourbaki 1998/99, exposés tions of thought’; the author, M. Paty, Applications 170, Cambridge University Press, 850-864, Astérisque 266, Société compares it with other concepts and tries Cambridge, 1998, 553 pp., £55, ISBN 0-521- Mathématique de France, Paris, 2000, 483 to connect them with the scientists’ inven- 62462-2 pp., FRF 450, ISBN 2-85629-090-6 tions. The ninth article presents the histo- This voluminous monograph is devoted to This volume comprises written versions of ry of the negative answer to Hilbert’s tenth the interplay between orthonormal expan- fifteen lectures at the Bourbaki Seminar problem on the solvability of diophantine sions and Banach space geometry. A the- during 1998-99. The topics covered are equations, to which the author, Y. ory of orthonormal expansions with vec- the following: recent proofs of the local Matiasevitch, contributed. In the final tor-valued coefficients is presented. Langlands Conjecture for GL(n) and of article, J. Bénabou studies the analogy Besides the classical trigonometric system, Kepler’s Conjecture on the densest sphere between the categories of ‘observable sets’ other orthonormal systems are considered packing in R3; quantum computing; the (introduced in the text) and ‘ordinary (Haar and Walsh functions, Rademacher classification of simple Lie algebras in sets’. functions and Gaussian random variables). characteristic p > 7; chaotic behaviour of The book has appeared in the collection Harmonic analysis is a starting point and the motion of inner planets in the solar L’écriture des mathématiques, directed by M. classical inequalities and special functions system; spin glasses; p-adic L-functions Serfati. It is aimed at a wide readership: lead to the study of orthonormal systems and p-adic integration; Brownian motion mathematicians, philosophers, historians, of characters on compact Abelian groups. with obstacles; finite subgroups of Lie teachers, and others interested in science The authors investigate numerical para- groups; Thurston’s uniformisation theo- and its development. (efa) meters that can be used to quantify certain rem; singularities of solutions of non-lin- properties of Banach spaces (such as a ear wave equations; holonomy groups; L2- P. Taylor, Practical Foundations of measure of non-Hilbertness of the space). methods in algebraic geometry; reso- Mathematics, Cambridge Studies in Advanced EMS December 2000 41 RECENT BOOKS Mathematics 59, Cambridge University Press, Distributive modules are characterised in Lazard’s description of cohomology alge- Cambridge, 199, 572 pp., £50, ISBN 0-521- several different ways, and are studied bras of uniform pro-p groups. The final 63107-6 simultaneously with uniserial and Bezout chapter is on finitely presented pro-p ‘Our system conforms very closely to the modules (each finitely generated submod- groups. way mathematical constructions have actu- ule is cyclic). In some cases the notions The treatment is accessible to graduate ally been formulated in the twentieth cen- coincide (for example, over local rings) or students and includes exercises and histor- tury. The claim that set theory provides are closely related (over semi-perfect ical and bibliographical notes. The clear the foundations of mathematics is only jus- rings). Right distributive rings are topological character of the subject is tified via an encoding of this system, and described when they are semi-perfect right explored in the development of the theo- not directly’, says the author in paragraph Goldie rings, right perfect or semi-perfect ry but is also well explained at elementary 2.2 after Chapter I on First order reason- right noetherian. It is proved that the level. (rb) ing’ and after the first paragraph of Krull dimension of left or right regular Chapter II on ‘Constructing the number modules over noetherian right distributive M. C. K. Yang, Introduction to Statistical system’. He does not use ‘the encoding’ rings is at most 1. Then semi-distributive Methods in Modern Genetics, Asian and he works with sets quite freely using modules are defined as