CONTENTS EDITORIAL TEAM EUROPEAN MATHEMATICAL SOCIETY

EDITOR-IN-CHIEF ROBIN WILSON Department of Pure Mathematics The Open University Milton Keynes MK7 6AA, UK e-mail: [email protected] ASSOCIATE EDITORS STEEN MARKVORSEN Department of Mathematics Technical University of Denmark Building 303 NEWSLETTER No. 39 DK-2800 Kgs. Lyngby, Denmark e-mail: [email protected] March 2001 KRZYSZTOF CIESIELSKI Mathematics Institute EMS Agenda ...... 2 Jagiellonian University Reymonta 4 Editorial - Bodil Branner ...... 3 30-059 Kraków, Poland e-mail: [email protected] KATHLEEN QUINN EMS Summer School 2000 ...... 5 The Open University [address as above] e-mail: [email protected] Statutes of the European Mathematical Foundation ...... 6 SPECIALIST EDITORS INTERVIEWS Raising Public Awareness of Mathematics ...... 7 Steen Markvorsen [address as above] SOCIETIES Maths Quiz 2000 ...... 7 Krzysztof Ciesielski [address as above] EDUCATION Tony Gardiner Interview with Bernhard Neumann ...... 9 University of Birmingham Birmingham B15 2TT, UK Interview with Sir Roger Penrose, part 2 ...... 12 e-mail: [email protected] MATHEMATICAL PROBLEMS Problem Corner ...... 18 Paul Jainta Werkvolkstr. 10 D-91126 Schwabach, Germany SIAM-EMS Conference ...... 21 e-mail: [email protected] ANNIVERSARIES Fondazione CIME Courses ...... 22 June Barrow-Green and Jeremy Gray Open University [address as above] Forthcoming Conferences ...... 23 e-mail: [email protected] and [email protected] and CONFERENCES Recent Books ...... 27 Kathleen Quinn [address as above] RECENT BOOKS Designed and printed by Armstrong Press Limited Ivan Netuka and Vladimir Sou³ek Crosshouse Road, Southampton, Hampshire SO14 5GZ, UK Mathematical Institute telephone: (+44) 23 8033 3132 fax: (+44) 23 8033 3134 Charles University Published by European Mathematical Society Sokolovská 83 ISSN 1027 - 488X 18600 Prague, Czech Republic e-mail: [email protected] and [email protected] NOTICE FOR MATHEMATICAL SOCIETIES ADVERTISING OFFICER Labels for the next issue will be prepared during the second half of May 2001. Vivette Girault Please send your updated lists before then to Ms Tuulikki Mäkeläinen, Department of Laboratoire d’Analyse Numérique Mathematics, P.O. Box 4, FIN-00014 University of Helsinki, Finland; e-mail: [email protected] Boite Courrier 187, Université Pierre et Marie Curie, 4 Place Jussieu INSTITUTIONAL SUBSCRIPTIONS FOR THE EMS NEWSLETTER 75252 Paris Cedex 05, France Institutes and libraries can order the EMS Newsletter by mail from the EMS Secretariat, e-mail: [email protected] Department of Mathematics, P. O. Box 4, FI-00014 University of Helsinki, Finland, or by e- OPEN UNIVERSITY mail: ([email protected]). Please include the name and full address (with postal code), tele- PRODUCTION TEAM phone and fax number (with country code) and e-mail address. The annual subscription fee Liz Scarna, Kathleen Quinn (including mailing) is 60 euros; an invoice will be sent with a sample copy of the Newsletter.

EMS March 2001 1 EMS NEWS EMS Committee EMS Agenda EXECUTIVE COMMITTEE PRESIDENT (1999–2002) 2001 Prof. ROLF JELTSCH Seminar for Applied Mathematics 4-6 May ETH, CH-8092 Zürich, Switzerland EMS Workshop on Applied mathematics in Europe, Berlingen (Switzerland) e-mail: [email protected] Contact: Rolf Jeltsch, e-mail: [email protected] VICE-PRESIDENTS Prof. LUC LEMAIRE (1999–2002) 11-12 May Department of Mathematics EMS working group on Reference Levels in Mathematics Université Libre de Bruxelles Conference on Mathematics at Age 16 in Europe C.P. 218 – Campus Plaine Venue: Luxembourg Bld du Triomphe Contact: V. Villani, A. Bodin or Jean-Paul Pier B-1050 Bruxelles, Belgium e-mail: [email protected], [email protected] or [email protected] e-mail: [email protected] Prof. BODIL BRANNER (2001–2004) 15 May Department of Mathematics Deadline for submission of material for the June issue of the EMS Newsletter Technical University of Denmark Contact: Robin Wilson, e-mail: [email protected] Building 303 DK-2800 Kgs. Lyngby, Denmark 19-21 June e-mail: [email protected] EMS lectures at the University of Heraklion, Crete (Greece) SECRETARY (1999–2002) Lecturer: Prof. George Papanicolaou (Stanford, USA) Prof. DAVID BRANNAN Title: Time Reversed Acoustics Department of Pure Mathematics Contact: George N. Makrakis, e-mail: [email protected] The Open University Walton Hall 9-25 July Milton Keynes MK7 6AA, UK EMS Summer School at St Petersburg (Russia) e-mail: [email protected] Title: Asymptotic combinatorics with applications to mathematical physics TREASURER (1999–2002) Organiser: Anatoly Vershik, e-mail: [email protected] Prof. OLLI MARTIO Department of Mathematics P.O. Box 4 15 August FIN-00014 University of Helsinki Deadline for proposals for 2003 EMS Summer Schools Finland Contact: Renzo Piccinini, e-mail: [email protected] e-mail: [email protected] Deadline for submission of material for the August issue of the EMS ORDINARY MEMBERS Newsletter Prof. VICTOR BUCHSTABER (2001–2004) Contact: Robin Wilson, e-mail: [email protected] Department of Mathematics and Mechanics Moscow State University 19-31 August 119899 Moscow, Russia EMS Summer School at Prague (Czech Republic) e-mail: [email protected] Title: Simulation of fluid and structure interaction Prof. DOINA CIORANESCU (1999–2002) Organiser: Miloslav Feistauer, e-mail: [email protected] Laboratoire d’Analyse Numérique Université Paris VI 24-30 August 4 Place Jussieu EMS lectures in Malta, as part of the 10th International Meeting of European 75252 Paris Cedex 05, France Women in Mathematics e-mail: [email protected] Lecturer: Michèle Vergne (Ecole Polytechnique, Palaiseau, France) Prof. RENZO PICCININI (1999–2002) Title: Convex polytopes Dipartimento di Matematica e Applicazioni Contact: Dr. Tsou Sheung Tsun, e-mail: [email protected] Università di Milano-Bicocca Via Bicocca degli Arcimboldi, 8 20126 Milano, Italy 1-2 September e-mail: [email protected] EMS Executive meeting in Berlin (Germany) Prof. MARTA SANZ-SOLÉ (1997–2000) Facultat de Matematiques 3-6 September Universitat de Barcelona 1st EMS-SIAM conference, Berlin (Germany) Gran Via 585 Organiser: Peter Deuflhard, e-mail: [email protected] E-08007 Barcelona, Spain e-mail: [email protected] 19-21 November Prof. MINA TEICHER (2001–2004) EMS lectures at Università degli Studi, Tor Vergata, Rome (Italy), jointly Department of Mathematics and Computer arranged by Tor Vergata and Roma Tre Science Lecturer: Michèle Vergne (Ecole Polytechnique, Palaiseau, France) Bar-Ilan University Title: Convex polytopes Ramat-Gan 52900, Israel Contact: Prof. Maria Welleda Baldoni, e-mail:[email protected] e-mail: [email protected] EMS SECRETARIAT 22-23 November Ms. T. MÄKELÄINEN Diderot Mathematical Forum 5 Department of Mathematics Title: Mathematics and telecommunications P.O. Box 4 Venues: Eindhoven (Netherlands), Helsinki (Finland) and Lausanne FIN-00014 University of Helsinki (Switzerland) Finland Contact: Jean-Pierre Bourgignon, e-mail: [email protected] tel: (+358)-9-1912-2883 fax: (+358)-9-1912-3213 2002 telex: 124690 1-2 June e-mail: [email protected] EMS Council Meeting in Oslo (Norway) website: http://www.emis.de 2 EMS March 2001 EDITORIAL ment to subscribe. I believe that all the above can be han- dled by the individual societies without too EditorialEditorial much difficulty. Bodil Branner (Lyngby, Denmark) What does the EMS offer? The purpose of the EMS is to ‘promote the EMS Vice-President (2001–2004) development of all aspects of mathematics in the countries of Europe, with particular President of the Danish Mathematical Society (1998–2002) emphasis on those which are best handled on the international level’. Better interaction Some of the society’s activities are of a During the EMS Council meeting in purely mathematical nature – for instance, Barcelona last July, some of the corporate summer schools, conferences, and lectures member societies called for better interac- are organised under the EMS umbrella, tion between the EMS and its member soci- mainly by different volunteers. The weak- eties. There is indeed a need for more ness is that the EMS, besides moral sup- direct interaction in between the biannual port, can offer at most symbolic economic Council meetings – the question is how this support, and so the work involved is not can be set up most effectively. very different from organising any other The best way to be informed about EMS such international mathematical activity; activities is through the EMS Newsletter, still, we hope that many will find it worth which is mailed to all member societies and doing so under the name of the EMS. The all individual members. In addition, since Diderot Mathematical Forums are of a differ- taking office the EMS president Rolf ent nature. These are workshops that take Jeltsch has started a tradition of sending place in three different cities simultane- out – by the end of each calendar year – a ously, featuring a special topic – the last letter to each individual member and one was Mathematics and Music and the next another letter to all corporate member is Mathematics and Telecommunications – with societies, describing the highlights of the the Newsletter. Some societies have three different themes, and with a round- previous year and the most important appointed such a person, others not yet. table discussion in which the three cities future undertakings. Also, more e-mails are linked (when possible) by electronic than ever before have been sent to mem- What can member societies do to make means. ber societies to inform, to ask for opinions the EMS more visible? Other activities need local support to be on certain questions, and to ask for help in Many societies publish a newsletter and of a truly European dimension. One spreading information of general interest have a homepage. We wish to encourage example is the announcement of vacant received by the EMS. I think we should all societies to publicise its EMS member- jobs in mathematics in Europe. Some of develop such activities even more in the ship, both in its newsletter and on the front these are collected by a number of member future. homepage of the society, and to advocate societies and can be found under Euro- For the EMS to be successful, it is highly individual membership. Math-Job at EMIS; however, to be really dependent on collaborations with member Individual membership of the EMS is useful, the list should be not only more societies and work done by individuals. usually obtained and paid through mem- complete, but also searchable. The EMS is Mathematicians working for the EMS are ber societies, but this may not be obvious to also a partner of Zentralblatt MATH. however all appointed as individuals, not the society’s members, and neither may it Through the LIMES (Large Infrastructures representing any society in particular. be obvious how to distinguish between the in Mathematics – Enhanced Services) pro- This gives a freedom which is of great society being a member and a person ject, funded by the EU, the database is value, and I have no doubt that this is the being a member. We encourage societies being improved in many ways. way it should be. Only for Council meet- to make it easy for their members to join One of the goals of the several distrib- ings do all member societies directly the EMS. However, the way in which each uted editorial units is to ensure a better appoint their delegates. Together with the society collects its EMS fees should be as coverage of the European mathematical elected delegates of individual members, easy as possible for that society, and may literature. One example where member they elect the president, vice-presidents differ from one society to another. societies can be helpful is to ensure that all and other members of the executive com- The most visible sign of EMS member- PhD theses are recorded in the database – mittee, and can express their opinions and ship is the receipt of the EMS Newsletter. this is done systematically in Germany, and have influence on the broader aspects of Societies can add new EMS members four ought to be followed up throughout the the work and the initiatives taken by the times per year, in mid-February, May, rest of Europe. Another activity is EULER various committees of the EMS. August and November, by mailing the (EUropean Libraries and Electronic The best idea would be to interact more additional list to Tuulikki Mäkeläinen. Resources in mathematical sciences), a for- with the delegates between Council meet- The individual benefits, besides the mer EU project that will be continued in ings, or with other appointed representa- Newsletter, include reduced conference fees some way. Institutions other than the core tives of the member societies. In the past, for EMS conferences and reduced prices of participants can join by making their the role of a society’s delegate has been certain books and journals (see the list at library resources available (for more details more-or-less restricted to participation in EMIS under ‘How to join the EMS’). see EMS Newsletter 38, December 2000). Council meetings. Indeed, some delegates Societies should also encourage mathe- The newest added activity is the EMF (the were mainly chosen because they were also matical departments to subscribe to the European Mathematical Foundation), speakers at the conference associated with EMS Newsletter: it is very cheap (60 euros founded in March 2001. Its main purpose the Council meeting. To make a more annually). As explained above, it is mainly is to run the EMS publishing house, which lively and valuable exchange of ideas through the Newsletter that the mathemati- will start to function this year. Again, between and during Council meetings, it cal community can be informed about the interaction with member societies is highly could be important for the EMS and its work of the EMS. Many mathematical needed. member societies to rethink the role of the departments subscribe to the AMS In order to expand such activities as the delegates and increase their responsibili- Newsletter, the Notices, and it should be above, the EMS will have to describe more ties. The EMS Newsletter too wishes to have just as natural to subscribe to the EMS precisely how member societies or other a person appointed by each society, Newsletter. Individual members of the EMS partners may contribute. The list of activ- responsible for transferring local news to can of course also encourage their depart- ities is by no means exhaustive. EMS March 2001 3 Asymptotic combinatorics with application to mathematical physics. European Mathematical Society SUMMER SCHOOL Euler International Mathematical Institute, St. Petersburg, Russia. 9th July – 25th July 2001

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

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

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

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

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

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

Local Organizing Committee A.Vershik, Ju. Neretin, K. Kokhas, E. Novikova

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

European Mathematical Society EMS NEWS same programme was carried out in prop- er generality to develop the natural frame- EMSEMS SummerSummer SchoolSchool inin work in continuous time. The tools used are general semi-martingale theory and functional analysis. PPrrobabilityobability TheoryTheory Michael Talagrand, research director at Saint-Flour, Cantal, France the CNRS (National centre of Scientific 17 August – 2 September 2000 Research) in France, taught us about Spin glasses. Under this name one denotes a Pierre Bernard (Blaise Pascal University, Clermont-Ferrand, France) number of stochastic models introduced by physicists. These models are purely math- Around one hundred mathematicians and this summer school provided the ematical objects of a remarkably simple from twenty countries, mainly young opportunity for a state-of-the-art review by and canonical character. They have been researchers, took part in the Year 2000 one of the main specialists in Europe. studied by physicists using non-rigorous Summer School in Probability Theory at Saint- The course dealt with the applications of methods, and predict a number of Flour. This was the thirtieth such summer stochastic analysis to mathematical extremely interesting behaviours that school; the series was founded in 1971 by finance. The basic economic concept potentially open new areas of probability Paul Louis Hennequin, and is well known underlying the modern approach to math- theory. The challenge is to say something by the international community in proba- ematical finance is the ‘principle of arbi- rigorous about this topic, and the purpose bility theory, statistics, and their applica- trage’. The theory was developed by sys- of the course was to introduce the basic tions. tematically elaborating this concept and its ideas and tools that have been developed The three series of lectures were chosen applications. The mathematical model of in this direction. to have applications in different fields, and a financial market is a stochastic process S The young participants were all invited provided advanced training in these areas. modelling the discounted price of a finan- to present their research work in 40- Each series consisted of ten 90-minute lec- cial asset (a ‘stock’); the time index can be minute presentations, and forty of them tures. either discrete or continuous. Trading on did so. The environment of the summer The lectures by Sergio Albeverio, the stock S is modelled by predictable school was very favourable for exchanges University of Bochum, were on Dirichlet processes H, describing the amount of between young researchers and more forms and infinite-dimensional processes, their stock held by an investor at a given time. experienced participants. theory and applications. These covered basic The basic object of study is the process of The help of the EMS was much appreci- theory: analytic and probabilistic tools; dif- gains and losses given by the stochastic ated, and I wish to thank all the sponsors fusion processes; jump processes; connec- integral H.S. of the Summer School: the European tions with systmes of stochastic ordinary The course was in two parts. In the first, Commission XII, Région Auvergne, differential equations and stochastic par- the underlying probability space was Départment du Cantal, City of Saint-Flour, tial differential equations; applications to assumed to be finite. In this elementary UNESCO, the Blaise Pascal University, stochastic dynamics (statistical mechanics, setting all the measure-theoretic difficul- and CNRS. quantum field theory and polymer ties disappear, and the basic themes and The next Saint-Flour Summer School physics); and analysis and geometry on ideas of mathematical finance can then be will be held from 8-25 July 2001, and the configuration spaces and applications. presented without unnecessary technicali- three lecturers will be Olivier Catoni Walter Schachmayer, University of ties: the basic relation between arbitrage (CNRS, Paris VI University): Statistical Vienna, gave an updated overview of The and equivalent martingale measures (the learning theory and stochastic optimization; mathematical probabilistic tools of finance and ‘fundamental theorem of asset pricing’), Simon Tavaré (University of Southern the mathematics of arbitrage; the applications pricing and hedging of derivative securi- California): Ancestral inference from molecu- here concern banks and finance and insur- ties in complete and incomplete markets, lar data; Ofer Zeitouni (Technion–Israel ance companies. This subject has been optimal investment and consumption Institute of Technology): Random walk in greatly developed in the past ten years, problems, etc. In the second part, the random environments: asymptotic results.

EMS March 2001 5 EMS NEWS SS TT AA TT UU TT EE SS of the European Mathematical Foundation

