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 NEWSLETTER No. 41 Building 303 DK-2800 Kgs. Lyngby, Denmark September 2001 e-mail: [email protected] KRZYSZTOF CIESIELSKI EMS Agenda ...... 2 Mathematics Institute Jagiellonian University Editorial - Carles Casacuberta ...... 3 Reymonta 4 30-059 Kraków, Poland EMS Summer School at St Petersburg 2001 ...... 4 e-mail: [email protected] KATHLEEN QUINN EAGER - ENI - EMS Summer School ...... 5 The Open University [address as above] e-mail: [email protected] Meeting of the EMS Council ...... 6 SPECIALIST EDITORS EMS Lecturer 2002 - Gianni Dal Maso ...... 8 INTERVIEWS Steen Markvorsen [address as above] Interview with Marek Kordos ...... 9 SOCIETIES Krzysztof Ciesielski [address as above] Interview with Graham Higman ...... 12 EDUCATION Tony Gardiner The Methodology of Mathematics, part 2 ...... 14 University of Birmingham Birmingham B15 2TT, UK Mathematical Societies: Norwegian ...... 17 e-mail: [email protected] MATHEMATICAL PROBLEMS Mathematical Societies: Estonian ...... 18 Paul Jainta Werkvolkstr. 10 Problem Corner ...... 20 D-91126 Schwabach, Germany e-mail: [email protected] Mathematics Education on the Web ...... 23 ANNIVERSARIES Forthcoming Conferences ...... 24 June Barrow-Green and Jeremy Gray Open University [address as above] Recent Books ...... 27 e-mail: [email protected] and [email protected] and CONFERENCES Designed and printed by Armstrong Press Limited Kathleen Quinn [address as above] Crosshouse Road, Southampton, Hampshire SO14 5GZ, UK RECENT BOOKS telephone: (+44) 23 8033 3132 fax: (+44) 23 8033 3134 Ivan Netuka and Vladimir Sou³ek Published by European Mathematical Society Mathematical Institute ISSN 1027 - 488X Charles University Sokolovská 83 The views expressed in this Newsletter are those of the authors and do not necessarily 18600 Prague, Czech Republic represent those of the EMS or the Editorial team. e-mail: [email protected] and [email protected] ADVERTISING OFFICER NOTICE FOR MATHEMATICAL SOCIETIES Vivette Girault Labels for the next issue will be prepared during the second half of November 2001. Laboratoire d’Analyse Numérique Please send your updated lists before then to Ms Tuulikki Mäkeläinen, Department of Mathematics, P.O. Box 4, FIN-00014 University of Helsinki, Finland; e-mail: Boite Courrier 187, Université Pierre [email protected] et Marie Curie, 4 Place Jussieu 75252 Paris Cedex 05, France INSTITUTIONAL SUBSCRIPTIONS FOR THE EMS NEWSLETTER e-mail: [email protected] Institutes and libraries can order the EMS Newsletter by mail from the EMS Secretariat, OPEN UNIVERSITY Department of Mathematics, P. O. Box 4, FI-00014 University of Helsinki, Finland, or by e- PRODUCTION TEAM mail: ([email protected]). Please include the name and full address (with postal code), tele- Liz Scarna, Kathleen Quinn phone and fax number (with country code) and e-mail address. The annual subscription fee (including mailing) is 65 euros; an invoice will be sent with a sample copy of the Newsletter.

EMS September 2001 1 EMS NEWS EMS Committee EMS Agenda EXECUTIVE COMMITTEE PRESIDENT (1999–2002) 2001 Prof. ROLF JELTSCH 15 November Seminar for Applied Mathematics Deadline for submission of material for the December issue of the EMS ETH, CH-8092 Zürich, Switzerland Newsletter e-mail: [email protected] VICE-PRESIDENTS Contact: Robin Wilson, e-mail: [email protected] Prof. LUC LEMAIRE (1999–2002) Department of Mathematics 19-21 November Université Libre de Bruxelles C.P. 218 – Campus Plaine EMS lectures at Università degli Studi, Tor Vergata, Rome (Italy) Bld du Triomphe Lecturer: Michèle Vergne (Ecole Polytechnique, Palaiseau, France) B-1050 Bruxelles, Belgium Title: Convex Polytopes e-mail: [email protected] Prof. BODIL BRANNER (2001–2004) Contact: Prof. Maria Welleda Baldoni, e-mail:[email protected] Department of Mathematics Technical University of Denmark 22-23 November Building 303 DK-2800 Kgs. Lyngby, Denmark Fifth Diderot Mathematical Forum e-mail: [email protected] Title: Mathematics and Telecommunications SECRETARY (1999–2002) Venues: Eindhoven (Netherlands), Helsinki (Finland) and Lausanne Prof. DAVID BRANNAN Department of Pure Mathematics (Switzerland) The Open University Contacts: Paul Urbach at Philips, Eindhoven ([email protected]); Olavi Walton Hall Nevanlinna, Helskinki University of Technology ([email protected]); Milton Keynes MK7 6AA, UK e-mail: [email protected] Gerard Ben Arous, Ecole Polytechnique Fédérale de Lausanne TREASURER (1999–2002) ([email protected]) Prof. OLLI MARTIO Department of Mathematics P.O. Box 4 2002 FIN-00014 University of Helsinki 9-10 February Finland e-mail: [email protected] EMS Executive Committee Meeting in Brussels (Belgium), at the invitation ORDINARY MEMBERS of the Belgian Mathematical Society and the Université Libre de Bruxelles Prof. VICTOR BUCHSTABER (2001–2004) Department of Mathematics and Mechanics Moscow State University 15 February 119899 Moscow, Russia Deadline for submission of material for the March issue of the EMS Newsletter e-mail: [email protected] Contact: Robin Wilson, e-mail: [email protected] Prof. DOINA CIORANESCU (1999–2002) Laboratoire d’Analyse Numérique Université Paris VI 24-28 February 4 Place Jussieu EMS Summer School in Eilat (Israel) 75252 Paris Cedex 05, France e-mail: [email protected] Title: Computational Algebraic Geometry and Applications Prof. RENZO PICCININI (1999–2002) Contact: Mina Teicher, e-mail: [email protected] Dipartimento di Matematica e Applicazioni Università di Milano-Bicocca Via Bicocca degli Arcimboldi, 8 1 March 20126 Milano, Italy Deadline for Proposals for 2003 EMS Lectures. e-mail: [email protected] Contact: David Brannan, e-mail: [email protected] Prof. MARTA SANZ-SOLÉ (1997–2000) Facultat de Matematiques Universitat de Barcelona 31 May Gran Via 585 EMS Executive Committee meeting in Oslo (Norway) E-08007 Barcelona, Spain e-mail: [email protected] Prof. MINA TEICHER (2001–2004) 1-2 June Department of Mathematics and Computer EMS Council Meeting, Oslo Science Bar-Ilan University Ramat-Gan 52900, Israel 3-8 June e-mail: [email protected] Abel Bicentennial Conference, Oslo EMS SECRETARIAT Ms. T. MÄKELÄINEN Department of Mathematics 2004 P.O. Box 4 25-27 June FIN-00014 University of Helsinki Finland EMS Council Meeting, Stockholm (Sweden) tel: (+358)-9-1912-2883 fax: (+358)-9-1912-3213 27 June-2 July telex: 124690 e-mail: [email protected] 4th European Congress of Mathematicians, Stockholm website: http://www.emis.de 2 EMS September 2001 EDITORIAL advisers were elected for this new series: Fabrizio Catanese and Ragnar Winther, and the first volume of the series, authored by EditorialEditorial Nigel Cutland, appeared as LNM 1751 dur- ing 2000. At present, two more volumes have Carles Casacuberta (Barcelona) been submitted, and four are in preparation.

(EMS Publications Officer, Chair of the Publications Committee) The Publications Committee Stewart Robertson retired and ended his term Since the foundation of the EMS in 1990, the of office as Chairman of the Publications Society’s publications strategy has been a Committee in April 1997. Marta Sanz-Solé source of lengthy debates, often with divided was elected Chair pro tempore and held this views and energetic standpoints. Yet, a lull position until the end of that year. In October has arrived when most of the former subjects 1997, terms of reference for the Publications of discussion have become realities, while Committee were formally approved; among exciting new ventures are about to start. The other things, it was stated that ‘The recently appointed Managing Director of the Publications Committee has as its main task EMS Publishing House, Thomas to formulate a strategy of publications of the Hintermann, starts his term of office in Society, to suggest and discuss ideas for its September 2001. Before looking into the implementation, and to supervise the devel- future, however, let me shortly review the opment of EMS publications’. The current past. Chair was appointed in 1998 and has been re- elected for 2001 and 2002. The other com- The first steps mittee members are the Publicity Officer The EMS Publications Committee started at (David Salinger), the Editor-in-Chief of JEMS the same meeting that the Society was found- Electronic versus traditional (Jürgen Jost), the Newsletter Editor (Robin ed. Stewart Robertson was elected as Chair. Articles published in JEMS become available Wilson), the Chair of the Electronic His first task was to launch the Newsletter, on EMIS, the Society’s website, two years after Publishing Committee (Bernd Wegner), and which has appeared regularly every three the printed version. By the middle of the past the Managing Director of the EMSph months since September 1991. It was initial- decade, debates about the impact and man- (Thomas Hintermann). This updated list of ly edited by David Singerman (Southampton) agement of electronic publications were ubiq- members was agreed by the Executive and Ivan Netuka (Prague). An editorial team uitous. Peter Michor, former EMS Secretary, Committee in March 2001, in accordance from the Glasgow Caledonian University, was strongly supportive of the new means of with the terms of reference. The most imme- represented by Roy Bradley and Martin storage and diffusion. He created EMIS in diate goal of the committee is to define its way Speller, undertook the production in 1996, 1995, in collaboration with FIZ Karlsruhe. of action for the years to come, obviously in still in connection with the Prague team. In The website was implemented by Michael Jost coordination with the activities of the EMS early 1999, Robin Wilson accepted the role of and Bernd Wegner; it contains general infor- Publishing House. Editor-in-Chief and built up the current edi- mation about EMS, currently compiled by torial team. The inner format was designed Volker Mehrmann, and a large Electronic The EMS Publishing House by Jan Kosniowski, from Armstrong Press Library giving free access to journals, mono- The old dream of making EMS a publishing (Southampton), while the cover design stems graphs and preprints. Bernd Wegner has force went through its decisive step soon after from Marie-Claude Vergne of IHÉS (Paris). dedicated an enormous amount of time and Rolf Jeltsch began his presidency in 1999. It is unanimously recognised that, after the ability to this and other initiatives involving The creation of the EMS Publishing House efforts and expertise of all the people EMS, such as the Mathematical Preprints (EMSph) was approved by the Executive involved, the EMS Newsletter has reached Server System, the EULER project, and the Committee in November 2000, after a maturity in its content and layout. EMS Publishing House. Most notably, the detailed proposal of an ad hoc committee. EMS is an active partner of Zentralblatt The EMSph is not financially dependent on The birth of JEMS MATH since 1997. The process to make the EMS, but it belongs to a newly created The desire to produce a learned journal was Zentralblatt a large European-based infra- European Mathematical Foundation. This is also as old as the Society itself, in spite of structure started with a French-German coop- a non-profit organisation, subsidiary of EMS, much controversy about the manner in which eration in 1995, highlighted by the creation with seat in Zurich; its statutes have recently it should materialise. A letter of intent was of the Cellule MathDoc in Grenoble. It is cur- been registered under Swiss law. According signed by EMS President Friedrich rently being cofunded by the European to these, the tasks of the Foundation include Hirzebruch and representatives of Springer- Commission under the LIMES project, which the establishment and supervision of the Verlag in December 1994. At that stage, the is coordinated by FIZ Karlsruhe and in which EMSph, the furtherance of the activities of the negotiations were conducted by Stewart EMS is a major partner. The EMS has a EMS, and support of activities of corporate Robertson and David Wallace. The journal Database Committee, chaired by Laurent member societies of the EMS. was to have a distinct European flavour and Guillopé, after John Coates, who chaired it The EMSph aims to provide a framework carry articles in as wide a variety of fields as from its creation in 1995 until 2000 and an where member societies can join efforts in possible. Nevertheless, still a long way was Electronic Publishing Committee, which has printing and distributing mathematical jour- needed until a contract was signed and the successively been chaired by Peter Michor nals. Two prospective meetings with editors first issue of JEMS appeared in January 1999, (since 1995) and Bernd Wegner (since 2001). of learned journals were held in August 1998 after the impulse given by EMS President (Berlin) and July 2000 (Barcelona), resulting Jean Pierre Bourguignon. Thanks are due to Other EMS publications in clearly positive reactions and several pre- Springer for the friendly collaboration and Originally, the text of EMS Lectures was to be liminary intents of partnership. The EMS joint promotion of the journal. The board of published in JEMS. This idea was later dis- wishes however to preserve the current diver- editors of JEMS consists of Jürgen Jost carded and a project came out, under the sity of mathematical literature in Europe and (Editor-in-Chief), Luigi Ambrosio, Gérard presidency of Jean-Pierre Bourguignon, to does not want to damage any existing coop- Ben Arous, John Coates, Helmut Hofer, produce a series of volumes with material eration schemes. As a non-profit body, the Alexander Merkurjev, and twenty-five associ- from the EMS Lectures, EMS Summer EMSph will fight to keep prices as low as pos- ate editors. As the Editor-in-Chief wrote in his Schools and Diderot Mathematical Forum sible; on the other hand, the EMSph has to 1998 Newsletter editorial, ‘JEMS aims at pre- events. This initiative led to a letter of agree- prove itself financially viable, and in the long serving the unity of mathematical thinking by ment with Springer in 1998, in which an EMS run it should enable further initiatives presenting profound and important advances subseries of Springer Lecture Notes in towards enhancing the visibility, strength and in both pure and applied mathematics’. Mathematics was originated. Two scientific influence of the EMS. EMS September 2001 3 EMS NEWS EMSEMS SummerSummer SchoolSchool atat StSt PPetersburetersburgg 20012001 Asymptotic combinatorics with application to mathematical physics

ular new results in mathematical physics (P. Deift), combinatorics (A. Borodin and R. Kenyon) and algebraic geometry (A. Okounkov). The impressions of the participants were very positive and most of them con- sidered the lectures to be very useful. The programme was over-full, with additional talks. A round table on current problems was organised, and the younger partici- pants took part in short discussions about their studies. The lectures and some addi- tional matter will be published in the spe- cial volumes of the proceedings of the school, to be edited by V. Malyshev and A. Vershik; Volume 1 will be published by Springer-Verlag, and Volume 2 by the Kluver Publishing House, in 2002.

Main lectures P. Biane, Asymptotics of representations of symmetric groups, random matrices and free cumulants A. Borodin, Asymptotic representation theory and Riemann-Hilbert problem M. Bozejko, Positive definite functions on Coxeter groups and second quantization The EMS Summer School 2001 and April, but there were a lot of applications of Yang-Baxter type NATO Advanced Study Institute at St after the deadline. However, many who E. Brezin, An introduction to matrix mod- Petersburg, Asymptotic combinatorics with had registered in time did not take part, els application to mathematical physics, attracted for different reasons, including some main P. Deift, Random matrix theory and com- the leading world-wide specialists in the lecturers. The support of the EMS allowed binatorics: a Riemann-Hilbert approach theory of integrable systems, random us to include more participants from East L. Faddeev, 3-dimensional solitons and matrix theory, the Riemann-Hilbert prob- Europe than was announced in the appli- knots lem and asymptotic combinatorics. Many cation. J. L. Jacobsen, Enumerating coloured tan- young mathematicians took part in the The scientific Committee was A. Vershik gles summer school and the school was espe- (Chair), L. Faddeev, E. Brezin, P. Deift, V. V. Kazakov, Matrix quantum mechanics cially important for them. One important Malyshev and O. Bohigos. and statistical physics on planar graphs features of the school was the presence of R. Kenyon, Hyperbolic geometry and the mathematicians and physicists simultane- Scientific report low-temperature expansion of the Wulff ously as lecturers and students. The Asymptotic combinatorics with applications to shape in the 3D Ising model school was successful and fruitful, and mathematical physics was devoted to some V. Korepin, Quantum spin chains and many new contacts between participants areas of mathematics and mathematical Riemann zeta function with odd argu- were established. More than half the par- physics that have been studied very inten- ments ticipants were less than age 35. sively in recent years. The idea was to get I. Krichever, τ-functions of conformal The preparation was started in the specialists in integrable systems, asymptot- maps autumn of 1999 after EMS Executive ic combinatorics, representation theory, V. Liskovets, Some asymptotic distribution Committee meeting in Zurich. An appli- random matrix theory and quantum field patterns for planar maps cation was sent to the NATO Scientific theory to give short courses of lectures on V. Malyshev, Combinatorics and probabil- Affairs Division in July 2000, and a grant the subject and attract young students to ity for maps on two-dimensional sur- was obtained in January 2001. A grant those areas. The list of lecturers included faces from the Russian Fund of Basic specialists of the highest level (P. Deift, E. M. Nazarov, On the Frobenius rank of a Researches was obtained in April 2001. Brezin, L. Faddeev, etc.). The school well skew Young diagram Information about the school was pub- confirmed the organisers’ initial idea on S. Novikov, On the weakly nonlocal lished in the AMS Notices, the EMS the fruitfulness of the interrelations Poisson and symplectic structures Newsletter, the INTERNET sites of NATO, between asymptotic combinatorics and A. Okounkov, Combinatorics and moduli the AMS Mathematics Calendar, the mathematical physics. Methods from the spaces of curves International Euler Mathematical Institute theory of integrable systems and matrix G. Ol’shanski, Harmonic analysis on big and the St Petersburg Mathematical problems, as well as the theory of the groups, and determinantal point Society. A poster was prepared by the Riemann-Hilbert problem together with processes EMS and sent to many Universities. asymptotic combinatorics and representa- L. Pastur, Eigenvalue distribution of uni- The opening session was at the Steklov tion theory, give very powerful tools for tary invariant ensembles of random Institute of Mathematics in St Petersburg, the solution of many old problems (fluctu- matrices of large order and all the lectures and seminars were ations of the eigenvalues of random matri- S. Smirnov, Critical percolation is confor- held at the Euler International ces, counting the number of coverings of mally invariant Mathematical Institute (part of the Steklov algebraic curves, universality of distribu- R. Speicher, Free probability and random Institute) and were supported by staff from tion of spacing and other statistics charac- matrices the Euler and Steklov Institutes. teristics of Young diagrams, etc.). We A. Vershik, Introduction to asymptotic The deadline for the registration was 1 emphasise some of the lectures on spectac- theory of representations 4 EMS September 2001 EMS NEWS There are also some direct flights from Europe to Eilat: you may check this possi- EAGER - ENI - EMS Summer School bility with your own travel agency. Computational Algebraic Geometry and Applications Occasionally there are charter flights from Eilat, Israel 24-28 February 2002 Europe to Ovda Airport (40 minutes from First announcement Eilat: no regular bus services, but taxis or transportation provided by the charter company). The school is an introduction on how to use southernmost point of Israel, a resort city From Eilat airport or from the Eilat cen- computer algebra systems such as SINGU- hugging the shores of the Red Sea, sur- tral bus station, take a taxi to the hotel, or LAR and MACAULAY2 and packages such rounded by the magnificent Edom walk. as SCHUBERT (intersection theory) for Mountain Range and characterised by its research in algebraic geometry and its crystal clear waters and year-round sun- Registration applications. The programme consists of shine. Its unique undersea vista, flora and The number of places in the school is limit- lectures and practical exercise sessions with fauna can be admired by boat, from a ed. If you are interested in coming, fax the a computer. breathtaking underwater observatory, or by form below by 15 September to (972-3)- The topics to be considered include: snorkelling or diving. The inland desert 5353325, att. Boris Kunyavski, indicating computer algebra systems, Gröbner bases landscapes are no less fascinating. Eilat is arrival/departure data and credit card and syzygies, ideal and radical member- well known for its mild winter climate. The number. ship, manipulating ideals and modules, weather in Eilat during February is warm Hilbert polynomials and Hilbert functions, during the day and cooler at night; tem- Financial support for local expenses elimination, computations in local rings peratures range from 12C to 25C, and it If you belong to EAGER you can apply to and Milnor numbers, homological algebra seldom rains. Participants who extend your node coordinator for support. (constructive module theory, Ext and Tor, their stay may like to join interesting day- Otherwise, please contact Boris sheaf cohomology, Beilinson monads), pri- trips from Eilat, inside Israel or over the Kunyavski at the Emmy Noether Institute: mary decomposition, normalisation, rings border to Jordan or Egypt (Moon Valley, [email protected] indicating age, academic of invariants, parametrisation, deforma- Akaba, Petra, Wadi Rum, Nuweiba, degree and institution where awarded, PhD tions, intersection theory, applications to Colored Canyon and Santa-Catarina). thesis title, name of advisor, nationality, special varieties, computer vision and cod- (Reservations can be made on the spot place of residence and affiliation. ing theory. through the hotel concierge.) If you are getting financial support from The course will be directed by Prof. The Summer School will take place in any of the above sources, please send your Wolfram Decker (Saarbrucken, Germany). Hotel Meridian. The hotel special confer- hotel receipts to the responsible party after There will be additional guest lectures by ence rates are $90 a day per person (FB) in the conference for reimbursement. Dr Jeremy Kaminski (Bar-Ilan, Israel). a double occupancy; the registration fee is $80. Organisation Computers More details will appear in the second Profs. Mina Teicher and Boris Kunyavski of Please bring your laptop with you. Two stu- announcement. the Emmy Noether Institute dents can use one computer, so if you coor- (Conference Secretary: Ms Chen Fireman) dinate with a fellow student you need bring Arrival just one computer between you. Before You can get to Eilat by bus from the central Sponsors coming you should download some soft- bus station in Tel Aviv (about 4.30 hours); EMS (European Mathematical Society), ware from the Internet – this will be made this is very economical and is the way we EAGER (European Algebraic Geometry precise in January 2002. recommend. Education and Research) and ENI (Emmy The standard way to arrive by air is to Noether Research Institute for Accommodation take a connecting flight to Eilat city airport Mathematics at Bar-Ilan University and the The site of the Conference, Eilat, is the from Ben-Gurion international airport. Minerva Foundation) To be sent by fax to (972-3)-5353325, att. Boris Kunyavski Hotel registration form EMS/EAGER Summer School JEMS Computational Algebraic Geometry Eilat, Hotel Meridian, 25-28 February 2002 Journal of the European Surname: Mathematical Society First (and other) name(s): Affiliation: Volume 3, number 3 of JEMS con- tains: Tel.: L. Birgé and P. Massart: Gaussian model selection Fax: R. Meyer: Excision in Entire Cyclic e-mail: Cohomology

Credit card number: Volume 3, number 4 of JEMS con- Arrival data (date and time): tains: Departure date: A. de Carvalho and T. Hall: Pruning theory and Thurston’s classi- Accompanying persons (with names, sex, ages, if under 16): fication of surface homeomorphisms O. Biquard and M. Jardim: Accommodation: single room Asymptotic behaviour and the moduli double room with ... space of doubly-periodic instantons Remarks:

