CONTENTS EDITORIAL TEAM EUROPEAN MATHEMATICAL SOCIETY EDITORS-IN-CHIEF MARTIN RAUSSEN Department of Mathematical Sciences, Aalborg University Fredrik Bajers Vej 7G, DK-9220 Aalborg, Denmark e-mail: [email protected] ROBIN WILSON Department of Pure The Milton Keynes MK7 6AA, UK NEWSLETTER No. 50 e-mail: [email protected] ASSOCIATE EDITORS December 2003 VASILE BERINDE Department of Mathematics, EMS Agenda ...... 2 University of Baia Mare, e-mail: [email protected] Editorial by Martin Raussen ...... 3 KRZYSZTOF CIESIELSKI Mathematics Institute Jagiellonian University EMS Executive Meeting ...... 4 Reymonta 4 30-059 Kraków, Poland Mathematical Articles for the general public ...... 5 e-mail: [email protected] STEEN MARKVORSEN Playing a trick on uncertainty by F. Thomas Bruss ...... 7 Department of Mathematics Technical University of Denmark Building 303 Integrability of Hamiltonian systems by Michèle Audin ...... 9 DK-2800 Kgs. Lyngby, Denmark e-mail: [email protected] Kolmgorov and Contemporary Mathematics by Beno Eckmann ...... 13 SPECIALIST EDITORS Proof on Broadway, Preuve a Bruxelles by Luc Lemaire ...... 14 INTERVIEWS Steen Markvorsen [address as above] SOCIETIES Interview with ...... 15 Krzysztof Ciesielski [address as above] EDUCATION The Moscow Mathematical Society ...... 17 Tony Gardiner University of Birmingham The Fourth European Congress of Mathematics ...... 20 Birmingham B15 2TT, UK e-mail: [email protected] Forthcoming Conferences ...... 22 MATHEMATICAL PROBLEMS Paul Jainta Werkvolkstr. 10 Recent Books ...... 25 D-91126 Schwabach, Germany e-mail: [email protected] Designed and printed by Armstrong Press Limited ANNIVERSARIES Crosshouse Road, Southampton, Hampshire SO14 5GZ, UK June Barrow-Green and Jeremy Gray telephone: (+44) 23 8033 3132 fax: (+44) 23 8033 3134 Open University [address as above] e-mail: [email protected] e-mail: [email protected] and [email protected] Published by European Mathematical Society CONFERENCES ISSN 1027 - 488X Vasile Berinde [address as above] RECENT BOOKS The views expressed in this Newsletter are those of the authors and do not necessari- Ivan Netuka and Vladimir Sou³ek ly represent those of the EMS or the Editorial team. Mathematical Institute Charles University Sokolovská 83 NOTICE FOR MATHEMATICAL SOCIETIES 18600 Prague, Czech Republic Labels for the next issue will be prepared during the second half of February 2004. e-mail: [email protected] Please send your updated lists before then to Ms Tuulikki Mäkeläinen, Department of and [email protected] Mathematics, P.O. Box 4, FIN-00014 , Finland; e-mail: [email protected] ADVERTISING OFFICER Vivette Girault INSTITUTIONAL SUBSCRIPTIONS FOR THE EMS NEWSLETTER Laboratoire d’Analyse Numérique Institutes and libraries can order the EMS Newsletter by mail from the EMS Secretariat, Boite Courrier 187, Université Pierre Department of Mathematics, P. O. Box 4, FI-00014 University of Helsinki, Finland, or by e- et Marie Curie, 4 Place Jussieu mail: ([email protected]). Please include the name and full address (with postal code), 75252 Cedex 05, telephone and fax number (with country code) and e-mail address. The annual subscription fee e-mail: [email protected] (including mailing) is 80 euros; an invoice will be sent with a sample copy of the Newsletter.

EMS December 2003 1 EMS NEWS EMS Committee EMS Agenda EXECUTIVE COMMITTEE PRESIDENT 2004 Prof. Sir JOHN KINGMAN (2003-06) 31 January Isaac Newton Institute Closing date for nominations for delegates to EMS Council to represent individual members 20 Clarkson Road Cambridge CB3 0EH, UK Contact: Tuulikki Mäkeläinen, e-mail: [email protected] e-mail: [email protected] 1 February VICE-PRESIDENTS Prof. LUC LEMAIRE (2003-06) Deadline for nominations for the EMS Prizes and the Prize, to be awarded at 4ecm Department of Mathematics Nominations for the EMS Prizes to 4ecm Organising Committee, Prof Ari Laptev, Department Université Libre de Bruxelles of Mathematics, Royal Institute of Technology, SE-100 44 Stockholm, Sweden. C.P. 218 – Campus Plaine Nominations for the Felix Klein Prize to EMS Secretariat, Ms Tuulikki Mäkeläinen, Bld du Triomphe Department of Mathematics, P.O. Box 4 (Yliopistonkatu 5), FIN-00014 University of Helsinki, B-1050 Bruxelles, Belgium e-mail: [email protected] Finland Prof. BODIL BRANNER (2001–04) Department of Mathematics 15 February Technical University of Denmark Deadline for submission of material for the March issue of the EMS Newsletter Building 303 Contact: Martin Raussen, e-mail: [email protected] DK-2800 Kgs. Lyngby, Denmark e-mail: [email protected] 20 February SECRETARY Deadline for proposals for EMS Lectures, EMS Joint Mathematical Weekends and EMS Prof. HELGE HOLDEN (2003-06) Department of Mathematical Sciences Summer Schools in fundamental and interdisciplinary mathematics Norwegian University of Science and Technology (see announcement in Newsletter 49, September 2003, page 4) Alfred Getz vei 1 Contact: Luc Lemaire, e-mail: [email protected] NO-7491 Trondheim, Norway e-mail: [email protected] 28 February TREASURER Executive Committee Meeting in Helsinki (Finland), at the invitation of the Finnish Prof. OLLI MARTIO (2003-06) Mathematical Society Department of Mathematics Contact: Helge Holden, e-mail: [email protected] P.O. Box 4, FIN-00014 University of Helsinki, Finland e-mail: [email protected] 25 June EMS Executive Committee meeting at Uppsala (Sweden) ORDINARY MEMBERS Prof. VICTOR BUCHSTABER (2001–04) Contact: Helge Holden, e-mail: [email protected] Steklov Mathematical Institute Russian Academy of Sciences 26-27 June Gubkina St. 8, Moscow 117966, EMS Council meeting at Uppsala (Sweden) e-mail: [email protected] Contact: Helge Holden, e-mail: [email protected] Prof. DOINA CIORANESCU (2003–06) Laboratoire d’Analyse Numérique or Tuulikki Mäkeläinen e-mail: [email protected] Université Paris VI 4 Place Jussieu 27 June - 2 July 75252 Paris Cedex 05, France 4th European Congress of Mathematics, Stockholm e-mail: [email protected] website: http://www.math.kth.se/4ecm Prof. PAVEL EXNER (2003-06) Department of Theoretical Physics, NPI Academy of Sciences 4-24 July 25068 Rez – Prague, Czech Republic EMS Summer School at Cortona (Italy) e-mail: [email protected] Evolution equations and applications Prof. MARTA SANZ-SOLÉ (2001-04) Facultat de Matematiques 15-23 July Universitat de Barcelona Gran Via 585 EMS Summer School at Bedlewo (Poland) E-08007 Barcelona, Spain Analysis on metric measure spaces e-mail: [email protected] Prof. MINA TEICHER (2001–04) 30 August - 3 September Department of Mathematics and EMS Summer School at Universidad de Cantabria, Laredo (Spain) Empirical processes: theory and statistical applications Bar-Ilan University Ramat-Gan 52900, Israel e-mail: [email protected] Cost of advertisements and inserts in the EMS Newsletter, 2004 EMS SECRETARIAT Ms. T. MÄKELÄINEN (all prices in British pounds) Department of Mathematics P.O. Box 4 Advertisements FIN-00014 University of Helsinki Commercial rates: Full page: £230; half-page: £120; quarter-page: £74 Finland Academic rates: Full page: £120; half-page: £74; quarter-page: £44 tel: (+358)-9-1912-2883 Intermediate rates: Full page: £176; half-page: £98; quarter-page: £59 fax: (+358)-9-1912-3213 telex: 124690 Inserts e-mail: [email protected] Postage cost: £14 per gram plus Insertion cost: £58 (e.g. a leaflet weighing 8.0 gram will website: http://www.emis.de cost 8x£14+£58 = £170) (weight to nearest 0.1 gram) 2 EMS December 2003 EDITORIAL editorship. I would like to thank him very much for his good wishes, for his support, and for the tremendous work he has invested in the Newsletter since 1999. EditorialEditorial Initially, the Newsletter was edited by David Singerman (Southampton) and Ivan Netuka History and Plans (Prague). An editorial team from Glasgow Caledonian University, represented by Roy Martin Raussen Bradley and Martin Speller, then took over from 1996 to 1998. As Carles Casacuberta explained in his editorial for issue 41, the Newsletter has Presentation reached maturity in its content and layout, but con- As the newly appointed editor of the EMS stant work will be necessary to maintain a journal Newsletter, I have assigned the task of writing the that is interesting for the members of the Society editorial for this ‘jubilee’ issue 50 to myself. I and as such an asset for the EMS. should perhaps start by introducing myself: I am an associate professor at Aalborg University, Future North Jutland, Denmark (100 km south of the From next year, you will have to cope with an edi- Northern tip of the peninsula). One of the reasons tor who is not a native English speaker. that I feel warmly about the EMS is that I have Fortunately, Robin Wilson has agreed to carry on been affiliated with several European countries. I work for the Newsletter as an associate editor, and Martin Raussen was born and raised in Germany, and studied the Society has promised to find some help with mathematics and computer science at the the necessary revisions of the English language for How can we find articles that are well suited for Universities at Saarbrücken and Göttingen, where contributions from outside the British Isles. the Newsletter? For the present issue, I was lucky. I earned a Ph.D. degree in 1981. During a year of When beginning my job as an editor, I had of The society’s committee on ‘Raising Public studies in Paris, I met my Danish wife, and it is her course to reflect on how I would like the Awareness’ has recently run a competition for the fault that I moved to Denmark and finally became Newsletter to develop in the future. My only expe- best articles on mathematical themes that have established at the (relatively new and reformed) rience in this business is my affiliation with the been published in a newspaper or a periodical for university at Aalborg. My research interests are in newsletter Matilde of the Danish Mathematical a general readership; see the report by the com- geometry and topology; in recent years, I have Society. It was founded in 1999, and we have only mittee’s chair Vagn Lundsgaard Hansen in this tried to work with applications in theoretical com- recently reached issue 18. I have been in the edi- issue. Some of the proposed articles report on top- puter science, notably concurrency theory. torial board from the very beginning, and have ics that are not very well known among many pro- mainly worked with the interview section. For cer- fessional mathematicians; one of these is included History tain issues and for the past year, I was the editor- in this issue; others will probably follow. And of Issue 50 of the Newsletter of the European in-chief, and I hope that I will be able to use some course, I have asked people I know for interesting Mathematical Society is of course an occasion to of my experience. surveys from their (electronic) drawers… reflect on the development of the Newsletter and In my view, the most important lesson is that a This strategy cannot be successful for a long on what it might look like in the future. newsletter - as any journal, for that matter - relies period. This is why I hope to be able to ‘steal’ first- The first newsletter appeared in September on a collaborative effort. When all members of an class papers from several of the excellent newslet- editorial board divide the tasks and do their best in ters of the national mathematical societies, and finding good articles, then there is every chance have them translated. I have written to the editori- for a good result that readers will appreciate. This al boards of the newsletters that I know of and newsletter is not a specialised research journal. On invited them to collaborate. Some of them have the other hand, its readership consists mainly of given very supportive answers, and I hope for professional mathematicians whose main profes- more to arrive soon. Let me use this opportunity to sional interest is mathematics. So, apart from invite you all to tell me about well-written and up- spreading information about European mathemat- to-date papers for a general professional mathe- ical affairs, we have to serve the reader with arti- matical readership. cles on the subject itself, on mathematical themes. Another idea is to develop the electronic side of Our aim is to include at least two feature articles in the Newsletter. There is already an electronic every issue - articles that inform the reader about archive covering the most recent issues, so that newer mathematical themes (including connec- you can read or download most of the content. I tions with other disciplines, of course) in a non- would like to expand this service to a collection of technical manner, often in the form of a survey. As searchable databases which, apart from the earlier, these will be complemented by articles on archive, contain the information from the Recent the history of mathematical developments and by Books and the Conferences sections and thus interviews with significant mathematicians. The make them more useful. When established over a layout of the Newsletter will be a concern, too, that range of years, users should be able to search for will be worked upon in collaboration with our reviews of books on a particular topic, to connect Southampton printer and designer. electronically to other reviews or to the publisher of a book that they are interested in. We will experiment with this service over the coming year in collaboration with the suppliers of the Recent 1991, almost a year after the foundation of the Book section from the Czech Republic; it will cer- society itself. In its list of contents, you can see tainly take a little time before the readers can ben- some of the items that would fill the later issues: efit from it. agendas, announcements of forthcoming confer- All these projects can only work out with the ences, newspaper articles, advertisements, and so active support from the editorial board of this on - but no genuine mathematics at all! The Newsletter, from those of the member societies Newsletter has developed a lot through the twelve and from individual members. Please send me years of its existence and many people have con- your contributions and your ideas, by ordinary or tributed to its contents (including some mathemat- by e-mail. Thank you in advance! ics) and to its look. The editor for the past five Martin Raussen [[email protected]], years, Robin Wilson, reported in his editorial in Jan Kosniowski (printer), Robin Wilson Department of Mathematical Sciences, Aalborg the last issue 49 about many initiatives under his & Martin Raussen University, Denmark. EMS December 2003 3 EMS NEWS

EMSEMS ExecutiveExecutive MeetingMeeting inin LisbonLisbon David Salinger, EMS Publicity Officer

The Executive Committee met on the afternoon look at venues and arrangements for the fifth of the 14th September, immediately after the European Congress of Mathematics. The report very successful Joint Mathematical Weekend was very positive and the Executive Committee of the Portuguese Mathematical Society and would recommend the proposal to the EMS the EMS. The Executive Committee might Council. have been gloomy, in view of the lack of suc- The schedule of the Fourth Congress in cess in bidding for funds from the 6th Stockholm was largely fixed (and all the infor- Framework Programme. In the event, so much mation is now on the website). Council would was going on that cautious optimism was the meet in Uppsala before the Congress. order of the day. Unfortunately, the EU would not support stu- The President had attended the very impres- dents attending the Congress. sive Kolmogorov meeting in Moscow and There was a good prospect of having further David Salinger would also represent the Society at the forth- mathematical weekend meetings in 2004 and coming meeting of Spanish, Portuguese and 2005 in collaboration with two of our member EMS-SIAM-UMALCA meeting in Chile and South American societies in Santiago de Societies. would participate in ICIAM2007 in Zürich. Compostela. A Euroscience Open Forum had been organ- The Raising Public Awareness Committee The Treasurer reported that, while expendi- ised for August 2004 in Stockholm, and the had awarded three prizes for popular articles ture was running as forecast, additional mem- EMS would make a proposal for a session, about mathematics. The Committee on Women in Mathematics had distributed its question- naire in Ukraine and reported its findings. The President had been encouraged by the willingness of mathematicians to join the EMS Scientific Advisory Panel, which appeared to work well in helping the Executive Committee judge the merit of proposals. EULER was continuing as a consortium reg- istered in Germany: The EMS wished to con- tinue being involved and would appoint two representatives. The Committee reaffirmed its support for Zentralblatt and the digital math library project. The Publishing House would soon produce its first books and journals. Then, news of the Newsletter itself. Martin Raussen was welcomed as the incoming Newsletter Editor. Robin Wilson was thanked for all the work he had put in. In his acknowl- edgement, Robin paid tribute to Jan Kosniowski’s work on the design, which had played a big part in the Newsletter’s success. In an extra item of business, the Committee agreed to support the Belgian Academy’s criti- The Executive Committee with guests and Portuguese hosts cism of the blind use of the Impact Factor as an bership would enable the Society to spend a lit- based on emphasising subjects where applica- automatic formula determining a part of tle more money than planned. It was agreed tions had necessitated the development of new research funding. that some of this would go to support planned mathematics. After the local organisers had been thanked, EMS meetings in 2004 that had not obtained The Society would be represented at the the meeting closed. EU funding. Fourteen mathematical institutes, members of ERCOM, had joined the Society. Individual membership had also increased, to around 2400: the President stressed the importance of individual membership in strengthening the voice of the Society. None of the applications to the EU Framework Programme had been successful, but one - the project to support Summer Schools and other meetings - had been placed on the reserve list. This gave some hope that the Society would succeed in the second round. The Committee would encourage the Summer Schools planned for 2004 to go ahead. (Subsequently, the Committee has been heart- ened to learn that four of the Summer Schools will run, thanks to the enthusiasm of the organ- isers.) A site visit had been made to Amsterdam to Secretaries Tuulikki Mäkeläinen and Helge Holden at work 4 EMS December 2003 EMS NEWS MathematicalMathematical ArticlesArticles forfor thethe generalgeneral publicpublic Report on the EMS article competition

Vagn Lundsgaard Hansen (Lyngby, Denmark)

Background FIRST PRIZE In the current climate for intellectual activities Professor Nuno Crato, Department of in general, and for mathematics and science in Mathematics, Instituto superior de Economia e particular, the future of a subject may strongly Gestão, Universidade Técnica de Lisboa, Rua depend on stimulating articles about its trea- Miguel Lupi 20, 1200 Lisboa, Portugal, for a sures and importance for society. In this con- three-part article Cibersegredos invioláveis text, it is widely accepted that writing articles (Unbreakable ciber-secrets), published in the about mathematics for a fairly general public Portuguese weekly newspaper Expresso, on 8, is a particularly hard task - but it is not impos- 22 & 29 September 2001. Vagn Lundsgaard Hansen sible, as witnessed by such articles appearing from time to time in many countries. In order Evaluation of the jury: article by Bruss is published in this to encourage authors to produce mathematical This is an excellent collection of three short Newsletter. The article can also be found in articles with wide appeal in an international articles for a general audience published in a German and English under “Publications” at: forum, the European Mathematical Society, weekly newspaper with large circulation. It is http://www. through its committee for Raising Public on a hot social and economic topic, where ulb.ac.be/facs/sciences/math/perso/bruss.html Awareness of Mathematics (RPA), has recent- advanced mathematics plays a crucial role, ly run a competition for articles that have even though it is not possible to explain the THIRD PRIZE appeared in a newspaper, or some similar gen- full technical aspects. Nevertheless, in the sec- Professor Sava Grozdev (co-authors Ivan eral magazine, in the home country of the ond article, the key mathematical idea is Derzhanski, Evgenia Sendova), Union of author. All languages that could be read by explained for a large audience, and in the Bulgarian Mathematicians, Acad. G. Bonchev more than one member of the RPA-committee third a clear message of the interdisciplinary Street, Block 8, 1113 Sofia, Bulgaria, for their were allowed. aspects of the topic is given, not forgetting article For those who think mathematics drea- By 31 December 2002, the deadline for sub- mathematics. It might be said that it is a some- ry, published in the Bulgarian daily newspaper missions, we had received 26 proposals from how standard article on cryptography of Dnevnik, 27 December 2001. 14 countries. which there are several similar around. The three-part article is, however, very well writ- Evaluation of the jury: The jury ten, interesting and amusing, and shows how This is a well-composed article with a positive The jury consisted of the members of the mathematics is useful for important issues in message concerning mathematics. It is written RPA-committee: life. This article has wide appeal and deals in a charming personal style and it contains Chris J Budd, (UK), with a timely topic in a style very suitable for beautiful pictures. It is accessible for almost Mireille Chaleyat-Maurel, (France), the scientific section of a good newspaper. anyone as there are no mathematical formu- Michele Emmer, (Italy), The prize winning articles by Crato can be las. This may make it superficial for the Andreas Frommer, (Germany), found in Portuguese and English at: expert, but it will undoubtedly attract the gen- Vagn Lundsgaard Hansen, Chair, (Denmark), http://pascal.iseg.utl.pt/~ncrato/EMS/ eral reader. Osmo Pekonen, (Finland), The prize winning article by Grozdev, José Francisco Rodrigues, (Portugal), SECOND PRIZE Derzhanski and Sendova can be found in Marta Sanz-Solé, (Spain). Professor F. Thomas Bruss, Dpt. de Mathéma- Bulgarian and English at: http://www.math. tiques, Université Libre de Bruxelles, Campus bas.bg/ml/iad/dremat/dmathen.html The jury had a similar dilemma to that of the Plaine, CP 215, B-1050 Bruxelles, Belgium, Links to the winning articles can also be poster competition for World Mathematical for his article Der Ungewissheit ein found at the URL containing the Internet Year 2000. Some articles were mathematical- Schnippchen schlagen (Playing a Trick on information on the results from the article ly very elegant and attractive to mathemati- Uncertainty), published in the magazine competition: http://www.mat.dtu.dk/people/ cians, but were probably not readable by a Spektrum der Wissenschaft, 6 June 2000, and V.L.Hansen/rpa/resultartcomp.html general reader. Other articles seemed too sim- a similar article in the daily German newspa- In addition to the winning articles, the jury ple, and though readable by a general reader, per Die Welt, 17 May 2001. wishes to mention the runner-up in the compe- were neither stimulating nor representative of tition. The article in question is an excellent mathematics. Evaluation of the jury: one on symmetry. But after thorough consid- After a careful selection, including evalua- This is an excellent article. It immediately eration, the jury felt that it was addressed to a tions of all proposals received, four proposals grabs the reader’s attention with a practical more mature mathematical audience than the were selected for the final round. problem, shows that this problem really is articles on mathematics you would expect to hard, inviting the reader to think what to do find in a newspaper. The prize winners about it. It also shows the reader that mathe- Based on the evaluations, and placing empha- matics is needed to solve the problem - without RUNNER-UP sis on the general interest and readability of an leaving them behind, and links the problem Directeur de la Rédaction Philippe Boulanger article, the RPA-committee of the EMS made with the rest of the mathematical universe. (on behalf of Professor Alain Connes), Pour la its recommendations for the prize winning Published in a high-level general scientific Science, 8, rue Férou,75278 Paris Cedex 06, articles to the Executive Committee of the journal, it is probably most appealing to edu- France, for the article Symétries (par Alain EMS, which accepted the recommendations at cated people. Connes), published in the French magazine its meeting in Lisbon, 13-14 September 2003. The English version of the prize winning Pour la Science, February 2002. EMS December 2003 5 EMS NEWS Evaluation of the jury: This is an original and masterly article at a high level, and a beautiful expression of the unity of EMSEMS SummerSummer SchoolsSchools mathematics. The message concerning mathe- matics is (among other ideas) that some notions 20032003 andand 20042004 of mathematics like symmetry have a universal value. The author knows the history of mathe- Two EMS Summer Schools have been held in 2003: Applications to the theory of risk measures in a matics and embeds his concepts in the historical Stochastic Methods in Finance, Bressanone dynamic context were suggested, with particular and logical development of ideas. Reading the (Italy) emphasis on the issues of time consistency of the article requires a solid mathematical back- The CIME-EMS Summer School: “Stochastic dynamic risk measures. ground but a university student of pure mathe- Methods in Finance” (July 6-13, 2003), was attend- Prof. Walter Schachermayer, Technical Univ. of matics will undoubtedly find it extremely inter- ed by 115 scientists and researchers, coming from all Vienna: Utility Maximization in Incomplete esting. The article is, however, not readable by continents: among them 85 were Europeans, 35 of Markets. a general reader without a good level of math- which Italians. This course was mainly focused on the maximization The aim of the School was to provide a broad and of the expected utility from terminal wealth in ematical knowledge and motivation: it is too accurate knowledge of some of the most up to date incomplete markets. A part of the course was dedi- abstract for the general public. There are inter- and relevant topics in Mathematical Finance. cated to the presentation of the stochastic model of esting applications of permutations to problems Particular attention was devoted to the investigation the market, with particular attention to the formula- in music, textile patterns and dancing, which of innovative methods from stochastic analysis that tion of the condition of No Arbitrage. Some results of would present the same material in a more rel- play a fundamental role in the mathematical model- convex analysis and duality theory were also intro- evant way and would have greater appeal. ling of finance or insurance: the theory of stochastic duced and explained, as they are needed for the for- Nevertheless, it is an extraordinary pleasant processes, optimal and stochastic control, stochastic mulation of the dual problem with respect to the set and valuable read for a mathematically educat- differential equations, convex analysis and duality of equivalent martingale measures. Then some ed person. theory. recent results of this classical problem were present- The outstanding and internationally recognized ed in the general context of semi-martingales finan- Other articles lecturers have contributed in an essential way to the cial models. There were also other valuable articles submit- development of the theory and techniques that con- Marco Fritelli, Università di Firenze (Italy) ted for the competition. I was particularly sur- stituted the subjects of the lectures. The financial ori- prised by information on the occurrence of the gin or motivation of the mathematical analysis Dynamical Systems, Porto (Portugal) numbers from 1 to 9 as the first significant digit undertaken was presented in a rigorous manner that The EMS Summer School in Dynamical Systems facilitated the understanding of the interface between was held at the Department of Pure Mathematics of of a number from everyday life (temperature, mathematics and finance. Great evidence was also the Faculty of Sciences at the University of Porto dates, prizes, quotations, etc.) given in an article put on the importance and the efficiency of the math- from June 30th to July 4th, 2003. It was proposed as on La loi de Benford. One meets many more ematical instruments for the formalization and the a 1-week training program for graduate students and numbers in daily life beginning with 1, 2, or 3, resolution of the given financial problem. Moreover, young researchers, leading to an understanding of than with 7, 8, or 9! All mathematicians surely the direct financial origin of the development of recent developments in Dynamical Systems and will appreciate a fascinating and interesting his- some theories of remarkable importance in mathe- . The organizing committee received torical article on Stefan Banach and the Lvov matics has emerged with clarity. 103 applications of which 63, mainly post-graduate Mathematical School. Finally, I will mention a Short summary of the five courses: students from 18 countries, were able to attend the well-written article on the incompleteness phe- Prof. Kerry Back, Univ. of St. Louis: Partial and School. nomenon in . Some of the asymmetric information. The invited lecturers (Luís Barreira, Jean-Marc articles mentioned might be of interest for the In the context of economic equilibrium, a survey of Gambaudo, Floris Takens, ), from readers of the Newsletter. incomplete and asymmetric information (or insider research centres on dynamical systems (the trading) models was presented. First, a review of fil- Technical University of Portugal, the University of More competitions? tering theory and stochastic control was introduced. Bourgogne in France, the University of in The RPA-committee was impressed by the cre- In the second part of the course some work on The and the Institute of Pure and ativity shown in many of the submitted articles. incomplete information models was analyzed, focus- Applied Mathematics in Brazil, respectively) are ing on Markov chain models. The last part was con- experts in this area, and their lectures included mate- It was a particularly difficult task to decide cerned with asymmetric information models, with rial from their recent publications. between articles with lesser depth for a broad particular emphasis on Kyle model and extensions The School was financially supported by the public and articles with greater depth for a more thereof. European Mathematical Society, through UNESCO- mathematically educated public. I think we Prof. Tomasz Bielecki, Illinois Institute of Techno- ROSTE (contract number 875.600.3), and by Centro found some very good and representative win- logy: Stochastic methods in credit risk modelling: de Matemática do Porto/Centro de Matemática ners, but there were also other deserving arti- valuation and hedging. Aplicada do Porto, through the Portuguese cles. This indicates that there is a good basis for A broad review of the recent methodologies for the Foundation for Science and Technology. More further competitions. management of credit risk was presented in this details and photos can be found on the web page The RPA-committee of the EMS might there- course: structural models, intensity-based models, http://www.fc.up.pt/cmup/sds/. fore arrange another competition fairly soon. To modelling of dependent defaults and migrations, Maria de Fatima Carvalho (Centro de Matemática avoid the difficulty of deciding between articles defaultable term structures, copula based models. da Universidade de Porto, Portugal) on mathematics for a general public and popu- For each model the main mathematical tools have lar articles on mathematics for an educated pub- been described in details, with particular emphasis EMS Summer Schools in 2004 lic, the RPA-committee suggests that the on the theory of martingales, stochastic control, Plans for four EMS Summer Schools are about to emphasis of a new competition should be given Markov chains. be set up currently: to popular articles on mathematics for the edu- Prof. Christian Hipp, Univ. of Karlsruhe: • Summer School on Evolution equations and Financial control methods applied in insurance. applications cated layman and professional mathematicians. The methodologies developed in modern mathemat- Cortona, Italy, July 4-24, 2004. Let me finally thank all of the authors who ical finance have found wide use also in the applica- • Analysis on metric measure spaces submitted articles for the competition. I am tions to the control and the management of the spe- Bedlewo, Poland, July 15-23, 2004. happy to say that the RPA-committee was very cific risk of insurance companies. In particular, in the • Empirical Processes: Theory and Statistical satisfied with the number of submissions and course it was shown how the theory of stochastic Applications with the generally good quality of the contribu- control and stochastic optimization can effectively be Santander, Spain, August 30 - September 3, tions. used and how it can be integrated with the classical 2004. insurance and risk theory. • Séminaire Européen de Statistique (SemStat): Vagn Lundsgaard Hansen [V.L.Hansen@ Prof. Shige Peng, Shandong Univ: Nonlinear expec- The Statistics of Spatio-temporal systems mat.dtu.dk] is chairman of the EMS-committee tations, nonlinear evaluations and risk measures. EURANDOM, Eindhoven, The Netherlands, on Raising Public Awareness of Mathematics. In this course the theory of the so-called ‘g-expecta- December, 2004. He is a Professor of Mathematics at the tions’ was developed, with particular attention to the Department of Mathematics, Technical following topics: backward stochastic differential Latest News: The EMS summer schools planned University of Denmark, Lyngby, Denmark. equations, F-expectation, g-martingales and theo- for 2004 and 2005 will receive funding from the rems of decomposition of E-supermartingales. EU Framework Programme. 6 EMS December 2003 FEATURE PlayingPlaying aa tricktrick onon uncertuncertaintyainty F. Thomas Bruss (Bruxelles)