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

Article 1: Name, seat and duration Institute of Technology in Zürich Report of the Board, and a budget. The The European Mathematical Foundation [“ETHZ”]. Board of Trustees will specify who is autho- is a Foundation in the sense of Article 80 of The Board of Trustees may appoint fur- rized to sign for the Foundation. The the Swiss Civil Code. It has its seat in ther members at its discretion, as specified Executive Committee will supervise the Zurich. The duration of the Foundation is in the by-laws of the Foundation. investment of the funds of the Foundation. not limited. The President of EMS will chair the Board of Trustees. Article 8: Changes to the Statutes of the Article 2: Purpose of the Foundation The Board of Trustees will appoint a Foundation The Foundation has the purpose of fur- Secretary. The Board of Trustees may, within the lim- thering Mathematics in all its aspects in The Board of Trustees will supervise the itations of Article 85 and 86 of the Swiss Europe (including scientific, educational management of the funds of the Civil Code, with a two-thirds majority of and public awareness). Foundation, and will ensure that the funds the present or represented members of the To this purpose the foundation can are used according to the aims of the Board of Trustees, decide on a total revi- undertake the following activities: Foundation. The Board of Trustees will sion or a partial revision of the a) the establishment and supervision of a meet at appropriate intervals, and at least Foundation’s Statutes. publishing house, whose mode of opera- once a year for an Annual Meeting. In that The Board must submit its decision to tion will be regulated by by-laws deter- Annual Meeting the financial statements the supervising body, the Swiss Ministry of mined by the Board of Trustees of the and budget of the Foundation will be the Interior, with a request for an appro- Foundation; approved. The Executive Committee of priate changement decree. b) furtherance of the activities of the the Board may call a meeting of the Board European Mathematical Society [“EMS”]; of Trustees at any time. Article 9: Termination of the Foundation in particular, the furtherance, dissemina- The board of trustees will establish by- Should the objectives of the Foundation no tion and popularization of mathematical laws on the details of the organization and longer be achievable for any reason, then knowledge, furtherance of exchange of the management. These by-laws can be the Board of Trustees may at its discretion ideas between mathematicians in Europe changed by the board of trustees within the decide to dissolve the Foundation, subject and between mathematicians in Europe purpose of the foundation any time and to the approval of the supervising body and other mathematicians; have to be approved by the supervising (the Swiss Ministry of the Interior). In the c) collaboration with the administration authorities. event of a dissolution of the Foundation, of the EMS; A member of the board of trustees will in its remaining funds, after payment of all d) support of activities of corporate principle receive no payment from the debts, may be given to organizations, foun- member societies of the EMS. Foundation. Expenses will be covered dations and institutions with the same or The Foundation may, on the basis of upon presentation of receipts. If some similar objectives to the present decisions made by its Board of Trustees, tasks are extraordinarily work intensive Foundation. extend all its activities towards further these can be paid for in an isolated case. tasks with similar objectives. In the name of the European Article 6: Agency for Auditing Mathematical Society Article 3: Foundation capital The Board of Trustees elects an indepen- President The foundation capital is Euro 10000. The dent external Agency for Auditing which Rolf Jeltsch foundation capital may be increased any- audits the books of the foundation on an time by additions from the founder or annual basis and submits its result to the third parties. Board of Trustees in a detailed report with EURESCO Conferences the motion for approval. In addition it has Article 4: Bodies of the Foundation to ensure that the Statutes and the purpose in Mathematics in 2001 The operating bodies of the Foundation of the foundation are adhered to. will be: The Agency for Auditing has to inform 18-23 August: Algebra and a) its Board of Trustees; the Board of Trustees on shortcomings Discrete Mathematics, Hattingen, b) the Executive Committee of the Board observed during their investigation. If Germany of Trustees; these are not corrected in a reasonable EuroConference on Classification, c) the managing director of the publish- time period, the agency has to inform the Non-Classification and Independence ing house; supervising body. Results for Modules, Groups and Model d) the Auditors of the Foundation. Theory Article 7: Executive Committee Article 5: Board of Trustees The Executive Committee of the Board Chairs: Rüdiger Göbel, Essen, and The Board of Trustees will consist of at shall comprise the following: Manfred Droste, Dresden least 4 members but not more than 12 * the Chairman of the Board of Trustees members. (or a nominee from amongst the Board 1-6 September: Number Theory Members of the Board of Trustees will members); and Arithmetical Geometry, include: * two nominees of the Board of Trustees, Acquafredda di Maratea (near * the current President of EMS; at least one of whom is also a member of Naples), Italy * the previous President of EMS, unless the Board of Trustees. EuroConference on Arithmetic he/she renounces this office; The Executive Committee is responsible Aspects of Fundamental Groups * the current Secretary of EMS; for the day-to-day running of the Chair: Anthony J. Scholl, Durham, * the current Treasurer of EMS; Foundation, the Annual Report of the * a representative of the Swiss Federal Board, the audited Annual Financial UK 6 EMS March 2001 EMS NEWS Raising Public Awareness Raising Public Awareness Maths Quiz 2000 ofof MathematicsMathematics The international competition Maths Quiz 2000 took place on 4-5 December. There were 378 An ARTICLE COMPETITION teams from around the world registered to play, Vagn Lundsgaard Hansen including 108 from the USA, 27 from Russia, and teams from Mongolia, Azerbaijan, the Articles in many ways, eral magazine, in the country of the Philipines and Thailand, to name a few. As the math displays, author, thereby providing some evi- competitors and organisers can testify, it was an says, dence that the article does catch the emotional experience and an excellent way to the EMS committee of RPA: interest of a general audience. Articles celebrate World Mathematical Year 2000. A competition surely may, for the competition shall be submitted It is now time to announce the winners and inspire the way, both in the original language (the pub- explain the background to this unique event. in which to pay, lished version) and in an English trans- The competition was conceived and designed at as we say, lation. The English version shall be the Centre de Recerca Màtematica in Barcelona attention to public awareness! submitted both in paper and electroni- by Manuel Castellet, Rafel Serra and Jaume cally. Aguadé. Designed as a real-time question-and- During World Mathematical Year There will be prizes for the three best answer competition to be played over the inter- 2000, several articles on mathematics articles of 200, 150 and 100 Euros, and net, with successive levels of scoring following addressing a general audience were the winning articles will be published in the Fibonacci sequence, it was played in one published throughout the world, and the EMS Newsletter. Other articles from continuous 24-hour session to guarantee fair- many valuable ideas for articles popu- the competition may also be published ness to players from all time zones. In this way larising mathematics have been gener- if space permits. Furthermore, it is it was also a test of endurance! ated. The newly appointed committee planned to establish a web-site contain- When Sun Microsystems offered to sponsor the for Raising Public Awareness of ing English versions of all articles from game with five magnificent prizes, as well as Mathematics of the European the competition approved by the RPA- donating the server needed to host it, Maths Quiz Mathematical Society (acronym RPA) committee. 2000 was born. Now that the game is over, the believes that it is vital that such articles By submitting an article for the com- server will be presented to a mathematics be written. In order to inspire future petition, it is assumed that the author department in a less-favoured region. articles with a mathematical theme and gives permission to translation of the Birkhäuser publishers and Wolfram Research to collect valuable contributions, which article into other languages, and for also showed interest, offering prizes of book deserve translation into many lan- possible inclusion into a web-site. vouchers and Mathematica licences. The Centre guages, the EMS wishes to encourage Translations into other languages will de Recerca Matemàtica offered a special prize to the submissions of articles on mathe- be checked by persons appointed by the team from the Catalan countries which matics for a general audience, in a relevant local mathematical societies achieved the best result in the competition. competition. The EMS is convinced and will be included on the web-site. To make the game fun to play it was necessary that such articles will contribute to rais- Articles should be sent before 31 to think carefully about the design (and the ing public awareness of mathematics. December 2001, to the chairman of the prizes!) and to have good questions. A commit- The RPA-committee of the EMS invites RPA-committee of the EMS: tee of five mathematicians (Jaume Aguadé, Joan mathematicians, or others, to submit Professor Vagn Lundsgaard Hansen, J. Carmona, Enric Nart, Pere Puig and Jaume manuscripts for suitable articles on Department of Mathematics, Soler) worked for months on the task of produc- mathematics. Technical University of Denmark, ing a large database of questions that were To be considered, an article must be Building 303, simultaneously hard, enjoyable and correct. At published, or about to be published, in DK-2800 Kongens Lyngby, Denmark. the technical end, the game was programmed by a daily newspaper, or some other gen- e-mail: [email protected] Rafel Serra and Waldo Mateo, with a group from Universitat Oberta de Catalunya, implementing the hardware and providing and configuring the internet connection for the server. JEMSJEMS The competition started at noon, Greenwich Mean Time, on Monday 4 December and ended The Journal of the European Mathematical Society (JEMS) is a joint publication of the exactly 24 hours later. The organisers, following European Mathematical Society and Springer-Verlag. There is one volume per year, con- its progress from the UOC, were consistently sisting of four quarterly issues; Volume 1 was published in 1999. JEMS includes research surprised by the high level of play sustained by papers in all areas of pure and applied mathematics. The Editor-in-Chief is Dr. Jürgen the competitors during this period and their Jost. extraordinary skill on really hard questions! JEMS is one component of the continuing effort of the European Mathematical The leading team was a group of young Society to promote joint scientific activities among the many diverse structures that char- researchers from the Universitat Autònoma de acterise European mathematics. Barcelona, led by Raul Fernandez, who achieved To look at the journal, use the website: a final score of 925 points. In second place came http://link.springer.de/link/service/journals/10097/index.htm an Italian team led by Andrea Rizzi from Pisa. To see the contents of the first two volumes, use the website: Third was a Brazilian team from IMPA in Rio de http://link.springer.de/link/service/journals/10097/tocs.htm Janeiro, lead by Krerley Oliveira, while the Individual members of the EMS can purchase JEMS at a special price of DM 52 in Central fourth position went to Martin Kollar and his and Eastern European, and DM 80 outside Central and Eastern European (plus carriage collaborators from the Comenius University in charges). The corresponding rates for institutional subscribers are DM 180 and DM 396. Slovakia. Positions 5, 6, 7 and 8 were taken by These subscription rates include both the printed and electronic forms of the journal. To Andrei Moroianu (currently in USA), Juan Pablo subscribe, please contact Springer-Verlag on the website: Rossetti (Dartmouth College, USA), Greg http://link.springer.de/link/service/journals/10097/subs.htm Martin (University of Toronto, Canada) and The contents of the most recent issue (Volume 3, No. 1) are as follows: Ricardo Perez-Marco (UCLA, USA). D. Mucci, A characterization of graphs which can be approximated in area by smooth graphs The CRM is now considering ways to make the M. Harris, A note on trilinear forms for reducible representations and Beilinson’s conjectures game MQ2000 available to the public. L. Ambrosio, V. Caselles and J.-M. Morel, Connected components of sets of finite perimeter and applications to image EMS March 2001 7 Simulation of Fluid and Structure Interaction

European Mathematical Society SUMMER SCHOOL Campus of the Faculty of Mathematics and Physics, Charles University, Prague 19th-31st August 2001

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

The level will be appropriate for graduate students and young researchers. Limited financial support will also be available for graduate students.

Speakers J. Ballmann, TH Aachen, Germany D. Causon, University of Manchester, UK M. Feistauer, Charles University, Prague, Czech Republic J. Felcman, Charles University, Prague, Czech Republic P. Hemon, Institut Aerotechnique, France P.Leyland, EPFL, Switzerland A. Quarteroni, EPFL, Switzerland M. Schäfer, TH Darmstadt, Germany L. Tobiska, University of Magdeburg, Germany J.Vierendeels, University of Gent, Belgium

Organizers P. Wesseling, Delft University of Technology, Netherlands L. Tobiska, Otto-von-Guericke-University, Magdeburg, Germany M. Feistauer, J. Felcman, Charles University, Prague, Czech Republic

Co-Organisers P. Chocholatý, Commenius University, Bratislava, Slovakia G. Stoyan, ELTE University, Budapest, Hungary J. Rokicki, Warsaw University of Technology, Poland

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

EuropeanEuropean MathematicalMathematical SocietySociety INTERVIEW InterviewInterview withwith BernharBernhardd NeumannNeumann interviewer: John Fauvel, Open University, UK

Bernhard Neumann was born in Berlin in 1909, and took his Dr. phil from the University of Berlin in 1932. He moved to England in 1933, taking a PhD at the University of Cambridge, and taught at the Universities of Wales, Hull and Manchester before moving to Australia in 1962, to create the mathematics department within the Institute for Advanced Studies of the Australian National University. This interview focuses on his life, work and experiences in Europe up to 1962.

Professor Neumann, I wanted to ask you first about the influence of your family on your mathematical interests and develop- ment. My home environment was a very strong influence. My father, Richard Neumann, was an engineer who worked for the elec- tricity company AEG (Allgemeine Elektrizitäts Gesellschaft) in Berlin, and we lived in a prosperous Berlin suburb where I went to school. It was one of his student textbooks that first introduced me to the fascination of mathematics, at the age of eleven or twelve: it was Stegemann and Kiepert’s textbook on differential and inte- gral calculus. Studying differential and integral calculus was in those days not gen- erally done until university—it had not migrated down to high school yet. My father showed this textbook to me, to show me the shape of some curve, I think, and it fired my imagination. I worked through the differential calculus from cover to cover, did all the exercises, and enjoyed it immensely. The second volume, on inte- gral calculus, I think I got stuck on after a while. I must have been eleven, but that interesting. Caesar was somewhat spoiled of my studies (1929-32). There were four certainly was a great influence. for us by having to translate word for word, professors in Berlin: Issai Schur, Richard At school I was thoroughly lazy, but very but here was something that was alive, and von Mises, Ludwig Bieberbach and Erhard good at mathematics, and that saved me, suddenly I got interested in Latin. My Schmidt. They made an excellent team, except on one occasion when my form Latin improved tremendously and I did each completely different in style, in lec- master sent for my mother and told her I rather well at the examination. turing style. Issai Schur was my was very close to not being moved up at the It was when I went to university that I Doktorvater, a distant but kindly and sup- end of the year. That would have been a began to read more books in Latin, read- portive man. terrible thing, for me and for the family, ing the authors we hadn’t got to in eight and my sister, who was four years older years’ study of Latin at school. I would Were you always primarily interested in than I, was delegated to see to it that I read them on the train journey, using a algebra? worked. I worked really hard under her tiny little dictionary in which I looked up I had really intended to become a topolo- guidance for something like six weeks, and the words I didn’t know. After a while I gist, influenced by Heinz Hopf, and my by then I had got into the habit of doing could read Latin with reasonable fluency. conversion to algebra was rather by enough work to get by. Among other things, I read engineering chance. In my sixth semester, towards the texts of Frontinus, from the end of the first end of my third year, there was a student And how did you enjoy your school days? century, about the Roman water supply. It seminar and I was the understudy to the I spent three years in primary school and sounds a very dry subject but I was fasci- person reporting on a paper about auto- nine years in high school. All in all I took nated by the author’s tables of pipes, their morphism groups by the Danish mathe- nine years of French, eight years of Latin, diameter, circumference & carrying capac- matician Jacob Nielsen. I realised that I and three years of English. The French ity, which he calculated by a simple formu- could reduce the number of generators he was uninspiring. The Latin I didn’t like— la. I taught myself calculating with Roman used from four to two, in general; so I I didn’t like the teacher because he insist- fractions and checked all the tables, which wrote it up and showed it to Heinz Hopf. ed on our working—until one day, I was in had been rather badly transmitted. He immediately suggested I might like to the fifth form by then, when he said we take my doctorate with this work, to which have covered what we have to do, let us Where did you go to university? I replied that the paper was too slight and read an author who is not on the syllabus. I spent my first two semesters (1928-29) at I was too young. Nonetheless, Hopf He chose a minor writer from the second the University of Freiburg, in the Black showed the work to Issai Schur, who sent century AD, Minucius Felix, an early Forest, but returned to the Friedrich- his assistant Alfred Brauer with the same Christian apologetic, and suddenly this was Wilhelms University in Berlin for the rest suggestion, to which I gave the same EMS March 2001 9 INTERVIEW response. In the next semester Schur sent then, and when we talked more I discov- or so he might ask me a question or two Brauer again to say “But Professor Schur ered that she too was anti-Nazi—Hitler about my research work, but never pointed wants you to take your doctorate with this had come to power at the end of January, me in any particular direction (indeed he work.” So I agreed. A little later Schur and it was now the end of February. So we did not interfere in my false starts). In fact, himself spoke to me and said that perhaps became friendlier and at Easter I invited at that time I knew a bit more about infi- I was right and it was rather thin, so would her to come for a walk with me in one of nite groups than he did. This didn’t last; I like to use the same methods to explore the forests on the outskirts of Berlin. After later he knew a tremendous amount about an additional topic in which he had the walk we came back and played table infinite groups. Anyway, the experience of become interested, that is now called the tennis. Suddenly she called me, by mis- being supervised by Philip Hall was very wreath product of groups. My suspicion is take—by deliberate mistake, perhaps—it pleasant. that Schur was thinking in that direction was certainly a highly significant slip in The oral examination for my thesis was because of a recent paper by Loewy in terms of German etiquette—the familiar over lunch in my supervisor Philip Hall’s which the idea had been adumbrated. I Du instead of the impersonal Sie. That led rooms. He was one examiner, and Max did the work in a fortnight, which doubled to my really falling in love with her. Newman was the other. They asked me the length of my thesis, and submitted it A little later I took her home to meet my two questions: did I prefer beer or wine secretly, as a surprise for my parents. family. My mother immediately took to with my lunch, and would I like my coffee I took my oral examination for the doc- her, and that became very important for black or white. I must have answered these torate in November 1931—my examiners us. questions to their satisfaction, for I were Issai Schur and Erhard Schmidt in received my PhD. mathematics, and for my subsidiary sub- 1933 was an eventful year in Germany. jects Peter Pringsheim in physics, and Yes, in August of that year I moved to Meanwhile Hanna was in Germany Wolfgang Köhler in psychology. My exam- Cambridge, in England. I knew I had no throughout these years? ination with Schmidt was intended to take chance in Nazi Germany, but my arrival in Hanna was in Germany over this period. forty minutes but we found the conversa- Cambridge was quite coincidental. A We had become engaged in 1934, and cor- tion so interesting that it took twice as school friend of mine knew I wanted to responded in secret through my parents long. I received the degree some months emigrate somewhere, I didn’t mind partic- and her old geography teacher. In 1936 later in July 1932, since I had to get copies ularly where, and rang up one day to say we managed to spend a fortnight together of my dissertation. I expected Issai Schur “Meet me in Amsterdam tomorrow.” I in Denmark, where I stopped off on my to publish my dissertation in Mathematische phoned home and my father, who was at way back from the ICM in Norway. Zeitschrift, of which he was founding editor, work, immediately arranged for me to get In that year, too, 1936, Hanna took her but he didn’t, and instead Otto a passport, which took a day to obtain. I teachers examination, the Staatsexamen. Blumenthal published it in Mathematische took a train to Amsterdam, met my school She had been warned by a good friend that Annalen. Then Jacob Nielsen, on whose friend there, who said Cambridge was the this would have involved her being ques- work it was based, reviewed it in best place for a mathematician, and saw tioned in the oral examination by Ludwig Zentralblatt. I was then 23, which was a very me on my way. So I arrived in Cambridge. Bieberbach on “political attitudes”, which young age to take a doctorate in mathe- Like so many of the emigrés at that time I was a new requirement. Because he sus- matics in Berlin. had a doctorate already, but unlike some I pected her of being friendly with Jews, he immediately registered to do a second doc- would certainly have failed her. So the oral Of the four full professors at Berlin, Ludwig torate in Cambridge. This was against the was arranged for the long summer vaca- Bieberbach became notorious later for his advice of G. H. Hardy, who said that a doc- tion, when Bieberbach was away. In the Nazi sympathies. Were there signs of this torate was unnecessary; what was impor- event it was conducted by Neiss, who was when you knew him? tant was to do good mathematics. Most of more sympathetic and reliable. But clear- One always had the feeling that this was a us ignored his advice, except for Hans ly it was impossible for Hanna to take her defensive move, because Bieberbach had Heilbronn, who indeed went on to do very doctorate at Berlin, as she wanted, so she been too friendly before with wives of well. moved to Göttingen to continue her Jewish colleagues—not in Berlin, but else- research under Helmut Hasse. She got on where—at least, that was the rumour. What was your experience of research at well there, but after three semesters she Perhaps he was concerned for his own Cambridge? saw the way the wind was blowing, aban- future, and that is why he became not just I was allocated to Philip Hall as my doned her research (it was only completed Nazi but ultra-Nazi. He was in fact quite research supervisor—I had no idea who a few years later, I think by André Weil), an inspirational teacher, whose effect was Philip Hall was, as the allocation was made and came to join me in England in July rather great in a number of ways on by someone else, I don’t know who. I start- 1938 where we got married. Hanna, my first wife. ed my research working on a topic which at the time was just too early, on rings of non- You had a job by this time? What happened after you took your doctor- associative polynomials; this problem was Yes, after completing my Cambridge doc- ate? only solved later, and not by me. I worked torate I stayed on there for a while. At that I continued to go to lectures and seminars, on it until Christmas 1934 and then I time Olga Taussky was in Cambridge, and also worked as an unpaid assistant in realised that it would not make for a PhD teaching a course on algebraic number experimental physics. Some time, it must thesis in the months remaining. Then I theory; this needed a preparatory course have been in January 1933, a friend intro- remembered something I had started a few which I was asked to give. At the end of duced me to a young student, Hanna von years earlier and had not really pursued, these lectures the students who had attend- Caemmerer. I thought her name sounded and looked at it again, and within a very ed gave their judgement of the lecturer in rather right-wing (and indeed her ances- few days I realised I had struck oil. I a letter delivered to one of the members of tors were Prussian officers), so thought lit- worked on it and worked on it and one day staff who was a confidential go-between. I tle more about it. in May 1935 I said to myself “I must stop got given a sealed letter with the results of But then I met her again at Mapha, the now and write up”, otherwise it would just the survey. I eagerly opened it, to find it Mathematisch-Physikalische go on and on and on. said: “No comment”. I was paid £10 for Arbeitsgemeinschaft, which was a wonder- I saw Philip Hall once a week, when I that course. The year after they offered ful institution within the university, set up would look him up in King’s College after me the same course again, for which they by Alfred Brauer, one of Professor Schur’s dinner. He would offer me a cigarette. We would pay £50. If I had stayed on giving assistants, and Hans Rohrbach. It had, then talked about everything under the that course I would be rich now. uniquely, its own rooms and its own sun. He was very well informed over a But by that time I was in Cardiff, in library, and there we would play chess, play wide range: about rhizomorphs in botany, Wales. I had applied for every job going, go, and discuss mathematics and other about the political situation in European and was interviewed by University College, things. Hanna was in her second semester countries, everything. At about ten o’clock Cardiff, who wanted a temporary assistant 10 EMS March 2001 INTERVIEW for three years and had shortlisted one onciled to our marriage which she had not Alan Turing was also on the staff at mathematician from the north of England, at first been at all enthusiastic about. Of Manchester. Was he someone you knew and myself. I got the job, I suspect because course my having produced three grand- there? I was not English. It was temporary children for her, with a fourth on the way, I did not know him well, but he showed me because they had a very bright student who was a great help in bringing her round. the computer they were developing, which had just finished her first degree and had was a matter of pink string and sealing gone off to Cambridge to take a PhD, and So after the war you didn’t return to Wales? wax, down in the basement. In the corner they wanted to keep an opening for her for No, I turned down their invitation to was a big column which was the memory. when she had finished. After those three return, since they did nothing for me dur- It was a strange contraption. He was the years the war had started, and things were ing the War, and instead took a lectureship brains behind it. He gave a number of very different. In fact, she didn’t come at University College, Hull. Shortly after- talks, from which it was quite clear that he back after three years but after four years, wards Hull offered a job also to Hanna, was a genius. He was not easy to under- by which time I was no longer in Cardiff, I who had her DPhil by now, supervised at stand, as he would stumble over the was in the army. Oxford by Olga Taussky. She stayed there words—he thought so much faster than he for twelve years, but in 1948 I was recruit- could speak—but it was very clear he was a What happened to you during the War? ed by Max Newman to join the mathemat- genius. There is no doubt about that. At first I continued to teach in Cardiff, liv- ics department at Manchester. ing there with Hanna and with my parents, It was in Manchester too, I understand, who had come to join us, after much per- Why did Max Newman lure you to that you developed your interest in the his- suasion, just before the outbreak of war. In Manchester? Of course, you had long tory of mathematics? May 1940 there were a lot of scare stories known him, as he was your PhD examiner I had an engineer friend who worked for in the press about enemy aliens and we at Cambridge. Ferranti in Manchester, B. V. Bowden were moved away from Cardiff, which was It wasn’t just that. What happened was (later Lord Bowden) who was very con- a port, and went to live in Oxford, safely that Hanna and I decided to organise a cerned with computers and their history, inland. After a few days I was interned, at meeting, a colloquium, of British mathe- and one day said to me that he had found first in Southampton and then in maticians in Hull. We hoped the London papers belonging to Ada Lovelace which Lancashire. By autumn of that year Mathematical Society would support it, but he had on loan from the family. These internees were being released if their for- some members were a bit lukewarm about were largely letters from Ada Lovelace to mer employer asked for them. My former the idea. In particular, our own professor her mathematics tutor, Augustus De employer, University College, Cardiff, of mathematics in Hull, George Steward, Morgan, with some of his replies. I read made no move at all, so I decided to join was rather dismissive of the idea of holding these papers and found them fascinating— the Army. When later Cardiff asked me to a colloquium of mathematicians in Hull. they included also notebooks from her come back I said “No”, since they did not The problem was that he had been daughter’s study with the first professor of support me and try to secure my release appointed quite young to the Chair in mathematics at Owen College, Manchester—and as they were in disarray I put some effort into arranging them in chronological order, using watermarks, internal evidence, and so on. I read as much as I could about her and De Morgan and their circles, including Augustus De Morgan’s son William de Morgan, who became famous for his tiles and also for writing novels. In fact, my father gave me a copy of William De Morgan’s first novel, Joseph Vance, to read on the journey when I went to Cambridge in 1933. Eventually the Lovelace papers went back to the fam- ily, but my son Peter made photocopies of them and I have kept up my interest in the history of that time. I later wrote a paper about this which appeared in the Mathematical Gazette.

