The Catalan Mathematical Society EMS June 2000 3 EDITORIAL

Home , Leo Corry, Ludolph van Ceulen, Oleg Viro, Renato Caccioppoli, Sandu Popescu

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. 36 DK-2800 Kgs. Lyngby, Denmark e-mail: [email protected] KRZYSZTOF CIESIELSKI June 2000 Mathematics Institute Jagiellonian University Reymonta 4 30-059 Kraków, Poland EMS News : Agenda, Editorial, 3ecm, Bedlewo Meeting, Limes Project ...... 2 e-mail: [email protected] KATHLEEN QUINN Open University [address as above] Catalan Mathematical Society ...... 3 e-mail: [email protected]

SPECIALIST EDITORS The Hilbert Problems ...... 10 INTERVIEWS Steen Markvorsen [address as above] SOCIETIES Interview with Peter Deuflhard ...... 14 Krzysztof Ciesielski [address as above] EDUCATION Vinicio Villani Interview with Jaroslav Kurzweil ...... 16 Dipartimento di Matematica Via Bounarotti, 2 56127 Pisa, Italy A Major Challenge for Mathematicians ...... 20 e-mail: [email protected] MATHEMATICAL PROBLEMS Paul Jainta EMS Position Paper: Towards a European Research Area ...... 24 Werkvolkstr. 10 D-91126 Schwabach, Germany e-mail: [email protected] Forthcoming Conferences ...... 28 ANNIVERSARIES June Barrow-Green and Jeremy Gray Open University [address as above] Recent Books ...... 34 e-mail: [email protected] and [email protected] and Personal Column ...... 42 CONFERENCES Kathleen Quinn [address as above] RECENT BOOKS Ivan Netuka and Vladimir Sou³ek Designed and printed by Armstrong Press Limited Mathematical Institute Unit 3 Crosshouse Road, Southampton, Hampshire SO14 5GZ, UK Charles University phone: (+44) 23 8033 3132; fax: (+44) 23 8033 3134 Sokolovská 83 18600 Prague, Czech Republic Published by European Mathematical Society e-mail: [email protected] ISSN 1027 - 488X and [email protected] ADVERTISING OFFICER NOTICE FOR MATHEMATICAL SOCIETIES Vivette Girault Labels for the next issue will be prepared during the second half of August 2000. Laboratoire d’Analyse Numérique Please send your updated lists before then to Ms Tuulikki Mäkeläinen, Department of Mathematics, Boite Courrier 187, Université Pierre P.O. Box 4, FIN-00014 University of Helsinki, Finland; e-mail: [email protected] et Marie Curie, 4 Place Jussieu 75252 Paris Cedex 05, France INSTITUTIONAL SUBSCRIPTIONS FOR THE EMS NEWSLETTER 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-mail: OPEN UNIVERSITY Please include the name and full address (with postal code), telephone and fax number (with coun- PRODUCTION TEAM try code) and e-mail address. The annual subscription fee (including mailing) is 60 euros; an Liz Scarna, Barbara Maenhaut invoice will be sent with a sample copy of the Newsletter.

EMS June 2000 1 EMS NEWS EMS News: Committee and Agenda EXECUTIVE COMMITTEE EMS Agenda PRESIDENT (1999–2002) 2000 Prof. ROLF JELTSCH 13-16 June Seminar for Applied Mathematics EMS Lectures by Prof. Dr. George Papanicolaou (Stanford, USA). ETH, CH-8092 Zürich, Switzerland ETH, Zürich (Switzerland): Financial Mathematics e-mail: [email protected] Contact: David Brannan, e-mail: [email protected] VICE-PRESIDENTS 17-22 June Prof. ANDRZEJ PELCZAR (1997–2000) EURESCO Conference in Mathematical Analysis at Castelvecchio Pascoli (Italy): Institute of Mathematics Partial Differential Equations and their Applications to Geometry and Physics Jagellonian University Organiser: J. Eichhorn, Greifswald (Germany), e-mail: [email protected] Raymonta 4 [This series of conferences is financed by the ESF.] PL-30-059 Krakow, Poland 3-7 July e-mail: [email protected] ALHAMBRA 2000: a joint mathematical European-Arabic conference in Granada (Spain), Prof. LUC LEMAIRE (1999–2002) promoted by the EMS and the Spanish Royal Mathematical Society Department of Mathematics Contact: Ceferino Ruiz, e-mail: [email protected] website: www.ugr.es/~alhambra2000 Université Libre de Bruxelles 6 July C.P. 218 – Campus Plaine Executive Committee Meeting in Barcelona (Spain) Bld du Triomphe 7-8 July B-1050 Bruxelles, Belgium EMS Council Meeting at Institut d’Estudis Catalans, Carrer del Carme 47, E-08001 Barcelona e-mail: [email protected] (Spain), starting at 10 a.m. SECRETARY (1999–2002) Contact: EMS Secretariat, e-mail: [email protected] Prof. DAVID BRANNAN 10-14 July Department of Pure Mathematics Third European Congress of Mathematics (3ecm) in Barcelona (Spain) The Open University e-mail: [email protected] website: www.iec.es/3ecm/ Walton Hall 24 July-3 August Milton Keynes MK7 6AA, UK EMS Summer School, in Edinburgh (Scotland): e-mail: [email protected] New analytic and geometric methods in inverse problems TREASURER (1999–2002) Organiser: Erkki Somersalo, Otaniemi (Finland), e-mail: [email protected] Prof. OLLI MARTIO 15 August Department of Mathematics Deadline for submission of material for the September issue of the EMS Newsletter P.O. Box 4 Contact: Robin Wilson, e-mail: [email protected] FIN-00014 University of Helsinki 17 August-2 September Finland EMS Summer School at Saint-Flour, Cantal (France): Probability theory e-mail: [email protected] Organiser: Pierre Bernard, Clermont-Ferrand (France), e-mail: [email protected] ORDINARY MEMBERS 28-30 August Prof. BODIL BRANNER (1997–2000) EMS Lectures by Prof. Dr. George Papanicolaou (Stanford, USA). Department of Maathematics University of Crete, Heraklion, Crete (Greece): Time Reversed Acoustics Technical University of Denmark Contact: David Brannan, e-mail: [email protected] Building 303 Autumn DK-2800 Kgs. Lyngby, Denmark Fifth Diderot Mathematical Forum on Mathematics and Telecommunications e-mail: [email protected] Date and programme to be announced. Prof. DOINA CIORANESCU (1999–2002) Contact: Jean-Pierre Bourguignon, e-mail: [email protected] Laboratoire d’Analyse Numérique 22-27 September EURESCO Conference at Obernai, near Strasbourg (France): Université Paris VI Number theory and Arithmetical Geometry: Motives and Arithmetic 4 Place Jussieu Organiser: U. Jannsen, Regensburg (Germany), e-mail: [email protected] 75252 Paris Cedex 05, France EURESCO Conference at San Feliu de Guixols (Spain): e-mail: [email protected] Geometry, Analysis and Mathematical Physics: Analysis and Spectral Theory Prof. RENZO PICCININI (1999–2002) Organiser: J. Sjöstrand, Palaiseau (France), e-mail: [email protected] Dipto di Matem. F. Enriques 30 September Universit à di Milano Deadline for proposals for 2001 EMS Lectures Via C. Saldini 50 Contact: David Brannan, e-mail: [email protected] I-20133 Milano, Italy Deadline for proposals for 2002 EMS Summer Schools e-mail: [email protected] Contact: Renzo Piccinini, e-mail: [email protected] Prof. MARTA SANZ-SOLÉ (1997–2000) 11-12 November Facultat de Matematiques Executive Committee Meeting in London (UK), at the invitation of the London Mathematical Universitat de Barcelona Society. Gran Via 585 15 November E-08007 Barcelona, Spain Deadline for submission of material for the December issue of the EMS Newsletter e-mail: [email protected] Contact: Robin Wilson, e-mail: [email protected] Prof. ANATOLY VERSHIK (1997–2000) P.O.M.I., Fontanka 27 2001 191011 St Petersburg, Russia 15 February e-mail: [email protected] Deadline for submission of material for the March issue of the EMS Newsletter EMS SECRETARIAT Contact: Robin Wilson, e-mail: [email protected] Ms. T. MÄKELÄINEN 10-11 March Department of Mathematics Executive Committee Meeting in Kaiserslautern (Germany), at the invitation of the Fraunhofer- P.O. Box 4 Institut für Techno- und Wirtschafts Mathematik. FIN-00014 University of Helsinki 9-21 July Finland EMS Summer School at the Euler Institute, St Petersburg (Russia): tel: (+358)-9-1912-2883 Asymptotic combinatorics and applications to mathematical physics and representation theory fax: (+358)-9-1912-3213 Organiser: Anatoly Vershik, e-mail: [email protected] telex: 124690 3-6 September e-mail: [email protected] 1st EMS-SIAM conference, Berlin website: http://www.emis.de Organiser: Peter Deuflhard, e-mail: [email protected] 2 EMS June 2000 SOCIETIES

TheThe CatalanCatalan MathematicalMathematical SocietySociety

Sebastià Xambó-Descamps

One of the 25 societies of the Institute of tiques and René Descartes’ La Geometria. Cangur and awarded special prizes to the Catalan Studies (IEC), the Catalan Among high school teachers, the Recull twelve best winners of the joint three com- Mathematical Society (SCM) today faces Cangur has a growing popularity; this book munities; the award ceremony for this year important challenges. Some of these stem is a collection of Cangur problems, updated will be in Barcelona in July, after the 3ecm. from its local contributions to World every year to prepare entrants for the The SCM is a member of the EMS and has Mathematical Year 2000, but the main one Mathematical Olympiad. reciprocity agreements with several other springs from the EMS Council’s decision at societies, among them the Real Sociedad Budapest in 1996 to select the SCM’s bid to Matemática Española (RSME) and the organise the Third European Congress of American Mathematical Society (AMS). Mathematics (3ecm). This takes place in The present SCM, with over 1000 mem- Barcelona from 10-14 July, in the Palau de bers, was founded in 1986 as one of the four Congressos (Congress Palace), with 1500 societies resulting from the splitting by the participants expected. IEC of the Catalan Society for Physics, That the SCM has such international Chemistry and Mathematics (SCCFQM). The responsibilities should interest mathemati- SCCFQM was founded in 1931, inside the cians and professionals who use mathemat- Science Section of the IEC (which itself dates ics, and also most Catalonians, since one of from 1907), and was initially structured in the goals of the Rio de Janeiro declaration three branches: Physics and Chemistry; was to promote mathematics as a basic ele- Mathematics and Engineering; and ment for world development. This fact, a Astronomy, Meteorology and Geophysics. natural consequence of the ingredients that The 1986 SCCFQM resulted from a restruc- have contributed to the present level of sci- turing in 1968, in which the third branch ence and technology, is today even more disappeared and the first was divided into pronounced due to the progressive digitisa- two, Physics and Chemistry, and a second tion of information and knowledge (includ- restructuring in 1973, in which the second ing fundamental aspects of the financial branch was also split into two, Mathematics world), particularly with regard to the oper- and Engineering. In the late 1970s and ations that regulate its storing, transmission early 1980s, the Mathematics section expe- and use. rienced considerable growth under the pres- According to its statutes, the SCM’s mis- Another initiative is the Évariste Galois idency of Manuel Castellet, who is currently sion includes: “to cultivate the mathematical prize for students, awarded each year. In President of the IEC and a member of the sciences; to spread its knowledge among the addition, every two years the IEC awards the EMS Council: there was a substantial Catalan society; to foster its teaching and Josep Teixidor prize for the best PhD thesis increase in the membership and of the num- research, both pure and applied; and to and the prestigious Ferran Sunyer i Balaguer ber of organised activities, such as the 1980 publish whatsoever works that fit these international prize for research memoirs. Spanish-Portuguese Meeting. goals”. What are the main activities in which The SCM carries out its activities in all the In the past twenty years, the SCM has the Society engages in order to fulfil these lands of the Catalan language and culture. been a nexus between researchers in the aims? Thus Cangur 1999 and 2000 were organized three mathematics schools (the University of The oldest is the organisation in jointly, for the first time in 1999, in Barcelona, the Universitat Autònoma de Catalonia of the regional contribution to the Catalonia, the Balearic Islands and the Barcelona and the Universitat Politècnica Mathematical Olympiad, including the Valencia Country, while this year the de Catalunya), high school teachers, and preparation sessions. Since 1996, it has also Trobada was held in Valencia; the previous mathematicians working in other organisa- organised the European ‘Kangaroo’ or two meetings were held in Barcelona and it tions. An institutional presentation to the Cangur, also called the Mathematical Feast. is hoped that the 2001 meeting will be in general public of World Mathematical Year This year, on 16 March, there were over Palma de Mallorca. Since mathematics is a 2000 was held on 7 March at the University 6000 participants from the top four levels of universal language, the small linguistic vari- of Barcelona, jointly promoted by the SCM, secondary education, with the collaboration ations between the local tongues, whether in the three mathematics schools and FEEM- (for the first time) of several city halls. the Cangur or the Trobada, are very pleasant, CAT, an organisation of secondary level Last March the Trobada Matemàtica when perceptible, for the participants. teachers of mathematics. These same insti- (Mathematical meeting) was held for the The Society also maintains close relations tutions also organised a celebration of third time. Taking place once a year, it is with the institutions on which it depends for WMY2000 at the Parliament of Catalonia in based around five or six interdisciplinary its existence. In particular, since most of the early June. lectures by mathematicians from Catalan- courses that are offered need computer To conclude, I hope that the Society can speaking lands. rooms and the Society has no such facilities, continue to play this integrating role, and The Society publishes two journals: the help from the mathematics schools has been that more and more people value it as an Bulletí twice a year, and the SCM Notícies indispensable. Another example is the rela- attractive professional platform to join. four times a year. It also publishes books, tionship with the Institut Joan Lluís Vives (a such as two classics recently translated into network of the 18 universities in Catalan- Sebastià Xambó-Descamps is President of the Catalan: C. F. Gauss’s Disquisicions aritmè- speaking lands), which in 1999 housed the Catalan Mathematical Society EMS June 2000 3 EDITORIAL

BarBarcelonacelona EditorialEditorial 3ecm:3ecm: ShapingShaping thethe 21st21st centurycentury

Marta Sanz-Solé (Barcelona)

In July Barcelona will host the Third and cultural poverty in which scientific mathematicians in an international scien- European Congress of Mathematics (3ecm). research was an unattainable luxury. Many tific forum is very unusual; few of them are As the most visible manifestation of the intellectual people decided, or were members of editorial committees of high- EMS, the European Congresses of forced, to choose exile as a way to be safe. level mathematical journals or take part in Mathematics aim to present the unity of Those who remained lived in an environ- scientific committees. Now, for the first mathematics in an interdisciplinary, plur- ment of isolation compared with what was time, the Catalan mathematical communi- al, and interactive framework with an inno- happening beyond the Pyrennees. Our ty is involved in the organisation of a large- vative format. They provide a forum for lack of freedom often led to introspective scale scientific event. This clearly provides the diffusion of mathematical knowledge attitudes, largely incompatible with the us with a wonderful opportunity to and for cooperation between mathemati- process of scientific activity. increase the visibility of mathematics in cians. However, the internal dynamism of a our country and to collaborate to spread The theme chosen for the 3ecm, Shaping society oppressed by a political regime, but the aims of the EMS. This is also both a big the 21st century, invites mathematicians to challenge for us and a heavy responsibility. think about the main streams that will More than three years ago we undertook guide the development of mathematics in the organisation of the 3ecm. We started the near future. The scientific programme with the Executive Committee as its kernel. reflects this purpose. Some of the activities Very soon we felt the need to enlarge the may also encourage us to discuss the social team of mathematicians working for the role of mathematics and evaluate its 3ecm. An organising committee was impact on the development of new tech- appointed and support committees for nologies in industry, in education and, specific tasks were created. In addition we more generally, in culture. As in former received the valuable contribution of col- ECMs there will also be an opportunity to leagues for special jobs. Many mathematics recognise the talent of brilliant young students have volunteered to assist during mathematicians through the award of EMS the week of the congress. prizes. The organisation of the 3ecm would have By a happy coincidence, the 3ecm occurs been impossible without substantial back- in World Mathematical Year 2000, ing from a wide variety of public and acad- declared by a resolution of UNESCO in emic institutions, private foundations and November 1997. In Spain the Congress is corporations. Thanks to the funds provid- considered the most important mathemat- ed by a long list of sponsors we have been ical activity of the year. able to implement the main objectives of More than eight years ago, the leaders of our organisational procedure: the Catalan Mathematical Society and – to provide financial support for young their collaborators made a determined bid researchers in mathematics, with special for the Congress. With effort and keen- with well-organised structures in clandes- attention to their professional situation ness, and with a background of self-confi- tinity, made it possible for us to recover. and their country of origin. dence and hope, our colleagues proposed From a timid attitude towards Europe we – to afford the scientific recognition pro- to host an ECM. Our bid was presented for are now completely integrated in it. This gramme designed by the EMS for excellent the first time in Paris in 1992 and for the complex social evolution has resulted in young European researchers in mathemat- second time, successfully, in Budapest in the normalisation of our activities in all ics. 1996. The selection of Barcelona as the site fields, and has led in particular to a rapid – to offer good facilities for an excellent for the 3ecm is a landmark in our history, development in science. Mathematics has scientific programme conceived by the the end of a peculiar political situation, been a part of this process. May I take this Scientific committee and the Round tables and a contribution to the normalisation of opportunity to express our gratitude to a committee. the status of mathematics in Catalonia and generation of mathematicians who suc- Attending a congress provides the in Spain compared with that of other ceeded in teaching mathematics to a high opportunity to meet old friends, make new European countries with a long tradition level and who trained research fellows ones and enjoy some free time. Barcelona, in this subject. under conditions that were not conducive the venue of the 3ecm, is an attractive des- Spain, and Catalonia as a part of it, has to enthusiasm. tination for cultural tourism. A privileged missed out on the creative progress in sci- A recent report by the Institute of geographical situation, rich architecture ence in the modern age. In the early 20th Catalan Studies makes an accurate quanti- and a large variety of museums focus the century there were many attempts to create tative and qualitative analysis of mathe- attention of a large number of visitors. The and promote institutions that fostered sci- matical research in the period 1990-96; its diversity of cultural activities is one exam- entific research – in particular, research in conclusion is that, in terms of production, ple of the dynamism of a society known for mathematics. These endeavours were sud- the scientific level in our country is compa- its open character, vitality and hospitality. denly and sadly stopped by a long and rable with that of other European coun- We are convinced that the participants of cruel civil war. After April 1939 our countries with a long tradition in mathematics. the 3ecm will enjoy a high-quality scientific try was immersed in conditions of material Nevertheless, the presence of Catalan event in a splendid environment. 4 EMS June 2000 EMS NEWS 3rd European Congress of Mathematics Shaping the 21st Century EMS Council Barcelona, 10 – 14 July 2000 Meeting, 7-8 July Summary of main events 2000 (For further details, and information about satellite conferences, see EMS Newsletter 34.) The EMS Council Meeting will The plenary lectures will be given by: take place at the Robbert Dijkgraaf (Amsterdam), Hans Föllmer (Berlin), Hendrik Lenstra (Berkeley/Leiden), Yuri Manin (Bonn), Yves Meyer (Cachan), Carles Simó Institut d’Estudis Catalans, Carrer (Barcelona), Marie-France Vignéras (Paris), Oleg Viro (Uppsala/St Petersburg), del Carme 47, E-08001 Barcelona Andrew Wiles (Princeton) (Spain), Parallel lectures will be given by: starting at 10 a.m. on 7 July Rudolf Ahlswede (Bielefeld), François Baccelli (Paris), Volker Bach (Mainz), Viviane Contact: EMS Secretariat, e-mail: Baladi (Paris), Joaquim Bruna (Barcelona), Xavier Cabré (Barcelona), Peter Cameron (London), Ciro Ciliberto (Rome), Zoé Chatzidakis (Paris), Gianni Dal Maso (Trieste), [email protected] Jan Denef (Leuven), Barbara Fantechi (Udine), Alexander Givental (California), Alexander Goncharov (Providence), Alexander Grigor’yan (London), Michael Harris (Paris), Kurt Johansson (Stockholm), Konstantin Khanin (Edinburgh/Moscow), Pekka Koskela (Jyväskylä), Steffen Lauritzen (Aalborg), Gilles Lebeau (Palaiseau), Nicholas Manton (Cambridge), Ieke Moerdijk (Utrecht), Eric Opdam (Leiden), Thomas Peternell (Bayreuth), Alexander Reznikov (Durham), Henrik Schlichtkrull Barcelona accommodation (Copenhagen), Bernhard Schmidt (Augsburg), Klaus Schmidt (Vienna), Bálint Tóth The Council meeting room is (Budapest) located at the CSIC Residence, where participants can be Mini-Symposia: housed. You can see some The speakers at each mini-symposium have been selected by the Chair (listed below). information about it in: Quantum chaology [Sir Michael Berry, Bristol], Computer algebra [Wolfram Decker, http://www.resa.es/home_ Saarland], Mathematics in modern genetics [Peter Donnelly, Oxford], String theory eng.htm and M-theory [Michael Douglas, Rutgers], Mathematical finance: theory and practice http://www.resa.es/csic/ [Hélyette Geman, Paris], Quantum computing [Sandu Popescu, Cambridge], Free castellano/RESIDENCIA.HTM boundary problems [José Francisco Rodrigues, Lisbon], Symplectic and contact geometry and Hamiltonian dynamics [Mikhail Sevryuk, Moscow], Curves over finite fields and The number of reservations is codes [Gerard van der Geer, Amsterdam], Wavelet applications in signal processing limited. [Andrew Walden, London] The address is: Residence Hall for Researchers, C/ Hospital, Round Tables 64, 08001 Barcelona, Spain. Mathematics teaching at the tertiary level What is mathematics today? tel: (+34)-93-443-86-10; The impact of mathematical research on industry and vice versa fax: (+34)-93-442-82-02 The impact of new technologies on mathematical research e-mail: [email protected] Building networks of cooperation in mathematics Javier Martinez de Albeniz How to increase public awareness of mathematics [[email protected]], Shaping the 21st century Treasurer of the SCM Contact AddressesCongress e-mail: Congress web site: http://www.iec.es/3ecm/or Mailing address: Societat Catalana de Matemàtiques, Institut d’Estudis Catalans Carrer del Carme, 47 E-08001 Barcelona, Spain Phone: (+34)-93-270-16-20 Fax: (+34)-93-270-11-80

EMS Secretariat Tuulikki Mäkeläinen Since the creation of the Society, the Secretariat of the EMS has been at the Department of Mathematics of the University of Helsinki and I have been in charge of it. My main objective has been to try to help the Executive Committee and others working for the EMS by taking care of some of the day-to-day routines. My activities include preparations with the President, Secretary and Treasurer for the Executive Committee and Council meetings; contacts with member societies; contacts with individual members; daily financial transactions, in consultation with the Treasurer; keeping track of various bureaucratic handlings; replying to queries. Since receiving a M.Sc. degree in Mathematics at the University of Helsinki I have been working with mathematics and mathematicians, with the International Congress of Mathematicians in Helsinki, the International Mathematical Union, with various mathematical journals, and now with the EMS. During these years I have met countless mathematicians. It was once suggested that I write a book Mathematicians in my life – I could, with a subtitle ‘with love’. Tuulikki Mäkeläinen, e-mail: [email protected] Tuulikki Mäkeläinen photo by Czes³aw Olech EMS June 2000 5 EMS NEWS

EMSEMS ExecutiveExecutive CommitteeCommittee meetingmeeting Bedlewo 24-25 March 2000

Committee members present were: Rolf Kaiserslautern be elected corporate members. the AMS and corporate members of the Jeltsch (President, in the Chair), David The European Mathematical Trust had EMS.” It will be proposed to Council that fur- Brannan (Secretary), Bodil Branner, Doina reported its dissolution and thus its withdraw- ther reciprocity agreements can be approved Cioranescu, Luc Lemaire (Vice-President), al from EMS membership. It was noted that by the Executive Committee. The first such Olli Martio (Treasurer), Andrzej Pelczar (Vice- President) and Marta Sanz-Solé. Apologies were received from Renzo Piccinini and Anatoly Vershik. Also in attendance (by invitation) were Jean-Pierre Bourguignon, Mireille Chaleyat-Maurel, Volker Mehrmann, Tuulikki Makelainen, David Salinger and Bernd Wegner.

Officers’ Reports The President announced that Volker Mehrmann had been elected Editor of the left-hand side of EMIS, and David Salinger had been elected EMS Publicity Officer. He had signed a sponsorship contract with MathDidaktik, made a presentation in Macao, and attended the AMU 2000 Pan-African Congress in Cape Town. J.-P. Bourguignon had attended a meeting in Denmark on digitising journals; it would be good for similar European activities to work together in a co- ordinated way. The Secretary was looking at remits of committees in cases where existing ones were available, and for possible remits for those that were lacking; he was directed to send the draft Institute of Mathematics, Bedlewo remits to Committee Chairs for comment. The Treasurer reported briefly on the financial statements for the year 1999, which there is a need for two separate classes for agreements are likely to be between the EMS were accepted and signed. Waivers and reduc- institutional membership, one for academic and the Australian Mathematical Society and tions of fees for certain corporate members and one for commercial institutes, and that the Canadian Mathematical Society. who had difficulty in paying their fees were this should be taken care of whenever the agreed. A proposal for the Society budget for Statutes are revised. EMS Committee matters the years 2001 and 2002 was approved and A reciprocity agreement between the EMS The procedure to elect members for EMS will be submitted to the Society Council meet- and the American Mathematical Society will committees was reviewed: The Executive ing in July 2000. be recommended to the EMS Council for Committee should elect the Chair, and the approval. It will contain the sentence: “This Chair suggest committee members for the Council Meeting in Barcelona: 7-8 July 2000 exchange agreement shall in no way interfere Executive Committee to approve. When the The delegates for the individual members for with existing reciprocity agreements between acceptance of the committee member has the period 2000-03 will be: Giuseppe Anichini, Vasile Berinde, Giorgio Bolondi, Alberto Conte, C. T. J. Dodson, Jean-Pierre Francoise, Salvador S. Gomis, Laurent Guillope, Klaus Habetha, Willi Jager, Tapani Kuusalo, Laszlo Marki, Andrzej Pelczar, Zeev Rudnick and Gerard Tronel. The terms of office of four existing members of the Executive Committee end at the end of the year 2000, but they are eligible for re-election. The Executive Committee will propose to Council that Oslo be the site for the Council meeting in 2002, with dates 8-9 June 2002, and will make a recommendation as to the location of the European Mathematical Congress in Summer 2004.