The European Mathematical Society is Paris….London ...; it is not likely that Mr. X grateful to Spektrum der Wissenschaft and and Mrs. Y know each other. Should you to Deutsche Mathematikervereinigung for perhaps try to push up the price by telling their kind permission to publish the English each of them how much interested the other version of this article, that was awarded the one is? Perhaps trying with Mr. X first? - second prize in the article competition of But no, you dismiss this idea, a man like the EMS-committee Raising Public Mr. X would hardly be impressed by this - Awareness of Mathematics. rather on the contrary. Trying with Mrs. Y perhaps? But then, the day she comes, Mr. If you have to decide between two X is already out of the game and can no alternatives without knowing which longer serve as a means of pressure. F. Thomas Bruss one is more favourable, then you And again you arrive at the same conclu- may quite as well flip a coin - sion as before: You may as well flip a coin Strategy: Think of an arbitrary num- Right? in order to decide. Perhaps you should sim- ber Z. Now uncover the first number, No, you can do better. ply make the deal with Mr. X to have at X, and choose this number if it is least your Sunday free! larger than Z, otherwise choose Y. You want to sell your house. Your ad “Sell for the best offer above 800,000 Euro” has Game with two cards Why should this strange strategy be better been running for weeks already in the news- Such situations in real life occur in many than choosing X or Y at random? paper. But now, next Sunday is the dead- different variations. A special offer in the Here is the proof: line. supermarket, a nice apartment, an attractive Recall X is the first number, Y the second. Two potential buyers have announced job offer, or even the woman or man for Let Min=min{X, Y} be the smaller and their definite interest. Mr. X from Paris life: One must so often decide without Max=max{X,Y} the larger one. There are called, saying that he will make an offer knowing whether something better is still to exactly three possibilities: exceeding 800,000 Euro but that he would come. like to see the house again next Saturday To clear the view for the problem, we (A) Both numbers X and Y are not before finalizing his exact offer. And then summarize the essence in terms of a little larger than Z, there is Mrs. Y who called from London to game: You ask your son and your daughter say essentially the same, except that she can to write, each one secretly, and not consult- Min Max Z come next Sunday only. Both Mr. X and ing each other either, one arbitrary number ______|______|_____|______Mrs. Y insisted that they would need your on his/her card. You point out that “arbi- irrevocable Yes or No on the very day of trary” means really as they want: Large, their visit. small, negative, decimal point, everything is (B) Z lies between X and Y (possi- You would have liked so much to find out allowed. Then they place their cards, face bly coinciding with the smaller more! If you only had been able to obtain an down, on the table. You can now turn over one), indication of what limit Mr. X and Mrs. Y the card of your son, inspect the number, were prepared to pay! and then decide whether you accept it. If Min Z Max However, all you got on the phone was a you refuse it, then you receive automatical- ______|______|______|______short laugh and something like “Please let ly your daughter’s card. Now both numbers me see the house again.” True business peo- are compared. If you have chosen the larger ple, both of them! You also have made number you win, otherwise you lose. (C) Both X and Y exceed Z. already your inquiries: Both are serious and The difference between these numbers is reliable, and both have the necessary funds. now without importance you just want to Z Min Max But then, it seems hard to guess who of the win. If the numbers happened to be the ______|______|______|______two could be expected to be the more inter- same, the game would be repeated, but this ested one. case is improbable. Further, if you think you According to the strategy, you choose the It is time to analyse the exact circum- may have some advantage from knowing number Y in case (A) and the number X in stances of your situation. Clearly, you will your children well, you can imagine them case (C). In these two cases you end up with have again the opportunity to point out the being replaced by others. Alternatively, one a random choice between the larger and the splendid features of your house. However, person may also fill in both cards. This real- smaller number, hence you win with proba- this will not change your dilemma: If you ly looks now like a purified game of chance bility ½. In case (B), however, you win with accept the offer of Mr. X you will lose the with win probability ½. certainty, because if X is the larger of the two one of Mrs. Y, and if you want to wait for But now the surprise: There is a strategy numbers you accept it, but if it is the smaller the offer of Mrs. Y you lose the one of Mr. with which you can increase your win prob- one, you refuse it. Thus your total win prob- X. This seems like gambling! You will lose ability above ½. It is based on an idea of ability is now the better of the two offers with probability Professor Thomas Cover (Stanford a c w= /2 + /2 + b, ½, won’t you? University). Let X and Y be the two differ- Another idea comes to your mind. ent numbers on the cards. where a, b, and c denote the unknown prob-

EMS December 2003 7 FEATURE abilities of the events (A), (B) and (C) the two numbers on the two cards and it is I existence would already be a true progress. respectively. One of these events is bound to who has to choose one. As before, I win if I However, at the current state of knowledge, I happen, of course, that is a+b+c=1, and choose the larger one. Suppose you would see no sufficiently safe foundation to attack hence like to decrease my win probability. What such a proof, not even to make this question should you do? sufficiently precise. (a+b+c) b 1 b w= /2 + /2 = /2 + /2. The answer is simple. You just choose two Is it not truly surprising that nobody can numbers that are very close to each other. optimise, in a strict mathematical meaning, Thus the win probability exceeds equal Take 6.123455 and 6.123456, say. My the sale of a house to two potential buyers? b chance by /2 > 0, because (B) can occur. advantage of using a Z-strategy is now hard- Given that we see so many impressive things [Remark: To make this precise it suffices to ly worth talking about because my chosen Z mathematics has achieved in our world, we choose Z according to any density that is will have little chance of falling between would agree that it is, wouldn’t we? Think of strictly positive everywhere on the real line.] these two numbers. other optimisation problems from modern How can you apply this strategy most skil- In real-world problems, things are often airplane engineering, for instance. Compared fully? Evidently you should try to make the different, however. Real-world strategies are to such problems, our little problem seems case (B) as probable as possible. This means developed by one party and typically not ridiculously simple. However, this is not you should choose Z such that it has the communicated to the other party. What dif- true. Airplane engineers have a huge time largest possible probability of falling ference does this make? advantage: They can work, routinely, with between X and Y. Since these two numbers To find this out, I made, several years ago, many methods for which the mathematical are unknown, no general recommendation a test with Vesalius College business stu- foundations have been well established for can be made. In concrete cases, however, one dents. Everybody in the audience received two to three centuries. This is not the case for may quite well have some ideas. two cards to write down his or her numbers, our little problem. and then I passed from one to the other to Such contrasts do exist in many fields of Optimal choice of a threshold make my choice. I had not mentioned Z- mathematics. Our house-selling problem is such a concrete strategies before. Is this a symptom of the eternal youth of a case. On the first card is the offer of Mr. X My score was 32 successes for 41 or 42 discipline? and you will not know the number on the participants. With a random choice we would Yes, I think it is. other card, the offer of Mrs. Y, when you expect some 21, and some three or four more must say Yes or No to Mr. X. The first dif- with a bit of luck. 32 however should not be Footnote: This is the author’s English translation ference to the card game with arbitrary num- explained by pure luck alone, as they knew. of his article Der Ungewissheit ein Schnippchen bers is that you know that both X and Y are Even the best students were puzzled. It is dif- schlagen published in Spektrum der Wissenschaft above 800,000. The second difference is that ficult to see what one does not expect. But (= German Edition of Scientific American). The the amount |X-Y| is now of real interest to you, dear reader, you probably guess correct- latter was based on the author’s Unerwartete you. ly: I had applied a Z-strategy, even a particu- Strategien published in Mitteilungen der An offer of 900,000 Euro or more would larly naive one. I had chosen Z = 0. Deutschen Mathematikervereinigung (German be nice, but is not very likely. On the other Why was this so successful? – I think it Mathematical Society.) The idea was instigated hand, if you accepted Mr. X’s offer of was because I could prepare the field for the by Cover’s Problem. (References below.) 801,000 Euro, say, then you would not suffer strategy: My remark “The numbers may also much regret if Mrs. Y would offer 802,000 be negative” had seemingly succeeded in Acknowledgement. The author would like to Euro. There is no point in trying to hedge being sufficiently casual. The fact is that thank Professor Lundsgaard Hansen for his com- against a loss of too modest a magnitude. numerous students made use of negative ments on the English version. Therefore it may be best to choose Z clearly numbers, and all those who had written down above 800,000 Euro, but then again not too just one negative number made me automat- References large. If you asked me what I would do: I ically a winner. F.Thomas Bruss, Unerwartete Strategien, Mitteilungen would toss a die and, for each eye of my This experiment shows that strategic think- der Deutschen Mathematikervereinigung, Heft 3 result, add 5,000 Euro to the amount 800,500 ing has no simple rules Some people preach (1998), 6-8. Euro. So, for instance, if I obtained 3, I would that the key to success in strategic behaviour F.Thomas Bruss, Der Ungewissheit ein Schnippchen choose Z=815,500. But, by all means, there is always narrowing down the adversary’s schlagen, Spektrum der Wissenschaft, Juni Heft is nothing special about this suggestion and field of action. However, this is not true. If (2000), 106-107. you may be much happier with your own we believe that our adversary does not expect Thomas M. Cover, Problem 2.5 : Pick the largest num- idea. our strategy it can be unwise to narrow down ber. In: Open Problems in Communication and Why toss a die? Why not simply fix the set of his or her options. The less one can Computation, Springer Verlag, New York. (1987). Z=820,000, say, if we feel this should be do, the more one thinks about each step. more or less in the right order of magnitude? Indeed, in our experiment, by allowing nega- F. Thomas Bruss [[email protected]] studied Apart from our probability argument there is tive numbers we did not narrow down but Mathematics in Saarbrücken (FRG), Cambridge, another reason: In such a game-theoretical actually enlarged the set of options for the and Sheffield (UK). He started his career as assis- situation, it is often better to be unpre- students. This probably helped to distract tant and first assistant in Namur (B) from where dictable. If we act in a predictable way, the from paying attention. he moved on to the United States, where he was other player may adjust his behaviour. Hence Visiting Associate Professor at UC Santa the introduction of a random component. A few words about mathematics Barbara, U of Arizona, and UCLA, successively. What is our strategy worth? - Definitely You have just learned to know a little prob- In 1990 he was appointed Professor at Vesalius more than the random choice, as we have lem in a field of mathematics, which, com- College of the Vrije Universiteit Brussels. Since seen. We cannot really quantify the advan- pared to other fields, is still in its infancy: 1993 he is Professor of Mathematics and tage compared with a random choice, but strategic thinking, as a part of probability the- Statistics at the Université Libre de Bruxelles. His some additional 10,000 Euro or so may quite ory. Even at this introductory level several research is in probability: limit theorems, branch- well be in it (in expectation.) questions are still open. For instance, does ing processes, probabilistic models, and optimal Let us go a step further and look again into there exist a strategy for the two card game stopping. He is fellow of the Institute of our two-card game. Now you and I play the which is generally more efficient than a Z- Mathematical Statistics and Feodor-Lynen-fellow game. Suppose you are the one who writes strategy? - The proof of existence or non- of the von-Humboldt Foundation.

8 EMS December 2003 FEATURE

IntegrabilityIntegrability Integrability of Hamiltonian systems ofof Mich`ele Audin (Strasbourg) HamiltonianHamiltonian SystemsSystems

Michèle Audin, Strasbourg

A satellite moves along its Integrable systems Let us carry out an entertaining ex- around the Earth. A circular orbit, periment. Rather than putting a satel- an apparently quiet revolution, never- A Hamiltonian system is a mechani- lite in orbit (which is complicated and theless accompanied with various mo- cal system governed by the celebrated expensive), we just play with a spin- tions: the satellite oscillates, rotates, Hamilton equations ning top. And we observe it spinning, turns upside down.  looking carefully at the motion of the ∂H ∂H The attitude, this is how physicists  end of its axis. q˙1 = ,...,q˙n = , call these motions – motions that can  ∂p1 ∂pn For those of you who do not have a obviously not be ignored. Let us spinning top at home, Figures 1 and 2  pretend that the satellite decides (de-  ∂H ∂H show the object and the result of the p˙1 = − ,...,p˙n = − . plorable attitude) to turn its back to the ∂q1 ∂qn experiment. The top is considered as Earth, the antennae and the cameras a rigid body with a fixed point (the now looking at the other side. The total energy H of such a system, point O at which it meets the horizon- q ,...,q This is obviously not what was in- a function of 1 n (think of po- tal plane) in a constant (vertical) grav- p ,...,p tended when the satellite was put into sitions) and 1 n (think of mo- itational field. The rigid body has an orbit. It is thus necessary to know how menta), is constant, and it does not de- axis of revolution in this case, so that, to describe and control these changes pend on time. It happens that other in addition to the total energy, the mo- in the attitude. Will the satellite always quantities are conserved as well. They mentum with respect to this axis is a perform the same motion? Will it wrig- are called first integrals. first integral. The end of the axis os- gle restlessly? If there are enough first integrals cillates between two parallel circles on The mechanical system governing (and if they commute, in a sense that an (ideal) sphere. I will not make more precise here), Li- the motion of the satellite is a Hamilto- Here is another easy experiment. We nian system. The question I just asked ouville showed, in the 19-th century, fix a ball at an end of a rod, the other can be made more precise and refor- that the differential system (Hamilton equations) can be solved by quadra- end of which is fixed: what we get is mulated as is this Hamiltonian system a pendulum, usually called a spheri- integrable?. A theorem of Morales and tures (computing integrals); this is why the system is said to be integrable. cal pendulum. Here again, the only Ramis can be applied to answer this force present is the gravitation. The question, using differential Galois the- momentum with respect to the vertical ory. The aim of this article is to give Examples (direction of the gravitation field) is a an idea of the statement of this theorem first integral. The ball turns, stuck be- and the ways to apply it. There are many well known mechani- tween two parallel circles on a sphere, cal systems that are integrable. as shown on Figure 4.

O

O

Figure 1: A spinning top Figure 2: The end of the axis Figure 3: A spherical pendulum

EMS December 2003 9 FEATURE

Figure 4: A trajectory of the Figure 5: A geodesic on a sur- Figure 6: A geodesic on an ellip- spherical pendulum (from face of revolution soid above)

You may have noticed the similar- orem), each visiting regularly a neigh- systems that are not integrable. The ities between these two experiments. bourhood of its initial point, and the most famous is the three-body problem A similar behaviour (oscillations in a motion is said to be quasi-periodic. dealing with three bodies (Sun-Earth- band) can be observed in many other Figure 7 shows a quasi-periodic mo- Moon) in gravitational interaction. It mechanical problems, as, for instance, tion that is drawn in the configuration is known that the two-body problem the motion of a free particle on a space (an annulus, on the sheet of pa- (Sun-Earth) is integrable. It was ac- surface of revolution or an ellipsoid. per), or in (in more dimen- tually to integrate this problem that A free particle goes the shortest way sions, on a torus) that looks very much Lagrange introduced the beginnings – this is why the solutions are the like the previous ones. The Arnold- of symplectic geometry and Hamilto- geodesics on the surface. Figures 5- Liouville theorem is more precise – it nian mechanics. Poincar´e showed that 6 represent a geodesic on a surface of states that these trajectories are linear in the three-body problem could not have analytic revolution and of an ellipsoid (respec- the sense of the affine structure of the enough first integrals ( in posi- tively). In the case of the surface of rev- torus, as shown in Figure 8. tions and momenta). olution, the momentum of the particle The method I explain here allows us with respect to the axis of revolution is to prove that this is still true if we ac- a first integral. For a generic ellipsoid Are all Hamiltonian sys- cept poles – namely, meromorphic first there is also a first integral, although integrals. this is less obvious (this is due to Ja- tems integrable? Some Hamiltonian systems are sus- cobi). pected to be non-integrable because A geometrical or dynamical expres- As we have seen, physics can provide we have not been able to find enough sion of Liouville integrability (as de- first integrals as the momentum with first integrals and, more seriously, be- fined above) is the regularity of the so- respect to an axis of revolution (this cause some experiments or numerical lutions. The motion described by an is what happens for a spinning top, a simulations show a chaotic behaviour integrable Hamiltonian system is very spherical pendulum, a free particle on that seems to be incompatible with the regular, the trajectories wind on tori a surface of revolution, ...) Arnold-Liouville theorem. This is the (this is part of the Arnold-Liouville the- There are also many Hamiltonian case for the H´enon-Heiles system:

Figure 7: Trajectory on an annulus Figure 8: Linear trajectories on a torus

10 EMS December 2003 FEATURE

0.03

0.02

0.01

0

-0.01

-0.02

-0.03 1.68 1.69 1.7 1.71 1.72 1.73 1.74 1.75 1.76

Figure 9: Chaotic behaviour of the H´enon-Heiles system

The H´enon-Heiles system is the Y : This is an Airy equation, the solutions Hamiltonian system defined by the of which, the Airy functions, are an- Y˙ =˙y − x˙ Hamiltonian alytic on the whole complex line, but = X(y(t)) − X(x(t)) no solution of which is a rational func- 1 =(dX)x(t)(Y (t)), up to order 1. H = (p2 + p2) tion or even an algebraic function. For 2 1 2 those who like formulas, Airy func- This linear differential equation in Y is 1 2 2 2 1 3 tions can be written + (Aq1 + Bq2 ) − q1q2 − λq2 , the variational equation. 2 3 ∞ Q(t)= cos(x3 ± xt)dx. where A, B and λ are parameters. It is The H´enon-Heiles example 0 known to be integrable for some values of these parameters. Figure 9 comes In the simple case of the H´enon-Heiles Differential Galois theory – in from Morales’ book and is supposed to system for A = B =0and λ =0, the about 195 words! illustrate the chaotic behaviour of this Hamiltonian is system. It shows part of the dynam- 1 The situation of a linear differential H = (p2 + p2) − q q2. ics (the dynamics of the Poincar´e map, 2 1 2 2 1 equation with polynomial coefficients, to be precise) for the parameters A = whose solutions are not rational, is B =0 λ =3/2 and : it seems to indi- The Hamiltonian system is reminiscent of the situation of an al- cate that the system is not integrable in gebraic equation with rational coeffi- this case. Using the method I am going q˙1 = p1, q˙2 = p2 cients and irrational roots. to explain, it is possible to prove that, 2 The base field is the field of mero- p˙1 =2q1q2, p˙2 = q1. in general, the H´enon-Heiles system is morphic functions on our trajectory – not integrable. C(t) It is quite easy to find solutions (this this is the field in the example of And the attitude? Is the attitude of a is an academic example!). We choose Airy. With the linear differential equa- satellite an integrable system? a trajectory that is a straight line: tion is associated a smallest differential field which contains all the solutions of the linear equation (the Picard-Vessiot q1 =0,p1 =0, q (t)=at − b, p (t)=a extension, the differential analogue of The Galois group, an ob- 2 2 the splitting field of an algebraic equa- a b tion). The differential Galois group is struction to integrability for some constants and . The varia- its group of automorphisms. The set of tional equation along one of these solu- solutions of a linear differential equa- Following a tradition that goes back, tions is the linear differential system tion is a vector space on which the Ga- at least, to the M´ecanique c´eleste of  lois group acts as a subgroup of the Q˙ = P , Q˙ = P , Poincar´e, let us consider the variational  1 1 2 2 corresponding linear group in the same equation, a linear differential equation  way as the Galois group of an algebraic  ˙ ˙ that describes the solutions that are in- P1 =2q2(t)Q1, P2 =0. equation is a subgroup of the symmet- finitesimally close to a given solution. ric group, which acts by permutation We assume that we know a solution We look only at the solutions for which of the roots. According to a theorem of ˙ x(t) of the Hamiltonian system. Let Q2 = P2 =0. The linear system Q1 = Kolchin, the differential Galois group is ˙ y(t) be another solution, that is very P1, P1 =2q2(t)Q1 is equivalent to the an algebraic group. close to x(t). Then we can write y(t)= differential equation In the Airy example, the solutions x(t)+Y (t) and, up to order 1, our (and, more precisely, their behaviour ¨ Hamiltonian system becomes linear in Q − 2(at − b)Q =0. at infinity) are intricate enough for the EMS December 2003 11 FEATURE

Galois group to be really huge: it is the the simple case of the H´enon-Heiles ex- She was investigating the integrabil- entire group SL(2; C). ample (A = B = λ =0), we have seen ity of the rigid body with a fixed that the Galois group is SL(2; C). This point. She asked herself under which group is not almost Abelian, and hence assumptions the solutions of the sys- The Morales-Ramis theorem the system cannot be integrable. How- tem would all be meromorphic functions 3 ever the case A = B =0,λ = 2 illus- of the time variable – this is called the The Galois group is the main character trated in Figure 9 is still open. A non- Painlev´e test – that is, their only sin- of a non-integrability theorem due to symmetric spinning top cannot lead to gularities are poles (no ramification, no Morales and Ramis, in a tradition that an integrable system either. logarithm, etc). goes back to Kowalevskaya, Poincar´e, Morales and Ramis have many ap- She proved that this property is satis- Painlev´e and, more recently, Ziglin. plications, some of which can be found fied in only three cases: when the fixed This theorem can be considered as an in Morales’ book (cf. reference below). point is the centre of gravity, when the analogue of the theorem on the solv- rigid body has an axis of revolution (in ability of equations by radicals. It these two cases, the system was known asserts that if a Hamiltonian system is The satellite to be integrable), and in a new case integrable, then the Galois group of the (now called the Kowalevskaya top). variational equation along any trajectory The main difficulty is not to find a par- She was in fact able to find an addi- must be almost Abelian (in the sense that ticular solution to start with. The Ga- tional first integral in this last case too. its neutral component is an Abelian lois group is an algebraic subgroup of GL(2n; C) n group). Notice that, if we neglect fi- the linear group , where is The relation between integrability nite groups, this is a stronger prop- the number of degrees of freedom of (in the Liouville sense) and the softness 2 erty than being solvable. Notice also the system. This number is in most of the singularities of the solutions is that this theorem, although it gives a of the examples considered here, which still not completely clear. The methods gives a subgroup of GL(4; C). Using of differential algebra used here give very powerful tool for proving non- 1 integrability, is much easier to prove the fact that the Hamiltonian is a first precisely such a relation, up to order . than the algebraic analogue: being an integral, this can be reduced to a sub- SL(2; C) algebraic group, the Galois group can group of , which allows us to Acknowledgments be attacked with infinitesimal tools – compute it. the statement is actually that its Lie al- However, the attitude of the satellite The spinning top in Figures 1 and gebra must be Abelian. in a circular orbit is a system with 3 2 was originally drawn by Raymond degrees of freedom, so that, after re- Seroul. Figure 9 was given to me by duction, we still have a subgroup of Juan Morales. I thank them, together Conclusion SL(4; C). This is too big: recall that the with Etienne´ Ghys who is responsible group needs only to be almost Abelian, for the fact that I wrote the first ver- so that it is not enough to find two sion of this paper, with the anonymous In concrete terms (if I dare say it): elements that do not commute. With referees he used to help me improving you choose a solution, you linearise the help of computer algebra (resp. an it, and with Martin Raussen and Robin the system along it, and you compute additional geometric argument), Del- Wilson for their help with the present the Galois group. If it is not (almost) phine Boucher (resp. the author of the version. Abelian, your original system was not present paper) were able to prove, in integrable. Notice that you should Mich`ele Audin [Michele.Audin@ 2003, that the attitude is not integrable. have found a solution, that is both sim- math.u-strabg.fr] studied mathemat- ple enough so that you are able to This does not prevent the satellite ics in Paris and Orsay. She has been a SPOT compute the Galois group, and compli- to take its beautiful pictures of professor at Universit´e Louis Pasteur cated enough so that the latter is not the Earth (free advertisement!), since, (Strasbourg, France) since 1987. Her re- (almost) Abelian. as the orbits, the attitudes can be cor- search is in geometry and topology, es- rected. pecially on the geometry of integrable systems. She is the author of several Applications The end research and student textbooks.

Nevertheless, this theorem does have The first approach to non-integrability numerous applications; for instance: In goes back to S. Kowalevskaya in 1889.

A few references

Details, information and references are given in: • Juan Morales’ book, Differential Galois theory and non-integrability of Hamiltonian systems, Progress in Math., Birkh¨auser, 1999; • the author’s book, Les syst`emes hamiltoniens et leur int´egrabilit´e, Cours Sp´ecialis´es, 8, Soci´et´e Math´ematique de France & EDP Sciences, 2001; • the references in these two books, and more recent references in other papers of the author are listed on the website: http:www-irma.u-strasbg.fr/˜maudin . 12 EMS December 2003 REPORT KolmogorovKolmogorov andand ContemporaryContemporary MathematicsMathematics International Conference, Moscow 2003 Address at the Opening Ceremony Beno Eckmann (ETH Zürich)

I heard the name Kolmogorov for the first time few could anticipate the importance that topologi- around 1936, when I was a beginning mathemat- cal ideas and all their ramifications would gain ics student. Here is the story. In a course by during the century – but Kolmogorov did! Thus George Polya (‘Aufgaben und Lehrsätze’), there we have to admire the really independent mind of Beno Eckmann were nice and amusing applications of probability, this young man, independent of fashion. In that including his favourite topic ‘random walks’. But part of the century, it was fashionable to work on their long voyage for several months on the Volga we were unhappy with the so-called definitions of Hilbert’s problems. Surely he did that too, in the and beyond in 1929 – what impressed me tremen- the probability concept. For Polya, intuition and axiomatisation of the applied field probability. In dously was the fact that Andrei took with him on application were more important. that context, let me remark that algebraic topology that long trip, apart from mathematics books, the Michel Plancherel (‘Plancherel formula’), who (homology, analysis situs) is not even mentioned Odyssey! The better I got to know Kolmogorov, was also one of our professors, told us to consult in Hilbert’s famous list of 1900: it was not consid- the more I realised that his cultural universality the 1933 Ergebnisse book by a certain Russian ered to be a respectable field for a long time. I went far beyond mathematics, into logic and foun- named Kolmogorov, where the concept of proba- heard this much later from Hermann Weyl, who dations, into arts, poetry, history and education. bility was put on a final rigorous axiomatic basis. told me that he had published two papers on alge- His human and humanistic universality enabled I still remember it like today: it was sensational, braic topology in Spanish around 1925 in a South him to be an extraordinary teacher. dramatic and enlightening. I am not certain that American Journal, because he did not want his His students are here, they can tell about that Plancherel really liked the new type of abstraction. colleagues to read them! better than I can. He inspired them to do mathe- He was more impressed by a very early construc- Clearly I wanted to meet this man Kolmogorov. matics according to its true nature and unity: tion from 1923 of Kolmogorov, a Lebesgue- But there was World War II and then the Cold abstract, valid within its strict context, universal, Fourier series diverging almost everywhere; he War. Communication by letter was slow if not and (precisely for that reason) eminently practical. did not tell us that the author was not even 20 impossible, and contacts were limited. Finally, I With the passing of each human being a mys- when he wrote that paper! But we, the young met Kolmogorov in 1954; he gave the famous ple- tery disappears from the world, a mystery that ones, were happy, and the logical foundation of nary lecture at the International Congress in nobody else will be able to rediscover [Friedrich probability from 1933 was crystal-clear, abstract, Amsterdam - about dynamical systems, the begin- Hebbel]. Words, and certainly my words, are inad- and at the same time relating everything to the ning of what would later become KAM theory. equate to describe the mystery of Andrei standard terminology and to concrete applications. As for myself, surely enough, I lectured about the Kolmogorov. With regard to our common profes- We just liked the new trend of axiomatic abstrac- cohomology of groups. sion we learn from him that mathematics is a man- tion. As for the name Kolmogorov, it also was Now things continue in much the same way: At ifestation of the free creative power of the human quite abstract for us - just a name. the next International Congress 1958 in Edinburgh mind and the organ for understanding the world In 1939, I got in real contact with Kolmogorov I heard Kolmogorov lecture, this time about func- through theoretical construction. And that it is part when I began my Ph D. work in topology under tional analysis (just a short communication), at the of the cultural tradition we have to transmit to the Heinz Hopf (‘Hopf algebras’ and ‘Hopf fiber- 1962 Congress in Stockholm about optimal next generation. ings’); not in personal contact, of course, but Hopf approximation of functions, and so on. No need to To achieve more we dare not hope, to achieve told me about the famous topology conference in continue, you all know what I want to say: I was less, we must not try. We are grateful to Andrei Moscow in 1935. At that conference, Kolmogorov lucky to know Kolmogorov and to see him devel- Kolmogorov for his outstanding contributions had introduced a very important new concept op into one of the truly universal mathematicians towards this goal. (independently of J. W. Alexander) – namely, of our time, covering all fields (with the exception cohomology: homology is due to Poincaré of number theory, as far as I know), original, cre- The centenary of Andrei N. Kolmogorov’s birth (around 1900), and cohomology is its dual. ative, deep and broad. has already been commemorated in the Topologists had not expected a product structure Contacts with Russia became easier during the Newsletter with an article by N. H. Bingman and for arbitrary complexes. They had been con- years 1956-61 when I was secretary of the an address by EMS president Sir John Kingman in vinced that a ring structure is possible only for International Mathematical Union. It was the time issue 49. manifolds. Well, cohomology in very different when we succeeded in making the Union truly Beno Eckmann [[email protected]] appearances has been taken over by so many peo- international, in spite of great political difficulties. is professor emeritus at the Institute for ple, more and more in close connection with alge- All Eastern European countries and also China Mathematical Research, ETH Zürich, bra, functional analysis, measure theory and so on: became members. Paul Aleksandrov was appoint- Switzerland. it is no longer known in general that the concept ed as a new mem- goes back to Kolmogorov. Although my thesis ber of the Execu- was about homotopy, cohomology became one of tive Committee of my favourite research topics and remains so. IMU. I would Could this be the same person as the proba- have preferred bilist? Believe it or not, Hopf said, it was! Not Kolmogorov, but only that; he told me about a young man, remark- I understood that able in every respect, of unusual physical strength, he was apolitical climbing mountains of over 4,000 metres altitude, in every respect. skiing enormous distances, swimming in ice-cold Of course we water - almost like Paul Aleksandrov. liked Paul Alek- Aleksandrov was a very good friend of Heinz sandrov; he told Hopf; their joint book on topology is well-known. us many stories During these early years Kolmogorov made about his friend many other important contributions to algebraic Kolmogorov - for topology, for various types of spaces. At that time example, about A.N. Kolmogorov P Aleksandrov EMS December 2003 13 REVIEW they wanted to - one of them came the first day with some pages of computations to understand which primes would be Germain Proof on Broadway, numbers. But mostly they wanted to under- stand the characters of the play, and were gen- uinely interested in our lives, problems and motivations. Preuve à Bruxelles In September, I went back to the theatre to discover that the director Jonathan Fox, the Luc Lemaire, Bruxelles actresses Valérie Marchant (Catherine), Isabelle Defossé (Claire) and actors Alexandre von Sivers (Robert) and Philippe In August 2001, I attended in New York a per- During my first visit to the theatre, it was Allard (Hal) had come up with tremendously formance of ‘Proof’, by David Auburn, and clear that each member of the staff present in good performance (which obtained the also read the text of the play. the building on that day found a (weak) deserved success of having the room packed It was a huge success, got the author an excuse to enter the room where I was to have every night and going for a second season). avalanche of awards (including a Pulitzer a good look at me. It was my turn to learn something new, prize) and ran for more than three years on When I met the director of the play and the about theatre. Indeed, having read the play Broadway. A film version is now in prepara- actors, they started the ball rolling by explain- and attended one performance already, I tion. ing candidly that having read the play they realised more than before the differences that Quite enthusiastic about it, I wrote a review wanted “to see and possibly to touch a mathe- can be made in a play by the direction, the for the EMS Newsletter (Issue 44, June 2002, matician in the flesh”. Then to see how I scenography and the acting. Many ideas (dif- p.22). would react, one actor asked me what the play ferent in Brussels and New York) were In June 2002, the theatre ‘Le Rideau de was really about, “since everything to know is brought in (without adding any line to the Bruxelles’ announced that they would trans- known in mathematics”. Although he did it text), and since I actually saw the play three late and produce the play in French in very well (he is an actor!) the trick did not more times, I could see it evolve as more ideas September. On an impulse, I wrote to the work and they passed to other questions. were introduced. It was of course a new expe- directors of the theatre (whom I didn’t know After four hours of discussion, I like to rience for me to attend the same play a num- at all), congratulating them on their initiative, believe that they concluded a mathematician ber of times over two years. Likewise, seeing adding some explanations about the obvious could be pretty normal after all. successive versions of the excellent transla- sources of inspiration of the author (Nash’s Their questions were interesting and could tion by Isabelle Anckaert showed me an life, Wiles’s proof of Fermat’s last theorem...) give us some idea of what we should explain approximation process, in which some ideas and offering assistance in case they had any when attempting to popularise mathematics. current in the US do not quite apply in Europe mathematical question. Since part of the play is about a long proof, and have to be well presented through the I don’t know if I really expected an answer, they wanted to understand what a proof is, translation. but as it happened they invited me to a num- how it looks like, in particular what the proof But the story doesn’t stop there. Indeed, it ber of meetings with everyone involved in the of Fermat’s last theorem could involve when was quickly agreed that before one perfor- play, asked me to check the translation of the the statement consists only of long multiplica- mance of the play, the director Jonathan Fox (short, but absolutely correct) mathematical tions and additions (not an easy question). and I would make a joint presentation, mostly parts of the play, to contribute a four pages The play describes briefly the way mathe- to high school students, of aspects of the play interview on mathematics and mathematicians maticians attack a problem - by different from the point of view of theatre work and for the programme, and even to produce some angles, by intuition, and not at that stage by mathematics. We planned for one such pre- attractive cohomology sequences as back- logical deductions. sentation - but ended with more than ten over ground for the pages. This is very well shown in Simon Singh’s two seasons. This was an extremely interesting and television programme Fermat’s Last This gave me the opportunity to speak enjoyable interaction, for many reasons. One Theorem, Horizon, BBC, 1996, which gave a about Nash and Fermat - but more to the point of the characters of the play, Robert, is good insight to the actors and the director. about the existence of mathematical research, inspired by the life of Nash: a genius who suf- The image usually given of mathematics is its enjoyment, the possible careers, to around fered a dramatic mental breakdown at 30, the final presentation: definitions, statements 1500 high school students, a rare opportunity. stopping short his remarkable development. and deductive proofs. We must realise that And by the way, this generation of students His daughter Catherine (in the play) is the this hides the most attractive part of mathe- who are now 17 years old mostly does not main character, and wonders if she might have matical work - namely the non-deductive and know about that old event - the proof of inherited his mathematical power or maybe non-logical search for the solution of a prob- Fermat’s theorem way back in 1994. When his mental fragility. The other two characters lem. After some time, the theatre people could we think of information and popularisation of are Hal, a former student of Robert currently talk ad lib on the common features of artistic mathematics to a young audience of potential teaching in a mathematics department, and and mathematical creation. Although it is mathematicians, we must remember they keep Claire, Catherine’s sister, a financial analyst clear they need not understand any of the changing and information does not percolate meant to give a totally external vision of soci- mathematics to impersonate the characters, through more than a couple of years. ety on the group of academic mathematicians. To conclude, if you ever get an impulse of The reference to Robert’s mental illness gave writing a letter about mathematics on such an to the theatre people the idea that a mathe- occasion, do it, by all means, and you are matician was necessarily skating on a thin line liable to meet other people of unbounded between genius and madness. enthusiasm.

from ‘Preuve’ at Bruxelles 14 EMS December 2003 INTERVIEW InterviewInterview withwith SolomonSolomon MarMarcuscus Interviewer: Madalina Berinde (University of Cluj-Napoca, Romania)