Do you think history of mathematics can Three octo- and nono-geraniums (as they called themselves) met again at the 2000 Christmas meet- help to stimulate and inform students’ ing of the British Society for the History of Mathematics, held in University College, London: the interest in learning mathematics? late Robert Rankin (left) with two refugees from Nazi Germany, Bernhard Neumann and Walter Well, for example, Charles Curtis’s book Ledermann. on the history of group representations will do a lot to introduce students to the when they could have done. Hull, and in about 1936 he had died, but work of Frobenius, Schur, Burnside, and so I served for two-and-a-half or three years no-one told him he had died, so he carried on. And my son Peter has done a lot of in the Pioneer Corps, an interesting out- on living until his nineties, when in fact he work in history, including a forthcoming door life building Nissen huts and the like, had been dead for many years before that. edition of the work of Burnside, which and when we were allowed to volunteer for So there was no mathematical colloqui- should help further in bringing mathemat- combat service I transferred to the Royal um in Hull, but meanwhile Max Newman, ics from the past alive to students today. Artillery where I had a slightly more math- who was putting a lot of energy into build- ematical role using surveying equipment ing up the department in Manchester, In 1962 Bernhard and Hanna Neumann and doing basic numerical work. Later I learned through his LMS contacts of our moved to Australia, where he took up the was in the Intelligence Corps. proposal which he thought an excellent Foundation Chair in mathematics at the After the fighting had stopped I volun- one. The upshot was that he invited me to Institute of Advanced Studies, Australian teered to go with a unit of the Intelligence join the faculty at Manchester, and the fol- National University in Canberra. Hanna was Corps to Germany, as I hoped to make lowing year, 1949, we were able to organ- appointed a Professorial Fellow there, and contact with Hanna’s family near Lübeck. ise the first British Mathematical shortly afterwards became head of the pure When I arrived with a kitbag full of food Colloquium, in Manchester. mathematics department in the School of tins, my mother-in-law became further rec- General Studies. EMS March 2001 11 INTERVIEW InterviewInterview withwith SirSir RRogeroger PPenrenroseose part 2 Interviewer : Oscar Garcia-Prada

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

In the early 1970s you discovered two people now call the ‘impossible triangle’ tilings, but when I spoke to him later he ‘chickens’ that can tile the plane in a way were things my father and I played around said he was aware of them. I suspect that that must be non-periodic. How did you with. Later, Maurits Escher incorporated it’s the kind of thing that puts you in a kind find these non-periodic – or perhaps I them in some of his pictures: Ascending and of frame of mind, so that when you see should say aperiodic – tilings? descending used the staircase and the water- something, you’re more receptive to it than Yes, aperiodic tile sets, I suppose, but the fall used the triangle. And he actually used you would have been otherwise. So yes, tilings are non-periodic. Tiling problems ours, because we sent him a copy of our I’m sure it’s true of me with Kepler that I have always been a doodling side interest paper. was more receptive of his kind of design . of mine, just for fun; if I got bored with I met Escher once, and left him a copy of what I was doing I’d try and fit shapes a puzzle I’d made which consisted of wood- These three-dimensional forms of your tiles together, for no particular scientific reason en pieces which he had to try and assem- have appeared in recent years, as quasi- – although I supposed that there was some ble. Well, he managed to do this all right, crystals. Did you ever anticipate such appli- connection with my interest in cosmology, and somewhat later when I explained the cations of your non-periodic tilings? in that there seem to be large structures in basis on which it was constructed he pro- Well I did, but I was overcautious I sup- the universe that are very complicated on a duced a picture called Ghosts – as far as I’m pose, because I certainly knew this was a large scale, whereas one believes that they aware it was his last picture, when he was theoretical possibility. But what worried should be governed by simple laws at root. quite ill – and it’s based on this tile I’d me was that if you ever tried to assemble So I tried to find a model where we have shown him – twelve different orientations these them you’d find it very hard, and simple structures that produce great com- of this shape. without kind of foresight it’s difficult not to plication in large areas; I had an interest in But that was just a sideline, an amuse- make mistakes. I sometimes gave lectures types of hierarchical design. ment really, and the way the tilings came on these tilings, and people asked me ‘does So I played around with such hierarchi- about was in two stages. I’m sure I owe a this mean that there’s a whole new area of cal tilings, where you form bigger shapes debt to Kepler, although I didn’t realise it crystallography’ – and my response would out of smaller ones; the bigger ones you at the time, because my father owned a be ‘yes, that’s true – however, how would produce have the same character but are book showing the picture that Kepler Nature produce things like this, because on a larger scale than what you just did. I designed which had a number of different they would require this non-local assem- also had an interest in Escher and his work tilings that he played with. Some of these bly?’. And it seemed to me that maybe you and met him on one occasion: I had pro- were of pentagons, and these tilings with could synthesise such objects with great dif- duced single tile shapes that would tile pentagons are very close to the tiling ficulty in the laboratory, but I didn’t see only in rather complicated ways, and shapes I produced later. how nature would produce them sponta- Escher himself used one of these in his last Now I was aware of these things because neously. picture. I’d seen them, but they were not what I Now I think, although people now thought of when I was producing my own. understand them better, the situation is What was the name of these tiles? The They just coloured my way of thinking, much the same. I still don’t think we know magic something? which must be rather similar to what hap- how they’re produced spontaneously, and That’s different: those are the impossible pened to Shechtman when he discovered there are different theories about how they objects. The staircase and the tribars that quasi-crystals. He didn’t think about my might come about – maybe there was something a little bit non-local, something basically quantum-mechanical, about those assemblies which I came to think is proba- bly true, but it’s not an area that people are agreed about – in fact, it’s not totally agreed that quasi-crystals are this kind of pattern, although I think its getting pretty well accepted now. I was first shown the physical objects, the diffraction patterns, by Paul Steinhardt at a conference in Jerusalem to do with cosmol- ogy. I was talking about general relativity and energy and he was talking about infla- tionary cosmology, and he came up to me and said ‘look, I want to talk to you about something nothing to do with this confer- ence’. He showed me these diffraction pic- tures that he’d produced, and it was quite startling but very gratifying – in fact, curi- ously enough, I wasn’t completely sur- prised. I suppose I felt that it must be right and nature is doing it somehow. Nature seems to have a way of achieving things in ways which may seem miraculous; this was A Penrose tiling just another example of that. 12 EMS March 2001 INTERVIEW Have other examples of your recreational time that there are things of a mathemati- So I started writing the book, expecting mathematics appeared in nature or cal nature that are outside computation; that by the time I’d finished I’d have some physics? that didn’t frighten me – it just seemed to clearer ideas on this. This happened to One thing I got from my father is that you me ‘all right, why not?’ Then I saw a tele- some degree – I was very ignorant about never draw a line between the two. He was vision programme where Marvin Minsky, lots of things to do with the brain when I like that – he did things for fun, which Edward Fredkin and others were making started, and I had to study to write the might be making children’s toys out of outrageous statements about how comput- chapters specifically on them, and in par- wood – puzzles, or gadgets connected with ers were going to exceed everything we ticular the idea of ‘brain plasticity’, that the his work. He’d make things like compli- could do. What they were saying was logi- connections between neurons can change cated slide rules that were supposed to test cal if we were indeed computers – but since and that these changes can take place some statistical results. He made a bust of I didn’t believe that, it seemed to me that quickly – it seemed to me that this was very one of his patients, I seem to remember, this was something they’d completely important, and that the new physics comes and then he spent his later years produc- missed – and not only they had missed it, in to regulate these changes. ing wooden models that reproduce them- I’d never seen it anywhere else. So as I’d selves. There was no line – you couldn’t previously considered writing a popular or So has this given you any clue as to what say what was recreational and what was semi-popular book on physics – something changes are needed in quantum mechanics? professional work. I thought I’d do at some stage in life, but A little, but the physical motivations are perhaps not until I retired – this provided largely independent. I did, however, think In 1989 you published a best-selling book, an opportunity: ‘All right, let’s explain about the needed changes in quantum ‘The Emperor’s New Mind’, where you’re things about physics and what the world is mechanics a little more seriously than concerned with computers and artificial like, as far as we know, but with a different before and I changed my views between intelligence, the mind, the laws of physics, focus: let’s explore the laws of physics to writing the two books. and many other things. What is the central see whether there’s scope for something of question you address in this book? a non-computational character’. I’d never You’re now referring to your second book I think I mentioned earlier that I had for- seen anybody put this forward as a serious ‘Shadows of the Mind’. Do you pursue the mulated a certain view while I was a gradu- viewpoint. Since it seemed to me that it same problems in this book? ate student. Before that, I’d been quite needed to be put forward, that’s what I did. Yes, but Shadows of the Mind had slightly sympathetic to the idea that we were all ambivalent purposes. Originally I started computers, but it seemed to me that So this was an attack on artificial intelli- to write it to address some of the points Godel’s theorem tells us that there are gence; but you also questioned the fact that that arose out of people’s criticisms and aspects of our understanding which you as quantum mechanics is incomplete, the misunderstandings of The Emperor’s New cannot encompass in a computational pic- understanding of physical laws has to pre- Mind, and in particular my treatment of ture. Nevertheless, I still maintain a scien- cede understanding the functioning of the Gödel’s theorem. I just didn’t expect the tific viewpoint that something in the laws mind? kind of vehement response that I got. I of physics allows us to behave the way we Yes, I’d felt that if there was something was very naïve, perhaps. I didn’t realise do, but that the laws of physics are much non-computational it needed to be outside that people would feel attacked in the way deeper. My view is that we know much less the laws that we presently understand in they did and would therefore respond by about them than many people would main- physics, because they seem to have this attacking me, while misunderstanding a lot tain. computational character. And it also of what I was trying to say. So I was quite prepared to believe that seemed to me that the biggest gap in our The main point that I was making about there was something outside computation. understanding is when quantum theory Gödel’s theorem was that if you have a sys- I’ve been interested in mathematical logic relates to large-scale objects, where the tem which you believe in and which you for a long time, and I’d known for a long rules of quantum mechanics give us non- believe might be usable as mathematical sense – they tell us that cats can proof, then you can produce a statement be alive and dead at the same which lies beyond the scope of the system, time, and so on, which is non- but which you must also believe in. Now sense; we don’t see our world there is an assumption here that the system like that. Yet quantum theory is consistent, something which I didn’t was supposed to be so absolutely bother to stress particularly, because it accurate, so why are we not seemed to me quite obvious that there’s no aware of the manifestations of point in using the system if you don’t think that theory on a large scale? It it’s consistent: the proof is no proof if it’s in seemed to me that the theory a system you don’t thoroughly believe in can’t be quite accurate and that and therefore don’t trust its consistency. there must be some changes that That seemed to me to be obvious, but I take place when it gets involved didn’t make those points strongly and so with large-scale objects. there are loopholes that people could I think I’d already thought this point to, which of course they did. So I felt was to do with when gravitation- it necessary to address these issues with a al phenomena start to become great deal more care in Shadows. It was not entangled with quantum effects meant to be a particularly long or popular – that’s when the changes start book; it was just addressing these points to appear, and there are good and was quite technical and complicated in reasons for believing that. So places, but in the process of writing this that is what I believed at the book, two things happened. time, and still believe; but when One of them was that I received a letter writing The Emperor’s New Mind I from Stuart Hameroff telling me about the didn’t really have any clear idea cytoskeleton whose structures inside cells I where in brain function quan- was totally ignorant of. But most people tum effects could start to who work in artificial intelligence didn’t become important and have an seem to know about them either; Marvin influence on large-scale effects, Minsky didn’t, as he told me afterwards. and where these new physical But it seemed to me that here was a com- processes that should be non- pletely new area for which it was much computable could come in. more plausible that quantum effects could EMS March 2001 13 INTERVIEW takes place, because it is dif- One can see how to use it, and these ideas ficult to maintain quantum I developed with Stuart Hameroff. We coherence on the large scale produced a number of suggestions about that one needs in order for how this idea could be carried forward, but these ideas to work. One again we found a great deal of opposition. needs to go beyond what can A lot of these ideas are clearly speculative, be done in any physics lab but I don’t think it is that speculative that today, and there is no physi- something like this is going on. It seems to cal experiment performed me it has got to be. Consciousness seems today that can achieve the to be such a different phenomenon from kind of quantum coherence the other things we see in the physical at the level that I would world that it’s got to be something very need in order for the kind of special, in physical organisation. I can’t phenomena to take place see that it’s really just the same old physics that I suspect are taking put together in more complicated systems. place all the time in our It’s got to be something of quite a different brains. So nature has been a character from other things that are lot cleverer than physics has important in the way the world operates. been able to be so far, but This new physics would be only occasional- why not? It seems to me ly employed in a useful way, and you have quite plausible that this is to have a very careful organisation that the case. The cytoskeleton, takes advantage of whatever is going on in and in particular micro- state reduction and channels it in a direc- tubules, seem to me to be tion which makes us operate. But it is very structures where quantum rarely actually taken advantage of in phys- coherence is much more ical systems, where most things don’t use plausible. So I changed the this phenomenon in any useful way at all. nature of this book. I want- ed to put in microtubules In these books you reveal yourself to be a and the cytoskeleton, and so philosopher. How have you come to terms I needed to learn a little bit with the Great Mystery? more about it and to express Well, there are lots of hugely unanswered why I think that’s important questions – there’s no doubt about that. in brain action. I’d certainly want to emphasize that even if The other thing which everything that Stuart Hameroff and I say be important. They are much smaller happened was that I somewhat shifted my turns out to be absolutely correct, it would structures than neurons and are much viewpoint on quantum state reduction, in not answer these questions. I hope we are more tightly organised structures. The relation to gravity. The viewpoint I’d held moving a little in the right direction most relevant of these were microtubules, for quite a long time, and which is towards answering those questions. where one has a much more believable expressed in The Emperor’s New Mind, is However, I think there’s very little place for coherent quantum phenomena to more or less that if you have too big a dis- progress towards answering the deep ques- take place. It is still hard to see how this crepancy between two states, then they tions of what is going on in mentality, who don’t superpose and the state we are, what is consciousness, why are we reduces. This discrepancy is to be here and why does the universe allow measured in terms of space-time beings who can be aware, or is there life geometry; I called it the one gravi- after death? Any questions of that nature ton criterion. It is to do with how seem to me to require us to know more many gravitons come into this dif- about what the world is like – we really ference between the two states. know very little. Now in work that I did subse- People say, and physicists often say: we quently, and also in work that had nearly have the solution to the grand theo- been done by others (particularly ry of the universe which is just around the Diósi, a Hungarian, and Ghirardi corner, the theory of everything. I simply in Italy) who had developed dif- just don’t believe that. I think there are ferent ideas in connection with major areas of which we have almost no quantum state reduction, it understanding at all, and it’s quite curious seemed to me that I needed to that one can have a view which seems to modify the view that I had before. encompass (at least in principle) most of I think it’s quite a significant mod- the things that you see around you, why ification. Basically, when you they behave this way and that way. have two states that are signifi- One of the major things that isn’t cantly different from each other, explained is simply ignored and just swept then their superposition becomes under the carpet by physicists generally, unstable, and there is a calculable which is quantum state reduction. They time scale involved in how long it say: quantum theory is a beautiful theory; takes for the superposition to it works perfectly well and describes how ‘decay’ to one state or the other. tiny particles behave. But, to put it blunt- The details are probably a bit too ly, it gives you the wrong answer. What the technical for here, but there is a theory tells us is that, for example, if you time scale, rather than an instan- had Schrödinger’s famous cat, the cat taneous reduction, and this time could very easily be put into a state of scale produces figures that are being alive and dead simultaneously. much more plausible – also, it is That’s simply wrong; it doesn’t do that. So easier to use and it may be much what is it? I mean, there is something big more relevant to brain action. missing from our view of the world, it’s 14 EMS March 2001 INTERVIEW huge and it’s not just a tiny phenomenon that changes our whole outlook. Now what is not something which is as deep, and it’s which we haven’t quite got because we’ve I am saying is that when we see how quan- not something I’ve devoted so much of my got to get the last decimal place right and tum theory is to be changed to accommo- life to. Even when I call it ‘twistor’, this is the coupling constant or something: it’s a date state reduction as a phenomenon, slightly inappropriate, because we don’t huge aspect of the way the world behaves with the measurement process as a phe- know what the theory is yet, the real theo- which we simply do not understand, and nomenon, we shall have to come to terms ry. But if I can answer your question in understanding it is, in my view, one small with a completely different way of looking that way, I think I’d say that the whole pro- step towards understanding what mentali- at what matter is and what the world is like. gramme of representing Einstein’s theory ty is. I think it must be part of it, but it is in terms of twistors is what I’m proudest of. not going to answer the question of men- Are you saying that this is a step that will I don’t know if you’d call that an answer to tality. We may well know what state reduc- help us to address questions that are typi- your question, because it’s not a theorem, tion is (and I expect we will know if we cally addressed, by some people at least, by it’s not one result, it’s a body of ideas. I don’t destroy ourselves first); then that the- religion? suppose in amongst those I would think ory will have as part of its nature some Well it will, I think so. But you see I’m the ‘non-linear graviton construction’ is completely different way of looking at the more… ‘tolerant’ isn’t quite the right word probably the thing which I feel most world from the way we have now. We’ve … I think I’m a little bit more supportive pleased about so far, but that is part of the already seen that happen in Einstein’s gen- of religious viewpoints than many totally programme, and I don’t regard that as an eral relativity, because before that we had scientific people would be. It’s not that I end-point; it’s something on the way. I Newton’s theory which told us how bodies believe in the dogma that is attached to hope that when one really understands attracted each other with forces and moved any of the established religions, because I how the Einstein equations can be incorpo- around and so on. It’s beautifully accurate: don’t. On the other hand, religions are rated into twistor theory, we’ll see some- Newton’s theory is incredibly precise. It trying to address questions which are not thing, a much broader picture of which tells you how the planets move around in addressed by science, particularly moral that is just part. But we don’t quite have their orbits to almost complete precision – issues. I regard morality as something with that fully yet. not quite, but almost. You might think it an absolute ‘platonic’ component which is only requires a little tiny modification to outside us. Is there really a platonic How do you actually work? How do you absolute notion of select a problem? I’ve seen you many times ‘the good’? I have in your lectures and seminars making pic- to say I’m inclined tures – do you visualise things that way? somewhat in that A lot is very visual, but not entirely visual. direction. I think Certainly some things I’ve done are not morality is not just entirely visual; for example, algebraic man made, but things aren’t. But I do find visual imagery there are things absolutely essential in my work, and that’s outside us which the way I do it usually. I often start to write we have nothing in something down, but it doesn’t help very the way of scientif- much. I may see a picture sometimes, but ic understanding even drawing it on a piece of paper doesn’t of at the moment, help because it’s not something you can but nevertheless express very well that way. It’s an image they are part of a which is hard to put down, but I do find big overall picture expressing things by drawing pictures rea- which maybe sonably accurately sometimes helps, but someday will come sometimes it’s an accuracy that’s very mis- together. So I leading. You’ve got to know what it means. mean these ques- I mean, the pictures are not really accurate tions are nowadays because I might draw a picture of some- considered not to thing which has got the wrong number of be part of science, dimensions and it’s really in a complex but they’re part of space rather than a real space, but you get religion, you see. a feeling for knowing what’s reliable and Well, as I say, I what isn’t in a picture. In some ways it’s a don’t believe in the relief when one can do a calculation. If you dogma of religion, can reduce something to a calculation you but I do think that can deal with it, but so few of the things I religion is groping do actually find their way into calculations, for something because that’s not the problem. The prob- which we don’t lem is a conceptual or a geometrical prob- know the answers lem – often conceptual: you have to look at to yet, and which is things in the right way. And that can be outside traditional quite hard sometimes. But I suppose what science. So I sup- I am probably best at, when it comes to pose what I think is mathematics, is the geometrical, where I that the whole sci- can visualise things well. I don’t find it entific enterprise very easy to work with complicated formu- Roger Penrose and Jean-Pierre Bourguignon, Durham 1982 must broaden its lae or analytic notions. It’s an uphill job scope and eventu- sometimes. To a certain extent, things in make it completely right, but that’s not ally change its character. analysis can be looked at geometrically. what happened. Einstein produced a the- ory that is completely different. Its struc- We have touched on only a fraction of your What are you working on right now? ture is utterly different from that of work but what is the result, theorem or the- Well, what I think about most of the time Newton’s theory, but it gives almost exact- ory that you are most happy with? when I’m not doing something else like ly the same answers. The philosophical Well, I’d have to say twistor theory. The trying to write up endless things one after framework of that theory is quite different. non-periodic tilings are nice, and they’re the other, is this problem about the The very nature of space is warped, and something to show to somebody. But this Einstein equations and twistors, which I EMS March 2001 15 INTERVIEW think is almost there or is very close to Something I’ve found slightly discouraging physics it’s to do with the expense of big being resolved; but it’s not completely about the way things go these days, which experiments, which means that you have to clear. I regard that as the major problem. is not the fault of the mathematicians or build bigger and bigger machines to look I also try to make progress with the quan- the physicists, is a problem of technology. at higher energies, and so on. I can see tum state-reduction problem – even with Although technology is a wonderful thing, why they do that, but since the experi- some possible experiment. it has the effect of magnifying fashions. It’s ments are so expensive, they require a lot so easy to communicate ideas from one of money and government backing and You’ve mentioned a number of mathemati- end of the world to the other instanta- huge organisations, and therefore you cians and physicists, but who is your most neously, and this means that the fashions must have committees to decide which to admired mathematician or physicist? have global control over what goes on. It support, and so on. Committees have to That’s a tough one – there are a lot of means that a lot of progress can be made in decide where the money’s going, so they them. I’m not sure there’s a single one as fashionable areas, but there’s something consult people who are considered to be a mathematician. I always had a tremen- which is lost in there, I feel. It’s a bit silly the experts in the relevant areas, and dous admiration for Riemann because he to hark back or think about things in this therefore the things tend to get locked in was a miracle man. As a mathematician I way, but when communications were much certain directions because the experts have think it probably would be him, I guess, but more difficult there were these little pock- got there, as they were the ones who were it’s not quite clear because on the physics ets of people working on different things. important in the development of the cur- side I’ve always thought of Galileo, Maybe I have this starry eyed view of what rent theories. It’s hard to break away from Newton, Maxwell and Einstein as being the it was like; it probably wasn’t like that. this. four most major figures. Of those, I sup- Fashions were still important in the old pose one would have to say that Newton days too, but somehow there’s something It becomes a political thing as well. was a more impressive figure than about the global nature of the fashions It does become political as well, because Einstein, no matter how impressive these days which I find a bit disturbing. money is involved. One doesn’t have the Einstein is. I mean, he’s had absolutely But it’s worse in physics than it is in math- free-ranging way of thinking about things wonderful ideas. General relativity is a fan- ematics, I think. I always used to think that that was there before. So I’m only express- tastic theory – of any physical theory that mathematics was immune from this kind of ing the negative points, because maybe the I’ve seen, I’d say put general relativity at thing. You would have these people work- positive ones are more obvious. Obviously the top because it’s amazing. But Newton ing away and developing their wonderful there’s huge activity – all over the world was such a powerful mathematician, and ideas in relative isolation, and occasionally you find people who work in areas and who Einstein was less so. Einstein had these getting together, and some things would have never before had a chance to think tremendous physical insights, but Newton spring from the coming together of differ- seriously about science. Now the internet had this great power as a mathematician as ent viewpoints. But now it’s much more as allows them to get involved. That’s all positive and I agree with that, but I’m just pointing out that there’s a downside also, which I find disturbing.

Are you questioning the way in which what’s important and what’s not are decid- ed? It is hard to advise them, you see, because they’re caught up in the system, and if young researchers want to get jobs after doing research, they must work in an area which is going to be recognised by the peo- ple who employ them. If I were talking from the point of view of science I’d say that in quantum state reduction there are some important problems: you see why quantum mechanics needs modification. But if they’re working on that, it’s regard- ed as marginal at best, and crackpot at worst. I could even say that with twistor theory: in mathematics it seems to have caught on, but as a physical theory – if you work in twistor theory you’ll find it hard to get a job. There are very few people in physics departments who know about this subject and consider it important.

