European Mathematical Society

December 2004 Issue 54

EMS Mathematical Weekend Prague 2004 p. 5

Feature Symmetries p. 11

Societies Polskie Towarzystwo Matematyczne p. 24

ERCOM ISSN = 1027 - 488X ICMS Edinburgh p. 30 NEWSLETTER CONTENTS EDITORIAL TEAM EUROPEAN MATHEMATICAL SOCIETY EDITOR-IN-CHIEF MARTIN RAUSSEN Department of Mathematical Sciences, Aalborg University Fredrik Bajers Vej 7G DK-9220 Aalborg, Denmark e-mail: [email protected] ASSOCIATE EDITORS VASILE BERINDE Department of Mathematics, University of Baia Mare, Romania NEWSLETTER No. 54 e-mail: [email protected] KRZYSZTOF CIESIELSKI Mathematics Institute December 2004 Jagiellonian University Reymonta 4, 30-059 Kraków, Poland EMS Agenda ...... 2 e-mail: [email protected] STEEN MARKVORSEN Editorial - Luc Lemaire ...... 3 Department of Mathematics, Technical University of Denmark, Building 303 EMS Executive Committee...... 4 DK-2800 Kgs. Lyngby, Denmark e-mail: [email protected] 2nd Joint Mathematical Weekend ...... 5 ROBIN WILSON Department of Pure Mathematics EMS Executive Committee Metting ...... 7 The Open University Milton Keynes MK7 6AA, UK Applied Mathematics and Applications of Mathematics ...... 9 e-mail: [email protected] COPY EDITOR: Symmetries - Alain Connes ...... 11 KELLY NEIL School of Mathematics Mathematics is alive... - Luc Lemaire ...... 19 University of Southampton Highfield SO17 1BJ, UK Polish Mathematical Society - Janusz Kowalski ...... 24 e-mail: [email protected] SPECIALIST EDITORS ERCOM - ICMS ...... 30 INTERVIEWS Steen Markvorsen [address as above] Problem Corner - Paul Jainta ...... 33 SOCIETIES Krzysztof Ciesielski [address as above] Personal Column ...... 37 EDUCATION Tony Gardiner Forthcoming Conferences ...... 38 University of Birmingham Birmingham B15 2TT, UK Recent Books ...... 41 e-mail: [email protected] MATHEMATICAL PROBLEMS Printed by Armstrong Press Limited Paul Jainta Werkvolkstr. 10 Crosshouse Road, Southampton, Hampshire SO14 5GZ, UK D-91126 Schwabach, Germany telephone: (+44) 23 8033 3132 fax: (+44) 23 8033 3134 e-mail: [email protected] e-mail: [email protected] ANNIVERSARIES Published by European Mathematical Society June Barrow-Green and Jeremy Gray Open University [address as above] ISSN 1027 - 488X e-mail: [email protected] and [email protected] The views expressed in this Newsletter are those of the authors and do not necessari- CONFERENCES ly represent those of the EMS or the Editorial team. Vasile Berinde [address as above] RECENT BOOKS Ivan Netuka and Vladimír Souèek NOTICE FOR MATHEMATICAL SOCIETIES Mathematical Institute Labels for the next issue will be prepared during the second half of February 2005. Please send your updated lists before then to Ms Tuulikki Mäkeläinen, Department of Charles University, Sokolovská 83 Mathematics and Statistics, P.O. Box 68 (Gustav Hällströmintie 2B), FI-00014 University of 18600 Prague, Czech Republic Helsinki, Finland; e-mail: [email protected] e-mail: [email protected] and [email protected] INSTITUTIONAL SUBSCRIPTIONS FOR THE EMS NEWSLETTER ADVERTISING OFFICER Institutes and libraries can order the EMS Newsletter by mail from the EMS Secretariat, Vivette Girault Department of Mathematics and Statistics, P.O. Box 68 (Gustav Hällströmintie 2B), FI-00014 Laboratoire d’Analyse Numérique University of Helsinki, Finland, or by e-mail: ([email protected] ). Please include Boite Courrier 187, Université Pierre the name and full address (with postal code), telephone and fax number (with country code) and et Marie Curie, 4 Place Jussieu e-mail address. The annual subscription fee (including mailing) is 80 euros; an invoice will be 75252 Paris Cedex 05, France sent with a sample copy of the Newsletter. e-mail: [email protected]