“The mathematical way of thinking is useful in matical education would be one of the aims of my any profession” life. I devoted several books and tens of articles to this topic, as can be seen in my C.V. Beginning with your school years, one can see (http://funinf.cs.unibuc.ro/~marcus). you have been in love with all kinds of human The main shortcomings of school mathemati- knowledge. Why then did you choose to enter cal education are: the absence of ideas, replaced the Faculty of Mathematics? by procedures; the lack of motivation in the way I discovered mathematics very late, just in the notions are introduced, problems are selected and summer before going to University. I happened to theorems are proved; insufficient attention paid to Solomon Marcus read something about non-Euclidean geometry, historical aspects; exclusive attention to formal and I was fascinated by the contrast between this correctness, lack of attention for understanding, everyone’, nor ‘popularising science, particularly and the intuitive perception of the world. I also for meaning, for ‘why’ questions; a very poor link mathematics’. Such slogans are misleading, they became impressed by the infinity of the sequence to other disciplines taught in high school; cur- may suggest that mathematicians deliberately of natural numbers, a fact that for me became an riculum and text books two or three times larger encode their message in a way that is not avail- object of contemplation and continuous reflec- than what could be effectively assimilated by nor- able for everyone, and then we need translators of tion. Philosophy was a possible choice to satisfy mal students; insufficient use of the ludic aspects mathematicians’ messages into ordinary lan- this curiosity, but I soon realized that only math- of mathematics; insufficient correlation with guage. As a matter of fact, there is a trend of ematics could give me the feeling of certainty I computer science; insufficient attention to the snobbery in some cases - of schizophrenia in was looking for. mathematical way of thinking and to its relevance other cases - pushing some authors to an exces- In the fall of 1944 I left the city of Bacau, in the everyday life. sive use of symbolism (Errett Bishop had an arti- where I had spent my first 19 years, and I came to As a consequence of all the aspects above, cle entitled ‘The schizophrenia of contemporary where I paid a visit to the Mathematics mathematics is not attractive for most high school mathematics’). The exacerbation of the syntactic Section of the Faculty of Science. I was students. Moreover, it is not perceived as a cul- component of mathematical language at the impressed by announcements including magic tural enterprise. Goethe’s remark “Mathe- expense of its semantic component is an impor- words such as ‘infinitesimal calculus’ and ‘trans- maticians are like French people. You tell them tant concern of mathematical education today – finite cardinal and ordinal numbers’. I took a pro- something, they translate it into their language remember Goethe’s remark that I have already visional decision to try mathematics. The first and you no longer recognize what you told mentioned. class I attended was, fortunately, with a professor them.” is still valid. Most intellectuals remember My main interest in addressing people with who decided not only my preference for mathe- from school mathematics nothing more than the respect to mathematics is to point out the cultural matics, but also my way of life, to be devoted to four fundamental operations with positive inte- aspect of mathematics, its genuine link with other mathematical research. His name was Miron gers. On the other hand, the trend of school math- fields of culture, be they scientific, artistic or Nicolescu. He was born exactly hundred years ematics to exaggerate the use of symbols, at the philosophical. It seems to me that one of the main ago, in 1903, and meeting him was a great expense of natural language, increases the gap reasons why mathematical education fails in ful- chance. But he was not the only one to have a between mathematical correctness and mathe- filling its task and makes of it, for most people, an decisive impact on me. , matical meaning. We need a complete change of unattractive topic, is the absence of a bridge to the Gheorghe Vranceanu, Dan Barbilian, Octav school mathematical curriculum and school universe of other disciplines and to human life as Onicescu, Alexandru Froda, Grigore C. Moisil, mathematical textbooks (which, in their present such. I will give some examples. Are we able to and other professors gave me form, are used more by teachers and less by stu- explain to high school students in what the math- many intellectual satisfactions and I often wrote dents, at least in Romania). But mathematical ematical way of thinking consists and how it is about their personalities; see for instance my education should be revised not only with respect involved in everyday life? Can we show the rele- book ‘From Romanian Mathematical Thinking’ to school. All professions need a mathematical vance of mathematical thinking outside mathe- (in Romanian, 1975), ‘Simion Stoilow’ (in col- training related to their way of thinking and a matics? This means, among other things, to be laboration with Cabiria Andreian; in Romanian, mathematical background. Mathematics has a able to separate mathematical thinking from 1982) and my articles in ‘Academica’, the journal kind of universality and any form of education mathematical symbolism, although this separa- of the . should exploit this fact. Combinatorial thinking, tion is possible only within certain limits; to give Trying to explain my option for mathematics, I recursive thinking, algorithmic thinking, step-by- examples of the use of mathematical symbolism think that my motivation in 1944 was that in step thinking, deductive thinking, inductive devoid of mathematical thinking and examples of mathematics the effort of memory is minimal, thinking, thinking by analogies, probabilistic mathematical thinking in the absence of mathe- while the feeling of certainty is maximal. No thinking are only a part of the many types of matical symbolism; to show, by relevant exam- other discipline can challenge mathematics in this mathematical thinking. They are essential not ples, how mathematical symbolism is born just respect. This was my thinking in 1944 and it is only in mathematics, not only in science, but also from the need to develop the mathematical way still so in 2003. in all aspects of life, even in the absence of math- of thinking. ematical jargon. This is exactly the direction in which mathe- As anyone can see, the unnatural conflict matical education fails, and this fact explains why between science, especially mathematics, and Tens of books, articles and delicious radio con- most intellectuals relate mathematics not to the humanities is something very common, at ferences prove your interest in making science human thinking, but to formulas whose rele- least at pre-university level education. Is there accessible to everyone. The importance of popu- vance, outside mathematics, is zero. Mathematics anything to be done in this respect? larising science, particularly mathematics, has is a potential bridge between different disciplines; When, as a university student, I began to discov- been outlined often in the last years. Are the this means it can be such a bridge to the extent to er the real face of mathematics and when I real- young generations of scientists aware of this? which mathematical education succeeds to prove ized how high-school mathematics transforms Are they prepared to face the difficulties of such it, to make it effective. The logarithmic, the expo- mathematics into a caricature of itself, I decided a task? nential and the polynomial functions cover a that the fight for a fundamental change in mathe- My aim is neither ‘making science accessible to large variety of processes in physics, chemistry, EMS December 2003 15 INTERVIEW geology, biology, psychology, linguistics and form research, but also persons who learned March 1925, in Bacau, Romania. Elementary social sciences; do high school students have the much mathematics and are able to transmit it to school and high school in Bacau. Graduation chance and the pleasure to contemplate the way young generations. But good teaching is not (1949) and PhD (1956) in Mathematics, mathematics brings such heterogeneous phenom- equivalent to popularisation, except in the case , Romania. Assistant, lec- ena into a unique framework? The answer is when clear presentation is considered as popular- turer, associate and full professor in Mathematics, unfortunately negative. isation. A good teacher is a kind of actor, he sim- University of Bucharest; emeritus professor ulates that he is discovering spontaneously the 1991. Corresponding member (1993) and full Research takes a lot of effort and time, and sci- things he is teaching, despite the fact that he has member (2001) of the Romanian Academy. entists gifted in so many fields as you are still already taught them about hundred times before. Research and teaching in mathematics, linguis- rare. Should popularising mathematics or the Indeed, it may happen that some university teach- tics, computer science, semiotics, mathematical , for example, remain the ers are not that successful in doing research. But, and computational linguistics and poetics, history complementary occupation of pure mathemati- in order to succeed in teaching, it is obligatory to and philosophy of science. One of the initiators of cians, or should special attention be paid to adopt a research attitude with respect to what you mathematical linguistics and poetics. Author of forming young scientists willing to take on this are teaching. about 30 volumes and more than 400 research job? papers. Hundreds of invited lectures in universi- There are now, mainly in Western countries, sci- Last but not least: It is obvious for anybody see- ties of most of the European countries, in entific journalists, specialized in the popularisa- ing the lists of your publications that you are a America, Asia and Oceania. tion of science. They have a special gift to trans- kind of scientific monster. How was that possi- form the scientific enterprise into a story, a narra- ble? Are there other ways in which you express Notes tive structure that sometimes takes the form of a your delight towards the seen and unseen world, [1] Simion Stoilow (1887-1961), Romanian mathe- detective story. It happens that such journalists besides writing and lecturing? matician (analysis and function theory), Professor are sometimes (ex) successful researchers. I I am far from being as productive as you suggest. at the University of Bucharest; Member of the appreciate their work, but I don’t think that this is My younger colleague Gheorghe Paun, who is Romanian Academy the main way to approach the dichotomy proper also my former student and PhD student, is more [2] Miron Nicolescu (1903-1975), Romanian mathe- mathematics - popularised mathematics. productive and, what is very important, highly matician (analysis and function theory), Professor Irrespective of its level, good mathematics should appreciated by the international community in the at the University of Bucharest, Member of the include a cultural dimension, referring to the field of mathematical computer science, as a pio- Romanian Academy, President of the Romanian ideas, the motivation, its relevance to other fields, neer of several fields in this area, the most recent Academy (1966-1975), Director of the Institute of the aesthetic aspect, a philosophical aspect, and being membrane computing. Mathematics of the Romanian Academy (1963- an historical aspect. But this cultural dimension Before writing and lecturing, my first job and 1973) should not be something added artificially, as a first source of pleasure and satisfaction is reading [3] Gheorghe Vranceanu (1900-1979), Romanian compensation of the aridity of mathematical for- and studying. It often happens to me that I regret mathematician (differential geometry and topolo- malism; it should be organically related to this that I cannot leave aside writing when I am gy), Professor at the University of Bucharest, formalism and essentially required by the text. pressed to write something in order to conform to Member of the Romanian Academy Ideas, motivation, historical, cultural and philo- a deadline. I would rather spend all my time read- [4] Dan Barbilian (1895-1961), Romanian mathemati- sophical context, relations to other fields, aesthet- ing so many interesting things spread in books, cian (number theory and algebra), Professor at the ical dimensions should belong to the process of journals, the internet etc. However, I always read University of Bucharest, also an appreciated understanding, so much neglected. If things are with pen and paper in front of me, because I need Romanian poet by the name of Ion Barb separated, if real mathematicians are only those to react to what I am reading and these reactions [5] Octav Onicescu (1892-1983), Romanian mathe- who make long calculations and manipulations of sometimes take the form of an article. So, reading matician (probability theory, mechanics), formulas, while popularisation mathematicians and writing stimulate each other, they need each Professor at the University of Bucharest, Member are those who learn how to package already exist- other, and each of them develops at the expense of the Romanian Academy ing mathematics, then people get a distorted of the other. But reading does not refer only to the [6] Alexandru Froda (1894-1973), Romanian mathe- image of mathematics. field of my research interests. I read not only matician (real analysis), Professor at the Mathematical education will succeed as a cul- mathematics; I also read philosophy, literature, University of Bucharest tural enterprise only when people discover in it social sciences, natural sciences, practically [7] Grigore Moisil (1906-1973), Romanian mathe- something to enjoy, to play with, to contemplate, everything. matician (both pure and applied mathematics), to relate to their way of seeing the world and of My belief is that apparently heterogeneous Professor at the University of Bucharest, Member considering their life. Mathematics was tradition- fields strongly interact, there is a unity of human of the Romanian Academy, one of the promoters ally taught by ignoring the evolutionary aspect knowledge and human creativity; if you don’t of computer science in Romania and the historical context. The trend of contem- take into consideration this fact, you risk getting [8] Alexandru Ghika (1902-1964), Romanian mathe- porary mathematical textbooks is to reduce the a fragmentary representation of reality. I can matician (analysis and function theory), Professor proportion of ordinary language, by replacing it illustrate this fact by some examples. For at the University of Bucharest, Member of the with formulas. The habit of leaving to the histori- instance, it is interesting to observe how a mathe- Romanian Academy an, to the populariser, the task of humanizing matical object such as the Möbius strip in the last [9] Cabiria Andreian-Cazacu (1928- ), Romanian mathematics is the wrong strategy. In Romania, a decades became a basic point of reference in mathematician (complex analysis), Professor at typical example of a person who successfully anthropology (Claude Levi-Strauss), in art (M. C. the University of Bucharest transformed many mathematical enterprises into Escher), in biology (Jesper Hoffmeyer) and in literary stories was Florica T. Campan. Authors many other fields. It is also interesting to observe Interviewer’s note: of this type have a role that should not be under- the spread of the metaphors related to Interviewing an outstanding personality of the estimated. However, the general problem of Mandelbrot’s geometry in all directions. Romanian scientific scenery is an honour and a big mathematical education is to train teachers who Gödel’s incompleteness theorem is another uni- challenge for anybody; in particular, when the person- are able not only to develop syntactic abilities, but versal paradigm characteristic for post-modern ality in question is Prof. Solomon Marcus. It all also to bridge mathematics with understanding, culture. becomes a great pleasure as well – for which I am very with natural and social sciences, with philosophy To be a mathematician today is a very demand- grateful. It remains to regret that a dialogue as the one and art, and to form the mathematical way of ing job and we should try to impart this feeling to above is too short to satisfy our curiosity. thinking as a tool occurring in any possible field. the young generations. In particular, we are oblig- Finally, I would like to join Professor Marcus when Many years ago, a great mathematician, ed to explain the way in which mathematical he confessed: “I hope to contaminate the readers of our , published an article thinking is universal and, as a consequence, very dialogue with at least a part of the passion we invested arguing for the need to separate the profession of useful for anybody, irrespective of their profes- in it.” the university researcher and that of the universi- sional interests. Mathematics is a part of the cul- ty teacher. One can understand that, to the extent tural heritage of mankind. Madalina Berinde to which we need more and more university [[email protected]] is a master teachers in mathematics, we are obliged to accept SHORT BIOGRAPHICAL NOTE degree student at ‘Babes-Bolyai’ University of as teachers not only those with a real gift to per- Professor Solomon Marcus was born on the 1st of Cluj-Napoca, Romania. 16 EMS December 2003 SOCIETY society. These two became its first presi- dent and vice president respectively, and The Moscow Mathematical V. Ya. Zinger (1836-1907) was elected as The Moscow Mathematical secretary. According to the Society’s statute, a sine qua non condition for mem- SocietySociety.. bership was a master or a doctorate in mathematical sciences or an important Part I publication. The Society consisted of only S.S. Demidov, V.M. Tikhomirov, T.A. Tokareva (Moscow) 14 members in the first year. Among the members let us mention the professors of the university, the astronomer F. A. This is the first part of an article on the his- As a consequence, an active mathemati- Bredikhin (1831-1904), the mathematician tory of the Moscow Mathematical Society. cal centre was formed in Moscow in the N.V. Bugaev, F.A. Sludskii (1841-1897), Part II will appear in the next issue of the 1860s. This was a period of exceptional known for his research in mechanics, the Newsletter. evolution of the Russian social life. Tsar physicist N. A. Lyubimov (1830-1898), as Alexander the well as professor A. V. Letnikov (1830- 1. The birth of the Society second, crowned 1898) from the In the beginning of the 19th century, math- in 1855, imple- Moscow ematical education at Kazan University mented cardinal Technological School, attained a high level of standard, compara- reforms that and K. M. Peterson ble to the best European Universities, due changed the pro- (1828-1881), a humble to primarily Lobatchevskii’s efforts. file of the coun- teacher of mathemat- During this period, the oldest Russian try’s life. The ics of the German University, Moscow University, reached most radical gymnasium (the Peter an equal standard only in the mid 1830s [1, reform was the – Pavel gymnasium in ch. 12]. In 1834, N. E. Zernov (1804 – abolition of the Moscow), who was in 1862), a student of Moscow University, serfdom in 1861. reality the most promi- and N. D. Brashman (1796-1899), a stu- He also intro- nent mathematician in dent of Vienna University and Vienna’s duced funda- this time in Moscow. Polytechnic Institute, who had previously mental changes P. L. Chebyshev, who worked at Kazan University, started to give in the national lived in St. lectures on pure and applied mathematics. educational sys- Petersbourg, actively These lectures were successful since both tem, in particular supported the Moscow Zernov and Brashman were excellent in university mathematical society teachers. Zernov’s ‘Reflexions on education. These from the beginning of Integration of Partial Differential changes were its existence. Equations’ (1837) (the first doctoral disser- implemented in At first, the tation in mathematics defended in Russia) the new founders of the was the first textbook on the integration of University Society proposed very partial differential equations in Russian Statute of 1863 modest goals. Thus, language, and for a very long period it was that increased we can read in the the manual for students of Moscow the representa- minutes of its first University. His treatise ‘Differential tion of the math- meeting that “the goal Calculus and Application to Geometry’ ematical sci- of the Society is mutu- (1842) won a special prize of the ences and recog- al cooperation in the study of mathematical Petersburg Academy of Sciences. Also nized the educa- N.D. Brashman Brashman’s manuals ‘Handbook on tional impor- sciences” [3, p. 239]. Analytical Geometry’ (1836) and ‘Stability tance of the sci- All mathemati- Theory of Solid and Fluid Bodies’ (1837) entific societies at the Universities. cal disciplines were divided among the were awarded this prize in 1835 and in The activities of the mathematical com- members of the society, who informed 1837 respectively. These two books were munity started to develop in this frame- their colleagues about the latest findings in the standard manuals for the university stu- work. The idea to found a mathematical their respective branches. In addition, the dents until the end of the century. society in Moscow was reconsidered. The members had to present the results of their Brashman was elected corresponding first society of that kind had been created own research during an obligatory monthly 1 member of the Petersburgian Academy of long before in 1810 [2]. But this society conference. Sciences [1, p. 220-222] for his research did not survive, as there did not exist at that However, the leaders of the Society achievements. time a sufficient number of active profes- quickly modified their goals in a more Nevertheless, the most important result sional mathematicians in Moscow to sup- ambitious framework. So, when they of Zernov’s and Brashman’s teaching was port its function. A similar attempt was invoked St. Petersbourg in January 1866 to the raising of the standard of mathematical made in the 1860s, by a group of mathe- secure the society, (which was ‘estab- education at Moscow University. maticians who lived in Moscow, most of lished’ a year later, under the name of Among their pupils we can mention the whom were affiliated with the university. ‘Moscow Mathematical Society’) official- well known mathematicians O. I. Somov Such a Society was finally founded in ly, they proposed a new statute. In its first (1815-1876; graduation in 1835), P. L. 1864. The minutes of its first meeting are paragraph they declared the following: Chebychev (1821-1894; graduation in dated 15 (27) September of the same year. “The goal of the organization of the 1841 - the most famous Russian mathe- This date constitutes the starting point of Society is to promote the development of matician of the second half of the 19th cen- the history of one of the oldest mathemati- the mathematical sciences in Russia” [3, p. tury), A. Yu. Davidov (1823-1886; gradua- cal societies. 240]. tion in 1849), and N. V. Bugaev (1837- N. D. Brashman and A. Yu Davidov took At the fourth conference, on 15 1903; graduation in 1859). the initiative for the foundation of this December 1864, the leaders of the Society decided that the reports, which were pre- EMS December 2003 17 SOCIETY sented in the meetings, merited publica- with another school which at that time was was maintained until 1930.3 tion. So in April 1865 they decided to start much more famous in Europe, namely the From the very beginning of his mandate the publication of a journal, which could Petersburgian school of P. L. Chebyshev, as president of the society, D. F. Egorov appear “twice a year in octavo” [4, c. 478]. (A. A. Markov, A. M. Lyapunov etc).2 actively began to ensure the normal func- The journal was named ‘Mathematicheskii The reason for this conflict was first of tion of the national Soviet mathematical Sbornik’ (Mathematical Collection). As all ideological disagreement, which to a community. The first step in this direction the founders of the Society considered certain extent determined the mathematical was to resume the ‘Mathematicheskii their main goal “to promote the develop- orientation of the two schools: on the one Sbornik’ which was interrupted in 1919. ment of Mathematical Sciences in Russia” hand positivism, liberal democracy and The 31st volume appeared in 1924. For it was natural to choose Russian as lan- antimonarchism, which were dominant many years Egorov corrected himself the guage. N. Brashman was in charge of the among the Petersburgian mathematicians, printer’s proofs. preparation of the first volume. The first on the other hand militant antipositivism, D. F. Egorov also wished to put an end to volume appeared in October 1866 and was passion for idealistic and even religious the long-standing conflict between the dedicated to his memory as the founder and philosophy, orthodox spirits and monar- principal mathematical schools. V. Steklov the first president of the Society died in chism among the Muscovite. was invited to sit on the editorial board of May 1866. This date marks the beginning This conflict marked the entire life of the the 32nd volume of the ‘Mathematicheskii of the publications of one of the most influ- Russian mathematical community until the Sbornik’, and therefore the mathematicians ential mathematical journals of the 20th 1930s. The conflict sometimes led to open from Petrograd-Leningrad began more century [3]. clashes, as for example in the fights in willingly to publish their papers in the 1887-1891 related to the works of acade- revived journal: A. S. and Ya. S. 2. The adolescence of the Moscow mician V. G. Imshenetskii, who was sup- Bezikovich G. M. Fikhtengoltc, and V. A. Mathematical Society ported by the Muscovites in his conflicts Fok (v.31) – I. M. Vinogradov (v.31, 42), In the beginning of the 20th century, the with A. A. Markov and A. N. Korkin, or in N. M. Gyunter (v.32, 35) – I. A. Lappo – Moscow Mathematical Society, which connection with S. Kovaleskaya’s work on Danilevskii (v.34) – S. L. Sobolev (v.38, started as a closed circle of professional the problem concerning the motion of a 40) - L. V. Kantorovich (v.41). mathematicians, expanded into a large and rigid body around a fixed point, attacked At the end of the 1920s, this review active scientific society. As we have by A. A. Markov; finally, in the fierce became the leading national journal pub- already mentioned, the society consisted of arguments of the latter with P. A. Nekrasov lishing papers of mathematicians from var- 14 members only in 1867; 13 of them lived on the questions of probability theory and ious scientific centers of the U.S.S.R. for in Moscow and only one (P. Chebyshev) its teaching in schools. All these conflicts example from Kazan (N. G. Chebotariov), lived in another city (in St. Petersbourg). In were discussed during the conferences of from Kiev (D. A. Grave, N. M. Krylov), 1913, on the eve of the First World War, the Moscow Mathematical Society, which from Tashkent (V. I. Ramanovskii), from the Society comprised 112 members. At had to be an arbiter. Rostov-on-Don (D. D. Mordukhai – this time, the geographical distribution of Boltovskoi) and from Odessa (M. G. its membership was wider: 34 members 3. Period of the ordeals Krein). lived in Moscow, 57 in various Russian The First World War, which began in In the spring of 1927, a pân, a Russian cities, and 21 were foreign members. 1914, became the starting point of the Congress of Mathematicians, was orga- The activity of the Society assumed a ordeals of the country. The Revolution of nized in Moscow on the initiative of the national character. Regarding its impor- 1917, which broke out at the height of the Society and the Institute of Mathematics tance for the life of the Russian mathemat- war and the ensuing civil war (1918-1920) and Mechanics at Moscow University with ical community, “the Society was number turned into a catastrophe for the entire sci- the active participation of D. F. Egorov. two after the Academy of Sciences”, as A. entific community. The cessation of the This Congress marked the beginning of P. Youshkevitch wrote [1, p. 317]. normal rhythm of the state, a disastrous sit- the normal social mathematical life in the Conferences were organized regularly, and uation due to the lack of provisions and U.S.S.R at large. During the Congress it we can evaluate the evolution of mathe- fuel, brought the professors to the edge of was decided to organize a national associa- matical studies in Moscow and the whole survival. Soon the old, the weak and the tion of mathematical institutions, in order Empire from the presentations, which were sick began to perish: N.E. Zhukovoskii and made there (the proceedings of these con- K.A. Andreev died in 1921, and B.K. ferences were published in Mlodzeevskii died in 1923. For the ‘Matematicheskii Sbornik’). younger and more active people, it was a Around this time, Moscow was trans- period to search for ways to survive. N. N. formed into a notable centre of mathemati- Luzin and his disciples escaped to cal studies in Europe, known first of all for Ivanovo-Voznesensk where more tolerable the scientific schools in applied mathemat- conditions for the professors were orga- ics (N. E. Zhukovskii (1847-1921), S. A. nized in a new Polytechnical Institute. Chaplygin), in differential geometry (K. The civil war ended in 1921, and the M. Peterson, B. K. Mlodzeevskii (1858- Bolsheviks started to restructure the coun- 1923), D. F. Egorov (1869-1931), and also try according to their ideology. The scien- in projective geometry (K. A. Andreev, A. tists began to return to Moscow, which had K. Vlasov), in number theory (N. V. become the capital of the state in 1918. Bugaev) and in functions of complex vari- Luzin, returning in 1920, found an active ables (P. A. Nekrasov (1858-1924). department and an active mathematical In the beginning of the 20th century, one society. D. F. Egorov, who had remained in of the most significant mathematical Moscow, had tried hard to maintain the schools of the past century was born in high level of mathematics. After B. K. Moscow – the school of the theory of func- Mlodzeevskii’s death in 1923, D. F. tions of a real variable. D. F. Egorov and Egorov who had been the vice president, N. N. Luzin (1883-1950) were its founders became the president of the Moscow and its first representatives. Mathematical Society. This custom, that The formation of the Moscow School the vice president was elected president took place in competition and in conflict after the president’s death or resignation, Matematicheskii Sbornik, Vol 1 18 EMS December 2003 SOCIETY to assemble the mathematical congresses School analysis and the history of mathematics in of the whole Union and to coordinate the 6. The Moscow Mathematical Society after Russia. activities of the mathematical communities the end of the Second World War V.M. Tikhomorov was born in 1934 and between congresses. 7. Conclusion graduated from the Faculty of Mathematics It was paramount for the leaders of the will be published in the next issue. and Mechanics of the M.V. Lomonosov Mathematical Society to normalize the . He is Chair of the country’s destroyed mathematical life and Literature Optimal Control Department of the M.V. to transform the national mathematical [1] Yushkevich A. P. History of mathematics in Lomonosov Moscow State University and a community into an international one. One Russia until 1917. Moscow: Nauka. 1968 (In member of the Russian Academy of Natural of the means was to change the editorial Russian). Sciences. His main interests are the theory of policy of ‘Mathematicheskii Sbornik’. Its [2] Tokareva T. A. The philomathical prologue of optimal control, the theory of functions, the editorial board, D. Egorov, editor on chief, the Moscow Mathematical Society. In: history of mathematics in Russia, and the and V. Kostitcyn, secretary, decided to Istoriko-matematicheskie Issledovaniya. 2nd history of variational calculus. publish papers not exclusively in Russian Ser. Vol. 7 (42). 2002. p. 39-61. (In Russian). T.A. Tokareva was born in 1950 and grad- as they had done before, but also in the [3] Demidov S. S. La revue ‘Matematicheskii uated from the Faculty of Mathematics and principal European languages – German, Sbornik’ dans les années 1866-1935. In: Mechanics of the M.V. Lomonosov Moscow French, Italian and English. At that time, Ausejo E., Hormigon M. (Eds.) Messengers State University. She is a researcher at the when Europe was not that rich in scientific of Mathematics: European Mathematical Department of History of Mathematics of the periodicals, this attempt was successful. Journals (1800-1946). Siglo XXI de España S.I. Vavilov Institute of the History of Among the collaborators of Editores, S.A. Madrid, 1993, p. 235-256. Science and Technology of the Russian Mathematicheskii Sbornik in the second [4] The documents on the history of the Moscow Academy of Sciences. Her main interests are half of the 1920s we can mention: É. Mathematical Society. In: Matematicheskii the history of algebra in the XVII century Cartan (v.34, 42), M. Fréchet (v.32) , J. Sbornik. 1889. v. 14. N3. p. 471-486. (In and the history of mathematics in Russia. Hadamard (v. 41), H. Hopf (v.37), S. Russian). Lefschetz (v.39), R. von Mises (v.41), E. Noether (v.36), R. Memke (v.36), W. Authors: 1 Such a regulation existed until the mid 1920s Sierpinski (v.31,36), L. Tonnelli (v.33). S.S. Demidov[[email protected]] was when the conferences of the society took As a result of such initiatives and activi- born in 1942 and graduated from the place twice a month. At the end of the 1940s ties, the Moscow Mathematical Society Faculty of Mathematics and Mechanics of such conferences took place weekly. headed by D. Egorov held the leading posi- the M.V. Lomonosov Moscow States 2 This conflict which already existed during tion in the country’s mathematical life in University. He is the Head of Department of Chebychev’s life (who maintained special the second half of the 1920s and gradually History of Mathematics of the S.I. Vavilov relationships with the Muscovite) was mani- took over a great number of functions as Institute of the History of Science and fested strongly after his death. the chief organizer in the national mathe- Technology of the Russian Academy of 3 Cf. the list of presidents before D. Egorov: matical community. Sciences and a professor at the Faculty of N. D. Brashman (1864-1866), A. Yu. Mathematics and Mechanics of the M.V. Davidov (1866-1886), V. Ya. Zinger (1886- Part II with Lomonosov Moscow State University. He is 1891), N. V. Bugaev (1891-1903), P. A. 4. The Soviet Government and the mathe- also vice-president of the International Nekrasov (1903-1905), N. E. Zhukovskii matical community Academy of the History of Science. His main (1905-1921), B. K. Mlodzeevskii (1921- 5. The birth of the Soviet Mathematical interests are the history of mathematical 1923). Can you spare books? Can you spare a little more? Herbert Fleischner, Tsou Sheung Tsun

In issue 44 of this newsletter (June 2002), the Committee for Developing Countries (CDC) launched the book donation scheme under the title ”Can you spare books?” and got a resounding response from our colleagues all over the world (see the CDC report in issue 46, December 2002). We are of course continuing this scheme, with encouraging support from the President and the EC, and with substantial material help from the ICTP at Trieste, plus a small grant from the London Mathematical Society. Thus we are still interested in books (even on an undergraduate level) and journals from colleagues or institutions, who no longer need those items. However, in our book donation scheme we also want to assist departments and colleagues in developing countries in their research in specific areas. Therefore, we would now like to ask a little more from our colleagues in the ”developed world”. From time to time we are asked by a scientific publisher to referee a book before they accept to publish it. As a ”sweetening” incentive they may offer an honorarium, either in cash or in books. The sum in cash usually serves the purpose of complicating our tax forms and not much else, and the sum in books, although often larger, serves more often than not to increase the number of books we have to donate to the EMS- CDC scheme when we retire ...However, donated now to a library which cannot buy many expensive books it will serve a good purpose (dete- riorating exchange rates are the daily nightmare in many developing countries). We are thinking of small groups, which are active in a specific research area, and whose libraries cannot afford specialized (and hence expensive) graduate texts or research monographs. Sometimes even half a dozen books will make a difference! Recently, we were able to persuade Princeton University Press to increase (by 30%) the honorarium paid to one of our colleagues, and we were thus able to send 8 classical and/or recently published books to a South American mathematics depart- ment. The important point here is that the recipient group can choose the books (from the publisher’s catalogue) most useful to them. So next time a publisher asks you to referee a book, please think of us. This may even help you to decide to do the refereeing in the first place, and thus serve the mathematics community twice at one go!