I remember reading in one of your papers Durham 1982 that twistor theory was an esoteric subject from that point of view. Do you still main- well as deep physical insights. And I’ve if you can go anywhere and they’re all tain this view? always had a soft spot for Maxwell, but also working on the same thing, which is rather Yes, it’s not much studied as a physical the- Galileo – I think I’ve always admired him discouraging. In physics it is particularly ory. I’m not unhappy from my own per- from a long time back. I don’t know – I’m like that. This applies to highly theoretical sonal point of view, because when it gets not trying to single out any one in particu- work where there is no stabilising input studied by lots of people it becomes too lar. Riemann as a mathematician I sup- from experiment. complicated to find out what they’re doing. pose, possibly more than Gauss I would I’d have to learn and understand their say. Archimedes also, I suppose, was a very Communication is obviously beneficial, but notation, which would probably be differ- impressive character; also Euler. the aspect you’re remarking on is actually a ent from mine – and that’s hard work. But negative one. if I know that nobody’s working on it, then What is your opinion about the direction in It is a negative one, I think. And there’s that’s fine – I’m not rushed, and I don’t which research in mathematics and physics another thing that goes along with this. have to think I’ve got to get in there before is going nowadays? It’s not relevant in mathematics, but in someone else does! 16 EMS March 2001 INTERVIEW That’s right. But what recent development Quasi-crystals are remarkable. High have hardly any time to read anything for in science at large has made the greatest temperature superconductors are amaz- fun, which I like to do. impact on you? ing, and so are developments in molecular Gosh, I’d say cosmology is one of the areas. biology. Some of the things that people If the day had 36 hours, what would you Gravitational lensing I worked on for a lit- are now learning about cells and about like to do? tle bit at one point; it’s an amazing thing. cytoskeletons and microtubules I find fan- If it had 36 hours, there should be a rule I’d say that astrophysics and cosmology are tastic. Partly this is because I didn’t know that the extra twelve were only allowed to exciting areas. what was going on in biology, and having be used for things that were not directly to got slightly involved in this subject I see do with one’s work. I enjoy reading when What aspects do you find most interesting? some remarkable things. I get the chance, and also going to the the- What I was talking about, gravitational atre, to films, and listening to concerts – all lensing. I find it interesting because it’s an I came across Schrödinger’s book ‘What is the things I’d love to do more of. But effect of Einstein’s general relativity, which life’, for which you wrote a preface. there are films that I never get the chance I think people thought was very hard to That’s right, and it mentions the aperiodic to see, including some wonderful ones that measure. It was the first thing that con- crystal ideas that he believes are at the are going around now. I used to read quite vincingly suggested that Einstein’s theory heart of life. a bit of science fiction, but I hardly have a might be right. The Eddington experi- chance to now. I sometimes read Michael ment showed the deflection of light of the Can you tell us something about this? Frayn; I find him tremendously funny. stars by the sun, but to see this effect on a My father was very taken with the idea, I cosmological scale, to see a galaxy focussed recall, and we had these mechanical What about music? Do you have a by another galaxy, would have seemed devices I mentioned earlier which repro- favourite composer? absolutely ridiculous. Nowadays it’s an duce themselves, and he then developed On the whole I prefer classical music, but I observational tool, people use it all the much more elaborate devices which he enjoy jazz too – my wife’s influence! My time – it’s a way you can tell the mass of an made of wood, little things with levers parents each had a favourite composer: for object by how much focussing it exerts on which copied themselves. He sometimes my father it was Mozart, who was his God, the image behind it, and it’s wonderful. referred to these things in Schrödinger’s and for my mother it was Bach. I eventu- It’s not just that it’s a powerful tool in cos- terminology as ‘aperiodic crystals’ – a crys- ally realised that Bach is really my mology, but because it uses something tal that went on for a while and then favourite by quite a long way; I’ve always which is close to my heart I suppose; that’s stopped growing, because that was how it regarded those two as a cut above the other why I like that one so much. came to an end. Life was to be thought of composers. Bach I think I see much more But that’s just one area; there are lots of in that way. So it had some influence on in, there’s something you can always go other things. I suppose the experiments him, and indirectly on me. But I don’t back to, and there’s more and more of it. on quantum entanglements (non-locality): know about the quasi-periodic tilings, But I think it’s the perfection in Mozart you can get two ends of the room, 12 whether that has much connection histori- that I find somewhat magical. I’ve always metres apart – well, nowadays it’s longer cally. rated him above Beethoven, who never than that (10 kilometres) – by these quan- had quite the same magic for me. I can see tum entanglement effects the two are con- What books are you reading at the moment? he had the power and originality, but nected through quantum mechanics, it’s Gosh, I never get a chance, though some- somehow there’s not quite the magic there. amazing. One knew it had to be there in times I have to review a book. I’m not Maybe even with Schubert there’s a bit of quantum theory, but it’s very impressive to reading one right now. It’s one of my magic that I don’t quite see in Beethoven. see that it’s real. regrets that I find myself so busy that I To go back further, I like Vivaldi and Purcell and others. But I also quite like some modern composers, such as Stravinsky, Prokofiev and Shostakovitch.

What do you consider the most profound development of the last century? Einstein’s general theory of relativity. I might have said quantum mechanics, but I think that that theory isn’t finished yet, because of the measurement paradox. I discuss these things in my new book, The Road to Reality, which should be published sometime early in 2002. It’s almost com- pleted, but still there’s work to be done on it. It’s about mathematics and physics and the profound relation between the two. In it I try to give my views of some of the pop- ular developments in modern theoretical physics – I may get myself into more trou- ble!

Finally, what will keep you busy in the future? I have many more books in mind, so I’ll keep writing. Also, I’ll keep working on twistor theory, as there’s a great deal to do in that subject. I have new ideas to do with quantum state reduction – even an experi- ment (FELIX) which I hope will be per- formed in space while I’m still alive! In addition to all this, I have a new son (Maxwell Sebastian – Max, for short) who Roger Penrose is the current Gresham Professor of Geometry, in London. Last year he met the was born on 26 May 2000 and will keep me earliest Gresham Professor, Henry Briggs. even busier! EMS March 2001 17 PROBLEM CORNER PPrroblemoblem CornerCorner Contests from Romania, part 3 Paul Jainta (Schwabach, Germany)

Competing against teams representing 81 countries, a team initiated in 1986 between the four counties of Maramures, of six Romanian high-school students won six medals at the Bistrita-Nasaud, Satu-Mare and Salaj, in the north-west of Fortieth International Mathematical Olympiad (IMO), held Romania, and alternates between the capital towns of the in their capital Bucharest in summer 1999. While the com- counties involved. The Moisil contest is open for pupils petition is between individuals, unofficial rankings placed from grades 5-12, and is run as a team-contest with teams of the Romanian team fourth, after China, Russia and 3 or 4 individuals for each grade, selected from the contes- Vietnam. tants who completed the county round of the National It’s no accident that Romania is exceptional in identifying Mathematics Olympiad. Thus, the Moisil competition mathematically able young persons. As Mark Saul, serves as additional screening for the national event. It is Bronxville School, N.Y., wrote in his foreword to a new col- mainly supported by the Ministry of Education and also by lection of problems consisting mainly of difficult problems the regional branch of the Romanian Society for from Eastern Europe: ‘Preparation for participation in a Mathematical Sciences and private sponsors. Diplomas and challenging contest demands a relatively brief period of prizes are awarded. Normally, the problems are chosen by intense concentration, asks for quick insight on specific an independent selection committee of experts outside the occasions, and requires a concentrated but isolated effort’. catchment area of the contestants. The rich Romanian Olympic ambience can serve as a Another tricky competition, similar to the Moisil contest, model for studying the conditions that go to make great is named after the Romanian mathematician, Traian Lalescu problem-solvers. For these young people an Olympiad (1882-1929). The Lalescu contest is for students from problem is an introduction, a glimpse into a world of math- grades 5-12 in one of the four south-west counties Arad, ematics not afforded by the usual classroom situation, for a Caras-Severin, Hunedoara and Timis. The Lalescu contest good Olympiad problem captures in miniature the process is organised by the Faculty of Mathematics, West University of creating mathematics: immersion in the situation, exam- of Timisoara. Here the problems are first-hand, or emerge ination of possible approaches, and the pursuit of various from prestigious journals such as Gazeta Matematica or paths to the solution. But struggling with tough questions Revista Matematica din Timisoara. The Lalescu competition often leads to fruitless dead ends. Grappling with a good has been running for fifteen years. problem provides practice in dealing with the frustration of The following contests form a small portion of what is working at material that refuses to yield, until a lucky solver available to youngsters hungry for mathematics: the Spiru catches sight of a silver lining that heralds the start of a suc- Haret – Gh. Vrânceanu contest takes place in the Eastern parts cessful solution. Like a well-crafted work of fiction, a good of Romania; the L. Duican competition is established in Olympiad problem tells the story of mathematical ability to Brasov, and the N. Paun event occurs in and around create which apprehends a good part of the real experience, Bucharest; the Gh. Mihoc tournament takes place in the and leaves the participant wanting still more. south-east of the country; the D. V. Ionescu – Gh. Lazar con- test occurs in the central region of Transylvania, the home The Grigore Moisil Contest In Romania, thanks to Vasile of Count Dracula. Berinde, North University of Baia Mare, a student can choose between nine intercounty competitions. The biggest From this contest scene I’ve compiled a cross-section of and most famous is the contest named after the great questions; readers are invited to compete with the leading Romanian mathematician Grigore Moisil (1906-73). It was junior minds of Romania.

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

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

124 Find the real numbers x1, x2, … , xn satisfying 2 2 2 √(x1 – 1 ) + 2√(x2 – 2 ) + …+ n√(xn – n ) = (x1 + x2 + …+ xn)/2 (Gazeta Matematica)

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

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

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

18 EMS March 2001 PROBLEM CORNER Solutions to some earlier problems

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

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

111 Let k be an integer and let p(x) be the polynomial p(x) = x1997 – x1995 + x2 – 3kx + 3k + 1. Prove that the polynomial has no integer roots, and that the numbers p(n) and p(n) + 3 are relatively prime, for each integer n.

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

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

Solution by Gerald A. Heuer, Moorhead, MN (USA); also solved by Pierre Bornsztein, J. N. Lillington, Dr Z Reut. The image is the interval [0.5, 2.5]. Since f is continuous and f(x + 2) = f(x) for all x, it suffices to show that the minimum and maximum values of f(x) on [–π,π] are 0.5 and 2.5, respectively. We shall show that f(x) is decreasing on [–π,–π/2], increasing on [–π/2, π/2] and decreasing on [π/2,π]. Since f(–π/2) = 0.5, f(π/2) = 2.5 and f(–π) = f(π) = 3/√2, the stated result then follows. For 0 ≤ x ≤ π,cos (x/2)= cos (x/2) andsin (x/2)= sin (x/2), so with u = x/2, and using the half-angle formulas √(1 + cos x) = √2 cos (x/2) and √(1 – cos x) = √2 sin (x/2), we get √2 f(x) = g(u) = (3 + 4 sin u cos u) / (cos u + sin u), 0 ≤ u ≤ π/2. Then g´(u) = [(cos u sin u)(1 + 4 sin u cos u)] / (cos u + sin u)2, which has the same sign as cos u – sin u. Thus g´(u) > 0 when 0 < u < /4, and g´(u) < 0 when π/4 < u < π/2. It follows that f(x) is increasing on [0, π/2] and decreasing on [π/2,π]. For –π ≤ x ≤ 0,cos(x/2)= cos(x/2) andsin(x/2)= –sin(x/2), so with u = x/2 again, √2f(x) = h(u) = (3 + 4 sin u cos u) / (cos u – sin u). Then h´(u) = [(cos u + sin u)(7 – 4 sin u cos u)] / (cos u – sin u)2, which has the same sign as cos u + sin u. Thus h´(u) > 0 for –π/2 < u < –π/4 and h´(u) < 0 for –π/4 < u < 0. It follows that f(x) is increasing on [–π,–π/2] and decreasing on [–π/2,0], establishing the claim.

EMS March 2001 19 PROBLEM CORNER 113 Let f(x) = ax3 + bx2 + cx + d be such that f(2) + f(5) < 7 < f(3) + f(4). Prove that there exist u, v such that u + v = 7 and f(u) + f(v) = 7.

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

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

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

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

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

In the remaining case a ≥ 1. Now r2 = 1 and s2 = a ≥ 1; r3 = a + 1 > 1 and s3 = a ≥ 1. Suppose that rk > 1 and sk ≥ 1. Then rk+1 = rk + sk > rk > 1 and sk+1 = rka ≥ rk > 1. It follows by induction that rk > 1 for all k > 2. Thus x is rational, and is therefore an integer.