EMS September 2001 5 EMS NEWS of the Executive Committee shall be elect- ed for a period of 4 years. Committee MeetingMeeting ofof thethe EMSEMS CouncilCouncil members may be re-elected, provided that consecutive service shall not exceed 8 Oslo: 1-2 June 2002 years. No current member has served on the Executive Committee for 8 years, so all First Announcement existing Committee members are in prin- ciple available for re-election. The EMS Council meets every second year. Election to Council of delegates of other It would be convenient if potential nom- The next meeting will be held in Oslo, categories of EMS members inations for office in the Executive Norway, on 1-2 June 2002, before the Abel Full EMS Members are national mathe- Committee, duly signed and seconded, Centennial Meeting in Oslo which begins matical societies, which elect 1, 2 or 3 del- could reach the Secretariat by 1 February on 3 June. The first session of the Council egates according to their size and 2002. It is strongly recommended that a meeting will start at 10 a.m. on 1 June, and resources. Each society is responsible for statement of intention or policy is enclosed will run all day with a break for lunch. The the election of its delegates. Each society with each nomination. second session will probably start at 9 or 10 should notify the EMS Secretariat in The Council may, at its meeting in Oslo, a.m. on 2 June, and may last most or all of Helsinki of the names and addresses of its add to the nominations received and set up the day with a break for lunch, depending delegate(s) no later than 1 February 2002. a Nominations Committee, disjoint from on the volume and complexity of the busi- As of 1 July 2001, there were about 47 such the Executive Committee, to consider all ness on the agenda. societies, which could designate a maxi- candidates. After hearing the report by mum of about 69 delegates. the Chair of the Nominations Committee Invitation to suggest business for the There is one associate EMS member: the (if one has been set up), the Council will Council Gesellschaft für Mathematische proceed to the elections to the Executive The Executive Committee is responsible Forschung. According to the Statutes: Committee posts. for preparing the matters to be discussed “delegates representing associate members If a nomination comes from the floor at Council meetings. Items for the agenda shall be elected by a ballot organized by during the Council meeting, there must be of this Council meeting should be sent the Executive Committee from a list of a written declaration of the willingness of (preferably by e-mail) as soon as possible, candidates who have been nominated and the person to serve, or his/her oral state- and no later than 1 February 2002, to the seconded, and have agreed to serve.” ment must be secured by the chair of the EMS Secretariat in Helsinki. The There are three institutional EMS mem- Nominating Committee (if there is such) Executive Committee will meet on 9-10 bers: Institut Non-Linéaire de Nice, the or by the President. It is recommended February 2002 to put together the Council Moldovian Academy of Sciences and the that a statement of policy of the candidates agenda. Mathematical Institute of the Serbian nominated from the floor should be avail- Delegates to the Council will be elected Academy of Sciences and Arts. According able. by the various categories of members, as to the Statutes: “delegates representing per the Statutes. institutional members shall be elected by a Accommodation arrangements ballot organized by the Executive Delegates to the Council meeting, who are Election to Council of representatives of Committee from a list of candidates who planning to attend the Abel Centennial ‘individual EMS members’ have been nominated and seconded, and Meeting, are advised that their accommo- A person becomes an individual member have agreed to serve.” dation arrangements should be made of EMS either through a corporate mem- The EMS Secretariat will contact the through the normal Abel Centennial ber, by paying an extra fee, or by direct society members in these three categories Meeting organisation arrangements. For membership. On 14 July 2001, there were directly in connection with their delegates. delegates to the Council who are not some 2247 individual members, and, attending the Abel Centennial Meeting, an according to our Statutes, these members Membership of the EMS Executive address for accommodation arrangements will be represented on Council by 23 dele- Committee will be provided later. gates. Nomination papers for these elec- The Council is responsible for electing the tions appear in this Newsletter. President, Vice-Presidents, Secretary, Secretariat: Ms. Tuulikki Mäkeläinen Treasurer and other members of the Department of Mathematics 15 delegates were elected for the term 2000-03, Executive Committee. The present mem- P. O. Box 4 so they will continue unless they inform the bership of the Executive Committee, FIN-00014 University of Helsinki Secretariat to the contrary by 31 December together with their individual terms of Finland 2001. They are Giuseppe Anichini office, is as follows. e-mail: [email protected] (Firenze, Italy), Vasile Berinde (Baia Mare, President Romania), Giorgio Bolondi (Milano, Italy), Professor David Brannan Alberta Conte (Torino, Italy), Chris Rolf Jeltsch (1999-2002) Secretary of the EMS Dodson (Manchester, UK), Jean-Pierre Vice-Presidents e-mail: [email protected] Françoise (Paris, France), Salvador Gomis Professor Luc Lemaire (1999-2002) (Alicante, Spain), Laurent Guillopé Professor Bodil Branner (2001-04) (Nantes, France), Willi Jägger (Heidelberg, Secretary Timetable for the Germany), Klaus Habetha (Aachen, Professor David Brannan (1999-2002) Germany), Tapani Kuusalo (Jyväskylä, Treasurer Council Meeting Finland), Laslo Marki (Budapest, Professor Olli Martio (1999-2002) Hungary), Andrzej Pelczar (Krakow, Members September 2001 Poland), Zeev Rudnick (Tel Aviv, Israel) Professor Victor Buchstaber (2001-04) Information on the Council meeting is and Gerard Tronel (Paris, France). Professor Doina Cioranescu (1999-2002) printed in the EMS Newsletter. A nomina- The mandates of 5 of the present 20 delegates Professor Renzo Piccinini (1999-2002) tion form for delegates of the individual end on 31 December 2001, and so elections Professor Marta Sanz-Solé (2001-04) members to Council is given. Suggestions must be held for their positions. They are Professor Mina Teicher (2001-04) for Council business and for Executive Manuel Castellet (Barcelona, Spain), The President may serve only one term of Committee membership are invited. George Jaiani (Tbilisi, Georgia), Marina office, so Rolf Jeltsch cannot be re-elected Letters are sent to full, associate and Marchisio (Boves, Italy), Vitali Milman (Tel as President. David Brannan and Renzo institutional members, as well as continu- Aviv, Israel) and Jan Slovak (Brno, Czech Piccinini have indicated that they do not ing delegates, giving information on the Republic). All of these can be re-elected, since currently wish to be re-elected. Council meeting. (Delegates are kindly they have served in this capacity for only 4 years. Under Article 7 of the Statutes, members requested to keep the Secretariat informed 6 EMS September 2001 EMS NEWS of their correct and up-to-date addresses.) Specifically, points for the agenda and sug- gestions for future members of the Executive Committee are invited. Election of Council Delegates representing the Individual members of the Society 2 November Deadline for nominations for delegates to Council of individual members. Nominations are required for Council del- All of these can be re-elected, since they December egates representing individual members of have served in this capacity for only 4 A ballot paper for delegates of individual the Society. On 14 July 2001, there were years. members to Council is sent to individual some 2247 individual members and, Nominations are therefore now sought members in the EMS Newsletter. The according to our Statutes, these members for 8 delegates to serve for the years 2002- venue and meeting times of the Council will be represented on Council by 23 dele- 05. With this notice in the Newsletter is a meeting are announced. gates. nomination form. Completed nomination 15 delegates were elected for the term forms must arrive at the Society’s office in 1 February 2002 2000-03, so they will continue unless they Helsinki by 2 November 2001. (A photocopy Close of voting for delegates to Council of inform the Secretariat to the contrary by of the nomination form is acceptable.) If there individual members. 31 December 2001. These delegates are: are more nominations than the allowed Giuseppe Anichini (Firenze, Italy) number of delegates, a postal election will February Vasile Berinde (Baia Mare, Romania) be held. Members will receive ballot forms Members elected as delegates to Council of Giorgio Bolondi (Milano, Italy) in the December 2001 Newsletter, which individual members are contacted by the Alberta Conte (Torino, Italy) must be returned by 1 February 2002. EMS Secretariat to inform them of their Chris Dodson (Manchester, UK) Nominated individuals must be individ- election, and to let them know the venue, Jean-Pierre Françoise (Paris, France) ual members of the Society, and they must meeting times and agenda of the Council Salvador Gomis (Alicante, Spain) be proposed and seconded by individual meeting. Laurent Guillopé (Nantes, France) members. The Society will pay subsistence Willi Jägger (Heidelberg, Germany) costs for them to attend the Council meet- March Klaus Habetha (Aachen, Germany) ings, if necessary, but it is not able to cover The results of the elections for delegates to Tapani Kuusalo (Jyväskylä, Finland) travel costs except perhaps in cases of par- Council of individual members are Laslo Marki (Budapest, Hungary) ticular hardship. announced in the EMS Newsletter. The Andrzej Pelczar (Krakow, Poland) Candidates for election are invited to venue, the meeting times, and the agenda Zeev Rudnick (Tel Aviv, Israel) submit with their nomination form a short of the Council meeting are given. Gerard Tronel (Paris, France) biography (not more than 200 words), A letter is sent to each delegate to Council, together with a statement of not more than containing the venue, meeting times and The mandates of 5 of the present 20 dele- 100 words in support of their candidature. agenda of the Council meeting. gates end on 31 December 2001, and so These will be circulated to the Society elections must be held for their positions. members with the ballot forms. A copy of April They are: the biography and statement can be sent as Final material for the Council meeting is Manuel Castellet (Barcelona, Spain) a text file by e-mail to the Secretariat at the sent to the delegates. George Jaiani (Tbilisi, Georgia) following e-mail address: makelain@cc. Marina Marchisio (Boves, Italy) helsinki.fi 1-2 June 2002 Vitali Milman (Tel Aviv, Israel) David Brannan Council meeting in Oslo. Jan Slovak (Brno, Czech Republic) EMS Secretary

NOMINATION FORM FOR COUNCIL DELEGATE [A photocopy of this nomination form is acceptable.]

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I certify that I am an individual member of the EMS and that I am willing to stand for election as a delegate of individual mem- bers to the Council.

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Completed forms should be sent to: Ms T. Mäkeläinen, EMS Secretariat, Department of Mathematics, P.O. Box 4, FIN-00014 University of Helsinki, Finland to arrive by 2 November 2001.

Note: While it is highly desirable that the Biography and Statement of candidates be received by e-mail (as text files), it is necessary that nomination forms are received in hard copy format to ensure the genuineness of signatures.

EMS September 2001 7 EMS NEWS Rational Mech. Anal. 139 (1997), 201-238. (with F. Murat) Asymptotic behaviour and cor- EMS Lecturer 2002 : Gianni Dal Maso rectors for Dirichlet problems in perforated domains with homogeneous monotone oper- The EMS Lecturer for 2002 will be vergence. In particular he developed, ators, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) Professor Gianni Dal Maso of the together with Giuseppe Buttazzo, several 24 (1997), 239-290. International School for Advanced Studies techniques to prove under different (with A. Braides) Non-local approximation of (SISSA) in Trieste, Italy. hypotheses that the gamma-limits of inte- the Mumford-Shah functional, Calc. Var. It is planned that he will visit two differ- gral functionals are still integral function- Partial Differential Equations 5 (1997), 293-322. ent locations in Europe to give the same als, and he used gamma-convergence tech- (with F. Murat, L. Orsina and A. Prignet) series of lectures to different audiences, niques to study the asymptotic behaviour Renormalized solutions of elliptic equations affording as many interested mathemati- of solutions to minimum problems with with general measure data, Ann. Scuola Norm. cians as possible the opportunity to attend strongly oscillating obstacles. Using the Sup. Pisa Cl. Sci. (4) 28 (1999), 741-808. and discuss the topics with him. EMS notion of capacity, he also gave a complete (with G. Alberti and G. Bouchitte) The calibra- members interested in organising such a characterisation of the sequences of obsta- tion method for the Mumford-Shah function- visit are invited to contact Professor Dal cle problems whose variational limit is still al, C.R. Acad. Sci. Paris I Math. 329 (1999), Maso directly (e-mail: [email protected]) by an obstacle problem. 249-254. mid-October, copying their e-mail to He later used these techniques to study, (with R. Toader) A model for the quasi-static Professor David Brannan (e-mail: d.a. with Umberto Mosco, the asymptotic growth of brittle fractures: existence and [email protected]). behaviour of the solutions of Dirichlet approximation results, Arch. Rational Mech. problems for the Laplace equation in per- Anal., to appear. Brief biography forated domains, and to determine the Professor Dal Maso was born in Vicenza, general form of their variational limits, as Italy, in 1954; in 1955 his family moved to well as the fine properties of the solutions Trieste, where he received his basic educa- of these limit problems. These results have Bringing tion. He was a student of the Scuola been extended, with different collabora- Normale of Pisa from 1973 to 1977, and tors, to the case of other linear and non- graduated in mathematics from the linear equations and systems. mathematicians University of Pisa in 1977, with Ennio De At present his main research interests Giorgi as his advisor. He was then a grad- are in free discontinuity problems. These uate student of the Scuola Normale di Pisa are variational problems where the func- into biology from 1978 to 1981, working with Ennio De tional to be minimised depends on a func- The Human Frontier Science Program is Giorgi on many problems connected with tion and on its discontinuity set, whose an international funding agency, sup- the theory of gamma-convergence that was shape and location are not prescribed. In ported by the G7 governments, the developed in those years. many cases the discontinuity set can be European Union and Switzerland. After serving as Assistant Professor of considered as the main unknown of the The HFSP supports interdisciplinary Mathematical Analysis in the Faculty of problem. Examples are given by the min- international collaborations in the life Engineering of the University of Udine imisation of the Mumford-Shah functional sciences, with an increasing focus on from 1982 to 1985, he moved to the in image segmentation, and by the mini- bringing scientists from various fields International School for Advanced Studies mum problems which appear in many vari- such as physics, mathematics, chem- in Trieste. He worked there as Associate ational models for fracture mechanics, istry, computer science and engineer- Professor of Mathematical Analysis from where the unknown crack is represented as ing, together with biologists, to open 1985 to 1987, and has been Full Professor the discontinuity set of the displacement up new approaches to understanding of Calculus of Variations since 1987. He vector, and the functional to be minimised complex biological systems. was awarded the Caccioppoli Prize in 1991 is the sum of the elastic energy and of an The HFSP promotes international and the ‘Medaglia dei XL per la integral on the discontinuity set, which collaboration through collaborative Matematica’ of the Accademia Nazionale represents the work done to produce the research grants and post-doctoral fel- delle Scienze detta dei XL in 1996. crack. lowships. Long-term and short-term At SISSA he has developed his research fellowships are available for scientists interests on gamma-convergence, Selected list of publications early in their research careers. Long- homogenisation theory and free disconti- An Introduction to Gamma-Convergence, term fellowships provide three years of nuity problems, and has supervised 19 Birkhäuser, Boston, 1993. support to obtain research training in PhD students working on these subjects. Asymptotic behaviour of minimum problems another country in a new research He is currently Head of the Sector of with bilateral obstacles, Ann. Mat. Pura Appl. area; the third year can be used in the Functional Analysis and Applications of (4) 129 (1981), 327-366. home country and under this condi- SISSA. Some necessary and sufficient conditions for the tion can be delayed for up to two years. convergence of sequences of unilateral convex The application deadline is Research interests sets, J. Funct. Anal. 62 (1985), 119-159. September each year. Short-term fellow- Professor Dal Maso started his research (with U. Mosco) Wiener criteria and energy ships provide travel and subsistence work in Pisa while Ennio De Giorgi was decay for relaxed Dirichlet problems, Arch. support to visits from 2 weeks to 3 developing the new notion of gamma-con- Rational Mech. Anal. 95 (1986), 345-387. months to acquire new techniques or vergence to deal systematically with the fol- (with G. Buttazzo) Shape optimization for establish new collaborations; there is lowing phenomena: the solutions of varia- Dirichlet problems: relaxed formulation and no deadline for this programme. tional problems depending on a parame- optimality conditions, Appl. Math. Optim. 23 Research grants support international ter may converge to the solution of a limit (1991), 17-49. collaborative projects. Teams of scien- problem even if the integrands of the func- (with J. M. Morel and S. Solimini) A variational tists wishing to apply for a grant must tionals to be minimised do not converge in method in image segmentation: existence submit a letter of intent via the HFSP any reasonable sense, or converge to a and approximation results, Acta Math. 168 web site. The next deadline for appli- limit integrand which is different from the (1992), 89-151. cations for letters of intent to submit integrand of the functional minimised by (with A. Garroni) New results on the asymptotic research grants is 30 March 2002. the limit of the solutions. Gamma-conver- behaviour of Dirichlet problems in perforated Further information can be obtained gence is a very efficient tool to tackle this domains, Math. Mod. Meth. Appl. Sci. 3 (1994), from the HFSP web site at kind of problems. 373-407. http://www.hfsp.org In his work in Pisa and Udine he studied (with I. Ambrosio and A. Coscia) Fine properties several problems related to gamma-con- of functions with bounded deformation, Arch. 8 EMS September 2001 INTERVIEW InterviewInterview withwith MarMarekek KKorordosdos (Editor-in-Chief of the Polish monthly Delta) interviewers: Krzysztof Ciesielski and Zdzis³aw Pogoda

‘Delta’ is a popular mathematical, physical and said: ‘Certainly. How much time do you astronomical monthly publication by the Polish want?’. I answered: ‘One week’. Then Mathematical Society, Polish Physics Society Sikorski told me: ‘Of course, you have one and Polish Astronomical Society. The first issue week, but tomorrow you must take part in was published in January 1974. Since then, a special meeting with physicists and, as Professor Marek Kordos from the Mathematics Editor-in-Chief, select the associate editor. Institute of Warsaw University has been the So, in fact I never agreed to be the Editor- journal’s Editor-in-Chief. in-Chief, but I probably didn’t have to, it was not necessary. In Poland, most mathematicians know I do not know how everything connected ‘Delta’, but it is less well known abroad. with Delta would have gone on if I hadn’t What is its purpose? found Tomasz Hofmokl. He was a wonder- The main purpose of Delta is to present ful man, a very good scientist and a very mathematics, physics and astronomy by good manager (this is rather rare), and an people who work on these sciences. We excellent partner. Tomasz was capable of believe that such scientists can present everything I couldn’t do and of doing their subjects in a clear way, and do not some things I could do and this was really believe in the picture of science that is pre- fine. My meeting with Sikorski took place sented by the professional journalists. The on 3 June 1973 and on 8 December 1973 true picture we obtain is the main advan- another meeting with the Boards of the tage of our point of view. However, many Mathematical and Physical Societies was scientists do not believe that they can speak planned. During this meeting we were to and write about their results for a general present our plans concerning Delta. audience: this is the main disadvantage. Marek Kordos However, we did not present the plans, we presented the first issue just printed. This How was ‘Delta’ created? enough to say that the authorities of the was a great surprise and perhaps for that The godfather of Delta was Professor Leon Polish Mathematical Society did not like it. reason the Boards liked it very much. In Jesmanowicz, who convinced Professor Then they were terrified, they did not practice, the official acceptance of the Roman Sikorski, President of the Polish know what to do and desperately looked for shape of Delta took place in September Mathematical Society, of the idea. The an editor. Eventually, my candidature was 1974 in Torun, during the General efforts of Sikorski and Tadeusz Iwianski, put forward by Andrzej Makowski, who Assembly of the Polish Mathematical Secretary of the Society, who was very thought that my previous work on the jour- Society, where there was a row. This row clever at fighting administrative formali- nal Wiedza i Zycie (Knowledge and Life) was was provoked by the godfather of Delta, ties, brought Delta into existence. If some- a sufficient recommendation. They were Leon Jesmanowicz himself, who said that body wanted to discredit Delta he could say short of time and nobody worried about any Delta was too effective as its covers were too that the event happened during the Gierek precise investigation. Sikorski phoned me colourful: perhaps because of that, some decade, so Delta, together with small ‘Fiat and asked if I could visit him – immediate- fools might buy Delta. He regarded such a 126p’ cars, Central Railway Stations, ly, if possible. Of course, I came to him at situation as bad. Such an opinion may Lazienki Roads, etc., form the same com- once and then I was offered the job of seem strange now, but in the 1970s Delta pany. From the very beginning of Delta, Editor-in-Chief of Delta. I asked for some was the most colourful journal sold in the idea of presenting only mathematics in time to think the proposal over. Sikorski Poland, I think. Delta was defended by the journal was given up, because the authorities decided that mathematics alone is not interesting enough to fill the whole journal. In fact, I am very surprised at such a point of view, but I think it is much better that Delta is not only a mathe- matical journal.

Where did the title come from? It was Jesmanowicz who invented the title. The idea was to put something mathemat- ical in the title and ∆ = b2 – 4ac is suppos- edly the best-known mathematical term.