direct sums of dis- Mathematics Series 3, Gordon and Breach, the axiom of comprehension: if X is a set and tributive ones. Rings over which all right Singapore, 2000, 247 pp., US$75, ISBN 90- ϕ[x] is a predicate on X, then {x: X ϕ[x]} is modules are semi-distributive, serial rings, 5699-134-5 also a set. Beginning with Chapter IV, his and rings such that each (finitely generat- The importance of genetics can be felt basic mathematical tool is category theory. ed) right module is serial, are charac- almost daily. The topics the author choos- The use of categories as a basis for mathe- terised. The author continues with the es are undoubtedly biased; as he explains: matics and the application of categorical study of tensor product and flat modules; ‘these are the topics I wanted to know notions and methods in logic and in com- the relation of this part to the topic is quite more about when I got into this field, and puter science (more precisely, the devel- vague. Some strong results are obtained I hope that many beginners will share the oping of logic and of computer science on for modules over right invariant rings same interest in them’. The topics include the basis of categorical notions and cate- (rings whose right ideals coincide with questions as how a gene is found, how sci- gorical methods) has been widely and two-sided ones). After that, the question of entists have separated the genetic and intensively examined during the last peri- the left-right symmetry of distributivity is environmental aspects of a person’s intel- od. The book presents a systematic expo- treated and distributive modules over ligence, how genetics has been used in sition of this topic, explaining its ideas and commutative rings are studied. Among agriculture so that domestic animals and summarising the corresponding results. other things it is shown that an endomor- crops are constantly improved, what a The author also aims to show how these phism ring of a distributive module over a DNA fingerprint is and why there are con- modern ideas develop the classical ones commutative ring is commutative. Finally, troversies about it, and how genes were and he presents many interesting histori- the question of preserving distributivity used to rebuild evolutionary history? cal facts. The book is intended for pro- when passing from a ring to another using The author believes he understands grammers and computer scientists, rather certain classical ring constructions is these questions and hopes that his readers than for mathematicians and logicians, but touched on. A remarkable fact is that a will not find gaps in how they are it can be useful for both groups. (vt) module M is distributive if and only if its answered. The book is written mainly for character module Hom (M,E), where E is statistics students and therefore some sta- A. Terras, Fourier Analysis on Finite an injective cogenerator, is End (E) dis- tistical background beyond elementary Groups and Applications, London tributive. statistics is assumed. The author hopes Mathematical Society Student Texts 43, The author does not separate lemmas, that a year of graduate study in most sta- Cambridge University Press, Cambridge, propositions and theorems, and the text tistical departments is sufficient back- 1999, 442 pp., £18.95, ISBN 0-521-45108- gives the impression of a homogeneous ground. (jant) 6 and 0-521-45718-1 mass of claims; orientation is very difficult. The main goal of this book is to consider Even though it is impossible to avoid all List of reviewers for 2000 finite analogues of symmetric spaces, such formal mistakes in such a large text, the The Editor would like to thank the following for as Rn and the Poincar upper half plane. number of inaccuracies in the book is their reviews this year: J. Andìl, J. Antoch, The author describes finite analogues of greater than one would expect. However, R. Bashir, J. Beèváø, M. Nìmcová- all the basic theorems in Fourier analysis, despite a few drawbacks, the book pro- Beèváøová, V. Beneš, L. Beran, J. Bureš, both commutative and non-commutative, vides