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

20 EMS March 2001 SIAM-EMS Applied Mathematics REGISTRATION FEES ❐ short and medium range meteorolo- SIAM / EMS Non- gy and oceanography; Members Members ❐ pollution transport in air, water, in our Early Registration (prior to 04/01/01) and soil; US$ 150,— US$ 200,— ❐ atmospheric chemistry, ozone hole; DM 300,— DM 400,— ❐ computational hydrology. Changing World EUR 150,— EUR 200,— Full Registration (after 03/31/01) 5.Nanoscale technology September 2-6, 2001 US$ 210,— US$ 260,— ❐ integrated optics, optical networks; Berlin, Germany DM 420,— DM 520,— ❐ quantum electronics and optics, EUR 210,— EUR 260,— general microwave technology; First SIAM-EMS Conference ❐ nanoscale techniques in medicine, Meals and accommodation expenses are porous ORGANIZED BY: EMS, SIAM and ZIB not covered. SIAM and EMS are trying to materials. raise special support for students and CHAIRS: Eastern European scientists. 6.Communication R. Jeltsch, G. Strang and P. 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The conference office prefers 7.Traffic PROGRAM COMMITTEE on-line registration and on-line submis- ❐ optimal periodic train scheduling, ❐ Vincenzo Capasso, Milano, Italy sion of abstracts. network planning; ❐ Peter Deuflhard, Berlin (chair) ❐ schedule synchronization; ❐ Heinz Engl, Linz, Austria TOPICS: ❐ discrete and continuous traffic flow ❐ Björn Engquist, Stockholm, Sweden 1.Medicine models; ❐ David Levermore, Maryland, USA ❐ medical imaging methods; ❐ traffic on-line simulation and con- ❐ Volker Mehrmann, Berlin ❐ computational assistence of surgery, trol, route guidance and planning; ❐ Bill Morton, Oxford, UK therapy planning; ❐ traffic assignment ❐ Stefan Müller, Leipzig, Germany ❐ hospital information systems; ❐ pharmacokinetics, tumor growth 8.Market and finance INVITED PLENARY SPEAKERS modelling; ❐ financial mathematics and statistics; ❐ Medicine: Alfio Quarteroni, I / CH ❐ artificial organs, immune system ❐ option pricing; ❐ Biotechnology: Michael Waterman, modelling; ❐ derivative trading, risk manage- USA ❐ infectuous disease control, ment; ❐ Materials Science: Jon Chapman, epidemic spreading; ❐ economic time series. 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Körnig ❐ Sigrid Wacker ❐ realistic modelling and simulation of erogenous systems, Konrad-Zuse-Zentrum Berlin (ZIB) composite materials, magnetic mate- ❐ variational principles, Takustr. 7 rial, polymers, glass, and paper; ❐ inverse problems, D-14195 Berlin-Dahlem ❐ crack propagation and further fail- ❐ stability and bifurcation analysis, Germany ure mechanisms; ❐ PDE computational finite element INTERNET: http://www.zib.de/amcw01 ❐ phase transitions, crystal growth, methods, E-MAIL: [email protected] superconductivity, and hysteresis; ❐ spatial and temporal homogeniza- ❐ control of phase transitions and tion, DEADLINES AND IMPORTANT solidification, modelling of ironmak- ❐ spatial statistics, DATES ing process; ❐ stochastic geometry, March 31 Deadline for early registration ❐ coupling of atomistic and continu- ❐ interacting particle systems, April 15 Deadline for submission of min- um models, quantum-classical ❐ stochastic analysis, isymposia proposals approximation and calculation. ❐ multiscale analysis and algorithms, May 31 Second Announcement Invited ❐ multigrid and domain decomposi- speakers and minisymposia 4.Environmental science tion, June 30 Deadline for submission ❐ climate and climate impact research, ❐ wavelets, of abstracts and posters stochastic climate modelling, inter- ❐ turbulence modelling. July 30 Program and collected abstracts mediate complexity modelling; EMS March 2001 21 CIME COURSES (b) Level sets and excursions of the Brownian sheet (5 lectures) Prof. Robert Dalang, Ecole Polytechnique Fondazione CIME Federal de Lausanne [robert.dalang @cpfl.ch] International (c) Weak convergence of set-indexed martin- gales (5 lectures) Mathematical Summer Center Prof. Gail Ivanoff, University of Ottawa (d) Set-indexed martingales and point processes (5 lectures) 2001 courses Prof. Ely Merzbach, Bar-Ilan University The Fondazione CIME will present four courses in 2001: three will be [[email protected]] (e) Stochastic geometry of spatially structured held at Martina Franca (Taranto, Italy); the other will be at Cetraro birth and growth processes: applications to poly- (Cosenza, Italy). mer crystallization processes (5 lectures) Prof. Vincenzo Capasso, Università di Information concerning these courses is presented below. Further infor- Milano [[email protected]] mation can be found on the Web server http://www.math.unifi.it/~cime/ (f) Local time and sample path properties of n-parameter processes. (5 lectures) Participants can also address their requests to: Prof. Marco Dozzi, Université de Nancy, E- Fondazione C.I.M.E. c/o Dipartimento di Matematica ‘U. Dini’ mail [email protected] }} Viale Morgagni, 67/A - 50134 Firenze, Italy tel: +39-55-434975 or +39-55-4237123 Course 3: 2-8 September in Martina Franca (Taranto): Optimal transportation fax: +39-55-434975 or +39-55-4222695 and applications e-mail: [email protected] Scientific direction: Prof. Luis Caffarelli, Direttore del CIME: Prof. Arrigo Cellina [[email protected]] University of Texas, Austin (USA) Segretario del CIME: Prof. Vincenzo Vespri [[email protected]] [[email protected]] Prof. Sandro Salsa, Politecnico di Milano, Italy [[email protected]] (a) The Monge-Ampere equation, optimal trans- Applications Lecture notes portation and periodic media (6 lectures) If you wish to attend any session you These are published as soon as possible Prof. Luis Caffarelli, University of Texas, should fill in an application to the C.I.M.E after the session. Austin [[email protected]] Foundation at the above address one (b) title to be announced (6 lectures) month before the beginning of each course Course 1: 2-10 June in Cetraro (Cosenza): Prof. L. C. Evans, University of California (not later than 20 July for courses starting Topological Fluid Mechanics Berkeley [[email protected]] in September). In your application you Scientific direction: Prof. Renzo L. Ricca (c) Shape optimization problems through the must specify your field of current research. (University College London, UK) [ricca@ Monge-Kantorovich equation (4 lectures) An important consideration in the accep- math.ucl.ac.uk] Prof.Giuseppe Buttazzo, Università di Pisa tance of applications is the scientific rele- (a) Topological methods in astrophysics (5 lec- [[email protected]] vance of the session to your field of inter- tures) (d) Geometric PDEs related to fluids and plas- est. Participation will be allowed only to Prof. Mitch A. Berger, University College mas (4 lectures) persons who have applied in due time and London (UK) [[email protected]] Prof. Yann Brenier, LAN-UPMC, 4 Place have had their application accepted. (b) Applications of knot theory (5 lectures) Jussieu, Paris VI [[email protected]] CIME will be able to provide partial sup- Prof. De Witt Sumners, Florida State (e) Mass transportation tools for dissipative port to some of the youngest participants. University (USA) [[email protected]. PDEs (4 lectures) Those who plan to apply for support fsu.edu] Prof. Cedric VILLANI, University of should mention this on the application (c) Elements of knot theory (5 lectures) Texas, Austin [[email protected]. form. Prof. Louis H. Kauffman, University of utexas.edu] Illinois at Chicago (USA) [kauffman Sites and lodging @uic.edu] Course 4: 9-15 September in Martina Martina Franca is a charming, beautifully (d) Topological methods in hydrodynamics (5 Franca (Taranto): Multiscale Problems preserved ancient city on the hills of Puglia lectures) and Methods in Numerical Simulation in southern Italy. Participants are lodged Prof. Boris Khesin, University of Toronto Scientific direction: Prof. Claudio Canuto, at the Park Hotel S. Michele; which has a (Canada) [[email protected]] Politecnico di Torino, Italy [ccanuto@ well-kept garden and a large swimming (e) Invariants and energy relaxation techniques polito.it] pool. The lectures will be at the Palazzo (5 lectures) (a) Multilevel methods in finite elements (6 lec- Ducale (City Hall), within a short walking Prof. H. Keith Moffatt, Isaac Newton tures) distance. Institute for Mathematical Sciences (UK) Prof. James H. Bramble, Texas A. & M. Cetraro is a beatiful location on the [[email protected]] University Tirrenian coast of Calabria in southern (f) From fluid knots to complex systems (5 lec- (b) Nonlinear approximation and applications Italy. The nearest train station is Paola, on tures) (6 lectures) the Roma-Salerno-Reggio line. Prof. Renzo L. Ricca, University College Prof. Albert Cohen, Laboratoire d’Analyse CIME activities are made possible thanks London (UK) [[email protected]] Numérique, Université Pierre et Marie to the generous support received from: Curie Paris VI Ministero degli Affari Esteri; The Course 2: 1-8 July in Martina Franca (c) Wavelet methods for operator equations (6 European Commission, Division XII, TMR (Taranto): Spatial Stochastic Processes lectures) Programme ‘Summer Schools’; Consiglio Scientific direction: Prof. Ely MERZBACH Prof. Wolfgang Dahmen, Institut für Nazionale delle Ricerche; Ministero (Bar-Ilan University) [merzbach@macs. Geometrie und Praktische Mathematik, dell’Università e della Ricerca Scientifica e biu.ac.il] RWTH Aachen Tecnologica; UNESCO-ROSTE, Venice (a) Mixing results for critical nearest particle (d) Variational multiscale methods, physical Office. systems via spectral gap estimates (5 lectures) models and numerical methods (6 lectures) Prof. Tom Mountford, University of Prof. Thomas J. R. Hughes, Stanford California, Los Angeles. University, USA 22 EMS March 2001 CONFERENCES Algorithmic Aspects of the Theory of Semigroups and its Applications; 4-8 June, School on Automata and FForthcomingorthcoming conferconferencesences Languages; 11-13 June, Workshop on Model Theory, Profinite Topology and Compiled by Kathleen Quinn Semigroups; 2-6 July, School on Semigroups and Applications; 9-11 July Workshop on Presentations and Please e-mail announcements of European 18-20: Third Postgraduate Group Geometry conferences, workshops and mathematical Theory Conference, Imperial College Sessions: each school consists of several meetings of interest to EMS members, to London, UK 5-hour courses held by prominent [email protected]. Announcements Theme: group theory researchers. The workshops include 50- should be written in a style similar to those Aim: the conference is aimed at post- minute invited lectures and a limited here, and sent as Microsoft Word files or as graduate students who are currently number of 20-minute talks on the spe- text files (but not as TeX input files). Space working towards obtaining a PhD in cific topics of the workshop, proposed permitting, each announcement will appear mathematics in an area that is related by the participants in detail in the next issue of the Newsletter to to group theory. It is meant to bring Languages: English and Portuguese go to press, and thereafter will be briefly students together in a less intimidating Programme and organising commit- noted in each new issue until the meeting atmosphere, and we encourage partici- tee: Gracinda M. S. Gomes (Lisbon, takes place, with a reference to the issue in pants to give a talk CAUL), Jean-Eric Pin (Paris VII, which the detailed announcement appeared Main speakers: Roger Carter CNRS), Pedro V. Silva (Porto, CMUP) (Warwick), Dan Segal (Oxford) Sponsors (confirmed): Centro de Álge- April 2001 Organising committee: Jason Fairley bra da Universidade de Lisboa, Centro and Alain Reuter de Matemática da Universidade do Sponsors: (provisional) London Porto, Centro Internacional de 2-6: Lévy Processes and Stable Laws, Mathematical Society Matemática, Fundação Calouste Coventry, UK Grants: (provisional) subsidy of regis- Gulbenkian, Fundação Luso-Americana Information: tration fee para o Desenvolvimento, Fundação Web site: Notes: participation is restricted to para a Ciência e a Tecnologia. http://science.ntu.ac.uk/msor/conf/Levy/ postgraduate students Proceedings: to be published 7-9: 16th British Topology Meeting, Deadlines: 16 March for registration Site: Observatório da Universidade de Edinburgh, UK Information: Coimbra (Almas de Freire, Coimbra, Information: e-mail: [email protected] or Portugal) Web site: ttp://www.ma.hw.ac.uk/icms/ [email protected] Grants: A limited number of scholar- current/britopol Web-site: http://www.ma.ic.ac.uk/~jtf98 ships will provide funding for those 9-12: British Mathematical pggtc2001/ participants in need of financial sup- Colloquium, Glasgow, UK port, particularly postgraduate students Information: May 2001 Deadlines: already passed Web site: www.maths.gla.ac.uk/bmc2001 Information: 15-21: Spring School in Analysis, e-mail: [email protected] Paseky nad Jizerou, Czech Republic May – July: Thematic Term on Web site: http://alf1.cii.fc.ul.pt/term2001 Information: Semigroups, Algorithms, Automata 27-2 June: Spring School in Analysis: e-mail: [email protected] and Languages, Coimbra, Portugal Function Spaces and Interpolation, Web site: http://www.karlin.mff.cuni.cz/ Theme: semigroups, algorithms, Paseky nad Jizerou, Czech Republic katedry/kma/ss/apr01/ss.htm [For details, automata and languages Information: see EMS Newsletter 38] Aim: to make Coimbra the gathering e-mail: [email protected] 17-20: EuroConference: Applications point of researchers in the subjects of Web site: http://www.karlin.mff.cuni.cz/ of the Macdonald Polynomials, semigroup theory and automata theory katedry/kma/ss/jun01/ss.htm Cambridge, UK during May, June and July, besides pro- [For details, see EMS Newsletter 38] Organisers: B. Leclerc, M. Nazarov, M. viding a basepoint for the development 28-1 June: Harmonic morphisms and Noumi, J.-Y. Thibon of joint research projects harmonic maps, Marseille, France Topics: Macdonald polynomials and Topics: algorithmic aspects of the theo- Information: related special functions in mathemati- ry of semigroups and its applications, e-mail: [email protected] cal physics, harmonic analysis, and rep- automata and languages, model theory, Web-site: http://beltrami.sc.unica.it/ resentation theory. In particular: clas- profinite topology and semigroups, harmor/ sical and quantum many-body prob- semigroups and applications, presenta- [For details, see EMS Newsletter 38] lems, integrable models in statistical tions and geometry physics and quantum field theory, Main speakers: J. Almeida (Porto), C. June 2001 quantum symmetric spaces, deforma- Choffrut (Paris VII), J. Fountain (York), tions of Virasoro and W-algebras, quan- S. Margolis (Bar-Ilan), L. Ribes tum Knizhnik-Zamolodchikov equations (Carleton), M. Sapir (Vanderbilt), M. 4-8: Conference on Algebraic Speakers: H. Awata, J.-F. van Diejen, Volkov (Ekaterinburg), T. Wilke (Kiel), Topology, Gdansk, Poland C. Dunkl, J. Faraut, P. Forrester, K. M. Branco (Lisbon), V. Bruyère (Mons), Theme: algebraic topology Hasegawa, G. Heckman, N. Jing, T. O. Carton (Marne-la-Vallée), A. Restivo Topics: homotopy theory, topology of Koornwinder, K. Mimachi, T. Miwa, S. (Palermo), T. Coulbois (Paris VII), H. manifolds, geometric group theory, Odake, E. Opdam, E. Rains, S. Straubing (Boston College), P. Trotter algebraic K-theory and homotopy theo- Ruijsenaars, J. Shiraishi, J. Stokman, V. (Tasmania), P. Weil (Bordeaux), K. ry in algebraic geometry Tarasov. Auinger (Vienna), M. Lawson (Bangor), Main speakers: Gregory Arone Site: Isaac Newton Institute for W. D. Munn (Glasgow), A. Pereira do (University of Aberdeen), Matthew Mathematical Sciences Lago (São Paulo), R. Gilman (Stevens Ando (University of Illinois at Urbana- Deadline: for receipt of applications, Inst. of Tech.), D. McAlister (DeKalb), Champaign), Mladen Bestvina already passed J. Meakin (Lincoln), S. Pride (Glasgow), (University of Utah), Alexander Information: N. Ruskuc (St. Andrews), B. Steinberg Dranishnikov (University of Florida), Web site: http://www.newton.cam.ac.uk/ (Porto) Nicholas Kuhn (University of Virginia), programs/sfmw02.html Programme: 2-11 May, School on Jack Morava (Johns Hopkins EMS March 2001 23 CONFERENCES University), Fabien Morel (Universite tions, economic models, stochastic mod- Theme: function theory, scientific com- de Paris VII,) Teimuraz Pirashvili elling and data analysis, variational putation, complex variables (including (Georgian Academy of Sciences) model for non-linear problems, mathe- generalisations like quaternions, etc.), Organising committee: Stanislaw matical models in biology approximation theory and numerical Betley (Warszawa), Tadeusz Kozniewski Languages: English and Russian analysis (Warszawa), Stefan Jackowski Call for papers: an abstract of a com- Main speakers: · Alan Beardon (UK), (Warszawa), Adam Przezdziecki plete typewritten page should include Percy Deift (USA), Richard Delanghe (Warszawa), Andrzej Szczepanski the title of the paper, names of all (Belgium), Toby Driscoll (USA), (Gdansk) authors, complete affiliations, address- Alexandre Eremenko (USA), Klaus Deadlines: for registration, 15 April es, e-mail. A collection of abstracts will Gürlebeck (Germany), Doron S. Information: be printed before the conference. The Lubinsky (South Africa), Andrew e-mail: [email protected] participants must send their abstracts (2 Odlyzko (USA), Lawrence Zalcman Web site: www.mimuw.edu.pl/~cat01 copies) in camera-ready format to the (Israel) 4-9: Fractals in Graz 2001, Analysis- Organising Committee by normal mail. Organising committee: Helmuth R. Dynamics-Geometry-Stochastics, Graz, We do NOT use electronic versions of Malonek (Aveiro), Nicolas Papamichael Austria abstracts (Cyprus), Stephan Ruscheweyh, Information: Programme committee: S. M. Ustinov, (Würzburg), Edward B. Saff, (South e-mail: [email protected] P. A. Jilin, G. G. Malinechkii, V. M. Florida) Web site: http://finanz.math.tu-graz. Ivanov, G. Yu Riznichenko, A. G. Sponsors: Fundação para a Ciência e a ac.at/~fractal/ Chenchov, V. N. Kozlov, A. V. Tecnologia, Sociedade Portuguesa de [For details, see EMS Newsletter 38] Zinkovskii, A. F. Andreev, Yu. N. Matemática, COSTED, The British 18-22: Fourth European Conference Bibikov, S. Yu Piliugin, N. N. Petrov, E. Council, Universidade de Aveiro UI&D on Elliptic and Parabolic Problems: N. Rozenvasser, I. V. Miroshnik, A. V. ‘Matemática e Aplicações’ (as of 31 Theory, Rolduc, Netherlands Timofeev, V. N. Fomin, N. E. October 2000) Information: Barabanov, Yu.V. Churin, E. K. Ershov, Proceedings: as with past CMFT meet- e-mail: [email protected], A. S. Matveev, V. B. Smirnova, A. I. ings, the conference proceedings of [email protected] Shipiliavui, A. V. Flegontov (all Russia) CMFT2001 will appear in a carefully Web site: http://www.math.unizh.ch/ Organising committee: conference refereed volume distributed by a noted rolducgaeta chair, George Osipenko, St Petersburg mathematics publisher. The style-file [For details, see EMS Newsletter 38] Technical Univ.; Valentin Zaitsev, cmft.sty and the description-file cmft- 18-23: Tools for mathematical model- Russian State Pedagogical Univ.; manual.ps will be available for down- ling, St. Petersburg, Russia Dmitrii Arseniev, Institute of loading from the homepage of CMFT Theme: mathematical modelling International Education Program. 2001 by 1 April. Both files are also Topics: mathematical modelling, Secretary, Lidiya Linchuk, St available from the local organizing applied mathematics, computer alge- Petersburg State Technical Univ. committee upon request bra, methods of approximation, dis- Sponsors: Russian foundation for Basic Site: University of Aveiro, Campus de cretisation and computation, design Research, Laboratory of Nonlinear Santiago, Aveiro techniques, numerical methods, parallel Analysis, St Petersburg Technical Univ., Deadlines: for final registration, pay- and distributed algorithms, computer Editorial Board of the Electronic ment of conference fee (US$150) modelling in dynamical systems, shad- Journal ‘Differential Equations and already passed; for payment of late con- owing, adaptive methods and integral Control Processes’, Institute of ference fee (US$180), 25 June; for sub- equations, mathematical models in biol- International Education Programs mission of papers for the proceedings, ogy, medicine etc., applications to Proceedings: it is suggested that refer- 15 October physics, electrotechniques and electron- eed Conference proceedings including Information: contact: H. R. Malonek, ics, dynamic economic models, general full versions of selected papers accepted Departamento de Matematica macro-economic models, market mod- for presentation will be published Universidade de Aveiro, Portugal els Site: St Petersburg State Technical tel./fax: +351-234-370359 /+351-234- Main speakers: N. Ajabyan (Armenia), University 382014 V. Aladjev (Estonia), A. Alawneh Deadlines: for abstracts and registra- e-mail: [email protected] (Jordan), M. Alrefaei (Jordan), M. tion forms, 15 April Web site: http://event.ua.pt/cmft2001/ Andramonov (Russia), I. Andrianov Information: contact Lidiya Linchuk, (Germany), V. Bakhrushin (Ukraine), V. MATHTOOLS’2001, Dept. of July 2001 Chiroiu (Romania), A. B. Dragut Mathematics, State Technical (Romania), J. Esquivel-Avila (Mexico), University, Polytechnicheskaya st., 29, S. Gavryushin (Russia), V. Ivanov St Petersburg, 195251, Russia, 1-6: Eighteenth British Combinatorial (Russia), H. Ji-Huan (China), D. fax: +7-812-5343314, +7-812-5527770. Conference, Brighton, United Khusainov (Ukraine), M. Kulshrestha e-mail: [email protected] Kingdom (India), M. Lopez Garcia (Spain), A. Web page: http://www.neva.ru/journal Information: Markov (Russia), R. Meyer-Spasche 19-22: Computational Intelligence, e-mail: [email protected] (Germany), A. Mikitinskiy (Russia), A. Methods and Applications (CIMA Web sites: Prykarpatsky (Poland), K. Pucher 2001), Bangor, UK http://www.maths.susx.ac.uk/Staff/JWPH (Austria), J. Purczynski (Poland), G. K. Information: / http://hnadel.maps.susx.ac.uk/TAGG/ Raju (India), A. Rasulov (Uzbekistan), e-mail: [email protected]; operat- Confs/BCC/index.html R. Ravindran (India), M. Razzaghi [email protected]; [For details, see EMS Newsletter 36] (USA), G. Sadovoi (Russia), V. Samovol [email protected] 3-7: Barcelona 2001 EuroPhD (Russia), M. Schahz (German), M. Web site: http://www.icsc.ab.ca/cima2001. Topology Conference, Bellaterra, Shamolin (Russia), M. Stojanovic htm Barcelona (Yugoslavia), A. Stulov (Estonia), M. [For details, see EMS Newsletter 37] Invited speakers: O. Cornea Veljovic (Yugoslavia), V. Vladimirov 25-28: Banach Algebras and (Université de Lille), N. P. Strickland (Ukraine), R. Volosov (Russia), M. Cohomology, Newcastle, UK (University of Sheffield), P. Salvatore Yunusi (Tajikistan) Information: (Università degli Studi di Roma), D. J. Sessions: equations of mathematical Web site: http://www.ncl.ac.uk/icbacn Green (Universität Wuppertal), S. physics, theory of control processes, 25-29: Cmft2001, Computational Schwede (Universität Bielefeld), S. group analysis of differential equations, Methods and Function Theory, Aveiro, Whitehouse (Université d’Artois), W. statistic modelling and integral equa- Portugal Chachólski (Yale University), P. Turner 24 EMS March 2001 CONFERENCES (Heriot-Watt University) Speakers: Christophe Breuil A. Dow (Charlotte, NC), I. Farah Scientific committee: Antonio Viruel (Université Paris-Sud), p-adic Hodge the- (Staten Island, NY), D. Fremlin (Essex, (Conference Coordinator) (Universidad ory, deformations and local Langlands, Bas UK), P. M. Gartside (Oxford, UK), G. de Málaga), Jérôme Scherer (Université Edixhoven (Université de Rennes 1), Godefroy (Paris), G. Gruenhage de Lausanne), Natàlia Castellana Modular forms, Galois representations and (Auburn), N. Hindman (Washington, (University of Aberdeen) local Langlands D.C.), I. Juhász (Budapest), H.-P. Site: Centre de Recerca Matemàtica Coordinators: Xavier Xarles, Adolfo Kuenzi (Cape Town), K. Kunen Information: Quirós (Madison, WI), D. Lutzer Web site: http://www.crm.es/b2001etc Site: Centre de Recerca Matemàtica, (Williamsburg, VA), W. M. Marciszewski 4-6: MathFIT workshop: The Information: (Warsaw), M. Megrelishvili (Ramat- Representation and Management of Web site: http://www.crm.es/tn2001 Gan), J. van Mill (Amsterdam), P. Uncertainty in Geometric 23-27: 20th IFIP TC 7 Conference on Nyikos (Columbia, SC), J. Pelant Computations, Sheffield, UK System Modelling and Optimization, (Prague), V. Pestov (Wellington), M. Information: Trier, Germany Reed (Oxford, UK), D. Repovs Web site: Information: (Ljubljana), M. E. Rudin (Madison, WI), http://www.shef.ac.uk/~geom2001/ e-mail: [email protected] D. Shakhmatov (Matsuyama), L. 5-7: British Congress of Mathematics Web site: http://ifip2001.uni-trier.de Shapiro (Moscow), S. Solecki Education, Keele, UK [For details, see EMS Newsletter 38] (Bloomington, IN), M. Tkachenko Information: (Mexico), V. Tkachuk (Mexico), S. Web site: http://www.bcme.org.uk August 2001 Todorcevic (Belgrade and Paris), V. V. 8-13: Second Workshop on Algebraic Uspenskiy (Athens, OH), S. Watson Graph Theory, Edinburgh, Scotland (York), Y. Yajima (Yokohama) Information: 5-18: BALTICON 2001, Banach Programme committee: Simon e-mail: [email protected] Algebra Theory in Context, Krogerup Donaldson (representative of IMU), B. Web site: http://www.ma.hw.ac.uk/icms/ Hojskole, Humlebaek, Denmark Balcar, M. Husek, J. Pelant, P. Simon, current/graph/index.html Information: V. Trnková [For details, see EMS Newsletter 38] e-mail: [email protected] Organising committee: B. Balcar, M. 9-22: European summer school Web site: Husek (Chairman), J. Pelant, P. Simon, ‘Asymptotic combinatorics with appli- http://www.math.ku.dk/conf/balti- V. Trnková, P. Holický, O. Kalenda, A. cation to mathematical physics’, St con2001/ Klíc, J. Coufal, R. Rohácková Petersburg, Russia [For details, see EMS Newsletter 38] Sponsors: International Mathematical Information: 5-18: Groups St Andrews 2001, Union, Mathematical Institute of Czech e-mail: [email protected] Oxford, UK Academy of Sciences, Department of Web site: www.pdmi.ras.ru/EIMI/2001/ Information: Groups St Andrews 2001, Mathematics of Prague Institute of emschool/index.html Mathematical Institute, North Haugh, Chemical Technology, Department of [For details, see EMS Newsletter 38] St Andrews, Fife KY16 9SS, Scotland Mathematics of Economical University 10-15: Advanced Course on e-mail: [email protected] and Faculty of Mathematics and Physics Symplectic Geometry of Integrable Web site: http://www.bath.ac.uk/~masgcs/ of Charles University. Hamiltonian Systems, Bellaterra, gps01/ Deadlines: for abstracts and registra- Barcelona [For details, see EMS Newsletter 36] tion, 31 May Speakers: Michèle Audin (Strasbourg), 12-18: 39th International Symposium Information: Lagrangian and special lagrangian sub- on Functional Equations, Denmark Web site: http://www.karlin.mff.cuni.cz/ manifolds, Ana Cannas da Silva (Lisboa), Organising committee: Peter Friis and ~toposym/ Symplectic toric manifolds, Eugene Henrik Stetkaer (Aarhus University) e-mail: [email protected] Lerman (Illinois), Symmetries of symplectic Information: 24-30: 10th International Meeting of and contact manifolds e-mail: [email protected] European Women in Mathematics, Coordinators: Carlos Curràs-Bosch, Web site: http://www.mi.aau.dk/isfe39/ Malta Eva Miranda 12-19: Summer School 2001 Information: contact Dr. Tsou Sheung Site: Centre de Recerca Matemàtica Homological Conjectures for Finite Tsun (EWM01), Mathematical Institute, Information: Dimensional Algebras, Nordfjordeid, 24-29 St. Giles, Oxford OX1 3LB, UK, Web site: http://www.crm.es/sgihs Norway fax: +44-1865-273583 15-20: Algorithms for Approximation Information: contact Øyvind Solberg, Web site: IV International Symposium, ([email protected], NTNU, http://www.maths.ox.ac.uk/~ewm01/ Huddersfield, UK Trondheim) 27-31: Equadiff 10, Czechoslovak Information: Web sites: http://www.mathematik.uni- International Conference on e-mail: [email protected] bielefeld.de/~sek/summerseries.html Differential Equations and their Web-site: http://www.math.ntnu.no/~oyvinso/ Applications, Prague, Czech Republic Http://Helios.Hud.Ac.Uk/A4a4/ Nordfjordeid/ Information: [For details, see EMS Newsletter 38] [For details, see EMS Newsletter 38] e-mail: [email protected] 17-23: Advanced Course on Global 19-25: 9th Prague Topological Web site: www.math.cas.cz/~equadiff/ Riemannian Geometry: Curvature and Symposium, International Conference [For details, see EMS Newsletter 38] Topology on General Topology and its Relations Speakers: S. Markvorsen (Denmark), to Analysis and Algebra, Prague, September 2001 Distance geometric analysis on Riemannian Czech Republic manifolds, M. Min-Oo (McMaster Topics: topology (mainly set-theoretical University), K-area, mass and assymptotic topology, continua, descriptive topolo- 2-6: First SIAM-EMS Conference, geometry gy, categorical topology, geometrical Berlin, Germany Coordinator: Vicent Palmer topology), topological groups and semi- Topic: Applied Mathematics in Our Site: Universitat Jaume I, Castelló de la groups, topology of Banach spaces, Changing World Plana topology and computer science, topo- Information: contact Peter Deuflhard Information: logical dynamics (Chair of the Local Organizing Web site: http://www.crm.es/geom2001 Main speakers: A. V. Arhangelskii Committee), SIAM/EMS 2001 18-28: Advanced Course on Modular (Moscow and Athens, OH), Z. Balogh Conference Office c/o Konrad-Zuse- Forms and p-adic Hodge Theory, (Oxford, OH), M. Bell (Winnipeg), A. Zentrum Berlin (ZIB), Takustr. 7, Bellaterra, Barcelona Bouziad (Rouen), D. Dikranjan (Udine), 14195 Berlin-Dahlem, Germany EMS March 2001 25 CONFERENCES fax: +49 (30) 841 85-107 Local organisers: Julio Rubio, Main speakers: Drazen Adamovic e-mail: [email protected] Departamento de Matemáticas y (Croatia), Anne-Marie Aubert (France), Web site: http://www.zib.de/amcw01/ Computación, Universidad de La Rioja, Damir Bakic (Croatia), Chongying 3-8: Sixth International Conference on Edificio Vives, Calle Luis de Ulloa s/n, Dong (USA), Detlef Gronau (Austria), Function Spaces, Wroclaw, Poland E-26004 Logroño (La Rioja, España) Boris Guljas (Croatia), Jorgen Topics: the topics are connected with Site: Ezcaray (La Rioja, Spain) Hoffmann-Jorgensen (Denmark), Jing- functional analysis, e.g. operator theo- Note: this is a Spanish forum with Song Huang (China), David R. Larson ry, interpolation, geometry, topology, international participation (USA), Bojan Magajna (Slovenia), Allen approximation in function spaces. Information: Moy (USA), Goran Muic (Croatia), Programme: plenary lectures (about 45 e-mail: [email protected] Pavle Pandzic (Croatia), Manos minutes) and short communications Web site: http://www.unirioja.es/dptos/ Papadakis (USA), Goran Peskir (not more than 20 minutes) dmc/eaca2001/ (Denmark), Gilles Pisier (France), Organiser: Institute of Mathematics, 12-15: EuroConference on Zoran R. Pop-Stojanovic (USA), Mirko Wroclaw University of Technology Combinatorics, Graph Theory and Primc (Croatia), Murali Rao (USA), Organising committee: M. Burnecki Applications, Bellaterra, Barcelona Ludwig Reich (Austria), David Renard (secretary), R. Grzaslewicz (chairman), Invited speakers: Noga Alon (Tel Aviv (France), Paul J. Sally (USA), Boris H. Hudzik, J. Musielak, Cz. Ryll- University), Peter Cameron (Queen Sirola (Croatia), Renming Song (USA), Nardzewski Mary and Westfield College), David David Soudry (Israel), Marko Tadic Site: Institute of Mathematics, Wroclaw Epstein (University of California), (Croatia), Mitchell H. Taibleson (USA), University of Technology Levon Khachatrian (University of David A. Vogan (USA), Guido Weiss Information: Bielefeld), Jiri Matousek (Charles (USA) Web site: www.im.pwr.wroc.pl/~fsp/ University), Kevin Phelps (Auburn Programme: series of lectures, single e-mail: [email protected] University), Vera T. Sós (Alfréd Rényi lectures and a limited number of short 5-7: LMS Workshop on Domain Institute of Mathematics), Carsten communications Decomposition Methods in Fluid Thomassen (Technical University of Organising committee: Hrvoje Mechanics, Greenwich, UK Denmark), Nick Wormald (University of Kraljevic (Croatia), Davor Butkovic Aim: to disseminate amongst graduate Melbourne) (Croatia), Murali Rao (USA), Damir students, researchers and industrialists Local organising committee: Gabor Bakic (Croatia), Pavle Pandzic the state-of-the-art domain decomposi- Lugosi (Universitat Pompeu), Fabra (Croatia), Hrvoje Sikic (Croatia), Zoran tion methods applied to various engi- Conrado Martínez (Universitat Vondracek (Croatia) neering problems related to fluid Politècnica de Catalunya), Marc Noy Site: Inter-University Center, mechanics. The workshop may be (Universitat Politècnica de Catalunya), Dubrovnik, Croatia served as a short intense course for Josep Rifà (Universitat Autònoma de Deadlines: to register and submit graduate students who will be exposed Barcelona), Oriol Serra (local abstracts, 1 April 2001 to review and overview lectures and cur- Chairman) (Universitat Politècnica de Information: rent research interests and results Catalunya) e-mail: [email protected] Inivited speakers: C. Bailey Site: Centre de Recerca Matemàtica Web page: http://www.math.hr/ (Greenwich), X.-C. Cai (Boulder), M. Information: ~congress/Dubrovnik01/ Garbey (Lyon), I. G. Graham (Bath), G. Web site: http://www.crm.es/comb01 18-22: Euro Summer School on Proper Lube (Göttingen), F. Nataf (Paris) 14-18: International Conference: Group Actions, Bellaterra, Barcelona Organisers: C.-H. Lai, M. Cross, K. A. Function Spaces, Proximities and Speakers: Guido Mislin (ETH Zürich), Pericleous, A. K. Parrott Quasi-uniformities, Italia, Caserta Equivariant K-homology of the classifying Grants: some scholarships are available [on the occasion of Som Naimpally’s 70 space for proper actions, Alain Valette to graduate students with priority given birthday] (Université de Neuchâtel), Topological to those from UK mathematics depart- Information: K-theory of group C*-algebras ments e-mail: [email protected] Coordinator: Carles Casacuberta Information: Web site: http://www.unina2.it/ Site: Centre de Recerca Matemàtica Web site: http://cms1.gre.ac.uk/ topological.sun/homesun.html Information: conferences 16-22: Conference of the Austrian Web site: http://www.crm.es/group- 12-14: Seventh Meeting on Computer Mathematical Society and the German actions Algebra and Applications (EACA-2001) Mathematical Society, Vienna, Austria 24-28: Fourth European Conference Topics: effective methods in algebra, [please note corrected dates] on Elliptic and Parabolic Problems: analysis, geometry and topology; algo- Plenary speakers: V. Capasso (Milano), Applications, Gaeta, Italy rithmic complexity; scientific computa- M. H. A. Davis (London), I. Ekeland Information: tion via symbolic- numerical methods; (Paris), H. P. Schlickewei (Marburg), M. e-mail: [email protected], development of symbolic-numerical Kreck (Heidelberg), N. J. Mauser [email protected] software; analysis, specification, design (Vienna), V. L. Popov (Moskau), T. Web site: http://www.math.unizh.ch/ and implementation of symbolic com- Ratiu (California), D. Salamon (Zürich), rolducgaeta putation systems; applications to sci- G. Teschl (Vienna), J.-C. Yoccoz (Paris), [For details, see EMS Newsletter 38] ence and technology D. Zagier (Bonn), G. M. Ziegler (Berlin) 24-28: III International Conference, Speakers: Hoon Hong (North Carolina Local organisation: Karl Sigmund Analytic and Probabilistic Methods in State University), Jacques Calmet (University of Vienna) Number Theory, Palanga, Lithuania (Universität Karlsruhe), Marc Chardin Information: Karl Sigmund, University [in honour of J. Kubilius] (Institut de Mathématiques, CNRS, of Vienna, Institute of Mathematics, Theme: analytic and probabilistic meth- Paris), Eladio Domínguez (Universidad Strudlhofgasse 4, 1090 Vienna; ods in number theory de Zaragoza), y Luis Español tel: +43 1 4277 506 02 or 506 12, fax: Organizing committee: B. Juodka (Universidad de La Rioja), Laureano +43 1 4277 9506 (Chairman), A. Laurincikas González-Vega (Univesidad de e-mail: [email protected] (Programme Director), E. Manstavicius, Cantabria), y Rafael Sendra Web site: http://www.mat.univie.ac.at/ (Vice-chairman), V. Stakenas (Vice- (Universidad de Alcalá) ~oemg/Tagungen/2001/index.html chairman) Programme: invited speakers will deliv- 17-26: Functional Analysis VII, Proceedings: to be published er one-hour lectures on current and Dubrovnik, Croatia Information: future trends in their research fields. A Topics: operator algebras, probability e-mail: [email protected] number of sessions for presentation of theory, harmonic analysis, representa- Web site: http://www.mif.vu.lt/ttsk/palan- 30-minute talks are planned tion theory ga.htm 26 EMS March 2001 RECENT BOOKS Fractal Geometry and Stochastics II, Progress in Probability 46, Birkhäuser, Basel, 2000, 292 pp., ISBN 3-7643-6215-4 This book includes the main contribu- RecentRecent booksbooks tions to the second conference on fractal edited by Ivan Netuka and Vladimír Sou³ek geometry and stochastics, held at Greifswald, Germany in 1998. The con- ference and its proceedings reflect recent rapid and interesting research develop- ments on the boundary between fractal geometry (of objects that show consider- able irregularities of magnitude) and probability theory. The topics covered are fractal dimen- sions of measures and sets, self-similar, self-affine sets and measures generated by iterated function systems, constructions of random fractals via contractions, random coverings, etc., dynamical systems and fractals, and harmonic analysis on frac- tals. The contributions range in style from survey papers to standard mathe- matical articles and an open problems collection. (jstep)