Membership matters It was agreed to recommend to Council that the European Society for Mathematical and Theoretical Biology (ESMTB) and ITWM David Brannan, Rolf Jeltsch, Bernd Wegner and Olii Martio 6 EMS June 2000 EMS NEWS been received, their name and address should It was agreed that information on the possi- Relations with Mathematical be put on EMIS. bility of obtaining funds for East and Central Institutions/Organisations The Summer School Committee will now be European mathematicians from the Society’s The 1st EMS-SIAM conference – on Applied responsible for the organisation of an annual Committee for the Support of East European Mathematics in a Changing World – has been series of EMS lectures. A call for proposals will Mathematicians should be more widely adver- planned to take place in Berlin from 3-6 tised. September 2001; the local organiser is Peter Posters produced for the EMS Poster Deuflhard, and Vincenzo Capasso, Heinz Competition for World Mathematical Year Engl, Bjorn Engquist and Stefan Mueller were 2000 will be shown in Barcelona during the nominated to represent the EMS on the EMS Congress. An EU-supported ‘Raising Programme Committee. Awareness in Science Technology Week’ will The President is discussing several possible take place throughout Europe from 6-12 forms of operation with the new President of November 2000. It was decided to establish an the African Mathematical Union. EMS committee to plan the participation of It was agreed that the EMS should apply for EMS in the ‘European Science and membership as a “large associated society” in Technology Week’, partly to replace the ICIAM. WMY2000 committee; and to support a proposal for a competition of papers for Raising Future Executive Committee meetings Public Awareness in Mathematics. 6 July 2000, in Barcelona Alberto Cabada was added to the 11-12 November 2000, in London Prof Bojarski, host of the meeting Committee for Developing Countries; Volker March 2001, at ITWN in Kaiserslautern Mehrmann was added to the Electronic Publishing Committee; and Vincenzo And finally ... be put on EMIS and in the Newsletter, with a Capasso, Jean-Paul Pier and Olli Martio had The President thanked the Polish Academy of deadline of 1 March in year N for year N+1; joined the Group on Relations with European Sciences, Polish Mathematical Society and proposals should contain the name of the per- Institutions. Bogdan Bojarski for their assistance in organ- son responsible and location(s) for the lec- The Executive Committee decided to inves- ising and supporting the meeting, and for tures. Exceptionally, in the year 2000 the tigate holding a meeting of editors of non- their tremendous hospitality. deadline should be 30 September for lectures commercial European mathematics journals

EMS and the International Mathematical Union

The EMS has been granted affiliate status in the IMU. According to the IMU Statutes, multi- national mathematical societies and professional societies can be affiliated with the IMU. The affiliated members have the right to participate in the IMU General Assembly, but have no voting rights. They have a right to submit proposals for joint activities. For information on the IMU, see, for instance, http://elib.zib.de/IMU to be given in 2001. during the Barcelona Congress. It was agreed to draft an EMS position The Publicity Officer agreed to update paper by early May for the French EU existing EMS publicity material prior to 3ecm, European Research Conferences Presidency which begins 1 July. and revised arrangements for the production (EURESCO) E. Mezzetti had accepted an invitation to of the annual EMS ‘Agenda’ poster were The European Science Foundation is invit- chair the Women and Mathematics approved. ing proposals for new conference series as Committee; her nomination of Polina well as the continuation of established Agranovich, Nicole Berline, Bodil Branner, Publication matters ones. It aims to consolidate at around 50 Kari Hag, Ina Kersten, Laura Tedeschini P. Massart has joined the group of Associate conferences per annum. The EURESCO Lalli, Irene Sciriha and Tsou Sheung-tsun as Editors of JEMS. programme supports series of top-level Committee members was accepted. It was agreed that the EMS Newsletter will in meetings devoted to the same general sub- Carles Broto and Rudolf Fritsch were added future provide a limited amount of free space ject. Normally the conferences in a series to the Summer Schools Committee member- for member societies, for their publications, take place every other year. ship. It was agreed that EMS will pay for the meetings, etc., with equal rights for the EMS If a conference is already supported by the production of posters for the EMS Summer in their Newsletters, and to remind the EMS European Commission, it can attract up to Schools, as a standing arrangement. Progress member societies of the recommendation to an extra 8000 euros. If not so supported, was reported on planning for summer schools use in their letterheads the phrase “member the sum can be up to 25,000 euros. in 2001 in St Petersburg and Prague. of the European Mathematical Society”. Deadline: 15 September 2000 I. Diaz, A. Quarteroni and Kjell-Ove Vivette Girault is now in charge of seeking Details from: www.esf.org/euresco or from Widman have agreed to join the Special advertising for the EMS Newsletter. European Science Foundation, Events Committee. The Diderot Forum on The question of publishing a small booklet EURESCO, 1 quai Lezay-Marnèsia, 67080 ‘Mathematics and Music’ had been very suc- on ‘Early History of EMS’ was discussed. Strasbourg Cedex, France cessful. EMS June 2000 7 EMS NEWS libraries and mathematics departments with technical assistance on computerised documentation. In 1999, the MDC ReportReport onon thethe LIMES-PLIMES-Prrojectoject released edbm/w3, the first module of the European Database Manager for Michael Jost and Bernd Wegner Mathematics software. In LIMES MDC will be in charge of the extensive development of this software-like integration of new LIMES is a project that will be guided by dition of excellence is under economic tools for search, identification, display, and the EMS and funded within the Fifth threat. Stimulation of usage for all kinds of update, the development and integration Framework Programme (FP5) of the research, as well as for funding organisa- of improved access control management, European Community. The acronym tions before decision making by initial sup- and it will help national access nodes with LIMES stands for Large Infrastructure in port of two national test sides. the development and installation of Mathematics – Enhanced Services. It will deal 3.Improved coverage and evaluation of improved update procedures. with the enhancement of Zentralblatt research literature via nationally distrib- Eidetica is the most recent spin-off com- MATH to a European database in mathe- uted editorial units, development of tech- pany of CWI. It was created by researchers matics. The project fits into the horizontal nologies for efficient database production, in the Interactive Information Engineering FP5 programme ‘Improving human exemplified by two further European theme, combining expertise in linguistics research potential and the socio-economic member states. and mathematical clustering of dependen- knowledge base’, providing and improving The LIMES project sets out to achieve cy networks for textual domains. Eidetica access to research infrastructures. The these goals by stimulation of usage (nation- will mainly develop new models and tools duration of the project is from April 2000 al access nodes and centres for dissemina- for identification of author and serial to March 2004. tion, including development of adequate names (two of the most requested features The partners of the LIMES project are: licensing models, and links with offers of from the user community), based on data- FIZ (Fachinformationszentrum Karlsruhe, complete documents); better coverage and mining techniques, and other improve- Germany), MDC (Cellule de Coordination more precise evaluation of literature ments of the retrieval process. Documentaire Nationale pour les through improved update procedures (for SIBA is the structure of the University of Mathématiques, France), Eidetica (The the data gathering, but also for data distri- Lecce, Italy, that coordinates, manages Netherlands), SIBA (University of Lecce, bution to access points) via a network of and develops the Telematics Information Italy), DTV (Technical Knowledge Centre distributed editorial units. Networked System for Research. SIBA coordinates the and Library of Denmark, Denmark), access to the database these days is mainly SINM project, which aims at developing a USDC (University of Santiago de via rapidly evolving WWW technologies, National Information System for Compostela, Spain), HMS (Hellenic with high potential for further develop- Mathematics which will allow the Italian Mathematical Society, Greece), TUB ment of innovative access and retrieval mathematical community to have easy (Technical University of Berlin, Germany), methods, and improvements in the quality access to the bibliographic and full-text with the EMS as the supervising society. of data (such as author and source identifi- resources available in the libraries of the The project coordinator will be FIZ. The cation through data-mining techniques). field, on-line and on CD-ROM. In LIMES contracts for the funding of the projects In contrast to the centralised American SIBA will establish an editorial unit for were signed in March 2000 and the work model, Zentralblatt MATH will be extend- Zentralblatt MATH in Italy, identifying started on 1 April. ed to a European research infrastructure and covering mathematical publications of with distributed sites (Editorial offices and the region, according to high reviewing Objectives Access nodes) in the member states. The standards and editorial policies. The objective of the project is to upgrade EMS, representing the scientific communi- DTV was established in 1942 as the cen- the database Zentralblatt MATH into a ty concerned, has promoted these ideas tral library of the Technical University of European-based world-class database for among the national societies and individ- Denmark (DTU) and the national library mathematics and its applications, by a ual members. for engineering and applied sciences in process of technical improvement and Since the EMS is member of Denmark. The role of DTV will be twofold: wide Europeanisation. Upgrading the Zentralblatt MATH’s Editorial Board, it is 1.Setting up and running an editorial unit. existing database, improving the present guaranteed that the results of the project The unit will be responsible for covering a system, and developing a new distributed will be available for usage in further number of journals and other sources, to system, both for the input and output of research infrastructure partnerships. be defined and agreed by the data, are necessary to allow Tools and methods developed during the DTU/MATH/DTV and the coordinator. Zentralblatt MATH to use the latest devel- project will be usable by project partners The unit will be engaged in experimenting opments and to anticipate future develop- on a royalty-free license basis. Further with modern electronic-reviewing meth- ments in electronic technology. This will results, documentation and information ods, especially with the results of the SIBA- make Zentralblatt MATH a world refer- will be made publicly available. UNILE system development in this area. ence database, offering full coverage of the Commercial exploitation of results 2.Establishing a raw data system for the mathematics literature world-wide, includ- beyond European research infrastructure editorial units. Following the agreement of ing bibliographic data, reviews and/or services will be considered. the publishers, this data will be made avail- abstracts, indexing, classification, excellent able for LIMES in two ways: search facilities, and links to offers of arti- The role of participants – for end-user current awareness purpos- cles, with a European basis. FIZ operates the central editorial office for es, through a special LIMES gateway Improvements will be made in three Zentralblatt MATH (de/). Founded in added to the Current Awareness System of areas: 1931, it is today the longest-running inter- the EMS and to the Zentralblatt MATH 1.Improvement of content and retrieval national abstracting and reviewing service database, facilities through sophisticated further in the field of pure, applied and industrial – for cataloguing reuse by the editorial development of the current data sets and mathematics. Zentralblatt MATH covers units through the export of raw catalogu- retrieval programs. the entire spectrum of mathematics and ing data to the Zentralblatt editorial sys- 2.Broader and improved access to the theoretical computer science, with special tem. database via national access nodes and new emphasis on areas of applications, and The role of USDC will be to propagate data distribution methods. In particular, provides about 60,000 reviews per year. As the use of the database in Spain and to improvement of access for isolated univer- a consequence of this role, FIZ will be facilitate the access to it, first by installing sities in regions in economic difficulties engaged in all work packages in the pro- a national site server and regularly updat- and in associated states of Central and ject. ing the database and new software releases, Eastern Europe where a mathematical tra- MDC provides mathematics research and second by setting up a task group for 8 EMS June 2000 EMS NEWS stimulating the usage of the database by member of that department and can look Spanish universities, research centres and back on a 25-year period of experience in EMS 2001 libraries. A network structure for informa- this position. TUB will cooperate with all EMS 2001 tion in mathematics will be developed, partners, providing them with continuous integrating Zentralblatt MATH with advice and caring about the evaluation of LECTURES - national information offers. This work will results. An important task for TUB will be be undertaken in cooperation mainly with the development of editorial tools. CALL FOR PROPOSALS CESGA (Centro de Supercomputacion de As a partner in the Editorial Board of Galicia) and RSME (Real Sociedad Zentralblatt MATH, the EMS has set up an The European Mathematical Society has Matemática Española). HMS, founded in Innovations Committee whose main task is been running a successful series of EMS 1918, aims to encourage and contribute to to define and enact innovative features and Lectures for some years. For example, in the study and research of mathematics and procedures in Zentralblatt MATH. The June this year Professor George its applications, as well as to improve EMS has two roles to play in the project: it Papanicolaou (Stanford, USA) will give mathematical education in Greece. HMS represents the end-user, and gives a pan- series of lectures over a period of four days will apply a methodology similar to that of European framework to the project as a at ETH in Zurich (Switzerland) on USDC in a different environment. whole. In both roles, it will be part of the Financial Mathematics and over three days TUB is one of the largest universities in Project management. at The University of Crete, Heraklion, Germany, offering curricula for the full Crete (Greece) on Time reversed acoustics. scale of engineering sciences. The mem- Michael Jost, Fachinformationszentrum The EMS Lectures may be in Pure or bers of its mathematics department have Karlsruhe, Abteilung Mathematik, Applied Mathematics or may span both traditionally cooperated with Zentralblatt Franklinsträsse 11, D-10587 Berlin areas. With this activity, the Society aims to MATH by serving as consultants for the Bernd Wegner, TU Berlin, Fachbereich encourage European mathematicians input to the database. In particular, the Mathematik, Sekr. MA 8-1, Strässe des 17 Juni (especially young mathematicians) to meet Editor-in-chief of Zentralblatt MATH is 135, D-10623 Berlin and study together current developments in Mathematics and its applications. The lectures should be given over several days in each of at least two locations, in order to give as many as possible the opportunity to attend. Costs of participation should be kept low, and (if possible) grants should be EMS SUMMER SCHOOLS – available to people from countries which EMS SUMMER SCHOOLS – cannot afford any financial support. The EMS will guarantee its moral support to the selected lecture series, and will pay for posters advertising the lectures within the CALL FOR PROPOSALS European mathematical community; it will CALL FOR PROPOSALS also do its best to help the organisers to raise funds, and is likely to offer some financial support to organisers for participants who are young or come from European countries with financial difficul- The European Mathematical Society has schools, and will pay for posters advertis- ties. been running a successful series of ing the summer schools within the Topics (which may be single or compos- Summer Schools for some years now. European mathematical community; it will ite) for the lecture series, the sites, and the Readers of the Newsletter will recall, for also do its best to help the organisers to organisers of the schools will vary from example, Newsletter reports on a 1996 raise funds. year to year to cover a wide range of the Summer School in Hungary on Algebraic subject. Geometry [issue 20] and a 1998 Summer Topics (which may be single or composite) The Society is now inviting proposals for at least School on Wavelets in Analysis and for summer schools, the sites, and the one Lecture Series for 2001. Proposals should Simulation in France [issue 29]. organisers of the schools are likely to vary contain at least: the topic (title and short from year to year to cover a wide range of description), names of the likely lecturer, the site, The series is intended to include at least the subject. the timing, conditions for participants and the two summer schools each year – preferably name and address of the organiser submitting at least one in Pure Mathematics and at The Society is now inviting proposals for at least the proposal. least one in Applied Mathematics. With two Summer Schools for 2002. Proposals should Please send proposals for series of EMS Lectures this activity, the Society aims to encourage contain at least: the topic (title and short in 2001 to: young European mathematicians to meet description), names of likely lecturers, the site, Professor D. A. Brannan, Faculty of and study together current developments the timing, anticipated costs, conditions for par- Mathematics and Computing, The Open in Mathematics and its applications. ticipants, organising committee membership, University, Walton Hall, Milton Keynes MK7 and name and address of the organiser submit- 6AA, United Kingdom The Society’s Summer School Committee ting the proposal. e-mail: ; fax: (+44) 1908 652140 will also consider sponsoring proposals for if possible by 30th September 2000. summer schools fully organised by other Please send proposals for summer schools in The Society would hope to decide on institutions. To meet the EMS expecta- 2002 to: proposals within a month or so. tions, each school should be at pre-doctor- Professor Renzo Piccinini, Department of al level, should last from 2 to 3 weeks, and Mathematics and its Applications, Università di have 100-200 participants – mainly gradu- Milano, Bicocca, Via Bicocca degli Arcimboldi Correction to Newsletter 35 ate students or young mathematicians 8, 20126 Milano, Italy; On page 18, at the end of Tony Gardiner’s coming from several European countries. e-mail: , fax: (+39) 02 6448 7705, article on Mathematics in English schools, Costs of participation should be kept low, phone: (+39) 02 6448 7751, the final line was inadvertently omitted. and (if possible) grants should be available if possible by 30 September 2000. The final sentence should have read: In to people from countries which cannot later years (ages 12-15) Number and algebra afford any financial support. The EMS will The Committee would hope to decide on may occupy 55% of the time, Shape, space guarantee its moral support to the selected proposals within a month or so. and measures 30%, and Handling data 15%. EMS June 2000 9 FEATURE

Hilbert’s Problems Asterisks denote the ten problems presented during the Paris lecture TheThe HilbertHilbert 1. * Cantor’s problem of the cardinal number of the continuum (continuum hypothesis) 2. * The compatibility of the arithmetical axioms prproblemsoblems 3. The equality of two volumes of two tetrahedra with equal bases and equal alti- tudes 4. The problem of the straight line as the shortest distance between two points 1900-2000 (alternative geometries) 1900-2000 5. Lie’s concept of a continuous group of transformations, without the assumption of the differentiability of the functions defining the group Jeremy Gray 6. * A mathematical treatment of the axioms of physics 7. * The irrationality and transcendence of certain numbers 8. * Problems of prime numbers (including the Riemann hypothesis) In 1900 David Hilbert went to the second 9. A proof of the most general law of reciprocity in any number field International Congress of Mathematicians 10. The determination of the solvability of a Diophantine equation 11. Quadratic forms with any algebraic numerical coefficients 12. The extension of Kronecker’s theorem on abelian fields to any algebraic realm of rationality 13. * A proof of the impossibility of solving any equation of the 7th degree by means of functions of only two arguments 14. A proof of the finiteness of certain complete systems of functions 15. A rigorous foundation for Schubert’s enumerative calculus 16. * The problem of the topology of algebraic curves and surfaces 17. The expression of definite forms by squares 18. The building-up of space from congruent polyhedra (n-dimensional crystallog- raphy groups, etc.) 19. * Determining whether the solutions of regular problems in the calculus of variations are necessarily analytic 20. The general problem of boundary values (variational problems) 21. * A proof of the existence of linear differential equations with a prescribed mon- odromy group 22. * The uniformisation of analytic relations by means of automorphic functions 23. Further development of the methods of the calculus of variations

the problems to which the mathematicians should axiomatisation of physics. But he gave turn in the future. With this, you might con- good reasons for caring about his prob- David Hilbert, around 1900 ceivably have people talking about your speech lems, and bound it all up with his inspiring even decades from now. optimism. In opposition to Emil du Bois- in Paris to give an invited paper. He spoke And if the Congress itself was something Reymond’s fashionable academic pes- on The Problems of Mathematics, to such effect that Hermann Weyl later referred to anyone who solved one of the 23 problems that Hilbert presented as entering the honours class of mathematicians. Throughout the 20th century the solution of a problem was the occasion for praise and celebration.

Hilbert in 1900 By 1900 Hilbert had emerged as the leading mathematician in Germany. He was famous for his solution of the major problems of invariant theory, and for his great Zahlbericht, or Report on the theory of numbers, published in 1896. In 1899, at Klein’s request, Hilbert published The foundations of geometry as part of the commemorations of Gauss and Weber in Göttingen. Hurwitz saw clearly that the implications of that little book reached far beyond its immediate field. As he put in a letter to Hilbert: You have opened up an immeasurable field of Hermann Minkowski Adolf Hurwitz mathematical investigation which can be called the “mathematics of axioms” and which goes far of a shambles, Hilbert’s written text simism, Hilbert insisted that in mathemat- beyond the domain of geometry. nonetheless made its impact, just as ics: We can know, and we shall know. Hilbert was therefore poised to lead the Minkowski had predicted. international community of mathemati- One reason is undoubtedly that, for Hilbert presents his problems cians. He consulted with his friends Hilbert, problem solving and theory for- Hilbert’s problems came in four groups. In Minkowski and Hurwitz, and Minkowski mation went hand in hand. Indeed, some the first group were six foundational ones, advised him to seize the moment, writing: of his problems are not problems at all, but starting with an analysis of the real num- Most alluring would be the attempt to look into whole programmes of research: Hilbert’s bers using Cantorian set theory, and the future, in other words, a characterisation of 6th problem, for example, calls for the including a call for axioms for arithmetic, 10 EMS June 2000 FEATURE and the challenge to axiomatise physics. The next six drew on his study of (algebraic) number theory, and culminated with his revival of Kronecker’s Jugendtraum, and the third set of six were a mixed bag of algebraic and geometric problems covering a variety of topics. In the last group were five problems in analysis – the direction that Hilbert’s own interests were going. He asked for a proof that suitably smooth elliptic partial differential equations have the type of solutions that physical intuition (and many a German physics textbook) suggest, even though it had been known since the 1870s that the general problem of that kind does not. He made a specific proposal for advancing the general theory of the calculus of variations. The Hilbert problems very quickly succeeded in getting young mathematicians to create the future that Hilbert had conjec- tured – not always accurately, however, as we shall see. The Russian mathematician Serge Bernstein travelled from Paris to Göttingen in 1904 to present his proof that, under the conditions Hilbert had stated, elliptic partial differential equations do have analytic solutions. It would take more than a single book to describe the results produced in this mushrooming field in the 20th century. The opportunistic and power-seeking Bieberbach was another man drawn to the Hilbert problems for the fame that they could confer. In 1908 he showed that there are only a finite number of crystallographic groups in a Euclidean space of any dimension, thus solving part of Hilbert’s 18th problem. His results con- firmed that there are 17 patterns of this kind for the Euclidean plane, and 219 patterns (or crystal structures) for Euclidean 3-dimensional space.

Hilbert and axiomatisation Hilbert himself did not work exclusively on the problems, and nor did most of his many students, who were instead often drawn in the early 1900s to the study of ‘Hilbert space’. After 1909, when his friend Title page of ICM Paris Proceeding Minkowski died, Hilbert became more and more interested in questions of axiomatisation and the foundations of mathematics. His interest in axiomatising physics was to lead to several lecture courses but few publications, and has accordingly been much misunderstood (see Leo Corry’s work for the new picture). Hilbert lectured regularly and well, and he promoted the view that an axiom was a fundamental idea from which many others followed. So axioms played a crucial role in organising theo- ries, as they had done in his Foundations of geometry. As the years went by, the consistency of arithmetic seemed more of a challenge. In 1917 Hilbert wrote: Since the examination of the consistency is a task that cannot be avoided, it appears necessary to axiomatise logic itself and to prove that number theory and set-theory are only parts of logic. He imposed high standards on the task. What was needed, he said, were proofs that: – every mathematical question be solvable Hilbert’s lecture from the Paris Proceddings in principle; EMS June 2000 11 FEATURE – every result be checkable; ematics. On the basis of this distinction of Kronecker’s Jugendtraum was mislead- – every mathematical question be decid- between valid formulae and their interpre- ing, but the Japanese mathematician able in a finite number of steps: this is the tations he called for a proof theory – truly Takagi, who had studied under Hilbert in decision problem which asks: can a given a remarkable idea. Göttingen in 1900 while writing his thesis statement be shown to be In the event Brouwer withdrew from the for the University of Tokyo, succeeded provable/refutable in a given theory, or is it contest, Weyl found that he needed classi- with the crucial generalisation to the independent? cal analysis for his work on Lie groups, and abelian case in 1920. In 1923 Emil Artin By 1922 Hilbert became locked into a the crisis passed, although Hilbert’s ideas bumped into Takagi’s work almost by acci- dispute with Brouwer, who believed that about proof theory did not convince the experts. But Hilbert continued, with his assistants Ackermann and Bernays, and in 1931 he published a forceful re-statement of his views on the occasion of his becom- ing an honorary citizen of his native Königsberg. The lecture ends with a moving affirmation of his deepest belief about mathematics: There are absolutely no unsolvable problems. Instead of the foolish ignorabimus, our answer is on the contrary: We must know, We shall know. Ironically, the day before Hilbert lectured, the young Austrian logician Kurt Gödel also lectured in Königsberg on his

L E J Brouwer Emil Artin the human mind was strikingly limited in dent and the work he did as a result its ability to deal with infinite sets. Hilbert’s enabled Hasse to solve Hilbert’s 9th hopes for the future of mathematics were Problem (calling for a general reciprocity further darkened by the fact that his best law) in 1927. Hasse had meanwhile also student, Hermann Weyl, also found solved Hilbert’s 11th problem, on quadrat- Brouwer’s ideas attractive, and for a time ic forms, in 1923. In 1926 Artin solved seemed willing to forgo certain mathemat- Hilbert’s 17th problem in a restricted, but nonetheless very general, case. Hilbert’s 7th Problem (to show that ab is an irrational transcendental number when a is Kurt Gödel algebraic and not equal to 0 or 1, and b is irrational and algebraic) was solved by the incompleteness theorem, the work that is Soviet mathematician A. O. Gelfond in popularly said to have killed Hilbert’s pro- 1934, and by Th. Schneider independent- gramme, even though, as Gödel said in the ly later the same year. famous paper: I wish to note expressly that [this theorem does] The Bourbaki connection not contradict Hilbert’s formalistic viewpoint. After the Second World War the struggle The problem is that finding such proofs for the heart of mathematics was won by has proved elusive. No agreed place to the pure mathematicians. In this context a stand has been found which compels uni- vigorous contest developed for the mantle versal assent (such as the elementary rules of Hilbert. Should it go to mathematical of logic) and which delivers all of set theo- logicians, or to applied mathematicians ry. On the other hand, powerful negative working in the tradition of Courant and results continued to accumulate. When Hilbert, or to the number theorists and Alan Turing showed in 1936 that the deci- algebraists? The most powerful advocate of sion problem is also unsolvable, the origi- the last of these views, both by word and Hermann Weyl nal hopes for Hilbert’s programme were deed, was Bourbaki. André Weil and Jean all in tatters. Dieudonné shared a view of mathematics ical arguments on philosophical grounds. that put Hilbert centre stage, although it Hilbert attempted to get round their argu- The Hilbert problems between the Wars was a Hilbert created in their own image. ments in 1922 by defining mathematics as The Hilbert problems themselves have They espoused the axiomatic method, a system of signs and distinguishing care- perhaps proved a more enduring part of which Dieudonné claimed fully between mathematics and meta-math- his legacy than Hilbert’s own work on has revealed unsuspected analogies and permit- ematics. Mathematics was to be identified mathematical logic. Those on number the- ted extended generalizations; the origin of the with the stock of provable formulae, while ory have turned out, given Hilbert’s own modern developments of algebra, topology and inference about the content was only interests, to have been particularly well group theory is to he found only in the employ- admissible at the level of a new meta-math- put. Ironically, Hilbert’s own formulation ment of axiomatic methods. 12 EMS June 2000 FEATURE André Weil made the connection to um hypothesis, Cohen did indeed settle problem. … When I was a young topologist, that Hilbert even more forcefully. In 1947 he Hilbert’s 1st problem. was a problem I really wanted to solve – but I quoted Hilbert: In the early 1970s the Russian mathe- could get nowhere. It was Gleason, and A branch of science is full of life, as long as it matician Yuri Matijasevich solved Hilbert’s Montgomery-Zippin, who solved it, and their offers an abundance of problems; a lack of prob- 10th problem (Is there a finite process solution all but killed the problem. What else is lems is a sign of death. Great problems, said Weil, furnish the daily bread on which the mathematician thrives. And turning to Hilbert’s famous list of problems he singled out the 5th problem (then still unsolved) on Lie groups, the Riemann hypothesis, and the problem of generalising the theorems of Kronecker’s Jugendtraum which ‘still escapes us, in spite of the conjectures of Hilbert himself and the efforts of his pupils’.

The Hilbert problems after the War Hilbert’s Paris address had brilliantly united problems with their theoretical context and so, for a generation, did Bourbaki. The stock of the Hilbert problems rose with theirs. The remaining ones acquired an extra cachet for having held out, and they too began to fall. The 5th problem, on characterising Lie groups, was solved through the work of Gleason and of Montgomery and Zippin in 1952. The 14th problem on rings of invariants was Martin Davis, Julia Robinson and Yuri Matijasevich which determines if a polynomial equation there to find in this direction? I can only think is solvable in integers?) in the negative, of one question: [but it] seems quite hard – but a using earlier papers by the American solution would have no application whatsoever, mathematician Martin Davis, Hilary as far as I can see. Putnam, and Julia Robinson. Julia Yet there are numerous examples Robinson had been close to solving this where the Hilbert touch has proved bene- problem herself, and a powerful collabora- ficial. The great development of algebraic tion developed between her and number theory was surely animated by Matijasevich, despite the Cold War. It fol- Hilbert’s problems. The topic of partial lows from their work that had Hilbert’s differential equations was re-opened by 10th problem been answered positively, Hilbert with his 19th problem, and much Goldbach’s conjecture (mentioned by of that rich theory can be traced back to Hilbert in Paris) would have been the work it inspired. Not only are the answered in the negative – a connection implications of the solutions and reformu- that Hilbert surely had not suspected. lations of the problems still to be worked Few of Hilbert’s problems have dwin- out, some of the problems are still alive. dled with the years. Perhaps the complete Russian mathematicians have recently solution of the 5th problem is a case in shown that in two cases the original ‘solu- point. In 1986 Jean-Pierre Serre said: tions’ were flawed, and have given differ- Still, it is true that sometimes a theory can be ent and rigorous accounts. Aspects of both killed. A well-known example is Hilbert’s 5th halves of the 16th problem (on real algebraic curves and on vector fields) are still open questions. Hilbert’s deepest vision, of the intricate dance of theory and problems in mathematics, is one that all mathematicians share, but few have articulated as well Andrew Gleason as he. interpreted geometrically by Zariski and then solved in the negative by Nagata in Further reading 1959. In the 1960s and 1970s mathemati- F. Browder (ed.), Mathematical Developments cal logicians turned back to the Hilbert Arising from Hilbert’s Problems, Symposia in Pure problems. A high point was reached with Mathematics, American Mathematical Society the award of a Fields Medal in 1966 to Paul 28, Providence, Rhode Island, 1976. Cohen for showing that the axiom of Leo Corry, ‘David Hilbert and the choice and the continuum hypothesis are Axiomatization of Physics (1894–1905)’, Archive independent of the other axioms of set for History of Exact Sciences 51 (1997), 83–198. theory. The axiom of choice is not one of Jeremy J. Gray, We must know, we shall know; a Hilbert’s problems, but in calling for a con- History of the Hilbert Problems, Oxford University sistent set of axioms for arithmetic, Hilbert Press, 2000 (to appear). had opened the way to similar analyses of all of mathematics, and indeed by estab- Jeremy Gray is a Senior Lecturer in the Department of lishing the independence of the continu- Paul Cohen Pure Mathematics, The Open University, UK. EMS June 2000 13 INTERVIEW

Where would you see the main emphasis in the programme? InterviewInterview withwith Topics to be mentioned (among many others) are medicine (computationally assisted surgery, immune system modelling), PPetereter DeuflharDeuflhardd biotechnology (density functional theory, (President, Konrad Zuse Zentrum, Berlin) drug design – not to be confused with designer drugs!), material sciences (com- interviewer: Rolf Jeltsch, EMS President (Zurich) posite materials, crack modelling and simulation), nanoscale technology (fast optical networks, quantum electronics), climate You will be organising the first EMS-SIAM Do you think that applied mathematics has research, communication (network simula- meeting in Berlin from 2-6 September 2001. changed in the last 20 years? tion and optimisation), traffic (route guid- The main theme is ‘Applied Mathematics in Yes, indeed, but this is too long a story. In ance and planning), finance (option pric- our Changing World’. How would you short: there has been a clear drive from ing, derivative trading – of course, not the describe what applied mathematics is? applicable mathematics to applied mathe- mathematical derivatives!), speech and In principle, most substantial mathematics matics. Today mathematics interferes with image recognition (filter analysis). (including pure mathematics) is applicable. unexpectedly many facets of our life, part- We now know that Hardy failed in aiming ly visible to the public, but mostly invisible. How many participants do you expect? at mathematics that would surely never be We aim at an attendance of from 500 to applicable – never say ‘never’. The ques- 1000 participants. tion is, whether applicable mathematics, when applied, is already applied mathe- Do you have any particular message you matics. It is my repeated experience that would like to give to potential participants? solving hard real-life problems usually Whatever you have contributed in terms of goes far beyond just taking mathematical mathematics or want to contribute to the tools off the shelf. Hence, ‘Applied changes of our world, come to the exciting Mathematics’, as I understand it, deals with city of Berlin and contribute your part to questions arising from quantitative prob- the conference community! lems outside mathematics and the corresponding development of new mathemat- To change topic, you are President of the ics for their solution. This characterisation Konrad Zuse Zentrum in Berlin. Some of certainly includes most parts of computa- our readers attended the ICM 1998 organ- tional mathematics (such as the design of ised by Martin Grötschel, also from this efficient algorithms in differential equa- institute, but might not know much about it. tions or discrete mathematics), but also Could you describe it briefly? modelling (such as the regularisation of Here I am tempted to refer to a recent arti- inverse problems or the homogenisation of cle in SIAM NEWS (March 1999) that I multi-scale problems), and visualisation, as wrote for just that purpose. Its title was ZIB far as it requires the development of new – computational mathematics as key to key algorithmic tools (such as feature extrac- technology. There you can find a lot of tion in fluid dynamics). details about ZIB’s organisation, mission, funding, and scientific achievements. Where do you see the differences between applied mathematics, industrial mathemat- And even mathematicians recognise that You have been director since 1986 when it ics and computational sciences and engi- fact. The beauty of a theorem is more and was founded. What have been the main neering? more compared with its use. Fortunately achievements in this period? Even though I have worked quite a bit on quite often beauty and use go nicely Yes, I founded ZIB in the scientific sense in really tough industrial problems, I am not together (I still love beauty). But beauty 1986, and Martin Grötschel joined ZIB in too fond of the definition of ‘industrial alone has a short life, as we all know. 1991. As for the main achievements in the mathematics’: the only distinguishing fea- past: there have been many extremely suc- ture in my definition (above) of applied Do you expect further changes in the cessful projects in Discrete Mathematics mathematics might be that these problems future? In what direction and with what and Indiscrete Mathematics (my subject!) arise in industry. Computational sciences speed? that have been pursued within ZIB. The and engineering, however, deserve some I hate to answer such questions, which typ- ZIB story has been a real success story so further thoughts: in my understanding, ically are asked at the end of a mathemati- far, I’m proud to say. Why don’t you just CSE does not say anything on the methods cian’s active professional life. I would pre- give our web address below? that are used to solve the problems, so this fer to continue to be active for a while. I see Perhaps the main project structure of field necessarily includes good, bad and a lot of changes, but would like to wait with our research may be worth mentioning indifferent. I use the term ‘scientific com- the answer. explicitly: we distinguish between long- puting’ to describe the discipline that addi- term projects (usually basic research, ori- tionally takes care of the efficiency of You also chair the scientific committee. ented towards the treatment of difficult methods applied to computational prob- How are these changes reflected in the pro- problem classes and funded by ZIB), medi- lems from sciences and engineering. gramme? um-term projects (usually publicly funded There is, however, a close mutual depen- The title ‘Applied Mathematics in our on the basis of competitive proposals), and dence between scientific computing and Changing World’ was deliberately chosen short-term projects (industrial problems). computational science: only if computa- by the initiators from SIAM and the EMS. This span from basic research via algo- tional science brings up interesting prob- The key idea is to focus on some (not all!) rithm development to real-life applications lems can scientific computing then work aspects of mathematics where it interferes has turned out to be quite fruitful over the on the methods and, in turn, help to raise with the typically fast changes in our real years. Behind this concept stands my firm the complexity barrier of tractable compu- world. Generally speaking, the fastest belief that genuine progress in the solution tational problems in science and engineer- changes arise in fields such as finance, net- of hard real-world problems typically ing. work technology, environment and life sci- requires thorough and creative basic ences. research. 14 EMS June 2000 INTERVIEW tion, this kind of work is still insufficiently Computing at the Free University of You have almost another decade of direc- valued in terms of a university career. This Berlin. And I still love teaching in close torship ahead of you. In which direction may partly have to do with the fact that at contact with the next generation. should ZIB evolve? some places software development is done Unfortunately, like other departments We are always on our way and have lots of without any theoretical component – an world-wide, we do not have enough stu- new ideas – enough for the years to come. attempt often prone to lead to rather dents. This is in extreme contradiction to short-sighted software solutions. Back to the excellent professional perspectives of ZIB is one of the German supercomputer your question. In principle, universities computational mathematicians that have centres, and may even be THE centre. Will could be good places for software design. worked in real applications. As for my per- supercomputing increase at ZIB and in I would like to make an additional sonal research topics: with the ZIB as Germany? remark in this context: without any real organisational background, I have been First of all, supercomputing should be application work, which will usually able to look at more complex computa- more than just computing on supercom- include software implementation, universi- tional problems that play a role in real life puters. At least, this is the spirit of super- ty careers have increasingly been slowed and require a larger group. So my activities computing at ZIB, as opposed to other down in recent years, and speeded up oth- in Berlin cover all kinds of differential comparable institutes. At ZIB, supercom- erwise. Just look at the many new universi- equations, far beyond the Munich or puting is part of our branch of Computer ty openings (and more to come in a few Heidelberg topics of ODEs. To give one Science. Secondly, as for the installation of years when the present generation leaves) example, I worked on adaptive multi-level supercomputers, Germany has a govern- in the fields of scientific computing or in finite element methods for PDEs, especial- mental road-map for the development of discrete mathematics. It seems that a good ly in cancer therapy and microwave tech- such centres. The trouble is that only very strategy mix is necessary for young people nology. few problems are especially well suited for to grade up successfully. By the way, we are supercomputers, and very few experts always looking for enthusiastic young grad- What are the research challenges you still know how to deal efficiently with them. As uates to join us. want to attack in the future? a consequence, supercomputing does not have a broad lobby, but must be regarded as a strategic issue for any industrial society – spoken in the direction of all the govern- ments of readers of this interview.