How did you become Editor-in-Chief? The editor had to be somebody who was involved in journalism in some way. At that time there was precisely one man in Poland who fulfilled the required condi- tion – a mathematician, who presented quizzes and logical puzzles on TV. However, this candidate was rejected when he presented his idea of the journal – let me not speak about the details, it is Marek Kordos (centre) with Krzysztof Ciesielski (right) and Zdzislaw Pogoda (left) EMS September 2001 9 INTERVIEW Professor Zofia Krygowska with the words: strongly recommend such articles, fre- International Congress of Mathematicians ‘Mathematics is colourful’. Such was the quently somebody agrees with me, so we in Warsaw beginning of Delta. fight to accept the articles regarded as triv- Yes, it attracted people mainly because of ial by others. However, one must be very our many colourful covers. Every partici- Who is Delta aimed at? careful here. In any article (easy or difficult pant of the Congress received a special We were given some suggestions about the – it does not matter), there should be issue of Delta in English, containing select- enough information. In my opinion, the ed articles from its first ten years. Some most difficult task in our editorial job is to articles into English can now be found on find articles that can be read easily read – the internet (www.mimuw.edu.pl/delta/). say, as a newspaper – not studied. Also, a selection of our articles will be pub- After the period of the authors from the lished in Catalonian, which will probably Polish Mathematical School came another be the beginning of an analogous journal generation came – a rather strange one. there. Articles from Delta are also fre- They regarded mathematical techniques as quently published in the Russian ‘Empire most important. They would prefer gath- of Mathematics’. ering different techniques (and perhaps never apply them) to something converse, Delta’s sold together with newspapers. How that is to fighting with problems without does this work? suitable techniques being learnt. This gen- Ah, this is very interesting. At first, we pub- eration (not all of them, of course) saw no lished 30000 copies of the journal and reason to inform a general audience about everything was sold. Later we jumped to their work. Fortunately, the next genera- 50000! However, ten years ago economic tion turned out to be more normal, and changes came, and several new journals young mathematicians are now very inter- and newspapers started. We believed that ested in their connections with the ‘non- many of our readers bought Delta because mathematical world’. they could not get Penthouse or Daily Rag. Now, we publish 4500 copies, which seems Somebody writes an article for ‘Delta’ and to be reasonable and is probably about the submits it. It is read, judged... number of fans of mathematics and physics ... and frequently accepted. In such a way in Poland. On the other hand, such num- we get many articles by scientists from dif- bers of copies are not easy to distribute and kind of readers. At the beginning, most ferent countries, and all continents! – from outstanding Polish mathematicians had an India, New Zealand and Vietnam, as well ambition to have an article published in as the USA, Great Britain and Germany. Delta. On the basis of what they wrote, it is But it sometimes happens that the difficult to point out the readers. Editorial Board does not want to publish a Nevertheless, I do not think that the read- submitted article. ers of Delta should be precisely charac- terised. Sometimes you suggest corrections, don’t I always explain why I do not want to you? popularise science professionally. Ah, yes, of course, but I think that we Somebody who does it professionally tries always do it in a reasonable way. Only once to take science in, to digest it and to show did I shorten an article, written by a pro- the final result to the audience. It is dis- fessor of physics. It originally had eleven gusting, isn’t it? I was always against such a pages and after my rewriting had 2 pages. method of popularising science. The The author was very grateful to me, he ‘speaking science’ is my ideal. I mean sci- liked it very much. I think it was the only entists should be able to speak about their article that was really changed. In fact, we work in the way understandable for others. interfere only in articles we want to pub- In my opinion, this is the only method if lish, and the author has always the right to we want science to be accepted. Scientists appeal. should inform others what they do and what for. If the subject is interesting for Comparing the first Editorial Board with them, they should know how to interest the present board, one can see that only you A caricature of Marek Kordos drawn by Leon others in it. This is the reason why Delta have remained. Of the present Committee, Jesmanowicz was always interested in particular authors, only Andrzej Maakowski was in the first not in particular readers. Scientific Committee… sell through a large number of news- Some time ago, Delta went through a bad Delta cannot be the main point in some- stands. period. After some years, elderly mathe- body’s CV. People come to Delta and later maticians stopped writing for it (usually for they go away. Professor Hofmokl, for Has Delta ever published original research the reasons such as illness or death) and instance, was the Assistant Editor of Delta, papers? so-called ‘young doctors’ became the later he was Chairman of the Department One summer day in 1980 Professor Karol majority of our authors. These people (not of Experimental Physics at Warsaw Borsuk called me and gave me a manu- all of them, of course) are probably the University, and at the end of his life was script. In his article he wrote about his now worst kind of authors: they want to write as the Head of Polish Internet. Besides scien- well-known theorem that there exists an quickly as possible about everything tific and academic careers, people leaving intrinsic isometry mapping n-dimensional they’ve managed to learn. They do not Delta often organise their own printing Euclidean space En onto a subset of the want anyone thinking: ‘Ah, they didn’t business. Why I didn’t I give up? I wanted space En+1 with arbitrarily small diameter. mention martingales? That means they do to several times, but there was never any The article was written in a style suitable not know what martingales are!’ Then, fool who agreed to take over the job. On for Delta. Borsuk wanted this theorem to be Delta became really too difficult. the other hand, it would be a pity to give it first published in Delta, because it acted for We still have discussions during editorial up. It seems that Sikorski sentenced me to imagination: in fact, if an alien from committee meetings. Some members of life imprisonment. Aldebaran could move in the Euclidean the committee want to reject the articles space of higher dimension, then the dis- which seem too easy and trivial. I always Delta had a stall during the 1983 tance from his feet to his nose might be the 10 EMS September 2001 INTERVIEW same as the distance from his feet to the mathematical paper by a secondary school Then the ladies told him: ‘Ah, it’s noth- Earth. Thus Delta was the first journal student… ing, really’. When, additionally, it turned which published this result. Yes, we do this together with the Polish out that the copy editor brought with him Mathematical Society, and consider only two copies of the best-seller Cezars’ lives You haven’t published only this journal, original results obtained by young people. and wanted to leave these books in the have you? The first competition took place in 1978. office, everything cleared up very soon. Delta has only 17 pages, so we had to pub- The winners are awarded Gold, Silver and Speaking of censorship, they also can- lish something else, as our writers wanted Bronze Medals and small amounts of celled a lot in the issue about ecology. to write much more. We used to publish money, but the main prize is that the win- three series of books: Delta’s library, Delta ners present their papers to the General Look back at 27 years of Delta. What do you presents and Read! Perhaps you will under- Assembly of the Polish Mathematical admire mostly? Any articles... stand: in this last series we published 26 Society. Also, the winning papers are pub- It is difficult to say which articles were the volumes. But first we published a book for lished in Delta. best: I cannot judge it in any sensible way. schoolchildren Can you wonder? which sold The level of that competition is really I would rather mention whole issues. I 200,000 copies! This is double the number high. Some papers presented here are especially liked three of them: the issues of copies of any most popular crime book later published in good professional math- on ecology and the 17th century, and a in Poland. We continued with a second ematical journals. The winner of the 1980 mini-monograph on complex numbers. book See it in another way, which was also a Competition is now a professor of mathe- Another valuable thing was a map of the best-seller. matics, while some others are on the way to sky. In 1985, we printed part of such a map These two books were based on Little professorships. Since 1995, Poland has on the back cover of each issue: twelve Delta. For some years, each issue of Delta participated in the European Competition parts put together formed a large umbrel- included a special column for younger for Young Scientists, organised by the la with the sky pictured. I have never heard readers, called Little Delta. In 1981 we gave European Community. There, papers are of anything similar to that. up this column and start publishing an presented by young students and school- independent journal Little Delta. After two children, aged 20 or less. All the mathe- years, its title changed to A glass and an eye, matical papers from Poland in that compe- the editors changed, and the journal was tition won in our Competition first, and edited by the group from Bialystok. 99 gained one Silver Medal, two Bronze issues were published, and A glass and an Medals and one special distinction in the eye existed for about ten years. Now we European Competition. These were all the have a Little Delta column again. We also prizes gained by mathematicians in the produced special TV programmes and European Competition. even some theatre and circus perfor- mances. What about the future of Delta? Our situation is complicated. It is similar to What about other columns in Delta? Bialowieza National Park in Poland. Two The oldest is Club 44, and is a challenge to rare species live in this National Park: the readers. In each issue some mathemat- large ones, European bisons and small ical and physical problems are published ones, Laxmann’s shrews. Bisons are at least and the readers are invited to send in solu- representative, but what are shrews are? tions. These are marked, and a competitor The existence of Delta depends on Warsaw who gains 44 points joins the club. Marcin University. As long as the University Kuczma, who edits and organises the com- regards Delta as useful, or at least good, petition, was presented with the David Delta will exist. This may be called ‘ecology Hilbert Award by the World Federation of outside biology’. In fact, there is not much National Mathematical Competitions for trouble with Delta. this column. For many years we had a column called Any problems with censorship? The cover of the book On different Laboratory at home. Edited by Jan Gaj, this Oh, yes, there were some problems at the geometries, written by Marek Kordos and was very popular with the readers and very beginning. In the first issue an article published in the series Delta presents slightly less so with their schoolteachers, about the Department of Nuclear Physics who sometimes did not know how to was going to be published. Professor Also, we made special glasses for stereo- explain the physics of the experiments. In Hrynkiewicz, a Krakow physicist, talked graphic pictures, which was a swindle ... ah, 1979 the astronomers joined Delta, and we about that Department and photographs you wanted a funny story about Delta, so started a new column Look at the sky. were included. In the article it was noted now you have one. In 1984 we wanted to We work in Warsaw, but have columns how large an area is taken by the print special stereographic pictures. These edited by colleagues from other cities. In Department. This piece of information pictures have to be printed in two colours, 1991-97 we had a column named Epsilon. turned out to be top secret information but when you look at them through special It appeared in 77 issues and was edited by and we were strictly forbidden to publish it. colour glasses you see three-dimensional our Krakow colleagues. They presented It was so silly, that when we were ordered pictures. The glasses should have been mathematics in ‘a light way’, with a special to cancel a sentence about this, we laughed enclosed with the issue of Delta. We had to sense of humour. Now we have about it very much. I believe that we produce glasses and we needed special Gammalimatias written by a Wroclaw math- laughed too much, because as a result we colour transparents. We needed money for ematician. forgot to cancel this piece of information. that – where from? No chance to get So, the issue was printed in 30000 copies money. So, we started looking for some- In 1986, the International Mathematical and this issue had no right to be distrib- body who would give us such transparents. Olympiad took place in Warsaw... uted. What to do? The copy editor of Delta In particular, we asked several institutions Yes, it was really quite interesting. A group visited the Main Censorship Office, then engaged in international trade. In one selected from the members of Club 44 took he fainted and fell into the arms of the cen- such agency, we were told: ‘It is possible’. part in the competition unofficially, but sor-ladies in the office. When the ladies They wrote to the foreign producer of such with the acceptance of the Jury. The result brought him back to consciousness he transparents saying that they were consid- of our group was similar to that obtained whispered tragically about the extremely ering buying transparents, and asked for by the official Polish team. large amount of money he would have to sample copies. These sample copies were pay as the one who accepted the issue for used for producing our glasses and the Each year you organise a competition for a publication. agency did not buy anything! EMS September 2001 11 INTERVIEW LookingLooking back:back: GrahamGraham HigmanHigman interviewers: ADRIAN RICE and ROBIN WILSON

ioned traditional algebra – determinants, matrices, and that sort of thing. But there was no group theory – except that you could do it as a special subject. In fact, you could take one or two special subjects – I took two: the theory of groups and differ- ential geometry.

What happened after that? I did a D.Phil. – straightaway; Whitehead was my supervisor. I suppose I was already interested in an academic career. My topic, Units in group rings, was designed to make me into an algebraist. But I was also interested in combinatorial topology at that time, as Whitehead was. He was a topologist – but he still had an interest in algebra of various kinds – more in linear algebra, I think, than in group theory. From his point of view, the point of study- ing units of group rings was because of their applications to K-theory.

When did you first get to know Philip Hall? I spent a year in Cambridge at the end of my D.Phil. course, and that’s when I met him. However, Cambridge was a shocking place to be at that time – the beginning of the War. It was too near the East Coast! I Graham Higman was Waynflete Professor of While you were an undergraduate, you also got to know Max Newman a little – but Mathematics at Oxford University from 1960 to helped to found the Invariant Society – the more so later, when I went to Manchester. 1984, following a period at the University of undergraduate mathematical society? He was interested in group theory and also Manchester. His contributions to pure mathe- Yes, that was Whitehead’s idea – it was the in logic. He contributed greatly to my matics have been mainly in group theory and first Oxford society for mathematics becoming partly a logician and partly an mathematical logic. undergraduates. I remember G.H. Hardy algebraist. giving the opening address in 1936 – on How did you become interested in mathe- round numbers. What did you do during the War? matics? I signed up as a conscientious objector, The chief reason was that my elder broth- and then drifted in to the meteorological er did chemistry! When I went to Oxford I office – partly because of that, I suppose. I came up on a natural science scholarship – began in Lincolnshire, and spent part of but since he’d read chemistry, I thought I’d the time in Northern Ireland, and also in better read mathematics. My brother had Gibraltar for a while. I didn’t do any math- already gone to Balliol College, so I went ematics – I was just a straightforward fore- there too. caster.

What sort of mathematics did you enjoy at Did you enjoy your work there? Oxford? Yes, as much as anyone enjoyed anything I was a born pure mathematician – I had in those days – more than I would have no respect for applied mathematics – but I done in other jobs. It exercised the mind. had to study it all the same to a certain extent. What happened after the War? Well, I can tell you why I became a mathe- Was there any pure mathematician who matician after the War. By then, I was particularly inspired you? totally disillusioned by my experiences in Well, Henry Whitehead, who was my tutor, the real world, and I decided that since I of course…a very good tutor. I didn’t knew nothing else I’d better stay in the Met realise how great he was – undergraduates Office. I was interviewed there, and at the don’t. He certainly wasn’t intimidating. end the interviewer asked me: ‘If it hadn’t But he had a sense of humour that was been for the War, what would you have rather different from mine, and his non- done?’ And I said, ‘I suppose I would have mathematical attitudes were also different. tried to climb the academic ladder.’ And But apart from that, we got on very well. G.H. Hardy’s notes on round numbers he said, ‘Well, with qualifications like However, I don’t think he took me serious- yours, you still could.’ And I thought, ly for the first couple of terms, because I’d To what extent was algebra taught at ‘Well, a nod’s as good as a wink to a blind opted to do mathematics on a natural sci- Oxford in the 1930s in the undergraduate horse, so I left.’ They offered me a job, but ence scholarship, and he had no idea syllabus? it wasn’t a particularly good one, so I whether I was any good. Well, there was a good deal of old-fash- decided to apply for academic jobs. 12 EMS September 2001 INTERVIEW Were academic jobs difficult to get at that 1955? looked for a new one. I succeeded A. G. time? In the first place I was very ambitious – I Walker. On the whole I think I was a rather No, they were actively looking for people – started applying for professorships that I conservative president. You have to remem- anybody with the right qualifications could was perhaps not qualified for, and also for ber that the London Mathematical Society get one. In fact, I was offered a job at other jobs – which rather unsettled me. But is a very much larger institution than ever Durham. I was interviewed at Durham and I think Henry Whitehead was partly respon- before, and most of that increase took place Manchester in the same week – Durham sible for my returning to Oxford, because after I was president. It was already begin- first – and they offered me a job. Since I he thought I would be wasted in certain ning when I was president – it was clear that knew that Max Newman was now at places that shall remain nameless. He it had to come, but I was a bit frightened of Manchester, I asked Durham if I could arranged a lectureship, which became a it. delay my decision until I’d heard from readership pretty quickly. Manchester. But they said ‘no’, so I turned Since you retired, have you kept up your Durham down and went to Manchester. Then Whitehead died suddenly, in 1960. interest in mathematics? Yes, it was a shock for everybody. He died I’m interested in the kinds of mathematics Who else was in Manchester at that time? in America – a long way away – at a meeting that people do. I’m interested in algebra. Jack Good and David Rees – Bernhard in Princeton. He was at a tragically young I’m interested in logic. And I’m particularly Neumann and Walter Ledermann came age. interested in seeing a greater number of later. people working on the borderline between algebra and logic. I’m interested, for Was Alan Turing still around? instance, in model theory, and the work I’ve He must have been around when I got done on almost free groups; this leads you there, but I didn’t meet him until I became to consider the language in which you can interested in his stuff. I think he was in the talk about these things. But above all, I’m department of experimental physics, since interested in going on doing mathematics, he had to have a laboratory. He was rather even if it isn’t very good mathematics, a kind man, but rather ‘up in the air’, because there’s nothing else that keeps me ‘loose’, one might say. He didn’t give the sane – relatively sane, at least. impression of being very organised. I worked a little with him, when he solved the Do you still keep abreast of current mathe- word problem – or nearly solved it. Henry Whitehead and friend matical developments? No, I don’t claim to keep abreast of current Were there any other colleagues at Whitehead had been the Waynflete Professor developments – I never did. I was never a Manchester with whom you worked, or were of Pure Mathematics at Oxford. Were you good person for knowing the literature – it you a solitary figure? the heir apparent – the natural successor? wasn’t my strong point. Others are far more No, I was never very solitary. David Rees, as I thought so, but whether anybody else did, widely read than I am. What I do is fiddle I said, and Trevor Evans – and Bernhard I don’t know! I applied, and I thought that around with problems that are probably of Neumann when he arrived. Bernhard I had a reasonable chance – and I was in fact no interest to anybody else. On the whole Neumann settled me down a great deal, appointed. I’ve been interested in things that other though it was Max Newman who I chiefly people haven’t been interested in. I’ve remember with gratitude. In the first place, During your time as Waynflete Professor never felt that my lines were popular. he took me at a time when I’d not done any from 1960-84, you built up a formidable serious mathematics for several years and I research school in algebra in Oxford. Is there a piece of your own work of which was rather doubtful of my capacity to do so Well, that is what I was there for, really. you are particularly proud? – he encouraged me a good deal. And the Perhaps the topologists had had it all their Oh yes – there’s no doubt whatever! The other thing about Max Newman was that he own way for too long! one thing that I hope to be remembered for was interested in various things – he gave a is the theorem on the necessary and suffi- course of lectures which I remember very Your time in Oxford coincided with all the cient condition on a finitely generated well, on logic and word problems. My inter- exciting work on the classification of finite group for it to be embeddable into a finite- est in that aspect of group theory springs simple groups. ly generated group with finite presentation. very much from him. Yes, but I was always on the periphery. I was In the first place, that is solely mine – more interested in infinite groups than nobody even so much as conjectured it The two prominent figures in your career, finite groups, and I was never a very good before. And in the second place it opens up Whitehead and Newman, were primarily conjecturer – I always thought there would vistas; for instance, an obvious corollary is topologists. be far more simple groups than there that there are universal finitely presented They were topologists primarily, but they turned out to be. I tried to construct them groups – that is, finitely presented groups were both mathematicians with a wide range – rather to characterise them. In fact, I which contain every finitely presented I hope people would describe me the same missed a great opportunity to discover the group as a subgroup. As far as I know, way. They were interested in the funda- Conway groups. I should have known bet- there’s no proof of this which doesn’t use mental group as a topological tool, and this ter than to go on footling along with finite my theorem. There are other things that led immediately to algebraic questions, of groups, but unfortunately I was really obsti- I’m proud of – but that’s the thing I think is course. nate about it. really important.

Have you noticed an increasing tendency During the 1960s you became President of Is there any mathematician from the past towards specialisation in recent years? the London Mathematical Society. Had you who you wish you had met? Well, I think that analysts have always spe- always been involved with the LMS? I suppose, Kurt Gödel. I have an immense cialised – it’s in their nature – they’re self- I’d been a member since just after the War. admiration for his work. sufficient, aren’t they, whereas nobody else I was on the Council for many years and was can afford to be self-sufficient. The D.Phil., assistant editor on the LMS Proceedings to And finally, how would you like to be remem- as an almost universal gateway to the pro- John Todd. When I was on the Council I bered? fession, wasn’t there in my day. But after attended all the meetings they had in As a great mathematician, of course – what the War, it became obvious that we needed London – except on one occasion when else is there? something like that. such a fog descended that I arrived just as the meeting was due to finish, and there We thank the Mathematical Institute and Prof A With such a stimulating environment in didn’t seem much point in going. Kosinski for supplying the photographs in this Manchester, why did you return to Oxford in In those days the outgoing president article EMS September 2001 13 FEATURE ARTICLE

TheThe MethodologyMethodology ofof MathematicsMathematics part 2 Ronald Brown and Timothy Porter The first part of this article appeared in the June 2001 issue.

We return to the last three questions at the that the history of mathematics is the his- * An abstract theory allows for simpler beginning of Part 1 of this article: tory of improved notation. This reflects proofs. This is a surprise, but is com- 4.What is the methodology of mathematics, and the finite nature of intelligence, which monly found to be true. The abstract what is the way it goes about its job? requires props and metaphors to help and theory allows for the distillation of essen- 5.Is there research going on in mathematics? If guide it. tials. It is of interest to know if a theo- so, how much? What are its broad aims or main Some symbols are in themselves rem or fact is true in the general situa- aims? What are its most important achieve- metaphors: examples are /, <, →, ∫. tion or only in the particular example. ments? How does one go about doing mathe- Others have acquired strong associations, The abstract theory allows for the matical research? so that we can use them as metaphors. removal of possibly irrelevant aspects. 6.What is good mathematics? Is there a future Symbols are able to express ‘with economy Generalisation and extension for mathematics? and precision’, to use words of A.N. This has some features in common with We re-emphasise that the tone of this Whitehead. The use of particular symbols abstraction, but usually applies differently. article is that of an address to students. is something that changes with time, as Thus a generalisation of the (3, 4, 5) right- We hope that its publication will encourage mathematicians become accustomed to, angled triangle is Pythagoras’ theorem, debate about how, as Dantzig [1] put it, to and appropriate, a new notation. while an extension is Fermat’s last theo- present mathematics ‘with its cultural con- In some cases, a notation, brought about rem, that the equation xn + yn = zn has no tent’ and not as ‘a bare skeleton of techni- by the laziness of mathematicians, leads to solutions for positive integers x, y, z, if n > calities’, with the danger of repelling a new theory. For example, expressions of 2. This has now been famously proved by ‘many a fine mind’. Serious and difficult the type (a11x1 + … + a1nxn; … ; am1x1 + … Andrew Wiles [7]. questions remain on the balance between + amnxn) get abbreviated over time to Ax, Proof this kind of teaching and the usual materi- and to allow for the correct manipulation The rigorousness of the notion of proof is al, and on assessment: students are rightly of this abbreviation the rules for matrices a particular feature of mathematics. It is inclined to believe that matters not are worked out. why mathematics is essential in engineer- assessed are thought to be unimportant. If To give an example close to the heart of ing, safety, physics, and so on. matters related to professionalism are not some of our research, the first author has The notion of proof, of validity, in math- discussed, then there is a danger that naive been concerned for many years as to ematics, is an aspect of the general ques- attitudes will prevail – such as that ‘good whether the linear notation for mathemat- tion: what is the notion of validity in an mathematics is precisely the mathematics ics is a necessity or a historical result, based area of study? Each area, from social sci- done by top mathematicians’. on the needs of printing. ences, economics, chemistry, biology, edu- The analysis of this linguistic point has cation, law, literature, and so on, has its What is the methodology of mathematics? led to a new kind of ‘higher-dimensional notion of validity, and the contrast and Here again is a subject that is rarely and algebra’, in which symbols are related not uses of this notion are of particular inter- not widely studied. There is the comment just to those to the left and to the right, but est. of Paul Erdõs [4] that mathematics is a also up and down or out of the page. This The question of what is acceptable as a means of turning coffee into theorems; algebra then becomes closer to, and more valid argument in mathematics is still sub- perhaps, though, this does not help the able to model, some geometric situations, ject to argument and discussion, particu- beginner too much. So let us look at some and this leads to the formulation and larly with the existence of very long proofs of the issues discussed in the books by P. proofs of new theorems, and to new calcu- (for example, 15000 pages [5]), and with Davis and R. Hersh, The mathematical expe- lations and insights. the use of computers for visualisation, rience [2] and Descartes’ dream [3], particu- Abstraction experimentation and calculation. larly the section of the first book on ‘Inner This is an essential part of mathematics, Existence of mathematical objects issues’. This deals with a number of and again is one part of what makes math- A great mathematician has urged that the themes. ematics incomprehensible to the general major problem of mathematical education Symbols public. is to teach the reality of mathematical The use of symbols and symbolic notations Mathematical structures are abstract. objects. What is this reality? In what way is one of the characteristics of mathemat- They are defined by the relations within do these objects exist? For example, ics, and one that puts off the general pub- them. They are thought of as non-sensual. Eternity, by John Robinson, is a symbolic lic. People will say they were able to do The advantages of abstraction are at least mathematics till it got onto x and y. three-fold: The manipulation of symbols according * An abstract theory codifies our knowl- to rules is still an important part of the edge about a number of examples, and craft of mathematics. We find we have to so makes it easier to learn their common teach people who wish to master (say) eco- features. Only one theory is needed, to nomics, but who are unable to deduce from replace a multiplicity. This codification x + 2 = 4 that x = 2. This makes it very exploits analogies, not between things difficult to understand the concepts of eco- themselves, but between the behaviour nomics. and relations of things. Finding these Very complicated relations can be analogies, the abstract theory that expressed symbolically in a way that can replaces a multiplicity, is an important hardly be conveyed in words. This econo- method in mathematics. my which symbols allow is improving con- * Once the theory is available, it may be tinually, as symbols are used in the denota- found to apply to new examples. This tion of advanced concepts and the rules of leads to the excitement and joy of: ‘That symbol manipulation are used to model reminds me of ...!’. For this new exam- the rules for the concepts. ple, a body of established theory is avail- It has been said, in an exaggerated way, able at the turn of a page. Eternity, by John Robinson 14 EMS September 2001 FEATURE ARTICLE sculpture, but is also the construction of a Is there research going on in mathematics? in 1964, suggested that while this method fibre bundle! (see [6]). Those who wish a practical test should look might appeal to young ambitious persons, These questions have been a matter of at the change in Mathematical Reviews since concentration on this might also not allow major interest to many philosophers of it started in 1940. This monthly journal them to develop the kind of mathematics mathematics, but their interest is perhaps contains abstracts of mathematical papers. most personal and characteristic to them- in the process of being downgraded. Roughly speaking, a few paragraphs are selves, because they are solving other peo- Mathematics is often about processes. The enough for a five-page paper. The growth ple’s problems. question of existence of a mathematical in numbers of pages over these years is Usually, though, one attacks smaller structure is maybe like asking whether the about eleven-fold: each month now about problems at the frontiers of knowledge, game of chess exists. Clearly it does not 400 large pages of abstracts of mathemati- problems to which less effort has been exist in the way that tables and chairs exist, cal papers are published. This is indeed devoted, and so where there is a greater but nonetheless it influences many lives the golden age of mathematics, both in likelihood of success. You will almost cer- and, to put it crudely, passes the cash test. quantity and quality. tainly have to study to find what has been (Does it earn money? The answer is clear- The aims of this research are at various done, what techniques are available, and ly: yes, for some – for example, world levels. One is the advancement in knowl- which you need to master. champions and makers of chess equip- edge about particular types of structures, It is helpful to have problems whose cri- ment.) that are already well defined. Another is teria for success are clear: the answer is yes The relation of mathematical concepts the introduction of the study of new struc- or no to some question. On the other and methods to processes is indicated by tures, as they have appeared and been hand, failure to provide a solution is then the way that memory of muscular action shown to be relevant. There are new rela- also clear cut, as is finding the problem too and rhythm are important aspects of math- tions between structures. There is the urge easy. Mathematicians need to build into ematical work. A lot of mathematics is con- to simplification, to find structures that their strategy plans for dealing with both cerned with the realisation and under- explain and help us to understand the way too little and too much success on the standing of the effect of repetitive process- particular structures behave in themselves problem at hand. es and methods. relative to other structures. Method 3: Relate different areas of knowledge Mathematicians are good at understand- What is difficult for the newcomer in the In this method you learn about the begin- ing and imagining moving things around, field, and for the general public, to under- nings of different areas, and find relations such as from one side of an equation to stand is how one goes about doing mathe- between them – so you fill in the gaps another, or changing a pattern in space. matical research. Here we give some between the peaks, while often the ‘top They use movements of their hands and pointers, by suggesting four ways of going people’ are occupied with building up the arms to convey what is happening. The about the job. There are certainly many peaks. The advantages of this method are objects and ideas of which mathematicians more, and individual researchers must in that you learn something of different talk are sometimes a kind of concatenation the end devise their own strategy for suc- areas, and in a useful way, since you have of a variety of such remembered processes. cess. It is also difficult to know how much to work to do the translations between the By contrast the representation of these one must know before starting on mathe- two areas. This is a good method for PhD ideas in writing is often bare and sparse, matical research: a famous answer to this theses, since a supervisor can often see the and this is part of the difficulty of learning particular question was: ‘Everything, or relation without having worked out the the use and application of these objects nothing.’ details; it also advances the general unity and ideas. On the other hand, it also Method 1: Apply a standard method to a stan- of mathematics. Another advantage is that allows for each person to make the inter- dard type of problem it gets you used to the idea of proving pretation and internalisation most appro- This method is probably a part of every small but useful results which help to fill in priate to themselves. successful research project, and has every the gaps and create the picture of what is Infinity guarantee of success, provided that one is going on. The taming of the infinite, or the enlarge- sufficiently skilful in the standard method. Method 4: Blue sky research ment of the imagination to include infinite Indeed, a common method of mathemati- Here you have some idea of a mathematics operations, is one of the joys of mathemat- cal research is to reduce a problem to one that ought to exist, and of its characteris- already considered. If the original prob- tics. You also have a few hints as to the lem is too difficult, then a standard strate- kind of materials from which the mathe- gy is to simplify the problem so that it matics ought to be made. The problem is becomes of standard type, before adding that proper mathematics requires defini- the complications that make it a new prob- tions, examples, propositions, theorems, lem. The general presumption might be proofs, calculations, and in the beginning that one can do only easy things. So the none of these exist – so they have to be method is to reduce a problem to a type assembled over a period of time. In what that can be seen to be easy. If in doubt, do order should this be done, and how impor- the obvious thing first. tant will the work be? This can hardly be Those who become skilful at applying judged until the theory is worked out, and standard methods may someday find that such a theory does not emerge, like Venus their skills apply to a problem no one else Anadyamene, fully formed from the sea. A has considered, and that this leads to new theory accumulates in a journey over a and important results. Much of the educa- period of years, and a gut feeling of the tion of a mathematician is concerned with importance of a line of investigation is nec- Eternity, by John Robinson acquiring the skills and knowledge appro- essary to motivate travel on a long road. priate to work in a chosen area. For decades we have both been working ics, and also one of its scandals. Are these Method 2: Attack a famous problem at the fron- on this kind of research, as well as on other infinite objects real? The surprise is that tiers of knowledge kinds. In the mid-1960s the first author these infinite, possibly unreal, objects can This is the strategy of going for a famous formulated the theme of ‘higher-dimen- be used to prove finite real things, and this problem at a peak of knowledge. The sional algebra’, as mentioned earlier. The again is an aspect of the mystery of the sub- advantage is that if you succeed, then you aim was that of an algebra more closely ject. Suppose for example that these infi- will become famous. It is more difficult to related to the geometry, and allowing a nite objects are used to prove the safety of assess your chances of success: you will more general type of composition. The a nuclear installation, or of an aircraft probably need some new ideas. expectation was that this algebra would landing system? What credence should be This seems to be the most ambitious yield some formulations and proofs of new placed on such a proof? These are real method for a young person. However, S. theorems, which would automatically lead issues. Ulam, in conversation with the first author to new methods of calculation. EMS September 2001 15 FEATURE ARTICLE This in the end has proved correct, with of introducing pupils and students to the Another danger is the growing reliance many people joining in the project. For a notion of good exposition, and even to ask on computers as a black box to give the long time (five years), though, all that them to compete, not in problem solving, answer, without any understanding of the could be said was that it was possible to but in producing expositions and exhibi- processes involved, or of the concepts that draw pictures that suggested that the ideas tions of mathematical principles and appli- are intended to be manipulated. So both would have to work. The problem was a cations. We have found the work on pro- the scope and the limitations of the com- lack of framework to express the algebra ducing a mathematical exhibition enor- puter fail to be understood, the mathemat- corresponding to the pictures, and to the mously instructive [8, 9]. ical basis is neglected and perhaps fails to geometry. This framework was built up be developed, and the computer may be gradually, and it became ever more amaz- Is there a future for mathematics? used in ways that are inappropriate, or ing to see how natural and fitting a way it There is a view that there is no more basic simply limited by the software design. It is was, once the ideas were thought about in mathematics to be found. This view is said that some engineering firms are dis- the ‘correct’ manner. Thus, as suggested comparable to the view of those who have pensing with their mathematical research by Wigner in the quotation given earlier, said that physics was ended, the basic prob- departments in favour of engineers manip- the aesthetic criteria for a proper theory lems having been solved. We feel, to the ulating software packages. Will this ensure were nicely satisfied, and the theory contrary, that mathematics is undergoing a the safety or reliability of the product, and became better than the vision that had revolution – a quiet one, but a revolution will it allow the use of the most advanced prompted it. nonetheless. This is occurring on two mathematical concepts? It has to be said that, paradoxically, the fronts. If these dangers are to be averted, then secret of success in research is the success- There is first the computational revolution. an increased understanding and apprecia- ful management of failure. For if you For computation with numbers, or for tion of the questions with which we started never fail, then it is likely that the tasks you graphics presentation, this revolution is are essential. There may be ways of speed- have set yourself are simply too easy. well known. Less well known publicly is the ing up the process of transfer from the Interesting research must have an element computer software that can manipulate conceptual foresight of the mathematician of risk. You need strategies for dealing symbols and axioms, and other software to the realisation in a scientific or techno- with situations when things go wrong: the that can carry out automated reasoning. logical application. To find them, we need problem may have proved too hard, or too In principle, these should give mathemati- in society a real understanding of the work easy. What comes next? Analysing the cians power to calculate and reason a mil- of mathematicians, and of the way mathe- reasons for failure, and comparing these lion-fold more than they can at present, matics has played a role in the society in reasons with the reasons for wanting to do and to deal with the complexities of sys- which we live. It is our responsibility to the this problem in the first place, becomes tems thought previously to be intractable. subject we love to find ways of developing instructive for future work. The prospective effect of these on the this understanding. teaching of mathematics has yet to be What is good mathematics? properly understood and assessed, Acknowledgements Many of the questions We would not like to attempt any final although a lot of work is in progress. The raised in this article were discussed with answer to this, but all of us should try and effect on research has already been consid- students of the final-year Maths in Context formulate some of the aspects we are look- erable and is likely to grow in its influence. course we ran together at Bangor, and also ing for. Indeed, as editors of journals, we A more subtle revolution is the conceptu- with first-year honours mathematics stu- have to make judgements on this question al revolution. The emphasis on mathemat- dents taking the course Ideas in Maths. The on a daily basis. For a new mathematical ics as the study of structures is finding its contributions of these students through paper we ask the questions: are the results mathematicisation in category theory, the discussions and essays have strongly influ- new? how far ahead of current literature do mathematical and algebraic study of struc- enced our thinking. We also wish to thank they go? is the paper clear and well writ- tures. Category theory has revealed new Roger Bowers and Brian Denton who have ten? are the authors familiar with current approaches to the basic concepts of math- run a course on Mathematics in Society at work in the field and aware the relation of ematics, such as logic and set theory, and Liverpool University. their results to the field? how surprising indeed has made respectable the idea that are the results? how elegant are the meth- the practice of mathematics needs not one References ods, and are any new methods introduced? foundation, as traditionally sought, but 1 T. Dantzig, Number: the language of science, Some of ‘the best’ mathematics is that alternative environments and a framework 1930, 2nd ed. 1954, Macmillan. which introduces new ideas and concepts for their comparison. These ideas are also 2. P. Davis and R. Hersh, The mathematical expe- that make the previously difficult easy. important for the progress of computer rience, Penguin, 1981. This contradicts an impression you may science, as (for example) in showing new 3. P. Davis and R. Hersh, Descartes’ Dream, have that mathematics is meant to be hard, approaches to data structures. Penguin, 1988. and is good for you partly for that reason, 4. http://www-history.mcs.st-and.acuk/history/Math like a cold bath. On the contrary, good Some dangers ahead ematicians/Erdos.html mathematics can (perhaps, should) be One of the pleasures of mathematics is the 5. S. Gorenstein, The longest proof, Scientific easy. It is just that often we do not know way it operates on various levels, which American. how to do this. The combination of appar- then interact. So the algebraic study of 6. John Robinson sculpture: for a discussion of ently simple arguments with a surprising mathematical structures has itself led to this, see http://www.cpm.informatics.bangor.ac.uk conclusion, perhaps with a surprising twist, new mathematical structures. Some of /sculmath/wake.htm is what we like best of all. these structures have had notable applica- 7. http://www-history.mcs.st-and.ac.uk/history/Math What is worrying is that many young tions in mathematics and in physics. ematicians/Wiles.html mathematicians go through their educa- Nevertheless, there are still many cur- 8. Bangor Maths exhibition group, Mathematics tion without the notion of ‘good mathe- rent dangers for mathematics. There is a and knots, Exhibition for the PopMaths Road- matics’ even being debated. Yet for any general lack of appreciation of what math- Show, 1989 (16 A2 boards); also a brochure, human activity, there is always the question ematicians have accomplished, and of the published by Mathematics and Knots, 1989, and of its value, both for society and personal- importance of mathematics. Some of this the website: http://www.cpm.informatics.bangor. ly. There is an argument that the teaching has come about through mathematicians ac.uk/ of a subject should reflect something of the themselves failing to define and explain 9. R. Brown and T. Porter, Why we made a values of the professionals in it; for exam- their subject in a global sense to their stu- mathematical exhibition, The Popularisation of ple, for a professional, it is not enough just dents, to the public, and to government Mathematics (ed. G. Howson and P. Kahane), to produce an answer, but it is important and industry. It is possible for a student to Cambridge, 1992. also to produce (if possible) a satisfying get a good degree in mathematics without explanation. any awareness that research is going on in School of Informatics, University of Wales, Thus we would argue for the advantages the subject. Dean St., Bangor, Gwynedd LL57 1UT, UK 16 EMS September 2001 SOCIETIES SocietiesSocieties CornerCorner