EMS December 2004 1 EMS NEWS EMS Committee EMS Agenda EXECUTIVE COMMITTEE 2005 PRESIDENT 1 February Prof. Sir JOHN KINGMAN (2003-06) Deadline for submission of material for the March issue of the EMS Newsletter Isaac Newton Institute Contact: Martin Raussen, email: [email protected] 20 Clarkson Road Cambridge CB3 0EH, UK 19-27 February e-mail: [email protected] EMS Summer School at Eilat (Israel) VICE-PRESIDENTS Applications to Geometry, Cryptography and Computation. Prof. LUC LEMAIRE (2003-06) Web site: http://www.cs.biu.ac.il/~eni/ann1-2005.html Department of Mathematics Université Libre de Bruxelles 16-17 April C.P. 218 – Campus Plaine EMS Executive Committee meeting, at the invitation of the Unione Matematica Italiana, in Bld du Triomphe Capri (Italy) B-1050 Bruxelles, Belgium Contact: Helge Holden: [email protected] e-mail: [email protected] Prof. BODIL BRANNER (2001–04) Department of Mathematics 25 June – 2 July Technical University of Denmark EMS Summer School at Pontignano (Italy) Building 303 Subdivision schemes in geometric modelling, theory and applications DK-2800 Kgs. Lyngby, Denmark e-mail: [email protected] 17-23 July EMS Summer School and European young statisticians’ training camp at Oslo (Norway) SECRETARY Prof. HELGE HOLDEN (2003-06) Department of Mathematical Sciences 11 - 18 September Norwegian University of Science and Technology EMS Summer School and Séminaire Européen de Statistique at Warwick (UK) Alfred Getz vei 1 Statistics in Genetics and Molecular Biology NO-7491 Trondheim, Norway Web site: http://www2.warwick.ac.uk/fac/sci/statistics/news/semstat/ e-mail: [email protected] 13-23 September TREASURER EMS Summer School at Barcelona (Catalunya, Spain) Prof. OLLI MARTIO (2003-06) Recent trends of Combinatorics in the mathematical context Department of Mathematics P.O. Box 4, FIN-00014 Web site: http://www.crm.es/RecentTrends/ University of Helsinki, Finland e-mail: [email protected] 16-18 September EMS-SCM Joint Mathematical Weekend at Barcelona (Catalunya, Spain) ORDINARY MEMBERS Prof. VICTOR BUCHSTABER (2001–04) 18-19 September Steklov Mathematical Institute EMS Executive Committee meeting at Barcelona Russian Academy of Sciences Gubkina St. 8, Moscow 117966, Russia e-mail: [email protected] 12-16 December Prof. DOINA CIORANESCU (2003–06) EMS-SIAM-UMALCA joint meeting in applied mathematics Laboratoire d’Analyse Numérique Venue: the CMM (Centre for Mathematical Modelling), Santiago de Chile Université Paris VI 4 Place Jussieu 2006 75252 Paris Cedex 05, France 3-7 July (to be confirmed) e-mail: [email protected] Applied Mathematics and Applications of Mathematics 2006 at Torino(Italy) Prof. PAVEL EXNER (2003-06) Department of Theoretical Physics, NPI 22-30 August Academy of Sciences International Congress of Mathematicians in Madrid (Spain) 25068 Rez – Prague, Czech Republic Web site: http://www.icm2006.org/ e-mail: [email protected] Prof. MARTA SANZ-SOLÉ (2001-04) 2007 Facultat de Matematiques 16-20 July Universitat de Barcelona 6th International Congress on Industrial and Applied Mathematics (iciam 07) at Zürich Gran Via 585 (Switzerland) E-08007 Barcelona, Spain Web site: http://www.iciam07.ch/ e-mail: [email protected] Prof. MINA TEICHER (2001–04) Department of Mathematics and 2008 14-18 July Computer Science Bar-Ilan University 5th European Mathematical Congress at Amsterdam (The Netherlands) Ramat-Gan 52900, Israel Web site: http://www.5ecm.nl 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 and Statistics P.O. Box 68 (Gustav Hällströmintie 2B) Advertisements FI-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)-9-1915-1426 Intermediate rates: Full page: £176; half-page: £98; quarter-page: £59 fax: (+358)-9-1915-1400 telex: 124690 Inserts e-mail: [email protected] Postage cost: £14 per gram plus Insertion cost: £58 (e.g. a leaflet weighing 8.0 grams will Web site: http://www.emis.de cost 8x£14+£58 = £170) (weight to nearest 0.1 gram) 2 EMS December 2004 EDITORIAL EditorialEditorial Towards a European Research Council (ERC) Luc Lemaire (EMS Vice-President, Bruxelles)