Should there be more supercomputer centres in Europe? There is a natural decay time of supercomputer technology, which more-or-less leads to an optimal number of such centres for each community. It depends on what you define as the community.

Currently most of these centres are funded nationally. Should the European Union go into the business of maintaining such centres? My personal opinion is that in the long run supercomputers will be a common European strategic issue, but in the short term the organisational details would kill the performance if an EU option were realised too early.

What is the relation between such centres in Europe and in North America, especially the USA? Like other such centres we have international connections both in research and Design of biomolecules in a 3D virtual laboratory at ZIB networking, such as in metacomputing. For example, we did computations about I still have a number of personal scientific the collision of neutron stars and rotating How can they maintain the software if PhD dreams. Let me just mention the next steps black holes (the numerical solution of students who did it have left the university? in the development of conformational Einstein’s equations of general relativity) Attempts have been made to establish net- dynamics, which will play a decisive role in simultaneously on four supercomputers in works to maintain software, but in my the design of biomolecules, or the further San Diego, Urbana, Munich and Berlin. opinion with doubtful success. I do not spread of topics in quantitative medicine know a really good remedy for this unde- which opens a new door to computational Let me ask some questions related to your sirable effect. On the other hand, redesign mathematics. It seems to be a coincidence personal research. In your early days in of software, which is necessary anyway after that some of my favourite topics appear on Munich and Heidelberg you worked on a certain period, is strongly supported by the list of topics of the SIAM-EMS confer- ordinary differential equations. You not some present languages like C++ or Java ence! only did theory but also software and which partly reduces pressure from this applied examples. How would you position issue. I am sure that with you as pivotal person in the development of software at universities? the organisation of the first EMS-SIAM There is no general answer to this kind of Have your style of working and the scientif- conference, it will be a success. question: some university groups take a lot ic topics chosen changed when moving from of care in developing software, some don’t. a purely university environment in For more information on the interviewee, see However, the short-term restrictions on Heidelberg to the ZIB, where you are also http://www.zib.de/deuflhard/ typical university jobs are a severe obstacle attached to the Freie Universität Berlin? for more information on the ZIB, see for any software developments. In addi- I still hold the Chair of Scientific http://www.zib.de/ EMS June 2000 15 INTERVIEW LookingLooking back:back: JarJaroslavoslav KKurzweilurzweil (Prague) interviewers: Jirí Jarník and Štefan Schwabik

Jaroslav Kurzweil was born on 7 May 1926 in brains, that all this was well known and to continue, and the teaching of some Prague. He graduated in mathematics in 1949 that it is impossible to find anything new subjects (such as history and literature) at the Charles University in Prague and start- in this direction. Later, as a student at sec- was prohibited. Students at higher grades ed his scientific career as a student of Professor were forced to work in Bohemia or in Vojtech Jarník in the metric theory of Germany, mostly in the war industry. The Diophantine approximations. Until 1951 he institutions of higher education resumed was at the Czech Technical University and was their activities immediately after the end then a research student at the Mathematical of the war, but the situation was not nor- Institute which became a part of the newly mal. First, the number of professors had founded Czechoslovak Academy of Sciences in decreased considerably: some had been Prague. After a research visit to Poznan with executed, some did not survive Nazi con- W. Orlicz he studied analytic operations centration camps, others had died by nat- defined in a Banach space with values in ural causes. Many younger teachers had another Banach space. He has been fully found other jobs. This caused a genera- employed at the Mathematical Institute of the tion gap in the community of university Academy since 1954. In 1955 he received his teachers, since people younger than 45 Candidate of Sciences degree and was appoint- years of age were quite an exception. ed Head of the Department of Ordinary Many former students deprived of the Differential Equations at the Institute. At that possibility of studying after the abolition time his work in ODEs started. His main inter- of Czech universities in 1939 desired to est was in stability theory, invariant manifolds complete their studies, and others applied and problems concerning the continuous who had graduated from secondary dependence of solutions on a parameter. In this schools during the war. last field he created a new definition of an inte- Thus the education in Mathematics I gral based on Riemann-type integral sums that received at the Faculty of Science from turned out to be equivalent to the Perron or 1945 to 1949 was rather unbalanced. I narrow Denjoy integral. This was indepen- attended excellent lectures and seminars dently discovered by Ralph Henstock, and since ondary school (‘gymnasium’) I used to in topology and number theory, but I the early 1960s has been studied intensively. In solve competition problems published by learnt nothing of differential equations, 1958 Kurzweil received his Doctor of Science the Union of Czechoslovak probability theory or numerical methods degree, and in 1966 was appointed Full Mathematicians and Physicists in the jour- (no computers were then available). I Professor of Mathematics. nal Rozhledy matematicko-fyzikální graduated with a thesis in metrical num- Soon after the revolution in 1989 Kurzweil (Horizons in Maths and Physics), intend- ber theory; it dealt with Hausdorff mea- was elected Director of the Mathematical ed for secondary school pupils. I read arti- sures of sets of real numbers that can be Institute of the Academy, and he has held this cles appearing in the Journal and also well approximated by rational numbers. I office till 1996. Since 1990 he has been some popular books, such as On the was initiated in this field by Vojtech Chairman of the Board for Accreditation, Apollonius problem, On Einstein’s theory of Jarník. Postgraduate studies had only appointed by the Government of the Czech relativity and gravitation, and Introduction to started in the early 1950s. After 1950 Republic to give approval to educational activ- group theory. In 1945 I completed my sec- some universities and research institutes ities of all institutions of higher education. ondary education. I briefly considered the were allowed to enrol students in doctoral Jaroslav Kurzweil has been Chairman of the study of medicine, having been influenced studies and offer them scholarships. The Union of Czech Mathematicians and Physicists by care I received in my childhood during preparation of the legislative framework since 1996. In 1999 he was reelected for the a long hospital stay after an accident. only started then, and so the first disserta- next three years. Nevertheless, I enrolled in the Faculty of tions might not have been defended until The deep trace of Kurzweil’s work and per- Science of Charles University in Prague to 1956. sonality in Czech mathematics is evident and study mathematics and physics. My deci- After a short period of teaching at the incontestable to everybody who has met him, sion was also affected by the fact that all Faculty of Mechanical Engineering of the either as a mathematician or simply as a man. secondary school maths appeared quite Czech Technical University I arrived in clear to me without learning it. 1951 as a postgraduate student What made you choose mathematics as the (‘Aspirant’) to the institute, which later subject of your studies? You became a university student in 1945, became the Mathematical Institute of the Mathematics has attracted me since my immediately after the end of World War II. Academy of Sciences. The generation gap childhood. For example, I remember that How did your studies continue at that was still strongly felt, there was a lot of when I was a pupil at primary school I time? improvisation and much was left to the asked adults questions of the type: ‘how The Nazi occupation in 1939-45 resulted initiative and efforts of individual stu- can we calculate the heights of a triangle in a drastic restriction of Czech school dents. In the initial period of communist if we know its sides?’. I wondered how it education. All Czech institutions of higher rule in Czechoslovakia it was difficult to was possible to construct the triangle and learning were closed from November travel even to politically allied countries, measure its heights, but I could not calcu- 1939 till the end of World War II. The but I was lucky enough to spend several late them. I had no luck and nobody number of secondary schools was months in 1953 studying in Poland. My helped me. I was told not to rack my reduced, the number of students allowed stay in Poznan with Professor Wladyslaw 16 EMS June 2000 INTERVIEW Orlicz was of extraordinary importance I remained, together with several other The above works, including Henstock’s for me. I succeeded, among other things, colleagues, closer to ODEs. We con- 1963 monograph, aroused interest, which to solve the problem of whether it is pos- tributed to control theory in its begin- led to the publication of several mono- sible to uniformly approximate a function nings and to the theory of differential graphs and a long series of papers. defined on a real Banach space by an ana- inclusions. The research in the theory of Numerous modifications of the original lytic function (defined by means of a invariant manifolds was based on a geo- definition were investigated, in particular power series). It appeared that in the class metrical approach; its results were for integration in multi-dimensional of uniformly convex spaces the affirmative applied to ODEs, and in particular to dif- spaces, characterisation of primitives, answer holds if and only if there exists a ferential equations with delayed argu- convergence of sequences of functions. In polynomial q such that q(0) = 0 and inf ment. Occasionally I was also engaged in particular, such modifications of the defi- q(x); |x| = 1 > 0. In the special case of the other topics in mathematical analysis. nition for multi-dimensional integration spaces Lp and lp this occurs if and only if p There is not much sense in presenting the were found which allow the usual trans- is even. details of these results. On the other formation formula and can be used to for- hand, I would like to mention another mulate a general Stokes theorem (for What prospects did you have after com- branch of research that stemmed from example, by W. Pfeffer in 1993). I joined pleting your doctoral studies? our work in the ODEs. this trend by a monograph in 1980 and by The institutions of higher learning were In classical theorems on the continuous a series of papers that I published with co- then introducing new specialisations and dependence of the solutions of the differ- authors in the years 1983-99. they also started to work with smaller ential equation x’ = f (x, t, e) on a para- Until recently, in connection with the groups of students; they therefore needed meter e, it is assumed that f is a continu- summation approach to integration, the younger experts for assistants’ jobs. Also ous function of the triple of variables (x, t, convergence of sequences of functions was research institutes took on qualified spe- e). In the Fifties cases were considered in investigated, but no attention was paid to cialists, so there was no lack of job offers. which the right-hand side of the differen- the topologisation of the space of inte- In the Fifties our country did not have tial equation contains terms of the type grable functions. As we know, S(P) inte- specialists in some branches of mathemat- cos t/e, f(x)/e cos t/e, so that with respect to gration is Perron integration and, as such, ics who would be able to communicate e it is not continuous in a neighbourhood is an extension of Lebesgue integration. and cooperate with specialists in non- of 0. Nevertheless, the primitive functions Therefore, if for f: [a, b] → R and mathematical sciences or with people with respect to t of such terms are e sin t/e, g:[a, b] → R, f = g a.e., and if f is S(P)- from ‘practical’ professions. The f(x) e2sin t/e, which are continuous. Thus a integrable, then g is S(P)-integrable and Mathematical Institute was supposed to natural question arises under what addi- the two functions have a common primi- act to improve this situation. This was why tional conditions the solutions of the tive. It is natural to study both the conver- the following working groups were found- equation depend continuously on the gence and the topology in the vector ed in it: parameter e, provided that we assume space Int S(P), which is the space of equiv- – probability theory and mathematical only that the primitive function to f with alence classes of S(P)-integrable functions. statistics respect to t is continuous. When I was A convergence of sequences with reason- – theory of PDEs and mathematical theo- looking for general principles governing able properties can be introduced in the ry of elasticity these problems I was working with space Int S(P), and we arrive at the fol- – theory of ODEs Lebesgue integrable functions and lowing problem: Does there exist a topol- – numerical methods Riemann sums. This provided the inspira- ogy Cal U on Int S(P) such that (Int S(P), – elementary mathematics. tion for a modification of the Riemann Cal U) is a complete topological vector When I finished my doctoral studies, integral; the modification of the usual space and every sequence convergent in the Director of the Mathematical Institute ‘epsilon-delta’ definition consists in the above sense is convergent in (Int S(P), Vladimír Kníchal offered that I should admitting for d not only positive constants Cal U) as well? remain in the Institute as a research work- but positive functions (called gauges). In view of the first requirement the er, allowing me to choose which group I I published the modified definition for topology may not be too coarse, by the lat- wanted to work in. I chose the group of the first time in 1957 and used it to intro- ter it may not be too fine. I have recently ODEs, which was gradually increasing and duce generalised differential equations proved that the answer is affirmative. in 1955 was established as a department. and establish results on the continuous However, if we add the condition that (Int Two topics for the start of the research dependence of solutions of generalised S(P), Cal U) is a locally convex space, then were given by Kníchal – the existence of differential equations on a parameter. In the answer is negative. periodic solutions, and Lyapunov func- this way the concept of S(P)-integrable I am author or co-author of 57 papers tions and stability. functions was obtained. which concern the above subjects, which The work on the first problem was start- It was also shown there that a real func- were published between 1957 and 1991. ed by a smaller group that ultimately tion g is S(P)-integrable if and only if it is formed an independent department deal- Perron integrable, and that the integral in What brought you to work in mathemat- ing with evolution problems in all their the sense of S(P)-integrability is the ics teaching? breadth. I started to work on the other Perron integral of the function g. (In the I got to mathematics teaching without any topic, and soon succeeded in proving in notation S(P), the symbol S corresponds effort. In the so-called socialist regime the the case of asymptotic stability that there to the approach by Riemann sums, while publishing of textbooks for elementary exists a Lyapunov function with the corre- P refers to Perron integration.) I regarded and secondary schools was directed cen- sponding properties and is of class C8, the definition as a contribution to the the- trally. There were publishing houses in even if the right-hand side of the differ- ory of the Perron integral; I did not deal both Prague and Bratislava, whose task ential equation is just continuous. Nice with it for about the next two decades, my was to publish textbooks which were then results concerning Lyapunov characterisa- main field of interest having been ODEs used in all schools throughout the coun- tion of stability under permanently acting and related problems. try. When the higher authorities decided perturbations and integral stability were The same definition was published that new textbooks for primary and sec- obtained by Ivo Vrkoc, who later started independently in 1961 by Ralph ondary schools were to be published, the research in stochastic perturbations and Henstock. In 1969 McShane published a two publishing houses divided the task of gradually formed a small group which general theory of integration in abstract preparing mathematical textbooks for now works in the theory of stochastic dif- spaces, which in the special case of inte- individual grades. Then the publishing ferential equations in infinite-dimension- gration on a one-dimensional interval houses started to seek for the authors and al spaces. reduces to another modification of my referees. I do not know what had to be definition. McShane proved that by this approved ‘up there’ before the publishers Thus we have arrived at your research modification we obtain exactly the were allowed to appoint the authors and activities ... Lebesgue integrable functions. referees. Other institutions involved in EMS June 2000 17 INTERVIEW the process were the Pedagogical school mathematics with explanations of Institute of the Academy of Sciences, and Research Institutes, both in Prague and the terms. The first edition appeared in I remained in this office till 1996. In the Bratislava. They published, in small print- 1981 and the Committee soon started to same year I was elected the Chairman of ings, the so-called experimental textbooks prepare an improved edition. It met reg- the Union of Czech Mathematicians and of various subjects. Those employed in the ularly on seminars, found authors for Physicists. Institutes were authors of most of these individual chapters and in 1989 it had As a consequence of the new law on experimental textbooks and were often prepared about 80 per cent of the manu- institutions of higher studies, passed in authors or co-authors of the standard script; only two chapters were missing – 1990, the government appointed the ones also. In the Seventies I was asked by unfortunately, the key ones concerning Accreditation Board that was entrusted the Director of the Mathematical geometry and computers. However, after with important tasks regarding institu- Institute, Jirí Fábera, to write a review of 1989 most of its members were chal- tions of higher learning. I acted as one of these textbooks. The review was lenged with more urgent tasks and the Chairman of this Board from 1990 to unfavourable, sometimes even biting. I Committee ceased to work. 2000. I have strictly applied the opinion was therefore surprised when the head of In 1987 the Union awarded me its high- that statistical and scientometrical data the Mathematical Department of the est distinction and elected me its hon- have an auxiliary character and that every Pedagogical Research Institute, Jana orary member. decision of the Accreditation Board Müllerová, thanked me and asked me to requires expert substantiation. In 1991 I give my critical opinion of future manu- scripts. The result was that for several years I served as an advisor to those authors of mathematical textbooks who showed interest. In the Seventies a very formal approach to elementary mathematics, based on naive set theory, was put through. First- grade pupils in primary schools (6-7 years old) learnt the union and the intersection of sets, and set diagrams became idols. Maybe the authors actually believed that they had discovered how to make maths attractive and accessible, but above all, they had enough influence to enforce this system all over the country. I was co- author of a letter addressed to the Pedagogical Research Institute and signed by about ten outstanding Czech mathematicians working at institutions of higher learning. We pointed out that the process of cognition of children proceeds along other lines, that abstractions have to be prepared by experience, and that they are the top and not the start of the cognitive process. The letter was not published, but was circulated and had a positive effect. New authors were appointed for some textbooks and the whole system was moderated. The first director of the Mathematical Institute (in the Fifties) was the famous topologist Eduard Cech. He believed that maths could be taught at secondary These stamp were issued in 1987 to commemorate the 125th anniversary of the Union of schools much better and more effectively. Czechoslovak Mathematicians and Physicists. The middle one features Kurzweil’s teacher Vojtech He had already organised meetings with Jarník (right). teachers before World War II, lectured and discussed with them, and even wrote We now come to the year 1989... was elected a member of the Scientific several secondary school textbooks him- One consequence of the generation gap Board of the Charles University and of self. It was this tradition which led to the in the mathematical community, which I the Scientific Board of the Faculty of foundation of a Department of Didactics mentioned in connection with the Nazi Mathematics and Physics. In 1994 I was of Mathematics in the Mathematical occupation, was that I and some of my among the founding members of the Institute. When the head of this contemporaries were entrusted very early Learned Society of the Czech Republic, Department Josef Horálek died in 1984, I with organisational and decision-making which continues a tradition going back to was put in charge of its activities; of course tasks. I served as a member of numerous the second half of the 18th century. I did not participate in the proper work of expert committees, and committees for the Department, but I willingly discussed academic and scientific degrees. From You lived during the periods when your with its members their experience, prob- 1956 to 1970 I was the Chief Editor of the country was governed by Nazis and then lems and plans. In 1985 I became mathematical journal now published by Communists. How do you see the pre- Chairman of the Committee for under the name Mathematica Bohemica. sent? Mathematical Terminology of the Union The year 1989 brought changes and It is indeed a miracle that both regimes, of Czechoslovak Mathematicians and hopes. The system of central directing was Nazi entirely and Communist except for Physicists. This Committee had earlier to be abandoned and self-government at Belorussia, have disappeared from prepared for publication a list of mathe- all levels was to be restored in all areas of Europe. Nonetheless, the introduction of matical terms and symbols, which went society, including education and science. each was made possible only due to a fail- mildly beyond the needs of primary and Naturally I desired to participate in bring- ure of Western civilisation and education. secondary schools. The aim of the ing about these changes. In 1990 I was This, in my opinion, is a warning for us as Committee was to prepare a dictionary of elected Director of the Mathematical well as for the future generations. 18 EMS June 2000 NEWS link to their web sites. One even has the Joint BMS-DMV meeting of the ACM (Agenda des option of asking for a timely e-mail Belgian and German reminder. conferences As the geographical coverage increases, Mathematical Societies more numerous and focused regions will 8-10 June 2001 be included. Apart from the present ten mathématiques) general subfields of mathematics that can The Belgian Mathematical Society and the This message is to announce a French ini- be selected for a focused search, each talk Deutsche Mathematiker Vereinigung will hold tiative on the occasion of World can present a list of MSC2000 codes that their first joint meeting at the University of Mathematical Year 2000. It is a web sys- can be used in the searches. The ACM Liege, 8-10 June 2001. 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All mathemati- correspondents are also needed; some are relevant information for its database. cians who have pre-registered this way will already available for France, Austria, The purpose of this message is also to automatically receive the Second Belgium, Germany, Italy and Canada. inform you that the whole concept, and Announcement of the meeting in October Mirrors will also be welcome everywhere software, is a gift of the French 2000. possible, including on EMIS. Mathematical Societies and community to At the core of ACM is a large database, their fellow mathematicians around the but the interesting concept is that the world; but for ACM to succeed, the contri- database gets created and maintained bution of everyone is now needed, and automatically, by querying the web pages most notably our European fellow of the seminar series, conferences or col- Societies. Société loquia. Of course, it requires the active One final note: ACM does not wish to Mathématique de cooperation of their organisers, but this is compete with other sites that propose straightforward for them: the URL of the similar information. On the contrary, it is France electronic web pages of the announcements have to willing to incorporate all information that mailing list be indicated to the ACM-master and these resides on these sites, with direct links to pages have to include some simple tags them. An example of such cooperation is that the ACM engine can recognise. This with ACMA, as noted above. The Société Mathématique de France procedure allows for instantaneous We sincerely believe that the ACM con- (SMF) maintains an electronic mailing list updates. Detailed explanations are avail- cept can play an outstanding role in fos- allowing each member to be periodically able on the ACM server, in English, tering the sense of a world-wide mathe- and freely informed of the SMF’s latest German, Italian, Spanish and French). matical community. At the dawn of the issues. On the user’s side, the search engine can third millennium, mathematics can again To sign up, send an e-mail message to be queried in these languages and can be be an example to all other disciplines on [email protected] without subject or signature customised to each user’s profile or pro- that front. and containing exclusively the line: files (which can be saved as Bookmarks or Thank you for all the help you can give to subscribe publications-smf full name (for Favorites). Simple and complex searches this idea, the care of which the SMF and example: subscribe publications-smf Dupont are possible, including geographical SMAI now put into your hands. Jean) regions and time periods. Setting up a Rolf Jeltsch, President of the EMS, e-mail: To unsubscribe to the mailing list, send profile is a little lengthy the first time, but [email protected] an e-mail message to [email protected] without well worth it because, once bookmarked, Mireille Martin-Deschamps, President of subject or signature and containing exclu- it performs a new search in one single the SMF, e-mail: [email protected] sively the line: step and keeps the date as relative, not Patrick Le Tallec, President of the SMAI, e- unsubscribe publications-smf absolute. The result of a search is a list- mail: [email protected] Beware: the body of your message should ing of all the seminar talks, symposia talks Alain Damlamian, Past-President of the contain only raw text (neither html nor and conferences in the database which SMAI, e-mail: [email protected] Word attachment). satisfy the criteria, with their title and a EMS June 2000 19 FEATURE AA majormajor challengechallenge forfor mathematiciansmathematicians The undervaluing of the role of mathematics in today’s society