interesting reading for the general mathe- The Norwegian matically interested public, and to give young and aspiring mathematicians a Mathematical Society chance to have their work printed. That Bent Birkeland problem found a temporary solution when Heegaard succeeded in obtaining funds The first attempt to create a mathematical for a series of pamphlets, Norsk Matematisk society in Norway was made in 1885 by Forenings Skrifter (Publications of the Sophus Lie, who was at that time professor NMF), where younger Norwegian mathe- in Oslo. This was a time when similar ini- maticians, including Øystein Ore, Thoralf tiatives took place in many European coun- Skolem, Trygve Nagel and Ragnar Frisch tries. The Moscow Mathematical Society (Nobel laureate in Economics, 1969) had was founded in 1864, London in 1865, and some of their early work published. the Finnish, French and Danish ones in Regrettably, this enterprise was discontin- 1868, 1872 and 1873, respectively. In ued in the 1930s for financial reasons. Norway, however, the mathematical com- The journal continued for 34 years. munity at that time was too small, and the Finally, in 1952, it was amalgamated with venture broke down when Lie moved to the corresponding journals in the other Leipzig in the following year. But a series Scandinavian countries to form two new of reforms in the high schools and at the periodicals, the Mathematica Scandinavica university in the second half of the 1800s for professional mathematics, and the (less Latin and Greek, more modern lan- Nordisk Matematisk Tidskrift (Normat, for guages, science and mathematics) led to a short), which aims at a broader audience marked expansion of that community, and and mainly prints work in the Poul Heegaard (1871-1948) a formal organisation became necessary. Scandinavian languages. Both of these are In particular, the need for a Norwegian professor of geometry in Oslo, was willing still active, under the joint auspices of the mathematical journal was felt. The diffi- to edit the journal. He had valuable expe- five Nordic mathematical societies culty was to find financial support for it, rience from editing the Danish (Danish, Icelandic, Finnish, Swedish and and to find persons able and willing to take Mathematical Journal for a couple of years. Norwegian). on the editorial work. Finally, on 2 November 1918 (incidentally, Another publishing venture of the Society In 1918 the time had come, and prelim- Heegaard’s birthday), the Norwegian was Sophus Lie’s collected works. The edi- inary discussions took place in the early Mathematical Society was born. Its pur- tors, Friederich Engel and Poul Heegaard, autumn. Arnfinn Palmstrøm, who at that pose was stated broadly as ‘connecting published the first volume in 1922, but for time worked as an actuary and who from mathematically interested persons from all financial and other reasons the seventh 1919 until his untimely death in 1922 was over the country’, and the first more spe- and last volume did not appear until 1960. cific task was to start a national mathemat- The NMF also published a series of ical journal. Professor Carl Störmer was small popular booklets in the early 1990s, elected the Society’s first president, while aimed mainly at college students, on Palmstrøm became its secretary and the themes varying from ‘number systems’ to more arduous task of editing the journal ‘Norwegian mathematicians’. was taken on by Heegaard for the mathe- matical side and Anton Alexander for the Present activities didactic one. Mathematics competitions for college stu- The ‘founding fathers’ were university mathematicians, leading schoolteachers, actuaries, officers (mainly from the geodet- ic service) and students. At least two of them came to be closely involved with the Society for more than sixty years: the num- ber-theorist Viggo Brun (1885-1978) became a university professor, while Fredrik Lange Nielsen (1891-1980) became a leader in the insurance world in Norway. The activities taken on were what one might expect: meetings, publications, and some lobbying for good mathematical causes. A few of these are described below.

Publications The first issue of the Norsk Matematisk Tidsskrift (Norwegian Mathematical Arnfinn Palmstrøm (1867-1922) Journal) appeared in 1919, opening, sadly, with the obituary of Ludvig Sylow, written Norway’s first professor of insurance math- by Thoralf Skolem. The first volume also ematics, secured financial support from contained contributions by the young and the major insurance companies. promising number-theorists Viggo Brun Government sources also responded posi- and Trygve Nagel. tively, and the Danish mathematician Poul The journal was intended to serve two Heegaard, who had just been appointed not-quite-compatible purposes: to provide Carl Störmer (1874-1957) EMS September 2001 17 SOCIETIES will be the Abel Bicentennial Conference 2002, of mathematics, and topics from the histo- which takes place in Oslo from 3-8 June. ry of mathematics were discussed at meet- The conference is arranged jointly by the ings of the Society. Norwegian Mathematical Society, the After the annexation of Estonia by the Norwegian Academy of Science and Soviet Union in 1940, the activities of sci- Letters and the Norwegian Mathematical entific societies were banned, and the last Council, with support from the meeting of the Academic Mathematical International Mathematical Union and the Society took place in November 1940. In European Mathematical Society. The con- June 1941 the new puppet Soviet govern- ference will present an overview of the ment in Tallinn ordered the University of mathematical heritage of Niels Henrik Tartu to reorganise the activities of eigh- Abel and, based on this heritage, will iden- teen functioning academic societies at the tify new mathematical trends for the 21st University (the Academic Mathematical century. There will be sessions on the his- Society included). In reply to this regula- tory of mathematics, algebraic geometry, tion, Professor Hans Kruus, Rector of the complex analysis, differential equations University, made a proposal to the govern- and non-commutative geometry. On this ment that the Academic Mathematical occasion the remaining unsold copies of Society be closed. The activities of other Abel’s works (edited by Sylow and Lie in societies were reorganised in accordance 1881) will be on sale, nicely bound in with the requirements presented to organ- leather. This should be of interest to both isations in the Soviet society. Because of mathematicians and bibliophiles. the German invasion of Estonia in July 1941, the Soviet administration in Tallinn Viggo Brun (1885 -1978) could not deal with the Rector’s proposal: the Academic Mathematical Society was dents have been part of the Society’s activ- 75 Years of the Estonian not closed down, but its activities were ities nearly constantly from the beginning. stopped. During the war Hermann For many years, starting in 1922, Crown Mathematical Society Jaakson, the most recent President of the Society, repeatedly tried to obtain permis- Prince Olav awarded a prize for the best Mati Abel solutions to a series of problems posed in sion to continue the Society’s activities, but the Journal. Later on, other sponsors took the new occupation system in Estonia over. For many years the Norwegian placed demands on the statutes of academ- telecommunications company Telenor has ic societies, with the result that the sponsored a mass contest in three stages, Academic Mathematical Society changed called the Abel Competition. The first its statutes several times during the war. stage involves several thousand students, After the war the activities of all former and successive eliminations bring it down academic societies were forbidden by the to about twenty participants in the final Soviet regime in Estonia. round. Of these, the best six are selected The idea of re-establishing the to take part in the International Mathematical Society circulated among Mathematical Olympiad. (A note on this Estonian mathematicians in the 1970s, but competition appeared in EMS Newsletter 32 the timing was not favourable. The first in June 1999.) attempt to re-establish the activities of the Another activity is the winter seminar Society in 1983 failed: the Estonian ‘Ski and Mathematics’, early in January. Communist Party leadership did not give This tradition was initiated in the 1960s by permission to re-open it. In spite of this Professor Karl Egil Aubert. The seminar refusal, Estonian mathematicians contin- was arranged regularly for half a dozen ued to seek ways to continue the Society’s years, then more intermittently, until it was The Academic Mathematical Society, pre- activities. resumed on a regular basis in 1997. It decessor of the Estonian Mathematical The re-opening conference of the takes place at a hotel in the mountains, the Society, was founded in Tartu 75 years ago. Academic Mathematical Society, under the programme being divided between out- The statutes of the Society were registered door activities before lunch and mathemat- by the Council of the University of Tartu ics in the afternoon. (There is not much on 23 February 1926, although the open- daylight after 3 p.m. at that time of the ing meeting of the Society with 68 found- year.) ing members present – mostly teachers and students from the University’s mathe- World Mathematical Year 2000 matics department – took place in the The international mathematical year was Festive Hall of the University on 21 March celebrated by the NMF and by others, with 1926. A lecture on ‘Archimedean and activities at the universities, in the schools Euclidean Sentences’ was delivered by and in the streets. Mathematically, the Professor Jaan Sarv. Professor Gerhard main event was a conference in Trondheim Rägo became the first President of the in January. Later in the year a large num- Society, and his successors as President ber of events were aimed at the schools and were Jüri Nuut (1927-32), Edgar Krahn the general public. On one occasion the (1932-36) and Hermann Jaakson (1936- first 5000 digits of ð were written along the 40). In addition to mathematicians from main street of Oslo! Tartu, several members of the Society came from other towns in Estonia. The Abel Bicentennial Conference The Academic Mathematical Society Niels Henrik Abel was born on 6 August played an essential role in the mathemati- 1802. His bicentennial will be the occasion cal life of pre-war Estonia. Interesting cur- for much celebration, aiming both at rent results in the field of exact sciences, mathematicians from home and abroad reports on research by members of the Mati Abel and Imre Csiszar and at society in general. The main event Society, issues connected with the teaching 18 EMS September 2001 SOCIETIES

Anniversary meeting of the society on 17 February 2001 new name of the Estonian Mathematical Academician Arnold Humal, the Arnold With over 400 members, the Estonian Society, took place with 118 founding par- Humal Annual Prize for a young mathe- Mathematical Society was a founding ticipants and 52 guests on 17 September matician was established. In 1995 the member of the European Mathematical 1987, once again in the Festive Hall of the Association of School Teachers of Society, and since 2000 has been a member University. Professor Ülo Lumiste, the Mathematics Teaching in Russian joined of the International Mathematical Union main initiator of the re-establishment of the Estonian Mathematical Society and as an associated organisation. the Society, opened the conference. The started functioning as a section of the On 17 February 2001, the Estonian participants elected the Board of the Society. Mathematical Society celebrated its 75th Society, the President of the Society (Ülo Every other year the Estonian anniversary with a conference in Tartu. Lumiste) and the Honorary President Mathematical Society organises ‘Days of Among the participants were Imre Csiszar (Academician Arnold Humal). Ülo Mathematics’ in Estonia, and the Union of (President of the Hungarian Mathematical Lumiste led the activities of the Society for School Mathematics holds the annual Society), Aleksanders Šostaks (President of two three-year terms until 1993. Mati Abel Summer and Winter Days of the Union the Latvian Mathematical Society), Ulf has occupied the Presidency since 1994. which are very popular with teachers of Perrson (President of the Swedish Immediately after its re-establishment, mathematics. In addition to workshops Mathematical Society), Wies³aw Zelazko the Society started its activities with the and report meetings, the Society helps to (President of the Polish Mathematical restored pre-war structure. The section on organise international and local confer- Society in 1984-86 and a foreign member school mathematics (from which the Union ences in mathematics, to publish mathe- of the Estonian Mathematical Society), and of School Mathematics was formed in matical books and a journal, to compose Hans-Olav Johannes Tylli (a member of 1989), and several working groups (on reference books and dictionaries in mathe- the Finnish Mathematical Society). Jonas computer science, mathematical terminol- matics, to organise open competitions and Kubilius (President of the Lithuanian ogy, history of mathematics, etc.) began to Olympiads in mathematics for high school Mathematical Society) and Anatoly Vershik work. It was decided to publish the year- students, and to train them for interna- (President of the St Petersburg books of the Society, to establish annual tional competitions in mathematics. An Mathematical Society) sent anniversary prizes for high-school final-grade students International Workshop on Topological congratulations. (for the best results in the Olympiads in Algebras and an essay competition ‘The We cherish friendly ties and cooperation mathematics) and for undergraduate stu- beauty, magic and pain of mathematics’ with mathematicians from other countries dents (for outstanding results in research). were held by the Society to celebrate World and are determined to expand our inter- On a proposal from the heirs of Mathematical Year 2000 in Estonia. national contacts.

EMS September 2001 19 PROBLEM CORNER PPrroblemoblem CornerCorner Contests from Romania, part 4 Paul Jainta (Schwabach, Germany)

The solution of problems … is the also led to the creation of remarkably art- ladder by which the mind ascends into ful problems aimed at differing levels of the higher fields of original research students. The variety of the questions, and investigation. Many dormant their suitability for a range of age groups minds have been aroused into activity and the nature of the challenge has been through the mastery of a single prob- of a consistently high standard since the lem. first competition in 1895. Benjamin Finkel and John Colaw, American Math. Monthly, 1894, The Gazeta Matematica Competition Vol. 1, no. 1, page 1 In January 1910 the renowned mathemat- ical journal Gazeta Matematica rose to the At the beginning of the twenty-first centu- fore by promoting mathematics competi- ry, elementary mathematics is undergoing tions and created a separate maths event two major changes. The first is in teach- nationwide. As a result the questions were ing, where one moves away from routine circulated to the remotest corners of the exercises and memorised algorithms country overnight. In this way a competi- towards creative solutions to unconven- tion was launched that embraced the tional problems. The second consists in whole Romanian area for the first time. spreading the culture of problem solving - The Annual Gazeta Matematica Contest especially in Eastern Europe. Romanian had a life-span of about 40 years, and was mathematicians have influenced both finally replaced by the National ability is the number of submitted solu- trends strongly, breaking new ground in Mathematics Olympiad in 1950. tions to the problems. establishing competitions for youngsters. However, the competition that was run by Another selection rule for candidates Following Paul Halmos, who said, 'prob- a mathematics journal had its eyes on par- wishing to become one of Romanian top lems are the heart of mathematics', they ticular benchmarks. Priority was given to young mathematicians invokes the results emphasised them in the classroom, in those participants who had corresponded achieved in national or international seminars, and in many publications of all well to the problems published in the jour- events during the previous year. But they kinds written to train their students to be nal over a given period (perhaps a year). have to get over two other hurdles to reach better problem-solvers and problem- The columns of Gazeta Matematica repre- the summit, the 'reasoning' round and a posers. sent a hotbed of demanding questions that stage that examines problem-solving It has been claimed that mathematics in cannot be solved in a short time, and thus strategies and a mastery of mathematical East Europe flourished under this periodical was an excellent vehicle for techniques. In August 1998, forty-eight Communism because it is less susceptible identifying highly talented young persons. students attended the final round, held in to political interference than other intel- At the start of the global economical Pitesti, undergoing two 4-hour papers. lectual pursuits, and consequently attract- race in the 1980s, Romanian authorities The top scorers are awarded prize money, ed many talented minds to its study. With again became aware of the significance of books, medals or diplomas. In 2000, the ungrudging admiration we observe that Gazeta Matematica in drafting new mathe- editorial department of Gazeta Matematica Romania has produced more than its fair matical blood. The old 'tool' has been conducted its twentieth annual contest. In share of top-quality mathematicians. revived, and the Gazeta Matematica the Western hemisphere no other publica- What makes Romania unusual compared Competition now runs annually in sum- tion does as good a job as the Romanian with other Western countries is the large mer. The competitors can choose between periodical, and in general the assortment proportion of its distinguished mathemat- problems for the Gazeta's contest and of mathematics events in Romania exceeds ical experts who have devoted time and tricky questions designed to prepare by far the range of contests elsewhere. energy to the younger generation of young people for mathematics competi- I wish to thank Professor Vasile promising scientists, through via their tions. Both types of problems are pub- Berinde, of the Department of problem-solving competitions and associ- lished in Romanian and English. In 1998 Mathematics and Computer Science in the ated training units. These Academicians about 2100 teasers of the first type were North University of Baia Mare, for provid- know the necessity for challenging the amassed, while in the second some 280 ing me with information for use in this cleverest students with difficult problems problems were devised for junior contes- corner. and teaching them the mathematical skills tants and roughly 900 challenging ques- This leads to the next set of questions, required for their solution. This involve- tions for senior contributors. A useful from the Gazeta Matematica Contest in ment by many mathematical experts has barometer of the increase in mathematical September 1997.
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22 EMS September 2001 MATHDI MAMATHDITHDI -- mathematicsmathematics educationeducation onon thethe WWebeb Gerhard König and Bernd Wegner