Herbert Fleischner [Herbert.Fleischner@ oeaw.ac.at] and Tsou Sheung-Tsun [[email protected]] are the Chair and the Vice-Chair of the EMS-committee on Developing Countries.

EMS December 2003 19 4ECM THETHE FOURFOURTHTH EUROPEANEUROPEAN CONGRESSCONGRESS OFOF MAMATHEMATHEMATICSTICS Ari Laptev (Stockholm)

Every four years, the (France) (http://www.math.kth.se/4ecm/prog European Mathe- ram/plenary.lectures.html). matical Society There will be 33 Invited Lectures in four par- (EMS) organizes a allel sessions covering innovative, multidisci- European Congress of plinary topics (http://www.math.kth.se/4ecm/ Mathematics. The program/invited.lectures.html). purpose of this major event of European One of the novelties of the 4ECM is the Mathematics is threefold: to present various organization of ‘Science Lectures’ where the new aspects of Pure and Applied Mathematics most relevant aspects of mathematics in sci- to a wide audience; to provide a forum for ence and technology will be discussed. So far, discussion of the relationship between mathe- the following speakers have accepted our invi- matics and society in Europe; to enhance co- tation: Richard R. Ernst (Switzerland, Nobel operation among mathematicians from all Prize in Chemistry 1991), Gerard ’t Hooft (The Ari Laptev European countries. Netherlands, Nobel Prize in Physics 1999), Walter Kohn (USA, Nobel Prize in Chemistry Institute for Industrial Mathematics in So far, there have been three European 1998), Martin Nowak (USA) and George Oster Kaiserslautern, is awarded to a young scientist Congresses of Mathematics: (USA). or a small group of young scientists (normally 1ECM in Paris, France, July 6-10, 1992. Another novelty will be information on the under the age of 38) for using sophisticated 2ECM in Budapest, Hungary, July 21-27, work of the EU Research Training Networks in methods to give an outstanding solution, which 1996. Mathematics and Information Sciences and is met with the complete satisfaction of indus- 3ECM in Barcelona, Spain, July 10-14, 2000. Programmes from the European Science try, to a concrete and difficult industrial prob- Theme: ‘Shaping the 21st century’. Foundation (ESF) in Physical and Engineering lem (http://www.math.kth.se/4ecm/felix.klein. Sciences (PESC). Twelve EU Research html). The Fourth Congress of Mathematics (4ECM) Training Networks and PESC projects from will take place in Stockholm, Sweden, June 27 Brussels and Strasbourg have been chosen by Grants to July 2, 2004. Without doubt it will be the the Scientific Committee and have already The budget of the 4ECM includes 100,000 major international mathematical event of the nominated their speakers. (http://www.math. EURO for covering the Congress expenses of year 2004. The theme of the Congress is kth.se/4ecm/program/european.nw.lectures.html) young researchers. About 200 grants will be ‘Mathematics in Science and Technology’. distributed. At least 20% of the participants The programme will be devoted to Pure and 4ECM Prizes to young mathematicians and should be early stage researchers. Since women Applied Mathematics and highlight the impor- the Felix Klein Prize are in minority in mathematics, some efforts tance of mathematics in scientific areas - There will be 10 EMS prizes of 5000 EURO will be made in order to encourage female themes like physics, biology, chemistry, infor- each to young mathematicians who have made researchers to attend. mation and computer science. The content will a particular contribution to the progress of include interesting mathematical problems that Mathematics. The Prize Committee is chaired Contributed Papers arise from applications in various scientific by N.N.Uraltseva (St.Petersburg). I would like About 20% of the participants should have fields. to remind readers that the deadline for 4ECM poster presentations. The Organizers under- The Organizing Committee, following phi- prize nominations is the 1st of February, 2004. stand that some participants of the Congress losophy and advice of the European (http://www.math.kth.se/4ecm/nomination.ecm might have the possibility of obtaining local Mathematical Society, aims at strengthening .html). The maximum age may be increased by funding if their posters are accepted. In order to the training aspects of the conference for young up to three years in the case of an individual speed up the process of acceptance, the orga- researchers by facilitating the possibility of with a ‘broken career pattern’. Mathematicians nizers have decided on the following: their attendance at the Conference. are defined to be ‘European’ if they are of Abstracts submitted before February 15, European nationality or their normal place of 2004, will be considered by the committee and 4ECM Scientific Programme work is within Europe. ‘Europe’ is defined to notification of its decision will be sent out The scientific programme of the 4ECM has consist of all countries parts of which are geo- before February 25, 2004. Alternatively, been formed by the Scientific Committee con- graphically within Europe or that have a corpo- acceptance of abstracts submitted before April sisting of 13 mathematicians of international rate member of the EMS based in that country. 20, 2004, will be notified before May 1, 2004. recognition. This committee is chaired by Prizes are to be awarded for the best work pub- Abstracts submitted after April 20, 2004, Lennart Carleson (Stockholm). The other lished before December 31, 2003. The Prize will not be considered (http://www.math. members are: Björn Engquist (Stockholm and Committee shall interpret the word ‘best’ using kth.se/4ecm/posters.html). Princeton, vice chairman), Noga Alon (Tel its judgment: e.g., it may refer to innate quality Aviv), Luigi Ambrosio (Pavia), Girard Ben or impressiveness, influence, etc. Nominations Satellite Conferences Arous (Lausanne), Boris Dubrovin (Trieste, may be made by anyone, including members of There will be about 10-15 satellite conferences. Moscow), Josi Luis Fernandez (Madrid), the Prize Committee or by the candidates them- Some of these will be a part of research activi- Ursula Hamenstädt (Bonn), Eduard J.N. selves. It is the responsibility of the nominator ties of EU Research Training Networks and Looijenga (Utrecht), Leonid Pastur (Kharkov, to provide all relevant information to the Prize ESF PESC programmes (http://www.math. Paris), Benoit Perthame (Paris), Caroline Committee, including a résumé and documen- kth.se/4ecm/list.html). Series (Warwick) and Andrzej Schinzel tation. The nomination for each award must be (Warsaw). accompanied by a written justification and a Organization The committee has chosen 7 plenary lectur- citation of about 100 words that can be read at The 4ECM will be hosted by the Royal eres: François Golse (France), Francesco the award ceremony. The prizes cannot be Institute of Technology in Stockholm (KTH) in Guerra (Italy), Johan Håstad (Sweden) Andrei shared. collaboration with Stockholm University (SU). Okounkov (USA, Russia), Oded Schramm The Felix Klein Prize, established in 1999 by KTH and SU are international institutions with (USA), Zoltán Szabó (USA), Claire Voisin the EMS and the endowing organization, the established research and educational exchanges 20 EMS December 2003 4ECM throughout the world, especially in Europe, Summary USA and South America. Their Mathematics The aim of this Congress is to highlight the Departments have a number of excellent math- importance of mathematics in different areas of ematicians and are among the leading technology and its place regarding other Departments in Europe. research subjects such as physics, chemistry, The Congress will take place at the biology and medicine. We expect that the University of Stockholm, which has all the 4ECM will show the importance of mathemat- required facilities. The plenary talks will be ics in modern life and will inspire to further held at Aula Magna (1200 sits) which is interaction between different scientific areas. equipped with the most modern facilities and Conferences like 4ECM are vital to the has a number of smaller rooms available the progress of research in mathematics. Such a 4ECM. There will also be some smaller rooms congress will stimulate young mathematicians available at KTH and SU. to attract long-standing problems and also to The 4ECM Organising Committee includes: solve new problems leading to many scientific Ari Laptev (KTH, Chairman), Torsten Ekedahl and technological discoveries. (SU), Christer Kiselman (Uppsala University), Anders Lindquist (KTH), Mikael Passare (SU), Ari Laptev [[email protected]] is the chair- Ulf Persson (Chalmers University of man of the 4ECM Organizing Committee. He Stockholm Technology, Göteborg), Kjell-Ove Widman earned his PhD-degree in 1978 at Leningrad (Institute Mittag-Leffler), Jon Larsson (KTH) makes Stockholm an ideal city for pedestrians. University. He is a specialist in the spectral and Mikael Johansson (KTH). It is easy to reach - Arlanda Airport handles theory of partial differential operators works The Stockholm Convention Bureau (SCB) some 225 international flights daily to and as professor and vice-chairman at the will take care of the practical necessary from thirty countries and four continents. Department of Mathematics, Royal Institute of arrangements such as hotel reservation, regis- Direct buses as well as the Arlanda Express Technology in Stockholm. He served as presi- tration of the participants, etc. SCB has a long Train connect Arlanda Airport with the City dent of the Swedish Mathematical Society in experience of more than 20 years in organizing Terminal in central Stockholm. the period 2001-2003. large congresses in Stockholm.