Jean-Pierre Bourguignon

In today’s society, mathematics is present function automatically, when as mathe- too often omit to make this point publicly more widely than ever before, but this is maticians we know how much one has to – in particular, to our students. In my opin- rarely acknowledged, even by mathemati- struggle to solve a problem or break it into ion, it is of fundamental importance that, cians. As a result, citizens need to be com- simpler pieces. In other words, if a mini- as early as possible (probably even before fortable with mathematical situations, so as mum of technique is required to function secondary school), students meet the idea to be helped with the ever-complex choic- in a mathematical environment, the mean- that mathematics is still in the making, and es they have to make. that thousands of mathematicians around This article arose from discussions with the world are working at solving outstand- many people inside and outside the ing and extremely varied problems. research mathematical community (in par- These misconceptions are encapsulated ticular, teachers) and the perception I in the unfortunate reputation of the word developed from them. For the most part, ‘abstract’, when it is precisely in abstracting mathematicians have not appreciated the from a situation what makes a phenome- large demand for mathematics in society non general that mathematics is at its best. and the consequences they must draw from Henri Poincaré gave a great definition of this. The starting point is to analyse what mathematics. For him, “faire des mathé- makes mathematics special among the sci- matiques, c’est donner le même nom à des ences and how it is perceived by those who choses différentes”. From that point of have to deal with it in some way – parents, view, opposing ‘concrete’ and ‘abstract’ can professionals, managers, and mathemati- be nonsense. Indeed, extracting from a cians themselves, as teachers or as complex situation what constitutes its core researchers. is what makes us really understand how Elaborating on the ordinary perception things work. We must therefore spend time of mathematics, I present what I see as a in making this process explicit, emphasise new relationship of mathematics to society, and the role it plays in the foundation of math- consider detailed implications on its inter- ematics. In my opinion, here lies one of the nal organisation, together with needs to be main roots of the quid pro quo that current- addressed in its development and teach- ly affects the relationship of mathematics ing. to society. Some other sources of misunderstanding 1.The ordinary perception of mathemat- ing of what we manipulate cannot be in connection with education A fact that ics ignored if we are to ensure a learning relates the study of mathematics in school Its role in basic teaching Around the world, process with long-lasting effects. to that of languages is that mathematical one’s first encounters with mathematics Some fallacies about mathematics For knowledge is accumulated, so that a lack of happen in elementary schools. Indeed, all some people, mathematics is just the language understanding at one stage affects the abil- countries consider mathematics to be part of the quantitative. They base their judge- ity to understand and progress later. This of the basic training that children must ment on the fact that mathematics enter- dependence is not as strong in other disci- have. The form it takes varies from one tains a special relationship with language; plines such as physics, history or geogra- country to the next, but basic operations this opinion is shared by some of our fellow phy. on numbers are always taught, together scientists. We mathematicians know how Another aspect that is largely country with fundamental geometric figures. wrong this is, and how much effort goes dependent is the role that mathematical Mathematics is also viewed everywhere as a into building concepts, establishing facts, performance plays in the selection process subject of its own. From this perspective we and following avenues we once thought at school. From this point of view, France cannot say that mathematics lies hidden. plausible but turn out to be dead ends. used to be an extreme case because of its When one digs deeper and considers the This widespread belief forces us to consid- dual system of higher education, split actual pedagogical practices, one realises er more carefully how mathematics inter- between the so-called ‘Grandes Écoles’ and that, although mathematics appears every- acts with other disciplines and what is the the universities, and the weight attributed where, the intellectual process on which exact nature of the mathematical models to mathematics in the entrance competi- mathematics is based is not always fully that appear ever more frequently. tions to the ‘Grandes Écoles’. presented. For example, the abstraction Second, mathematics is seen as a dead sub- A partial and brief conclusion: It is clear process that lies at the heart of mathemat- ject. One of the most frequent questions to me that the image mathematicians give to the ics is often hidden behind routines that that professional mathematicians hear general public (an image that is amplified by the have value but are only one side of the from laypersons and from media people, media) needs to be worked at to make it more in coin. The same can be said of the research is: “What can you do in a subject where line with what they actually do. This requires a process. Mathematics is sometimes pre- everything is known?” So many examples considerable and sustained effort on the part of sented as an activity in which one has to can be used to prove the contrary that we mathematicians. 20 EMS June 2000 FEATURE and of the status of this reality in relation ciently pertinent about infinity to be able 2.What is the nature of mathematics? with ordinary ‘sensible’ reality. On this to handle major questions connected with In this section I shall concentrate on a few broader issue, mathematicians have differ- it. This was the birth of modern analysis. topics that make mathematics special as a ent views: from the ‘Platonists’ at one end Think that, in spite of formidable evidence science. It is my opinion that mathemati- of the spectrum who believe that they ‘dis- imposed on mathematicians by everyday cians have not worked enough on this front cover’ new territories and new facts of the experience, the early part of the 19th cen- and are not always in a good position to mathematical world, to the ‘intuitionists’ at tury saw the birth of new geometries, gen- ‘make a case’ for their needs. This implies the other end who view mathematical con- eralising the well-established Euclidean that more thought needs to be given to the structions as purely human scaffoldings geometry, and opening the way to a more objective basis for the misconceptions dis- based on consensus opinions among a radical broadening of the scope that cussed in the previous section. Last but not restricted community – in other words, enabled general relativity to take ground. least, we must take a historical perspective that mathematics is ‘invented’. I have no Think that major problems concerning when discussing these issues, because doubt that the vast majority of mathemati- prime numbers and Diophantine equa- mathematics has a long history (maybe the cians are closer to the Platonist viewpoint tions have been solved one after another, longest of all sciences), and because the sit- than to the other one, or at least that ‘they thanks to numerous technical and more uation is changing very rapidly. So what is spend a good portion of their professional conceptual achievements. Think that ran- so special about mathematics? time behaving as if they were’, as André dom quantities can now be properly esti- A special relationship to language To Weil once put it. mated on the basis of a few precisely address this question we must realise that The possibility of using today results analysed assumptions. Many of these revo- the way one expresses mathematics evolves established centuries ago is another aspect lutions were the basis for major changes in with time. The now well-accepted idea that of the special relation that mathematics the organisation of society: we come back mathematics can be totally formalised was has with truth. It gives it the possibility of to this later. Why are we so shy? stated as a general paradigm only at the transcending civilisations, and makes its end of the 19th century. It is largely a transmission much easier since it is a priori 3. A new relationship to society philosophical stand and has little in com- less ideology dependent. This founds its This part is the most controversial, since I mon with the actual work done by most claim for universality. Of course, this is all dare to offer some perspectives for the mathematicians or teachers. based on the role that proofs play in math- future. It may be good to start by taking a We can view this evolution as the culmi- ematics, in spite of their changing appear- last look at the evolution of our discipline nation of the need to use a precise language. ance with time. Making this dimension in the 20th century, in order to identify the Since mathematicians like to borrow their accessible to students is of course the major changes that I claim force us to continue. vocabulary from ordinary language, we challenge that teachers face at all levels. Mathematics and mathematicians in the must realise that there is a price to pay – This gives us mathematicians a special 20th century In the 20th century, mathe- namely, that students may resent it as a obligation towards the history of our disci- matics, as many other sciences, enjoyed form of robbery, while we would rather pline, and requires that it be properly unsurpassed development. Many new view it as a form of poetry. More attention incorporated in our teaching, both as doc- results were obtained – some very spectac- should be devoted to explaining why we so umented proof of its evolution, and of the ular, such as the solution of the Fermat appropriate the common language, and progress it has made. Facts accessible in problem – but the real change was in the why we would be in trouble without it. We the past to a very restrictive group of great finer web of results and the extraordinary should behave as responsible citizens and minds can be, and are, correctly assimilat- broadening and ramification of areas that explain what we make of common ed today by large crowds of students, transformed our discipline. At the end of resources. After all, are we not forcing our because new concepts and new techniques the 20th century, the yearly production of students to live double lives, depending on have been developed in the meantime. research articles had reached the 60,000 whether we are talking in a mathematical Here lies for me the best antidote against mark, when in the 1950s there had been context or not? This is not easy and should the spreading of the devastating belief that only 5000. This is undoubtedly the be identified as such. mathematics is a dead subject. Let me note mechanical effect of the growth of the The constraint of expressing mathemat- in passing that efforts in the direction just mathematical community throughout the ics by a well-controlled language is certain- mentioned will also give us stronger argu- past century. Whereas some 250 partici- ly one reason for seeing mathematics only ments to support our worldwide struggle pants took part in the 1900 Paris Congress, as a language. We require so much atten- for keeping good libraries. To do present- the century ended with the Berlin tion to this matter, and for good reasons, day mathematics, we need access to docu- Congress that hosted 4000. Today, the that we should carefully explain why, ments that may be a century old or more, number of mathematicians actively describe those mathematical activities that and this need will stay with us in the future. engaged in research is estimated at about are not affected by this practice, and iden- Another consequence of the relationship 50,000. At the same time we should keep tify times when forcing oneself to stick of mathematics to truth, also worth point- in mind that the number of biologists is strictly to a formalised language is a hand- ing out in connection with education, is the now estimated at about 1 million. icap. This particular difficulty appears following. A person who has understood a In a more qualitative vein, the 20th cen- both for advanced students and for begin- mathematical fact becomes able to chal- tury witnessed the generalisation of the use ners, and may be one of the major hurdles lenge his or her teacher, and his or her of the axiomatic approach. The distance it that young researchers must pass. classmates. This experience of winning by allows to take from the immediate reality A special link with truth Confronted with exercising the mind is of the utmost surrounding us has also some drawbacks – a mathematical statement, a mathemati- importance in training critical minds and namely, that of providing arguments for cian’s goal is to prove that it is ‘true’. What giving them access to autonomous think- people who say that mathematics has does this mean? Of course, now that math- ing. This of course needs to be thought deserted the real world. Symmetrically, it ematicians agree to work in the context of through in relation to the very diverse also frees mathematicians from relating a potentially completely formalised theory, social environments in which schools are what they do to real-life problems. its meaning can be none other than ‘the operating. Nevertheless, there is no doubt in my mind statement can be deduced from the basic that this newly gained freedom is a fantas- axioms agreed upon’. In fact, if we are to The unbelievable success story of mathemat- tic advantage, provided that we do not iso- address this question in the context of the ics throughout history Mathematicians tend late ourselves and pretend that we have relations of mathematics to society, we are to be shy about their field, even though the nothing to say and nothing to learn from forced to take it in a broader, more philo- history of mathematics is one of the most other scientists, from engineers, and from sophical, perspective because we have to extraordinary success stories of the human society in general. confront it with reality. This amounts to adventure. Think that at the end of the Before leaving this question of the deciding whether there is a mathematical 17th century precise mechanisms were axiomatic method, it is important to men- reality of which this statement is a part, designed in order to say something suffi- tion that it also had an extraordinary EMS June 2000 21 FEATURE impact on the teaching of mathematics at objects in everyday life We now come to If this reliance on communications has an advanced level, allowing us to teach what seems the most important aspect of the positive aspect of abolishing distance, much more quickly to a wider group of stu- the change occurring at the end of the last it also makes us dependent on the quality dents. In that sense, it has greatly con- century – the omnipresence of mathematical of information collected by various kinds tributed to the development of the mathe- objects in everyday life. of sensors that can be taught more easily to matical enterprise. The impact it has had This phenomenon is also connected with send falsified information than our eyes, on the teaching at lower levels has been the new possibilities offered by computers, our fingers, etc. diversely appreciated but, if some experi- but other aspects of modern society that Where is mathematics around us? We ences were really negative, the status quo have no relation whatsoever with mathe- now give some explicit examples of the was no more acceptable, for the very rea- matics play a major role in it. Every day we mathematics around us. We make no claim sons that we are presently examining. use many objects that incorporate some to be exhaustive, and there is certainly Coupling mathematics with powerful part of mathematics, and often recent some personal bias. computing tools Already by the 1970s, new ones. Most of the time even we mathemati- In modern societies complex systems are means of computation had appeared in cians do not realise it, because the mathe- everywhere, to the extent that this feature the form of powerful computers, exhibit- matical component is not immediately vis- can even be seen as one of their character- ing first a fantastic number-crunching abil- ible and pinning it down requires some istics. One needs to regulate these mecha- ity but later also unexpected wonderful thinking. nisms, and this requires a new type of visualisation capacities. These develop- The most obvious case is that of the knowledge that is basically mathematical. ments have continued at an unabated pocket calculator, but computers also The management of such systems has pace. Can one measure the types of impact exhibit impressive mathematical ability; important consequences for the develop- that the advent of computers has had on their use is widespread, and we are con- ment of society. mathematics? I propose to organise them fronted with them everywhere. Less evi- Biological systems provide many examples in four families: dent examples are CD-players, TV sets, of complex systems that we must come to – with computers, we can immediately cars, etc. This proliferation has been made terms with. This domain became one of the deal with huge data sets, with the result that possible thanks to the widespread use of great adventures of the second part of the some problems have now become accessi- high-technology products produced 20th century since the discovery of some ble, where earlier mathematical methods cheaply. fundamental mechanisms of living organ- were not feasible; this is typical with statis- The notion of industrial product Today’s isms such as DNA structure. Genomics, tical data. Here, the key question is to society is dominated by the notion of industrial which comes out of this, requires handling learn how to extract relevant information product. This means in particular that many huge amounts of information, some of it from huge data sets, and this is a mathe- objects and structures that surround us discrete (the digits of the 4-letter alphabet matical problem. have been designed in a well-defined man- coding of DNA) and some of it geometrical – these new machines, which handle only ner. Objects have to fulfil a definite task, (being able to act on the DNA requires one finite data, force us to reconsider the relations and their mere existence is due to the fact to know very precisely the geometric acces- between the finite and the infinite, about con- that a market for them has been identified. sibility to the proper sequence). vergence of processes. We can teach a com- Usually this requires optimising the object Mathematical tools to deal with such prob- puter how to handle very large numbers, or the structure in various ways, a step lems are not obviously available, and are thereby making some areas of mathemat- involving a mathematical process that ear- likely to require new mathematical devel- ics, such as number theory or dynamical lier would have been a purely hardware opments around the question of how to systems, amenable to experimentation. problem. structure large data sets. Here too, the net result is a considerable This goes further. Even if one is interest- Telecommunication systems have already had broadening of what is accessible to mathe- ed in keeping an industrial product longer a wide impact on the way that the whole maticians; as an example, we can name than was planned by its inceptor, one soon society functions. Again, when we consider cryptography, again an activity whose encounters problems because (and this is objects through which we communicates, nature is completely mathematical. especially true for high-tech products) they besides the fact that one punches numbers – machines have the capacity to repeat a sim- quickly become incompatible with their on a keyboard, there is no hint that there ple algorithm a very large number of times. The environment – they cannot be used any is some mathematics behind them. It looks example of Julia sets and fractals may have more. This compatibility problem poses at as though the chips, the laser (as in CD- been overused, but we have to learn how to a large scale the question of keeping the Roms) and the wires are the whole story, exploit their attractiveness and beauty memory of the past in proper form and, when in reality the design of the network properly; this has changed the view we after what we said earlier, this is no minor plays an equally important role, and this is have on some structures, transforming concern for mathematicians. The notion of where mathematics enters. The perfor- their pathological appearance into a real an industrial product incorporates its lim- mance of a telecommunication network is richness that has provided natural scien- ited life span. measured through its capacity for not tists with more models relevant to some Towards a communications society altering the information it carries. This can practical situations. Claiming that we are entering a society often be done through error-correcting – computers open new horizons to the numer- dominated by communications is not orig- codes, which encapsulate sophisticated ical treatment of sophisticated equations, a typ- inal. Here, we consider the consequences algebraic properties – that is, purely math- ical example being that of weather fore- of this change from a mathematical point ematical knowledge. casting. of view. Another side of telecommunications has By opening new domains to mathemat- First, more information is made avail- to do with the quantity of data to be trans- ics, changing our views on complexity, and able to a broader section of the public, mitted. Given a given physical system giving new tools to mathematicians to solve leading usually to too much information. designed to carry data, it may be necessary their problems, the impact of new comput- To learn how to navigate in such a world is to compress the data (typically, images) so ing tools is therefore considerable, even not easy, since one needs in particular to as not to exceed the capacity of the net- without mentioning the very deep mathe- know whether a piece of information is work. Designing compression and decom- matical problems posed by machines and reliable. In view of the quantity it is clear pression algorithms is another challenging by networks connecting them. This is that most of the available information can- problem, in which (for example) wavelet undoubtedly one of the new frontiers of not have been properly checked. If one theory, a truly mathematical modern mathematics. What was once considered by wants citizens to have control of the facts adventure, has had an important impact. mathematicians as a threat provides a on which to base their judgements, one The Global Positioning System (GPS) is great opportunity to revitalise the disci- needs to train them already at school to do another example of a modern tool that pline and make it relevant to many more that. This makes exposure to critical rea- must incorporate very advanced mathe- situations. soning more important, and training to matics in order to reach the accuracy that The omnipresence of mathematical distinguish bias a necessity. widens its use. Indeed, in order to improve 22 EMS June 2000 FEATURE precision from 10 metres to 10 centime- port systems rely on such networks. which special training is now offered at a tres, corrections that take care of effects Indeed, when we are waiting for a bus, a number of institutions of higher education. accounted for by the theory of relativity train or a plane, we focus our attention on must be considered. For GPS again, opti- the material carrier, but the important (but 4. Final words mising the choice of satellites from which hidden) fact is that this object is part of a From all the above challenges concerning one estimates the position on the earth whole network. All this relies on sophisti- the image of mathematics, the most serious requires the use of sophisticated mathe- cated graph theory. seems to me the need to make it evident that matics, such as discrete group theory. Optimisation of forms hides a lot of mathe- mathematics is a science and alive. The rest Finally, there is still another side to matics. We often use objects whose shapes should follow. problems posed by transmission of data – have been designed using advanced math- In my opinion, we mathematicians must their security. In many cases, for one rea- ematical tools, most of the time by extrem- agree to put more consideration into, and to son or another, one needs to be sure that ising some functional – just think of aero- show more curiosity for, what is happening the information is kept confidential all the dynamic forms of cars or planes to min- around us. We also have to agree to say way through. Encoding it is the solution, imise air resistance, and thus to improve more about our activities. This requires provided that the code cannot be broken fuel efficiency. Many other examples can taking more initiatives towards the general by a person different from the one that is be given that involve purely design criteria, public. We should be helped in that there supposed to receive it. Here again, one has lower production costs, or improve a mate- are many mathematical products around to deal with discrete data, and discrete rial strength. us. It is our duty to exhibit what is mathe- mathematics is a part of mathematics that, An interesting example is given by opti- matical about them. in many countries, has not yet been given mising the places where the body of a car Secondly, we have many opportunities to the same priority as the a priori more noble is linked to its frame. In choosing them make people dream. Mathematics still part using real numbers. properly (for example, at nodes of the remains a great creative adventure. New chal- Data collection and analysis is yet another basic harmonic vibrations of the body), one lenges are ahead of us. The need in our area where we get used to new ways of can minimise the transmission to the discipline for great minds remains as functioning and may miss the underlying inside of the car of vibrations caused by pressing as it has been throughout history. mathematics. Indeed, at the supermarket, bumps on the road. To achieve such goals, we need to find we have got used to no longer seeing the Yet another example, again taken from appropriate ways to connect ourselves with cashier type the price of goods we buy, the car industry, has to do with the shaping teachers in schools. In the long run, this is thanks to the systematic use of barcodes. of windshields, which are usually curved the only channel through which these new The net gain for the shop is not only the surfaces. A difficult question is to deter- trends will be passed on to the new gener- time saved in typing but the automatic mine the shape of the windshield when it is ation. The future of mathematics lies in the management of the reserves. Here again made flat, before one ‘forms’ it by heating hands of those younger people that our genera- the recognition process on which the bar- it again to make it deformable. tion will prove able to attract to mathematics. code system is based is purely mathemati- Banking products and insurance Many of cal. the above examples come from different A version of this article was presented at a Statistics and polls We read statistics in areas, and point to one fact that makes ‘Symposium on the Mathematics of the XXth newspapers every day, and to be able to mathematics different from other sciences century’ held in September 1998 in decipher what they say and what they omit – namely, that there is no industrial sector Luxembourg. has become of the greatest importance in that can easily identify itself with the disci- today’s world. The fact that we can collect pline. and deal with large data gives us more Recently, this may have changed with opportunities, but citizens need a basic the new trends taken by the banking and More on Dutch training in this field in order to allow them insurance industries. There are several to detect when they are being manipulat- reasons for this change. We have already mathematics and π ed. encountered one of them – the capacity to Other information that we now find reg- collect and quickly analyse large data sets. Peter Stevenhagen ularly in the news media are polls. We are Another is linked to the new telecommuni- The previous issue of the Newsletter had an exposed to many of them, and in this cation possibilities, and the fact that today article about the oldest national mathe- respect we regularly have the feeling that there is really one world financial market. matical society in existence, the Wiskundig information drawn from them is definitely Yet another is deeper, and has to do with Genootschap. As with British stamps, there improper. Since they have become a nec- the evolution of the banking activity which is no need for an explicit reference to its essary part of the process by which people actually functions more and more like an country of origin. form their opinions, they are very impor- insurance. A typical example of this An editorial error crept into the expla- tant to democracy. Knowing how to esti- change is given by derivatives and options, nation of wiskunde, the Dutch word for mate the error margin and how polls can the promise to buy a certain good at a mathematics. Many foreigners relate be falsified is therefore essential. prize determined by a well-defined mecha- wiskunde to the German word Wissenschaft, Using appropriate rigorous mathemat- nism, based on information that will be and incorrectly infer that wiskunde is Dutch ics, it is possible to prove whether a statis- available later. The key point is to estimate for science, which has the Latin scire for to tical claim is correct or not. It is probably the risks that one takes by making such a know or, in German, wissen as its root. The in this area that a contribution towards deal. One must use sophisticated proba- Dutch have wetenschap for Wissenschaft, and critical thinking through mathematics is bilistic models in order to evaluate the use weten for to know. Weten and wissen really possible, and at the same time indis- indices on which the price is based and derive from the same root, which also pro- pensable. which depend on random parameters. The duced wits in English and the Dutch word Automatics and robotics Many complex sys- kind of stochastic processes that are wis, meaning sure or certain. This is the tems are run using control parameters. involved turn out not to belong to any of word that Simon Stevin used in coining the Automatic pilot systems that run planes are the standard categories that have been word wiskunde. Thus, wiskunde literally examples of systems where control theory extensively studied in connection with translates as surology – a non-existent word plays an important role. processes met in the natural sciences, but coined by the American mathematician In fact, we are confronted with many rather with the general theory of stochastic Barry Mazur from Harvard. other automated systems in our everyday processes, a body of knowledge long con- As the article pointed out, a ‘new tomb- life: lifts, TV and telephone satellites, cars, sidered highly theoretical if not esoteric. stone’ for Ludolph van Ceulen containing etc. Scheduling, the theory that ensures These developments have led banks and ‘his’ 35 decimals of π will be unveiled in the that a given network can provide the ser- insurance companies to hire mathemati- Pieterskerk in Leiden. For ceremonial rea- vice it is designed for, also relies on appro- cians, and have given birth to a new branch sons, the date of the event has now been priate mathematical theory. Most trans- of our discipline, financial mathematics, in advanced to 5 July 2000. EMS June 2000 23 EMS NEWS EMS Position paper on the Commission’s communication: Towards a European research area Luc LEMAIRE (Bruxelles)