ample material for anyone interested E. Calda, K. Èuda, A. Drápal, V. Dupaè, including the Poisson summation formula in the topic. (pruz) J. Dupaèová, J. Eisner, A. Elashvilli, M. and the Selberg trace formula. One moti- Engliš, E. Fašangová, M. Feistauer, E. vation for this study is to prepare the J. S. Wilson, Profinite Groups, London Fuchs, S. Hencl, J. Hurt, M. Hušek, M. ground for understanding the continuous Mathematical Society Monographs New Series Hušková, T. Kepka, M. Klazar, J. theory by developing its finite model. The 19, Clarendon Press, Oxford, 1998, 284 pp., Kopáèek, O. Kowalski, J. Kratochvíl, M. book is written in such a way that it can be ISBN 0-19-850082-3 Kruík, M. Kríek, M. Kulich, P. Lachout, enjoyed by non-experts, such as advanced This book presents the study of a nice class M. Loebl, J. Lukeš, J. Málek, J. Malý, P. undergraduates, beginning graduate stu- of infinite groups that are built up from Mandl, M. Markl, J. Matoušek, D. dents, and scientists from outside mathe- their finite homomorphic images and, Medková, J. Milota, J. Mlèek, K. Najzar, matics. Several applications are included: indeed, appear in the classical literature as J. Nekovár, J. Nešetril, I. Netuka, the construction of graphs that are good Galois groups of algebraic field exten- Z.Pluhar, P. Pyrih, Š. Porubský, J. Rohn, expanders, reciprocity laws in number the- sions. Although profinite groups attract T. Roubíèek, P. Ruièka, P. Simon, J. ory, the Ehrenfest model of diffusion ran- the attention of abstract group theorists we Slovák, V. Souèek, J. Trlifaj, V. Trnková, dom walks on graphs, and vibrating sys- also meet them as topological quotients of J. Tuma, J. Vanura, J. Veselý, L. Zajíèek. tems and chemistry of molecules. (ae) compact groups. All of these are on the staff of the Charles The book starts with topological prelim- University, Faculty of Mathematics and A. Tuganbaev, Distributive Modules and inaries and introductory chapters giving Physics, Prague, except: J. Eisner, M. Related Topics, Algebra, Logic and the basics of completions, Sylow theory Engliš, M. Kruík, M. Kríek, M. Markl, Applications Series 12, Gordon and Breach, and Galois theory. These chapters are fol- D. Medková and J. Vanura (Mathematical Amsterdam, 1999, 258 pp., US$95, ISBN 90- lowed by the study of modules over group Institute, Czech Academy of Sciences), M. 5699-192-2 algebras for a profinite group, with coeffi- Nìmcová-Beèvárová and Š. Porubský This book is not based on a few main the- cients in a profinite commutative ring. (Technical University, Prague), J. Nekovár orems, but is rather a collection of miscel- The advanced part of the book explains (Cambridge University, UK), E. Fuchs and J. laneous results bearing on the central the cohomology theory of profinite Slovák (Masaryk University, Faculty of notion of a distributive module. groups. Among the main results is Natural Sciences, Brno). 42 EMS December 2000 INDEX EMS Lectures 36-9 Societies EMS Newsletter EMS Position Paper: Towards a European Dutch Mathematical Society [Wiskundig research area (Luc Lemaire) 36-24 Genootschap] 35-12, 36-23 Index for 2000 EMS Committee for Women and Danish Mathematical Society (Bodil Mathematics 37-15, 38-4 Branner) 35-14 The numbers 35–38 refer to the issue numbers, EMS Poster Competition 37-19 Catalan Mathematical Society (Sebastià in March, June, September and December, Xambó-Descamps) 36-3 respectively; the second number is the page num- Feature articles London Mathematical Society (Adrian ber. Jeremy Gray: The Hilbert problems 1900- Rice) 37-28 2000 36-10 L’Unione Matematica Italiana (Giuseppe Editorials Jean-Pierre Bourguignon: A major chal- Anichini) 38-26 Vagn Lundsgaard Hansen (WMY2000 lenge for mathematicians 36-20 President) 35-4 Philip Maini: Mathematical modelling in 2000 anniversaries Marta Sanz-Solé (3ecm organiser) 36-3 the biosciences 37-16 Sonya Kovalevskaya (June Barrow- Olli Martio (EMS Treasurer) 37-3 Aatos Lahtinen: The pre-history of the EMS Green) 35-9 Anatoly Vershik (EC retiring member) 38-14 Eugenio Beltrami (Jeremy Gray) 35-11 