The project of giving the responsibility of dis- ance, the Commission made the explicit request tributing research grants to a European Research that for the next financial period starting in 2007, Council (ERC) is currently the subject of intense the budget of research handled by the EC should discussions in scientific and political circles in be doubled. The budget of the Framework Europe. The EMS supports this project, and Programme (around four billion euros per year) encourages all its members to promote it at polit- would thus be supplemented by an equivalent ical and scientific levels in their countries. budget, a large part of which would be attributed Briefly put, the ERC would be both an advi- to the ERC. this initiative. It will also act as a reflection plat- sory council and a funding agency at the At this stage, the research ministers of Europe form for the development of science in Europe in European level, with the specific aim of devel- support the project, but the finance ministers will the future. oping fundamental research in all disciplines. have to include it in their global perspectives. Its web site is http://www.initiative-science- europe.org A two-year history Basic principles The idea of an ERC, catering for fundamental For the time being there is reasonable agreement What you can do? research in Europe, was officially launched at an (at all levels) on the principles to be implement- Right now we are in a position of great uncer- international conference in Copenhagen in ed in the ERC. tainty. The idea of an ERC has moved from October 2002, during the Danish EU presidency. The ERC would be in charge of basic, funda- nothing to a fully-fledged project, with backing An expert group was created by the Danish mental, scientist-driven research. Indeed, it is from the European Commission and (probably) Minister for Science, Technology and (finally!) admitted that this type of research is a the Council of European Ministers. However, Innovation, Helge Sander, under the leadership necessary investment with a long-term perspec- whether the necessary budget will be made of Federico Mayor, and the group made explicit tive that also has to be supported at the European available is doubtful. A strong political argument proposals in December 2003. level. has to be developed by scientists, in particular in Meanwhile, thanks to the role taken by the Researchers would apply for grants from the the larger states of the EU, which at this stage do European Life Science Forum (ELSF) and other ERC, and the selection would be based only on not approve of the idea of increasing the EU organisations of life scientists and to the scientific quality, by a rigorous peer review budget, thereby preventing the doubling of the unbounded energy of Luc Van Dyck, the scien- process. research budget. The next few months will be tific community caught on to the idea and There would be no notion of “juste retour” for crucial, and a determined action is required in formed an informal platform of discussion, the the states funding the ERC through their EU the member states whenever an opportunity aris- Initiative for Science in Europe (ISE), and had contributions, and no criteria other than the sci- es. further debates on the aims and the means of entific excellence, which in itself provides the The idea that basic research is key to the future such a structure. They quickly had the idea to added value for Europe. success of Europe is now better acknowledged, expand their vision to all sciences, including A scientific council would be appointed, and but further efforts are needed to convince the social sciences and humanities, and naturally be put in charge of managing the ERC without larger states that part of the scientific activities mathematics. political interference, for instance on the choice are better handled at a European level, and that a At the time, the idea was wild: the European of subjects. financial effort to make this possible at the right Union’s activities were limited to research The ERC should be funded thanks to the level must be made now. It is almost sure that the applicable to industrial development, with a increase in the part of the EU budget going to European Parliament will have a say in the component of training through the Marie Curie research, and absolutely not by diminishing the process, and at this moment very few of the actions, and it was believed that the EU could corresponding amounts from those of the nation- members have formed an idea on the project. It not do anything else without violating the sub- al research agencies. may be a good time to approach them, and dis- sidiarity principle, implying that basic research The budget should be commensurate with the cuss this issue with them. This could be a good should remain national. However, at a meeting ambitions of the project. Current discussions starting point for further contacts since, as math- in Dublin in October 2003, Achilleas Mitsos, (but not by those who will have to find the ematicians, we could make our case in the con- Director General with the EC Directorate money!) mention amounts between one and four text of a comprehensive project where all sci- General (DG) Research, announced that the billion euros per year. ences are involved, but nevertheless of promi- Commission supported the idea of an instrument A serious problem is the risk of oversubscrip- nent significance for us, in view of the almost for basic research and, for this purpose, would tion. In the EU Framework Programmes, the rate evanescent support now available for mathe- request a specific credit line from the EU budget. of success of applications is sometimes down to maticians in the Framework Programme. The question of budget is of course a major 5%, which leads to the discouragement of the problem, even though the EU member states best scientists. The scientific council of a future References committed themselves to reach the Barcelona ERC will have to handle this problem, but avoid More information on the project and its evolu- objective (3% of the National Product to be artificial rules resulting in the exclusion of some tion can be found in the ELSF brochure invested in research - 1% public and 2% private proposals. It may think of putting a limit to the http://www.elsf.org/elsfbrochures/elsferc03.pdf, funding) by 2010. However, if in 2002 all states size, rotating subjects or using other means - but and the site http://www.elsf.org/elsfercc.html agreed to have a much increased research budget overcoming this difficulty without perverting the provides access to a large collection of docu- by 2010, most decreased theirs within a few project will be necessary. ments. months! A letter of support for the ERC plan, signed by Still, rowing against many currents, the group EMS and ISE representatives of 52 associations, was published pursued its effort in favour of establishing an The ISE started informally with a dozen in August in Science, see http://www.initiative- ERC, with the effect that it now appears as a real European associations. On October 25th 2004, science-europe.org/forms_maps/Science.pdf and solid perspective. alongside a meeting held in Paris, it was more A web search on ERC and ISE will show that Clearly, the European Commissioner for officially created, and the EMS is one of its the debate on ERC is developing, and that the Research, Philippe Busquin, has played an members. In fact, it is the only body representing advisory board of the Commission (EURAB) is important role in this process by introducing the mathematicians in this structure. very much in favour of the project, but sadly that idea of a European Research Area and promot- The ISE was created around the idea of sup- some scientific bodies in large member states ing it vigorously inside the EU. Under his guid- porting the ERC and will continue monitoring refuse to give up some of their national power.