The following is the EMS’s response to the (1) A mathematical research team at a European support for research has an European Commission Communication given location is generally smaller, and immediate effect on the improvement of ‘Towards a European research area’, issued in research on a given subject is geographi- mathematical qualifications in the man- January 2000 (see the website cally more dispersed. That the quality of power of the future. It is the appropriate http://www.europa.eu.int/comm/research/area. moment to point out that the use of peo- html). The text was sent to Commissionner ple mastering an advanced mathematical Busquin, who received an EMS delegation (R. training has broadened enormously. Jeltsch, J.-P. Bourguignon and L. Lemaire) on Besides finance, we can mention telecom- 23 May, for a constructive exchange of ideas munication as a whole, the drugs indus- on the EMS document. try, material design and so on. (5) In Central and Eastern Europe and in The European Mathematical Society has the Third World the state of research in analysed the communication and related mathematics is often better than in the documents, and welcomes the new orien- other sciences, for relatively low cost ini- tations that it describes proposing a new tiatives have often allowed the creation of stage of development for European centres of excellence; without being Research. It wishes to contribute to the exhaustive, we could name some centres open debate initiated by Commissioner that are at the highest world level: the Busquin, bearing in mind its own IMPA in Rio de Janeiro, the Banach European mission and its direct connec- Centre in Warsaw, the Tata Institute in tion with research and training activities Bombay. Some institutes in Vietnam and throughout the countries of the Union, Africa can also be mentioned. the Associated States and all other coun- mathematical research is not necessarily Cooperation in mathematics with such tries of the European continent. directly dependent on a concentration of countries should not be seen as a one-way heavy resources is certainly an advantage aid to development, but as a true Contents – but requires that this fact be taken into exchange between active centres, from I. Some special features of mathematics account when setting up programmes. which EU groups can benefit directly. II. Fundamental research and its role in This doesn’t mean that no support at all is industrial development needed for mathematics, and computa- II. Fundamental research and its role in III. Human capital tional mathematics and scientific compu- industrial development IV. Centres of excellence tations will request investments – e.g. as The European Mathematical Society is V. Infrastructures and electronic databases large infrastructures. happy to see the importance of funda- VI. Central and Eastern Europe and Third (2) The unity of mathematics has always mental research underlined at various World made itself felt with considerable force, places of the Communication. VII. E.U. programmes and procedures and at the end of the 20th century the The private sector mostly invests in VIII. Benchmarking interplay of mathematical ideas between short-term applied research, and this IX. Summary of recommendations the different sub-domains had probably implies that the public bodies – States and reached an all-time high in intensity. European Union – must bear the major I. Some special features of mathematics There is no doubt that this phenomenon responsibility of long-term investment in Although the E.M.S. wishes to raise some sets mathematics apart from many other fundamental research. points pertaining to aspects of research in branches of science which have split up All opinions converge today on the fact all fields, it feels the need to point to into multiple specialities, each with its that the time-lag before a discovery can be some particular features of mathematical specific culture and sociology. A direct put to practical use is getting shorter and research, notably in Europe. consequence is that the task of a panel of shorter. But the time before commerciali- Some of these characteristics should mathematicians having to evaluate sation is getting short because the numer- have some bearing on the organisation of research is much easier, the existence of ous (and often cheap) innovations that the E.U. programmes. Let it be clear that factions and of schools in conflict with one come up everyday from the world of we do not call for a blind increase in another remaining exceptional. research are more quickly sorted out, and mathematics funding, or for adopting (3) The distance between fundamental not because the analysis of needs has rules that would favour mathematics, but mathematics and applications is relatively motivated the discovery. We are in an era for an increased flexibility in the defini- short and is getting shorter both in time where the precise demand is not shaping tion and the running of the programmes and in content. In some cases, mathemat- all the markets. that would leave all opportunities for ical subjects developed for reasons purely A very significant example will illustrate excellent proposals to be considered in all internal to mathematics have suddenly this. The necessary scientific basis that fields, with best possible efficiency. led to concrete applications, in sharp con- gave rise to the explosion of the commu- Different research traditions indeed trast with the traditional pattern of neigh- nication society (the internet, ...) is not require different frameworks, and we bourhood relationships: pure science – tied to discoveries that were motivated by believe that our comments do not apply applied science – applications. By way of that development, but by very theoretical only to mathematics. example, we might mention the use of results, obtained outside the pursuit of Mathematical research in Europe is stochastic processes in finance, or of num- commercial aim and following the initia- recognised as among the best in the ber theory in cryptography and system tive of the scientists (bottom-up world: several indicators place it ahead of security. approach). that of North America and of Asia. Other (4) In mathematics there is an extremely Indeed, the laser was built as a physical characteristics that distinguish it from sis- close relationship between research and curiosity in the sixties (the production of ter sciences, and in particular from the training. In fact, most researchers in coherent light). It has now become a nec- experimental sciences, are the following: mathematics are also teachers and essary ingredient of fast transmission 24 EMS June 2000 EMS NEWS along optical fibres – as well as being used excellence do exist in Europe but they With their present rules, the rate of suc- every day when we listen to compact discs, have less success, for the simple and basic cess has become too small to make choic- use code bars and have surgery per- reason that they do not have enough posi- es optimal, and the lack of flexibility formed. tions available! make them less adapted to sciences like Likewise, number theory was studied The E.U. can have a strong effect on mathematics, where at each node groups for centuries as an ultimate amusement of this problem provided it continues its are usually small. Thus, a more flexible mathematicians – but it led to cryptogra- efforts in the appropriate form. It is in rule should allow the inclusion of more phy and error correcting codes, without particular extremely important that small groups, without necessarily increas- which safe transmission on the internet young researchers after their PhD have ing the global budget of a network. and e-commerce would not exist. the opportunity to enlarge their perspec- Moreover, the panel of experts should be Also, the developments in non-linear tive, by gaining access to centres of high given much more freedom, including the analysis, like wavelets in image process- level outside of narrowly defined scientif- possibility to award less money than ing, have preceded the use of sophisticat- ic projects. requested or intervening to suggest the ed non-linear models in high technology Practically, it would first be important merger of two applications. industry. to create and impose a tax-free status for The freedom of action of the panel has In many commercial developments of the European postdoctoral researchers in been abruptly suppressed at the begin- the future, we should expect to observe mobility paid by the budget of the ning of the Fourth Framework the same pattern: scientific discoveries of Commission, with the hope that States Programme, and in mathematics the rules different fundamental fields being quite would follow that example with their own lead to a dramatic decrease of the cost- unexpectedly transformed into innovative fellowships. For the moment, the situa- efficiency of the programme. new tools, creating a new market. tion varies from country to country and it What can the E.U. do in this context? is ridiculous to see that the same III.3 The demography of researchers and Naturally, it must support the develop- researcher is twice as expensive in professors in European universities ment of industrial technology in an inte- Belgium as in the U.K. (in fact, in should be studied in detail for every field, grated way, to avoid negative effects of Belgium the E.U. will have to pay 180.000 to get a fairly precise estimate year by internal competition. Bef per month, the researcher receiving year of future needs. For mathematics, But it must mostly create a European 60.000!). A European tax-free status this was done a few years ago by the area of scientific freedom, where the sci- would overnight double the number of E.M.S. This study should be updated, and entists will create new science following positions offered by the Commission. would show some strong variations in the their own approach. It must then encour- Secondly, we propose to define a new number of retirements over different age or produce risk capital sufficiently (optional) format for a postdoctoral fellow periods (still following the effect of World quickly when a discovery can lead to in mobility – namely, two years in another War 2, of the growth in the sixties, the cri- unexpected applications. In short, E.U. country, then one year in their coun- sis in the seventies, and so on). Europe has to give a chance to the unex- try of origin (for the moment, this exists An ambitious policy is badly needed. It pected. This goes against the long-estab- only for researchers from regions in eco- should insure, by an adequate number of lished idea of very close control on pro- nomic difficulties). doctoral and postdoctoral positions, that grammes and choices made a priori. Such a move would greatly improve the the future professorial positions in uni- attractiveness of Europe for the following versities will be awarded to high-class III. Human capital reasons. researchers. In this age of science and technology, it is For the moment, the time gap between obvious that the future of Europe will completion of the PhD and the possibility III.4 A serious technical problem arises in depend crucially on the level of its of getting a permanent position is not the definition of Marie Curie training researchers, and in fact of all its work- easily covered by postdoctoral positions in sites. Indeed, they are open only to doc- force. Energetic measures are thus Europe – and as a result many have to go toral students and not postdoctoral ones. required to encourage the pursuit of sci- to the U.S., where they often remain. In certain fields, this corresponds to well- ence, improve the training of the The 2+1 positions would be much identified needs, but not in others (like researchers and fight the brain drain to more attractive: the researcher could mathematics). We propose to open these other continents. completely concentrate on his/her work sites to both doctoral and postdoctoral while abroad, without having to prepare researchers, thus allowing for optimal III.1 In mathematics, the crux of the mat- multiple applications for the following choices by the institutes in each field. ter is probably the question of availability year. Back home typically during year 3, of postdoctoral positions, and the E.M.S. he/she would be in a better position to III.5 Finally, anyone who has invited to wants to underline the very positive seek permanent employment. Europe a scientist from outside the E.U. results already achieved by the E.U. Finally, it happens from time to time knows that the visa problem is a major through its Network and Fellowship pro- that a bright researcher is offered simul- headache and a huge loss of time. In grammes. It calls for a substantial taneously a 2-year E.U. fellowship and a some cases, a foreign scientist invited and increase of means in these activities permanent job at home, to take effect paid by a government will not be able to (together with other changes described in immediately. The choice is obvious – but come, because the same government will section III 2 below), which would prevent not optimal for the development of not provide him/her with a visa. having to reject excellent proposals, as is European science. The employer (e.g., now the case in the network category. University) will not usually agree to give IV. Centres of excellence Why are postdoctoral positions so an immediate leave of absence. If on the The E.M.S. supports strongly the important for the future of science in other hand, a third year were offered to Commission’s view that centres of excel- Europe? We all observe that scientists in work in the same place, it could be anoth- lence are an important component of Europe are often better trained than in er matter. European research, well positioned for the U.S. – up to the doctoral level. This Altogether, these 2+1 positions would efficient and justified E.U. support. considerable investment in education not be seen as an obstacle to a local However, those centres cannot be iden- having been done, Europe often offers no career, but on the contrary could become tified (as in the Communication of the good opportunity for further employ- highly attractive positions, well recog- Commission) by their capacity to produce ment, leaving to the U.S. the benefit of nised in a Curriculum. industrially applicable discoveries. We hiring highly competent scientists. As a note however that the accompanying doc- result, the doctoral and postdoctoral U.S. III.2 The networks are also important ument: Action for “centres of excellence” programmes work with a majority of non- components of the postdoctoral pro- with a European dimension gives better U.S. citizens. gramme – well integrated with the effec- criteria, and a list of existing centres A number of undisputed centres of tive development of research. showing a broader vision. EMS June 2000 25 EMS NEWS As mentioned above, centres of excel- effectively need access to the whole of the Thus we advocate strong relations with lence in mathematics exist in Europe at literature for his/her work. Central and Eastern Europe, on the basis the highest level, and are by no means On the other hand, the price of sub- of scientific partnership at high level. The inferior to U.S. centres like the I.A.S. in scriptions to journals, in particular pub- very low cost of living there can be used as Princeton or the M.S.R.I. in Berkeley. lished by commercial companies, is rising an advantage, to organise activities more Seven of these centres have in fact unreasonably: increases of 10 to 20% per cost-efficiently. already constituted a network of year are the norm, 60% often happens. Concerning the countries of Latin European dimension, pre-tailored for the Thus, the average university library America, Asia and Africa, Europe could at scope of a European Research Area. The becomes unable to acquire the basic nec- this stage develop strong scientific rela- European Post-Doctoral Institute is essary tool for its mathematics tions, which would be beneficial in the indeed constituted by the Institut des researchers. long run. Indeed, these countries do not Hautes Etudes Scientifiques (Bures-sur- As a consequence, some U.S. enterpris- usually cherish the idea of being absorbed Yvette, France), the Isaac Newton es. have embarked on an ambitious pro- under the umbrella of powerful neigh- Institute for Mathematical Sciences gramme of digitising the existing scientif- bours (U.S.A. for Brazil, Japan for Korea, (Cambridge, UK), the Max-Planck ic literature, in order to distribute it elec- ...). Mathematically, some centres in these Institut für Mathematik (Bonn, tronically. This effort is funded by private countries are at the highest level, and a Germany), the Max-Planck Institut für foundations, but access will be reserved to programme of cooperation would be very Mathematik in den Naturwissenschaften institutes paying a heavy annual fee, like- efficient. Currently, cooperation is only (Leipzig, Germany), the Erwin ly to remain out of reach for most funded on specific themes defined by the Schrödinger Institut (Vienna, Austria), European universities. Commission (pollution, forest, health, ...), the Institut Mittag-Leffler (Djursholm, For these reasons, the E.M.S. presses missing the necessity for these countries Sweden) and the Banach Center (Warsaw, the European Commission to take strong to be integrated in the future society of Poland). action aiming at developing electronic knowledge. Various European states have The aim and function of this E.P.D.I. is databases and libraries, including the strong historical and cultural links with so much in line with the E.U. programme steps of developing the data and its distri- developing countries, and these links that its case should be examined by the bution. A more precise analysis should could form the basis of future coopera- appropriate scientific panels. In the past, show whether Europe should catch up tion. strict rules on different E.U. programmes with the digitising initiated in the U.S. by made the EPDI not eligible – usually to competition or by partnership. In the lat- VII. E.U. programmes and procedures the regret of scientific panels who rated it ter case, a sufficiently strong position Some rules currently enforced by the as excellent. must be reached before hoping for a real Commission do not allow for maximal Other networks in mathematics have partnership. By strong action, we mean in efficiency of the programmes. It seems also established themselves at the highest particular that the Commission support essential to identify and modify them. level, either in the E.U. programmes or be granted not only for access to infra- outside, and we wish to mention here the structure, or for research related to the VII.1 The rule of subsidiarity, pushed to Network MACSI ( Mathematics, comput- development of the infrastructure, but for some extreme, has sometimes been inter- ing, simulation in Industry), initiated by direct funding of the installation itself. preted as saying that the Commission will ECMI and ECCOMAS. This does not contradict any principle not fund research itself, but only In order to allow for a choice of optimal of subsidiarity, once it is acknowledged European added value. This is a self- centres and networks, the E.M.S. insists that the infrastructure is a truly European defeating interpretation which we strong- once more on the importance of flexibili- one and could just not be conceived at the ly regret. The European Space of ty in the rules: a network of centres of scale of one state. Research must be in itself a European excellence need not be very large, need We note also that the E.M.S. has the endeavour, and thus must be funded with- not be founded on a very specialised project of establishing itself as a out administrative restrictions. topic, and need not consist of large labo- Publishing House, which would work as a In the present programme, examples ratories, although of course it can. non-profit organisation. The European can be shown in which proposers, panels Finally, all calls for proposals should be Commission might think of supporting and administrators of the programme widely open to competition – science in this kind of endeavour at the initial stage, agree that a realisation is worthy of sup- Europe is not limited to a list of centres for the long-term benefit of the scientific port, yet the rules prevent it. fixed once and for all. community. For infrastructure, such restrictions were described in Section V. V. Infrastructures and electronic data- VI. Central and Eastern Europe and For the programme ‘Raising Public bases Third World Awareness’, the description of the pro- The E.M.S. gratefully acknowledges the As indicated above, very high-level math- gramme gives the impression that it will fact that the definition of a large infra- ematics research is pursued both in be easier to fund a round table of discus- structure was enlarged in the Fifth Central and Eastern Europe and in vari- sion than an actual concrete realisation Programme, thereby opening the way to a ous developing countries. In Central and (the latter being pushed in the category of successful application of support of the Eastern Europe, a high mathematical accompanying measures). comprehensive European database of level has been maintained as a tradition, mathematical literature, Zentralblatt- and using the fact that restricted access to VII.2 The E.U. procedures, applications MATH. expensive material over a short period of and contracts are absurdly complicated, Further support for electronic database time did not bring irreversible damage. notably because they aim at covering in and libraries will be essential for the This tradition is under threat in some extreme detail many different situations development of mathematics, and Europe countries, simply because economic prob- in the same format. It seems that all doc- must reach the level of an unavoidable lems have brought down the salaries of uments were drawn up by wary lawyers, partner in the world competition in this university teachers below a minimal level and not by scientists. The smallest appli- domain. – thus producing a massive exodus to the cation is the object of 50 to 80 pages of A special feature of that branch is that west. instructions on the Cordis website. A sim- documents are never obsolete: a brilliant It is of the interest of the whole of ple shift of a few months in the execution proof by Bernhard Riemann or Elie Europe to preserve its high and long-last- of a project requires a full amendment to Cartan can be of extreme contemporary ing tradition of mathematical research in the contract. importance, because ideas remain valid all countries. To disrupt it in the East Quite often, the questions asked in the and the tradition of mathematics publica- would irreversibly damage a remarkable application form constitute a barrage tion does not include repetition of past school of talent, which Europe will need between the proposer and the scientific results. Thus a present day researcher will in the near future. panel. This is in part due to a restrictive 26 EMS June 2000 EMS NEWS view of subsidiarity: instead of simply say- the assessor from a major element of as equal scientific partners, ing what research he plans to do, for the judgement. Moreover, if an assessor is 6. E.U. programmes include mathemati- panel to read, the proposer will have to suspected of helping his or her friends, it cal collaboration with researchers of answer a series of questions, distinguish- is clear that these friends will tell him they developing countries, where a good ing training, research, added European are behind the application. So nothing is mathematical level has been reached. value, externalities, ... It has gone so far gained but very much is lost in that 7. all Commission procedures be enor- that some high-level scientists do not process. mously simplified, to avoid possible apply any more, whereas other hire the neglect of programmes by the best sci- (paid) service of consulting agencies, who VII.6 To summarise this section, we entists; this applies to all steps of the specialise in knowing how to transform a reduce our recommendations to three process, from the initial applications to scientific project into a ‘viable’ E.U. appli- simple mottoes: more flexibility, fewer the final reports. cation. rules, less paperwork. 8. the scientific panels be given more freedom to judge, amend or make sug- VII.3 The scientific panels play, and must VIII. Benchmarking gestions about the proposals. play, an essential role in the peer review- The E.M.S. certainly approves of bench- ing of proposals. They must examine the marking exercises, but not to the extent Moreover, it would be extremely useful proposals and have an actual meeting to that it uses up a substantial part of E.U. to: exchange information and opinions. To funding and of scientists’ time. – allow both doctoral and postdoctoral ask for separate marks out of 100 and Moreover, the benchmarking described researchers in the Marie Curie training then to make a simple average (a process compares best practice in various E.U. sites; unfortunately used by INTAS) transforms countries, whereas the initial aim of the – go back to the previous system of the choice into a lottery, different experts programme seems to be to catch up on appointments for panel members, to having different scales for their marks. U.S. efficiency, where organisation of be chosen jointly by the present mem- The composition of panels must be research is concerned. bers and the Commission staff; handled with great care. At the beginning Maybe a benchmarking of practice with – suppress the idea of evaluating anony- of the Fifth Programme, a short sentence the U.S. would simply confirm that their mous proposals; suddenly appeared in the texts, restrict- efficiency is related to more freedom, less – stress the importance of easy visa pro- ing the choice of panel members by the control, less planning, more facilities to cedures for recognised researchers. E.U. staff to scientists who had filled in an patent a discovery, and simplified proce- application form. dures to start a company, and these could The result could have been expected: in be major guidelines for Europe. World Mathematical Year many disciplines, the top-class scientists did not agree to fill in a long form in IX. Summary of recommendations Stamps order to be included in a database of The European Mathematical Society rec- potential experts; on the other hand, ommends that 26,000 people applied, producing a list 1. the Commission take strong initiatives In EMS Newsletter 35 we featured the impossible to handle. to develop in Europe a large area of Belgian stamp issued to commemorate We propose to go back to the preceding fundamental research, preserving a World Mathematical Year 2000. Further procedure: the panel secretaries together ‘bottom-up’ approach leaving the door stamps have since been issued by with the present panel members make open for the unexpected discoveries Luxembourg, Slovakia and Argentina. proposals, taking into account the level of that could shape tomorrow’s society. Two of these are shown below: the the experts and their fields. Proposals are Concurrently, administrative simplifi- Luxembourg stamp features the WMY also made from the outside, and the panel cations and encouragement of risk cap- 2000 logo, Fermat’s last theorem, Stokes’ secretary and the director of the pro- ital should allow for quick exploitation theorem, the Riemann zeta-function and gramme make a final choice. They can of these discoveries. the decimal expansion of p, while the also call personally some well-known sci- 2. more flexibility be introduced in the Argentinian stamp depicts the infinity entists and ask them to take part in the definition of E.U. programmes, so that sign. If any reader can send the Slovakian panel. sciences with different traditions and stamp to the Editor, it will appear in the Anomalies in the way a panel member format can be efficiently supported. next issue. behaves are monitored by the E.U. staff, 3. the Improving Human Capital pro- and in case a problem arises the panel gramme be maintained and amplified, member is not appointed again. with the following specific suggestions: Our experience of the mathematics and – the creation of a tax-free status for doc- information science panel is that it has toral and post-doctoral European evolved well, with competent guidance of researchers; panel secretaries at high level. – the creation of ‘2+1 fellowships’, with In case a panel were to concentrate two years in another country and a around a specific subfield, it is up to the one-year return fellowship; E.U. administration (directors and panel – a study of the demography of universi- secretaries) to make the corrections. ty professors in each field, and adequate measures to insure a supply of VII.4 As already mentioned, panels high-level researchers when necessary; should be given more freedom by allow- – the development of the idea of centres ing added flexibility to the rules. of excellence and networks of such centres, with flexible rules and definitions, VII.5 In some E.U. programmes, part of allowing for the different ways of dif- the first application must be anonymous, ferent fields. in the sense that the scientific programme 4. the Commission firmly support the must be described but the name of scien- development of European electronic tists in charge must be deleted. This pro- databases and libraries, to allow access cedure may give an impression of impar- to the necessary scientific information tiality, but in fact it is absurd. The level of all over the continent. expertise of the researchers is the main 5. the Central and Eastern European factor in the success of a programme, and mathematicians be integrated rapidly to delete that information is to deprive into all the Commission programmes, EMS June 2000 27 CONFERENCES http://www.mat.ub.es/~logica/news.html or http://www.crm.es/ FForthcomingorthcoming conferconferencesences [For details, see EMS Newsletter 34] 9–15: Mathematical Methods for Protein compiled by Kathleen Quinn Structure Analysis and Design, Taranto, Italy 10–14: 3rd European Congress of Please e-mail announcements of European confer- Mathematics (3ecm), Barcelona, Spain ences, workshops and mathematical meetings of inter- 3–7: ANTS IV Algorithmic Number Theory Information: contact Societat Catalana de est to EMS members, to [email protected]. Symposium, Leiden, the Netherlands Matemátiques, Carrer del Carme 47, E-08001 Announcements should be written in a style similar to Information: Barcelona, Spain those here, and sent as Microsoft Word files or as text e-mail: [email protected] tel: (+34)-270-16-20, fax (+34)-93-270-11-80 files (but not as TeX input files). Space permitting, Web site: http://www.math.leidenuniv.nl/ants4/ e-mail: [email protected] each announcement will appear in detail in the next [For details, see EMS Newsletter 34] Web site: http://www.iec.es/3ecm/ issue of the Newsletter to go to press, and thereafter will [For details, including satellite conferences, see be briefly noted in each new issue until the meeting 3–7: Functional Analysis Valencia 2000, Second Announcement in EMS Newsletter 34] takes place, with a reference to the issue in which the Valencia, Spain detailed announcement appeared Information: contact: K. D. Bierstedt or J. 10–14: ICMS Workshop: Dynamical Systems, Bonet, Univ. Paderborn, FB 17, Math., D- Edinburgh, UK July 2000 33095 Paderborn, Germany or Universidad [satellite meeting of the International Congress Politècnica de Valencia, Departamento de in Mathematical Physics, 17-22 July, London, Matemática Aplicada, E-46071 Valencia, Spain UK] 2–7: 6th International Conference on p-Adic e-mail: [email protected] Information: Analysis, Ioannina, Greece Web site: http://math-www.uni-paderborn. e-mail: [email protected] Information: contact A. K. Katsaras, Dept. of de/VLC2000 Math., Univ. of Ioannina, 45110, Ioannina, [For details, see EMS Newsletter 32] 10–14: International Conference on Greece, Noncommutativity – Geometry and tel: (+30)-651-98289, fax: (+30)-651-46361 3–9: Euro-Summer School on Mathematical Probability, Nottingham, UK e-mail: [email protected] Aspects of Evolving Interfaces, Madeira, [satellite meeting of the International Congress Web site: http://www.uoi.gr/conf_sem/p-adic Portugal in Mathematical Physics, 17-22 July, London, [For details, see EMS Newsletter 34] Information: UK] e-mail: [email protected] Information: 2–15: NATO Advanced Study Institute 20th Web site: http://maei.lmc.fc.ul.pt e-mail: [email protected] Century Harmonic Analysis – a Celebration, [For details, see EMS Newsletter 34] Tuscany, Italy 10–14: IUTAM Symposium on Free Surface Information: 4–6: Catop 2000, Fribourg, Switzerland Flows, Birmingham, UK Web site: http://www.cs.umb.edu/~asi/analy- Scope: categorical topological methods Information: sis2000 Information: Web site: Web site: http://www.unifr.ch/math/catop2000/ http://www.mat.bham.ac.uk/research/iutam. htm 3–5: International Conference on Monte [For details, see EMS Newsletter 34] [For details, see EMS Newsletter 33] Carlo and Probabilistic Methods for Partial Differential Equations, Monte Carlo, Monaco 4–7: 2nd International Conference on 10–15: 2nd World Conference on Information: Mathematical Methods in Reliability, Mathematics and Computers in Mechanical e-mail: [email protected] Bordeaux, France Engineering, Vouliagmeni, Greece Information: contact Dr. Valentina Nikoulina, Information: 3–5: Rencontre de Probabilistes, Lectures on Université Victor Segalen - Bordeaux 2, Web site: http://www.softlab.ntua.gr/~mastor/ Probabilistic Topics in 3D Fluids, Barcelona, Statistique Mathematique, UFR MI2S, B.P. 69 mcme2000.htm Spain 33076 Bordeaux Cedex, FRANCE, Speaker: Franco Flandoli (Università di Pisa) tel: (+33) 5 57 57 10 70 & 5 57 57 14 25, 10–15: 2nd World Conference on Aim: the meeting will be devoted to some ax: (+33) 5 56 98 57 36 Mathematics and Computers in Physics, advanced lectures on probabilistic methods for & (+33) 5 57 57 12 63 Vouliagmeni, Greece 3D fluids. These will present the state of the art e-mail: [email protected], Information: in a subject that lies at the interface between [email protected] Web site: http://www.softlab.ntua.gr/~mastor/ fluid dynamics and probability theory. Two Web site: http://www.mass.u- mcp2000.htm points of view will be considered: the study of bordeaux2.fr/MI2S/MMR2000/ stochastic Navier-Stokes equation and the mod- [For details, see EMS Newsletter 34] 10–15: 4th World Conference on Circuits, elling of vortex structures by probabilistic Systems, Communications and Computers, methods 5–7: Scandinavian Workshop on Algorithm Vouliagmeni, Greece Organisers: the probability teams of the Theory, Bergen, Norway Information: University Paris 13 and the University of Information: Web site: http://www.softlab.ntua.gr/~mastor/ Barcelona e-mail: [email protected] cscc2000.htm Information: Web site: http://www.ii.uib.no/swat2000 Web site: 5–8: Ordinal and Symbolic Data Analysis 13–14: Computational Challenges for the http://orfeu.mat.ub.es/~gaesto/fluid.htm (OSDA 2000), Brussels, Belgium Millennium, Cambridge, UK Information: Information: 3–7: ALHAMBRA 2000, Granada, Spain Web site: http://www.ulb.ac.be/sciences/ Web site: http://www.ima.org.uk/mathematics/ Information: contact ALHAMBRA 2000 ulbmath/osda2000 confmillennium.htm Conference eurocongres Avda. Constitución, 18 [For details, see EMS Newsletter 35] [For details, see EMS Newsletter 35] - Blq.4 E-18012 - Granada, Spain tel: (+34)-958-209-361, fax: (+34)-958-209-400 6–8: 6th Barcelona Logic Meeting, Barcelona, 16–21: Advanced Research Workshop on e-mail: [email protected], eurocongres Spain Finite Geometries, UK @mx3.redestb.es Information: Information: Web site: http://www.ugr.es/local/alhambra 2000 e-mail: [email protected] Web site: [For details, see EMS Newsletter 34] Web site: http://www.maths.susx.ac.uk/Staff/JWPH/ 28 EMS June 2000 CONFERENCES Programme and invited speakers: the School 17–20: IUTAM Symposium 2000/10 29–4 August: Curves and Abelian Varieties is in two parts, and participants may attend Diffraction and Scattering in Fluid Mechanics over Finite Fields and their Applications, either or both. and Elasticity, Manchester, UK Anogia, Crete 4–5 August: instructional course on new numer- Information: contact Professor David [For details, see EMS Newsletter 35] ical methods for Navier-Stokes equations, Abrahams, Department of Mathematics, organised by J. Miller (Dublin) and G. Shishkin University of Manchester, Oxford Road, 30–5 August: ICOR 2000, Innsbruck, Austria (Ekaterinburg) with A. Hegarty (Limerick) and Manchester M13 9PL, UK, E. O’Riordan (Dublin City). 6–9 August: work- tel: (+44)-161-275-5901, 31–3 August: 3rd Conference of Balkan shop with invited lectures from J. R. King fax: (+44)-161-275-5819 Society of Geometers, Bucharest, Romania (Nottingham), V. Entov (Moscow), A. R. Davies e-mail: [email protected] Information: contact V. Balan, University (Aberystwyth), P. G. Drazin (Bristol), P. Web site: http://www.keele.ac.uk/depts/ma/iutam/ Politehnica of Bucharest, Department of Clarkson (Kent), B. Straughan (Glasgow), E. [For details, see EMS Newsletter 33] Mathematics I, Splaiul Independentei 313, RO- Mansfield (Kent), A. D. Craik (St. Andrews), P. 77206, Bucharest, Romania, Lynch (Met Eireann), L. Crane (Dublin). 17–21: Colloquium on Semigroups, Szeged, fax: (+401) 411.53.65 Lectures are scheduled for mornings and early Hungary e-mail: [email protected] evenings, leaving afternoons free for discus- Information: [For details, see EMS Newsletter 33] sions and interactions e-mail: [email protected] Language: English 31–4 August: Numerical Modelling In Organisers: Alastair Wood (Dublin City 17–21: 9th International Conference on Continuum Mechanics (Theory, Algorithms, University), Sir Michael Berry (Bristol) Fibonacci Numbers and their Applications, Applications), Prague, Czech Republic Sponsors: the Institute of Numerical Luxembourg-City, Luxembourg Information: Computation and Analysis. Support from the Information: e-mail: [email protected] EU under the Euro Summer School e-mail: [email protected] Web site: http://www.karlin.mff.cuni.cz/ Programme has been applied for katedry/knm/nmicm2000 Site: lectures will be held in the former 17–22: Colloquium on Lie Theory and [For details, see EMS Newsletter 35] Parochial School, in the very room where Applications, Vigo, Spain Stokes received his primary education Information: contact I Colloquium on Lie 31–4 August: Workshop on Partial Notes: because of the capacity of the scenic and Theory and Applications, E. T. S. I. Differential Equations: Thermo, Visco and historic venue, attendance is restricted to 50. Telecomunicación, Universidad de Vigo, Elasticity, Konstanz, Germany Accommodation will be in farmhouse bed-and- 36280 Vigo, Spain Information: local organiser: breakfast and self-catering house in the parish. tel: (+86) 81 21 52 // (+86) 81 24 45, [email protected] Meals will be taken communally in the fax: (+86) 81 21 16 // (+86) 81 24 01 Web site: http://www.mathe.uni-konstanz. Parochial Hall. e-mail: [email protected] de/~racke/announ/ws2000.html Deadlines: for application forms and payment, Web site: http://www.dma.uvigo.es/~clieta/ [For details, see EMS Newsletter 35] 7 July 2000 [For details, see EMS Newsletter 33] Information: contact Carmel Reid or Alastair August 2000 Wood, School of Mathematical Sciences, Dublin 17–22: International Congress on City University, Dublin 9, Ireland Mathematical Physics, London, UK 2–9: Summer School on Mathematical tel: (+353) 1 7045293, Information: Physics (emphasis on Quantum Field fax: (+353) 1 7045786 Web site: http://icmp2000.ma.ic.ac.uk/ Theory), Sandbjerg Manor, Denmark Web site: Information: http://webpages.dcu.ie/~wooda/stokes/stokes 19–26: 3rd World Congress of Non-linear Web site: http://www.maphysto.dk/events/ 2000.html Analysts (WCNA-2000), Catania, Italy e-mail: [email protected] or 2–18: Rings, Modules and Representations – [email protected] 22–28: New Mathematical Methods in Constanta 2000, Constanta, Romania Continuum Mechanics, Anogia, Crete Information: 6–9: ISSAC 2000 International Symposium on Information: Address: University of Stuttgart, Mathematisches Symbolic and Algebraic Computation, St e-mail: [email protected] Institut B/3, Pfaffenwaldring 57, 70550 Andrews, Scotland [For details, see EMS Newsletter 35] Stuttgart, Germany Information: fax: (+49)-711-685-5322 Web site: http://gap.dcs.st-and.ac.uk/issac 2000/ 23–31: ASL European Summer Meeting e-mail: [email protected] (Logic Colloquium 2000), Paris, France stuttgart.de 8–12: XVIII Nevanlinna Colloquium, Information: Web site: http://web.mathematik.uni- Helsinki, Finland e-mail: [email protected] stuttgart.de/~ovid Information: Web site: http://lc2000.logique.jussieu.fr [For details, see EMS Newsletter 35] e-mail: [email protected] [For details, see EMS Newsletter 33] Web site: http://www.math.helsinki.fi/ 24–29: SIAG OP-SF Summer School 2000, 3–5: Recent Development in the Wave Field ~analysis/NevanlinnaColloquium/ Laredo, Spain and Diffuse Tomographic Inverse Problems, [For details, see EMS Newsletter 33] Theme: orthogonal polynomials and special Edinburgh, UK functions Information: 12–17: Ninth International Colloquium on Information: e-mail: [email protected] Numerical Analysis and Computer Sciences e-mail: [email protected] or [email protected] with Applications, Plovdiv, Bulgaria 4–9: Stokes’ Millennium Summer School, Sessions: Acceleration of convergence, 24–3 August: EMS Summer School, New Skreen, County Sligo, Ireland Numerical simulation, Numerical approxima- Analytic and Geometric Methods in Inverse [Third in a series of conferences organised to tion, Numerical methods in complex analysis, Problems, Edinburgh, UK honour the life and work of Sir George Gabriel Numerical methods in linear algebra, Interval Information: contact Erkki Somersalo, Helsinki Stokes at his birthplace ‘within the sound of the arithmetic, Numerical algebraic or transcen- University of Technology, Finland Atlantic breakers’] dental equations, Mathematical programming, e-mail: [email protected] Theme: Navier-Stokes equations optimisation and variational techniques, Topics: computational methods and applica- Numerical analysis for ordinary differential 25–30: Colloquium on Differential Geometry tions, including asymptotic and perturbation equations, Numerical analysis for partial differ- and its Applications, Debrecen, Hungary methods, as well as numerical analysis. There ential equations, Computer arithmetic and Information: will also be talks on history and applications to numerical analysis, Computer aspects of Web site: http://pc121.math.klte.hu/diffgeo meteorology and engineering numerical algorithms, Parallel and distributed EMS June 2000 29 CONFERENCES algorithms, Concurrent and parallel computa- 2–9: International Conference on Topology mance computing, dual reciprocity method tions, Computer networks, Discrete mathemat- and its Applications, Ohrid, Macedonia and basis functions ics in relation to computer science, Computer Invited speakers: P. J.Collins (University of Programme: see the web site aided design, Theory of data, Programming, Oxford), J. Dydak (University of Tennessee, Organizer: Wessex Institute of Technology, Image processing, Pattern recognition, Knoxville), A. Koyama (Osaka Kyoiku Southampton, UK Communication systems, Information systems, University), Y. Lisica (Russian University of Sponsors: International Society of Boundary Manufacturing systems, Data base, Software Peoples Friendship, Moscow), L. Rubin Elements (ISBE) technologies, Software engineering, (University of Oklahoma, Norman), J. M. R. Deadlines: submit papers as soon as possible Applications in mechanics, physics, chemistry, Sanjurjo (Universidad Complutense, Madrid), Note: Wessex Institute of Technology is also biology, technology and economics M. Scheepers (Boise State University, Idaho) organising BETECH 2001 – 14th International Chairman of the organising committee: Programme committee: Prof. Yuri Lisica Conference on Boundary Element Technology, Professor Drumi Bainov, P.O. Box 45, Sofia (Moscow), Prof. Jose Sanjurjo (Madrid), Prof. 12–14 March 2001 in Orlando, Florida, USA. 1504, Bulgaria Nikita Shekutkovski Abstracts should be submitted as soon as possi- tel: (+359) 2437343, Local organising committee: Nikita ble. See the web site at http://www. fax: (+359) 29879874 Shekutkovski (president), Biljana Krsteska, wessex.ac.uk/conferences/2001/ e-mail: [email protected] Liljana Babinkostova Information: contact Susan Hanley, Call for papers: abstracts of 300-500 words Conference Secretariat, BEM 22, Wessex 17–3 September: EMS Summer School in should be sent no later than 1 July to the mail- Institute of Technology, Ashurst Lodge, Probability Theory, Saint-Flour, Cantal, ing address of the conference: ICTA2000, Ashurst, Southampton SO40 7AA, UK France Institute of Mathematics, Faculty of Natural tel: (+44) 238 029 3223, Information: contact: P. Bernard, Laboratoire Sciences and Mathematics St.Cyril and fax: (+44) 238 029 2853 de Mathematiques Appliquées, Univ. Blaise Methodius University, Skopje, Macedonia, or e-mail: [email protected] Pascal, F-63177 Aubiere, by e-mail to: Web-site: http:/www.wessex.ac.uk/ tel/fax: (+33) 4 73 40 70 64 [email protected]. If the conferences/2000/bem22/ e-mail: [email protected] abstract is sent electronically in TeX format, [For details, see EMS Newsletter 34] then it should also be sent in compiled form (as 4–6: Mathematics of Surfaces, Cambridge, a .jep file ready for printing) or printed UK 19–25: Discrete and Algorithmic Geometry, Sponsor: Faculty of Natural Sciences and Information: Anogia, Crete Mathematics at St. Cyril and Methodius Web site: http://www.ima.org.uk Information: University, Skopje, Macedonia e-mail: [email protected] Information: 4–8: FGI2000 French-German-Italian [For details, see EMS Newsletter 35] Web site: http://www.pmf.ukim.edu.mk/mathe- Conference on Optimisation, Montpellier, matics/ icta2000.html France 20–23: 3rd International Workshop on Information: contact: Bernard Lemaire, Scientific Computing in Electrical 3–6: 31st European Mathematical Psychology Mathématiques, Université de Montpellier II, Engineering SCEE–2000, Warnemünde, Group Meeting (EMPG 2000), Graz, Austria Place Eugène Bataillon, 34095 Montpellier Germany Information: cedex 05 Information: e-mail: [email protected] e-mail: [email protected] Web site: http://www.SCEE-2000.uni- rostock.de Web site: Web site: http://www.math.univ-montp2.fr/ http://psyserver.kfunigraz.ac.at/empg2000/ [For details, see EMS Newsletter 34] 21–25: International Association for [For details, see EMS Newsletter 35] Mathematics and Computers World Congress 4–15: Spatial Structures in Biology and (IMACS 2000), Lausanne, Switzerland 3–10: Noncommutative Geometry, Taranto, Ecology: Models and Methods, A Information: contact Prof. Robert Owens, Italy Biomathematics Summer School, Taranto, IMACS Congress 2000, DGM-IMHEF-LMF, Italy Swiss Federal Institute of Technology, CH- 4–6: BEM 22 – 22nd International Information: 1015 Lausanne, Switzerland Conference on the Boundary Element Web site: tel: (+41)-21-693.35.89, Method, Cambridge, UK http://www.mat.unimi.it/~miriam/ESMTB/ fax: (+41)-21-693.36.46 Aim: major new advances are made in the MARTINA-SS/summer_school.html e-mail: [email protected] boundary element field each year and this con- Web site: http://imacs2000.epfl.ch ference provides a way of disseminating the 5–7: Quantitative Modelling in the [For details, see EMS Newsletter 32] current research within the international scien- Management of Health Care, Salford, UK tific and engineering community. The world Information: 27–1 September: 9th Summer St Petersburg conference on boundary element methods Web site: http://www.ima.org.uk/ Meeting in Mathematical Analysis, St originated in 1978 and continues to act as the mathematics/conferences.htm Petersburg, Russia main focus for major developments in the tech- [For details, see EMS Newsletter 34] Information: nique. The proceedings are always published in Web site: www.pdmi.ras.ru/EIMI/2000/ book form and constitute a permanent record 5–8: Brno Colloquium on Differential and analysis9/index.html of the development of the method over the last Difference Equations, Brno, Czech Republic two decades Topics: qualitative theory of differential and 30–2 September: Innovations in Higher Topics: dynamics and vibrations, fracture difference equations and applications Education 2000, Helsinki, Finland mechanics and fatigue, inelastic problems, Aim: to bring together people working in the Information: composite materials, plates and shells, contact above mentioned areas e-mail: [email protected] mechanics, geomechanics, material processing Main speakers: O. Dosly (Czech Rep.), I. Gyori Web site: http://www.helsinki.fi/inno2000 and metal forming, soil and soil structure prob- (Hungary), L. Gorniewitz (Poland), T. Kusano lems, electrostatics and electromagnetics, bio- (Japan), I. Kiguradze (Georgia), M. Marini September 2000 mechanics, fundamental principles, computa- (Italy), S. Schwabik (Czech Rep.) tional techniques, advanced formulations, Programme: invited survey lectures, short com- 1–4: Constantin Caratheodory Congress, refinement and adaptive techniques, sensitivity munications, posters and extended abstracts Evros, Greece analysis and shape optimisation, inverse prob- Language: English Scope: measure theory, function theory, partial lems, applications in optimisation, industrial Programme committee – advisory board: P. differential equations and their applications applications, thermal problems, fluid dynamics, Drabek, J. Jaros, I. Kiguradze, F. Neuman, S. Information: fluid flow, groundwater flow, interfacial and Schwabik e-mail: [email protected] free surface flow, transport problems, wave Organizing committee: M. Bartusek (chair- propagation problems, acoustics, high perfor- man), Z. Dosla, O. Dosly, A. Lomtatidze, J. 30 EMS June 2000 CONFERENCES Vosmansky Web site: http://www.ima.org.uk/mathematics/ will become available at http://at.yorku.ca/cgi- Proceedings: to be published in 2001 conferences.htm bin/amca/cadq-01 Abstracts: will be distributed at the beginning [For details, see EMS Newsletter 34] Programme committee: T. Banchoff (USA), J. of the Colloquium P. Bourguignon (France), S. Donaldson Site: Santon hotel in the beautiful area of the 12–15: Imaging and Digital Image (England), J. Eells (England), S. Gindikin Brno dam lake, available by city transport Processing: Mathematical Methods, (USA), M. Gromov (France), O. Kowalski Conference fee: USD 170 (before 31 July), Algorithms and Applications, Leicester, UK (Czech Republic), M. Mezzino (USA), S. USD 190 (later); this covers the Registration fee Information: Novikov (USA), M. Pinsky (USA), A. Ros (usual conference expenses, proceedings and Web site: http://www.ima.org.uk/mathematics/ (Spain), S. Salamon (England), L. Vanhecke social programme) and the package of board conferences.htm (Belgium), J. Wolf (USA) and lodging for the whole period of the confer- [For details, see EMS Newsletter 34] Organising committee: M. Fernández (chair- ence man, Spain), L. C. de Andrés (Spain), L. A. Deadline: for abstracts and registration, 15 12–15: IWOTA – Portugal 2000 International Cordero (Spain), A. Ferrández (Spain), R. June 2000 (later applications may be accepted) Workshop on Operator Theory and Ibáñez (Spain), M. de León (Spain), M. Macho- Information: Applications, Faro, Portugal Stadler (Spain), A. Martinez Naveira (Spain), L. e-mail: [email protected] Information: contact F.