Publicity The Fourth European Congress of ICME-10ICME-10 Mathematics has its home page at http://www.math.kth.se/4ecm/ updated regu- The 10th International Congress on Mathematical Education larly. It contains an electronic registration (ICME-10) will be held July 4-11, 2004 in Copenhagen, Denmark. form, the programme of the congress, call for More details on the programme, including lists of themes, topics, contributed papers, call for grants, call for prize activities, etc., are available at http://www.icme-10.dk nominations, etc. The aim of the congress is to present the current states and trends The 4ECM Organizing Committee has pub- in and research at all levels of the educa- lished a poster, which has been sent to all tional system and to gather practioners and researchers from all Mathematical Departments in Europe and over the world. A more general objective of the congress is to sup- major Mathematical Departments in USA, port the development of mathematics teaching in order to meet essential needs in society, such Canada, Australia, Japan, China, and South as securing the recruitment for mathematically based professional training and securing math- America among others. ematics competences in general education in order to develop and maintain democracy. It has been agreed that the European The ICME congresses are held every fourth year under the auscpices ot the International Mathematical Society Publishing House will Commission on Mathematical Instruction (ICMI). The first ICME was held in 1969 in Lyon, be responsible for publishing the 4ECM France. Proceedings. ICME-10 in Copenhagen 2004 is expected to gather around 4000 researchers in mathemat- The opening ceremony will take place at ics education, mathematics teachers, and other interested parties from over 100 different coun- Aula Magna on the 28th of June, 2004. tries. The congress will maintain and develop the ICME traditions but will also introduce a Stockholm’s City Hall will be available for the number of new elements to the scientific programme. The scientific programme is planned by Conference dinner on the 29th of June, 2004. the International Programme Committee (IPC) consisting of 21 members from all over the world. Registration There is a possibility to have an advance regis- The main components of the scientific programme will be: tration. The registration form is available at § Plenary Activities, including six lectures http://www.math.kth.se/4ecm/registration.html § ICME-10 Survey Teams § Regular Lectures About Stockholm § Topic Study Groups Stockholm, the capital of Sweden, is a beauti- § Discussion Groups ful town with an excellent infrastructure. It is § Thematic Afternoon also a renowned conference city, and especial- § Workshops and Sharing Experiences Groups ly in the months of June-July (the time of the § Posters and Round Tables Congress) it provides many attractions for the § National Presentations. participants. The Organizing Committee is planning to arrange a number of excursions Further details are available at http://www.icme-10.dk. around Stockholm. Deadline for early registration is February 28, 2004. Since 1901 Stockholm has been the venue of Amongst the characteristic features of the ICME-10 programme will be ample opportuni- the Nobel Prize Ceremony - one of the most ties for the participants to discuss, develop, and share ideas and experiences in formal and prestigious of all events and - indeed the city informal settings, as well as the interaction between researchers and practitioners, and makes a fitting venue with a high standard of research and teaching practice. facilities and service. Public transport is Participation in ICME-10 will provide optimal opportunities for both formal and informal extremely efficient and convenient, with a interaction with colleagues across the spectrum of mathematics education. In addition to the choice of buses, trains and an underground sys- excursion on Thursday, July 8, the cultural and social events of ICME-10 will take place dur- tem. ing the congress in connection with the lunch breaks and the happy hours. Further details of Hotels, museums and restaurants in the city the social programme can be found at http://www.icme-10.dk. are generally within walking distance, which EMS December 2003 21 CONFERENCES (Università degli Studi Roma Tre), E. G. Houston (University of North-Carolina, Charlotte), L. Salce (Università degli Studi di Padova), P. Zanardo (Università degli Studi di Padova). ForthcomingForthcoming conferencesconferences Organising Committee: F. Girolami, G. Picozza & F. Tartarone (Università degli Studi Roma Tre) compiled by Vasile Berinde (Baia Mare, Romania) Sponsors: INdAM-Istituto Nazionale di Alta Matematica, with the contributions of the Mathematical Departments of the Universities of Padova and Roma Tre. Please e-mail announcements of European con- Information: www.ima.org.uk/mathematics Main Speakers: Visit the web site for an updat- ferences, workshops and mathematical meetings [For details, see EMS Newsletter 49] ed list of speakers. of interest to EMS members, to one of the follow- Information: ing addresses [email protected] or e-mail: [email protected]; [email protected]. Announcements April 2004 Website: http://www.mat.uniroma3.it/ should be written in a style similar to those here, seminari/conferenze/cortona2004.htm and sent as Microsoft Word files or as text files 5-7: 5th International Conference on 30 - June 6: SPT2004 - Symmetry and (but not as TeX input files). Space permitting, Modelling in Industrial Maintenance and Perturbation Theory each announcement will appear in detail in the Reliability - Impacting on Practice, University Cala Gonone (Sardinia, Italy) (follows SPT96, next issue of the Newsletter to go to press, and of Salford, UK SPT98, SPT2001 and SPT2002 conferences) thereafter will be briefly noted in each new issue Aim: To provide a forum for discussion of tradi- Information: Website http://www.sptspt.it until the meeting takes place, with a reference to tional and innovative modelling approaches to [For details, see EMS Newsletter 49] the issue in which the detailed announcement improve the performance of plant, products and appeared. processes Topics: Maintenance (inspection, overhaul, pre- June 2004 January 2004 ventive maintenance), fault diagnosis and condi- tion based maintenance, reliability, risk analysis and risk management, reliability-centred-mainte- 2-4: Mathematical problems in Engineering 19-28 : Advanced Course on Ramsey Methods nance, warranty modelling, life-cycle costing and and Aerospace Sciences, The West University in Analysis, Bellaterra (Barcelona) capital replacement, logistics in maintenance and of Timisoara, Romania (ICNPAA2004) Information: reliability, human factors assessment languages Information: e-mail: [email protected]; http://www.crm.es/RamseyMethods (if other than English) Website: www.icnpaa.com e-mail : [email protected] Organizers: The Institute of Mathematics and its [For details, see EMS Newsletter 49] [For details, see EMS Newsletter 49] Applications Organising committee: Dr. P. Scarf (University 7-11: Conference on Poisson Geometry, of Salford), Dr. W. Wang (Salford), Prof. M. Luxembourg City, Grand-Duchy of February 2004 Newby (City University, London) Luxembourg Proceedings: to be published Information: http://www.cu.lu/Poisson2004 [For details, see EMS Newsletter 49] 2-13 : Advanced Course on Contemporary Location: University of Salford, Manchester, Cryptology UK 16-23: 5th International Conference on Campus Nord, Universitat Politècnica de Information: Functional Analysis and Approximation Catalunya e-mail: [email protected] Theory (FAAT 2004), Acquafredda di Information: Website: www.ima.org.uk Maratea, Potenza, Italy http://www.crm.es/ContemporaryCryptology Aim: The meeting is devoted to some significant e-mail : [email protected] 19-24: CHT-04 International Symposium on aspects of contemporary mathematical research [For details, see EMS Newsletter 49] Advances in Computational Heat Transfer; on cruise ship in functional analysis, operator theory and approximation theory, including the applications 23-28, 2004: C*algebras and Elliptic Theory, MS Midnatsol between Kirkenes and Bergen, of these fields in other areas such as partial dif- Bedlewo, Poland Norway ferential equations, integral equations, numerical Topics: K-theory of C*-algebras, index theory, Information: analysis and stochastic analysis. One of the algebras of pseudodifferential operators on singu- e-mail: [email protected] major aims of the conference is to bring together lar manifolds, infinite Grassmannians and Website: http://cht04.mech.unsw.edu.au mathematicians working in the above topics in Fredholm pairs, deformation quantization [For details, see EMS Newsletter 49] order to spur interdisciplinary collaborations and Organising Committee: B. Bojarski, G. £ysik exchanges of results and techniques. and A. Weber (Warszawa); A. Mishchenko and May 2004 Main topics: Banach spaces, Banach lattices, E. Troitsky (Moscow) function spaces, (positive) linear operators, semi- Scientific Committee: M. Atiyah, P. Baum, D. groups of (positive) linear operators, evolution Burghelea, R. Melrose, A. Mishchenko, V. 30-June 5: Commutative rings and their mod- equations and stochastic analysis, approximate Nistor, B.-W. Schulze, N. Teleman, E. Troitsky, ules, Incontro INdAM, Cortona, Italy quadratures and integral equations, approxima- S. L. Woronowicz Aim & topics: The aim is to bring together tion processes in abstract spaces and in function Fees: 220 Euros full board researchers in the areas of commutative ring the- spaces, approximation by (positive) operators, Sponsors: International Mathematical Banach ory and module theory. The main emphasis of interpolation, polynomial approximation, con- Center, Research Training Network “Geometric the workshop is on almost perfect rings and their structive approximation, orthogonal polynomials. Analysis” modules; tilting torsion theories, tilting and Confirmed invited speakers: E. Behrends Grants: available for young participants (full cotilting modules on commutative rings; integer (Berlin), B. Bojanov (Sofia), A. L. Brown board) valued polynomials; multiplicative ideal and (London), N. Jacob (Swansea), N. Kalton Deadline: 15 January module systems, star and semistar operations, (Columbia), A. Kroò (Budapest), W. Luxemburg Gabriel-Popescu localizing systems; Krull and Information: contact A. Weber or G. £ysik, (Pasadena), F. Marcellan (Madrid), G. e-mail: [email protected] Mori domains; chain conditions and prime spec- Milovanovic (Nis), G. Monegato (Torino), B. Website: http://www.impan.gov.pl/~calgebra trum; Pruefer domains and their generalizations; Silbermann (Chemnitz), V. Totik (Szeged), G. factorization and divisibility properties, decom- Vinti (Perugia), L. Weis (Karlsruhe), Y. Xu position of ideals, class groups. Young (Eugene, U.S.A.). March 2004 researchers approaching these areas are welcome. Scientific Program: plenary lectures (50 min.), Programme Committee: M. Fontana selected section lectures (30 min.) and short com- (Università degli Studi Roma Tre), L. Fuchs 31-April 2: Quantitative Modelling in the munications (15-20 min.). (Tulane University, New Orleans), S. Gabelli Management of Healthcare IV, Salford, UK Organizing Committee: F. Altomare, A. 22 EMS December 2003 CONFERENCES Attalienti, M.Campiti, L. D’Ambrosio, S. grants for registration and accommodation Elsner (Germany), G.H. Golub (USA), J. Gro Diomede, G. Mastroianni, D. Occorsio, M. G. addressed to young researchers. (Germany), C.R. Johnson (USA), J. Kunert Russo Sponsors: Center for Studies in Deadlines: For applications for financial support, (Germany), T. Ledwina (Poland), E. P. Liski Functional Analysis and Approximation Theory April 18, 2004; For registration and payment, (Finland), P. Major (Hungary), R.J. Martin (UK), of the University of Basilicata (Potenza, Italy), May 15, 2004 V. Mehrmann (Germany), J. Tiago Mexia the National Research Group ‘Operator Theory, Organiser: Centre de Recerca Matemàtica, (Portugal), H. Monod (France), M.D. Perlman Semigroups and Applications to Evolution Barcelona (USA), PSSNVP Rao (India), W. Ratajczak Equations and Approximation Problems’ (PRIN Location: Universitat Pompeu Fabra, Barcelona (Poland), D. von Rosen (Sweden), Bikas K. – Cofin. 2003-05), the Departments of Information: Sinha (India), G. Tusnady (Hungary), and H. Mathematics of the University of Basilicata e-mail: [email protected] Yanai (Japan) (Potenza, Italy), the University of Bari (Italy), Website: Format: Invited talks and contributed presenta- and the University of Lecce (Italy), the http://www.crm.es/MathematicalFoundations tions Universities of Bari (Italy) and of Basilicata Sessions: There will be plenary lectures and two (Italy), the Department of Economics of the 26-July 1: 7th International Conference of parallel sessions. University of Bari (Italy), the National Group for The Mathematics Education into the 21st Call for papers: If you wish to present a con- Mathematical Analysis, Probability and their Century Project, Ciechocinek, Poland. tributed presentation, please submit an extended Applications (G.N.A.M.P.A.), the Basilicata Theme: The Future of Mathematics Education abstract (up to two pages). Tourism Board. Aim: to provide an international overview of Organizers: Stefan Banach International Location: Hotel Villa del Mare, Acquafredda di innovative ideas and materials for the teaching of Mathematical Center, Warsaw, Committee of Maratea, Potenza, Italy mathematics in schools Mathematics of the Polish Academy of Sciences, Information: http://www.dm.uniba.it/faat2004; Topics: Mathematical modelling, technology, Warsaw, Faculty of Mathematics and Computer http://www.dm.unile.it/faat2004 equity, teacher training Science, Adam Mickiewicz University, Poznan, Main speakers: I. Meznik (Czech Republic), A. Institute of Socio-Economic Geography and 18-23: Mathematical Foundations of Learning Rogerson (Poland) Spatial Management, Faculty of Geography and Theory, Universitat Pompeu Fabra, Barcelona Format: Keynote lectures, round table plenary, Geology, Adam Mickiewicz University, Poznan, Aim: The scope of the meeting includes all paper presentations, workshops and an open Department of Mathematical and Statistical aspects of the theoretical analysis of machine forum of ideas Methods, Agricultural University, Poznan learning techniques for prediction and other data Languages: English and a parallel programme in Programme committee: R. William Farebrother analysis problems. The main objective is to Polish (Shrewsbury, UK), S. Puntanen (Tampere, explore connections between learning theory and Organizers: The Mathematics Education into the Finland; chair), G. P. H. Styan (Montreal, many other areas of theoretical computer science, 21st Century Project Canada; vice-chair), and H. Joachim Werner mathematics and statistics. The program will Programme committee chairpersons: A. (Bonn, Germany) include sessions on such topics as empirical Rogerson (Poland), F. Mina (Egypt) Organising committee: J. Hauke, A. processes and concentration inequalities, local Organising committee chairperson: Margaret Markiewicz (chair), T. Szulc, and W. Wolynski - theory of Banach spaces, approximation theory, Fryska (Poland) Poland game theory, nonparametric statistics, etc. The Proceedings: to be published as hard copy and Proceedings: a special issue of Linear Algebra main goal of the meeting is to bring together on our conference website and Its Applications devoted to this workshop (probably, for the first time) a diverse group of Grants: available for students, teachers and par- Location: the Mathematical Research and mathematicians and theoretical computer scien- ticipants from countries in a difficult economic Conference Center of the Polish Academy of tists working on these problems. situation Sciences, Bedlewo near Poznan Coordinator: G. Lugosi (Universitat Pompeu Deadlines: to be advised in the First Grants: probably partial support for PhD stu- Fabra) Announcement due on 30 October, 2003 from dents Scientific Committee: P. Bartlett (University of the e-mail address below Deadlines: for registration, 31 May 2004; for California, Berkeley), Information: please e-mail: abstracts, 30 June 2004 E. Giné (University of Connecticut), V. [email protected] for all information Information: Koltchinskii (University of New Mexico, Website (of previous conferences only): e-mail: [email protected] Albuquerque), G. Lugosi (Universitat Pompeu http://math.unipa.it/~grim/21project.htm Website: http://matrix04.amu.edu.pl/ Fabra), S. Mendelson (Australian National University), V. Milman (University of Tel Aviv), 24-27: International Conference on Nonlinear S. Smale (University of California, Berkeley) July 2004 Operators, Differential Equations and Main speakers: V. Milman (University of Applications (ICNODEA-2004), Cluj-Napoca, Tel Aviv), S. Smale (Toyota Technological 26-31: 6th World Congress of the Bernoulli Romania Institute, Chicago and University of California, Society and the 67th Annual Meeting of the Information: e-mail: [email protected] Berkeley) Institute of Mathematical Statistics, Barcelona Website: http://www.math.ubbcluj.ro/ Tutorial speakers: S. Boucheron (Université (Spain) ~mserban/confan.htm Paris-Sud, Orsay), N. Cesa-Bianchi (Université Information: e-mail: [email protected] [For details, see EMS Newsletter 49] degli Studi di Milano), V. Kurková (Academy of Web site: http://www.imub.ub.es/events/wc2004/ Sciences of the Check Republic), P. Long [For details, see EMS Newsletter 49] (Genome Institute of Singapore), S. Mendelson September 2004 (Australian National University) Invited speakers: S. Ben-David (Cornell August 2004 8-11: Dixièmes journees montoises d’informa- University), R. DeVore (University of South tique theorique, à Liège (Tenth Mons theoreti- Carolina), L. Devroye (McGill University), R. cal computer science days, in Liège) Dudley (MIT), D. Foster (Wharton, University of 18-21: The Thirteenth International Workshop on Matrices and Statistics, in Theme: Some aspects of theoretical computer Pennsylvania), Y. Freund (Columbia University), science and discrete mathematics related to com- S. Hart (The Hebrew University of Jerusalem), Celebration of Ingram Olkin’s 80th Birthday, Bedlewo, near Poznan, Poland binatorics on words (in the broad sense). M. Ledoux (Université Paul-Sabatier, Toulouse), Scopes: This conference is widely open to young N. Linial (The Hebrew University of Jerusalem), Theme: matrices and statistics Aim: to stimulate research and to foster the inter- researchers. Notice that English and French are P. Massart (Université Paris-Sud), A. Naor the two official languages of the meeting. (Microsoft Research), M. Opper (Aston action of researchers in the interface between sta- tistics and matrix theory Topics: on words (including alge- University), A. Pajor (Université de Marne-la- braic and algorithmic aspects), all aspects of for- Vallée), A. Pinkus (Technion, Israel), G. Scope: to present methods of linear algebra with statistical origin or possible applications in statis- mal languages theory, variable length codes, Schechtman (Weizmann Institute of Science), A. automata theory and verification. Tsybakov (Université Paris 6), R. Vershynin tics Topics: Applications of linear algebra in statis- Main Speakers: J. Cassaigne, D. Caucal, C. (University of California, Davis), V. Vovk Frougny, T. Helleseth, S. Langerman, F. Neven, (Royal Holloway), J. Wellner (University of tics Main speakers: I. Olkin (USA), T. W. Anderson M.-F. Sagot. Washington), D. Xuan Zhou (City University of Call for papers : Please check the webpage. Hong Kong), J. Zinn (Texas A&M University) (USA), J.K. Baksalary (Poland), R. Bru (Spain), C.M. Cuadras (Spain), P. Druilhet (France), L. Organisers: J. Berstel, V. Bruyere, P. Lecomte, Grants: There will be a limited number of M. Rigo. EMS December 2003 23 CONFERENCES Location: Institute of Mathematics, University of Liege (Belgium). Grants: Some financial support for young scien- tists is expected, see the conference website for Personal column updated information. Personal column Deadline: 1st June for submission of a paper, 1st Please send information on mathematical awards and deaths to the editor. August for registration. Information: e-mail : [email protected]; Website: www.jm2004.ulg.ac.be Awards 20-24: 12th French-German-Spanish Conference on Optimization, European Prize in Combinatorics awarded Avignon, France A European Prize in Combinatorics has been established by the European Research and Aim: The French-German-Spanish Conference Training Network COMBSTRU and by the centre DIMATIA to recognise excellent contribu- on Optimization in 2004 is organized by the tions in Combinatorics by young researchers at most 35 years of age. The prize will be award- ‘Group of Nonlinear Analysis and Optimization’ of the University of Avignon. It will be held in ed biannually in conjunction with the meeting EUROCOMB. The award was funded with the Avignon, France, in September 2004. This contribution of private companies, DIMATIA and COMBSTRU. Conference is the 12th of the series of French- The first prize was presented at the European Combinatorial Conference EUROCOMB’03 German meetings which started in Oberwolfach held in Prague last September. It is combined with a monetary award of 2500 Euros. The Prize in 1980 and was continued in Confolant (1981), Committee for the first edition consisted of Jaroslav Nesetril, Chair (Prague), Vera T. Sos Luminy (1984), Irsee (1986), Varetz (1988), (Budapest) and Alexander Schrijver (Amsterdam). The prize recipients can be found in the col- Lambrecht (1991), Dijon (1994), Trier (1996), Namur (1998), Montpellier (2000), and Cottbus umn below. (2002). Since 1998, the conference has been organized under the participation of a third Hélène Exnault and Eckart Viehweg (Essen) have received a Gottfried Wilhelm-Leibniz- European country. In 2004, the guest country will Prize 2003 of the German Research Foundation for their joint work on algebraic and arithmetic be Spain. The conference will in particular pro- geometry. mote the contacts between researchers of the three involved countries and provide a forum for Gerhard Huisken (Potsdam) has received a Gottfried Wilhelm-Leibniz-Prize 2003 of the sharing recent results in theory and applications of optimization. However, scientists from other German Research Foundation for his work on surface evolutions and on foliations. countries are also encouraged to participate. Topics: Smooth and nonsmooth continuous opti- Daniela Kühn (Berlin) and Deryk Osthus (Berlin) have been awarded the first European mization problems, numerical methods for math- Prize in Combinatorics for an extensive collection of results in core graph theory devoted to ematical programming, optimal control and cal- the study of graph minors and random structures, particularly in their relation to Hadwiger’s culus of variations, differential inclusions and Conjecture. set-valued analysis, stochastic optimization, mul- ticriteria optimization, game theory and equilibri- um concepts, optimization models in finance and Rupert Klein (Berlin) has received a Gottfried Wilhelm-Leibniz-Prize 2003 of the German mathematical economics, optimization techniques Research Foundation for his work on turbulence phenomena. for industrial applications. Contributions on other issues related to optimization are also welcome. Andrzej Komisarski (£ódz) has received the Prize of the Polish Mathematical Society for Plenary speakers: A.Ben-Tal (Israel), young mathematicians. E.Carrizosa (Spain), E.Casas (Spain), Lachand- Robert (France), J-B.Lasserre (France), Y.Nesterov (Belgium), U.Rieder (Germany), Rafal Latala and Krzysztof Oleszkiewicz (Warsaw) have been awarded the Banach Great R.Tichatschke (Germany), S.Tijs (The Prize of the Polish Mathematical Society for their research papers. Netherlands), F.Troeltzsch (Germany), E.Zuazua (Spain) Alain Plagne (Palaiseau) has received the first European Prize in Combinatorics for extensive Scientific committee: F. Bonnans (France), J.- work in combinatorial number theory and for the solution of several open problems employ- B. Hiriart-Urruty (France), F. Jarre (Germany), ing techniques on the borderline between combinatorics and number theory. M. Lopez (Spain), J.E. Martinez-Legaz (Spain), H. Maurer (Germany), S. Pickenhain (Germany), A. Seeger (France), M. Thera (France) Wies³aw Pleœniak (Kraków) has been awarded an honorary doctorate at Format: Contributions are solicited for presenta- the Université de Toulon et du Var. tion at the conference. Each accepted paper will be allotted a 30-minute talk (including discus- Hans-Peter Seidel (Saarbrücken) has received a Gottfried Wilhelm-Leibniz-Prize 2003 of the sion). The conference language is English. German Research Foundation for his work on computer graphics. Besides the title of the proposed contribution, a short abstract (of at most 200 words) is also Piotr Sniady (Wroc³aw) has received the Kuratowski Prize. required. Deadline to propose a contribution is March 25 , 2004. Acceptance or refusal notice to authors Michael Szurek (Warsaw) has been awarded the first Marcinkiewicz Prize of the Polish will be given by April 1, 2004. Mathematical Society for students’ research papers. Information: e-mail: [email protected]; Pawe³ Wilczynski (Kraków) has received the Dickstein Great Prize of the Polish Website: http://www.fgs2004.univ-avignon.fr Mathematical Society for his achievements in popularizing mathematics. 23-26: 4th International Conference on Applied Mathematics (ICAM-4), Baia Mare, Deaths Romania (previous editions in 1998, 2000 and 2002) We regret to announce the deaths of: Information: e-mail: [email protected]; [email protected] Web site: http://www.ubm.ro/site-ro / Adam Bielecki (10 June 2003) facultati/departament/manifestari/icam4/index.ht ml Stanis³aw Lojasiewicz (14 November 2002) [For details, see EMS Newsletter 49] 24 EMS December 2003 RECENT BOOKS (Hooley). Exceptional in this aspect is the paper ‘G. H. Hardy as I knew him’ by R. A. Rankin. All papers certainly fulfil the editors’ RecentRecent booksbooks hope that a separate publication can help to stimulate the interest in the presented topics or edited by Ivan Netuka and Vladimír Souèek (Prague) related areas. All of them give the reader an up-to-date information on the development of Books submitted for review should be sent to duction to the theory of Lie groups and homo- basic ideas which paved the road to the the following address: geneous spaces. Classical examples are described major achievements in the subject Ivan Netuka, MÚUK, Sokolovská 83, 186 75 explained. The last part contains special topo- so that this collection give an integrated pic- Praha 8, Czech Republic logical and geometrical problems (connectivi- ture on main streams in contemporary number ty of matrix groups, description of maximal theory. As such, it can be recommended to I. Bajo, E. Sanmartín, Eds.: Recent tori in compact Lie groups, semi-simple fac- active number theorists and also to a general Advances in Lie Theory, Research and torisation, roots systems, Weyl groups and mathematical audience. (spor) Exposition in Mathematics, vol. 25, Dynkin diagrams). The book is a nice elemen- Heldermann, Lemgo, 2002, 398 pp., 44 euros, tary introduction to the theory of Lie groups. E. R. Berlekamp, J. H. Conway, R. K. Guy: ISBN 3-88538-225-3 (jbu) Winning Ways for Your Mathematical Plays This volume contains updated versions of II, second edition, A K Peters, Natick, 2003, selected contributions at the First Colloquium V. C. Barbosa: An Atlas of Edge-Reverse 277—473 pp., US$39, ISBN 1-56881-142-X on Lie Theory and Applications, held in Vigo, Dynamics, Research Notes in Mathematics The book under review is the second of four Spain in the year 2000. There are altogether 421, Chapman & Hall/CRC, Boca Raton, parts of the second edition of the book, which 23 contributions on the theory of Lie groups 2001, 372 pp., £56,99, ISBN 1-58488-209-3 was successfully published a quarter of a cen- and algebras. Three of them were related to This book is a reference volume on the tury ago. However, the second edition con- series of lectures at the Colloquium. The first dynamics of scheduling by edge reversal tains some new parts. The book treats a lot of paper, written by D. V. Alexeevsky and A. F. (SER), i.e., on transformations of acyclic games with their winning strategies, and it is Spiro, describes flag manifolds and homoge- graphs by changing sinks (nodes with arcs ori- full of pictures and diagrams for reader’s com- neous CR structures. The first part of the ented inward) to sources (nodes with arcs ori- fort. Although the topic belongs to recreation- paper contains an introduction to the geometry ented outward). The author first introduces al mathematics, all problems are studied in a of flag manifolds with a short description of scheduling by edge reversal as a mechanism to very precise way. I am not quite sure, whether the structure theory of semisimple Lie alge- rebuild communication routes in computer it was a good idea to divide the book into four bras and parabolic subalgebras. The second networks after topological changes. Other parts. Volume II contains many references to part describes the theory of compact, homoge- possible applications of this formalism are Volume I. Hence it is difficult to read it, if the neous Levi non-degenerate CR manifolds and given. In particular, the author discusses a reader is not familiar with the first volume. their classification. The second paper by Yu. description of resource-sharing systems (like (rc) A. Brailov and A. T. Fomenko deals with Lie the well known dining philosophers problem) groups and Hamiltonian systems as. It consists and networks of automata (e.g., Hopfield N. Berline, C. Sabbah, Eds.: La fonction again of two parts. The first part contains a neural networks or Bayesian networks). The zéta, École Polytechnique, Palaiseau, 2003, review of integration methods for special next topics are the basic properties of SER and 193 pp., 18 euros, ISBN 2-7302-1011-3 classes of Hamiltonian systems on Lie groups, an enumeration of the SER state space. The This book contains three texts surveying prop- symmetric spaces and homogeneous major part of the book consists of a repository erties of Riemann’s zeta function from differ- Riemanniaan spaces. The second part contains of graphical representations of SER basins of ent viewpoints. J.-B. Bost explains a proof of new results in the singularity theory of inte- attraction for some selected graphs. In partic- the prime number theorem based on the theo- grable Hamiltonian systems. The third paper ular, all graphs on six nodes, all trees on seven ry of the Fourier transform for distributions. P. is written by M. Scheunert and contains an nodes, and all rings on three to eight nodes are Colmez offers a panorama of arithmetic prop- introduction to the cohomology of Lie super- covered. The book is easy to read and under- erties of the zeta function (and more general algebras and some applications. The cohomol- stand thanks to many examples. It is intended Dirichlet series), ranging from polyloga- ogy of colour Lie algebras is introduced, and to be a repository of results and data. Thus, rithms, transcendence results and polyzetas to some classical results on Lie algebra coho- proofs for many results are omitted, but a modular forms, p-adic measures and the p- mology are generalized to this setting. There comprehensive set of references where these adic zeta function. Ph. Biane sketches the are also applications of the theory to the study proofs can be found is given. (rbar) heuristic relationship between the distribution of formal deformations of the enveloping of zeroes of ζ(1/2+it), the statistical properties algebra of a colour Lie algebra and to M. A. Bennett, B. C. Berndt, N. Boston, H. of eigenvalues of random unitary matrices , as Hochschild cohomology. (jbu) G. Diamond, A. J. Hildebrand, W. Philipp, well as a link between the zeta function and Eds.: Surveys in Number Theory, A K Peters, random walks and Brownian processes. (jnek) A. Baker: Matrix Groups. An Introduction Natick, 2002, 63 pp., US$30, ISBN 1-56881- to Lie Group Theory, Springer 162-4 N. L. Biggs: Discrete Mathematics, second Undergraduate Mathematics Series, Springer, The present collection of papers contains 14 of edition, Oxford University Press, Oxford, London, 2002, 330 pp., 16 fig., 34,95 euros, 72 papers published separately in three vol- 2002, 425 pp., £26,50, ISBN 0-19-850718-6, ISBN 1-85233-470-3 umes under the title Number theory for the ISBN 0-19-850717-8 This book is designed as an introduction to the Millennium and presented at the Millennium A well known definition says that a textbook theory of Lie groups by means of matrix sub- Conference on Number Theory held at is a book such that everybody thinks he can groups of the real or complex linear group. Urbana-Champaign. Thirteen of the papers are write a better one. Biggs’ Discrete The main examples treated are the special lin- devoted to concrete mathematical topics relat- Mathematics is an exception – not only for its ear groups, orthogonal and special orthogonal ed to the ‘simple’ or multiple Riemann zeta wide range of topics and its clear organization groups, unitary groups, symplectic groups and function (Huxley, Matsumoto), the Riemann but notably for its excellent style of explana- the Lorentz group. In particular, relations hypothesis (Balazard), normal numbers tion. Many examples and exercises contribute between complex matrix groups and real (Harman), arithmetical aspects of the theory to a better understanding. The book can be matrix groups are discussed. The author also of curves (Poonen, Perrin-Riou), Diophantine recommended to students and all those who treats various algebraic, analytic and topolog- approximation (Tijdeman), the Pell equation are interested in this field of mathematics. N. ical properties (e.g., norms, metric, compact- (H.Williams), expansion of a given function L. Biggs wrote a wonderful textbook. (ec) ness and group actions). The second part con- into a continued fraction (Lorentzen), tains a study of algebras, quaternions, quater- Waring’s problem (Vaughan and Wooley), J.