With the transition of the traditional jour- tries will catch up. World-wide coverage is nal Zentralblatt für Didaktik der Mathematik also becoming more complete, as can be (ZDM) to a web service, a more systematic seen for Mexico and Argentina. Moreover, European extension of this originally cooperation with ERIC in the US provides German-based enterprise took place. The an excellent representation of US- most prominent sign of this is the involve- American publications in MATHDI. ment of the EMS in the editing of the web As a consequence of the involvement of versions, accompanied by the installation the EMS in MATHDI, it has been possible of some European backbones, helping with to apply the same search software to this the management of the input (local and service as for its big brother Zentralblatt national), for the ZDM-database MATHDI MATH. This software was made available (MATHematics DIdactics). Indeed, the from the partner MDC of Zentralblatt combined printed service ZDM, providing MATH in Grenoble, and it is continually survey articles and a documentation part, improved there. As a result, a good inter- has been split between the database face and a selection of search menus of dif- MATHDI and an electronic journal which ferent levels of expertise are available. Bernt Wegner with Einstein publishes the ZDM-surveys. Both offers Search results are obtained in TeX-source are easily accessible through EMIS, with its code, but additional views such as displays stand-alone installation. Since the data- more than 40 mirrors. in Postscript can be chosen. Linking facil- base is comparatively small and spe- ities, such as hypertext links to authors, are cialised, the price is kept low so that indi- MATHDI on the web made available and such facilities are grow- viduals may consider buying the CD-ROM As mentioned above, MATHDI developed ing. Classification codes are assigned reg- for their private installation. from the literature documentation activi- ularly, according to an extended scheme ties in ZDM; its general aim is thus to pro- for didactics and its reduced version in The electronic version of ZDM vide an information service and referenc- MSC 2000, and the free trial access for Efforts to keep the pricing of ZDM at a low ing tool for education in mathematics and Zentralblatt MATH has been copied for level led to the decision to separate the computer science, which was the tradition- MATHDI. Anyone can do searches without part dealing with information on publica- al role of this part of ZDM. At present, having a subscription, but in the latter case tions from the journal-like part that pro- MATHDI is the largest and most up-to- only the first three answers are displayed: vides complete articles. Moreover, date world-wide database service in this just go to the URL http://www.emis.de and acknowledging the growing importance of field. Even before it was available on the click on the MATHDI-box. electronic media and taking into account web, this part of ZDM had been available The subjects covered by MATHDI are the reduced production costs for electron- electronically via STN for quite a long wider than a non-experienced user might ic versions, it was simultaneously decided period, under the English subtitle imagine. Information is provided on that the full articles should be offered in an International Reviews on Mathematical research in mathematical education, electronic journal. Since traditional sub- Education. Searches in MATHDI can thus methodology of didactics of mathematics, scribers should not be ignored, this part is be extended back over a period of 25 mathematical instruction from elementary offered as a dual journal as long as there is years. school to university teaching and teacher sufficient interest in the print version. The input for MATHDI has been taken training, elementary mathematics and its As mentioned above, survey articles on from as many relevant documents as possi- applications, education-relevant popular education in mathematics and computer ble, and currently journal articles from mathematics, education in computer sci- science will mainly appear here, and it has more than 400 world-wide journals are ence, basic pedagogical and psychological been no problem to convince authors invit- covered. These are complemented by issues for mathematics and general educa- ed to write an article that they should nor- information on textbooks and other mono- tion in science, and administrative issues mally deliver their article in some not-too- graphs, dissertations, conference papers, such as curricula and teaching pro- sophisticated TeX-dialect. The journal is curricula, software and teaching aids in the grammes. offered through FIZ Karlsruhe, but there is theory and practice of education in mathe- There is thus a variety of users for which an agreement with ElibM in EMIS that matics and computer science. As is com- MATHDI should be an important tool. back volumes will be made freely accessi- mon practice with literature information Almost everyone thinks of professionals in ble. This is already the case, and those databases, all publications are announced didactics for mathematics and computer who are interested are invited to look at by bibliographic data, reviews or abstracts, science, but it is also directed at all inter- EMIS and click on the journal at the end of and some additional information. The ested in research and education in these the journals box in EMIS under the elec- general texts and abstracts are usually in areas, at instructors and lecturers, and at tronic library. Frequent users may then English, although exceptionally they may teachers in all types of schools where math- enter a bookmark to their directories. appear in French, German or Spanish. ematics is relevant. The information avail- This change of ZDM is a consequence of At present the contents of MATHDI able from MATHDI is a basic infrastructure the transition of publications to new comprise more than 100,000 entries, with for the professional work of educational media, and profits a lot from the support an annual increase of 6000 entries; this technologists, curriculum experts, and pol- of the EMS. It furthermore underlines the includes about 21,000 reviews from rele- icy-makers in education and educational engagement of the EMS in the develop- vant publications in the US. Current cov- administrators. On the formal side, it also ment of education in mathematics. erage of European publications is improv- offers an important tool for librarians and ing, by means of a more reliable acquisi- information specialists. Addresses: Gerhard König, ZDM, FIZ- tion and handling of national offers As an additional attractive offer, MATH- Karlsruhe, 76344 Eggenstein-Leopoldshafen, through co-operational units; here, in par- DI is distributed on CD-ROM. The most Germany [[email protected]]; ticular, Portugal, Spain, Italy, Greece, recent update is MATHDI 2001, and in Bernd Wegner, TU Berlin, Fakultät II, Institut Hungary and Yugoslavia should be men- spite of missing functionality for some link- für Mathematik, 10623 Berlin, Germany [weg- tioned; hopefully, other European coun- ing facilities, it is a good solution for a [email protected]]. EMS September 2001 23 CONFERENCES Research Council and Isaac Newton Institute for Mathematical Sciences Grants: limited funds (worth 250 pounds per FForthcomingorthcoming conferconferencesences person) to support the attendance of mathe- maticians or research students in UK universi- ties Compiled by Kathleen Quinn Site: Heriot-Watt University Deadlines: for registration, 1 October Information: Please e-mail announcements of European confer- Programme committee: Sinisa Borovic e-mail: [email protected] ences, workshops and mathematical meetings of inter- (Belgrade), chairman Web site: www.ma.hw.ac.uk/icms/current/cqis est to EMS members, to [email protected]. Organizing committee: Svetomir Minic Announcements should be written in a style similar to (Belgrade), chairman; Obrad Cabarkapa those here, and sent as Microsoft Word files or as text (Belgrade), secretary. The organisation this February 2002 files (but not as TeX input files). Space permitting, year will be coordinated by the Military each announcement will appear in detail in the next Academy of the Yugoslav Army 1 February-30 April: Special Research issue of the Newsletter to go to press, and thereafter Sponsors: Yugoslav Army, Yugoslav Trimester on Dynamical Systems, Pisa will be briefly noted in each new issue until the meet- Operational Research Society, Faculty of Topics: non-uniformly and partially hyperbolic ing takes place, with a reference to the issue in which Transport and Traffic Engineering, Mihajlo systems, quasi-periodic orbits, holomorphic the detailed announcement appeared Pupin Institute, Economics Institute, Faculty of dynamics and foliations, interaction between Economics, Faculty of Mining and Geology, dynamical systems and biology, interaction September 2001 Faculty of Organizational Sciences, Faculty of between dynamical systems and physics (includ- Mechanical Engineering and Faculty of ing celestial mechanics) Mathematics Main speakers: M. Brunella (CNRS Dijon), A. 27-30: Acoustics and Music: Theory and Proceedings: refereed proceedings of selected Chenciner (Universit de Paris VII), D. Applications 2001 (AMTA 2001), Skiathos papers will be published prior to the confer- Dolgopyat (Pennsylvania State University), C. Island, Greece ence and available to all participants Favre (CNRS, Paris), R. Ferriere (Ecole Information: Site: Military Academy, Neznanog junaka 28, Normale Paris), G. Forni (Princeton Web site: http://www.worldses.org/wses/calen- Belgrade, Yugoslavia University), A. Giorgilli (Milano Bicocca), V. dar.htm Deadlines: already expired Yu. Kaloshin (Princeton University), R. 27-30: Mathematics and Computers in Information: contact SYM-OP-IS 2001 Krikorian, J. Laskar (CNRS-IMC Paris), P. Le Biology and Chemistry 2001 (MCBC 2001), Organising Committee, Neznanog junaka 28, Calvez (Paris XIII), C. Liverani (Roma ‘Tor Skiathos Island, Greece VP 2102-4, 11002 Belgrade Vergata’), S. Luzzatto (Imperial College Information: e-mail: [email protected] London), S. Marmi (Udine and SNS Pisa), J. Web site: Web site: http://www.vj.yu/symopis/ Mather (Princeton University), G. Moreira http://www.worldses.org/wses/calendar.htm 6: 1st Annual Arf Lecture, Ankara, Turkey (IMPA), J. Palis (IMPA), J. Rivera (SUNY), D. 27-30: Mathematics and Computers in [in memory of Cahit Arf] Sauzin (CNRS, Paris), S. Smirnov (KTH Business and Economics 2001 (MCBE 2001), Speaker: Prof. Dr. Gerhard Frey (Essen) Stockholm), L. Stolovich (Université Paul Skiathos Island, Greece Title: Bauer Groups and Data Security Sabatier,Toulouse), M. Viana (IMPA), J.-C. Information: Organisers: Department of Mathematics at Yoccoz (College de France), L. S. Young Web site: Middle East Technical University, Ankara, and (Courant Institute) http://www.worldses.org/wses/calendar.htm Turkish Mathematics Foundation Programme committee: S. Marmi (Italy), J. 27-30: Automation and Information: Theory Advisory board: M. G. I. Keda, R. Langlands Mather (USA), J. Milnor (USA), J. Palis (Brasil), and Applications 2001 (AITA 2001), Skiathos and P. Roquette J.-C. Yoccoz (France) Island, Greece Organising committee: C. Carminati (Italy), G. Information: December 2001 Da Prato (Italy), M. Giaquinta (Italy), S. Marmi Web site: (Italy) http://www.worldses.org/wses/calendar.htm Sponsors: Istituto Nazionale di Alta 2-8: International Centre for Mathematical Matematica, Scuola Normale Superiore October 2001 Sciences Workshop on Classical and Site: Scuola Normale Superiore (Piazza dei Quantum Integrable Systems and their Cavalieri 7, Pisa, Italy) Symmetries, Edinburgh, UK Grants: a limited number of grants supporting 1-5: Aspects of Hyperbolic Geometry, [satellite workshop of the Isaac Newton preferably long-term visits (at least one month) Fribourg, Switzerland Institute for Mathematical Sciences programme will be available for PhD students and Post Invited speakers: include J. Anderson on Integrable Systems (July to December Docs. Applications should be made as soon as (Southampton), M. Boileau (Toulouse), M. 2001)] possible Bourdon (Lille), B. Bowditch (Southampton), S. Theme: the emphasis will be on quantum inte- Information: Buyalo (St. Petersburg), J.-L. Cathelineau grable systems and the symmetry approach, e-mail: [email protected] (Nice), C. Drutu (Lille), J. Dupont (Aarhus), U. where recent progress has been rapid and Web site: http://www.math.sns.it/degiorgi/dynsys/ Hamenstädt (Bonn), G. Knieper (Bochum), U. intense 11-15: Neural Networks and Applications Lang (ETZ Zürich), G. Martin (Auckland), J.-P. Aim: to analyse current developments and to (NNA ‘02), Interlaken, Switzerland Otal (ENS Lyon/Orléans), J. Parker (Durham), encourage the exchange of ideas between Information: F. Paulin (ENS Paris), N. Peyerimhoff researchers in the field of integrable systems Web site: (Bochum), A. Reid (Austin), J.-M. Schlenker and those working in representation theory, http://www.worldses.org/wses/calendar.htm (Toulouse), B. Stratmann (St. Andrews), E. string theory and other areas of modern math- 11-15: Fuzzy Sets and Fuzzy Systems (FSFS Vinberg (Moscow) ematics and physics ‘02), Interlaken, Switzerland Scientific Board: Christophe Bavard (Uni. Main speakers: expected to include John Information: Bordeaux I), Gérard Besson (Uni. Grenoble I), Cardy (UK), Ivan Cherednik (USA), Patrick Web site: Ruth Kellerhals (Uni. Fribourg), Viktor Dorey (UK), Jonathan Mark Evans (UK), http://www.worldses.org/wses/calendar.htm Schroeder (Uni. Zürich) Kentaro Hori (USA), Michio Jimbo (Japan), 11-15: Evolutionary Computations (EC ‘02) Site: Department of Mathematics, University of Hitoshi Konno (Japan), André Leclair (USA), Interlaken, Switzerland Fribourg, Switzerland Jean-Michel Maillet (France), Barry McCoy Information: Information: (USA), Eugene Mukhin (USA), Maxim Nazarov Web site: e-mail: [email protected] (UK), Nikita Nekrasov (France), Francesco http://www.worldses.org/wses/calendar.htm Web site: http://www.unifr.ch/math/conference Ravanini (Italy), N. Reshetikhin (USA), Ryu 2-5: SYM-OP-IS 2001 XXVIII Yugoslav Sasaki (Japan), Jun’ichi Shiraishi (Japan), Symposium on Operations Research (SYM- Evgueni Sklyanin (UK), Feodor Smirnov March 2002 OP-IS), Belgrade (France), Kanehisa Takasaki (Japan), Gerard Topics: applications of operations research Watts (UK), Alexei Zamolodchikov (France) 18-19: Workshop on Under- and Over-deter- methods in organisational, technical, techno- Organising committee: Ed Corrigan (York), mined Systems of Algebraic and Differential logical, economic and other systems Chris Eilbeck (Heriot-Watt), Tetsuji Miwa Equations, Karlsruhe, Germany Programme: will be available by 21 September (Kyoto), Robert Weston (Heriot-Watt) Theme: algebraic or differential equations, Languages: Serbian and English Sponsors: Engineering and Physical Sciences 24 EMS September 2001 CONFERENCES computer algebra, numerical analysis Topics: combinatorial number theory, combi- Organizing committee: Léopold Simar (UCL), Scope: computational approaches to under- natorial methods in convexity, combinatorial Chair, Jean-Jacques Droesbeke (ULB), Claude and over-determined systems. Submissions are methods in harmonic analysis, concentration of Cheruy (INS), Michel Denuit (UCL), Catherine expected on theory and applications, algo- measure, geometric inequalities, Ramsey meth- Vermandele (ULB), Bernadette Govaerts rithms and software. The workshop will be of ods in Banach spaces (UCL), Thérèse Lekeux (ULB), Claudia an interdisciplinary nature; the intention is to Main speakers: Keith Ball (UC London), Lemoine (UCL), Sophie Malali (UCL) bring together researchers from many different Franck Barthe (Marne la Valle), William Grants: for students, statisticians from develop- fields in order to foster communication Beckner (U Texas, Austin), Béla Bollobás ing countries between different communities (Cambridge and Memphis), Anthony Carbery Deadlines: for registration at lower fee, 15 Topics: all aspects of under- and over-deter- (Edinburgh), Michael Christ (U of California, March; for abstracts, 15 January mined systems: completion, exact or approxi- Berkeley), Apostolos Giannopoulos (U of Crete, Information: mate solutions, structure analysis, symbolic Heraklion), Gil Kalai (Hebrew University of e-mail: [email protected] and/or numerical treatment, Gröbner or invo- Jerusalem), Michel Ledoux (Toulouse), Ted Web site: www.stat.ucl.ac.be/JSBL2002 lutive bases for polynomial or differential sys- Odell (Texas A&M), Imre Ruzsa (Hungarian 25-3: XXII International Seminar on Stability tems, differential algebraic equations (DAEs), Academy of Science), Gideon Schechtman Problems for Stochastic Models (SPSM) and symmetry analysis, applications in all fields of (Weizmann Institute), Terence Tao (UCLA), Seminar on Statistical Data Analysis (SDA), mathematics or sciences Christoph Thiele (UCLA) Varna, Bulgaria Call for papers: downloadable from the con- Programme: the conference will take the form Topics of SPSM: limit theorems of probability ference web site of a series of courses, each comprising 2, 3 or 4 theory, theory of probability metrics, asymptot- Programme committee: J. Calmet (Karlsruhe, one-hour lectures. The courses are structured ic statistics, limit theorems for stochastic Workshop Chair), V. P. Gerdt (Dubna, Chair), so that the more basic material is presented processes, queueing theory, applications of W. M. Seiler (Mannheim, Chair), J. Apel during the first week, leading to more probability theory, insurance and financial (Leipzig), G. Carra Ferro (Catania), G. advanced lectures in the second week mathematics Czichowski (Greifswald), L. Lambe Organising committee: Tony Carbery Topics of SDA: robust statistics, statistical algo- (Bangor/Rutgers), E. L. Mansfield (Edinburgh), Mike Christ (UC Berkeley), Tim rithms, application of stochastic models in the (Canterbury), B. Mourrain (Sophia-Antipolis), Gowers (Chair, Cambridge), Vitali Milman (Tel industry P. J. Olver (Minneapolis), E. Pankratiev Aviv), Terry Tao (UCLA) Organizing Committee: Vladimir Zolotarev (Moscow), V. P. Shapeev (Novosibirsk), J. Sponsors: EC Framework V and the (Russia, Chair of SPSM), Dimitr Vandev Tuomela (Joensuu) Engineering and Physical Sciences Research (Bulgaria, Chair of SDA) Proceedings: submissions are not formally ref- Council (EPSRC) of the UK Site: International Home of Scientists ‘Frederic ereed and may be submitted later elsewhere. Site: the James Clerk Maxwell Building of the Joliot Curie’ Informal proceedings will be distributed to par- University of Edinburgh Information: ticipants. Authors of outstanding submissions Grants: some funding available to assist EU Web site: http://stabil.fmi.uni-sofia.bg will be invited to contribute to a special issue of nationals who are under the age of 35. ICMS 27-29: Spring School on Frobenius Manifolds the AAECC journal also has funds from the LMS to assist UK in Mathematical Physics, Enschede, The Site: University of Karlsruhe mathematicians. Please refer to the website for Netherlands Deadlines: for submission, 1 December; for full details Information: registration, 15 February Deadlines: for registration, 16 December e-mail: [email protected] Information: Information: 27-31: 6th Congress of SIMAI, Chia Laguna, Web site: e-mail: [email protected] Sardinia http://iaks-www.ira.uka.de/iaks-calmet/ADE Web site: www.ma.hw.ac.uk/icms/meetings/ Topics: applied mathematics and the applica- 21-22: Eighth Rhine Workshop on Computer 2002/cama February 2002 tions of mathematics in industry, technology, Algebra, Mannheim, Germany environment and society Aims: to serve as a regional forum for May 2002 Information: researchers in the field, and in particular to Web site: offer an opportunity to young researchers and http://www.iac.rm.cnr.it/simai/simai2002 newcomers to present their work 13-17: 34th Journées de Statistique, French Topics: all aspects of computer algebra, from Statistical Society, Brussels and Louvain-la- June 2002 theory to applications and systems Neuve, Belgium Call for papers: downloadable from the con- Themes: statistical analysis of functional data, ference web site actuarial and financial econometrics, resam- 4-13: 3rd Linear Algebra Workshop BLED Programme committee: H. Kredel pling methods, non-parametric inference and 2002, Bled, Slovenia (Mannheim, Workshop Chair), W. M. Seiler modelling, voice and writing recognition, epi- Main theme: the interplay between operator (Mannheim, Chair), M. Bronstein (Sophia demics, genomics as well as mathematical sta- theory and algebra Antipolis), R. Buendgen (Boeblingen), J. tistics Programme: talks in morning sessions, work in Calmet (Karlsruhe), J. Della Dora (Grenoble), Topics: sequential analysis, multivariate robust smaller groups in the afternoons A. Cohen (Eindhoven), J. C. Faugere (Paris), V. analysis, classification, applications of copulas Programme committee: R. Drnovšek (SLO), L. P. Gerdt (Dubna), M. MacCallum (London), D. in insurance and finance, change point detec- Grunenfelder (CAN), T. Košir (SLO), M. Mall (Zurich), E. L. Mansfield (Canterbury), T. tion, censored and missing data, Gibbs sam- Omladiè (SLO), H. Radjavi (CAN) Recio (Santander), M. Schlichenmaier pling, frontier estimation, extreme value theo- Organizing committee: R. Drnovšek (SLO), T. (Mannheim), W. K. Seiler (Mannheim), T. ry, stochastic finance, data and files merging, Košir (SLO), M. Omladiè (SLO) Sturm (Passau), C. Traverso (Pisa), W. Werner the 150th anniversary of the first International Site: Hotel Golf, Bled (Heilbronn), F. Winkler (Linz), E. Zerz Congress of Statistics, statistical software for Deadlines: for registration, 1 December (Kaiserslautern) text analysis, algebraic methods in time series Information: Proceedings: submissions are not formally ref- analyses, non-standard mathematical methods e-mail: [email protected], ereed and may be submitted later elsewhere. in statistics and regression analysis, latent vari- Web site: http://www.ijp.si/ftp/pub/STOp/law/ Informal proceedings will be distributed to par- ables models, hierarchical Bayes models, sto- 5-9: Conference in Honour of Hans Wallin, ticipants chastic models in telecommunication, large Umea, Sweden Site: University of Mannheim panels of time series data, deconvolution prob- Themes: 1) potential theory, function spaces, Deadlines: for submissions, 1 December; for lems, stochastic processes in functional spaces, approximation theory and related topics; 2) early registration, 15 February; late registration data depth, wavelets and time series, chemo- mathematics education at higher fee until start of conference metrics, environmetrics, statistics in astronomy, Aim: to bring together researchers in some of Information: statistics and language, continuous time sto- the areas in which Hans Wallin has been work- Web site: http://www.uni-mannheim.de/RWCA chastic processes, spatial and directional statis- ing, in order to get a picture of some current 26-4 April: International Centre for tics, risk theory, image processing and neural research on these areas. We hope that the con- Mathematical Sciences EuroSummer School networks ference will be useful for a comparatively broad and Instructional Conference on Languages: French and English audience, and we especially encourage Ph.D. Combinatorial Aspects of Mathematical Programme committee: Marc Hallin (ULB), students to participate Analysis, Edinburgh, UK Chairman Lucien Birgé (Paris 6), Henri Main speakers: A. Ambroladze (Sweden), D. Aim: to instruct young mathematicians in top- Caussinus (Toulouse), Christian Genest (Laval), Broomhead (Great Britain), L. Carleson ics involving combinatorial ideas in mathemati- Irène Gijbels (UCL), Ludovic Lebart (ENST- (Sweden), Z. Cieselski (Poland), S. Janson cal analysis (10-day course) CNRS), Bernard Ycart (Paris 5) (Sweden), D. S. Lubinsky (South Africa), P.