-O. Speck, Ugarte (Spain) Web site: http://www.math.muni.cz/cdde Departamento de Matematica, Instituto Sponsors (provisional): Universidad del Paìs, Superior Tecnico, U.T.L., 1049-001 Lisboa, Vasco-Euskal Herriko Unibertsitatea, Aula 5–16: Advanced Course on Algebraic Portugal Documental de Investigaciòn, Ayuntamiento de Quantum Groups, Bellaterra, Spain tel: +351-21-8417095, Bilbao-Bilboko Udala, Bilbao Iniciativas Information: fax: +351-21-8417598, Turisticas, Burdinola, Canon, Casco Viejo, e-mail: [email protected] e-mail: [email protected] Codorniù, Colegio Mayor Miguel Unamuno, Web site: http://crm.es/quantum Web site: http://www.ualg.pt/cma/iwota Deia, European Mathematical Society, [For details, see EMS Newsletter 35] Euskaltel, La Fortaleza, Gobierno Vasco-Eusko 10–17: Summer School on Geometry of Jaurlaritza, Iberia, Iparlat-Kaiku, Jagoleak, Quiver-Representations and Preprojective 13–15: International Conference of the Royal Kodak, Kukuma, Metro Bilbao, El Mundo, Algebras, Isle of Thorn, UK Statistical Society, Reading, UK Norbega Coca-Cola, Panda Software, Radio Information: contact Karin Erdmann, Information: Nerviòn-Telebilbao, Radio Popular, Real Mathematical Institute, University of Oxford, e-mail: [email protected] Sociedad Bascongada de Amigos del Paìs, Real Oxford OX1 3LB, UK Sociedad Matemàtica Espaõla, Seur, Sociedad e-mail: [email protected] 15–18: Physical Interpretations of Relativity Estatal España Nuevo Milenio, Staedtler, Web site: http://www.mathematik.uni- Theory, London, UK Urrestarazu, Wolfram Research Inc. bielefeld.de/~sek/summerseries.html Information: Proceedings: to be published [For details, see EMS Newsletter 34] e-mail: [email protected] Site: the buildings of Facultad de Ciencias Economicas y Empresariales (Avenida 11–13: Optimal Discrete Structures and 18–22: International Data Analysis Lehendakari Agirre 83, Bilbao) Algorithms – ODSA 2000, Rostock, Germany Conference, Innsbruck, Austria Grants: probably support for participants from Scope: the emphasis will be on the interactions Information: countries in a difficult economic situation and between several aspects of discrete mathemat- e-mail: [email protected] young mathematicians ics, mathematical optimisation, and theoretical Web site: Deadlines: for registration, 30 June; for computer science http://www.statistik.tuwien.ac.at/ida2000/ abstracts, 31 May Topics: combinatorial optimisation and algo- Information: rithms on discrete structures, extremal prob- 18–23: International Congress on Differential e-mail: [email protected] lems in posets, design theory, coding theory, Geometry, Bilbao, Spain Web site: www.ehu.es/Gray cryptography, graph theory, complexity theory [in memory of Alfred Gray (1939-98)] Invited speakers: C. J. Colbourn (Burlington), [some details have changed since the announcement 18–27: 8th Workshop on Stochastic and P. Frankl (Tokyo), J. R. Griggs (Columbia), C. in EMS Newsletter 35] Related Fields, Famagusta, North Cyprus T. Hoang (Thunder Bay), D. Kreher Theme: differential geometry Information: (Houghton), A. Pott (Magdeburg), J. P. Spinrad Topics: special Riemannian manifolds, homo- Web site: http://mozart.emu.edu.tr/workshop (Nashville), I. Wegener (Dortmund), G. Ziegler geneous spaces, complex structures, symplectic (Berlin) manifolds, geometry of geodesic spheres and 19–22: Fractal Geometry: Mathematical Programme: nine invited plenary lectures (55 tubes and related problems, geometry of sur- Techniques, Algorithms and Applications, minutes) and contributed talks (25 minutes) in faces, computer graphics in differential geome- Leicester, UK parallel sessions try and Mathematica Information: Organisers: A. Brandstädt, K. Engel, H.-D. Main speakers: T. Banchoff (USA), R. Bryant Web site: http://www.ima.org.uk/mathematics/ Gronau, R. Labahn (Rostock) (USA), H. Ferguson (USA), T. Friedrich confractalgeometry.htm Proceedings: a special issue of Discrete Applied (Germany), K. Grove (USA), S. Gindikin (USA), [For details, see EMS Newsletter 34] Mathematics with papers related to the confer- A. Huckleberry (Germany), D. Joyce (England), ence and refereed according to the usual pro- M. Mezzino (USA), V. Miquel (Spain), E. Musso 19–22 SCAN 2000: 9th GAMM-IMACS cedures (Italy), R. Palais (USA), M. Pinsky (USA), A. Ros International Symposium on Scientific Information: (Spain), D. Sullivan (USA), I. Taimanov Computing, Computer Arithmetic and e-mail: [email protected] (Russia), J. Wolf (USA) Validated Numerics, Karlsruhe, Germany Web site: http://www.math.uni-rostock.de/odsa Programme: plenary lectures, a video confer- Information: ence, oral communications, a poster session, Web site: http://www.scan2000.de/ 11–14: International Colloquium in Honour and a round table discussion “Where does the of Professor Michel Mendès, Bordeaux, geometry go? A research and educational per- 22–23: International Colloquium on France spective” Mathematics and Art, Maubeuge, France Information: Language: English Aims: (1) to allow to artists to present works Web site: http://www.math.u-bordeaux.fr/ Call for papers: we are using the facilities of inspired by mathematics and/or using mathe- ~stan/Colloque/MMF.html Atlas Mathematical Conference Abstracts. If matics; (2) to permit mathematicians to present [For details, see EMS Newsletter 35] you wish to present an oral communication or a and discuss works that can inspire mathematics poster, please submit an extended (up to two (tilings, all kinds of surfaces and topological 11–15: Boundary Integral Methods: Theory pages) abstract (either plain ASCII or TeX) via objects, trajectories); (3) to debate the use of art and Applications, Bath, UK http://at.yorku.ca/cgi-bin/amca/ submit/cadq-01. as a tool to help the teaching of mathematics, Information: Abstracts accepted by the organising committee and to compare experiences and projects EMS June 2000 31 CONFERENCES Advisory committee: R. Brown (Bangor), J. (Oxford); Rick W. Sharpe (Toronto), A Survey of Sciences and Economics, Levico Terme Bochnak (Amsterdam), C.-P. Bruter (Paris 11), Infinite Dimensional Lie Groups (Trento), Italy M. Chaves (Porto), M. Emmer (Roma 1), T. C. Information: Aim: this conference is the seventh of a series Kuo (Sydney), Richard Palais (Brandeis), V. Web site: devoted to evolution equations. The purpose is Poenaru (Paris 11) http://fisica.ist.utl.pt/~jmourao/om/omix/ to bring together experts in the field of evolu- Proceedings: to be published om00b.html tion equations in order to discuss the state of Information: contact C. P. Bruter, the art of the theory and the specific applica- fax: 01 60 92 51 43 October 2000 tions to different classes of problems arising e-mail: [email protected] from applications 7–10: International Conference on Topics: ranging from functional analytic meth- 22–27: EURESCO Conference on Geometry, Mathematical Modelling and Computational ods for partial differential equations to specific Analysis and Mathematical Physics: Analysis Experiments (ICMMCE), Dushanbe, problems arising in applications, with particu- and Spectral Theory, San Feliu de Guixols, Tajikistan lar emphasis on non-linear evolution equations, Spain Information: semigroup methods and partial differential Information: Web site: http://www.tajnet.com/ equations, stochastic evolution equations, appli- http://www.esf.org/euresco/00/pc00127a.htm cations to mathematical physics and industry, [For details, see EMS Newsletter 35] 15–21: DMV Seminar on the Riemann Zeta applications to mathematical biology Function and Random Matrix Theory, Speakers: S. Albeverio , H. Amann, S. Anita, J. 22–27: EURESCO Conference on Number Mathematisches Institut Oberwolfach, Antoniou, W. Arendt , J. P. Aubin , A. V. Theory and Arithmetical Geometry: Motives Germany Balakrishnan, V. Barbu, P. W. Bates, C. Batty, and Arithmetic, Obernai (near Strasbourg), Information: Prof. Dr Matthias Kreck, Ph. Benilan , P. Cannarsa , J. Van Casteren, Ph. France Universität Heidelberg, Mathematisches Clement, T. Coulhon , M. Crandall, G. Da Organiser: U. Jannsen (Regensburg) Institut, Im Neuenheimer Feld 288, 69120 Prato, M. Demuth , O. Diekmann, G. Dore, J. Information: available from Heidelberg, Germany Escher, A. Favini, G. Goldstein, J. Goldstein, M. Web site: http://www.esf.org/euresco Gyllenberg, K. Hadeler , M. Iannelli, K. Kunish 15–21: DMV Seminar on Motion by , M. Langlais , A. Lasota, Y. Latushkin, S. O. 22–28: 4th International Conference on Curvature, Mathematisches Institut Londen, G. Lumer, V. P. Maslov, J. Milota, E. Functional Analysis and Approximation Oberwolfach, Germany Mitidieri, R. Nagel, J. van Neerven, F. Theory (4th FAAT), Acquafredda di Maratea , Information: Prof. Dr Matthias Kreck, Neubrander, E. Obrecht , S. Oharu, B. de Potenza, Italy Universität Heidelberg, Mathematisches Pagter, I. Prigogine, J. Pruss, A. Pugliese, S. Information: contact any of the organising Institut, Im Neuenheimer Feld 288, 69120 Romanelli, M. Rockner, W. Schappacher, B. W. committee or visit the website Heidelberg, Germany Shultze, G. Simonett, J. Sprekels, Th. Sturm, K. Web site: Taira, L. Tubaro, A. Valli, V. Vespri , G. Webb, http://www.dm.uniba.it/maratea/FAAT2000. 20–22: One Hundred Years of the Journal L. Weis, A. Visintin, Y. Zabczyk htm L’Enseignement Mathématique, Geneva, Programme: mainly morning and afternoon [For details, see EMS Newsletter 35] Switzerland sessions; also a poster session and opportunity Topics: on the occasion of the centennial of for workshops 25–27: International Workshop on Automated the journal L’Enseignement Mathématique (a lead- Scientific committee: S. Albeverio (Bonn), H. Deduction in Geometry, Zurich, Switzerland ing journal in the internationalization of math- Amann (Zurich), W. Arendt (Ulm), P. Benilan Information: ematics education at the beginning of the 20th (Besancon), M. Crandall (Santa Barbara), G. Da e-mail: [email protected] century), it was felt appropriate to hold a sym- Prato (Pisa), J. Goldstein (Memphis), G. Lumer posium with the aims of looking at the evolu- (Mons and Brussels), F. Neubrander (Baton 26–30: XI Biannual Conference of ECMI The tion of mathematics education over the last Rouge), S. Oharu (Hiroshima), J. Prüss (Halle), European Consortium for Mathematics for century and identifying some guidelines and G. Webb (Nashville), L. Weis (Karlsrue) Industry, Palermo, Italy trends for the future, taking into account, Local organizing committee: M. Iannelli, A. Information: among other sources, the documents, debates Pugliese, L. Tubaro, A. Valli, A. Visintin Web site: http://www.ecmi.dk/ and related papers that appeared in Sponsors: European Commission, DGXII, L’Enseignement Mathématique. The emphasis of Human Potential Programme, High level 29–2 Oct: IXth Oporto Meeting On the symposium is on secondary education (stu- Scientific Conferences; C.I.R.M. (Centro Geometry, Topology And Physics, Oporto, dents in the age range of about 12 to 18 or 19 Internazionale per la Ricerca Matematica, Portugal years). In addition to proposing a reflection on Trento-Italy); I.N.D.A.M. (Istituto Nazionale di [series formerly known as the Oporto Meetings the history of mathematics education and the Alta Matematica, Roma–Italy); University of on Knot Theory and Physics] evolution of mathematics and its teaching and Trento, Mathematics Department Aim: to bring together mathematicians and learning in the 20th century, the symposium Information: contact Mr. A. Micheletti – physicists interested in the inter-relation also gives the opportunity of a gathering of Secretary of CIRM, Centro Internazionale per between geometry, topology and physics and to some of the main actors, during the last la Ricerca Matematica Istituto Trentino di provide them with a pleasant and informal decades, in mathematics education as consid- Cultura 38050 Povo (Trento) environment for scientific interchange ered from an international perspective tel: (+39)-0461-881628, Themes: this year the focus will be on differen- Organizers: ICMI, University of Geneva fax: (+39)-0461-810629 tial geometry and algebraic geometry Programme committee: Daniel Coray e-mail: [email protected] Programme: mainly short courses, of approxi- (Switzerland), Fulvia Furinghetti (Italy), Hélène Web site: http://alpha.science.unitn.it/cirm/ mately three lectures each, given by the main Gispert (France), Bernard R. Hodgson speakers, and a limited number of seminars (Canada), Gert Schubring (Germany) November 2000 (more details later). The talks are at the Information: advanced graduate or postdoctoral level, and e-mail: [email protected] 12–18: DMV Seminar on Computational should be of interest to all researchers wishing Web site: http:/www.unige.ch/math/EnsMath Mathematics in Chemical Engineering and to learn about recent developments in the over- Biotechnology, Mathematisches Institut lap between geometry, topology and physics 30–3 November: Clifford Analysis and its Oberwolfach, Germany Main speakers: Maks. A. Akivis (Jerusalem) Applications, NATO Advanced Research Organisers: Peter Deuflhard (Berlin), Rupert and Vladislav V. Goldberg (Newark), The Workshop, Praha, Czech Republic Klein (Potsdam/Berlin) and Christof Schütte Geometry of Submanifolds with Degenerate Gauss Information: (Berlin) Mappings (joint course); Joseph Landsberg Web site: http://www.karlin.mff.cuni.cz/ ~clifford Information: Prof. Dr Matthias Kreck, (Toulouse), Representation Theory and the Universität Heidelberg, Mathematisches Geometry of Projective Varieties; Dietmar Salamon 30–4 November: Evolution Equations 2000: Institut, Im Neuenheimer Feld 288, 69120 (Zurich), Floer Homology; Simon Salamon Applications to Physics, Industry, Life Heidelberg, Germany 32 EMS June 2000 CONFERENCES 12–18: DMV Seminar on Characteristic Karl-Olof.Lindahl”@msi.vxu.se (USA), Orthogonal designs and space-time codes for Classes of Connections, Riemann-Roch Web site: http://www.msi.vxu.se/aktuellt/ wireless communication; L. A. Goldberg (UK), Theorems, Analogies with e-Factors, konferenser.html Sampling and counting unlabelled structures; B. Mathematisches Institut Oberwolfach, Mohar (Slovenia), Graphs on surfaces and graph Germany December 2000 minors; M. S. O. Molloy (Canada), Graph colour- Organisers: Spencer Bloch (Chicago) and ing with the probabilistic method; J. G. Oxley Helene Esnault (Essen) 18–20: 5th International Conference on (USA), The interplay between graphs and matroids; Information: Prof. Dr Matthias Kreck, Mathematics in Signal Processing, Coventry, J. A. Thas (Belgium), Ovoids; spreads and m-sys- Universität Heidelberg, Mathematisches UK tems of finite classical polar spaces; D. R. Woodall Institut, Im Neuenheimer Feld 288, 69120 Information: contact Pamela Bye, The Institute (UK), List colourings of graphs Heidelberg, Germany of Mathematics and its Applications, Catherine Programme: hour-long talks by the main Richards House, 16 Nelson Street, Southend- speakers and parallel sessions with 20-minute 27–1 December: Foundations of Probability on-Sea, Essex SS1 1EF, England talks on any relevant topic and Physics, Vaxjo, Sweden fax: (+44) 1702 354111 Call for papers: see the web site – full version Aim: to reconsider foundations of probability e-mail: [email protected] up in July 2000 theory in connection with foundations of [For details, see EMS Newsletter 34] Organizers: J. W. P. Hirschfeld, R. P. Lewis physics (quantum as well as statistical) Sponsor: British Combinatorial Committee Topics: the following problems will be dis- January 2001 Proceedings: papers based on the main talks cussed: will be published in a book available at the start (A) Creation of the conventional probability 28–3 February: 2001 XXI International of the conference; the publisher is probably theory: Kolmogorov’s axiomatics, 1933 (Why Seminar on Stability Problems for Stochastic Cambridge University Press. Papers based on and How?). Is probability a measure or a fre- Models, Eger, Hungary other talks can be submitted to the further pro- quency? A role of ergodicity. Why did [Dedicated to the 70th jubilees of M. Arató ceedings, probably to be published in a volume Kolmogorov’s model become dominating in (Debrecen, Hungary) and V. M. Zolotarev of Discrete Mathematics mathematics and physics? (Moscow, Russia), Honorary Chairmen of the Site: University of Sussex, (B) Creation of probabilistic foundations of Seminar] http://www.sussex.ac.uk quantum mechanics: Is quantum probability a Topics: limit theorems in probability theory; Deadline: for submission of abstracts, 30 April measure, frequency, potentiality or degree of characterisations of probability distributions 2001 my belief? The great diversity of opinions. Why and their stability; theory of probability metrics; Information: does everybody use finally Kolmogorov’s mea- stochastic processes and queueing theory; actu- e-mail: [email protected] Web sites: sure-theoretical approach? arial and financial mathematics; applications in http://www.maths.susx.ac.uk/Staff/JWPH/ (C) Probability in statistical physics, The role of informatics and computer sciences; applied sta- http://hnadel.maps.susx.ac.uk/TAGG/Confs/ ergodicity, Individual and statistical ergodicity. tistics BCC/index.html (D) Einstein-Podolsky-Rosen paradox, Bell’s Language: English inequality, locality, reality, probability. Programme committee: Chairmen: Z. Daróczy August 2001 Conventional probabilities and Bell’s inequality and G. Pap (Debrecen, Hungary), V. Yu. (via Bell, Clauser, Horne, Shimony, Holt, Korolev (Moscow, Russia). Members: A. 5–18: Groups St Andrews 2001, Oxford, Aspect, Home, Selleri, d’Espagnat, Fine, Rastal, Balkema (Amsterdam), A. Benczúr (Moscow), England Greenberger, Zeilinger,…). Thermodynamical V. E. Bening (Moscow), A. V. Bulinski [Sixth in the series of Groups St Andrews approach (via de Broglie, Lochak, de Muynck). (Moscow), L. Csôke (Eger), S. Csörgô (Szeged), Conferences] Quantum fluctuations, stochastic dynamics. I. Fazekas (Debrecen), J. Fritz (Budapest), V. V. Speakers: M. D. E. Conder (Auckland), P. W. Von Mises frequency approach and Bell’s Kalashnikov (Moscow), J. Kormos (Debrecen), Diaconis (Stanford), P. P. Palfy (Budapest), M. inequality. Fluctuating probabilities, modified V. M. Kruglov (Moscow), P. Major (Debrecen), P. F. du Sautoy (Cambridge), M. R. Vaughan- Bell’s inequality. Quantum probabilities. G. Michaletzky (Budapest), J. Mijnheer Lee (Oxford) Negative probabilities. p-adic probabilities. p- (Leiden), T. Móri (Budapest), E. Omey Programme: short courses of lectures given by adic model of fluctuating reality. (E) Quantum (Brussels), A. Plucinska (Warsaw), L. Szeidl the speakers above, and a full programme of computers and information and probability. (Pécs), M. van Zuijlen (Nijmegen) one hour invited lectures and short research (F) Paradoxes of probability theory (in particu- Secretaries of the organising committee: A. presentations lar, violations of the law of large numbers or V.Kolchin (Moscow), S. Baran (Debrecen) Topics: all aspects of group theory. The short ergodicity). Randomness, complexity and foun- Sponsors (provisional): Steklov Mathematical lecture courses are intended to be accessible to dations of probability theory and physics Institue of the Russian Academy of Sciences postgraduate students, postdoctoral fellows, Invited Speakers: S. Albeverio (Bonn Univ.), (Moscow), Institute of Mathematics and and researchers in all areas of group theory H. Atmanschpaher (Inst. of Psychology, Informatics of the Universtiy of Debrecen Accommodation: is booked at St Anne’s and Freiburg), L. Ballentine (Simon Fraser), J. (Hungary), Károly Eszterházy College of Somerville Colleges, Oxford, England. Lectures Bricmont (FYMA), A. Holevo (Steklov Math. Education (Eger) will be held in the Mathematical Institute, Inst.), S. Gudder (Denver Univ.), P. Lahti Proceedings: to be published in the Journal of Oxford (Turku Univ.), T. Maudlin (Rutgers), A. Mathematical Sciences Scientific organising committee: Colin Shimony (Boston Univ.), J. Summhammer Deadline: for registration and for abstracts, 15 Campbell (St Andrews), Patrick Martineau (Atominst. Wien) October 2000 (Oxford), Peter Neumann (Oxford), Edmund Organisers: L. Accardi (Rome), W. De Muynck Information: Robertson (St Andrews), Geoff Smith (Bath), (Eindhoven), T. Hida (Meijo University, e-mail: [email protected] or Brian Stewart (Oxford), Gabrielle Stoy (Oxford) Japan), A. Khrennikov (Sweden), V. Maximov [email protected] Site: University of Oxford (Belostok Univ., Poland), L. Accardi (Univ. Web site: Information: Groups St Andrews 2001, Rome-2, Italy), W. De Muynck (Univ. of http://neumann.math.klte.hu/ ~stabil Mathematical Institute, North Haugh, St Eindhoven, the Netherlands), T. Hida (Meijo http://bernoulli.mi.ras.ru Andrews, Fife KY16 9SS, Scotland University, Japan), A. Khrennikov (Växjö e-mail: [email protected] University, Sweden), V. Maximov (Belostok July 2001 Web site: http://www.bath.ac.uk/~masgcs/gps01/ Univ., Poland) Organising institution: International Center 1–6: Eighteenth British Combinatorial 27–31: Equadiff 10, Czechoslovak for Mathematical Modeling [in physics, tech- Conference, Brighton, United Kingdom International Conference on Differential nique and cognitive sciences], Vaxjo University, Aim: the promotion of all branches of combi- Equations and their Applications, Prague, Sweden natorics within the UK and around the world Czech Republic Deadline: 1 October 2000 Main speakers: M. Aigner (Germany), The Information: Information: Penrose polynomial of graphs and matroids; I. Web site: http://www.math.cas.cz/ e-mail: send abstracts and application forms to Anderson (UK), Cyclic designs; A. R. Calderbank EMS June 2000 33 RECENT BOOKS contrast to the mathematics that the reader may have learnt at school. The book is dedicated to Martin Gardner, the famous author of the RecentRecent booksbooks Mathematical Games column in Scientific edited by Ivan Netuka and Vladimír Sou³ek American for many years. The name of this mathemagician and pied puzzler guaran- tees that the book will bring enjoyment to Books submitted for review should be sent to line of the Brelot and Bauer axiomatic a wide range of readers. (ec) the following address: potential theory. Further chapters are Ivan Netuka, MÚUK, Sokolovská 83, 186 75 devoted to a deeper study of conic mea- C. Cercignani and D. H. Sattinger, Praha 8, Czech Republic. sures, zonoforms, adapted spaces and Scaling Limits and Models in Physical other subjects. Processes, DMW Seminar 28, Birkhäuser, A. Balog, G. O. H. Katona, A. Recski and It would have been useful if the author Basel, 1998, 190 pp., DM58, ISBN 3-7643- D. Szász (eds.), European Congress of had also mentioned the notable results of 5985-4 and 0-8176-5985-4 Mathematics, Budapest, 1996, Vols. I, II, N. Boboc and A. Cornea from the theory This book, written by experts on their Progress in Mathematics 168, 169, of lower semicontinuous cones, or the research areas, is in two parts. In the first Birkhäuser, Basel, 1998, 334 and 402 pp., results of J. Bliedtner and W. Hansen con- part, Scaling and Mathematical Models in each DM159, ISBN 3-7643-5497-6 and 3- cerning the simpliciality of certain cones of Kinetic Theory, Carlo Cercignani provides a 7643-5496-8 superharmonic functions in potential the- well-written quick introduction to pertur- The tradition of European Congresses of ory. bation and scaling methods in the mathe- Mathematics started in 1992 in Paris and This book is readable by anybody with a matical theory of the Boltzmann equation. from the beginning they have been major basic knowledge of functional analysis. (jl) The second part, Scaling, Mathematial events in European mathematical life. Modelling, and Integrable Systems, was writ- These volumes present the second one in M. Berger, Riemannian Geometry During ten by David Sattinger, and presents an Budapest in 1996. The main lectures were the Second Half of the Twentieth Century, introduction to dispersive waves and a given by N. Alon, Randomness in discrete University Lecture Series 17, American derivation of the associated completely mathematics; D. McDuff, Symplectic topology; Mathematical Society, Providence, 2000, 182 integrable systems, as the Korteweg de J. Kollár, Low degree polynomial equations; A. pp., US$34, ISBN 0-8218-2052-4 Vries and non-linear Schrödinger equa- S. Merkurjev, K-theory and algebraic groups; This is a new edition of the survey which tions, via a careful treatment of the inverse V. Milman, High-dimensional convexity theo- appeared for the first time as Jahresbericht scattering theory for the Schrödinger ry; and St. Müller, Microstructures, geometry der Deutschen Mathematiker-Vereinigung, operator. and the calculus of variations. There were ten Band 100, Heft 2, 1998. It starts with a his- Both parts contain interesting historical prize winners announced at the Congress, torical review devoted to Riemannian comments and a huge bibliography. The some of whose lectures can be found in geometry up to 1950 ‘from Gauss and book can be recommended to non-special- these volumes: W. T. Gowers, Geometry of Riemann to Chern’. The next six chapters ists in mathematics, physics and engineer- Banach spaces; A. Huber, Extensions of cover the following basic topics: pinching ing who desire a quick insight into these motives; D. Kramkov, Statistics and mathe- problems, curvature (sectional, Ricci, areas. (jmal) matics of finance; J. Matoušek, Geometric set scalar) of a given sign, finiteness, compact- systems; and L. Polterovich, Symplectic geom- ness, collapsing, space forms, Einstein S. B. Cooper and J. K. Truss (eds.), Sets etry. The volumes conclude with records of manifolds, the Yamabe problem, spectra and Proofs and Models and Computability, five round tables during the Congress and eigenfunctions, period geodesics, geo- London Mathematical Society Lecture Notes (electronic literature; mathematical desic flow, volumes, systols, holonomy Series 258, 259, Cambridge University Press, games; women and mathematics; public groups, Kähler manifolds, spinors, har- Cambridge, 1999, 436 and 419 pp., each image of mathematics; education). Written monic maps, generalisations of £29.95, ISBN 0-521-63549-7 and 0-521- versions of invited lectures in both volumes Riemannian geometry, submanifolds and 63550-0 provide a very good overview of recent geometric measure theory. These two volumes contain articles by substantial progress in a broad variety of Because the survey is devoted almost invited speakers at Logic Colloqium ’97 (the mathematical fields and should be of inter- exclusively to global aspects of major international meeting of the est to any working mathematician. (vs) Riemannian geometry, the Riemannian Association of Symbolic Logic); all authors manifolds are usually supposed to be com- are specialists in their respective fields. In R. Becker, Cônes convexes en analyse, pact, or at least complete. The exposition these volumes, the reader can obtain com- Travaux en cours 59, Hermann, Paris, 1999, is systematic and careful, yet also elegant prehensive information about the current 245 pp., ISBN 2-7056-6384-3 and charming, written with a pedagogical state of research in almost all branches of At the end of the 1950s, Gustav Choquet mastery that is characteristic of the author. mathematical logic. developed his theory of integral represen- The book includes a short introduction, The first volume contains papers from a tations for compact convex subsets in local- comments on the main topics, 33 pages of large variety of problems concerning sets ly convex spaces, and also for locally com- bibliography, and subject and author and proofs. We mention here ordinal pact convex cones. Later he published sev- indices. (ok) analysis, forcing large cardinals, covering eral papers generalising his theory to the properties concerning sets and construc- case of weakly complete convex cones. The E. Berlekamp and T. Rodgers (eds.), The tive logic, intuitionistic propositional logic, book contains a postlude where the master Mathemagician and Pied Puzzler, A. K. complexity of the propositional calculus, Choquet himself explains the history and Peters, Natick, 1999, 266 pp., £22, ISBN 1- mathematical truth, and infinite-time the value of the theory he introduced. 56881-075-X Turing degrees concerning proofs. The author starts with basic properties This book is a collection of nearly forty As well as classical problems of model of weakly complete convex cones and articles from recreational mathematics, theory (the topological stability conjecture introduces the main objects (extremal written by foremost puzzlers, magicians or relative categoricity in abelian groups) points, hats, topologies on certain cones, and mathematicians. It contains many and computability (computability and conic measures, maximal measures, cones interesting puzzles, games, and mathemat- complexity), the reader can find in the sec- bien-coiffé, the Cartier-Fell-Meyer theo- ical problems (for example, is it possible to ond volume effective model theory, non- rem and its generalisation). He then intro- fit unit squares on an n × n sheet of graph standard models of arithmetic or logic and duces several applications of the theory paper labelled with 0s and 1s so that every decision making. (k³) (proofs of the Bochner-Weil theorem and neighbourhood is odd). What most of of Bernstein’s theorem based on the these articles have in common is that they R. G. Douglas, Banach Algebra Techniques Choquet theory) and presents a short out- show the ‘magic face’ of mathematics, in in Operator Theory, Graduate Texts in 34 EMS June 2000 RECENT BOOKS Mathematics 179, Springer, New York, 1998, variational inequality. The physical moti- ial from algebra, linear algebra and com- 194 pp., DM98, ISBN 0-387-98377-5 vation in plasticity is explained in the text. plexity theory. There is a 30-page list of This book is the second edition of the 1972 The last contribution, on degenerate ellip- references, a list of notation and an index. original version and remains unchanged, tic equations in ordered Banach spaces The monograph is a welcome survey of a except that several mistakes and typo- and applications (P. Taká³), is concerned fast-growing and topical area of modern graphical errors have been corrected. with quasi-linear elliptic boundary value mathematics and computer science. (jtrl) There is also a brief report with references problems with Dirichlet boundary condi- on the current state of the double-starred tions in bounded domains in Rn, including E. A. Gavosto, S. G. Krantz and W. open problems. the p-Laplacian. The questions of exis- McCallum (eds.), Contemporary Issues in The book begins with basic results in tence and uniqueness of solutions, Mathematics Education, Mathematical Banach space theory (main examples, lin- Fredholm alternatives, and comparison Sciences Research Institute Publications 36, ear functionals, weak topologies, the principles are studied and compared with Cambridge University Press, Cambridge, 1999, Hahn-Banach and open mapping theo- the linear (p = 2) case. Systems of equa- 174 pp., £12.95, ISBN 0-521-65471-8 and rems, Lebesgue and Hardy spaces). The tions are also studied. In the last section 0-521-65255-3 second chapter is devoted to Banach alge- the abstract part-metric method in an The problem of how to teach mathematics bras (also the Gelfand-Mazur theorem and ordered topological cone is introduced has become extremely important recently. the Gelfand transform). Further chapters and illustrated on an elliptic system. (efas) Many things have changed. Undoubtedly, explain topics in the geometry of Hilbert computers have had a serious influence on spaces and operators on Hilbert spaces J. von zur Gathen and J. Gerhard, Modern the teaching process. Further, most of our and C*-algebras (normal and self-adjoint Computer Algebra, Cambridge University students major in non-mathematical fields operators, the Gelfand-Naimark theorem, Press, Cambridge, 1999, 753 pp., £29.95, such as biology, medicine, econometrics, the spectral theorem and functional calcu- ISBN 0-521-64176-4 technology, etc. To understand their main lus, weak and strong operator topologies, This monograph is devoted to modern subject, students must sometimes know W*-algebras, the Fuglede theorem). One algorithms for solving classical algebraic some advanced parts of mathematics, such chapter is devoted to compact and problems. These algorithms form the basis as partial differential equations or the fast Fredholm operators and index theory (the of various contemporary computer algebra Fourier transform. The usual courses of Calkin algebra, the Fredholm alternative, systems, and are presented, together with mathematics are limited and the teacher representations of the C*-algebra of com- their complexity analysis, with the neces- cannot present the necessary background pact operators). The last two chapters deal sary theoretical background and accompa- with complete proofs. Even in calculus with Hardy spaces (Beurling’s theorem, nied by numerous applications. This courses for students of mathematics it may the F. and M. Riesz theorem, the maximal approach makes it possible for the reader not be possible to include all proofs at the ideal space of H8, the Gleason-Whitney to comprehend even the rather sophisti- usual level of rigour. theorem, abstract harmonic extensions, cated recent methods. Readers can also This booklet contains writings from the maximal ideal spaces) and with Toeplitz improve their understanding by solving Conference on the Future of Mathematics operators (the spectrum and invertibility some of the numerous exercises and Education at Research Universities held in of Toeplitz operators, localisation, research problems, following each chapter. Berkeley in 1996, and is divided into four Widom’s result on the connectedness of The text is divided into five parts, named parts: Mathematics education at the university, the spectrum). At the end of each chapter after the founders of the respective areas. Case studies in mathematics education, The there are notes suggesting additional read- In Part I (Euclid), the Euclidean algorithm, debate over school mathematics education, and ing and a large number of problems of var- its modular forms, and the subresultant Reports from the working groups. The papers ious difficulty. theory for polynomials are examined. stress the importance of motivation in The book is accessible to anybody with a Applications include the Chinese teaching mathematics (including its histor- basic knowledge of analysis, general topol- Remainder Algorithm, Diophantine ical background) and in the presentation ogy, measure theory and algebra. (jl) approximations, and decoding of BCH of surveys of advanced topics. It is also codes. Part II (Newton) deals mainly with important to consult teachers of other sub- P. Drábek, P. Krej³í and P. Taká³, the fast arithmetic based on the Fast jects; for example, it is quite surprising to Nonlinear differential equations, Research Fourier Transform (FFT). The Karatsuba learn of the level of interest in mathemat- Notes in Mathematics 404, Chapman & and Strassen algorithms, as well as fast ics among some biology professors, who Hall/CRC, Boca Raton, 1999, 196 pp., £30, Chinese remaindering and the fast say that they never met a differential equa- ISBN 1-58488-036-8 Euclidean algorithm, are presented. There tion in their research that could be solved The three papers in this volume are is a special chapter on the applications of analytically. The mathematical community adapted versions of a series of lectures FFT to image compression. Part III polarises on what is the appropriate bal- given by the editors at the Seminar in (Gauss) deals with polynomial factorisa- ance between theory, technique and appli- Differential Equations, Chvalatice, Czech tion. Various algorithms are presented in cations. This book presents the views of Republic, in 1998. They present the full detail (such as the Zassenhaus one and several mathematics instructors and pro- authors’ recent research in a way that is a polynomial-time algorithm for factorisa- fessors on effective mathematics teaching. also comprehensible to PhD students tion in Q[x]). The main ingredients are Some authors include useful examples to working in differential equations. The pre- Hensel lifting and short vectors in lattices. demonstrate their ideas. sentations are based on recent papers and Part IV (Fermat) presents the two basic This book is valuable and can be recom- preprints and also contain new results tasks of algorithmic number theory: test- mended to mathematicians who are willing The first contribution, on non-linear ing for primality and the factoring of inte- to use the experience and advice of their eigenvalue problems and the Fredholm gers. There is a special chapter on applica- colleagues for improving the teaching alternative (P. Drábek), studies the eigen- tions to cryptography, such as the RSA process. (ja) values of the p-Laplacian, solvability of the cryptosystem. Part V (Hilbert) deals with elliptic boundary value problem with right- three more advanced topics: the Gröbner R. E. Gompf and A. I. Stipsicz, 4- hand side, and points to the connection bases of ideals in polynomial algebras Manifolds and Kirby Calculus, Graduate with the optimal Poincaré inequality. Most (including complexity analysis of the Studies in Mathematics 20, American of this paper deals with the one-dimen- Buchberger algorithm), symbolic integra- Mathematical Society, Providence, 1999, 558 sional case. tion of rational functions, and symbolic pp., US$65.00, ISBN 0-8218-0994-6 The second text, on evolution variation- summation. As applications, Gröbner This book gives an excellent introduction al inequalities and multi-dimensional hys- proof systems and Petri nets are consid- into the theory of 4-manifolds and can be teresis operators (P. Krej³í), is devoted to a ered, as well as the determination of all strongly recommended to beginners in discussion on the influence of the geome- spatial conformations of cyclohexane. this field. It is carefully and clearly written; try of the convex constraint on analytical The monograph has an appendix sum- the authors have evidently paid great properties of the solution operator of a marising the necessary background mater- attention to the presentation of the mater- EMS June 2000 35 RECENT BOOKS ial, and the prerequisites are not too high. (the Riesz representation theorem, duals with solutions at the end of the book. A postgraduate student will probably not of Hilbert spaces, weak convergence and This is a useful book on large deviations. It have to look very often to other sources. Hilbert bases) and a brief theory of Lp- can be used as a text for advanced PhD stu- The exposition contains many really pret- spaces (also duals, the Radon-Nikodým dents with a really good background in ty and interesting examples and a great theorem, weak convergence and convolu- mathematical analysis and probability the- number of exercises; the final chapter is tion). The second chapter is dedicated to ory. (mhusk) then devoted to solutions of some of these. operator theory (operators on Banach and I think that this type of presentation makes Hilbert spaces and their spectra, opera- D. H. Hyers, G. Isac and T. M. Rassias, the subject more attractive and its study tional calculus on Hermitian operators Stability of Functional Equations in easier. We add for young mathematicians and compact self-adjoint operators, Several Variables, Progress in Nonlinear that the theory of 4-manifolds has devel- Fredholm theorems, kernel operators). A Equations and Their Applications 34, oped very quickly in the last two decades, comprehensive final part of the book is an Birkhäuser, Boston, 1998, 313 pp., DM178, and can currently be considered as a very introduction to the theory of distributions. ISBN 0-8176-4024-X and 3-7643-4024-X promising field of study. (jiva) Multiplication, differentiation and convo- This book presents the current state of lution of distributions are studied in a fair solutions of one of Ulam’s problems: is an A. Hajnal and P. Hamburger, Set Theory, amount of detail. There are chapters on ε-homomorphism of metric groups close London Mathematical Society Student Texts the fundamental solutions of a differential to some homomorphism? The affirmative 48, Cambridge University Press, Cambridge, operator, with classical examples of the answer for the Cauchy functional equation 1999, 316 pp., £16.95, ISBN 0-521-59344- Laplacian, the heat and wave operators on a Banach space has been well known for 1 and 0-521-59667-X and the Cauchy-Riemann operator. As a long time (Hyers, 1941). This question This textbook on set theory is based on applications, several regularity results – in can be extended in various ways – for courses given by András Hajnal in Eötvös particular, those concerning the Sobolev example, how does the answer depend on Loránd University in Budapest. It divides spaces W1,p(Rn) – are proved. In the final the structure of the group (without com- into two parts. part the Laplacian on an open subset of Rn mutativity the result is not true)?, under Part I, Introduction to set theory, explains is studied (the Dirichlet problem, classical what conditions is an ε-homomorphism a the basics in an almost naive way, but with and variational formulations, eigenvalues homomorphism (so-called superstability) enough rigour to give a proof, not only a of the Dirichlet Laplacian, and the heat or a homomorphism plus something claim, of (say) the result that there is no set and the wave problems). ‘small’ (stability modulo ...)