38-3 The Hilbert Problems (Jeremy Gray) Short articles 36-10 Introducing . . . A. D. Gardiner: Mathematics in English John Napier (John Fauvel) 38-24 WMY2000 team 35-5 schools 35-16 New members of the Executive Paul Jainta: Problem Corner 35-20, 37-30 Summer Schools and Conferences Committee 38-5 Ulf Rehmann: The price spiral of mathe- Oberwolfach programme 2001 35-19 matics journals 38-29 Forthcoming conferences (Kathleen EMS News Hans J. Munkholm: Joint AMS- Quinn) 35-25, 36-28, 37-32, 38-31 Message from the EMS President (Rolf Scandinavia meeting 38-11 EMS Summer School in Edinburgh Jeltsch) 35-3 M. Joswig & K. Polthier: Digital models (Erkki Somersalo) 38-10 Executive Committee meetings and computer assisted proofs 38-30 EMS-SIAM Joint Conference on Applied (Bedlewo, Barcelona and London) Agenda des conferences mathématiques 36-19 Mathematics 37-14 36-6, 37-5, 38-6 3rd European Congress of Mathematics Interviews Book reviews (3ecm) 36-5, 37-8, 37-10, 38-11 Lars Gårding 35-6 Recent books (Ivan Netuka & Vladimír Prizes awarded at 3ecm 37-12 Peter Deuflhard 36-14 Soucek) 35-32, 36-34, 37-36, 38-35 World Mathematical Year 2000 Jaroslav Kurzweil 36-16 Book review by Sir Michael Atiyah 36-40 35-4, 38-12 Martin Grötschel 37-20 EMS Council 35-3, 37-6 Bernt Wegner 37-24 Miscellaneous Report on the LIMES-Project 36-8 Sir Roger Penrose 38-17 WMY2000 stamps EMS Summer Schools and Conferences Vadim G. Vizing 38-22 35-18, 36-27, 37-4, 37-27, 38-13 36-9, 37-14

MATHEMATICAL REVIEWS Associate Editor Applications and recommendations are invited for a full-time position as an Associate Editor of Mathematical Reviews (MR), to commence as soon as possible after 1 April 2001, and no later than 1 July 2001. The Mathematical Reviews division of the American Mathematical Society (AMS) is located in Ann Arbor, Michigan, not far from the campus of the University of Michigan. The editors are employees of the AMS; they also enjoy many privileges at the University. At present, MR employs fourteen mathematical editors, about six consultants and a further sixty non-mathe- maticians. MR’s mission is to develop and maintain the AMS databases of secondary sources covering the published mathe- matical literature. The chief responsibility is the development and maintenance of the MR Database, from which all MR-relat- ed products are produced: MathSciNet, the journals Mathematical Reviews and Current Mathematical Publications, MathSciDisc, and various other derived products. The responsibilities of an Associate Editor fall primarily in the day-to-day operations of selecting articles and books suitable for coverage in the MR database, classifying these items, determining the type of cover- age, assigning those selected for review to reviewers, editing the reviews when they are returned and correcting the galley proofs. An individual with considerable breadth in both pure and applied mathematics is sought; preference will be given to those applicants with expertise in the broad area of applied mathematics, and in particular in one or more of the following areas: numerical analysis (Section 65) or mathematical economics and life sciences (91, 92). The ability to write good English is essential and the ability to read mathematics in major foreign languages is important. It is desirable that the applicant have several years’ relevant academic (or equivalent) experience beyond the Ph.D. The twelve-month salary will be commensurate with the experience the applicant brings to the position. Interested appli- cants are encouraged to write (or telephone) for further information. Persons interested in taking extended leave from an aca- demic appointment to accept the position are encouraged to apply. Applications (including curriculum vitae, bibliography, and name, address and phone number of at least three references) and recommendations should be sent to Dr Jane E. Kister (Executive Editor), Mathematical Reviews, P.O. Box 8604, Ann Arbor, MI 48107-8604, USA (e-mail: tel: (+1)-734-996-5257; fax: (+1)-734-996-2916. The closing date for applications is 1 February 2001.