-M. Bismut, S. Goette: Families Torsion nionic symplectic groups, Clifford algebras pure and mixed exponential sums (Cochrane and Morse Functions, Astérisque 275, and spinor groups and their special properties. and Zheng), authomorphic forms (Winnie Li), Société Mathématique de France, Paris, 2001, The third part can be considered as an intro- or to primes in arithmetical progressions 293 pp., FRF 350, ISBN 2-85629-109-0 EMS December 2003 25 RECENT BOOKS The relation between classical Ray-Singer tor- phic sections of certain vector bundles over nice introductory text for graduate as well as sion and Reidemeister torsion was studied by Pn(C). The paper by M. A. Guest is an undergraduate students interested in the sub- many mathematicians (including J. Cheeger, overview over contemporary results and prob- ject. (mpok) W. Müller, J. Lot, M. Rothenberg, J.-M. lems in finite-dimensional Morse theory, Bismut, W. Zhang and U. Bunke). In this using Grassmannians as the main motivating S.-Y. A. Chang, P. C. Yang, K. Grove, J. G. book, the authors construct an equivariant ver- examples. N. Hitchin wrote a paper on a key Wolfson: Conformal, Riemannian and sion of the theory of analytic torsion, includ- concept both for mathematics and theoretical Lagrangian Geometry. The 2000 Barrett ing a suitable normalization of analytic tor- physics – the Dirac operator on spin mani- Lectures, University Lecture Series, vol. 27, sions forms. They show that equivariant tor- folds. Particular attention is devoted to the American Mathematical Society, Providence, sion forms in higher degrees are essentially case of low dimensions; it includes a discus- 2002, 84 pp., US$19, ISBN 0-8218-3210-7 invariant under a deformation of the consid- sion of Higgs bundles and of magnetic The book contains three survey articles based ered flat connection. They evaluate equivari- monopoles. The contribution by S. M. on lectures delivered at the University of ant torsion forms (up to coboundaries) for Salamon on Hermitian geometry discusses Tennessee as J. H. Barrett Memorial Lectures. fibrations satisfying suitable assumptions and several aspects of the theory of complex struc- The first part (Partial Differential Equations they prove a formula for equivariant torsion tures, depending on the existence of compati- related to the Gauss-Bonnet-Chern Integrand forms in the case of unit sphere bundles. Tools ble Riemannian metric. It contains also a dis- on 4-manifolds by S.-Y. A.Cheng and P. and methods used in the book include super- cussion of the Goldberg conjecture and the Yang) is devoted to four dimensional confor- connections and Chern-Simons theory, the theory of connections on vector bundles relat- mal geometry. Conformally invariant curva- (extended) de Rham map, the Witten deforma- ed to these geometric structures. The paper by tures, invariant operators and partial differen- tion of the de Rham operator and instanton J. Seade is an expository paper on indices of tial equations on 4-dimensional manifolds are calculus, local families index techniques, the vector fields and characteristic classes for sin- studied. In particular, the authors introduce Berezin integration and localization of esti- gular varieties. This collection of eight articles the Weyl curvature tensor and Q-curvature mates using finite propagation speed. The is prepared by former students of Brian Steer, and they study conformal compactification of book contains a full presentation of results and it is dedicated to him. The list of research a complete non-compact locally conformally announced earlier by the authors in publications of B. F. Steer is presented in an flat four-manifold with integrable Q. It Compt.Rend.Acad.Sci. (vs) appendix. The book contains rich and interest- includes also a study of properties of the sec- ing material covering a broad part of geometry ond elementary symmetric function σ2(A) , T. S. Blyth, E. F. Robertson: Further Linear and topology which is explained in an accessi- where A is the conformal Ricci tensor. The Algebra, Springer Undergraduate ble way. It is without any doubt an excellent second part by K. Grove (Geometry of, and Mathematics Series, Springer, London, 2002, book for graduate students as well as for math- via, Symmetries) is devoted to a study of 230 pp., 34,95 euros, ISBN 1-85233-425-8 ematicians from other fields interested in properties of the isometry group of The book is a continuation of the authors’ these topics. (jbu) Riemannian manifolds and its geometry and Basic Linear Algebra published in the same topology. Several known examples are pre- series. After a summary of the contents of that B. J. Cantwell: Introduction to Symmetry sented, the Alexandrov geometry of orbit volume, the authors proceed to inner product Analysis, Cambridge Texts in Applied spaces is described and studied using symme- spaces and elements of direct sum decomposi- Mathematics, Cambridge University Press, tries. In the last section open problems and tions of linear spaces. Then they come to the Cambridge, 2002, 612 pp., £35,95, ISBN 0- conjectures are stated. The third part (The heart of the book, the primary decomposition 521-77183-8, ISBN 0-521-77740-2 geometry of Lagrangian immersions into sym- theorem. This theorem is subsequently applied The book is an introductory text on symmetry plectic manifolds by J. G. Wolfson) is devot- to prove the Jordan form theorem and various analysis based on Lie group theory. After an ed to a study of Lagrangian immersions and canonical forms for real and complex matri- interesting historical preface, the author intro- their invariants and to the problem of mini- ces. There is also a section on dual spaces and duces basic tools for symmetry analysis. He mizing volume among Lagrangian cycles another one on bilinear and quadratic forms. shows how dimensional analysis can be inside a Lagrangian homology class. (jbu) The authors also included a section on the use applied to deduce some laws of physics. of MAPLE in linear algebra calculations. Symmetry analysis is then applied to several S. K. Chatterjee: Statistical Thought: A Besides numerous well-chosen examples scat- special situations. The following two chapters Perspective and History, Oxford University tered throughout the text, the reader can also contain a basic introduction to systems of Press, Oxford, 2003, 416 pp., £65, ISBN 0-19- enjoy short biographical profiles of twenty ordinary differential equations, first order par- 852531-1 one eminent mathematicians associated with tial differential equations and to classical It is not easy to describe the contents of this the subject. (jtu) Hamiltonian mechanics. The next chapter book in a few words. Generally speaking, the contains a precise definition of one-parameter author describes the course of development of M. B. Bridson, S. M. Salamon, Eds.: Lie groups; the author also describes basic statistical concepts from the perspective of the Invitations to Geometry and Topology, tools needed later on: Lie series and Lie alge- present state of the subject. Some chapters of Oxford Graduate Texts in Mathematics 7, bras. The following chapters show how Lie the book emphasize the philosophical context Oxford University Press, Oxford, 2002, 327 group techniques can be applied to a study of and the others are devoted to the history of pp., £37,50, ISBN 0-19-850772-0 several problems: ordinary differential equa- selected statistical and probabilistic methods. The purpose of the book is to present several tions, partial differential equations as well as Anyway, the reader is assumed to have a suf- topics in active areas of geometry and topolo- several special problems in fluid mechanics ficient knowledge of mathematical statistics. gy. Three papers are related to topology and (boundary layer models, incompressible The book is divided into two parts. Part I discrete groups. The paper by A. J. Berrick Navier-Stokes equations, compressible Euler called Perspective is oriented to a philosophi- contains a review of results and conjectures equations and a certain model of turbulence). cal background of statistical thought and an about perfect groups. In particular, acyclic The calculation of the determining equations interpretation of probability. Part II called groups are studied. The contribution by M. R. of the group is a tedious job. In order to help History describes the birth of probability the- D. Bridson relates geometry and the word the reader, the author attached to the book a ory, fundamental ideas (including the central problem. It includes a discussion of the Filling CD containing a Mathematica-based software limit theorem, maximum likelihood, informa- theorem and the Dehn function. In the paper for determining these equations as well as a tion criteria, outliers, robustness, and many by M. C. Crabb and A. J. B. Potter, the reader limited tool for solving them. The last part of others), and the role of outstanding scientists can find a description of the Fuller index in the the book is devoted to Lie-Bäcklund transfor- like Pascal, Bernoulli, Bayes, Laplace, Gauss, setting of equivariant fiberwise stable homo- mations (and applications to conservation Poisson, Galton, Pearson, Student, and Fisher. topy theory. The other five papers are related laws) and Bäcklund transformations (and However, this volume is neither a book on the to contemporary problems in differential applications to the Burgers potential equation history nor on the philosophy of statistics. To geometry. M. Eastwood and J. Sawon wrote and to the Korteweg-de Vries equation). The illustrate the subject of the book, I would like an article describing the Borel-Weil construc- book contains a large number of solved prob- to mention some details about sufficiency tion of finite-dimensional holomorphic repre- lems and exercises, which help to understand described in it. All statisticians know that this sentations of GL(n,C) on spaces of holomor- the theory. It can be recommended as a very concept was introduced by R.A. Fisher in 26 EMS December 2003 RECENT BOOKS 1922. It is less known (see p. 255) that Fisher phase space. The metaplectic group is a dou- tains a systematic treatment of the Floer discovered the principle of sufficiency when ble cover of the symplectic group. A study of homology groups of a homology 3-sphere. he solved a problem raised by the astronomer its representations is used in a treatment of the The second part of the book starts (Chapt. 6) and physicist Eddington in a 1914 book on Schrödinger equation for a class of with a description of the relation between the astronomy (Which of the two given estimators Hamiltonians and for a definition of certain Floer homology groups and the invariants of of standard errors has a better performance?). Feynman path integrals. (vs) 4-manifolds defined by Yang-Mills instan- But in fact, sufficiency was used already in tons. Chapt. 7 includes a description of a prod- 1860 by the American statistician Simon E. DiBenedetto: Real Analysis, Birkhäuser uct structure on the Floer homology groups Newcomb (see p. 254), who observed that in a Advanced Texts, Birkhäuser, Boston, 2002, and a discussion showing how the Floer sample X1,…,Xn with replacement from 485 pp., 99,81 euros, ISBN 0-8176-4231-5 groups fit into topological field theory for a {1,…,N}, the statistic max Xi in some sense Every teacher of mathematical analysis has special class of 4-manifolds. In the last chap- summarizes information from the complete seen many books on real analysis, first as a ter, further possible research directions are sample. The book can be recommended to student and later as a lecturer. I am not sure described. The book gives a nice account of teachers and students who are interested in whether it would be completely fair to say that the theory of an interesting topic in contempo- philosophical principles of statistics and in the the book under review is the best book on real rary geometry and topology. It can be strong- history of probability and statistics. (ja) analysis I have ever seen, but it is certainly a ly recommended to anybody interested in new good candidate for this position. I am sure I ideas coming from recent important interac- F. Collot: Construction of a Well-Ordering would like to have this book in my suitcase in tions between mathematics and modern theo- on the Continuum: Consequences for the case I would have to spend several years on a retical physics. (jbu) Continuum Hypothesis, Editions deserted island. The book covers virtually Européennes, Paris, 2001, 128 pp., ISBN 2- everything (in real analysis) that a teacher can E. B. Dynkin: Diffusions, Superdiffusions 908082-16-0 dream of or that a gifted undergraduate or and Partial Differential Equations, AMS The subject of the book is characterized in the PhD. student might need for his/her studies Colloquium Publications, vol. 50, American preface as an attempt, which “is perhaps a and further research. The text is a deep self- Mathematical Society, Providence, 2002, 236 very particular point of the mathematical sci- contained exposition of all important features pp., US$49, ISBN 0-8218-3147-7 ences, but it is simple and uses no elaborated of real analysis involving just about the right This is a book on the interplay between linear notion”. The author presents a proof of the amount of necessary abstraction and side-trips and semi-linear, elliptic and parabolic differ- continuum hypothesis (CH) from . to various fields of application, such as func- ential equations on one side, and the theory of The framework is not usual Zermelo-Fraenkel tional analysis, harmonic analysis, function diffusions and superdiffusions on the other set theory (ZFC). As informally indicated, a spaces, Sobolev embeddings, interpolation side. Superdiffusion can be viewed as a diffu- ‘missing’ axiom is added to ZFC, which theory, PDEs, potential theory, etc. Moreover, sion of a ‘cloud’ of particles, obeying suitable makes it possible to present a proof of CH. it covers a number of topics, which appear rules of branching (i.e. birth and death of the The author discusses particular ‘missing’ rarely in introductory textbooks but which are particles). There are integral formulas resem- axioms of set theory (e.g., the axiom of pro- absolutely indispensable in modern studies of bling - and greatly generalizing - the classical jective determinancy) and he adds various mathematical analysis. To name just a few, let Feynman-Kac path integrals. The author is a comments on the obtained results. (jmlc) me mention covering theorems, the Hausdorff leading expert in the field; he played a key measure, the non-increasing rearrangement of role in the development of probability theory S.B. Cooper, S.S. Goncharov, Eds.: a function, the Marcinkiewicz interpolation since the fifties. The book gives a detailed Computability and Models: Perspectives East theorem, Radon measures, the Rademacher treatment of the progress achieved by him and and West, The University Series in theorem, the Calderón-Zygmund decomposi- his collaborators in the last 12 years. An intu- Mathematics, Kluwer Academic Publishers, tion, the Fefferman-Stein inequality, etc. The itive explanation of some of the main ideas is New York, 2003, 375 pp., US$135, ISBN 0- material is presented in a truly delightful way. given in the introduction. Part 1 (Parabolic 306-47400-X Sufficient motivation for the investigation is equations and branching exit Markov systems) The book contains 15 articles from the classi- given, and theorems are illustrated with a contains chapters on linear PE and diffusions, cal theory of computability, the computability plenty of examples throughout the text. Each BEM systems, superprocesses, semilinear par- of aspects of familiar mathematical structures chapter is endowed with a `Problems and abolic equations and superdiffusions. Part 2 and so called ‘recursive model theory’. Some Complements’ section, which give the reader (Elliptic equations and diffusions) contains articles are surveys of work inadequately cov- plenty of further opportunities to exercise and chapters on linear EE and diffusions, positive ered elsewhere, some bring important new to notice tiny links between different subjects. harmonic functions, moderate solutions of the results, with pointers to a wider context. The (lp) equation Lu=ψ(u), stochastic boundary values contributors, all internationally recognised of solutions, rough trace, fine trace, the Martin experts in their fields, have been associated S. K. Donaldson: Floer Homology Groups capacity, null sets and polar sets. In the appen- with the three-year Research Project in Yang-Mills Theory, Cambridge Tracts in dices, the reader can find facts on martingales ‘Computability and Models’. This project has Mathematics 147, Cambridge University and elliptic differential equations. The book helped to transform the fragmented European Press, Cambridge, 2002, 236 pp., £50, ISBN combines both probabilistic and analytic tools scene into a lively community of researchers; 0-521-80803-0 with a very high skill; it summarizes the it has helped to overcome an earlier splitting This book is based on a series of seminars on results achieved in an important area, which of recursion theory between schools in the the topic held at Oxford University. It has two progressed significantly in recent years. East and West. (akuc) different goals. Firstly, it is an exposition of (mzahr) Floer’s original work. Secondly, the author M. A. de Gosson: The Principles of develops further aspects of the theory, which C.-A. Faure, A. Frölicher: Modern Newtonian and Quantum Mechanics: The did not appear in the literature before. The Projective Geometry, Mathematics and its Need for Planck’s Constant, h, Imperial Floer homology groups are new topological Applications, Kluwer Academic Publishers, College Press, London, 2001, 357 pp., £50, invariants of three-dimensional manifolds. Dordrecht, 2000, 363 pp., £89, ISBN 0-7923- ISBN 1-86094-274-1 They fit very nicely into a broader scheme 6525-9 This book is written for both physicists and inspired by topological quantum field theo- Opening a book on projective geometry, we mathematicians. The topics treated include ries. Intuitively speaking, they are middle expect an investigation of objects occurring in Newtonian mechanics, semi-classical dimensional holomogy groups of the infinite projective space. We expect to meet sub- mechanics, (non-relativistic) quantum dimensional space of connections modulo spaces, quadrics, algebraic subvarieties, dif- mechanics and its Bohmian interpretation. gauges. There are very important relations of ferential submanifolds, and many other The main tool in the book is symplectic geom- the Floer homology groups with invariants of objects. The book under review is not of this etry. A study of symplectic rigidity leads to a four dimensional manifolds, instantons and type, and this explains perhaps, why it carries semi-classical quantization scheme and to the Yang-Mills theory. The introduction describes the title Modern Projective Geometry. The Maslov index. A use of a general Leray index motivations for the theory and its evolution. main aim of the book is to introduce the cate- leads to a definition of a wave form on the The first part of the book (Chapt. 2 - 5) con- gory of projective geometries. This means that EMS December 2003 27 RECENT BOOKS the authors’ goal is to look at projective by A. S. Cattaneo, G. Felder and L. Society, Providence, 1994, 611 pp., US$51, geometries not only from inside, but also from Tomassini. Finally, D. Tamarkin gives a proof ISBN 0-8218-1538-5 outside. They adopt a synthetic definition of a of the Etingof-Kazhdan theorem on quantiza- The first book is a new edition of a volume projective geometry, and this definition has a tion of Lie bialgebras using the chain operad published by Academic Press under the same fundamental influence on the style of the of little disks. (vs) title in 1978. An appendix entitled ‘Some book. We find deep relations between projec- Details’ (consisting of 13 pages of clarifying tive geometries and other mathematical struc- D. A. Harville: Matrix Algebra: Exercises footnotes) has been added. We recall the con- tures. First of all, a relation to lattice theory, and Solutions, Springer, New York, 2001, 271 tent of the book for the younger generation: i.e., the category of projective spaces is equiv- pp., 39,95 euros, ISBN 0-387-95318-3 Elementary Differential Geometry, Lie alent with the category of projective lattices. The author has collected well over 300 exer- Groups and Lie Algebras. Structure of Secondly, a relation to closure spaces (and to cises from his earlier book Matrix Algebra Semisimple Lie Algebras, Symmetric Spaces, matroids, in particular), i.e., an equivalence From a Statistician’s Perspective into a sepa- Decomposition of Symmetric Spaces, between the category of projective geometries rate volume and he added solutions. The book Symmetric Spaces of the Noncompact type, and a category of certain closure spaces. The also contains extensive and detailed sum- Symmetric Spaces of the Compact Type. general approach leads us also to other geome- maries of the relevant terminology and nota- Hermitian Symmetric Spaces, Structure of tries, e. g. affine geometries, hyperbolic tion. It is thus accessible to anyone familiar Semisimple Lie groups, The Classification of geometries, and Möbius geometries. But this with basic concepts of matrix theory and lin- Simple Lie Algebras and Symmetric Spaces. look at projective geometries from outside ear algebra. It will be very useful for any The second book starts with geometric does not mean that we do not find information teacher of a linear algebra course as a source Fourier analysis on spaces of constant curva- about these particular geometries. The book is of exercises of various levels of difficulty. ture and then develops integral geometry and written according to an excellent plan, and it Besides the standard topics covered in any Radon transforms on Euclidean and two-point surely represents a milestone in the develop- course of linear algebra there are also sections homogeneous spaces. The principal chapters ment of projective geometry. The text is orga- on generalized inverses, matrix differentia- deal with invariant differential operators and nized very carefully and each chapter is fol- tion, Kronecker products, minimization of a spherical function theory. lowed by many exercises. It is a great advan- second degree polynomial subject to linear The third book is a continuation of the sec- tage of the book that it requires very modest constraints, and the Moore-Penrose inverse. ond book, presenting more recent material. It prerequisites. Hence, it can be recommended The second to last section contains various treats Radon transforms for various double already to undergraduate students in the first applications of the spectral decomposition of fibrations, the main example being the sets of year of their study. On the other hand, I expect symmetric matrices. (jtu) points and of horocycles in a symmetric space that also professional mathematicians will (p.145). Later chapters deal with a geometric appreciate it. (jiva) W. Haussmann, K. Jetter, M. Reimer, Eds.: Fourier transform on symmetric spaces (non- Recent Progress in Multivariate compact and compact) and range theorems for G. Halbout, Ed.: Deformation Quantization, Approximation: 4th International Radon and Fourier transforms with applica- IRMA Lectures in Mathematics and Conference, Witten-Bommerholz (Germany), tions to differential equations. The final chap- Theoretical Physics 1, Walter de Gruyter, 2000, International Series of Numerical ter deals with eigenspace representations asso- Berlin, 2002, 236 pp., 34,95 euros, ISBN 3-11- Mathematics, vol. 137, Birkhäuser, Basel, ciated with homogeneous spaces. The first 017247-X 2001, 265 pp., 98 euros, ISBN 3-7643-6505-6 book contains a complete table of contents. The contributions to this book are based on This volume presents main results of the (ok) lectures presented at the joint meeting of Fourth International Conference on mathematicians and theoretical physicists at Multivariate Approximation held at Witten- C. W. Henson, J. Iovino, A. S. Kechris, E. Strasbourg on deformation quantization. A Bommerholz in 2000. The book starts with an Odell: Analysis and Logic, London description of the contents can be found in a article by M. Reimer and H. Schwetlick: J. W. Mathematical Society Lecture Note Series short introductory paper by G. Halbout. The Schmidt in Memoriam. Jochen W. Schmidt 262, Cambridge University Press, Cambridge, survey paper by G. Dito and D. Sternheimer (1931 - 2000) was twice selected by the math- 2003, 267 pp., £29,95, ISBN 0-521-64861-0 includes a treatment of the Kontsevich formal- ematical community as a leading mathematics This book contains three large articles, based ity theorem and its description from the point reviewer of the German science foundation on minicourses presented by the authors dur- of view of deformations of algebras over oper- and participated in the preceding Bommerholz ing the conference ‘Analyse & Logique’ at ads. A short note by G. Dito contains a dis- Conferences on Multivariate Approximation. Mons, Belgium, in 1997. These papers are: cussion of deformation quantization of covari- The proceedings contains nineteen selected, Ultraproducts in Analysis, by C. W. Henson ant fields. The paper by B. Fedosov considers peer-reviewed contributions covering the fol- and I. Jovino, Actions of Polish Groups and deformation quantization on a symplectic lowing main topics: interpolation and approx- Classifications Problems by A. S. Kechris and manifold, a canonical trace on the algebra of imation on compact sets, Kergin interpolation; On Subspaces, Asymptotic Structure, and quantum observables and a variation formula error asymptotics; radial basic functions; ener- Distortion of Banach Spaces; Connection with for the trace density. The quasi-Hopf algebras, gy minimizing configurations on the sphere; Logic, by E. Odell. In each case, the title neat- the Drinfeld twist, quantum affine elliptic quadrature and cubature formulae, harmonic ly describes the contents of the corresponding algebras, deformed double Yangians and their functions near a zero; blending functions; section. Every paper follows the standard pat- relations are topics treated in the contribution frames and approximation of inverse frame tern of a conference minicourse: It assumes a by D. Arnaudon, J. Avan, L. Frappat and E. operators. The list of publications and further very low-level familiarity with the subject, Ragoucy. The paper by S. Waldmann data on J. W. Schmidt are enclosed in an explains basic notions and crucial examples, describes recent results on the representation appendix. (knaj) then takes a short path to quite recent deep theory of the star product algebras arising in theorems. Many times, proofs are omitted; deformation quantization. The survey paper S. Helgason: Differential Geometry, Lie instead, emphasis is given to the meaning of by C. Roger contains a discussion of proper- Groups, and Symmetric Spaces, Graduate theorems and their interrelations. Each paper ties of the Lie algebra of vector fields with Studies in Mathematics, vol. 34, American has its own extensive bibliography. One does vanishing divergencies and possibilities for its Mathematical Society, Providence, 2001, 641 not need to be a specialist in analysis to find (generalized) deformations. Abelian deforma- pp., US$69, ISBN 0-8218-2848-7 this book a worthy item in the library. (psim) tions of ordinary algebras of functions on S. Helgason: Groups and Geometric (possibly singular) manifolds and related Analysis, Mathematics Surveys and H. Iwaniec: Spectral Methods of Harrison cohomology are discussed in the Monographs 83, American Mathematical Automorphic Forms, second edition, paper by Ch. Frønsdal. The relation between Society, Providence, 2000, 667 pp., US$56, Graduate Studies in Mathematics, vol. 53, Toeplitz algebras and star-product algebras is ISBN 0-8218-2673-5 (reprint of the 1984 edi- American Mathematical Society, Providence, described in a paper by L. Boutet de Monvel. tion) 2002, 220 pp., US$49, ISBN 0-8218-3160-7 A construction of star-products on Poisson S. Helgason: Geometric Analysis on The book under review is the second edition manifolds and a connection with the Fedosov Symmetric Spaces, Mathematical Surveys and of the monograph ‘Introduction to the spectral construction in symplectic case are reviewed Monographs 39, American Mathematical theory of automorphic forms’ (the first edition 28 EMS December 2003 RECENT BOOKS was published at Biblioteca de la Revista find an introduction to the theory of partial excellent description of the contemporary the- Matemática Iberoamericana, Madrid, 1995) differential equations. The author focuses on ory of Riemann surfaces; it can be recom- by the same author. It reflects the fact that the elliptic equations and systematically develops mended to all mathematicians interested in the book has grown out of lectures given by the the relevant existence schemes, always with a field. (jbu) author in Spain. Automorphic forms play a view toward nonlinear problems. It includes central role on the border between analytic maximum principle methods (particularly N. V. Krylov: Introduction to the Theory of number theory, algebra, analysis and geome- important for schemes), Random Processes, Graduate Studies in try; hence it is not an easy task to write a com- parabolic equations, variational methods, and Mathematics, vol. 43, American Mathematical prehensive and readable introduction to this continuity methods. The book also develops Society, Providence, 2002, 230 pp., US$35, classical subject. The book under review the main methods for obtaining estimates for ISBN 0-8218-2985-8 demonstrates the author’s mastery in both a solutions of elliptic equations; Sobolev space The author, one of the leading researchers in conceptual and a pedagogical direction; it theory, weak and strong solutions, Schauder the field of stochastic control, presents an gives a very nice introduction to real analytic estimates, and Moser iteration. Connections introductory and, at the same time, advanced automorphic forms and their applications in between elliptic, parabolic and hyperbolic text aimed at undergraduate students majoring number theory. Two introductory chapters equations are explored, as well as the connec- in probability and statistics. The theory of ran- deal with the hyperbolic metric and eigen- tion with Brownian motion and semigroups. dom processes, an extremely vast and com- functions of the Laplacian on the upper half The book can be used for a one-year course on plex part of mathematics, is laid out gradually plane, and with Fuchsian groups. The next partial differential equations. Having some and in a manner that respects the student’s five chapters form the core of the book. The experience in teaching PDEs, I am always mathematical background; the presentation is reader is made familiar with the basic facts curious to see a new textbook in the field. I rigorous and, importantly, allows him/her to concerning cusp forms, Eisenstein series, and have found the book by J. Jost very well writ- appreciate the true probabilistic flavour of the their meromorphic continuation based on the ten. The concentration on elliptic equations formal theory. The main topics of the book are Selberg method, using the Fredholm theory of creates a new possibility for an exposition of Wiener process, stationary processes, infinite- integral equations. Then the author devotes his the main features of evolution equations. Both ly divisible processes and the Itô stochastic attention to spectral theory, Kloosterman sums the ‘Preface’ and the ‘Introduction’ are rather calculus that includes the strong theory of sto- and trace formulas. The Selberg trace formula helpful - it could be useful for the reader to chastic differential equations. There are some is applied to the problem of distribution of come back to them from time to time to put methodological novelties to be found in the eigenvalues. The next chapter considers a geo- the ideas together. I share the author’s opinion text that deserve to be mentioned: The Itô sto- metric application, the lattice point problem in that his book helps “in guiding the reader chastic integral which is introduced at a very the hyperbolic upper half plane of complex through an area of mathematics that does not early stage (Chapter 2) is viewed as a particu- numbers. The book provides a very readable allow a unified structural approach, but rather lar case of the integral with a random orthog- textbook on the spectral theory of automor- derives