? Similar ques- of all sets. All standard material for an The book is equipped with a rich collections can be asked for multiplicative func- undergraduate course is here. This part tion of exercises, with notes containing tionals on a Banach algebra, for polynomi- n ends with cardinal exponentiation, König’s examples and counter-examples, applica- al functionals on a Banach space (∆h f = theorem and the Hausdorff formula. A 40- tions to test the reader’s understanding of 0), for inequalities instead of equalities page appendix follows, where an axiomat- the text, and digressions to introduce new (e.g. ε-convex functions), for set-valued ic treatment is developed, with emphasis concepts. Answers are presented at the maps, etc. Recent results on Ulam’s other on the logical background (interpretation end. It is readable by anybody with a basic problems concerning the stability of sta- of the language, absoluteness of formulas knowledge of analysis (differential calculus tionary points or local minimum points are and relative consistency). in several variables, integration theory also described. Part II, the major part of the book, con- with respect to a measure, the basics of The authors present both classical and tains Hajnál’s beloved subject, infinitary Banach space theory and the topology of recent results clearly and in a self-con- combinatorics. Stationary sets, ∆-systems, metric spaces). It can be recommended to tained fashion. Since only a basic knowl- Ramsey’s theorem, partition calculus, set graduate students and will help more edge of algebra and functional analysis is mappings, square-bracket arrows, large advanced students and researchers required for understanding much of this cardinals, saturated ideals – all these are improve their knowledge of some parts of text, it is accessible for graduate students. presented in considerable detail. The cul- functional analysis. (jl) (jmil) mination is Shelah’s theorem on the power ℵ 2 ω. Although this is no longer a text for F. den Hollander, Large Deviations, Fields V. Ivrii, Microlocal Analysis and Precise undergraduates, the clarity of presentation Institute Monographs 14, American Spectral Asymptotics, Springer Monographs and the choice of the easiest proofs contin- Mathematical Society, Providence, 2000, 143 in Mathematics, Springer, Berlin, 1998, 731 ues. A very fine feature of the book is its pp., US$49, ISBN 0-8218-1989-5 pp., DM178, ISBN 3-540-62780-4 exercises. Each chapter contains about ten The subject of this book is large deviation The main topic of this comprehensive of them, usually demanding more than a theory. It corresponds to a graduate course monograph (more than 700 pages) con- mechanical application of theorems taught by the author at the Fields Institute cerns asymptotic formulae for the number proved in the text. Give this book fifteen for Research in Mathematical Sciences in N(λ) of eigenvalues of an operator A, which minutes’ attention – you will find that you Toronto in 1998. The book divides into do not exceed λ, and corresponding pre- must buy it! (ps) two parts. cise remainder estimates. Part A (Chapters I–V) describes the the- The first part of the monograph devel- F. Hirsch and G. Lacombe, Elements of ory, while Part B (Chapters VI–X) presents ops the necessary tools of micro-local Functional Analysis, Graduate Texts in applications. Chapters I and II contain the analysis for h pseudo-differential and h- Mathematics 192, Springer, New York, 1999, classical basic large deviation theory for Fourier integral operators, and contains 393 pp., DM98.00, ISBN 0-387-98254-7 i.i.d. random variables. Chapter III pre- the main theorems on propagation of sin- This book concerns that part of functional sents general definitions and theorems in a gularities. The second part contains semi- analysis that does not require advanced more abstract context. Chapter IV contains classical spectral asymptotics (as h → 0+) topological tools and does not cover topics a discussion of large deviations for Markov for various types of operators of their fam- like locally convex spaces or cornerstones chains, while Chapter V deals with large ilies. In particular, there are chapters of functional analysis (the open mapping deviations for random sequences with devoted to the Schrödinger and Dirac and closed graph theorems and the Hahn- moderate dependence. operators with strong magnetic field (the Banach theorem). The first part of the A number of interesting applications of coupling constant µ being another para- book is devoted to function spaces and the above general theory are presented in meter). The third part contains theorems their duals. It includes chapters on the Part B; for example, there are applications describing estimates of the spectrum which space of continuous functions on a com- to statistical hypothesis testing (Chapter are then applied in many special and inter- pact set (the Stone-Weierstrass and Ascoli VI), random walk environment (Chapter esting cases in the fourth part of the book theorems), on locally compact spaces and VII), heat conduction with random sources to get asymptotic formulae for spectrum in Radon measures (Daniell’s approach to and sinks (Chapter VIII), polymer chains corresponding cases. integration theory, Radon measures), rudi- (Chapter IX) and interacting diffusions The book is well organised, is written in ments from the theory of Hilbert spaces (Chapter X). There are many exercises, a careful and systematic way, and contains 36 EMS June 2000 RECENT BOOKS a huge amount of unpublished new mater- knowledge of calculus is required. This affine blocking sets, maximal arcs, and ial. The reader is expected to have a good book is based on an undergraduate sum- unitals). K. Metsch’s article is on Bose- background in the theory of pseudo-differ- mer course. It may provide an interesting Burton type theorems for finite projective, ential and Fourier integral operators. (vs) first reading on high analysis at an ele- affine and polar spaces. M. Dyer and C. mentary level. (efas) Greenhill focus on approximate sampling J. Jost, Riemannian Geometry and from combinatorially defined sets and Geometric Analysis, Universitext, Springer, V. Lakshmikantham and S. G. Deo: approximate counting using Markov chain Berlin, 1998, 455 pp., DM 78, ISBN 3-540- Method of Variation of Parameters for Monte-Carlo methods, viewed from the 63654-4 Dynamic Systems, Series in Mathematical perspective of combinatorial algorithms. This is the second edition of a textbook Analysis and Applications 1, Gordon and C. J. Colbourn, J. H. Dinitz and D. R. based on the material of a three-semester Breach Publishers, New Delhi, 1998, 317 pp., Stinson are concerned with applications of course given by the author at the £54, ISBN 90-5699-160-4 combinatorial designs to communication, University of Bochum. It includes material Many results on differential equations cryptography and networking; their exten- needed for the study of geometrical and whose linear parts are perturbed by (small) sive bibliography has 159 items. (jnes) analytic properties of Riemannian mani- non-linearities are based on so-called vari- folds. ation of parameters formulae. These for- G. F. Lawler and C. N. Lester, Lectures on At the beginning is a nice introduction mulae can also be extended to certain Contemporary Probability, Student to Riemannian geometry. In the first four cases where there are no adequate linear Mathematical Library 2, American chapters, all basic material for the subject parts. In this way, the method of variation Mathematical Society, Providence, 1999, 95 is collected and presented in an under- of parameters is closely related to modifi- pp., US$19, ISBN 0-8218-2029-X standable way. The fifth chapter develops cations of the Lyapunov second method This well-written booklet is based on lec- Morse theory and the sixth chapter con- (Lyapunov-like functions), comparison tures and computer labs for undergradu- tains a description of symmetric spaces, principles and monotone iteration tech- ates, held by the authors at the IAS/park offering a rich source of important exam- niques. This book brings together various City Mathematics Institute. The subjects ples in Riemannian geometry. In the sec- aspects of these procedures for ordinary include random walks (simple and self- ond part of the book, the author returns to differential, integro-differential, delay, dif- avoiding), Brownian motion, card shuf- the structure of geodesics on Riemannian ference, impulse differential, and stochas- fling, Markov chains, Markov chain manifolds using analytical tools (geodesics tic differential equations. Applications are Monte-Carlo, spanning trees, computer are treated as critical points of the energy given to stability, controllability, periodic simulation of random walks and models in functional). In addition, existence and fur- solutions, and more generally to boundary finance. The booklet is divided into 13 ther properties of harmonic maps between value problems. There are also brief sec- chapters (lectures) and includes a number Riemannian manifolds are intensively tions devoted to abstract differential equa- of exercises. The reader is assumed to be studied, with special attention to the two- tions and more special equations. This familiar with the basic elements of proba- dimensional case. book partly overlaps other books written bility. The authors try to present topics In this second edition, some new and by the first author, but has the advantage that are accessible to advanced undergrad- very interesting material is added. Seiberg- of collecting similar ideas together. (jmil) uates and to show the appeal of parts of Witten theory has recently found many modern probability, and make the lectures spectacular applications in geometry and J. D. Lamb and D. A. Preece (eds.), very attractive. To avoid long calculations the topology of manifolds. The author first Surveys in Combinatorics, 1999, London they rely either on intuition or on results of collects basic material for this theory (spin Mathematical Society Lecture Note Series 267, simulations. The emphasis is on a proba- geometry, the Dirac operator), and then Cambridge University Press, Cambridge, 1999, bilistic style of thinking. (mhusk) treats Seiberg-Witten equations as an ana- 298 pp., £24.95, ISBN 0-521-65376-2 logue of the Ginsberg-Landau equations. This volume contains the nine invited talks Tan Lei (ed.), The Mandelbrot Set, Theme Each chapter ends with appendices called presented at the 17th British and Variations, London Mathematical Society ‘Perspectives’, containing further results Combinatorial Conference, held at the Lecture Notes Series 274, Cambridge and directions for possible research. The University of Kent in 1999. A special fea- University Press, Cambridge, 2000, 365 pp., book will be very useful for readers from ture is W. T. Tutte’s article, The coming of ISBN 0-521-77476-4 many areas of mathematics. (jbu) matroids, which covers the development of This volume provides a systematic exposi- the theory of matroids from the point of tion of current knowledge concerning the R. Knobel, An Introduction to the view of one of the masters. Mandelbrot set, and presents the latest Mathematical Theory of Waves, Student The remaining eight survey articles research in complex dynamics. Topics dis- Mathematical Library 3, American deal with graph theory (4), finite geome- cussed include the universality and local Mathematical Society, Providence, 1999, 196 tries (2), probability and complexity (1) connectivity of the Mandelbrot set, para- pp., US$23, ISBN 0-8218-2039-7 and combinatorial designs (1). C. bolic bifurcations, critical circle homeo- This book provides an introduction to the Thomassen surveys parity arguments and morphisms, absolutely continuous invari- basic terminology and methods for solving problems in graph theory and their con- ant measures, and matings of polynomials, and analysing partial differential equations nection to the cycle space, and proves a along with the geometry, dimension and that describe wave phenomena – travelling new result on odd K4-subdivisions in 4- local connectivity of Julia sets. In addition and standing waves, the Fourier method, connected graphs. R. Thomas discusses to presenting new work, this collection the method of characteristics, shock and recent and not-so-recent excluded minor presents important results hitherto unpub- rarefaction waves, and the viscosity theorems for graphs, connected to linkless lished or difficult to find in the literature. method for conservation laws. Deeper embeddings, Hadwiger’s conjecture, The book contains a collection of six- mathematical concepts are mentioned Pfaffian orientations, and other topics. N. teen papers (four in French). The preface without proof, but are well illustrated with C. Wormald surveys results and methods presents an excellent historical back- simple models (such as the superposition for random graphs with restricted vertex ground of the topic. The papers are principle on the one-dimensional wave degrees, stressing results for regular grouped according to the properties dis- equation, and shock waves and viscosity graphs. J. Pach describes some results cussed (universality of the Mandelbrot set, solution on the traffic-flow problem). The from geometrical graph theory where quadratic Julia sets and the Mandelbrot derivation of models from physical laws is edges are represented in the plane by set, Julia set of rational maps, foundation- also given in some cases (the sine-Gordon straight-line segments, and considers some al results). The book will be of interest to equation, vibrating string, conservation generalisations to hypergraphs (here sim- graduate students in mathematics, physics law). The book also contains elementary plices replace segments). S. Ball gives an and mathematical biology, as well as to computer experiments (such as graphing overview of the recent method in finite researchers in dynamical systems and the analytically obtained solution) and geometry where objects are represented by Kleinian groups. (pp) exercises. To understand the text only a polynomials (problems concern nuclei, EMS June 2000 37 RECENT BOOKS H. van Maldeghem: Generalized Polygons, This book contains no mathematical of complex zeros) and Chapter 6 Monographs in Mathematics 93, derivations of statistical methods, but it (improved inclusion methods for polyno- Birkhäuser, Basel, 1998, 502 pp., DM218, helps the reader to understand statistical mial complex zeros). Various possibilities ISBN 3-7643-5864-5 and 3-8176-5864-5 methodology and carry out necessary cal- for parallel implementations of inclusion This book gives a contemporary survey of culations. The data used in the book is methods are presented in Chapter 7. results in the theory of generalised poly- available on a diskette in ASCII format. Numerical stability of iterative processes is gons. The approach used is mostly geo- There also exist supplements for the discussed in Chapter 8, and the book con- metrical in nature, and several algebraic instructor and supplements available for cludes in Chapter 9 with numerical com- results are also proved in a geometric way. purchase by students. (ja) putation of curvilinear integrals. The bibli- A generalised polygon (the notion is due to ography has 183 items. Tits) is a geometry (an incidence structure) A. N. Parshin and I. R. Shafarevich (eds.), This book demonstrates the usefulness that generalises the notion of an ordinary Algebraic Geometry V, Encyclopaedia of of complex interval arithmetic for solving polygon. This notion is then related to Mathematical Sciences 47, Springer, Berlin, numerical problems arising in complex another combinatorial structure called a 1999, 247 pp., DM159, ISBN 3-540-61468- analysis. It may be of interest both to spe- building and also to more general geome- 0 cialists and graduate students. In particu- tries, such as partial geometries, partial The main topic of this volume is the struc- lar, the part of the book that discusses quadrangles, near polygons, etc. Another ture theory of Fano varieties. A smooth inclusion of polynomial zeros (Chapters 4- type of structure used to describe gener- projective variety is a Fano variety if its anti- 6) may be found useful by specialists in alised polygons is a finite group of a spe- canonical divisor is ample. Examples of control theory. (jroh) cial type. Fano varieties in dimension 2 are the so- Each chapter starts with an introduc- called del Pezzo surfaces, and many other M. Picardello and W. Woess (eds.), tion, containing motivation and history examples of Fano varieties can be found Random Walks and Discrete Potential related to its content. We find a classifica- here. Theory, Symposia Mathematica 39, Cambridge tion and description of the classical and In the first part of the book, the basic University Press, Cambridge, 1999, 361 pp., mixed quadrangles, classical hexagons, properties of Fano varieties and results £16.95, ISBN 0-521-77312-1 and non-classical hexagons and octagons. concerning defining equations of varieties This book contains the proceedings of the There is a chapter in which the coordinati- related with Fano varieties are presented. conference on ‘Random walks and discrete sation of polygons is studied, and another The classification problem for algebraic potential theory’, held in Crotona (Italy) in chapter contains a description of homo- varieties was recently developed thanks to June 1997. It covers the interplay between morphisms of polygons and their auto- the Mori theory of minimal models and its the behaviour of a class of stochastic morphisms groups. Moufang polygons are extension from surfaces to higher-dimen- processes (random walks) and structure investigated in great detail in several chap- sional varieties. A minimal model is theory. Written by leading researchers, this ters. The final chapter is devoted to topo- defined as a normal projective variety with collection of invited papers presents links logical polygons and contains results from a numerically effective canonical divisor. with spectral theory and discrete potential algebraic topology. The book contains a The Mori conjecture says that any irre- theory, and includes probabilistic and lot of material and is complements the ducible algebraic variety over an algebraic structure theoretic aspects. Its interdisci- monograph by Payne and Thas, Finite closed field of characteristic 0 is birational- plinary approach spans several areas of Generalized Quadrangles, Pitman Research ly equivalent either to a minimal model or mathematics, including geometrical group Notes in Mathematics 110, 1984. (jbu) to a fibration over a variety of smaller theory, discrete geometry and harmonic dimension with the general fibre being a analysis. The book will be of interest to J. T. McClave and T. Sincich, Statistics Fano variety. This book gives a very good researchers and postgraduate students, 8e, Prentice Hall, New Jersey, 2000, 848 pp., survey of known results on Fano varieties, both in mathematics and statistical physics. £25.99, ISBN 0-13-022329-8 and 0-13- and also includes contemporary results (pp) 022574-6 and possible generalisations. There is a list This is an undergraduate introductory of open questions and problems. (jbu) R. C. Read and R. J. Wilson, An Atlas of text in probability and statistics. The chap- Graphs, Oxford Science Publications, ters devoted to probability contain such M. S. Petkovi³ and L. D. Petkovi³, Clarendon Press, Oxford, 1998, 454 pp., £75, classical elementary topics as events, sam- Complex Interval Arithmetic and its ISBN 0-19-853289-X ple spaces, conditional probability, and Applications, Mathematical Research 105, This is an unusual book, somewhat in the discrete and continuous random variables. Wiley-VCH, Berlin, 1999, 284 pp., DM 198, spirit of the Atlas of finite simple groups, Statistical chapters describe confidence ISBN 3-527-40134-2 although on a simpler and less technical intervals, tests of hypotheses, analysis of Real interval arithmetic was introduced in level (and in a smaller format). It contains variance, linear regression and model the 1960s as a tool for handling uncertain- some 10000 pictures of graphs, tables of building, contingency tables, and non- ties in the data, round-off errors and com- the number of graphs with given proper- parametric statistics. The book contains putations in finite precision arithmetic. ties, and extensive tables of properties of plenty of examples and exercises based on Complex interval arithmetic, introduced depicted graphs. Apart from its artistic very rich collections of real data. ten years later, is still much less known. value it will be of use to graph theory spe- Numerical processing of data is illustrated The present book is the first monograph cialists, chemists and combinatorialists in on the TI-83 Graphic Calculator with ref- devoted entirely to this subject and its var- general. It is certainly the most massive erences to the well-known statistical soft- ious applications. amount of graph data in existence. ware packages SPSS, Minitab and SAS. The two basic types of complex interval A small remark: for such an exhaustive This textbook has become very popular. arithmetic (rectangular and circular) are atlas perhaps the list of ‘special graphs’ Its eighth edition includes a new chapter introduced in Chapter 1. In Chapter 2, (listed on pp. 263-265) is somewhat too on multiple regression, and two other methods for estimating range of complex short. (jnes) chapters have been rewritten. The authors functions are studied with help of so-called include relevant recent results, such as the circular centred forms. Chapter 3 is devot- J. Robertson and W. Webb, Cake-Cutting Wilson adjusted confidence interval for a ed to best approximations of the range of Algorithms: Be Fair If You Can, A. K. population proportion, which is based on a complex functions by discs; in particular, Peters, Natick, 1998, 181 pp., £26, ISBN 1- paper published in 1998. On the other the functions eZ, Zn and Z1/k are studied in 56881-076-8 hand, the authors could have mentioned detail. A typical application of complex Since the 1950s the fair division of a cake that the Tukey method for multiple com- arithmetic is the inclusion of polynomial has been the popular subject of many eso- parisons was generalised for unequal sam- zeros. This problem is first handled in teric papers, as well as of serious mathe- ple sizes, since the generalisation is avail- Chapter 4 for single zeros, using interval matical research: we seek ways to divide a able in many statistical packages and slope methods, and its exposition contin- cake ‘fairly’ among several people in such belongs to the most sensitive methods. ues in Chapter 5 (simultaneous inclusion a way that each person is ‘satisfied’ with the 38 EMS June 2000 RECENT BOOKS piece received. Chapter 4, starting with an abstract frame- Σ ⊂ Rk. The number of limit cycles that We can trace the roots of the problem to work and continuing with optimisation bifurcate from a periodic limit set Γ, called the Polish group (Banach, Knaster), and problems with controls on Lebesgue the cyclicity of Γ in Xλ, plays a crucial role. the problem was presented by H. spaces. Much attention is paid to first- The cyclicity depends only on the germ Steinhaus in 1947. Since then the cake-cut- order optimality conditions for the relaxed (unfolding) of Xλ along Γ and a general ting problem has blossomed and has been problem and to integral and localised conjecture is that it is finite for any analyt- widely popularised. The book also includes (Pontryagin-type) maximum principles. ic unfolding. Hilbert’s 16th problem is notes on other similar problems (dividing Four examples are analysed: the optimal associated with the question: ‘Prove that beach-front property, the trimming algo- control of non-linear dynamical systems, for any n ≥ 2, there exists a finite number rithm, dividing an estate, a risk-free bet) elliptic and parabolic partial differential H(n) such that any polynomial vector field while other well-known ones (the problem equations and the optimal control of the of degree ≤ n has less than H(n) limit of the Nil, the problem of similar regions, Hammerstein integral equation. The treat- cycles’. There is a positive answer to this the bisection problem, the ham sandwich ment of the first example (dynamical sys- problem in Chapter 2, but – as the author problem, the border problem) are left out. tems) also includes its numerical approxi- adds – an algorithm for finding the bound The purpose of the book is to discuss mation and computational results. H(n) is unknown. the cake-cutting problem, to explain the The programme of Chapters 5 and 6 is the The author recalls in Chapter 1 the meaning of a ‘fair share’, and to offer algo- relaxation in the variational calculus in the Poincaré-Bendixson theory, and in rithms for its solution. The first few chap- scalar (Chapter 5) and the vectorial case Chapter 2 studies limit periodic sets and ters provide a survey of the problem while (Chapter 6). The author begins with con- their cyclicity, concentrating on quadratic later ones contain proofs of previously vex σ-compactifications of Sobolev spaces vector fields. The single vector fields are introduced problems. Topics in the book W1,p and constructs relaxation schemes for studied in Chapter 3, using the desingu- include fairly dividing a cake (two and the reflexive case (e.g., problems of larisation theory. In Chapter 4 regular more persons), pieces or crumbs, how martensitic microstructure in elasticity, p limit cycles, in Chapter 5 elementary many cuts are needed?, unequal shares, > 1), as well as the non-reflexive case (e.g., graphics, and in Chapter 6 non-elemen- moving knife algorithms, the Stromquist the non-parametric minimal surface prob- tary graphics, are considered and studied algorithm, disagreement, strong fair divi- lem, p = 1). Weierstrass-type optimality by methods of differential and analytic sion, envy-free division, exact division, conditions are derived for the relaxed geometry, using the Bautin ideal and applying graph theory, impossibility theo- problem, and its stability, correctness and asymptotic methods. (jeis) rems, Ramsey partitions and fair division. well-posedness are investigated. The theo- Exercises are included at the end of each retical results for p > 1 are accompanied I. P. Stavroulakis and S. A. Tersian, chapter. The book is readable by anybody with a numerical approximation theory, Partial Differential Equations. An with a basic mathematical knowledge and including optimality conditions for Introduction with Mathematica and can be enjoyed by both students and approximate solutions and sample calcula- Maple, World Scientific, Singapore, 1999, researchers. (jl) tions. 297 pp., £23, ISBN 9-8102-3891-6 The full strength of convex compactifi- This book provides an elementary intro- T. Roubí³ek: Relaxation in Optimization cations is explored in Chapter 7, dealing duction to the classical theory of first and Theory and Variational Calculus, de with the relaxation in game theory, where second-order partial differential equa- Gruyter Series in Nonlinear Analysis and both convexity and compactness are need- tions. Linear problems are studied and the Applications 4, Walter de Gruyter, Berlin, ed to ensure the existence of solutions. explicit formulae for problems with homo- 1997, 474 pp., DM298, ISBN 3-11-014542- The chapter contains two examples: games geneous and non-homogeneous boundary 1 with dynamical systems and elliptic games, conditions are provided. The contents of Having an integral functional as the argu- the former also with some numerical the book coincide, for example, with the ment of ‘direct methods’ that prove exis- results. first part of the monograph by L. Ewans, tence of the minimum relies on lower The book, containing many original Partial Differential Equations, AMS, 1998 semi-continuity of this functional, which is results, has a large and bibliography and (where, however, a more detailed discus- closely related to convexity properties of commentary, as well as reviews of classical sion and comments are given); the sepa- the integrand. In many cases, a lack of con- and standard facts from topology, optimi- rate chapters of the book can also be found vexity results in oscillating minimising sation, functional and mathematical analy- in other (cited) textbooks. Here, perhaps, sequences (or their gradients), whose weak sis (Chapter 1) which makes it fairly self- more convenient examples for students is limits are not minimisers of the problem. contained. It is a comprehensive and use- given. In the late 1930s, L. C. Young showed that ful source of information for researchers Although the book has a promising sub- the non-existence of minima of some func- and for graduate students working in opti- title, the usage of Mathematica consists only tionals can be overcome by enlarging the mal control theory, the calculus of varia- of drawing graphs (of insufficient quality) set of competing curves. These ‘gener- tions or the optimisation of distributed of solutions, and there are very few (only alised curves’ can be described by means of parameter systems. (mkr) two?) applications of Maple throughout the probability measures, now widely known as whole book. (jmal) ‘Young measures’ and the procedure R. Roussarie, Bifurcations of Planar (enlarging) known as ‘relaxation’. Vector Fields and Hilbert’s Sixteenth D. W. Stroock, Probability Theory, an In Chapter 2 the author deals with an Problem, Progress in Mathematics 164, Analytic View, Cambridge University Press, abstract and original theory of suitable Birkhäuser, Basel, 1998, 204 pp., DM118, Cambridge, 2000, 536 pp., £18.95, ISBN 0- convex and (σ)-compact envelopes of ISBN 3-7643-5900-5 521-66349-0 topological spaces. This is made specific in In this book the author studies bifurca- This is a new (paperback) edition of the Chapter 3 where the main emphasis is on tions of limit periodic sets in families X? of book first published in 1993 and reviewed convex σ-compactifications of Lebesgue planar vector fields depending on a para- in EMS Newsletter 14 (1994), p.36. The new spaces. It is shown that Young measures meter λ in Rk, where the phase space is R2 edition is described as ‘revised’, but the and their generalisations can be viewed as or in general a surface of genus 0. As an alterations are minimal – splitting some special cases of these compactifications, example, polynomial vector fields sections into two and adding a couple of and various topological properties of par- Xλn (x, y) = P(λ, x, y) δ/δx + Q(λ, x, y) δ/δy new exercises. (vd) ticular compactifications are studied in with polynomials P, Q of degree ≤ n are detail. The rest of the chapter is mainly considered. The problem under consider- K. Taira, Brownian Motion and Index devoted to approximation theory and ation is the change of behaviour of a vector Formulas for the de Rham Complex, numerical analysis of general convex com- field when λ varies Xλ – more precisely, the Mathematical Research 106, Wiley-VCH, pactifications, enabling efficient computer change of singular elements (singular Berlin, 1998, 215 pp., DM198, ISBN 3-527- implementation of relaxed problems. points and isolated periodic orbits – limit 40139-3 Optimal control theory is treated in cycles), when λ crosses a bifurcation set It is a well-known fact that cohomology EMS June 2000 39 RECENT BOOKS groups of a manifold (with values in R) can An important part of the book is devoted be represented by means of the de Rham to the study of homogeneous vector bun- Book Review by complex and harmonic forms. Similar facts dles. Some conditions for their flatness are for relative cohomology groups are less proved, and the classification of flat bun- Sir Michael Atiyah standard. This book offers a nice systemat- dles over parallelisable manifold is given. Conversation on Mind, Matter and ic treatment of these questions from an The author also treats deformations of Mathematics by Jean-Pierre Changeux and analytic point of view. The main tools used compact parallelisable manifolds and the Alain Connes edited and translated by M B De are some PDE techniques, with an addi- structure of the related cohomology Bevoise Princeton University Press, 1995 260 tional insight given by interpretation in group. As an important example, complex pp., £19.95, ISBN: 0 691 08759 8 terms of Brownian motion and probability nilmanifolds are studied and it is shown theory. The first four chapters are that they are always built from complex preparatory and contain basic facts needed tori admitting a certain kind of complex from differential geometry, functional multiplication. The author also presents analysis, Markov processes, Fourier inte- algebraic groups for which any invariant gral operators and spaces of currents. The holomorphic function invariant with calculus of pseudo-differential operators is respect to Zariski dense discrete subgroups used in the main part of the book for a is constant. treatment of interior elliptic boundary This book can be recommended to all value problems and for proofs of the main readers interested in the topics covered. index theorems. The book is well organ- (jbu) ised and is written in an understandable way with many intuitive explanations and W. Woess, Random Walks on Infinite Does mathematics have an existence inde- remarks. (vs) Graphs and Groups, Cambridge Tracts in pendent of our physical world? Do mathe- Mathematics and its Applications 78, maticians discover theorems, rather than K. Thomsen, Limits of Certain Cambridge University Press, Cambridge, 2000, invent them? Such questions have exer- Subhomogeneous C*-Algebras, Mémoires de 334 pp., £40, ISBN 0-521-55292-3 cised the minds of philosophers and math- la Société Mathématique de France 71, Société The topic of random walks is situated ematicians since the time of Plato, and Mathématique de France, Paris, 1997, 125 somewhere between probability, potential many books have addressed the issues. pp., ISBN 2-85629-064-7 theory, harmonic analysis, geometry, What is unusual and topical about this The main topic of this booklet is the struc- graph theory and algebra. The beauty of book is that it records a dialogue between ture of a certain type of simple unital C*- the subject stems from linking, both in the a mathematician, Alain Connes, and a biol- algebras. Typical C*-algebras considered way of thinking and in the methods ogist, Jean-Pierre Changeux. Each is an here are algebras of matrix-valued contin- employed, of different fields. The main international authority in his field and uous maps defined on the interval [0, 1] or theme of this book is the interplay between both are Professors at the Collège de on the unit circle T, with special properties the behaviour of a class of stochastic France, so that they speak from first-hand in a finite sequence of distinguished points processes (random walks) and discrete experience and the intellectual level is as well as inductive limits of a sequence of structure theory. The author considers high. As a neuro-physiologist Changeux is finite direct sums of such algebras. The Markov chains whose state space is particularly interested in the structure and main result is a classification theorem for a equipped with the structure of an infinite organisation of the brain. He is intrigued special type of such algebras and its exten- locally finite graph, or as a particular case, by the capacity of the brain to handle the sion to the non-unital case. Proofs of the of a finitely generated group. The transi- kind of sophisticated intellectual abstrac- main theorems use several different previ- tion probabilities are assumed to be adapt- tions represented by mathematics, and his ously developed methods and are quite ed to the underlying structure in some way dialogue with Connes is an attempt to get technical. The book is suitable for special- that must be specified precisely in each to grips with this problem. ists in the field. (vs) case. Most mathematicians would agree that From a probabilistic viewpoint, the their subject has evolved from the tradi- J. Winkelmann, Complex Analytic question is what impact the particular type tional disciplines of arithmetic and geome- Geometry of Complex Parallelizable of structure has on various aspects of the try. We have built an elaborate superstruc- Manifolds, Mémoires de la Société behaviour of the random walk. Conversely, ture on these twin foundations and so the Mathématique de France 72-73, Société random walks may also be seen as a useful nature of ‘mathematical reality’ can be Mathématique de France, Paris, 1998, 219 tool for classifying, or at least describing, adequately dealt with by posing basic ques- pp., ISBN 2-85629-070-1 the structure of graphs and groups. Links tions about the integers 1, 2, 3 ... and about The main subject of this book is the study with spectral theory and discrete potential the Euclidean geometry of triangles, cir- of complex parallelisable manifolds – com- theory are also discussed. cles, etc. Connes is an unabashed Platonist. plex manifolds that are quotients of com- This book will be essential reading for For him there is an ‘archaic mathematical plex Lie groups by discrete subgroups. all researchers in stochastic processes and reality’ represented by objects such as cir- Special attention is paid to quotients by related topics. According to the author, cles or integers, which has an existence, discrete subgroups that are cocompact or this book is not self-contained, by the independent of experience or of the at least of finite covolume. As a special nature of its topic. Another question may human mind. One practical consequence case, arithmetic groups are studied and a be whether it is usable by graduate stu- of such a belief, already adopted by NASA, criterion for cocompactness of arithmetic dents. Here let us quote the author: ‘It has is that mathematics provides the best form subgroups is presented. been my experience that usually, this must of potential communication with any The corresponding lattices in complex be taken with caution, and is mostly true extra-terrestrial intelligence. Lie groups are discussed and their basic only in the presence of a guiding hand that Any mathematician must sympathize properties are presented. The special cases is acquainted with the topic. Proofs are with Connes. We all feel that integers, or studied here are complex tori, and it is sometimes a bit condensed, and it may be circles, really exist in some abstract sense proved that they are the only quotients that even students above the student level and the Platonist view is extremely seduc- admitting a Kähler metric. Complex ana- will need pen and paper when they want to tive. But can we really defend it? Had the lytic subspaces of quotients and their work through them seriously – in particu- universe been one-dimensional or even Kodaira dimension are discussed. lar because of the variety of different meth- discrete it is difficult to see how geometry Properties of holomorphic mappings and ods and techniques that I have tried to could have evolved. It might seem that automorphism groups of parallelisable unite in this text. This does not mean, with the integers we are on firmer ground, manifolds are investigated; it seems that however, that parts of this book could not and that counting is really a primordial holomorphic mappings are often automat- be used for graduate or even undergradu- notion. But let us imagine that intelligence ically equivariant. ate courses.’ (jant) had resided, not in mankind, but in some 40 EMS June 2000 RECENT BOOKS vast solitary and isolated jelly-fish, buried quently produced marvellous formulae by and faster than humans at certain kinds of deep in the depths of the Pacific Ocean. It some unknown mental process which could calculation, but in most important respects would have no experience of individual neither be described nor repeated. I think they are still a long way behind. For exam- objects, only with the surrounding water. Connes would accept that such formulae ple, computers are not yet serious contrib- Motion, temperature and pressure would do not become an accepted part of mathe- utors to mathematical theory. Perhaps by provide its basic sensory data. In such a matics until others have understood them analysing the structure of mathematics we pure continuum the discrete would not and verified their validity. may learn something about the way the arise and there would be nothing to count. Our collective consciousness may not be brain operates. In particular a key feature Even more fundamentally, in a purely sufficiently ideal for Connes but it is the of mathematics is its hierarchical nature. static universe without the notion of time, world where all ideas live, not just mathe- Examples of patients with specific brain causality would disappear and with it that matical ones. damage show that particular levels of rea- of logical implication and of mathematical Where does this point of view leave the soning are associated with definite parts of proof. Connes actually alludes to this dichotomy between creation or discovery the brain. philosophical dilemma in the context of in mathematics? By resisting the embrace For Changeux, the comparison between relativistic cosmology. of the Platonic world have we lost the pos- the structure of mathematics and the struc- It may be argued that such “gedanken sibility of making “discoveries’”? Is every ture of the brain can be looked at both universes’” are not to be taken seriously. theorem man-made? Not at all. Since our from the evolutionary aspect and in terms Our actual universe is a given datum and concepts are based on the physical world of function. In evolutionary terms, the the inevitable background of all intelligent we can discover facts about these concepts brain has had to create a hierarchy of lev- discussion. But this is tantamount to con- experimentally. For example the formula els that adequately reflect the physical ceding that mathematics has evolved from 3 × 5 = 5 × 3 environment and the challenges it poses. the human experience. Man has created can be discovered by setting out 15 objects Man has been the ultimate winner of the mathematics by idealising or abstracting in a rectangular array. One can then move evolutionary process and his brain has the elements of the physical world. The num- on to the more general algebraic formula structure needed to produce mathematics. ber 2 for example represents the common a × b = b × a, Would a different neurological solution attribute of all pairs of objects that we have where a and b represent arbitrary integers. have led to a different kind of mathemat- encountered, in the same way as the word In a similar way the famous formula of ics, or does mathematics depend only on “chair” represents what is common to all Pythagoras the functional capacity of the brain, not on the individual items of furniture that we sit a2 + b2 = c2 its biological mechanism? on. Admittedly chairs are not so precisely relating the length of the sides of any If one views the brain in its evolutionary and unambiguously defined as numbers: is right-angled triangle was no doubt discov- context then the mysterious success of a three-legged stool a chair? Language is ered experimentally. So man creates the mathematics in the physical sciences is at inherently more fuzzy or open-ended than concepts of mathematics but he discovers least partially explained. The brain mathematics, but mathematics can proper- the subsequent connections between them. evolved in order to deal with the physical ly be viewed as a special kind of language. The reason he can have it both ways is that world, so it should not be too surprising This applies not only to the nouns or mathematics is firmly rooted in the real that it has developed a language, mathe- objects but also to the verbs or processes world. matics, that is well suited for the purpose. such as addition and subtraction. The legally minded reader may object The sceptic can point out that the struggle Connes will protest that there are thou- that I have slurred the difference between for survival only requires us to cope with sands of different languages reflecting the an empirical discovery, always concerned physical phenomena at the human scale, history and culture of a particular people, with the specific, and a proof, concerned yet mathematical theory appears to deal whereas mathematics is universal and with the general. I plead guilty, but only on successfully with all scales from the atomic unique. Surely this gives it a special status? a technicality and I hope the judge will be to the galactic. Perhaps the explanation This can be debated at length, but a short lenient. Sometimes a discovery will carry lies in the abstract hierarchical nature of answer is that the diversity of languages with it the seeds of a formal proof, as with mathematics which enables is to move up disguises a fundamental structural similar- the example that 3 × 5 = 5 × 3. and down the world scale with comparative ity, which is why dictionaries help us to Sometimes one has to work harder as ease. translate. Moreover different mathemati- Pythagoras had to do. But if possession is The internal workings of the mathemat- cal notations, such as the Roman numeral nine-tenths of the law, discovery is nine- ical mind were well described by Jaques system, do exist but western civilisation has tenths of the proof. Hadamard who distinguished three stages produced a uniformity which was not In his dialogue with Connes, Changeux in attacking a problem: preparation, incu- inevitable. keeps hitting the Platonist rock. As a hard- bation and illumination. What these corre- In describing mathematics as a lan- headed experimental scientist Changeux spond to neurologically is a challenging guage it is important to emphasise that a wants to identify mathematics with what question which generates a lively discus- language is not merely a set of words and actually goes on in the brain. For him this sion between Connes and Changeux. Is grammatical rules for producing coherent is the only reality and the only place where there for example a Darwinian element in sentences. Words mean something and mathematics exists. Connes disputes this the search for successful ideas? Henri they relate to our experience. In a similar extreme attitude and prefers to say that Poincaré argued that the subconscious way a mathematical statement has a mean- mathematical reality (which exists else- mind generates random ideas and “illumi- ing and one which, I would argue, rests where) is reflected in the neurological nation” occurs when one of these is select- ultimately on our experience. processes of the brain. To confuse the two ed. For a Platonist like Connes mathematics is like identifying a piece of literature or The original Socratic dialogues were lives in some ideal world. I find this a diffi- music with the ink and paper on which it is artificially constructed to present a coher- cult notion to grasp and prefer to say more recorded. It is hard to disagree, but fasci- ent view. The dialogue between Connes pragmatically that mathematics lives in the nating questions remain. and Changeux is quite different. It is the collective consciousness of mankind. I am The brain is frequently compared to an recording of real-life arguments where the not here embracing any obscure metaphys- electronic computer and artificial intelli- speakers are frequently at cross-purposes ical notion, but merely observing that gence is now a branch of computer science. and operate in different planes. For the there are two essential components of But, as the dialogue brings out, this analo- reader this can be irritating but it also mathematics. In the first place it deals with gy has serious limitations and may in fact encourages him to become involved and concepts and abstract processes which live be obscuring some of the essential features. frame his own answers, as I have endeav- in the conscious mind of the individual As Connes notes, the speed of transmission oured to do! mathematician. Secondly, it must be com- of signals in the brain is vastly less than the Sir Michael Atiyah, OM, was formerly President municable to other mathematicians. The corresponding speeds in modern comput- of the Royal Society and Master of Trinity famous Indian genius Ramanujan fre- ers. As a result computers are much better College Cambridge. EMS June 2000 41 PERSONAL COLUMN Erich Novak (Erlangen-Nürnberg) John Toland, FRS (Bath) has been is the first winner of the Prize for awarded an honorary degree from Achievement in Information-Based Queen’s University, Belfast. Complexity. PPersonalersonal ColumnColumn Jack van Lint (Eindhoven) received Juan-Pablo Ortega and Tudor the honorary degree of Doctor of Ratiu (Lausanne) have been award- Mathematics on 24 March 2000, ed the 1999 Ferran Sunyer i Balaguer from the University of Ghent. We list below information about some Prize (1998) by the Italian Prize by the Institut d’Estudis appointments, awards and deaths that Mathematical Union in recognition Catalans for their monograph enti- Don Zagier (Bonn) has been award- have occurred during late 1999 and of his theoretical research in the tled Hamiltonian Singular ed the Chauvenet Prize by the early 2000. Since this list is inevitably Calculus of Variations. Reduction. The awards will be pre- Mathematical Association of incomplete, we invite you to send appro- sented on 10 July during the 3ecm in America for his outstanding exposi- priate information from other countries Persi Diaconis (Stanford) was elect- Barcelona. tory article ‘Newmans short proof of to Barbara Maenhaut [b.m.maen- ed to hold the first UK Hardy the prime number theorem’ in the [email protected]] or to your Country Fellowship from May-August 2001, James C. Robinson has been award- American Mathematical Monthly. representative (see Issue 34) for inclu- at Queen Mary and Westfield ed a Royal Society University sion in the next issue. Please also send College, London. Research Fellowship to work in the any items you feel should be included in Mathematics Institute, Warwick. Deaths future Personal Columns. Matthias R. Gaberdiel has been We regret to announce the deaths awarded a Royal Society University Guido Sanguinetti (Genova) was of: Appointments Research Fellowship to work in the awarded the Franco Tricerri Prize George K. Batchelor, FRS (30 Department of Applied Mathematics (1999) by the Italian Mathematical March 2000) Paul Garthwaite (Aberdeen) and and Theoretical Physics, Union for his doctoral work in dif- Mary Bradburn (31 January 2000) Chris Jones have been appointed Cambridge. ferential geometry. Donna M. Carr (February 2000) Professors in the Statistics David G. Crighton, FRS (12 April Department at the Open University, Mikhail Gromov (Paris and New Jean-Pierre Serre (Paris) has been 2000) and Pia Larsen, Karen Vines and York) was awarded the 1999 Balzan awarded the 2000 Wolf Prize in Lord Halsbury (14 January 2000) Catriona Queen have been appoint- Prize for Mathematics, awarded by Mathematics, for his many funda- Timothy Swan Harris (22 July ed Lecturers. the Fondazione Internazionale mental contributions to topology, 1999) Premio E. Balzan. algebraic geometry, algebra and Nathan Jacobson (5 December Robert Hunt (Cambridge) has been number theory; he shares this prize 1999) appointed Deputy Director of the Alice Guionnet (Orsay) has received with Raoul Bott. Bertha Jeffreys (18 December 1999) Isaac Newton Institute for the 1999 Oberwolfach Prize for out- Jürgen Kurt Moser (17 December Mathematical Sciences. standing research in stochastics. Volker Strassen (Konstanz) has 1999) been awarded the 1999 Georg Peeter Muursepp (3 November Robert MacKay and Andrew Stuart Nicholas J. Higham (Manchester) Cantor Medal by the Deutsche 1999) have been appointed Professors at has been awarded a Royal Society Mathematiker Vereinigung. John A. Nohel (1 November 1999) Warwick University; John Rawnsley Senior Research Fellowship. Edwin J. Redfern (8 January 2000) and Colin Rourke have been pro- Alain-Sol Sznitman (Zürich) has Robert F. Riley (4 March 2000) moted to Personal Chairs; Claude Tom Hoeholdt (Technical been awarded the 1999 Line and Gottfried T. Ruttimann (11 October Baesens, Peter Topping and Balasz University of Denmark) has been Michel Loève International Prize in 1999) Szendroi have been appointed elected an IEEE Fellow, from 1 Probability. Ken Stroud (3 February 2000) Lecturers. January 2000, for “fundamental Jeffrey D. Weston (10 March 2000) contributions to the theory, analysis Awards and decoding algorithms of algebraic geometry codes”. Luigi Ambrosio (Scuola Normale Superiore of Pisa) was awarded the Peter Hydon (Guildford, Surrey) Renato Caccioppoli Prize (1998) by has been promoted to Reader in the Italian Mathematical Union for mathematics. his contributions to the Calculus of Variations and Partial Differential Kurt Johansson (Royal Institute of Equations. Technology, Stockholm) and David Wilson (Microsoft Research) have F. Bethuel and F. Helein have been been awarded the Rollo Davidson awarded the 1999 Fermat Prize in Prizes for 2000. Mathematics for ‘several important contributions to the theory of varia- Marco Manetti (University of Roma tional calculus which have conse- “La Sapienza”) has been awarded quences in physics and geometry. the Giuseppe Bartolozzi Prize (1999) by the Italian Mathematical Union The French Académie des Sciences for his contributions to the theory of has awarded special mathematical algebraic surfaces. prizes to Laurent Clozel (Orsay), Pierre Colmez (Paris), Sylvestre Bernard Maurey (Paris) has been Gallot (Grenoble), Laurent Manivel awarded a State Prize by the French (Saint Martin d’Hères) and Académie des Sciences for funda- Wandelin Werner (Orsay). mental work on Banach spaces.