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EMS December 2000 43 PERSONAL COLUMN PPersonalersonal ColumnColumn

We list below information about some appoint- of National Mathematics Competitions ments, awards and deaths that have occurred in Ludwig Elsner (Bielefeld) has been award- (WFNMC). the past few months. Since this list is inevitably ed the Hans Schneider Prize in Linear incomplete we invite you to send appropriate Algebra by the International Linear Jean-Pierre Serre (Paris) has been award- information to the Editor [[email protected]. Algebra Society (ILAS). ed an Honorary Doctorate by the uk] or to your Country representative (see Issue University of Durham. 34) for inclusion in the next issue. Please also Athanassios Fokas (London) has been send any items you feel should be included in awarded the Naylor Prize for 2000 by the Ian Stewart (Warwick) has been awarded future Personal Columns. London Mathematical Society for substan- the Institute of Mathematics and its tial contributions to the theory of inte- Applications Gold Medal for 2000 for Awards grable systems. exceptional service to mathematics and research Nigel Hitchin (Oxford) has been awarded Semyon Alesker (Israel), Raphael Cerf, the Sylvester Medal of the Royal Society of Vera Sós (Budapest) has been elected as an Emmanuel Grenier, Vincent Lafforgue, London for contributions to geometry. Honorary Fellow of the Institute of Paul Seidel and Wendelin Werner Combinatorics and its Applications. (France), Dominic Joyce and Michael Sir Tony Hoare (Cambridge) has been McQuillen (UK) and Stefan Nemirovski awarded an Honorary Doctorate by John Toland (Bath) has been awarded the (Russia) were awarded EMS prizes at the Oxford Brookes University. Senior Berwick Prize for 2000 by the Third European Congress in Barcelona; London Mathematical Society for an out- details of their work can be found in EMS John Howie (St Andrews) has been award- standing piece of research. Newsletter 37. ed an Honorary Doctorate by the Open University, UK. Hendrik Van Maldeghem (Ghent) has Pierre Auger (Lyon), Gérard Bricogne been awarded a Hall Medal by the Institute (Orsay) and Thibauld D’Amour (IHES) Laurent Lafforgue (Paris) has been award- of Combinatorics and its Applications. have been elected to membership of the ed a Clay Research Award by the Clay Académie des Sciences (Paris). Mathematics Institute for work on the Benjamin Weiss (Jerusalem) has been Langlands programme elected as a foreign honorary member of John Ball (Oxford) has been elected as a the American Academy of Arts and foreign member of the Académie des Jacques-Louis Lions (Paris) has been Sciences. Sciences de Paris. awarded the Lagrange Prize by ICIAM.

Grigory Barenblatt (Russia) has been Terry Lyons (Oxford) has been awarded Deaths awarded the Maxwell Prize by ICIAM (The the Pólya Prize for 2000 by the London International Council for Industrial and Mathematical Society for fundamental We regret to announce the deaths of: Applied Mathematics). contributions to analysis and probability. Slawomir Biel (1 September 2000) Richard Borcherds (Cambridge) has been Robert MacKay (Warwick) and Paul awarded an Honorary Doctorate by the Townsend (Cambridge) have been elected Florent J. Bureau (28 June 1999) University of Birmingham. Fellows of the Royal Society of London. V. N. Fomin (23 February 2000) Elisabeth Busser and Gilles Cohen have Stefan Müller (Leipzig) has been awarded been jointly awarded the d’Alembert Prize the Collatz Prize by ICIAM. Rainer Hettich (23 July 2000) for 2000 by the Société Mathématique de France. Helmut Neunzert (Kaiserslautern) has Aubrey Ingleton (28 June 2000) been awarded a SIAM Pioneer Prize by Mark Chaplain (Dundee), Gwyneth ICIAM. Frank Leslie (15 June 2000) Stallard (Milton Keynes), Andrew Stuart (Warwick) and Burt Totaro (Cambridge) Hilary Ockendon (Oxford) has been Thomas Lippold (10 June 2000) have been awarded Whitehead Prizes for awarded an Honorary Doctorate by the 2000 by the London Mathematical Society. University of Southampton. Cyril Offord (4 June 2000)

Michele Conforti (Padua) has shared the Sir Roger Penrose (Oxford) has been Jean-Marie Painvin (July 2000) 2000 Delbert Ray Fulkerson Prize for a awarded the Order of Merit. Membership paper on the decomposition of balanced of this order is limited to 24 people; anoth- Ian Sneddon (4 November 2000) matrices. er current holder is Sir Michael Atiyah. Terence Stanley (15 October 2000) Alain Connes (Paris) has been awarded a Sergei Pereversev (Ukraine) has been Clay Research Award by the Clay awarded the 2000 Prize for Achievement in Dirk Struik (21 October 2000) Mathematics Institute for revolutionising Information-based Complexity for many the field of operator algebras and invent- outstanding contributions to the area. Ion Suliciu (24 November 1999) ing modern non-commutative geometry. Istvan Reiman and János Suranyi Andreas Tamanas (15 August 2000) Simon Donaldson (London) has been (Hungary) and Francisco Bellot Rosado elected as a foreign associate of the US (Valladolid) have been awarded Paul Erdos Lyndon Woodward (12 June 2000) National Academy of Sciences. National Awards of the World Federation 44 EMS December 2000