its fascination from the multitude and onal measure as the integrator. Stochastic inte- phic forms, not only for those willing to enter diversity of approaches and methods...” (oj) gration in this generality is extensively used. this fascinating subject but also for those who The spectral representation of trajectories of need some help to orient themselves in the J. Jost: Compact Riemann Surfaces. An stationary processes and a representation of theory. For the latter group of readers, two Introduction to Contemporary Mathematics, trajectories of infinitely divisible processes appendices provide some necessary back- second edition, Universitext, Springer, Berlin, through jump measures are examples. About ground material from classical analysis and 2002, 278 pp., 24 fig., 39,95 euros, ISBN 3- 130 exercises (hints available) accompany the special functions. (spor) 540-43299-X bulk of mathematical reasoning, some of them The main topic of this book is the theory of being used in the main text. The reviewer’s G. J. O. Jameson: The Prime Number compact Riemann surfaces and their connec- opinion is that the book meets the advertised Theorem, London Mathematical Society tions to other areas of mathematics (two- purposes and may well serve as an introduc- Student Texts 53, Cambridge University dimensional differential geometry, algebraic tion to the modern theory of stochastic Press, Cambridge, 2003, 252 pp., £18,95, topology, , the calculus of processes. (jste) ISBN 0-521-89110-8 variations and the theory of elliptic partial dif- This is a leisure introduction into basic analyt- ferential equations). The discussion includes H. W. Kuhn: Lectures on the Theory of ic properties of Dirichlet series, which culmi- three fundamental theorems: the Riemann- Games, Annals of Mathematics Studies, no. 7, nates in proofs of the prime number theorem Roch theorem, the Teichmüller theorem and Princeton University Press, Princeton, and Dirichlet’s theorem on primes in an arith- the uniformization theorem. One of the main Oxford, 2003, 107 pp., £17,95, ISBN 0-691- metic progression. The book is accessible tools used throughout the book is the theory of 02772-2, ISBN 0-691-02771-4 even to undergraduate students - the prerequi- harmonic maps. The book can be also taken as This remarkable publication is based on sites include standard courses in real and com- a nice introduction to nonlinear analysis Harold Kuhn’s lectures on the theory of games plex analysis but hardly any number theory. applied to geometry. The first chapter contains held in 1952 at the Princeton University. It The first two chapters collect background background material from topology (e.g., fun- was supposed to appear in the Annals of material on Abel’s summation and elementary damental group and covering spaces). In the Mathematical Studies in 1953 but the author properties of Dirichlet series. Chapter 3 treats second chapter, Riemann surfaces are studied decided to extend the contents substantially (two version of) a Tauberian theorem relating from the point of view of two-dimensional and to postpone the publication. However, the the behaviour of Dirichlet’s series f(s) = Riemannian geometry. It includes a discussion intended extension for games of more than -s ∑n≥1a(n)n and the function A(x) = ∑n≥1a(n). of curvature, the Gauss-Bonnet theorem, of two persons did not take place and finally, The prime number theorem is obtained as a special Riemann surfaces which are quotients with the delay of 50 years, Princeton special case for f(s) = -ζ′(s)/ζ(s). These results of the Poincaré upper half plane with a hyper- University Press published the original text. are sharpened in Chapter 5 to include esti- bolic metric, and of conformal structures on As a whole, the book provides interesting and mates for error terms and for zero-free regions tori. The third chapter is devoted to the study valuable author’s insights on the theory of of ζ(s). Chapter 4 is devoted to Dirichlet’s the- of the Dirichlet principle and harmonic maps. games, a new mathematical discipline devel- orem and Chapter 6 to an ‘elementary’ proof Teichmüller theory is presented in the fourth oping rapidly a half a century ago. Matrix of the prime number theorem. Several appen- chapter, the topological structure of games are presented in Section 2 along with dices list background results from real analy- Teichmüller spaces is described and the uni- necessary results from convex analysis and an sis, others discuss numerical calculations of formization theorem for compact Riemann interesting survey of alternative proofs of the π(x) and historical background. (jnek) surfaces is proved. The last chapter contains minimax theorem. The next section is devoted an algebraic geometry approach to Riemann to games in the extensive form and the last J. Jost: Partial Differential Equations, surfaces; homology and cohomology groups section deals with the games on unit square Graduate Texts in Mathematics, vol. 214, of Riemann surfaces are introduced and their and includes also basic concepts of measure Springer, New York, 2002, 325 pp., 59,95 relations to forms are given. The main topic is theory and probability. The book contains euros, ISBN 0-387-95428-7 the Riemann-Roch theorem and the Abel the- many illuminating and motivating examples. The book is intended for students wishing to orem on elliptic functions. The book gives an Numerous exercises appear at well chosen EMS December 2003 29 RECENT BOOKS places of the text to support reader’s under- K. H. Parshall, A. C. Rice, Eds.: dent of mathematics (statistics, finance, etc.) standing. (jdup) Mathematics Unbound: The Evolution of an know in order not to be frustrated by an International Mathematical Research advanced and rigorous course on probability S. Majid: A Quantum Groups Primer, Community, 1800-1945, History of theory? An obvious answer is that the stan- London Mathematical Society Lecture Note Mathematics, vol. 23, American Mathematical dards on abstract integration and construction Series 292, Cambridge University Press, Society, Providence, 2002, 406 pp., US$85, of a measure are not enough when nowadays, Cambridge, 2002, 169 pp., £24,95, ISBN 0- ISBN 0-8218-2124-5 even at the undergraduate level, topics such as 521-01041-1 The book is an excellent work on the history martingales and Brownian motion are often This book is based on lectures of the author on of national and international mathematical included. The text is written in an economic the subject and it keeps the style of oral lec- communities and on the process of its interna- way, but space is left also for heuristics and tures in a nice and pleasant manner. The book tionalization in the period between 1800 and history with the aim to introduce the reader to is divided into three parts. Basic notions (Hopf 1945. The book is divided into 18 chapters probabilistic inventiveness and thinking. algebras, dual pairings, actions and coactions, written by 20 authors (T. Archibald, E. Integration is developed first via extended lin- the quantum plane, quasitriangular Hopf alge- Ausejo, J. E. Barrow-Green, A. Brigaglia, J. ear functionals on spaces of measurable func- bras and its ribbon version, the quantum dou- W. Dauben, S. E. Despeaux, D. D. Fenster, H. tions (in Appendix A); then follow basic prob- ble), together with a well chosen set of basic Gispert, I. Grattan-Guinness, J. J. Gray, M. abilistic topics like independence, condition- examples used throughout the book are pre- Hormigón, O. Lehto, J. Lützen, L. Martini, K. ing, martingales and Fourier transforms. The sented in the first part. Representation theory H. Parshall, A. C. Rice, Ch. Sasaki, S. L. central limit theorems, convergence to and its famous applications in knot theory are Segal, R. Siegmund-Schultze and Y. Xu). Brownian motion, strong representations, cou- treated in the second part using the point of Each chapter describes the evolution of math- plings and the law of iterated logarithm view of braided categories (including ematical communities, research and education receive a deep and detailed treatment to pro- (co)module categories, q-Hecke algebras, trends in relation to the political and econom- vide the reader with a solid information on quantum dimension, algebras in monoidal cat- ical situation in Europe (France, Britain, more recent developments in the Gaussian egories, braided groups and braided differenti- German, Italy, Spain) and in America and branch of probability. The text includes also ation). Applications of these methods to ordi- Asia (China and Japan). Special attention is sections that may be considered really nary Hopf algebras are described in the last paid to the American, Chinese and Japanese advanced (the martingale characterization of part of the book. The reader finds here a dis- mathematical communities as well as to the Brownian motion, option pricing via equiva- cussion of (double) bosonisation, Serre rela- situation in German after 1933. The book lent martingale measures and the disintegra- tions, R-matrix methods, Hopf algebra factor- describes the role of the most important tion procedures being a sample of them). The izations and Lie bialgebras, together with European mathematicians (including Ch. formal mathematical material is accompanied applications to Poisson geometry. Three prob- Hermite, G. Mittag-Leffler, C. Arzelà, L. by solved exercises and by a collection of fre- lems sets can be found at the end of the book. Dickson), the significance of the most impor- quently challenging problems. A really useful The book is self-contained and supposes only tant European mathematical journals (for book for everybody who faces students inter- a basic knowledge of algebra and linear alge- example, Journal de mathématiques pures et ested in modern probability without a prelim- bra. Many intuitive comments and informal appliquées, Acta Mathematica), the creation inary knowledge of measure and integration remarks, a well chosen set of main examples of national mathematical societies (e.g., theory. (jste) used systematically in the book and a clear Société mathématique de France, Circolo and understandable style make the book very mathematico di Palermo), the development I. R. Porteous: Geometric Differentiation. comfortable and useful for students as well as towards the creation of the international con- For the Intelligence of Curves and Surfaces, for mathematicians from other fields. (vs) gress of mathematicians and the establishment second edition, Cambridge University Press, of the International Mathematical Union. A Cambridge, 2001, 333 pp., £24,95, ISBN 0- R. J. McEliece: Theory of Information and lot of references to published as well as 521-81040-X, ISBN 0-521-00264-8 Coding, Encyclopedia of Mathematics and its unpublished sources and many historical and This book presents an introduction to the local Applications 86, second edition, Cambridge bibliographical notes appear in each chapter; differential geometry of curves and surfaces in University Press, Cambridge, 2002, 397 pp., hence the book can influence future investiga- Euclidean space. All standard parts of the the- £60, ISBN 0-521-00095-5 tions of interesting historical questions con- ory (plane curves, curves in three-dimension- This book is a self-contained introduction to nected with the internationalization of al space studied by Frenet frame methods and basic results in the theory of information and research-level mathematics. The book can be surfaces in three-dimensional space) are coding. It has three parts. The introduction recommended not only to historians of mathe- included but there are also some additional contains a short and elementary overview matics but also to professional mathemati- topics. Both the style of the book and its con- leading the reader to basic concepts of coding. cians, teachers and students who want to tent are inspired by ideas and the work of R. The first part is devoted to the main results of understand the creation of international math- Thom and V. I. Arnold on singularity theory. Shannon’s mathematical theory of communi- ematics. (mnem) Several topics related to singularities such as cation. After a reference section collecting umbilics, cusps, probes and ridges and ribs for technical results about entropy, mutual infor- F. Pierrot: K-Théorie et Multiplicités dans surfaces are presented in details. Topics dis- mation, etc., the author develops techniques L2(G/Γ), Mémoires de la Sociéte cussed in the book include moreover plane that are necessary to prove Shannon’s funda- Mathématique de France 89, Sociéte kinematics, curves on the unit sphere, multi- mental results, the channel and source coding Mathématique de France, Paris, 2002, 85 pp., linear forms and their applications in geome- theorems. The first part ends with a survey of FRF 150, ISBN 2-85629-119-8 try, probes and contacts. In surface theory, advanced results. The second part is devoted The author proves a generalization of Atiyah’s ridge and rib points as well as umbilics are to a study of specific codes, which can be used L2-index theorem, from which he deduces a K- studied very intensively. The discussion of for channel and source coding. It deals with theoretical version of Langlands’ theorem on parabolic and subparabolic lines on a regular linear codes in general, with cyclic codes, the multiplicity of discrete series representa- surface is interesting. Some concepts and BCH, Reed-Solomon, and related codes. tions occuring in L2(G/Γ). Using recent results results of V. I. Arnol’d regarding curves on S2 There are also chapters on convolutional of V. Lafforgue, the author then deduces a and its extension to curves and surfaces on S3 codes and on variable-length source coding. generalization of Langlands’ results to a larg- can be found in the last chapter. There are The second part ends with a survey of er class of groups. (jnek) many historical notes and comments through- advanced topics, as well. Appendices contain out the book as well as some curiosities. It is basic information on probability theory, con- D. Pollard: A User’s Guide to Measure a very good and interesting introduction to vex functions, Jensen’s inequality, finite Theoretic Probability, Cambridge Series in differential geometry of curves and surfaces, fields, and path enumeration in directed Statistical and Probabilistic Mathematics, which can be recommended to anybody inter- graphs. A prior knowledge of these areas is Cambridge University Press, Cambridge, ested in the subject. (jbu) not a necessary prerequisite but it will certain- 2002, 351 pp., £60, ISBN 0-521-80243-3, ly make it easier to follow the presentation. ISBN 0-521-00289-3 A. Pressley, Ed.: Quantum Groups and Lie (jtu) Which parts of measure theory should a stu- Theory, London Mathematical Society 30 EMS December 2003 RECENT BOOKS Lecture Note Series 290, Cambridge The book is also recommended to practition- lowing: A detailed formula for the pair corre- University Press, Cambridge, 2002, 234 pp., ers. (jh) lation function is given as an expan- £27,95, ISBN 0-521-01040-3 sion in the variable h, (the expectation < > is This book is based on the LMS Symposium on M. Schatzman: Numerical Analysis: A given in the above mentioned Gibbs measure). quantum groups held at Durham in 1999; it Mathematical Introduction, Clarendon The result is rather general and greatly contains papers based on lectures presented Press, Oxford, 2002, 496 pp., £24,95, ISBN 0- improves earlier results, due to the author and there. There are two surveys papers presenting 19-850852-2, ISBN 0-19-850279-6 others. The essential tool used in the proof, the content of two lecture series. The first one This book provides an elegant mathematical introduced already in the previous works by (written by S. Ariki) contains an introduction introduction to numerical analysis summariz- Helffer and Sjöstrand, is the formula express- to cyclotomic Hecke algebras using the lan- ing some of the required knowledge from lin- ing the pair correlation function in terms of guage of Fock spaces and their relation to the ear algebra, Fourier analysis, functional the ‘Witten Laplace’ operator (defined as the study of solvable lattice models in statistical analysis and partial differential equations. It Hodge Laplacian with the help of the corre- mechanics. The second one (by P. Etinghof covers many basic ideas of modern numerical sponding creation/ annihilation operators). and O. Schiffmann) contains a description of computing, with an emphasis on methods and The problem is then translated to the study of the classical and quantum dynamical Yang- . The book is divided into four spectral properties (bottom of the spectrum) of Baxter equation and its applications to the the- parts. Part I starts with a guided tour on float- such an operator, more precisely of its zeroth ory of integrable systems and to representa- ing number systems and machine arithmetic. and first part, applying a method called the tion theory. The book contains moreover 11 The exponential and logarithms are construct- ‘Grushin (Feshbach) approach’ by the author. papers on various topics in the field. They are ed from scratch to present a new point of view The book is divided into twelve short chapters written by E. Beggs (group doublecross prod- on well-known questions. Part II consists of 5 and two appendices. Chapter 0 and 1 intro- ucts), R. Carter and R. Marsh (canonical chapters. It starts with polynomial approxima- duce the problem and formulate the main bases), V. Chari and A. Pressley (modules of tion (polynomial interpolation, mean-square result (and its generalization); the remainder quantum affine sl2), B. Drabant (balanced cat- approximation, splines). It deals moreover develops the details of various steps of the egories and Hopf algebras), K. R. Goodearl with Fourier series, providing the trigonomet- proof (reshuffling of the creation/ annihilation (quantized primitive ideal spaces), I. Gordon ric version of least-square approximations, operators, investigation of higher order (quantum groups at roots of unity), J. Ding and with one of the most important numerical Grushin problems, asymptotics of the correla- and T. J. Hodges (YB equation for operators algorithms, the fast Fourier transform. Part III tions, etc.). The method developed in the pre- on function fields), S. Majid (twisting of (Chapters 9-12) relates to numerical linear sented book is an important contribution to a quantum groups), I. M. Musson (finite quan- algebra. This part is important because opera- subject which has great importance in statisti- tum groups and pointed Hopf algebras), D. tion counts are the limiting factor for any seri- cal physis and quantum field theory and which Parashar and R. J. McDermott (quantum and ous computation. Chapters 9-11 deal with was up to now treated in the literature by quite Jordanian deformations) and H. Wenzl (tensor direct and iterative methods for the solution of different methods, like cluster expansions. category and braid representations). This is an linear systems of equations, with an emphasis (mzahr) interesting collection of papers in the field. on operation counts. Chapter 12 presents (vs) orthogonality methods for the solution of lin- E. M. Stein, R. Shakarchi: Complex ear systems and introduces the QR decompo- Analysis, Princeton Lectures in Analysis II, B. Rustem, M. Howe: Algorithms for Worst- sition. Part IV (Chapters 13-18) treats a selec- Princeton University Press, Princeton, 2003, Case Design and Applications to Risk tion of non-linear complex problems: the 379 pp., £35, ISBN 0-691-11385-8 Management, Princeton University Press, numerical computation of eigenvalues and The book is the second volume of a ‘minis- Princeton, 2002, 389 pp., £52, ISBN 0-691- eigenvectors of a square matrix (the power eries’ of four books forming a course of analy- 09154-4 method, the QR method), solution of nonlin- sis. While the first three volumes contain Risk management topics are treated in this ear equations and systems, ordinary differen- introductory Fourier analysis, complex analy- book in a way that is different from what the tial equations (single-step and linear multistep sis and measure and integration theory (with reader might think of first. World-wide expe- schemes), and the numerical analysis of some Hilbert space theory), the last one covers rience during the last decade shows, that it is partial differential equations. Each chapter some parts of functional analysis, probability perhaps more realistic to consider worst possi- contains several exercises. The examples are theory, etc. The main idea behind the series is ble scenarios in the financial world than just to carefully selected and illustrate many impor- to explain these parts in a unified way accent- perform inference based on more optimistic tant ideas in the field. The book does not ing their mutual interplay. The book under criteria. The roots of the presented ideas and assume any previous knowledge of numerical review contains (together with traditional methods arise in part from the theory of methods and is written for advanced under- parts, like the Cauchy theorem and its conse- games, in particular from the minimax princi- graduate students in mathematics. It will also quences, meromorphic functions, entire func- ple. This seems to be the leitmotif. The title of be useful for scientists and engineers wishing tions or conformal mapping) also chapters on the book does not really describe the content to learn whether mathematics can explain why zeta functions and the prime number theorem, of the book and may be misleading. There is their numerical methods work - or fail. (knaj) elliptic functions, and on applications of theta much more in the book than just algorithms. functions. Two appendices deal with asymp- Most of the key concepts of financial mathe- J. Sjöstrand: Complete Asymptotics for totics and simple connectivity and with the matics are clearly explained and discussed in Correlations of Laplace Integrals in the Jordan curve theorem. The style of exposition the text. Nevertheless, the fundamental idea of Semi-classical Limit, Mémoires de la Société is similar to the classical book by Rudin. The ‘worst-case’ is thoroughly kept through the Mathématique de France 83, Société book contains numerous exercises (165) and whole text. Let us describe just a sample from Mathématique de France, Paris, 2000, 104 problems (50) with hints (some of the prob- a broad area of topics covered in the book’s pp., FRF 150, ISBN 2-85629-097-3 lems are rather demanding). The authors hope eleven chapters: computing of saddle points, The book summarizes progress achieved by that the book(s) will be accessible to students numerical experiments with minimax algo- the author and others (V. Bach, T. Jecko) in interested in diverse disciplines (mathematics, rithms, strategies for securities’ hedging, sim- the study of the correlation decay for Laplace physics, engineering, finance), at both the ulation studies, asset allocation problems, integrals of the type ∫exp{- φ(x)/h}dx in the undergraduate and graduate levels. (jive) asset-liability management including immu- low temperature regime, i.e., for small values nization, and currency management. In the of h. The variable x is taken from RΛ, where Λ G. Tourlakis: Lectures in Logic and Set bonanza of books on mathematical finance is either a d-dimensional torus, or a finite sub- Theory, vol. I: Mathematical Logic, and/or financial mathematics, the book under set of an infinite lattice. The function φ(x) is Cambridge Studies in Advanced Mathematics review is surely an exception. The presented assumed to satisfy a certain condition of posi- 82, Cambridge University Press, Cambridge, ideas may be exploited both in theory and tive definiteness for its second derivative near 2003, 238 pp., £47,50, ISBN 0-521-75373-2 practice. Strongly recommended to anyone its (unique) minimum point. The case of ‘one G. Tourlakis: Lectures in Logic and set with interest in financial mathematics and potential well’, with ‘nearly quadratic’ behav- Theory, vol. II: Set Theory, Cambridge keen to acquire non-standard pieces of knowl- iour around the minimum point is thus consid- Studies in Advanced Mathematics 83, edge of standard and other financial problems. ered. The main result of the book is the fol- Cambridge University Press, Cambridge, EMS December 2003 31 RECENT BOOKS 2003, 575 pp., £65, ISBN 0-521-75374-0 results. Many classical results on colourings (mnem) These two volumes present a basic course on not directly related to the Four Colour mathematical logic and set theory. The con- Theorem are left out as well. Anybody seri- V. V. Yaschenko, Ed.: Cryptography: An tents are standard, including first order lan- ously interested in graph colourings would Introduction, Student Mathematical Library, guages, axioms and rules of inference, basic thus need to look for other sources for further vol. 18, American Mathematical Society, metatheorems, the Löwenheim-Skolem theo- reading. On the other hand, the book provides Providence, 2002, 229 pp., US$39, ISBN 0- rems, Gödel’s completeness theorem, the a neatly wrapped up history of the Four 8218-2986-6 compactness theorem, and full proofs of both Colour Conjecture; it is moreover published in A very clearly written introduction to cryptog- Gödel’s incompleteness theorems in the first the year of its 150th anniversary. As such it raphy. The presented material is illustrated volume; the axioms of Zermelo-Fraenkel set will please every reader. And last but not with many examples and notes, so that the text theory, the axiom of choice, the natural num- least, it is a thin book, which will not eat much can be recommended to every novice in the bers, partially ordered sets, ordinal and cardi- space in your library; the contents to weight subject. The reader finds information on all nal numbers with their elementary arithmetic, ratio is well above the average. (jk) useful basic topics and results as well as com- the constructible universe and elements of ments outlining research topics. The book forcing in the second volume. Extreme care R. Wilson: Four Colours Suffice, Penguin, consists of 6 chapters (the English edition paid to details; most of the material is London, 2002, 262 pp., £12,99, ISBN 0-713- does not contain Chapter 6 of the Russian explained twice, first at an informal level and 99670-6 original, entitled ‘Computer and Crypto- then formally; plenty of footnotes are added to The Four Colour Problem was one of the most graphy’), which are written by different sharpen the reader’s understanding. This famous problems of mathematics of the 20th authors and which can be read independently approach (together with a verbatim repetition century. The problem even attracted the atten- of each other to a certain extent. For instance, of 57 pages from Volume 1 in Volume 2) may tion of the general public. Even the solution of the last Chapter 6 is based on problems be appreciated by a student, who wishes to the problem was accompanied by doubts, and offered at the Russian ‘Cryptography find every dot on every i. But it consumes a lot it was the starting point for a discussion about Olympiads for High School Students’ since of space and consequently makes it impossi- the nature of mathematical proofs. The pre- 1991. Its understanding requires almost no ble to include more advanced chapters: the sent book accurately describes the origin and knowledge beyond standard high school rationals and the reals are not constructed, the colourful history of the problem and vari- mathematics. The previous five chapters of except for the ∆-system lemma, infinitary ous proof attempts. The book is written for a the booklet provide the reader with the most combinatorics is not introduced, the basics of general public; the recent development (such basic notions and techniques used in the con- Borel and of projective sets is missing and as work of Robertson, Seymour and Thomas) temporary symmetric and asymmetric cryp- widely used principles, like MA, are never is only briefly mentioned. It is a very good tosystems. All notions like one-way functions, mentioned. (psim) popular book. Its lively and accessible form one-way trapdoor functions, key exchange will pave the way also to non-experts and to a protocols, or zero-knowledge protocols are R. A. Wilson: Graphs, Colourings and the non-mathematical audience. (jnes) discussed in a highly lucid way. Attractive Four-colour Theorem, Oxford Science themes like coin flipping over the phone, Publications, Oxford University Press, B. H. Yandell: The Honors Class. Hilbert’s basic algorithmic number theory (including Oxford, 2002, 141 pp., £19,50, ISBN 0-19- Problems and Their Solvers, A K Peters, RSA, factorisation and primality), 851062-4, ISBN 0-19-851061-6 Natick, 2002, 486 pp., US$39, ISBN 1-56881- questions for some number-theoretical algo- 141-1 rithms or threshold secret sharing schemes are The theory of graph colouring has flourished This book is devoted to the history of not missing. The book is well-motivated and over decades (in fact, already for one and a Hilbert’s problems and their solutions. The can be recommended as first reading to every- half centuries) with the planar map colouring collection of 23 mathematical problems pre- body interested in cryptography. (spor) problem having been its corner stone. sented by at the ICM in Paris in Considered first as a riddle, the Four Colour 1900 became soon a guiding inspiration for List of reviewers for 2003 Conjecture had soon turned into a nightmare mathematicians in the 20th century. The book The Editors would like to thank the fol- for discrete mathematicians and graph theo- under review starts with a description of lowing for their reviews this year. rists. But it was a nightmare with a very posi- Hilbert’s biography and the background of his J. Andel, A. Barkhudarjan, R. Barták, R. tive influence on the development of graph work and his problems. Then the author Bashir, M. Becvárová-Nemcová, M. Beneš, theory. Through numerous attempts and false describes the achievements and success of L. Beran, B. Maslowski, L. Bocek, J. proofs, it has motivated new notions, new the- people who devoted their lives to solve Bureš, E. Calda, R. Cerný, M. Cihák, T. orems and new theories. And finally its solu- Hilbert’s problem. The first part is about prob- Cipro, J. Dupacová, J. Drahoš, A. Drápal, tion, the proof that turned the Four Colour lems 1, 2 and 10 (set theory and real num- M. Engliš, M. Ernestová, E. Fašangová, M. Conjecture into the Four Colour Theorem, is bers), the second part explains problems 3, 4, Feistauer, J. Fiala, S. Hencl, D. Hlubinka, one of the strong concrete links between dis- 5 and 6 (geometry, specific areas), the third J. Hurt, M. Hušek, J. Jelínek, O. John, P. crete mathematics and computer science. The part describes problems 7, 8, 9, 11 and 12 Kaplický, M. Klazar, P. Knobloch, O. proof depends heavily on computers, and no (number theory), the fourth part is devoted to Kowalski, D. Král, J. Král, A. Kucera, P. human being can check all details and cases, problems 14, 15, 16, 17 and 18 (modern alge- Kurka, J. Lukeš , J. Málek, J. Malý, J. which is a blessing in the eyes of (most) com- bra, geometry and graph theory) and the last Milota, J. Mlcek, K. Najzar, J. Nekovár, J. puter scientists and rather a weird feature in part discusses problems 13, 19, 20, 21, 22 and Nešetril, B. Novák, L. Pick, M. Plátek, A. the eyes of (some) mathematicians. The book 23 (analysis). In all cases, the mathematical Pultr, P. Pyrih, Š. Porubský, L. Poul, D. under review presents a concise monograph substance of these problems, their solutions or Praák, Z. Prášková, M. Pokorný, T. on graph colouring results connected to the the attempts to solve them, short biographical Roubícek, P. Ševera, P. Simon, P. Four Colour Theorem. It describes ideas of information on mathematicians involved and Somberg, V. Soucek, J. Spurný, J. Stará, early attempts, points out mistakes in pub- interesting historical aspects are described. J. Štepán, J. Trlifaj, J. Tuma, P. Valtr, J. lished false proofs and gives a quite legible This part of the book can also be appreciated Vanura, J. Veselý, M. Zahradník, K. outline of ideas behind the computer-aided by readers who are not mathematicians. An Zvára, J. emlicka. proofs. On the way to concluding sections, appendix contains Hilbert’s lecture delivered several classical results are presented, includ- at the Second International Congress of All of the above are on the staff of the ing the Euler formula for plane graphs, and Mathematicians in 1900 (in English) and Charles University, Faculty of Mathematics theorems of Heawood, Brooks and Vizing. reprinted from the Bulletin of the American and Physics, Prague, except: The last section contains a detailed and useful Mathematical Society 8(1902). Ample notes, M. Engliš, B. Maslowski and J. Vanura exposition of the discharging method. A cer- an extensive selected bibliography and an (Mathematical Institute, Czech Academy of tain drawback of the book is the lack of pre- alphabetical index are included. The book can Sciences), Š. Porubský (Technical sentation of recent new results and directions be recommended to everybody who wants to University, Prague), P. Ševera (University in graph colouring - an area, which is very know more about the birth of modern mathe- of Bratislava, Slovakia), J. Nekovár rich both in applications and theoretical matics in the first half of the 20th century. (University Paris VI, France). 32 EMS December 2003