Yves Colin de Verdière (Grenoble) Alun Morris (Aberystwyth) received has been awarded the Prix Ampère an OBE and Andrew Wiles FRS de l’Electricité de France by the (Princeton) received a knighthood French Académie des Sciences, for in the British New Years Honours fundamental contributions to spec- List. Sir Roger Penrose FRS tral theory. (Oxford) has been appointed to the Order of Merit. Joachim Cuntz (Münster) was awarded the 1999 Gottfried Leibniz Stefan Müller (Leipzig) and Dieter Prize, given by the Deutsche Lüst (Potsdam) have been awarded Forschungsgemeinschaft, for work the 1999 Gottfried Leibniz Prize, in functional analysis. awarded by the Deutsche Forschungsgemeinschaft, for work Riccardo De Arcangelis (Napoli) in functional analysis. was awarded the Calogero Vinti

42 EMS June 2000 POSTDOCTORAL POSITIONS IN MATHEMATICS IN THE UNIVERSITY OF AVEIRO

The Research unit Mathematics and its Applications of the University of Aveiro accepts applications for post-doctoral research positions in the area of Functional Analysis until June 30th, 2000. Each position has a duration of at most one year and shall start no sooner than September 1st, 2000, at a date convenient for both parties.

The areas of research are - Non-Standard Methods in Analysis and Topology; - Function Spaces on Irregular Domains; - Convolution Type Operators and Diffraction Theory. The successful applicant is supposed to work as a member of one of the teams of the Group of Functional Analysis in one or more of these topics.

Candidates should have a PhD – preferably obtained within the last 5 years – in an area relevant to at least one of the subjects mentioned above and shall be selected on the basis of the following documents: 1. a letter of intent, 2. curriculum vitae, 3. official proof of PhD award, 4. copy of two research works (preferably in Portuguese, English, French or Spanish) 5. copy of passport.

The above documents must be sent, by registered mail, to Secretaria do Departamento de Matemática (c/o Prof. António Caetano) Universidade de Aveiro 3810-193 AVEIRO Portugal

For more information, please contact [email protected]

EMS June 2000 43