EMS September 2001 25 CONFERENCES Mattila (Finland), E. B. Saff (USA), J.-O. Gitik (Tel Aviv), Volker Halbach (Konstanz), Stromberg (Sweden), A. Teplayev (USA), O. Bakhadyr Khoussainov (Auckland), Steffen September 2002 Bjorkqvist (Finland), G. Gjone (Norway), M. Lempp (Madison, WI), Toniann Pitassi Niss (Denmark) (Tucson, AZ), Thomas Scanlon (Berkeley, CA), 4-7: International Conference on Dynamical Programme committee: Alf Jonsson, Johan Ralf Schindler (Wien), Patrick Speissegger Methods for Differential Equations, Lithner, Kaj Nyström, Tord Sjödin, Peter (Madison, WI), Katrin Tent (Würzburg), Lev Valladolid, Spain Wingren Beklemishev (Moscow/Utrecht), Steven Cook Theme: the focus is on those recent advances in Organising committee: Margareta Brinkstam, (Toronto, ON), Olivier Lessmann (Chicago, topological methods and ergodic theory which Jan Gelfgren, Tord Sjödin, Britt-Marie Stocke IL), Simon Thomas (Piscataway, NJ) are relevant to the analysis of ordinary differen- Site: Umea university, Umea, North of Sweden Sessions: Computability theory [organisers: tial equations, partial differential equations and Deadlines: for registration, 31 October Steffen Lempp (Madison, WI), Manuel functional equations, as well as on their applica- Information: Lerman (Storrs CT), Andrea Sorbi (Siena)], tions to science and technology e-mail: [email protected] Frank Stephan (Heidelberg), Iskander Scientific committee: Amadeu Delshams Web site: http://www.math.umu.se/aktuellt/ Kalimullin (Kazan), Sebastiaan Terwijn (Spain), Russell Johnson (Italy), Rafael Obaya HWkonferens.htm (Amsterdam), Charles McCoy (Madison, WI), (Spain), Rafael Ortega (Spain) 10-16: Aarhus Topology 2002, Aarhus, Vasco Brattka (Hagen), Ivan Soskov (Sofia), Organising committee: Ana I. Alonso, Sylvia Denmark Elias Fernandez-Combarro Alvarez (Oviedo); Novo, Carmen Núñez, Rafael Obaya, Jesús Rojo Theme: algebraic topology Non-monotonic logic [organisers: Karl (all Universidad de Valladolid, Spain) Main speakers: include Raoul Bott (USA), Schlechta (Marseille), Krister Segerberg Main speakers: L. Diaz (Brazil), A. Jorba Ralph Cohen (USA), Yakov Eliashberg (USA), (Uppsala)], Nick Asher (Austin, TX), (Spain), U. Kirchgraber* (Switzerland ), P. Jesper Grodal (USA), Karsten Grove (USA), Alexander Bochman (Holon), Dov Gabbay Kloeden (Germany), R. Krikorian (France), Y. Lars Hesselholt (USA), Mike Hopkins (USA), (London), Daniel Lehmann (Jerusalem), David Latushkin (USA), R. de la Llave (USA), R. Wolfgang Lück (Germany), Mike Mandell Makinson (Paris), Rohit Parikh (New York, Markarian (Uruguay), W. de Melo* (Brazil ), J. (USA), Fabien Morel (France), Bob Oliver NY), Renata Wassermann (São Paulo); Set the- A. Rodríguez (Spain), G. R. Sell (USA), Y. Yi (France), Erik K. Pedersen (USA), Zoltan Szabo ory [organisers: Alessandro Andretta (Torino), (USA) (* to be confirmed) (USA), Ulrike Tillmann (UK), Vladimir Turaev Sy Friedman (Vienna)], David Aspero Proceedings: selected papers from the confer- (France) (Barcelona), Doug Burke (Las Vegas, NV), ence will be published in a special issue of Organising committee: Johan Dupont (Chair), James Hirschorn (Helsinki), Ilijas Farah (New Journal of Dynamics and Differential Equations University of Aarhus; Hans Jørgen Munkholm, York, NY), Rene Schipperus (Beer-Sheva), Site: Castillo de la Mota (Medina del Campo), SDU, Odense University; Lars Hesselholt, MIT, Paul Larson (Toronto, ON), Su Gao Valladolid, Spain USA; Lisbeth Fajstrup, Aalborg University (Pasadena, CA) Grants: a limited number of financial grants are Site: University of Aarhus Programme committee: Klaus Ambos-Spies available for graduate and doctoral students Deadline: for registration, to be announced at (Heidelberg), Sam Buss (San Diego CA), Zoé Deadline: for pre-registration and submission our web site Chatzidakis (Paris), Alekos Kechris (Pasadena, of abstracts, 28 February Information: CA, Chair), Peter Koepke (Bonn), Peter Information: Web site: http://www.imf.au.dk/AT2002/ Komjath (Budapest), Manuel Lerman (Storrs, e-mail: [email protected]. 17-19: BEM 24, 24th International CT), Vann McGee (Cambridge, MA), Wolfram Web site: http://wmatem.eis.uva.es/~dmde02/ Conference on Boundary Element Methods Pohlers (Münster), Michael Rathjen (Leeds), February 2003 and Meshless Solutions Seminar, Sintra, Krister Segerberg (Uppsala), Boris Zilber 5-7: 4th IMACS Symposium on Mathematical Portugal (Oxford) Modelling, Vienna, Austria Organiser: Wessex Institute of Technology, Organising committee: Manfred Burghardt Aim: to give scientists and engineers using or UK and the University of Coimbra, Portugal (Bonn), Justus Diller (Münster), Peter Koepke developing models or interested in the develop- Sponsors: International Society of Boundary (Bonn), Benedikt Löwe (Bonn), Michael ment or application of various modelling tools Elements (ISBE) and the International Journal Möllerfeld (Münster), Wolfram Pohlers an opportunity to present ideas, methods and of Engineering Analysis with Boundary (Münster, Chair), Andreas Weiermann results and discuss their experiences or prob- Elements (Münster) lems with experts of various areas of specialisa- Deadline: for papers, 15 January Sponsors: Association of Symbolic Logic tion Information: contact Conference Secretariat , Information: Scope: theoretic and applied aspects of the var- BEM02, Wessex Institute of Technology, Web site: http://www.math.uni-muenster.de/ ious types of mathematical modelling (equa- Ashurst Lodge, Ashurst Southampton, SO40 LC2002/ tions of various types, automata, Petri nets, 7AA, UK, tel: 44 (0) 238 029 3223, fax: 44 (0) 10-11: Colloquium Logicum 2002, Münster, bond graphs, qualitative and fuzzy models, etc.) 238 029 2853 Germany for systems of dynamic nature (deterministic, e-mail: [email protected] [satellite conference of Logic Colloquium 2002] stochastic, continuous, discrete or hybrid with Web site: http://www.wessex.ac.uk/conferences/ Main speakers: Toshiyasu Arai (Hiroshima), respect to time, etc.). Comparison of modelling 2002/be02/index.html Joan Bagaria (Barcelona), Andre Nies approaches, model simplification, modelling 27- 3 July: Fifth International Conference on (Chicago, IL), Martin Otto (Swansea), Charles uncertainties, port-based modelling, and the Curves and Surfaces, Saint-Malo, France Parsons (Harvard), Anand Pillay (Urbana- impact of items such as these on problem solu- Information: Champaign, IL), Michael Rathjen (Leeds), tion, numerical techniques, validation, automa- e-mail: saint-malo@imag. Johan van Benthem (Amsterdam/Stanford CA) tion of modelling and software support for Web site: &http://www-lmc.imag.fr/saint-malo/ Organizing & scientific committee: Justus modelling, co-simulation, etc. will be discussed Diller (Münster), Peter Koepke (Bonn), in special sessions as well as applications of July 2002 Benedikt Loewe (Bonn), Wolfram Pohlers modelling in control, design or analysis of sys- (Münster, Chair), Christian Thiel (Erlangen), tems in engineering and other fields of applica- Wolfgang Thomas (Aachen), Andreas tion. Presentations of modelling and simulation 16-22: 7th International Spring School: Weiermann (Münster) software and a book exhibition will be organ- Nonlinear Analysis, Function Spaces and Organizer: DVMLG ised Applications (NAFSA 7), Prague, Czech Sponsors: Deutsche Vereinigung für Organiser: Division for Mathematics of Control Republic Mathematische Logik und für Grundlagen der and Simulation (E114/3) at Vienna University of Information: Exakten Wissenschaften (DVMLG) Technology. e-mail: [email protected] Information: Chair: of IPC, Univ. Prof. Dr. Inge Troch Web site: http://www.math.cas.cz/~nafsa7 Web site: http://wwwmath.uni-muenster.de/ Site: Vienna University of Technology [For details, see EMS Newsletter 40] LC2002/ Deadlines: for submission of abstracts, 15 May 25-30: Wireless and Optical Communications 2002; for notification of authors, 15 October August 2002 (WOC ‘02), Miedzyzdroje, Poland 2002, for full paper, 1 December 2002 Information: Information: Web site: Contact Univ. Prof. Dr. Inge Troch, Vienna 3-10: Logic Colloquium 2002 (ASL European http://www.worldses.org/wses/calendar.htm University of Technology, Wiedner Summer Meeting), Münster, Germany 25-30: Nanoelectronics, Nanotechnologies Hauptstrasse 8 - 10 A-1040 Wien, Austria, tel: Main speakers: Jeremy Avigad (Pittsburgh, (NN’02), Miedzyzdroje, Poland +431-58801-11451, fax: +431-58801-11499 PA), Arnold Beckmann (Münster), Tim Information: e-mail: [email protected] Carlson (Columbus, OH), Robert Constable Web site: Web site: (Ithaca, NY), Kosta Dosen (Toulouse), Moti http://www.worldses.org/wses/calendar.htm http://simtech.tuwien.ac.at/ MATHMOD

26 EMS September 2001 RECENT BOOKS of perturbation analysis of continuous optimi- sation problems. The framework of the book is abstract: the optimisation problem consid- ered is parametrised by a parameter varying RecentRecent booksbooks in a Banach space, theoretical results are for- mulated for Banach spaces, and the duality of edited by Ivan Netuka and Vladimír Sou³ek Banach spaces is the main tool for proofs. The book investigates the continuity and dif- ferentiability of the optimal value and the set Books submitted for review should be sent to the fol- sentation of elementary informal set theory; of all optimal solutions with respect to the lowing address: the chapters are: Sets, Functions, Relations, parameter. Ivan Netuka, MÚUK, Sokolovská 83, 186 75 Infinite and Finite Sets. Part 3 introduces The book has seven chapters. The intro- Praha 8, Czech Republic. such basic structures as groups, partially ductory chapter describes the topic’s relation ordered sets, lattices, positive integers, ratio- to other fields, such as non-linear program- H. Bass and A. Lubotzky, Tree Lattices, nal and real numbers, and presents computa- ming, optimal control and variational inequal- Progress in Mathematics 176, Birkhäuser, Boston, tions with finite sets. The text has over 400 ities. Chapter 2 contains the background 2001, 233 pp., DM 108, ISBN 0-8176-4120-3 exercises, with hints for selected ones. material needed for a full understanding of and 3-7643-4120-3 The presentation of the subject is sufficient- the text: basic functional analysis, duality in This is an advanced book on geometrical ly precise, and motivations of introduced Banach spaces, recession cones, directional methods in group theory and combinatorial notions are discussed. The book can be seen differentiability of a function, tangent cones, group theory. The authors study groups of as a solid base of general mathematical the basic elements of multi-function theory, autormorphisms of locally finite trees stress- notions and methods. (jml) properties of convex functions, and conjugate ing parallels with the theory of Lie groups. (Fenchel) and Lagrangian duality. The third Applications to combinatorics and number D. Bao, S.-S. Chern and Z. Shen, An chapter discusses first- and second-order opti- theory are given. In a sense, this is a continu- Introduction to Riemann-Finsler Geometry, mality conditions for the optimisation prob- ation of the classical monograph of J. P. Serre, Graduate Texts in Mathematics 200, Springer, lems; this is based on Lagrangian duality Trees (Springer, 1980), and indeed the book is New York, 2000, 431 pp., DM 98, ISBN 0-387- extended to a generalized Lagrangian. dedicated to Serre ‘in admiring tribute’. (jnes) 98948-X Chapter 4 is the main part of the book, giving This book offers the most modern treatment a comprehensive study of stability and sensi- V. I. Bernik and M. M. Dodson, Metric of the topic and will attract both graduate stu- tivity analysis of an optimisation problem Diophantine Approximation on Manifolds, dents and a broad community of mathemati- parametrised by a parameter varying in a Cambridge Tracts in Mathematics 137, Cambridge cians from various related fields. The authors Banach space. Subsections present a first- University Press, Cambridge, 1999, 172 pp., start with a short but informative historical order differentiability analysis of the optimal £27.50, ISBN 0-521-43275-8 exposition and with a guide to the contents of value and a discussion of the quantitative sta- This book deals with metric Diophantine the book. They show how the original moti- bility of optimal solutions and Lagrange mul- approximations on smooth manifolds embed- vation of the topic came from physics and tipliers, followed by a second-order analysis in dable in a Euclidean space. The text starts mention other fields of applications, such as Lipschitz and Hölder stable cases. A special with an overview of basic problems and results ecology and biology. subsection treats second-order analysis in on Diophantine approximation in one dimen- Whereas a Riemann structure is a smooth function spaces. The fifth chapter brings sion, and the necessary analytic background family of inner products, a Finsler structure additional materials and applications, includ- on such manifolds needed for transition to can be viewed as a smooth family of general ing a discussion of variational inequalities, metric aspects of the Diophantine approxima- Minkowski norms along a manifold. One of non-linear programming, semi-definite pro- tion on manifolds. In the second chapter, the main goals of this book seems to be to gramming and semi-infinite programming. they extend Khintchine’s and Groshev’s theo- answer the following question: to what extent Chapter 6 is relatively self-contained, and con- rem on simultaneous approximation to cer- can the basic notions and theorems from glob- siders optimisation problems based on partial tain manifolds, and prove a conjecture of A. al Riemann geometry be generalised to differential equations, such as the Dirichlet Baker related to his previous extension of Finsler structures, either general ones, or spe- problem, optimal control of a semi-linear Sprindžuk’s theorem. The next three chap- cialised ones? elliptic equation, the state-constrained opti- ters are devoted to Hausdorff dimension, The exposition is based mainly on the so- mal control problem and the obstacle prob- especially to different techniques developed called Chern connection and its curvature, lem. The final chapter gives bibliographical for upper and lower bounds of the associated which are studied in detail in the first part of notes and references for additional reading. null sets. Chapter 6 contains an account of p- the book. Through this approach, one learns The monograph contains the present state adic Diophantine approximation on mani- that the following items make sense in a gen- of knowledge of the topic. The text is written folds. The final chapter deals with various eral Finsler geometry: Schur’s lemma, the in a clear and correct manner and is arranged applications, such as the wave equation and generalised Gauss-Bonnet theorem, Jacobi in order to help the reader. The book can be the rotation number. Each chapter ends with fields, the Hopf-Rinow theorem, index form recommended for graduate students, notes containing material for further reading and the Bonnet-Myers theorem, cut and con- researchers and practitioners in optimisation and historical comments. Especially valuable jugate loci and Synge’s theorem, the Cartan- theory. (pl) are those connected with results spread in Hadamard theorem and Rauch’s first theo- journals within the former Soviet Union. rem. These topics are studied in the second X. Buff, J. Fehrenbach, P. Lochak, L. This book can be recommended not only to part of the book. The final part of the book is Schneps and P. Vogel, Espaces de modules des those interested in number-theoretic aspects, devoted to special Finsler spaces, and is con- courbes, groupes modulaires et théorie des but also to those whose interests lie in topics nected with the names of H. Akbar-Zadeh, F. champs, Panoramas et Synthéses 7, Société related to dynamical systems. The book is Brickel, A. Deicke and Z. Szabó. One chapter Mathématique de France, Paris, 1999, 143 pp., essentially self-contained and very readable. illustrates the general concepts in pure FRF 120, ISBN 2-85629-073-6 (šp) Riemannian geometry. This book consists of lecture notes from a Full credit is given to the traditional three-day workshop. The first two parts (Élé- E. D. Bloch, Proofs and Fundamentals. A First Japanese, Romanian and other schools of ments de géométrie des espaces de modules Course in Abstract Mathematics, Birkhäuser, Finslerian geometry, as well as to the mathe- des courbes, and Groupoïdes fondamentaux Boston, 2000, 424 pp., DM 108, ISBN 0-8176- maticians who made isolated contributions to des espaces de modules en genre 0 et cate- 4111-4 and 3-7643-4111-4 the topic. The authors seem anxious to be fair gories tensorielles tressées) are devoted to the This book presents an elementary abstract as concerns citations. The book includes an structure of Teichmüller space of Riemann basis of mathematics in three natural parts: impressive 393 exercises and some examples surfaces, while the third, more or less inde- logic, basic set notions and methods, and basic using Maple. (ok) pendent, part (Invariants de Witten- mathematical structures; the names of these Reshetikhin-Turaev et théories quantiques parts are: Proofs, Fundamentals, Extras. J. F. Bonnans and A. Shapiro, Perturbation des champs) is devoted to ‘quantum group’ Part 1 is an explanation of elementary Analysis of Optimization Problems, Springer invariants of links and 3-dimensional mani- notions of first-order logic and an exposition Series in Operations Research, Springer, New York, folds. Topics of these sections intersect in the of how to use them to produce logically cor- 2000, 601 pp., DM 139, ISBN 0-387-98705-3 notion of braided or strict monoidal category. rect arguments and proofs. Part 2 is a pre- This monograph presents a compact overview In the first part we find a definition of EMS September 2001 27 RECENT BOOKS