A. J. Berrick and M. E. Keating, An Introduction to Rings and Modules with K-theory in View, Cambridge Studies in Advanced Mathematics 65, Cambridge University Press, Cambridge, 2000, 265 pp., Books submtted for review should be sent to the chapter on buildings, BN-pairs and Tits £35, ISBN 0-521-63274-9 following address: systems. This is an introductory text for beginning Ivan Netuka, MÚUK, Sokolovská 83, 186 75 The exposition is often quite terse and graduate students who intend to move on Praha 8, Czech Republic. the development of structures outside the to study algebraic K-theory. The funda- group theory is done only to a minimal mentals of ring and module theory are M. Aschbacher, Finite Group Theory, extent, which makes applications within developed from scratch, accompanied by Cambridge Studies in Advanced Mathematics group theory possible. The purpose of numerous examples and exercises. 10, Cambridge University Press, Cambridge, the book was to lay foundations for the After presenting the basics, the authors 2000, 304 pp., £19.95, ISBN 0-521-78145- proof of the CFSG, and the book finishes devote a chapter to direct-sum decompo- 0 and 0-521-78675-4 with a chapter on finite simple groups, sitions and short exact sequences (includ- The second edition of Aschbacher’s Finite including an introduction to CFSG and to ing a section on IBN-rings). The follow- Group Theory differs from the first edition its proof. (ad) ing two chapters on (non-commutative) in three aspects. The first edition con- noetherian and artinian rings include the tained many misprints: using it for an G. Bachman, L. Narici and E. classical Wedderburn-Artin theory. In occasional reference I was aware of about Beckenstein, Fourier and Wavelet this part, an interesting construction is twenty misprints in those parts I read with Analysis, Universitext Springer, New York, presented of a stably free non-free right greater care (some sixty pages). Of these 2000, 505 pp., DM119, ISBN 0-387- ideal over the two-variable polynomial twenty misprints all but two have disap- 98899-8 ring over any skew-field which is not a peared, and the second edition seems This excellent book is intended as an field. This is in sharp contrast with the much more reliable. Some of the exercis- introduction to classical Fourier analysis, validity of Serre’s conjecture which is well es are quite difficult. The second edition Fourier series, Fourier transforms and known in the field case. has an additional thirty pages devoted to wavelets, for students in mathematics, The final two chapters deal with solutions of the more important and physics and engineering. The text Dedekind domains and modules over more difficult exercises. The author includes many historical notes to place them. The class group is introduced and found his extension of Bender’s proof of the material in a cultural and mathemati- the primary decomposition theorem is the solvable signaliser functor theorem to cal context. The topics are developed proved for finitely generated torsion odd primes too complicated, and he slowly for a reader who has never seen modules. Besides general theorems, decided to return to Bender’s original them before, with a preference for clarity explicit computations are also presented; result which covers only abelian 2-sub- of exposition in stating and proving for example, in Section 5.3, the factorisa- groups. The new proof is not much short- results. tion type of pO is completely determined er, but is presented in more detail and is The first three chapters present requi- when p is a prime and O is the ring of easier to follow. site background material, and these could integers in the quadratic extension Q(√d). The book covers most of the standard be read as a short course in functional The class group of Q(√d) is explicitly com- basic material (series, semi-direct, central analysis. The standard features of puted for small values of d. (jtrl) and wreath products, linear representa- Fourier analysis – Fourier series, Fourier tions and characters, extensions, presen- transform, Fourier sine and cosine trans- A. J. Berrick and M. E. Keating, tations, classical groups, Coxeter groups forms – are developed in Chapters 4 and Categories and Modules with K-theory in and root systems, extremal p-groups, 5. More recent developments, such as the View, Cambridge Studies in Advanced transfer), and emphasises themes that are discrete and fast Fourier transforms and Mathematics 67, Cambridge University Press, relevant to the Classification of Finite wavelets, are covered in the last two chap- Cambridge, 2000, 361 pp., £35, ISBN 0- Simple Groups (e.g., extra-special ters. Each section concludes with a set of 521-63276-5 groups). The more advanced topics exercises with hints for many of these. This book is an introduction to category include Thompson factorisation, fusion, (knaj) theory, with particular emphasis on mod- central extensions, and various theorems ule categories. The exposition starts with on complements. The book also has a C. Bandt, S. Graf and M. Zähle (eds.), basic material on categories, functors and EMS March 2001 27 RECENT BOOKS natural transformations; this includes The authors offer convexity as a tool there are three appendices with a bibliog- first facts on functors, categories and uni- which allows for transition from classical raphy and two indices. versal objects. Next, the authors deal to modern analysis, from smooth to non- The book is nicely written and provides with additive and abelian categories, and smooth problems, from linear to non-lin- an important source for university stu- introduce the Grothendieck group of a G- ear optimisation, and from functions to dents interested in mathematics and its exact category. There is also a chapter on multi-functions. They present a wide history. (mnem) the tensor product of modules and the spectrum of results on convex analysis in change of ring functors. Extn and Torn finite-dimensional spaces, their applica- E. B. Burger and M. Starbird, The Heart are introduced (with homological details tions and some extensions. The well- of Mathematics. An Invitation to Effective left as exercises). Chapter 4 deals with a known results on duality theory and the Thinking, Springer, New York, 2000, 646 fundamental application of category the- classical necessary and sufficient condi- pp., DM130, ISBN 1-55953-407-9 ory to modules: the Morita equivalence tions for non-linear optimisation are This book is a wonderful and fascinating theory. Another important construction, included in a succinct way and their non- excursion into the realm of mathematics the direct and inverse limits, is explained smooth versions are developed. The and the world of creative and effective in Chapter 5. The next chapter deals roles of polyhedral sets and functions, of thinking. The authors explain the math- with the localisation of rings and mod- fixed-point theorems, and of variational ematical ideas and results in such a way ules. Localisation is first treated in terms inequalities are explained. The lucid dis- that the reader can discover the basic of rings and modules of fractions, and cussion of the place of the assumed finite- strategies of thought and use them in real then more generally, as a quotient functor dimensionality indicates possible general- life. In spite of this, the excursion is with the universal property. The final isations. Eigenvalue optimisation is an made as pleasant as possible, with many chapter deals with completions and with example of a less frequent type of appli- references to history, philosophy and the local-global methods. Particular atten- cation. Algorithms for solving optimisa- fine arts, without a lot of computation. tion is paid to applications to lattices over tion problems are not the subject of the The book can be read by teachers, stu- Dedekind domains. The rings and mod- book. dents and anyone who is not afraid of ules of adèles are introduced, and projec- The text consists of short sections, each thinking and who wants to discover what tive modules over lattices are charac- accompanied by an extensive set of exer- mathematics really is. (ec) terised. The book may be viewed as a cises marked into three categories: easy continuation of Vol. 65 (see above), but it illustrative examples, more advanced R. P. Burn, Numbers and Functions. Steps is of independent interest as well. (jtrl) examples and additional theory, demand- into Analysis, Cambridge University Press, ing examples and peripheral theory. The Cambridge, 2000, 356 pp., £19.95, ISBN 0- F. Blanchard, A. Maass and A. Nogueira style is not strictly formal, with definitions 521-78836-6 (eds.), Topics in Symbolic Dynamics and included in the text without numbering, The main purpose of this textbook is to Applications, London Mathematical Society and with proofs and technical details help students with the transition from Lecture Note Series 279, Cambridge mostly omitted or left as exercises. (The high-school calculus to mathematical University Press, Cambridge, 2000, 245 pp., overall ratio is 45:55 in favour of exercis- analysis as it is taught at universities. £24.95, ISBN 0-521-79660-1 es.) Through several hundreds of exercises, Symbolic dynamics studies iterations of This is an up-to-date, well-written students are led from practical examples continuous selfmaps of spaces of symbolic advanced text which aims at a broad to the theoretical foundations of rigorous spaces. These spaces are equipped with mathematical audience. The general analysis. Solutions of all the exercises are the Cantor (product) topology and are requirements correspond to a master’s given at the end of each chapter. homeomorphic to subspaces of the level in pure or applied mathematics, and The topics presented in this book Cantor middle-third set. Symbolic selected sections and exercises may serve include mathematical induction, dynamics originated in the study of geo- as an underlying text for advanced mas- sequences and series of real numbers, desics on surfaces with negative curva- ters’ courses on optimisation. With a functions, differentiation and integration ture. It is used for coding smooth trans- detailed list of named results, the book and sequences of functions. Since the formations and has important connec- can be used as a compendium and a book is written in a very comprehensible tions with computer science. source of reference. (jdup) but exact way, it may serve as an excellent This volume contains the courses given guide through the basic course of mathe- at the CIMPA-UNESCO Summer School C.-P. Bruter, La construction des nom- matical analysis at university. Unlike on Symbolic Dynamics and its bres, Histoire et épistémologie, Ellipses, standard textbooks, most of the contents Applications held in Temuco, Chile, in Paris, 2000, 237 pp., ISBN 2-7298-7959-5 are presented in the exercises, which pro- January 1997. The volume is divided into This interesting book is a combination of vide the student with enough motivation eight chapters, each corresponding to a mathematics and its history with history. for the definitions and theorems. Each course given at Temuco and devoted to The author carefully describes the long chapter contains a summary of the pre- an active area of symbolic dynamics or history of mathematical thinking on the ceding theory and a few historical notes. applications to ergodic theory or number creation and extension of the concept of a Each item in the bibliography is com- theory. The authors and topics are V. number, up to the nineteenth century. mented on in a few words, so that even a Berthé, Sequences of low complexity: auto- The author prefers to present the most beginner can easily find his/her way to the matic and Sturmian sequences, B. Host, significant problem in the history of num- appropriate materials. (jvy) (= J. Vybiral) Substitution subshifts and Bratteli diagrams, bers so that the reader can appreciate the M. Boyle, Algebraic aspects of symbolic history of mathematics at first hand. This N. L. Carothers, Real Analysis, Cambridge dynamics, B. Kitchens, Dynamics of Zd book helps the reader to understand the University Press, Cambridge, 2000, 401 pp., actions on Markov subgroups, Z. Coelho, reason for the creation of the positive £19.95, ISBN 0-521-49749-3 and 0-521- Asymptotic laws for symbolic dynamical sys- integers, integers, rational and irrational 49756-6 tems, V. Bergelson, Ergodic theory and numbers, complex numbers, quaternions, This book presents a course on real analy- Diophantine problems, Ch. Frougny, Number octonions, and the so-called Clifford sis. It consists of three parts: metric (and representation and finite automata, R. numbers (elements of Clifford algebras). normed) spaces, function spaces, and Labarca, A note on the topological classifica- For each type of number, the author Lebesgue measure and integration (on tion of Lorenz maps on the interval. (pkur) explains how they were discovered and the real line). It does not suppose constructed, their symbolic and geometri- detailed knowledge of a large amount of J. M. Borwein and A. S. Lewis, Convex cal representations, their properties and material: a basic course of analysis Analysis and Nonlinear Optimization. uses in geometry and physics. The last (including deeper facts such as the Theory and Examples, CMS Books in chapter contains a philosophical answer Bolzano-Weierstrass theorem, complete- Mathematics 3, Springer, New York, 2000, to the question of what the word ‘number’ ness, a certain familiarity with ε-δ-tech- 273 pp., DM139, ISBN 0-387-98940-4 really means. At the end of the book, niques, the Riemann integral, etc.) is suf- 28 EMS March 2001 RECENT BOOKS ficient to read the book. The exposition resource constraints. Chapter 2 is the His result was preceded by a systematic is clear, easily understandable, and writ- counterpart of Chapter 1 when taking investigation of a (partial) monoid which ten in a way accessible to non-specialist communication costs into account. is generated by such transformations of students. Good examples treated in suffi- Chapter 3 is devoted to cyclic scheduling, binary terms that correspond to an LD- cient depth (Cantor function, Peano which constitutes a transition from task expansion at a certain term position. The curve, nowhere-differentiable continuous graph scheduling to loop nest scheduling, identities that are fulfilled within this functions) and a great number of exercis- the main subject of this book. Chapter 4 monoid led Dehornoy to consider a con- es (484, 231 and 280 in the three above- provides mathematical foundations for nection with the braid group, and this mentioned sections) will be appreciated the core chapter of the book, Chapter 5, connection led to many further discover- by those who use the book for the self- which deals with loop nests and loop par- ies, some of which are concerned just with study. The author also includes some his- allelisation algorithms. the braid group. torical remarks and comments, more than The readers are assumed to have only This book presents the fruits of 15 pages of references, and symbol and basic mathematical skills and some famil- Dehornoy’s research up to 2000. It is topic indices. The book will be enjoyed iarity with standard graph and linear pro- divided into three parts. Part A establish- by students: in several hundred pages it gramming algorithms. However, due to es the connection to braids, and then contains almost all that a non-specialist its high specialisation, this book is orien- studies divisibility in a wide class of mathematician should know about real tated more towards graduate students monoids that includes the braid monoid analysis on R. (jive) and researchers interested in loop nest (and which is later shown to contain also a parallelisation than to undergraduate stu- monoid that is induced by LD-expan- C. Cheverry, Systèmes de Lois de dents or the general mathematical public. sions). The applications to braids include Conservation et Stabilité BV, Mémoires de (jkrat) new linear orderings and a new normal la Société de Mathématique de France 75, form. Part B is devoted to a study of free Société Mathématique de France, Paris, 1998, W. A. de Graf, Lie Algebras: Theory and LD-systems. It starts by showing that any 106 pp., Ffrs150, ISBN 2-85629-072-8 Algorithms, North-Holland Mathematical two LD-equivalent terms can be expand- This short book considers questions con- Library 56, North-Holland, Amsterdam, ed to the same term. This observation is nected with the existence of weak solu- 2000, 393 pp., US$118, ISBN 0-444- then refined in various ways. First, a tions of the Cauchy problem for strictly 50116-9 (right) normal form for free LD-systems is hyperbolic systems of conservation laws. The theory of Lie algebras is ample, with given, and then various results concern- While the classical results give the exis- explicit constructions and concrete algo- ing the monoid that is induced by LD- tence in the large for data which are small rithms (Levi decomposition, branching expansions are obtained. These results in both the supremum and the total vari- rules, Hall-Shirshov and Grbner bases, then point to an extension of the braid ation norms, this book tries to relax the etc.). The present book contains a stan- group that is so important for Dehornoy’s assumption of the smallness of the total dard course in Lie algebras and practical- proof of acyclicity. Part C is concerned variation. It is shown that under some ly all existing algorithms of this theory. with connections with set theory and with additional assumptions on the growth The approach taken by the author simpli- certain finite LD-systems that appear nat- and (genuine) non-linearity of the flux fies proofs of some important theorems urally in this context and that yield a function, the total variation on intervals and makes them more transparent and number of formidable open problems. of fixed appropriate length is decreasing. clear. Moreover, since current research The book is written very carefully and As a consequence, the compactness prop- in the theory of Lie algebras requires requires almost no preliminary knowl- erties of the solution operator are proved, using computers, such an exposition facil- edge. Nevertheless, many readers will and the existence in the large for period- itates understanding and practical use of probably regard it as quite difficult, since ical initial data is proved in the case when computational methods for solving con- it penetrates deeply into structures that the BV-norm of these data is large but crete problems. The author of this book are far from conventional mathematical BV-norm by period remains small. is also the author of the sub-package Lie interests. The book seems to me to be a This book will be appreciated both by algebras in the programme package GAP. major achievement: its only weakness is specialists in the field and by (postgradu- This circumstance has enabled him to cre- perhaps a rather restricted attention to ate) students interested in basic questions ate a book which will be useful for the results of other authors active in the field. connected with the solvability of systems experts, as well as for interested Readers who have already come across of hyperbolic conservation laws. (mrok) researchers from other fields of mathe- some LD-systems (such as symmetric matics and mathematical physics. (ae) spaces) should be warned that the book is A. Darte, Y. Robert and F. Vivien, mainly about non-idempotent LD-sys- Scheduling and Automatic P. Dehornoy, Braids and Self- tems, in contrast to various idempotent Parallelization, Birkhäuser, Boston, 2000, Distributivity, Progress in Mathematics 192, LD-systems that are induced by reflection 264 pp., DM158, ISBN 0-8176-4149-1 and Birkhäuser, Basel, 2000, 623 pp., DM178, and knot colouring. (ad) 3-7643-4149-1 ISBN 3-7643-6343-6 This book deals with a very specific sub- In 1986 Richard Laver published a short A. M. Etheridge, An Introduction to ject: an automatic parallelisation of nest- remark that connected non-trivial ele- Superprocesses, University Lecture Series ed loops in a sequential computer pro- mentary embeddings in set theory to sys- 20, American Mathematical Society, gram. The core of the book is a detailed tems with one binary operation, in which Providence, 2000, 187 pp., US$33, ISBN 0- description of both classical and recent the left-distributive law x . (y . z) = (x . y) . 8218-2706-5 state-of-the-art techniques in this quite (x . z) is satisfied (the LD-systems). An Measure-valued processes (also known as specialised field. The book also covers LD-system induced by a non-trivial ele- superprocesses), as treated by this mono- the necessary preliminaries that are mentary embedding exhibits a certain graph, constitute an important and fre- essential for understanding the mathe- property (the acyclicity) which implies its quently visited topic of modern probabil- matical background of the techniques freeness (with respect to left-distributivi- ity theory. The basic example and a presented. ty). However, for some time there was no source of motivation for the underlying The book consists of five chapters, of direct proof that a free monogenerated mathematical research, the Dawson- which the first three deal with uni-dimen- LD-system is acyclic. This was a rather Watanabe superprocess (also called sional problems (task graph systems and strange situation, since non-trivial ele- super-Brownian motion) is known as a decomposed cyclic scheduling) and the mentary embeddings imply the existence limit of branching Brownian motions and last two with multi-dimensional systems of large cardinals. recently became a principal tool in math- (recurrent equations and loop nests). Patrick Dehornoy started to contribute ematical population dynamics with the Chapter 1 covers classical results on task to the theory of LD-systems in the late random size of the population governed graph scheduling with no communication 1980s, and in 1994 he published a proof by the Feller diffusion. Among the top- cost under both unlimited and limited of acyclicity that avoids large cardinals. ics in this book are branching Brownian EMS March 2001 29 RECENT BOOKS motion, the DW superprocess as a martin- 1-functions, absolutely measurable func- been suggested in the literature; this book gale problem, the duality theorem for DW tions, continuous functions of bounded offers more than 1000 different types. processes, the Fleming-Viot superprocess variation, continuously differentiable The approximately 4000 illustrations, and its relation to the DW superprocess, functions and approximate derivatives. prepared specially for this book, comple- discrete approximations, superprocesses Chapters on Lebesgue equivalence, densi- ment the text. The book is arranged in partial differential equation theory, ty topology and the Zahorski classes are alphabetically for easy reference and and catalytic and competing species mod- included. The second part, Mappings and extensive cross-referencing is used els. measures on Rn, is concerned with one-to- throughout. This monograph is aimed at graduate one transformations defined on intervals In addition to a detailed description of students and probabilists interested in in Euclidean space Rn. There are chap- each type of information graphics, the stochastic analysis who, the reviewer ters on bi-lipschitzian homeomorphisms, book includes: construction details such believes, will appreciate it as a deep and lengths of non-parametric curves, exten- as grids, symbols, text, lines, axes, leg- technically accessible summary of the sions of homeomorphisms and construc- ends, etc.; features that might mislead developments in the field during the last tions of deformations. The final chapter viewers such as scale breaks and perspec- decade. (jstep) in this part contains a discussion of the tives; specific applications such as break- connection between Blumberg’s theorem, even graphs, population pyramids, can- M. V. Fedoryuk (ed.), Partial Differential Baire spaces and self-homeomorphisms dlestick charts, and quality control charts; Equations V, Encyclopaedia of Mathematical of dense subsets. The last part of the and general terminology, such as vari- Sciences 34, Springer, Berlin, 1999, 247 pp., book is devoted to Fourier series, and able, class interval, cell, stub, coordinate, DM158, ISBN 3-540-53371-0 includes such topics as the behaviour of etc. This book is a collection of six papers by Fourier series, uniform and absolute con- The audience for this book will not be V. M. Babich, N. S. Bakhvalov, M. V. vergence, Bohr theorem, convergence of from those who specialise in pure mathe- Fedoryuk, A. M. Il’in, V. F. Lazutkin, G. Fourier series after change of variable, matics. However, I can imagine many P. Panasenko, A. L. Shtaras, and B. L. and absolutely measurable functions. applied statisticians, engineers, people Vainberg, published originally in 1988 by The appendix contains a supplementary from business and/or newspapers etc., for VINITI, Moscow. This edition presents material including Baire, Borel and whom this reference could represent an an English translation of these papers by Lebesgue sets and functions, Hausdorff important addition to the library. (jant) J. S. Joel and S. A. Wolf. dimension, non-parametric length and Each article considers contemporary area, and approximately continuous H. Hida, Modular Forms and Galois techniques in the study of properties of maps. Brief discussions of the historical Cohomology, Cambridge Studies in Advanced partial differential equations – in particu- background and other relevant matters Mathematics 69, Cambridge University Press, lar, the asymptotic behaviour of solutions; are contained in remarks to some chap- Cambridge, 2000, 343 pp., £42.50, ISBN 0- for example, one of the articles studies ters. 521-77036-X the equations with rapidly oscillating The monograph contains many results The groundbreaking work of Wiles (com- solutions, using the Maslov canonical of the authors. To read it with under- pleted by Taylor-Wiles) on the modulari- operator technique. Another article deals standing, an advanced knowledge of real ty of Galois representations and elliptic with the asymptotic expansion of solu- analysis and topology is needed. (jl) curves was one of the most significant tions of exterior boundary problems for events in number theory in the 1990s. hyperbolic equations, as the time G. Grätzer, Math into LaTeX, Birkhäuser, The main result can be summarised as an approaches infinity. The remaining arti- Boston, 2000, 584 pp., DM108, ISBN 0- isomorphism R = T between a Galois-the- cles consider problems connected with 8176-4131-9 and 3-7643-4131-9 oretic object R (a universal deformation semi-classical asymptotics of eigenfunc- This book is written by a mathematician ring) and a modular object T (a suitable tions, the higher-dimensional WKB for mathematicians and/or for other sci- Hecke algebra). The range of technicali- method, boundary-layer problems, and entists (physicists, engineers,...) who use ties involved in this work is enormous, averaging methods (the methods of many mathematical formulas in their which is why it is not easy to give a com- homogenisation). texts. The core of the book consists of prehensive introduction to the subject. As a whole, the book presents a collec- typesetting such formulas, and all the This book is based on a series of graduate tion of papers dealing with particular top- practical aspects of LaTeX are discussed. courses given by the author at UCLA. It ics in the theory of asymptotic and bound- For typesetting mathematics, the use of does not aim at maximum generality, but ary behaviour of solutions of partial dif- the AMS packages is strongly recom- concentrates on the proof of R = T in the ferential equations. It cannot be consid- mended by the author. In 1999 the ordinary case with trivial tame level. ered to be a textbook on PDEs but will be American Mathematical Society released The book consists of five chapters. valued mostly by specialists working in version 2.0 of the AMS packages; the Chapter 1 gives a brief introduction to the field. (mrok) third edition of the book covers the the subject, an overview of the book and a changes made in this release. In addi- reinterpretation of class field theory in C. Goffman, T. Nishiura and D. tion, this third edition contains three new terms of the Hecke algebra for GL(1). Waterman, Homeomorphisms in Analysis, chapters: Chapter 12 Books in LaTeX is Chapter 2 is devoted to basic results in Mathematical Surveys and Monographs 54, devoted to typesetting large documents in the representation theory of profinite American Mathematical Society, Providence, LaTeX; the other new chapters are TeX, groups, including deformation theory of 1997, 216 pp., ISBN 0-8218-0614-9 LaTeX, and the Internet and Putting LaTeX representations and pseudo-representa- This book describes the interesting role on the Web. The latter explains how to tions. Chapter 3 is the heart of the book. played by homeomorphisms in real func- publish LaTeX articles, reports and books It covers the following: an adelic descrip- tion and measure theories. More precise- on the world wide web. (mdont) tion of classical modular forms and Hecke ly, consider a function f from a certain algebras; p-adic Hecke algebras; Galois family F. What can be said about the class R. L. Harris, Information Graphics: A representations with values in Hecke f ° h: h is a homeomorphism on the Comprehensive Illustrated Reference, algebras; deformations of unramified domain of f?. Here there is a typical Oxford University Press, Oxford, 2000, 448 local Galois representations; Taylor-Wiles example of the Maximoff theorem: let f pp., £32.50, ISBN 0-19-513532-6 systems and the proof of R = T in a spe- be a real function on the interval (0, 1); This is an excellent reference for those cial case. then there is a homeomorphism h: (0, 1) wishing to describe data using graphs (of The key part of the proof is a construc- to (0, 1) such that f ° h is a derivative if any nature). The book covers the full tion of a Taylor-Wiles system of Hecke and only if f belongs to the Baire class 1 spectrum of charts, graphs, maps, dia- algebras, which is in turn based on calcu- and has the Darboux property. grams, and tables, ranging from the most lations involving duality theorems for The first part of the book deals with the basic to very specialised ones. I was sur- Galois cohomology. The latter is the sub- one-dimensional case and classes of Baire prised how many types of graphs have ject matter of Chapter 4; the proofs close- 30 EMS March 2001 RECENT BOOKS ly follow Milne’s book Arithmetic Duality tions have more universal validity in the S. G. Krantz and H. R. Parks, The Theorems. Chapter 5 gives, first, a reinter- study of non-linear waves. Geometry of Domains in Space, Birkhäuser pretation of the equality R = T in terms The second part of the book presents Advanced Texts, Birkhäuser, Boston, 1999, of a ‘generalized class number formula’ the basic concepts of the Whitham theory 308 pp., DM108, ISBN 0-8176-4097-5 and relating the special value of the L-func- of modulations, which describes the evo- 3-7643-4097-4 tion L(1, Ad(f)) and the order of a suitable lution of the parameters of periodic solu- This book serves as a comprehensive Selmer group (still in the ordinary case). tions. This theory serves as a basis for the treatment of domains in space and The final sections give a few new results development of some modern techniques emphasises the interaction between on p-adic Hecke algebras and deforma- of the treatment of non-linear periodic analysis and geometry. Usually, analysis tion rings. waves. The concepts are then again on Euclidean spaces is based on a study of This book will be useful to graduate stu- applied to the Kortweg-de Vries and classical groups (rotations, dilations and dents and researchers in number theory. Schrödinger equations and some more translations) acting on the space. Here Having gone through Chapters 1-3 applications are added. the authors focus on domains in space (backed up, if necessary, by Chapter 4), At the end of each chapter a few exer- and on properties that arise from the the reader can proceed to the original cises are provided to help the reader intrinsic geometry of the domain. work of Wiles or to its detailed exposition understand the topic. Solutions to the The first chapter contains a brief pre- in the book Modular Forms and Fermat’s exercises are given in appropriate detail sentation of basic facts on smooth func- Last Theorem (eds. G. Cornell, J. at the end of the book. Some of the math- tions, defining functions and measure Silverman and G. Stevens). (jnek) ematics used throughout the book is theory. The next chapters are devoted to explained in the appendix. Each chapter rudiments of differential geometry and A. A. Ivanov, Geometry of Sporadic is followed by bibliographic remarks with measure theory (rectifiability, Minkowski Groups I: Petersen and Tilde Geometries, useful comments. The book is especially content, a space-filling curve, covering Encyclopedia of Mathematics and Its valuable for the author’s attempt to inter- theorems, domains with finite perimeter, Applications 76, Cambridge University Press, connect the mathematical approach with the area and co-area formulas and Cambridge, 1999, 408 pp., £45, ISBN 0- its applications to physics, and it will Rademacher’s theorem). A short chapter 521-41362-1 undoubtedly be of great interest to both on Sobolev spaces deals with restriction This is the first volume of a two-volume mathematicians and physicists. (tf) (= T. and trace theorems and domain exten- set on Petersen and tilde geometries. Furst) sion theorems. The following chapters There is an infinite family of tilde geome- are devoted to smooth mappings, Sard’s tries associated with non-split extensions R. Korn and E. Korn, Option Pricing and theorem, Stokes’ theorem, the Whitney of symplectic groups over the two-ele- Portfolio Optimization, Graduate Studies in theorem, various kinds of convexity, the ment field. Besides them, there are 12 Mathematics 31, American Mathematical Krein-Milman theorem, Steiner sym- exceptional Petersen and tilde geome- Society, Providence, 2001, 253 pp., US$39, metrisation, the isoperimetric inequality, tries. These geometries are related to ISBN 0-8218-2123-7 quasi-conformal mappings and Weyl’s some sporadic simple groups, including Modelling financial markets by proba- theorem on eigenvalue asymptotics. the Monster group. The first volume pre- bilistic diffusion models has received a The book is understandable with a sents a construction of each of the excep- considerable acceleration since the Nobel good knowledge of multi-variable tional geometries; these constructions prize Black-Scholes formula for the fair advanced calculus and linear algebra, and also provide an independent proof of the pricing of the (European) options proved can be used as a text for graduate stu- existence of the corresponding automor- to be applicable to real trading processes. dents. The monograph also offers phism groups. Various applications, in As a result, many other types of very com- pleasent reading to anybody who enjoys particular to the representation theory of plex derivative have been used by finan- the interplay between analysis and geom- the Monster, are given. The book is cial markets, each of them calling for an etry. (jl) intended for researchers in finite groups, original application of the stochastic Ito geometries and algebraic combinatorics. calculus. This monograph offers a mathe- G. R. Krause and T. H. Lenagan, Growth (jtu) matically sophisticated treatment of the of Algebras and Gelfand-Kirillov following subjects: modelling security Dimension, Graduate Studies in Mathematics A. M. Kamchatnov, Nonlinear Periodic prices, trading strategies and wealth 22, American Mathematical Society, Waves and Their Modulations, World process, principal properties of the con- Providence, 1999, 212 pp., US$39, ISBN 0- Scientific, Singapore, 2000, 383 pp., £33, tinuous-time market model, option pric- 8218-0859-1 ISBN 981-02-4407-X ing based on the replication principle, This book is based on invited lectures at The book presents mathematical treat- arbitrage bounds, and the construction of the Euroconference held under the same ment of non-linear waves. Although optimal portfolios by the martingale and title at Bielefeld university in September there has lately been great progress in the stochastic control methods. The pricing 1998. The main topic is the role of infi- mathematical approach to these phenom- methods are developed for European, nite length modules in the representation ena, the techniques involved are quite dif- American and exotic options. theory of algebras. The book consists of ficult. It is the aim of the author to pre- The treatment of financial topics is 23 survey papers written by leading sent some of the theory clearly through accompanied by mathematically rigorous experts in the field. applications to physics. excursions into stochastic analysis. The The scene is set in a survey by Ringel The first part of the book consists of a authors explain and, where possible, containing many illuminating examples. thorough treatment of a few important prove results such as the Ito formula, the There follow papers on algebraically com- examples of equations, where the essen- Girzanov theorem and the Bellman prin- pact modules (by Huisgen-Zimmermann tial features of the theory are shown. The ciple. The relevant mathematics is insert- and Prest), decomposition theory (Eklof, theory is backed up by appropriate phys- ed to the text when the results are first Facchini, Göbel and Pimenov-Yakovlev), ical background and motivation. The needed. Theory is presented as far as dimension theory (Bavula, Lenagan and most important examples in this section possible without an application of the Schröer), functor categories (Kuhn and are the Burgers and Kortweg-de Vries Doob-Meyer theorem. Powell), homological methods equations, which describe waves on shal- The book is meant as lecture notes and (Martsinkovsky, Schwartz and Smalo), low water, wave breaking, and non-linear the authors claim it to be accessible to all localisation and moduli spaces (Rickard effects of viscosity. The non-linear those familiar with a basic course in prob- and Schofield), modular representations Schrödinger equation, which describes ability. The reviewer does not completely of finite groups (Benson and Carlson), electromagnetic pulses in dispersive accept this claim, but has no reservations and tameness (Bautista, Krause, Lenzing media, is also discussed in detail in this about the exceptional quality and useful- and Zwara). section. The author shows that the ness of the text. (jstep) This book is indispensable for anyone Kortweg-de Vries and Schrödinger equa- interested in current trends and methods EMS March 2001 31 RECENT BOOKS of representation theory. (jtrl) theories of semi-Markov processes and toric varieties over Z – a natural generali- mixing processes, the asymptotic consoli- sation of projective spaces. Such varieties S. Levy, The Eightfold Way: The Beauty of dation of state spaces, the statistical meth- and line bundles on them can be Klein’s Quartic Curve, MSRI Publications ods of bootstrap, of accelerated life mod- described in purely combinatorial terms. 35, Cambridge University Press, Cambridge, els, of point processes, Bayesian models The corresponding complex analytic line 1999, 331 pp., £35, ISBN 0-521-66066-1 and software models, and so forth. bundles admit natural hermitian metrics; The Klein quartic is the complex curve in This volume contains a selection of however, these are not C8 in general. As the complex projective plane given by the invited and contributed papers in the a result, the author first develops a gener- simple equation x3y +y3z+z3x = 0 and it above topics that describe the state-of- alisation of arithmetic intersection theory has many remarkable and amazing prop- the-art in their disciplines. The 31 chap- of Gillet-Soulé, in which one can make erties. Felix Klein discovered it in 1879, ters are grouped into eight parts: General sense of the first arithmetic Chern class and since then it has appeared in many approach, Probability models and related for line bundles with ‘integrable’ hermit- branches of mathematics (complex analy- issues, Asymptotic analysis, Statistical ian metrics. Applications of this theory to sis, geometry, algebraic geometry, num- models and data analysis, Methods com- toric varieties include a formula for the ber theory and topology). It has an mon to reliability and survival analysis, height of a hypersurface in terms of a exceptionally high degree of symmetry. Software reliability, Statistical inference generalised Mahler measure and an arith- The Hurwitz theorem asserts that the and Asymptotic methods in statistics. For metic version of a theorem of Bernstein- automorphism group of a curve of genus more details, see http://www.mass.u-bord Konshnirenko. (jnek) g has at most 84(g - 1) elements; the Klein eaux2.fr/MI2S/MMR2000 (jant) quartic is one of two curves where this V. P. Maslov and P. P. Mosolov, upper bound is reached. H. Lüneburg, Die euklidische Ebene und Nonlinear Wave Equations Perturbed by In this book, the reader can find an ihre Verwandten, Birkhäuser, Basel, 1999, Viscous Terms, de Gruyter Expositions in English translation of Klein’s original 207 pp., DM49,80, ISBN 3-7643-5685-5 Mathematics 31, Walter de Gruyter, Berlin, paper, together with several review This book is a nice introduction to the 2000, 329 pp., DM298, ISBN 3-11- papers on the role of the Klein quartic in theory of affine and projective planes that 015282-7 the further development of mathematics. can be described by two- and three- This book is an extended and revised ver- The article by H. Karcher and M. Weber dimensional vector spaces over some sion of the Russian original publication, describes its geometry, derives its classical fields. Special attention is paid to con- translated from Russian to English by M. equations and computes its Jacobian. Its nections between properties of the planes A. Shishkova. It deals with the classical role in number theory is treated in the (including finite ones) and the fields, e.g. system of equations describing the motion paper by N. D. Elkies. A short paper by Desargues’ or Pappus planes and the cor- of a one-dimensional compressible gas – A. M. Macbeath reviews automorphism responding fields. The book also con- the one-dimensional Navier-Stokes equa- groups of Riemann surfaces. Two papers tains a description of the theory of conic tions, the continuity equation and state by A. Adler treat more general questions sections, collineations and metric proper- equation, and the equations similar to concerning rings of invariants of repre- ties of conics. In the final chapter, a geo- these. The authors obtain sharper a priori sentations of certain finite groups, and metrical characterisation of the real plane estimates than is usual in this context, prove Hirzebruch’s earlier results for cer- among all affine planes is given. The and using these estimates, they are able to tain symmetric Hilbert modular surfaces. book deals with basic questions of ele- give necessary and sufficient conditions A historical review by J. Gray (first pub- mentary geometry and can be especially for the existence of smooth solutions of lished in the Mathematical Intelligencer) is appreciated by teachers. (lbic) the classical system for vanishing viscosi- reprinted here. ty. Various possibilities on the form of A sculpture inspired by properties of A. Mader, Almost Completely modelling equations and their solution Klein’s quartic was created by H. Decomposable Groups, Algebra, Logic and are discussed, including the form of the Ferguson and since 1983 has attracted Applications Series 13, Gordon and Breach viscosity perturbation, boundedness of visitors at the Mathematical Sciences Publishers, Singapore, 2000, 354 pp., non-linearities, etc. Research Institute in Berkeley. The book US$90, ISBN 90-5699-225-2 This book provides a skilled insight starts with W. Thurston’s speech at the The structure of rank one torsion-free into the contemporary problems connect- occasion of its inauguration and includes abelian groups – the subgroups of the ed with the solvability of the specified an essay written by the sculptor. (vs) additive group of rationals – is well known class of partial differential equations, and and completely described in the terms of can be recommended both to specialists N. Limnios and M. Nikulin (eds.), types. Arbitrary direct sums (finite or in the field and to young mathematicians Recent Advances in Reliability Theory. infinite) of rank one groups are called beginning their mathematical career in Methodology, Practice, and Inference, completely decomposable groups, and PDEs. (mrok) Statistics for Industry and Technology, their structure is also well known. A tor- Birkhäuser, Boston, 2000, 514 pp., DM178, sion-free group is called almost complete- A. S. Matveev and A. V. Savkin, ISBN 0-8176-4135-1 and 3-7643-4135-1 ly decomposable if it is a finite extension Qualitative Theory of Hybrid Dynamical This book discusses reliability, asymptotic of a completely decomposable group. Systems, Control Engineering, Birkhäuser, theory of processes (with or without Using regulating subgroups, regulators Boston, 2000, 348 pp., DM148, ISBN 0- switching), semi-Markov processes, mix- and near-isomorphisms and collecting 8176-4141-6 and 3-7643-4141-6 ing processes, asymptotic consolidation of known results, the monograph presents a Hybrid dynamical systems combine con- state spaces, bootstrap, accelerated life systematic study of the structural proper- tinuous and discrete dynamics and models, Bayesian models, software relia- ties of this class of groups. (lbic) involve both continuous and discrete state bility. variables. A well-known instance of a Conceiving reliable systems is a strate- V. Maillot, Géométrie d’Arakelov des var- hybrid system is a dynamical system gic issue for any industrial society – and iétés toriques et fibrés en droites inté- described by a set of ordinary differential this has been also the theme of the most grables, Mémoires de la SMF 80, Société equations with discontinuous or multi-val- important international conference in the Mathématique de France, Paris, 2000, 129 ued right-hand sides. Such mathematical past couple of years, MMR’2000, held in pp., FRF150, ISBN 2-85629-088-4 models can be used to describe various Bordeaux in July 2000, in which more Arakelov geometry combines algebro- engineering systems with relays, switches, than 350 people from all over the world geometric invariants of an arithmetic and hysteresis. In the linear control area, participated. variety X, a regular flat scheme of finite a typical example of a hybrid system is The main topics covered are stochastic type over Z with hermitian geometry of that which is created when a continuous- models and methods useful for reliability, the complex manifold X(C). time plant described by differential equa- including the asymptotic theory of In this monograph the author studies tions is controlled by a digital regulator processes (with or without switching), the Arakelov geometry of smooth projective described by difference equations. These 32 EMS March 2001 RECENT BOOKS types of systems are studied in modern gence. The remainder of the book devel- processes, allowing sufficient space for control engineering courses under the ops concepts of statistical inference. The results on additive functionals and Martin name of computer-controlled systems, or author deals with sufficiency, complete- boundary. sampled-data systems. ness and ancillarity (including Basu’s the- Volume 2, Itô Calculus, presents per- The book is primarily a research mono- orem), point estimation (method of haps the most attractive topic in modern graph that presents some recent research moments and of maximal likelihood, probability (stochastic analysis) to inter- in a unified fashion. The book consists Rao-Blackwell theorem, Lehmann-Scheff, ested and mathematically cultured audi- mainly of the authors’ original results and theorems), tests of hypotheses, confi- ences, in a manner based on both intu- is essentially self-contained. The main dence interval estimation (including mul- ition and formal mathematical reasoning. concepts are differential automata, cyclic tiple comparisons), Bayesian methods, The text differs from its previous editions linear differential automata and switched likelihood ratio tests, large-sample infer- in having many new examples and proofs. single server flow networks. The authors ence, and two-stage procedures for sam- The authors justify the volume’s title by study asymptotic behaviour of these mod- ple size determination. providing many examples that help read- els – in particular, the existence of limit The only prerequisite is a one-year ers to develop their calculation abilities. cycles. (pkurk) course of calculus. In individual chapters Chapter IV covers the standard theory of the reader finds numerous examples, and the stochastic integral (of a previsible V. Maz’ya, S. Nazarov and B. at the end of each chapter a long list of process with respect to a continuous semi- Plamenevskij, Asymptotic Theory of exercises is presented. The text contains martingale) up to the level of the semi- Elliptic Boundary Value Problems in interesting historical remarks and the martingale local time as an occupation Singularly Perturbed Domains, I, Operator author gives selected biographical notes density. Chapter V is devoted to the Theory Advances and Applications 111, on some exceptional contributors to the strong and weak theory of stochastic dif- Birkhäuser, Basel, 2000, 435 pp., DM268, development of statistics in an appendix ferential equations, and such other topics ISBN 3-7643-6397-5 and 3-7643-2964-5 to the book. as pathwise uniqueness and uniqueness in This book is devoted to the development This is a nice textbook. The text is writ- distribution, and martingale problems and applications of asymptotic methods ten in a very understandable way and the are treated with an additional emphasis to boundary value problems for elliptic explanation is illustrated with many care- on one-dimensional SDEs. The subject of equations in singularly perturbed fully selected examples. The book can stochastic differential geometry receives domains. The first volume contains Parts also be helpful as a supplementary text in here an original and clear exposition. I-IV, in which boundary value problems courses for graduate students. It is a plea- Chapter VI extends the stochastic inte- with perturbations near isolated singular- sure to read and I can recommend it to gration theory to local martingales with ities of the boundary of the domain are students and teachers of mathematical trajectories that are right continuous with studied. The second volume contains statistics. (jand) limits on the left, the general Meyer theo- Parts V-VII, which deal with other kinds rem and Itô excursion theory being the of perturbations (problems with perturba- L. C. G. Rogers and D. Williams, topics whose presentation the reviewer tions of the boundary of singular mani- Diffusions, Markov Processes and considers the highlights of the volume. folds, problems in thin domains, and Martingales, Vols. 1, 2, Cambridge The monograph as a whole is warmly problems with rapid oscillations of the University Press, Cambridge, 2000, 386 and recommended to post-PhD students of boundary of domain or coefficients of dif- 480 pp., £22.95 and £24.95, ISBN 0-521- probability and will be welcomed as a ferential operators). In Part I the authors 77594-9 and 0-521-77593-0; First and sec- good and reliable reference. (jstep) discuss boundary value problems for the ond editions published by John Wiley & Sons Laplace operator. Part II is devoted to 1979, 1994, reissued in 2000 by Cambridge J.-E. Rombaldi, Thémes pour l’agrégation the study of general elliptic boundary University Press. de mathématique, EDP Sciences, Les Ulis, value problems. Parts III and IV deal These books present a systematic treat- 1999, 205 pp., Ffrs160, ISBN 2-86883- with expansion of functionals over solu- ment of topics on the title of the mono- 407-8 tions of boundary value problems and graph and differ significantly from their This is a thorough and mathematically eigenvalues in the asymptotic series. In first edition. neat textbook on matrix theory. The Part V the authors study boundary value Volume 1, Foundations, is divided into algebraic and topological properties of problems in domains perturbed near three chapters, each adopting its own the matrix vector algebra (over both the multi-dimensional singularities of the individuality as far as the purpose and the real and complex numbers) are presented boundary. The behaviour of solutions of style of presentation are concerned. in a self-contained manner, starting with boundary value problems in thin domains Chapter I (Brownian motion, Gaussian a condensed treatment of the normed is investigated in Part VI. Part VII deals and Lévy processes) is designed as an vector spaces that can be easily followed with elliptic boundary value problems introduction that the reader both theo- by any reader familiar with the elements with oscillating coefficients or boundary rems and heuristics and reasons why the of linear algebra. Standard results of clas- of domain. (dmed) topics are worth studying. Chapter II has sical matrix theory are proved comfort- been transformed to a consistent treat- ably via the endomorphism interpreta- N. Mukhopadhyay, Probability and ment of what could be called a ‘handbook tion, and applications to systems of linear Statistical Interference, Statistics: Textbooks of probabilistic measure theory for young differential equations are delivered care- and Monographs 162, Marcel Dekker, New probabilists’. Its major subjects are the fully with respect to possible numerical York, 2000, 665 pp., US$95, ISBN 0-8247- elements of measure and probability the- problems. Each chapter is accompanied 0379-0 ory, measure and probability theory on by a collection of relevant solved exercis- This book covers standard topics belong- Lusin spaces, the Daniell-Kolmogorov es. ing to a two-semester introductory course theorem, and discrete- and continuous- The textbook is meant as a teaching of mathematical statistics. The first four time martingales. The authors treat these text at first- or second-year graduate level chapters introduce basic concepts such as subjects in a highly up-to-date manner; to provide a background for a course on probability and conditional probability, for example, the continuous-time martin- numerical analysis, for example. Some expectation, variance, moments, and gale theory is presented for the processes readers might miss applications to linear moment generating functions of the com- with trajectories that are continuous on programming or Markov chains. (jstep) mon discrete and continuous distribu- the right with limits on the left, and the tions, multivariate random variables and sophistications as augmented filtrations D. B. Shapiro, Compositions of Quadratic their characteristics, some standard prob- are discussed with rigorous mathematical Forms, De Gruyter Expositions in ability inequalities, and functions of ran- details. Chapter III (mostly identical with Mathematics 33, Walter de Gruyter, Berlin, dom variables, with a brief discussion on the first edition) is a semi-advanced treat- 2000, 417 pp., DM248, ISBN 3-11- χ2, t, and F-distributions. Chapter 5 is ment of Markov processes. It presents a 012629-X devoted to concepts of stochastic conver- neat treatment of Feller, Lévy and Ray A composition formula of size [r, s, n] is a EMS March 2001 33 RECENT BOOKS sum-of-squares formula of the type grammed motion controlled systems; this ‘Constructing the Number System’. He 2 2 2 2 2 2 (x1 +x2 + … + xr ).(y1 +y2 + … +ys ) is a controlled system governed by the ini- does not use ‘the encoding’ and he works 2 2 2 = z1 +z2 + … +zn , tial-value problem for a system of, in gen- with sets quite freely (using the axiom of where X = (x1, x2, … , xr) and Y = (y1, y2, eral non-linear, ordinary differential comprehension: if X is a set and Φ[x] is a … , ys) are systems of indeterminates and equations dy/dt = f(t, y, w). The informa- predicate on X, then {x: X Φ[x]} is also a each zk = zk(X, Y) is a bilinear form in X tion for synthesising a stabilising control set). From Chapter IV, his basic mathe- and Y. The first part of the book, w can be extracted from (some compo- matical tool is category theory. The use ‘Classical compositions and quadratic nents of) the response y, either continu- of categories as a base of mathematics and forms’, is devoted to classical sizes [r, n, n] ously, or in specific time instances only, the application of categorical notions and and the second one, ‘Compositions of size considering possibly a measurement methods in logic and in computer science [r, s, n]’, gives some partial answers to the error. (more precisely: the developing of logic possible sizes [r, s, n] and the dependence Chapter 1 treats the case of continuous- and of computer science on the basis of on the field of coefficients using the ly receiving information, for both so- categorical notions and categorical meth- methods of algebraic topology, combina- called stationary and non-stationary sys- ods) has been widely and intensively torics and linear algebra and geometry. tems (usually called autonomous and non- examined recently. The book presents a (lbic) autonomous systems). Reduction to a systematic exposition of this topic, canonical form, Lyapunov’s function, explaining its ideas and summarising the A. Schinzel, Polynomials with Special relay stabilisation, or feedback control, corresponding results. The author also Regard to Reducibility, Encyclopedia of are the methods used. Information about aims to show how these modern ideas Mathematics and Its Applications 77, y received in discrete-time instances develop the classical ones and he presents Cambridge University Press, Cambridge, (combined possibly with continuously many very interesting historical facts. 2000, 558 pp., £60, ISBN 0-521-66225-7 received information) is analysed in The book is intended for programmers As the title indicates, the book deals with Chapter 2. Chapter 3 deals with various and computer scientists, rather than for reducibility aspects of polynomials. It variants of the linear/quadratic optimal mathematicians and logicians, but it can covers mainly those reducibility results control problem. Finally, Chapter 4 is be useful for both groups of potential that hold over large classes of fields; in devoted to the so-called orbit stabilisation readers. (vt) particular, results valid only over the of variable-structure systems. rational field, or over finite and local The book is based mostly on results A. Terras, Fourier Analysis on Finite fields, are not included. The book also from the Russian school (about 85% of Groups and Applications, London contains many related results (e.g. Ritt’s the references are published in Russian), Mathematical Society Student Texts 43, theorems, or the Mahler measure). a large part (about 30%) being the Cambridge University Press, Cambridge, The titles of the chapters are: Arbitrary author’s original research. The book, 1999, 442 pp., £18.95, ISBN 0-521-45108- polynomials over an arbitrary filed (80 requiring only a three-year university 6 and 0-521-45718-1 pp.), Lacunary polynomials over an arbi- course in mathematics, is designed both The main goal of this book is to consider trary field (109 pp.), Polynomials over an for researchers in control theory and for finite analogues of the symmetric spaces algebraically closed field (62 pp.), graduate students. (trou) such as Rn and the Poincaré upper half- Polynomials over a finitely generated plane. The author describes finite ana- field (52 pp.), Polynomials over a number M. Stern, Semimodular Lattices: Theory logues of all the basic theorems in Fourier field (75 pp.), Polynomials over a and Applications, Encyclopedia of analysis, both commutative and non-com- Kroneckerian field (91 pp.). Mathematics and its Applications 73, mutative, including the Poisson summa- Since polynomials touch on a huge vari- Cambridge University Press, Cambridge, tion formula and the Selberg trace formu- ety of topics, it is hard to expect that such 1999, 370 pp., £50, ISBN 0-521-46105-7 la. One motivation for this study is to a book could be completely self-contained The theory of semimodular lattices is the prepare the ground for understanding and simultaneously cover a broad spec- part of lattice theory with most connec- the continuous theory by developing its trum of results. Nevertheless, the book tions to other areas of mathematics – in finite model. The book is written in such can be read fluently, and the eventual particular, to discrete mathematics and a way that it can be enjoyed by non- definitions and results from the theory of combinatorics, and especially to matroid experts such as advanced undergradu- polynomials needed to follow the exposi- theory. Other parts have relations to ates, beginning graduate students, and tion are collected in ten appendices. For group theory – namely, to the structure of scientists outside of mathematics. Several other standard results the reader is subgroup lattices. Other connections are applications are included, such as the referred to standard textbooks on real to the foundations of geometry. The construction of graphs that are good and complex analysis or the theory of author develops the theory of semimodu- expanders, reciprocity laws in number algebraic numbers. lar lattices and their generalisations, and theory, the Ehrenfest model of diffusion The book is written in a lucid style and presents applications to the areas of random walks on graphs, and vibrating contains many notes and cross-references mathematics mentioned above. systems and chemistry of molecules. (ae) to recent improvements on stated results The book will be valuable to that makes the book very versatile. For researchers, not only in lattice theory but instance, the last appendix (written by U. also in discrete mathematics, combina- Zannier) contains Bombieri and Zannier’s torics and general algebraic systems. (jtu) proof of one of the author’s conjectures. Also very helpful are the indices of defin- P. Taylor, Practical Foundations of itions and theorems. The bibliography Mathematics, Cambridge Studies in consists of over 300 items. The book can Advanced Mathematics 59, Cambridge also be recommended as a resource for University Press, Cambridge, 1999, 572 pp., anybody interested in reducibility prob- £50, ISBN 0-521-63107-6 lems or in related facets of the theory of “Our system conforms very closely to the polynomials. (šp) way mathematical constructions have actually been formulated in the twentieth E. Ya. Smirnov, Stabilization of century. The claim that set theory pro- Programmed Motion, Stability and Control: vides the foundations of mathematics is Theory, Methods and Applications 12, Gordon only justified via an encoding of this sys- WORLD MATHEMATICAL YEAR 2000 and Breach Publishers, 2000, 265 pp., tem, and not directly”, says the author in Last October an Italian WMY2000 stamp US$110, ISBN 90-5699-256-2 paragraph 2.2 (page 72), after Chapter I was issued. It features the WMY2000 logo, This monograph develops stabilisation entitled ‘First order reasoning’ and after a fractal pattern and Archimedes’ sphere and optimal stabilisation of so-called pro- the first paragraph of Chapter II entitled inside a cylinder 34 EMS March 2001 NewNew MembersMembers We welcome the following new members who have joined in the past year.