Teichmüller space Tg,n of Riemann surfaces of famous facts in the topic in just 150 pages. genus g with n distinct labelled points and Twenty modules (chapters) are followed by C. J. Colbourn and A. Rosa, Triple Systems, also two alternative descriptions of this space – many exercises, formulated as lemmas or the- Oxford Mathematical Monographs, Clarendon in terms of hyperbolic metrics and by sub- orems with many hints, and remarks and Press, Oxford, 1999, 560 pp., £80, ISBN 0-19- groups of PSL(2, R). The space Tg,n is topolo- questions. 853576-7 gised by the Fenchel-Nielsen coordinates The book contains the Dirichlet and This book presents the current knowledge which are then used to describe a compactifi- Hurwitz theorems, continued fractions, the about triple systems, collecting together com- cation CMg,n of the corresponding moduli Markoff spectrum, the Pell equation, Liouville mon themes and providing an accurate por- space Mg,n by adding ‘degenerate’ points. The and Roth’s results, elliptic curves, the geome- trait of an incredible variety of problems and try of numbers and its application to simulta- results. Representative samples of major decorated Teichmüller space T’g,n, intimately related to graph complexes of Kontsevich, is neous diophantine approximations, uniform styles of proof techniques are provided. The also studied. distribution, and p-adic analysis. Many proofs book is intended primarily for readers with a The central object of the second part is the are omitted (the Mordell and Mazur theo- basic knowledge of combinatorial design the- rems), some are only sketched (the Roth theo- ory. completed Teichmüller groupoid T’0,n, that is, the profinite completion of the fundamental rem), while some are presented with full After a historical background, the first three groupoid of the moduli space CM based at a proofs (the Lagrange theorem, the Minkowski chapters explain in detail the basic material 0,n convex body theorem, the linear forms theo- on constructions and existence of triple sys- neighborhood of ‘most degenerate’ configura- rem and Kronecker’s theorem). The book tems. The next five chapters describe topics tions of CM . This groupoid admits a natur- g,n includes an index and references for further connected with triple systems: isomorphism, al action of the Galois group Gal(Cl(Q)/Q). reading, and can be warmly recommended to enumeration, subsystems and automorphisms. The Grothendieck-Teichmüller group GT, ≥ students and university teachers. (bn) Chapters 9-23 treat a number of challenging isomorphic for n 4 to the automorphism problems on triple systems in detail. Chapters group of T’ , is also introduced. The cele- 0,n M. Capiñski and T. Zastawniak, Probability 24 and 25 provide a guide to two related class- brated conjecture that Gal(Cl(Q)/Q) is iso- Through Problems, Problem Books in es of triple systems in which the triples contain morphic to GT is then formulated. Mathematics, Springer, New York, 2001, 257 pp., ordered pairs. A comprehensive bibliography The third part explicates how a ribbon DM 109, ISBN 0-387-95063-X on triple systems is provided. (jj) braided monoidal category induces a link This book is intended to accompany an invariant or, more generally, an invariant of undergraduate course in probability; the only J. B. Conway, A Course in Operator Theory, pairs (M, L) of a 3-manifold M and a link L prerequisite are basic algebra (including ele- Graduate Studies in Mathematics 21, American embedded in M. Particular examples of these ments of set theory) and calculus. A brief sur- Mathematical Society, Providence, 1999, 372 pp., braided monoidal categories are provided by vey of the terminology and notation of set the- US$49, ISBN 0-8218-2065-6 representations of quantum groups. The rest ory and calculus is provided at the beginning This book is a continuation of the author’s of this part indicates how (in some cases) these of the book. previous text A Course in Functional Analysis invariants are equivalent to topological quan- The body of the book is divided into twelve (Springer, 2nd ed., 1990). Its aim is to cover tum field theories. chapters. Chapter 1 concerns elements of central topics of operator theory in a form The concise and self-contained exposition is modelling of random experiments. Chapters that is accessible to graduate students. primarily aimed at non-specialists and gradu- 2-6 are devoted to classical probability spaces The book is divided into eight chapters. ate students, but a specialist might also find and related combinatorial problems, fields The first two chapters, devoted to C*-algebras useful information and proofs that are diffi- and σ-fields of sets, finitely and countably and normal operators, overlap Chapters 8 cult to locate in the existing literature. (mm) additive probability. Chapter 7 focuses on and 9 from the previous edition, but they conditional probability and independence, make the book more independent. The theo- D. Bump, Algebraic Geometry, World Scientific, Chapter 8 concerns random variables and ry of C*-algebras is continued in Chapter 5, Singapore, 1998, 218 pp., £26, ISBN 9-810- their distribution functions, while Chapter 9 where irreducible representations and posi- 23561-5 offers problems on their expectations and tive maps are examined. As an application, This book is designed as a text for a one-year variances and is devoted to conditional expec- the dilation theorem is proved. Chapter 3 dis- course in basic algebraic geometry at the grad- tations. Charateristic functions are the subject cusses ideals of bounded operators in a uate level. It is divided into two parts. of Chapter 11, while Chapter 12 contains Hilbert space, especially the ideal of compact The first part is devoted to the general the- problems connected with the laws of large operators. Chapter 6 deals with a study of ory of affine varieties over an algebraically numbers and the central limit theorem, with Fredholm operators and compact perturba- closed field. The theory of affine varieties is an emphasis on consequences and applica- tions. The Weyl-von Neumann-Berg theorem systematically and clearly explained in the tions. Each chapter is divided into theory on almost diagonalisation of normal operators first part of the book, and some prerequisities (including basic notions and theorems), prob- is proved here. There are two central chap- from algebra are included. The dimension lems, hints and solutions. Hints are given for ters: Chapter 4 examines non-normal opera- and products of affine varieties are studied all problems and fully worked solutions for the tors like isometries in particular shifts, and and a section is devoted to theory of projective majority of them. All problem sections some deep connections between operator the- varieties. The second part is devoted to the include expository material. (mahu) ory and analytic functions are shown; Chapter theory of algebraic curves, and the main topic 7 treats von Neumann algebras and their clas- is the theory of complete non-singular curves. S. Choi, The Convex and Concave sifications. The final chapter explores rela- The questions studied are the ramification Decomposition of Manifolds with Real tions between sets of operators and their com- problem, extensions and completions of a Projective Structures, Mémoires de la Société mon invariant subspaces (reflexive and field with respect to a valuation, differentials, Mathématique de France 78, Société Mathématique hyper-reflexive operators). residues and the Riemann-Roch theorem. de France, Paris, 1999, 102 pp., FF 150, ISBN 2- This book is written in a very readable style. Special attention is paid to the theory of ellip- 85629-079-5 In spite of the many recent results included, a tic curves and their properties and to the zeta This small book is devoted to a study of prop- reader is not lost in technical explanations, function of a curve. As mentioned by the erties of real manifolds with a flat projective since the main ideas and comments are simul- author, the most significant omission is inter- structure. The author introduces a notion of taneously given. Many exercises make this section theory. Each chapter has a set of exer- i-convexity, generalising the ordinary notion book convenient for independent study, with cises illustrating its content and helping a bet- of convexity, and he proves a decomposition cross-references throughout. The whole text ter understanding of the theory. As a whole, theorem showing that a real projective mani- is well organised, rendering the book suitable the textbook offers a good introduction to fold of dimension n is either (n–1)-convex or for anybody interested in the above topics. algebraic geometry. (jbu) can be decomposed into simpler real projec- (jmil) tive manifolds. Special attention is paid to E. B. Burger, Exploring the Number Jungle: A the three-dimensional situation, as motivation A. Croft, R. Davison and M. Hargreaves, Journey into Diophantine Analysis, Student for such a study came from the study of Engineering Mathematics, with CD, Pearson Mathematical Library 8, American Mathematical hyperbolic 3-manifolds. The methods devel- Education, London, 2001, 969 pp., £39.99, Society, Providence, 2000, 151 pp., US$20, oped are applied to a classification of radiant ISBN 0-130-26858-5 ISBN 0-8218-2640-9 affine three-dimensional manifolds. The This book was written to serve the mathemat- This book provides an excellent survey of book brings a systematic and ordered treat- ical needs of students of a first course in engi- Diophantine analysis. It is surprising how the ment of this new field, which was not previ- neering, primarily for students of electronic, author is able to explain many interesting and ously available in book form. (vs) electrical, communication and system engi- 28 EMS September 2001 RECENT BOOKS neering. most useful and authoritative secondary M. Dummett, Elements of Intuitionism, Oxford The book has two main aims. The first is to sources and other types of primary source Logic Guides 39, Clarendon Press, Oxford, 2000, provide an accessible and readable introduc- (texts, manuscripts, correspondence, etc.). 331 pp., £60, ISBN 0-19-850524-8 tion to engineering mathematics at the degree Anyone who wishes to become acquinted with The first edition of this book appeared in level; the second is to encourage the integra- the history of mathematics can begin with this 1977. The author’s intention is to give basic tion of engineering and mathematics. impressive bibliographical database. (mnem) information about the fundamental ideas of The first three chapters include a review of intuitionism, and especially to clarify two such some important functions and techniques M. Dimassi and J. Sjöstrand, Spectral ideas underlying intuitionistic mathematics. from previous courses. These chapters con- Asymptotics in the Semi-Classical Limit, The first is a general theory of meaning for a tain a review of algebraic techniques, engi- London Mathematical Society Lecture Note Series mathematical language, according to which neering functions and trigonometric func- 268, Cambridge University Press, Cambridge, the only thing that can make a statement true tions. The next chapters include descriptions 1999, 227 pp., £ 24.95, ISBN 0-521-66544-2 is an intuitively acceptable proof, representing of many topics: coordinate systems, sequences This book is based on a course given by the a certain kind of mental construction. The and series, vectors, matrix algebra, complex authors at various universities in France. Its second is the concept of infinite ‘effective’ numbers, differentiation, integration, Taylor main theme is an application of methods of sequences, which are developed in Chapters 3 polynomials, Taylor and Maclaurin series, microlocal analysis to spectral problems in (Choice sequences and spreads), 4 (The for- ordinary differential equations, the Laplace semi-classical limit. Main notions are briefly malism of intuitionistic logic) and 5 (The transform, difference equations and the z- reviewed (local symplectic geometry, self- semantics of intuitionistic logic). transform, Fourier series and Fourier trans- adjoint operators), and the authors then Philosophical remarks form Chapter 7. Some form, functions of several variables, vector cal- develop the basic theory of h-pseudodifferen- parts from the first edition are revised – for culus, line and multiple integrals, statistics tial operators and a functional calculus for example, the account of Brouwer’s proof of and probability. There are four appendices them). WKB methods are used for the con- the Bar theorem, as well the treatment of gen- and an index. As a supplement, an interactive struction of local asymptotic solutions of the eralised Beth trees. CD testing and assessment package is includ- Schrödinger operator and the method of sta- This book provides a comprehensive pre- ed. (jkof) tionary phase is explained. There are discus- sentation of the subject and can be read with- sions of tunnel effects, asymptotic expansions out special preliminary knowledge. (jml) P. R. Cromwell, Polyhedra, Cambridge for the trace and of spectral results in various University Press, Cambridge, 1999, 451 pp., special situations. Each chapter ends with his- J. Fauvel, R. Flood and R. Wilson (eds.), £32.50, ISBN 0-521-55432-2 and 0-521- torical remarks, useful comments and recom- Oxford Figures: 800 Years of the Mathematical 66405-5 mendations for further reading. The book Sciences, Oxford University Press, Oxford, 2000, This book comprehensively documents many can be used for a one-semester course on the 296 pp., £35, ISBN 0-19-852309-2 ways that polyhedra have appeared in the his- topic. (vs) This book chronicles the development of tory of mathematics and the sciences. It is an mathematical research and studies at Oxford unusual book as it combines the style of a his- S. Dineen, Complex Analysis on Infinite University from its foundation to the 20th torical essay with a description of scientific Dimensional Spaces, Springer Monographs in century. It is a story of the intellectual and achievements and goals, and it aims to prove Mathematics, Springer, London, 1999, 543 pp., social life of the mathematical community (the basic results and theorems. The figures are DM 179, ISBN 1-85233-158-5 community of professors and students and the nice, ranging from Kepler and Dürer to mod- This book contains a comprehensive study of wider scientific community in Britain and ern times, and are complemented by a schol- properties of holomorphic functions on open throughout the world). arly exposition. To quote one of the reviews: subsets of infinite-dimensional complex topo- The authors describe those parts of mathe- ‘it is a labor of love’ and successfully comple- logical vector spaces. The first two chapters matics that were covered in lectures and how ments earlier books, most notably those of are devoted to a study of polynomials in an they were treated at Oxford, the content of Grünbaum. In some places the writing is a bit infinite-dimensional setting. They are intro- examinations and how they were realised, vague: for example, while disscussing the duced using multi-linear maps and tensor what and how was changed during more than proof of the 4-colour theorem, the author products, their relation to geometric concepts 800 years (the transformation of the mathe- describes Appel and Haken’s attempt without of Banach space theory is discussed and the matical curriculum and the role of mathemat- mentioning the recent work of Robertson, duality theory for polynomials is developed. ics in British society). The aspects of the his- Seymour and Thomas. This is a book for a Chapter 3 introduces basic definitions of holo- tory of mathematics in the periods of medieval broad mathematical and scientific audience. morphic maps between infinite-dimensional Oxford, renaissance Oxford, the mid-17th (jnes) spaces, studies their Taylor and monomial century, the Newtonian school, Georgian expansions and introduces main topologies Oxford, the mid-19th century and the 20th J. W. Dauben (ed.), The History of on the space of holomorphic maps on an open century are described so that the reader can Mathematics from Antiquity to the Present: A set. The next two chapters contain the central see how mathematics was developed century Selective Annotated Bibliography, Revised edi- theme of the book – a comparison of the main by century. There are stories about many tion on CD-ROM, American Mathematical Society, three topologies on the space of holomorphic well-known and sometimes surprising figures, Providence, 2000, US$ 49, ISBN 0-8218-0844- maps. To understand conditions under which such as Robert Boyle, Christopher Wren, 3 some of them coincide is a complicated ques- Edmond Halley, Percy Bysshe Shelley, This is not a book, but a CD-ROM. This sec- tion whose answer needs a surprising variety Charles Dodgson (Lewis Carroll), John Wallis, ond CD-ROM edition is the revised and of tools. T h Isaac Newton, Thomas Hornsby, Henry Smith updated first edition, which was published in final chapter is devoted to extension proper- and James Joseph Sylvester. 1985 by Garland Publishing in New York. It ties of holomorphic maps in the infinite- Appendices containing lists of the holders is an unusual and comprehensive guide to the dimensional situation. Each chapter ends of the Savilian, Sedleian, Waynflete, Rouse history of mathematics, containing some 4800 with exercises containing additional material, Ball and Wallis Chairs since their foundations, bibliographical entries with annotations; of and many of them are commented on in the and an index of names, are included at the these 2800 are entirely new and many of the Appendix. end of the book. The book is very well written remaining ones have been updated. A special feature of the book is a compre- and beautifully illustrated. It can be recom- Thirty-eight historians of mathematics from hensive and detailed description of the histo- mended to anyone interested in the history of ten countries have participated as contribut- ry of the subject and of its individual results, mathematics and the history of teaching. ing editors. They present the best database contained in the notes ending each chapter (mnem) with the best introductions for all topics. The and in the appendix. The author’s efforts in sections are devoted to Egyptian, Babylonian this respect add a special value to the book. D. Gardy and A. Mokkadem (eds.), and Greek mathematics, the Arabic, Latin and The reader is supposed to know basic complex Mathematics and Computer Science. Hebrew traditions, European mathematics in function theory, topology and Banach space Algorithms, Trees, Combinatorics and the 15th, 16th, 17th and 18th centuries, mod- theory; a knowledge of several complex vari- Probabilities, Trends in Mathematics, Birkhäuser, ern mathematics in the 19th and 20th cen- able theory is helpful, but not necessary. The Basel, 2000, 340 pp., DM 148, ISBN 3-7643- turies, mathematics in Africa and the Orient, book is very well written and can be recom- 6430-0 and women in mathematics. Each section mended to mathematicians working in the These proceedings of a colloquium held in contains major works on any given topic or field as well as those from other fields inter- Versailles in September 2000 consist of 28 ref- period, accompanied with critical descriptions ested in the subject. (vs) ereed research papers on diverse mathemati- of these works. The authors emphasise the cal problems, mainly motivated by computer EMS September 2001 29 RECENT BOOKS science. With a few exceptions, the questions ing to anybody interested in the subject. following points: user interface (front end), belong to probability theory or combinatorial The text is divided into six chapters and actual calculator (kernel) and additional enumeration, and the topics include estimat- enlarged by two introductory prefaces written macros (packages). ing various statistics of random trees, random by J. Gray. The first preface (just 1 page) Everything is complemented by examples that generation of words, probabilistic analysis of describes the historical background and are kept at a simple mathematical level and algorithms (QUICKSORT, genetic algo- Hadamard’s connections with Russian mathe- are largely independent of special technical or rithms, routing in a network, an algorithm for matics. The second one gives a very useful scientific applications. Emphasis is placed on rendezvous in graphs, calculating the station- brief account of the history of the theory of solving such standard problems as equations ary vector for discrete-time stochastic automa- automorphic functions from 1880 to 1930. and integrals, and on graphics. After working ta, universal prediction algorithm for mixing The Hadamard chapters range from realisa- through this course, readers will be able to sources), Markov chains, random walks, tions of the Lobatchevskian plane, the most solve problems independently and to find branching processes, large deviations for important properties of properly discontinu- additional help in the on-line documentation. polling systems, general enumeration tech- ous subgroups, and Fuchsian and Kleinian Depending on their interests and needs, com- niques, enumerating certain paths in lattices, functions, to the uniformisation of algebraic pleting the first two parts of this course may be asymptotic analysis of coefficients of multivari- curves and solutions of ordinary differential sufficient, as they include the most important ate generating functions, and 0-1 laws. A good equations with algebraic coefficients. calculations and graphics functions. The third 3-page overview of the contents is given in the This booklet provides interesting reading part is more technical, and the fourth part preface. (jmat) for anybody interested in the theory of auto- introduces programming in Mathematica. morphic functions, and this English edition The CD-ROM can be used with MacOS, J. R. Giles, Introduction to the Analysis of will certainly be of interest not only to this Windows 95/98/NT or Unix. Up-to-date Normed Linear Spaces, Australian Mathematical group of working mathematicians. (šp) information and any corrections to the book Society Lecture Series 13, Cambridge University can be accessed at http://www.ifm. Press, Cambridge, 2000, 280 pp., £19.95, ISBN M. Hazewinkel (ed.), Handbook of Algebra, ethz.ch/\~{}kaufman. (mbr) 0-521-65375-4 Vol. 2, Elsevier, Amsterdam 2000, 878 pp., This book presents a basic course in function- US$177.50, ISBN 0-444-50396-X A. G. Kulikovskii, N. V. Pogorelov and A. al analysis. The text is very readable and This handbook is the second part of a com- Yu. Semenov, Mathematical Aspects of offers a detailed explanation of the subject. prehensive guide through modern algebra. Numerical Solution of Hyperbolic Systems, Some recent results are also included. The whole project is divided into nine sec- Monographs and Surveys in Pure and Applied The book starts with basic properties of tions, and the second volume partially covers Mathematics 118, Chapman & Hall/CRC, Boca normed linear spaces (including the Schauder six of them. Section 2 contains a treatment of Raton, 2001, 540 pp., £63.99, ISBN 0-8493- basis), classes of examples and the theory of category theory, some parts of homological 0608-6 Hilbert spaces. The next chapters deal with and homotopical algebras and model-theoret- This book presents various methods and tech- spaces of continuous linear mappings (includ- ic algebras. Section 3 presents the theory of niques for the numerical solution of hyperbol- ing Banach algebras), the analytic form of the commutative rings and algebras, associative ic systems of partial differential equations, and Hahn-Banach theorem, reflexivity and subre- rings and algebras and the deformation theo- treats a number of problems with important flexivity (including a complete proof of the ry of rings and algebras. In Section 4 varieties applications. Bishop-Phelps theorem), the open mapping of algebras and Lie algebras are explained. The book consists of seven chapters. In and closed graph theorems and the uniform Section 5A is devoted to groups and semi- Chapter 1 the basic concepts and notations boundedness principle. Further chapters are groups and Section 6C deals with the repre- are introduced. Chapter 2 is concerned with devoted to various types of continuous linear sentation theory of continuous groups. Part E the formulation of basic approaches to the mappings (conjugate mappings, adjoint oper- is devoted to abstract and functorial represen- numerical solution of quasi-linear hyperbolic ators, projection and compact operators), tation theory. (lbi) systems, both in the conservative and non- main properties of spectra of linear and com- conservative forms. The methods of Godunov, pact operators (a proof of Lomonosov’s theo- C. Hillermeier, Nonlinear Multiobjective Courant-Isaacson-Rees, Roe and Osher are rem on invariant subspaces) and spectral the- Optimization: A Generalized Homotopy treated, and attention is paid to higher-order ory (culminating with spectral theories for Approach, International Series of Numerical schemes with reconstruction and limiting pro- normal, compact and hermitian operators). Mathematics 135, Birkhäuser, Basel, 2001, 135 cedures. The next chapters are devoted to The text includes many exercises, both ele- pp., DM 88, ISBN 3-7643-6498-X particular mechanical problems. Chapter 3 mentary and advanced. An appendix contains This book motivates and surveys the princi- deals with gas dynamics equations and the the set theory results used in the text (Zorn’s ples and classical methods of multi-objective solution of Euler equations equipped with var- lemma, the Schröder-Bernstein theorem and optimisation, including a recent stochastic ious state equations. In Chapter 4, shallow the Hamel basis). At the end of the book there approach and concentrating on the presenta- water equations are considered, and Chapter are historical notes showing the development tion of a new generalised homotopy approach. 5 is devoted to numerical solution of MHD of functional analysis from the late 19th cen- The homotopy methods require that all problems. Chapter 6 is an attempt to outline tury. Two late significant advances concern- functions in the optimisation problem are problems of solid dynamics that are governed ing (Ekeland’s) variational principles and twice continuously differentiable. They have by hyperbolic systems. Finally, Chapter 7 Asplund spaces are treated. been analysed mostly in the one-dimensional introduces the notion of non-classical discon- The book is addressed mainly at senior parameter case, whereas the proposed tinuity, discusses its various aspects and treats undergraduate and beginning postgraduate approach allows a multi-dimensional para- several applications. students, who are assumed to be familiar with metrisation useful for the solution of multi- This book is a substantial addition to the elementary real and complex analysis, linear objective optimisation problems. The set of existing literature, particularly because it con- algebra and the theory of metric spaces. (jl) efficient points is examined from the view- tains a number of applications. It will be of point of differential topology, and the sug- interest to students and researchers in fluid J. Hadamard (J. J. Gray and A. Shenitzer gested numerical algorithm is applied both to dynamics and continuum mechanics and in (eds.)), Non-Euclidean Geometry in the Theory an academic example and to optimisation various fields of physics. It contains a number of Automorphic Functions, History of problems dealing with the design and opera- of figures and examples and will be useful for Mathematics 17, American Mathematical Society, tion of industrial systems. The book will inter- specialists dealing with practical computation. Providence, 1999, 95 pp., US$19, ISBN 0-8218- est applied mathematicians and engineers. Although the word ‘Mathematical’ appears in 2030-3 (jd) the title, it is not written in a mathematical In the 1920s J. Hadamard wrote a survey on style. The results are not formulated as theo- automorphic functions, in connection with the S. Kaufmann, A Crash Course in rems and the mathematical theory of numeri- preparation of an edition of the collected Mathematica, Birkhäuser, Basel, 1999, 200 pp. cal schemes as convergence and error esti- works of N. I. Lobachevski. It was translated (CD-ROM included), DM 48, ISBN 3-7643- mates is not mentioned. (mf) by A. V. Vasiliev into Russian and edited by B. 6127-1 and 0-8176-6127-1 S. Lang, Fundamentals of Differential A. Fuks and published in 1951. The text is This book, and the accompanying Geometry, Graduate Texts in Mathematics 191, now available in a very readable English edi- Mathematica notebooks on the CD-ROM, give Springer, New York, 1999, 535 pp., DM 109, tion, prepared by Abe Shenitzer. Most of the the reader the basics of Mathematica in a com- ISBN 0-387-98539-X results in the booklet are stated without proof, pact form. The book itself is basically a print- There are many books on the fundamentals of and are not intended for novices. However, it out of the Mathematica notebooks contained differential geometry, but this one is quite can be recommended as supplementary read- on the CD-ROM. The author discusses the exceptional; this is not surprising for those 30 EMS September 2001 RECENT BOOKS who know Serge Lang’s books. The aim is to cally oriented textbooks, the emphasis is on lectures on the subject, but basic facts are present the fundamentals shared by differen- concrete examples, and the author’s explana- proved. Reading of the book requires only a tial topology, differential geometry, and dif- tion proceeds from specific to general, so that basic background and there are a lot of exam- ferential equations. students can follow the motivation of general ples illustrating the theory. The authors suc- The various notions of the smooth calculus mathematical concepts and constructions. ceed in explaining the essentials in just 100 on manifolds form the core of the book as The author avoids rigorous statements and pages, and this charming book can be recom- usual, but there are two distinct guidelines vis- proofs of theorems, referring instead to stan- mended to anybody interested in the modern ible; the coordinate-free treatment of all dard mathematical texts. Thus the reader development in the mathematics arising from proofs and main theorems, and the emphasis needs only basic knowledge of calculus and mathematical physics. (vs) on the developments in global analysis and linear algebra and some experience with geometry after the 1960s. These features fit studying mathematical texts. K. Peeva, H.-J. Vogel, R. Lozanov and P. well together, since much of the recent inter- By tradition, games are classified as either Peeva, Elsevier’s Dictionary of Mathematics, est heads towards various aspects of an infi- cooperative or non-cooperative, although the Elsevier, Amsterdam, 2000, 997 pp., US$209.50, nite-dimensional character, while the coordi- author considers this dichotomy imperfect, ISBN 0-444-82953-9 nate-free approach leads naturally to differen- since almost every conflict has an element of This is a useful guide for readers, writers and tial geometry modelled on Banach spaces cooperation and almost all cooperations have translators and all specialists exploring the (manifolds without further structure), self- an element of conflict. multilingual scientific terminology in English, dual Banach spaces (pseudo-Riemannian The book is divided into seven chapters. German, French and Russian. The dictionary structures), and Hilbert spaces (Riemannian The explanation begins with the concept of contains nearly 12000 entries with almost structures). Although these models are not Nash equilibrium in non-cooperative games. 5000 cross-references. The terminology cov- general enough to include many standard In Chapter 1 the author shows that a non- ers arithmetic, algebra, geometry, set theory, infinite-dimensional objects, such as the cooperative game can have several Nash equi- discrete mathematics, logic, linear algebra, spaces of all smooth mappings between finite- libria. The next chapter gives three criteria calculus, differential equations, vector alge- dimensional manifolds, Lang’s treatment pro- enabling us in such situations to distinguish bra, field theory, probability and statistics, vides the most conceptual way that complete- qualitative properties of different Nash equi- optimisation, numerical methods, mathemati- ly recovers and generalises (but also simpli- libria; in particular, various concepts of equi- cal programming, modern algebra, computer fies) the more usual expositions with x1,…,xn, librium stability are introduced and com- algebra, category theory, applied mathemat- i dx1,…,dxn , Γ jk , etc. pared. Chapters 3 and 4 are devoted to coop- ics, the theory of automata and formal lan- To indicate the wide area covered in about erative games; the author introduces coopera- guages, the theory of games and commonly 500 pages, we list the chapter titles: differen- tive games in strategic form and the corre- used entries in computer architecture. (in) tial calculus; manifolds; vector bundles; vector sponding Nash bargaining solution, charac- fields and differential equations; operations teristic function games, Shapley value and the G. M. Phillips, Two Millenia of Mathematics. on vector fields and differential forms; the Shapley-Shubik index. In Chapter 5, the con- From Archimedes to Gauss, CMS Books in theorem of Frobenius; metrics; covariant cept of the well-known Prisoner’s Dilemma is Mathematics 6, Springer, New York, 2000, 223 derivatives and geodesics; curvature; Jacobi described and a closely related concept, the pp., US$49.95, ISBN 0-387-95022-2 lifts and tensorial splitting of the double tan- so-called ‘Cooperator’s Dilemma’, is intro- This book is an extended collection of inter- gent bundle; curvature and the variation for- duced. Using the latter concept, the author esting mathematical topics from number the- π mula; an example of seminegative curvature; investigates cooperation within the context of ory and analysis – such as Archimedes and , automorphisms and symmetries; immersions a noncooperative game. Chapter 6 is devoted the discovery of exponential and logarithmic and submersions; volume forms; integration to population games; this chapter shows that functions, Napier and Briggs’s logarithms, of differential forms; the Stokes theorem; games are valuable tools in a study of both ani- Newton’s interpolation polynomial, finite and applications of the Stokes theorem; the spec- mal and human behaviour in some non-coop- other differences, the Euclidean algorithm, tral theorem. The book is designed rather for erative conflict situations. Chapter 7 contains Fibonacci numbers, prime numbers, Gauss’s graduates, although it is explicitly based only some concluding remarks, whose aim is to congruences, and Diophantine equations) – on elementary calculus, topology and linear give an objective evaluation of the usefulness ranging over two millennia. It does not pre- algebra. It can be warmly recommended to a of games for studying real conflict situations. tend to be a comprehensive history of mathe- wide audience. (jslo) Each chapter concludes with exercises, matics of this period. whose solutions are presented in an appendix. In five chapters (From Archimedes to E. H. Lieb and M. Loss, Analysis, 2nd edition, Another appendix contains an explanation of Gauss, Logarithms, Interpolation, Continued Graduate Studies in Mathematics 14, American the tracing procedure suggested by Harsanyi. fractions, More number theory), the author Mathematical Society, Providence, 2001, 346 pp., (kzim) shows that many interesting and important US$39, ISBN 0-8218-2783-9 results in mathematics have been discovered The first edition of the book appeared in 1997 T. Miwa, M. Jimbo and E. Date, Solitons, by ordinary people and not only by great (for its review, see EMS Newsletter 25, Cambridge Tracts in Mathematics 135, Cambridge geniuses. Each chapter includes the history of September 1997). In this second edition, University Press, Cambridge, 2000, 108 pp., £ 25, its topic with an interpretation of the mathe- besides correcting misprints, the authors have ISBN 0-521-56161-2 matical problems. The book shows how and added a new Chapter 12 containing a discus- The simplicity of the structure of the set of why some results in mathematics have been sion of a semi-classical approximation for the solutions of linear equations is due to the fact discovered or obtained, by following in the Schrödinger equation, using the Glauber that it is a linear vector space. Nothing com- steps of well-known mathematicians who dis- coherent states and various bounds on eigen- parable applies to non-linear equations. covered them. It is a useful source of mathe- values and their sums. In addition, Chapter 8 Nevertheless, for certain classes of non-linear matical material for teachers, undergraduate has been extended (compactness criterion, PDEs, such as KP-hierarchy of the famous KdV students, students and the vast numbers of and Poincaré, Nash and logarithmic Sobolov equations, there is an extraordinary substitute amateurs who love mathematics. (mnem) inequalities) and there are further additions in for a missing linear structure: the set of solu- Chapter 1 (integration using simple functions) tions is an orbit of an infinite-dimensional Lie L. Polterovich, The Geometry of the Group of and Chapter 6 (Yukawa potential). The num- group, and this symmetry makes a description Symplectic Diffeomorphisms, Lectures in ber of exercises has been increased. This of the set of solutions possible. The book con- Mathematics, Birkhäuser, Basel, 2001, 132 pp., attractive book can be highly recommended tains a discussion of these equations from DM 44, ISBN 3-7643-6432-7 for its style, clarity and interesting choice of many complementary point of views (symme- The main topic of the book is a discussion of a Ω material. (vs) tries of KdV equations, Lax form of equations, role of the group Ham(M, ) of Hamiltonian integrable systems, Hirota equations and ver- diffeomorphisms of a symplectic manifold Ω M. Mesterton-Gibbons, An Introduction to tex operators, bosonic and fermionic Fock (M, ) in geometry and classical mechanics. Game-Theoretic Modelling, Second edn., Student spaces, the Boson-Fermion correspondence, Under suitable assumptions on M, the group Ω Mathematical Library 11, American Mathematical transformation groups of equations and tau Ham(M, ) is the connected component of the Society, Providence, 2000, 368 pp., US$39, ISBN function, infinite-dimensional identity of the group of all symplectic diffeo- 0-8218-1929-1 Grassmannians, Young diagrams and charac- morphisms. In mechanics it is just the group This is the second updated edition of a suc- ters, Hirota equation as Plücker relations). of admissible motions. A basic question relat- ∈ Ω cessful textbook. It is an introduction to game The treatment is based on ideas of M. Sato, ed to elements f Ham(M, ) is how to evalu- theory and applications from the perspective developed by M. Kashiwara and the authors. ate the minimal amount of energy required to of a mathematical modeller. Unlike theoreti- The book has the pleasant spirit of informal generate the given diffeomorphism f. The EMS September 2001 31 RECENT BOOKS answer can be obtained using a distance The book is recommended to specialists in One exact statement of this type consists of defined on the group Ham(M, Ω) by means of global optimisation, mathematical program- inequalities of Poincaré type. The validity of the canonical biinvariant metric on Ham(M, ming and convex analysis, as well as to engi- Poincaré inequalities is a property of the space Ω) which is a solution of a certain natural neers and specialists in mathematical model- (M,µ), which holds if there exist positive con- variational problem on (M, Ω). The theory is ling. (jvy) stants C and k such that based on geometrical properties of the group ∫B |f(x) – a|dµ(x) ≤ Cr ∫kB Dε f(x) dµ(x), Ham(M, Ω), developed first by Hofer in 1990. D. G. Saari, Chaotic Elections! A for any ball B = B(x, r), any function f defined In recent years these questions have been Mathematician Looks at Voting, American on kB (with mean value a over B) and any pos- intensively studied in the framework of sym- Mathematical Society, Providence, 2001, 159 pp., itive ε < r. There are many recent treatments plectic topology, using the Gromov theory of US$23, ISBN 0-8218-2847-9 that consider the Poincaré inequalities as a pseudo-holomorphic curves, Floer homology Two events motivated the author to write this starting axiom. It is then important to under- and the theory of symplectic connections. book: the 2000 U.S. Presidential election and stand in which situations this postulate may The book is a good introduction to the topic the election debate about its outcome in hold. As fractal-like structures, Ahlfors regu- and offers a description of recent develop- Florida, and the final resolution, in some of lar spaces of dimension s are considered: these ments in Hofer geometry and its applications the author’s recent papers, of a 200-year-old are spaces on which the s-dimensional in dynamics and ergodic theory. It can be rec- mathematical problem concerning the source Hausdorff measure can be taken as the refer- ommended for mathematicians and physicists and explanation of the paradoxes and prob- ence measure µ. Among them BPI spaces are interested in problems related to symplectic lems of voting procedures. The purpose of studied as an appropriate generalisation of geometry and mechanics. (jbu) the book is to explain what can go wrong in self-similarity. An important class of spaces is elections, and why. based on Laakso’s construction, where the key S. Rigot, Ensembles quasi-minimaux avec con- The book is divided into six chapters. step consists in taking the product of an trainte de volume et rectifiabilité uniforme, Chapter 1 has an introductory character, and Ahlfors regular space of dimension s–1 with Mémoires de la Société Mathématique de France contains a critical analysis of the voting proce- the line interval. Another source of spaces 82, Société Mathématique de France, Paris, 2000, dure employed in the 2000 U.S. Presidential comes from geometric structures such as 104 pp.,FRF 150, ISBN 2-85629-086-0 election; in this chapter Arrow’s Impossibility Heisenberg groups (or, more generally, In this volume, interesting results are Theorem is presented. In Chapter 2, the Carnot spaces). Besides the validity of achieved concerning a useful generalisation of author explains the importance of the choice Poincaré’s inequalities, various other mapping the classical isoperimetric problems. In par- of election procedure. Although little interest properties are investigated, such as measure- ticular, the regularity of quasi-minimising sets is usually devoted to this problem, the choice mapping properties, rigidity of the structure for a variational problem related to isoperi- of election procedure can substantially influ- under mappings, and continuous families of metric inequality is investigated. ence the outcome of the elections. This is mappings. Tangential properties of Lipschitz The object of study is a measurable set G demonstrated on examples both from the and regular mappings are also studied, satisfying recent past (U.S. Presidential elections in the including rectifiability and its generalisations. ∫ χ ≤∫ χ ∆ |grad G | | grad H | + g(|G H|), 1990s) and from earlier (such as the 1860 This book provides a thorough discussion for each measurable test set H with relatively election in which Lincoln won the Presidency on the current state and perspectives of the compact symmetric difference G ∆ H and |H| with a majority vote in the Electoral College). topic. The visionary aspect is emphasised. finite and equal to |G| (integration over the Some general results concerning chaotic elec- There are many questions and conjectures in whole of Rn). Here g: (0, ∞) → (0, ∞) is fixed tion outcomes and their consequences are the text, but fewer theorems. Proofs, where so that g(t) = o(t(n-1)/n) as t → 0. It is shown presented in Chapter 3, while Chapter 4 included, are rather indicated than elaborated that a suitable representative of the set G has analyses the so-called strategic behaviour of in detail. This volume is a great source of the Ahlfors regular boundary, and that the voters. This behaviour means that the voters inspiration and a guide for students and numbers of connected components of G and do not vote ‘sincerely’ according to their real researchers looking for new areas to study. Rn\G are estimated by an universal constant. preferences, but try by a different choice to (jama) Moreover, when g(t) is of the form Ctp, then achieve a better total election result indirectly α from their subjective point of view. In R. Siegmund-Schultze, Rockefeller and the C1, -partial regularity of the boundary is α Chapter 5, the author discusses a provoking Internationalization of Mathematics Between obtained, and for n ≤ 7, full C1, regularity fol- question concerning the conditions under the Two World Wars, Science Networks/Historical lows. In the final chapter, the above results which the outcome of an election with a cho- Studies 25, Birkhäuser, Basel, 2001, 341 pp., DM are applied to the (physically motivated) prob- sen voting procedure reflects the real will and 170, ISBN 4-7643-6468-8 lem of the minimisation of preferences of the voters. The final chapter This book is the first detailed study of the brief n-1 ∂ ∫ E(G) = H ( G) + GxG K(x – y) dx dy, generalises some experience from the analysis but very influential role played by the where Hn-1 is Hausdorff measure and K is an of elections and investigates a more general Rockefeller family in the field of mathematics. integrable kernel with compact support. concept of aggregation, which is involved both It is based on extensive research by the (jama) with various election procedures, and also in Rockefeller Archive Center in the archives of other areas far beyond voting, such as the Harvard University and the Universities of A. Rubinov, Abstract Convexity and Global choice of power indices, non-parametric sta- New Hampshire, Göttingen and New York Optimization, Nonconvex Optimization and its tistics and various probability assertions. City. Applications 44, Kluwer Academic Publishers, The book is not a research monograph; its In the first chapter the author comments on Dordrecht, 2000, 490 pp., £135, ISBN 0-7923- reading requires little more than high-school the process of internationalisation and mod- 6323-X mathematics, since the author wanted to ernisation of mathematics and the conditions The contents of this book can be divided into inform practitioners involved with designing for international scientific collaboration two parts. The first part presents the notion practical voting or aggregation rules (political, between the First World War and the Nazi dic- of abstract convexity and the basic and economic or technical) what may happen tatorship after 1933. The second chapter advanced theorems relating to this topic. whenever any group makes a choice under describes the beginning of the International These results are then applied to the study of given rules. (kzim) Education Board, a central role of the some classes of functions and sets. This Fellowship Program in the careers of the mod- approach includes elements of monotonic A. Semmes, Some Novel Types of Fractal ern generation of mathematicians (S. Banach, analysis and elements of star-shaped analysis, Geometry, Oxford Mathematical Monographs, A. Weil, B. L. van der Waerden, etc.) and the as well as a study of quasi-convex functions. Clarendon Press, Oxford, 2001, 164 pp., £49.95, support of the Rockefeller Foundation for new The results of the first part are then used in ISBN 0-19-850806-9 mathematical publications and for the foun- the study of global optimisation, which is the Let (M, d) be a metric space with a doubling dation of new mathematical journals. The subject of the second part of the book. The measure µ. If A is a subset of M and f: A → R third chapter analyses the comparative devel- classical notions of Lagrange and penalty is a function, then the quantity Dε f(x) = ε -1 opments of mathematics in Europe and USA functions are presented and extended. New sup{|f(y) – f(x)|: y ε A, d(x, y) ≤ ε}with a small ε in the 1920s and 1930s. The fourth chapter methods of global optimisation based mainly > 0 measures oscillation of f near the point explains the practice of the Fellowship on solvability results for inequality systems x, just as the ordinary gradient does in the Programs of the International Education and optimisation of the difference of abstract classical Euclidean setting. A ‘decent calculus’ Board between 1923 and 1928 and of the convex functions are then developed. The is possible on a space, if one can estimate Rockefeller Foundation after 1928. Here the final chapter is devoted to the numerical large-scale quantities by integrating micro- reader can find the criteria for selections of aspects of global optimisation. scopic measurement oscillations. Fellows, the Fellowship programs, some 32 EMS September 2001 RECENT BOOKS Fellowship lists, etc. The rise of Soviet which actual data are met with the suggestion that here. Many special and interesting examples (Russian) mathematics and problems of its other data, if they were collected, might show some- of PDEs are discussed, and their solutions are collaboration with the Rockefeller Foundation thing else to believe it to have any value as an argu- described at the end of the book. This book is and its political self-isolation are commented ment. ‘‘Statistics on the table, please,’’ can be my sole a good source for anybody interested in PDEs, on. The fifth chapter shows the role of reply. differential geometry, Lie group theory and Rockefeller’s help in the foundation of new To give a flavour of the essays, I quote a few related fields. (jbu) scientific institutes, mainly in Europe. It sentences from the chapter Stigler’s Law of includes the foundation of the new Eponymy: For ‘‘Stigler’s Law of Eponymy’’ in its Tian-Xiao He, Dimensionality Reducing Mathematical Institute in Göttingen, the simplest form is this: ‘‘No scientific discovery is Expansion of Multivariate Integration, Institute of Henri Poincaré in Paris, the named after its original discoverer.’’ … Thus in the Birkhäuser, Boston, 2001, 225 pp., DM 156, Mathematical Institute in Djursholm in field of mathematical statistics it can be found that ISBN 0-8176-4170-X and 3-7643-4170-X Sweden and the School of Mathematics of the Laplace employed Fourier Transforms in print This book discusses a technique for numerical Institute for Advanced Study in Princeton, before Fourier published on the topic, that Lagrange integration by using dimensionality-reducing and their role in promoting cooperation in presented Laplace Transforms before Laplace began expansions (DRE) to reduce a higher-dimen- science. The sixth chapter shows the changes his scientific career, that Poisson published the sional integral to lower-dimensional integrals, of programmes of the Rockefeller Foundation Cauchy distribution in 1824, … and that Bienaym‚ with or without a remainder. Some important after 1933 and during the first years of the stated and proved the Chebychev Inequality a and common aplications of DRE include the Second World War. The seventh chapter decade before and in greater generality than construction of boundary-type quadrature for- gives a very short review of Rockefeller sup- Chebychev’s first work on the topic. One might mulas (BTQF) and asymptotic formulas for port for mathematics between 1945 and 1950. also expect that ‘Gaussian distributions’ were oscillatory integrals. The book concludes with seventeen appen- known before 1809 when Gauss associated it The book is organised as follows. Chapters dices. The first fourteen are letters and with the least squares method. In fact, Gauss 1 and 2 discuss the construction of DREs and reports of distinguished mathematicians, himself cites Laplace who dealt with this dis- BTQFs with various degrees of algebraic pre- workers or clerks of the Rockefeller tribution in 1774; moreover, both Gauss and cision. In Chapters 3 and 4, the author uses Foundation, illustrating Rockefeller’s contri- Laplace knew a 1733 publication by A. de DREs to approximate oscillatory integrals, bution to science, and mathematics in particu- Moivre, which is now considered as the origin and establishes their corresponding numerical lar. The next three appendices contain lists of of the ‘Gaussian’ (or ‘normal’) distribution. By quadrature formulas. Chapter 5 demon- Fellows in mathematics up to 1945, the way, the title of the next essay is ‘Who dis- strates how to construct DREs over a complex Guggenheim Fellows in mathematics up to covered Bayes’s Theorem?’ region, by using the Schwarz function and the 1945 and mathematicians from Europe who All the essays are very interesting, and some Bergman kernel. The final chapter examines were supported by the Rockefeller of them should be a compulsory part of the how the solutions of certain differential equa- Foundation Emergency Fund. This book can general education of every statistician – in tions can be used to construct exact DREs, and be recommended to everybody who wants to particular, the chapters devoted to the history how, conversely, some DREs can be utilised to know more about the birth of modern and of the maximum likelihood method and the derive a scheme of the boundary element international mathematics during the first invention of the least squares method. This method, used for evaluating the numerical half of the 20th century. (mnem) book can be warmly recommended to anyone solution of a boundary-value problem of a interested in probability and statistics and in partial differential equation. K. A. Sikorski, Optimal Solution of Nonlinear their history. (ja) This book will be a useful guide for a wide Equations, Oxford University Press, Oxford, range of readers in pure and applied mathe- 2001, 238 pp., £39.50, ISBN 0-19-510690-3 O. Stormark, Lie’s Structural Approach to matics, statistics and physics, and can also be The aim of this book is to review the state of PDE Systems, Encyclopaedia of Mathematics and used as a textbook for graduate and advanced the art in methods for optimal (or nearly opti- its Applications 80, Cambridge University Press, undergraduate students. (knaj) mal) solution of non-linear problems. In par- Cambridge, 2000, 572 pp., £70, ISBN 0-521- ticular, the following problems are consid- 78088-8 J. L. Walker, Codes and Curves, Student ered: finding roots of non-linear equations, This monograph is a comprehensive introduc- Mathematical Library 7, American Mathematical approximation of fixed points (of both con- tion to geometric methods for the study of sys- Society, Providence, 2000, 66 pp., US$15, ISBN tractive and non-contractive functions) and tems of partial differential equations. The 0-8218-2628-X computation of the topological degree. The results presented here concern local solutions This booklet arose from a series of lectures methods are required to be ‘robust’, and guar- of systems. The geometric approach is based held at the Institute for Advanced Study in antee that the computed and exact solutions on ideas developed by Sophus Lie, Elie Cartan Princeton. It consists of seven chapters and differ by a prescribed tolerance for a specified and Ernest Vessiot. Any system of PDEs can three appendices, written in a clear and ‘con- class of problems. The notion of optimality is be considered as a submanifold in the corre- versational’ style and oriented towards a gen- developed by means of information-based sponding jet bundle, which is equipped with a eral audience. The reader obtains an intro- complexity theory, which creates the least canonical contact pfaffian system or dually duction to classical coding theory, particularly overhead for the particular method. with a canonical system of vector fields. The its algebraic geometric branch. This book is self-contained. Each chapter is problem of solving the system is transported The first two chapters include the basic provided with exercises and detailed annota- into the problem of finding integral manifolds notions and results of coding theory: tions that encourage further reading. The of the pfaffian or vector field system. Hamming’s distance and weight, dimension, prospective audience of the book ranges from In the book, the duality between these con- etc.; all introduced notions are illustrated researches in computational complexity to cepts is described, and the Frobenius and using basic types of codes as error detecting or practitioners in numerical computation. For Cartan local existence theorems are proved. correcting, linear, cyclic, Reed-Solomon or the latter group, the book represents an alter- The notions of involutivity and prolongations ISBN. The illustrations are carefully selected native view to the classical local convergence of a given system of PDEs are introduced and and are naturally embedded into the exposi- analysis of non-linear iterative techniques. (vj) the results obtained are applied to special tion. There follow basic results on the dimen- first- and second- order PDEs. Another sion and on absolute and asymptotic bounds S. M. Stigler, Statistics on the Table. The object related to a system of PDEs on M is the (Plotkin or Gilbert-Varshanov) on the maxi- History of Statistical Concepts and Methods, contact transformation, a local diffeomor- mum number of codewords. In the introduc- Harvard University Press, London, 1999, 488 pp., phism of M that preserves the system. The tory geometric part, the reader learns the £30.95, ISBN 0-674-83601-4 family of all contact transformations form a basics of algebraic curves (Chapter 3), their This book is a collection of twenty-two essays, (Lie)-pseudogroup of local diffeomorphisms singularities and genus (Chapter 4) and func- divided into five parts: Statistics and social sci- of M. For the special first-order PDE system tions and divisor theory on curves (Chapter 5); ence, Galtonian ideas, Some seventeenth-cen- there is a special Lie pseudogroup,called a these are necessary for the last two chapters, tury explorers, Questions of discovery, and local Lie group, whose structure plays an which are devoted to the description of Questions of standards. The author is well interesting role in the study of solutions of the Goppa’s construction of algebraic geometry known for his papers dealing with the history system. The Cartan theory of Lie codes (Chapter 6) and the Tsfasman, Vladut of statistics; these papers were published in pseudogroups is also explained, together with and Zink bound for asymptotically good codes prominent journals and now serve as basic the equivalence problem. Using the Drach (Chapter 7). Appendices review basic algebra- material for most of the essays. The title of classification, an arbitrary PDE system can be ic notions, such as groups, rings, ideals and the book is borrowed from a letter of Karl reduced to a first- or second-order in one finite fields, and collect some important topics Pearson: I am too familiar with the manner in unknown, and both cases are studied in detail in coding theory that are not covered in the EMS September 2001 33 RECENT BOOKS text. by classical treatises on these subjects, such as Chapman & Hall/CRC, Boca Raton, 2000, 554 As already mentioned, the book is written in D. Ruelle’s book Thermodynamic formalism: the pp., ISBN 1-58488-059-7 a lucid style. It contains worthwhile exercises mathematical structures of classical equilibrium sta- This book is in the form of a handbook, pro- scattered through the text, and can be warmly tistical mechanics (Addison-Wesley, 1978). The viding tables and comprehensive lists of defin- recommended to all interested to learn about first four chapters of this book deal with the tions, concepts, theorems and formulae in the basics of classical coding theory. (šp) concepts of ergodicity, entropy (of a dynami- probability and statistics. Its emphasis is on cal system) and the Perron-Frobenius theo- basic statistics, as taught in most statistical Jianhong Wu, Introduction to Neural rem. Chapters 5 and 6 deal with conformal courses, but also covers many advanced topics Dynamics and Signal Transmissions Delay, De repellers and iteration of quadratic polynomi- from statistics and probability theory, such as Gruyter Series in Nonlinear Analysis and als; in particular, the Hausdorff dimension of non-parametric statistics, quality control, Applications 6, Walter de Gruyter, Berlin, 2001, Julia sets of these mappings is disscused. experimental design, Markov chains, martin- 182 pp., DM 99, ISBN 3-11-016988-6 Finally, Chapter 7 introduces the notion of a gales, resampling, queueing theory, self-simi- This book is a textbook for graduate and phase transition and applies some tools of this lar processes and the elements of stochastic senior undergraduate students. The mathe- theory (in particular that of Legendre trans- calculus. Almost every table is accompanied matics used is ‘elementary’, in the sense that it form) to a further study of the above prob- by a textual description and at least one exam- requires only basic knowledge of the theory of lems. ple that uses a value from the table. Most con- matrices and of ODEs. It is not fully self-con- There are few texts on the relationships cepts are illustrated with examples and step- tained, as it also makes use of theorems from between statistical mechanics and the theory by-step solutions. One section is devoted to literature, quoted without proof. of iterated quadratic maps in a plane. This ‘classic and interesting problems’, where the The contents of the book are as follows. short book successfully helps to fill this gap. It reader finds formulations and solutions of a Chapter 1 outlines the basics of neuroscience is clearly written and gives an excellent intro- number of well-known probability problems, – in particular, the structure of a single neu- duction to the subject. (mzah) such as Buffon’s needle problem, Bertrand’s ron and of a network of neurons, and the box paradox, Simpson’s paradox and the sec- mechanism of neural signal transmission. D. Zwillinger and S. Kokoska, Standard retary problem. The book also contains infor- Chapter 2 introduces a general mathematical Probability and Statistic Tables and Formulae, mation on electronic resources. (mah) model describing the dynamics of neural net- works. In Chapter 3, models of simple net- works that can perform some elementary functions (storing, recalling and pattern recognition) are analysed. Chapter 4 aims at a global analysis of neural networks from the viewpoint of the existence and stability of equilibria: two important approaches, via Liapunov functions and monotone dynamical systems, are developed. The final, and longest, chapter focuses on how the dynamics of a neural network is altered when allowing for time-delayed terms; the problems addressed here include delay-independent stability, delay-induced periodic/chaotic oscil- lations and delay-induced change of the domain of attraction. (dp)