Australia Bertrand, Pierre Fritz, Gregor Peccati, Lorenzo Knopfmacher, John Chaudouard, Pierre-Henri Gagelmann, Ursula Petronio, Carlo Chipot, Michel Grötschel, Martin Regazzini, Eugenio Austria Clement, Emmanuele Hatala, Michael Ricca, Renzo Schmidt, Klaus Cordier, Stephane Herber, T. Schiavi, V. A. Teichmann, Josef Coudrais, Jacques Jäger, Willi Sernesi, Edoardo Croisille, Jean-Pierre Korte, Ulrike Simonetti, Maura Belgium Delaruelle, Christian Lügger, Joachim Turrini, Cristina Bellemans, M. J. Di Maio, Pierre Morgenstern, Brigitte Vaio, Franco Bergh, Michel Dias, Frederic Rack, Heinz-Joachim Visintin, Augusto Borrey, Sabine Diener, Francine Schmidt, Wolfgang Botterman, Stefaan Donato, Patrizia Trapp, Monika Japan Kalinde, Ibert De Reffye, Jerome Ueberberg, Johannes Kajiwara, Joji Kuijken, Elisabeth Di Menza, Laurent Weber, Gerhard-W. Luyckx, D. Diener, Marc Weber, Horst Latvia Mielants, Wim El Badia, Abdellatif Winkler, Jörg Swylan, E. Plastria, Frank Foucher, Francoise Quarta, Lucas Fournier, Yvan India Netherlands Sebille, Michel Gallot, Sylvestre Sharma, A. K. Lenstra, H. W. Toint, Philippe Gilquin, Herve Lübke, Martin Warrinnier, Alfred Giovangigli, Vincent Israel Venema, R. Guillaume, Anne Belenkiiy, Ari Benin Guillas, Serge Berg, Y. New Zealand Houehougbe, Antoine Intissar, Abdelkader Braun, Amiram Glynn, David Iooss, Gerard Rudnick, Zeev Botswana Jacob, Christine Poland Cowen, R. Jaffard, Stephane Italy Kubarski, Jan Joly, Pierre Amadori, Debora Canada Kassel, Christian Argiolas, Roberto Portugal Brunner, H. Keller, Bernard Barletta, Elisabetta Bigotte Almeida, Maria Emilia Guastavino, Marc Kold Hansen, Soeren Bernardi, Claudio Castro, Luis Lasry, Jean Michel Bischi, Gian Italo Clemente, Maria Czech Republic Launay, Genevieve Bonanzinga, Vittoria Florencio Fidalgo, Carla Isabel Bohac, Zdenek Laurencot, Philippe Botteri Fattore, Maria Luisa Marado Torres, Delfim Francu, Jan Le Roux, Alain Yves Brenti, Francesco Fernando Lemay, Robert Calabri, Alberto Marques Cerejeiras, Paula Denmark Leruste, Christian Capparelli, Stefano Cristina Fuglede, Bent Maisonobe, Philippe Ceccherini Silberstein, Tullio Melo Margalho, Luis Manuel Hoholdt, Tom Massot, Marc Ciarabelli, Federico Mendes Lopes, Margarida Lind Jensen, Allan Merindol, Jean-Yves Ciliberto, Ciro Maria Nilsson, Dennis Pelletier, Michele Corti, Beatrice Monteiro Paixao, Miguel Jorge Penot, Jean-Paul Dedo, Ernesto Pinto, Joao Finland Picavet, Gabriel De Mari, Filippo Silva, Jorge Nuno Astola, Laura Richard, Denis Di Carpegna, Ranieri Granlund, Seppo Sablonniere, Paul Diomede, Sabrina Russia Heili, Matti Schuman, Bertrand Marc Emmer, Michele Borichev, Andrey Huuskonen, Taneli Short, Hamish Fedeli, Alessandro Rappoport, Juri Hyry, Eero Simonot, Francois Ferrara, Massimiliano Jussila, Tapani Terpolilli, Peppino Ferrari, Fausto Spain Laine, Marko Thieullen, Michele Frosini, Patrizio Aguilera Venegas, Gabriel Nevanlinna, Veikko Tollu, Christophe Gabelli, Stefania Alias, Luis J. Peltonen, Petri Treimany, Catherine Galletto, Dionigi Andradas Heranz, Carlos Pirinen, Aulis Werner, Wendelin Germano Crosa, Bruna Berenguer Maldonado, Maria Pohjolainen, Seppo Wirtz, Bruno Grimaldi, Renata Isabel Salli, Arto Isopi, Marco Bermejo Diaz, Isabel Stenberg, Rolf Germany Lanteri, Antonio Berriochoa, Elias Suuntala, Sauli Ansorge, Rainer Luciano, Claudia Ines Ernesta Casanovas Pujadas, Xavier Vuorinen, Matti Ballmann, Josef Mantegazza, Carlo Castells Oliveres, Eugeni Benner, Peter Manzi, Gilberto Castro Jimenez, Francisco France Carstensen, Carsten Marino, Maria Corinna Jesus Angeniol, Bernard Czech, Ingo Sigurd Mennella, F. Chavarriga Soriano, Javier Perthame, Benoit Forster, Brigitte Miranda, Annamaria Cubedo Cullere, Marta