M. Zinsmeister, Thermodynamic Formalism amd Holomorphic Dynamical Systems, SMF/AMS Texts and Monographs 2, American Mathematical Society, Providence, 1999, 82 pp., US$19, ISBN 0-8218-1948-8 This book links two different parts of mathe- matics and physics: the formalism of statistical physics and the theory of holomorphic dynamical systems. It is intended for researchers from other fields, and is inspired

34 EMS September 2001 EMS September 2001 35 The 2002 Ferran Sunyer i Balaguer Prize Citation Checker and Cross Reference Link Generator Ferran Sunyer i Balaguer (1912-1967) was SUBMISSION OF THE MONOGRAPHS a self-taught Catalan mathematician who, Monographs should preferably be typeset offered by Zentralblatt MATH in spite of a serious physical disability, was in TEX. Authors should send a hard copy BERND WEGNER very active in research in classical of the manuscript and two disks, one with For some time Zentralblatt MATH has Mathematical Analysis, an area in which he the DVI file and one with the PS file offered the facility of exporting BIB- acquired international recognition. Each (PostScript), and enclosing an accompany- TEX-files to users’ machines, from the year in honour of the memory of Ferran ing letter to the Ferran Sunyer i Balaguer hit list obtained by retrieving informa- Sunyer i Balaguer, the Institut d’Estudis Foundation. Submissions should be sent tion from the Zentralblatt MATH data- Catalans awards an international mathe- before 1 December 2001 to the following base. This has proved to be a useful matical research prize bearing his name, address: tool for authors preparing lists of refer- open to all mathematicians. This prize was Centre de Recerca Matemàtica (IEC) ences for their articles. Responding to awarded for the first time in April 1993. 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Karlovich (TU ficient to pinpoint the single item monograph will be published in Chemnitz-Zwickau) and (Ukrainian Academy looked for – in particular, when all four Birkhäuser Verlag’s series Progress in of Sciences), Carleson Curves, Muckenhoupt fields are used. In the rather unlikely Mathematics, subject to the usual regula- Weights, and Toeplitz Operators, Progress in case that more than three items match tions concerning copyright and author’s Mathematics 154. the search criteria, the three most rights. § Juan J. Morales-Ruiz (Universitat Politècnica recent items are displayed. If the § The submission of a monograph implies de Catalunya), Differential Galois Theory and requested item does not belong to the acceptance of all of the above condi- Non-integrability of Hamiltonian Systems, them, then the standard retrieval forms tions. Progress in Mathematics 179. for Zentralblatt MATH may provide fur- § The name of the prize-winner will be § Patrick Dehornoy (Université de Caen), ther help for those with a subscription announced in Barcelona in April 2002. Braids and Self-Distributivity , Progress in to this service. Mathematics 192. The search result offers precise stan- SCIENTIFIC COMMITTEE § Juan-Pablo Ortega and Tudor Ratiu (École dardised bibliographic data to be The winner of the prize will be proposed Polytechnique Fedérale de Lausanne), exported by users into their own refer- by a Scientific Committee consisting of: Hamiltonian Singular Reduction (to be pub- encing systems. The addition of the § H. Bass (University of Michigan) lished). Zentralblatt MATH annotation number § A. Córdoba (Universidad Autónoma de § Martin Golubitsky (University of Houston) will add a unique identifier for the arti- Madrid) and Ian Stewart (University of Warwick), The cle, which will be suitable as a persistent § W. Dicks (Universitat Autónoma de Symmetry Perspective (to be published). link to the electronic offer of this publi- Barcelona) cation and may serve as a tool for refer- § P. Malliavin (Université de Paris VI) For further information on the Ferran ence linking to a wider range than that § A. Weinstein (University of California at Sunyer i Balaguer Foundation, see provided by DOI, for example. Berkeley) Web: http://www.crm.es/info/ffsb.htm Société Française de Statistique French Statistical Society 34th Journées de Statistique May 13-17, 2002

The 34th Journées de Statistique of the Société Française de Statistique will be held in Brussels and Louvain-la-Neuve in Belgium. This event is jointly organized by the Institutes of Statistics of the Université libre de Bruxelles and of the Université catholique de Louvain. The main topics include statistical analysis of functional data, actuarial and financial econometrics, resampling methods, non- parametric inference and modelling, voice and writing recognition, epidemics, genomics as well as mathematical statistics. In addi- tion to these main topics, a number of invited sessions will focus on various subjects. The conference is open to all persons interested in statistics, affiliated to universities, national institutes of statistics, business or industry. For further information, please consult : JSBL 2002 Institute of Statistics, Université catholique de Louvain 20 Voie du Roman Pays, B-1348 Louvain-la-Neuve, Belgium Telephone : +32 10 47 43 54 -Fax : +32 10 47 30 32 36 EMS September 2001