EMS March 2001 35 de Prada Vicente, Maria Milan Lopez, Francisco Switzerland Maltese, George Angeles Mira Ros, Jose Manuel Aeschbach, Fritz Michaelis, Walter Diez Fernandez, Honorato Montes Lozano, Antonio Baur, Karin Milea, Constantin Escoriza Lopez, Jose Morales Bueno, Rafael Burger, Marc Robson, Robby Fagella, Nuria Munoz Martinez, Jose Vicente Buser, Peter Sagle, A.A. Fortes Ruiz, Inmaculada Narvaez Macarro, Luis Grote, Marcus Saxton, Ralph Gago Couso, Felipe Nunez Gomez, Marian Gutknecht, Martin Schötzau, Dominik Garay Bengoechea, Oscar Nuno Ballestros, Juan Kappeler, Thomas Vogtmann, Karen Jesus Perez-Fernandez, F. Javier Kaup, Burchard Garcia Garcia, Gloria Perez Fernandez, Javier Lang, Urs United Kingdom Garcia Hernandez, Jose Luis Perez Gonzalez, Fernando Luethi, Hans-Jakob Baesens, C. N. Garcia Rodriguez, Domingo Perez-Martinez, Eugenia Mazzola, Guerino Darbyshire, P. M. Gimenez, Philippe Ramos Alonso, Pedro Antonio Mislin, Guido Ferguson, W. A. Girela Alvarez, Daniel Salvador Alcaide, Adela Perret, St‚phane Fokas, A. S. Godoy Malvar, Eduardo Sanchez-Pedreno Guillen, Quarteroni, Alfio Grant-Ross, Peter Gomez-Perez, Javier Salvador Ratiu, Tudor Hannabuss, Keith Gonzalez Enriquez, Cristobal Sanmartin Carbon, Maria Rutishauser, Therese Hou, Zhanyuan Gonzales Jimenez, Santos Esperanza Strebel, Ralph Kelly, Clive Gracian Rodriguez, Enrique Sanz Gil, Javier Stuart, Ch. Kendall, Wilfrid Gual Arnau, Jose Joaquin Torres Cabrera, Maria Wuergler, Urs Kisil, Vladimir Indurain Eraso, Esteban Begonya Zehnder, Eduard Maenhaut, Barbara Jimenez Alcon, Francisco Velez Melon, Maria Piar Nijhoff, Frank Laliena Clemente, Jesus Turkey Pickup, D. A. Llovet, Juan Sweden Alpay, S.O. Pouroutidou, S. Lobillo, Francisco Javier Andersson, Mats Erik Zhang, P. Robertson, E. F. Maestre Vera, Manuel Brzezinski, Juliusz Testermann, Donna Martin Cabrera, Francisco Dahlgren-Johansson, Olga U. S. A. Wellington-Spurr, Gabrielle Martinez, Juan Fainsilber, Laura Fairweather, Graeme Williams, David Martinez Ansemil, Jose Maria Goos, Thomas Fossum, Robert Wilson, Robin Mata Hernandez, Agueda Persson, Leif Guidotti, Patrick Wotherspoon, C. I. Maureso Sanchez, Montserrat Roos, Jan-Erik Lawvere, William Wright, H.C. Miana, Pedro Jose Sjögren, Peter Lee Keyfitz, Barbara

36 EMS March 2001 The Institut des Hautes Études Scientifiques, located in Bures-sur-Yvette (France), hosts each year 200 to 250 mathematicians and theoretical physicists from all over the world and for various peri- ods (2 or 3 days up to 1 or 2 years).

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

EUROPEAN PROGRAMME In February 2000, the European Commission acknowledged the IHÉS as a Large European Research Infrastructure centre. A contract has thus been signed with the Commission aiming to develop and promote access to the Institute for European researchers (European Union and associated countries). This contract, valid through 2002, endows the Institute with financial support of 450,000 Euros over a 3 year period. These funds will be used to support approximately 50 visitors doing research in mathematics or theoretical physics and originating from countries of the European Union or associated countries. More information about conditions of application through the European programme are available on http://www.ihes.fr/APPLICATION/ProgEC/europe2A.html Applications are reviewed and selected by the IHÉS Scientific Committee twice a year: in January and in September.

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

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

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

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

Starting date of the fellowships September or October 2002.

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

Deadline for applications: December 31, 2001.

Information IHÉS - 35, route de Chartres, F-91440 Bures-sur-Yvette (FRANCE) tel: +33 1 6092 6600 o fax: +33 1 6092 6669 o e-mail: [email protected] or website: www.ihes.fr

Institut des Hautes Études Scientifiques 35, route de Chartres - 91440 BURES-SUR-YVETTE (FRANCE) TEL: +33 1 6092 6667 FAX: +33 1 6092 6669 WEB SITE: www.ihes.fr