EMS December 2004 3 EMS NEWS International Student Assessment). In his opinion the EMS has a very important role in promoting mathematical NewNew membersmembers research and supporting the European mathematical community. The EMS also gives considerable attention to increasing ofof thethe the level of understanding by the public of the nature and importance of mathematics. EMSEMS executiveexecutive committeecommittee The decline in the number of mathematics students in some European countries is one of his main concerns. He believes we need to continue and expand our efforts to -At its meeting in Uppsala (Sweden) in Giorgi. He has also been a visiting profes- attract good students into our subjects by June this year, EMS council elected three sor at the University of Marlyland, diversifying graduate programs and pro- new members to the societies executive University of Syracuse, University of moting excellence in the teaching of committee for a four-year period begin- Paris VI and the University of Umea. He mathematics. His main hobby is singing ning in January 2005. Victor Buchstaber became a full professor at the University Neapolitan songs. (Newsletter 38) was reelected. Thanks of Naples in 1980. were given to Bodil Branner, Marta Sanz- Klaus Schmidt received a PhD from the Solé and Mina Teicher who leave the com- University of Vienna in 1968. In 1969 he mittee after several years of service. went to Great Britain, first as a one-year post-doc at the University of Manchester, Olga Gil Medrano received a Doctorado then as a lecturer at Bedford College, from the Universitat de Valéncia in 1982 University of London, and in 1973/1974 and a Doctorate from the Université de he joined Warwick University, where he Paris VI in 1985. She has been associate was promoted to professor in 1983. In professor at the Universitat de Valéncia 1994, he returned to the University of since 1986. Vienna.

His research interests include: homogenization of linear and nonlinear partial differential operators, relaxation of integral functionals of calculus of variations, existence and regularity theory for solutions of variational problems, reverse Holder inequalities, weak minima of variational Her research interests lie in differential integrals, regularity theory of Jacobians geometry and global analysis (in particu- and quasiconformal mappings. lar, variational problems on Riemannian He has organized several conferences in manifolds). She has visited several univer- Italy and France, and he is a member of the sities, while working in cooperation with editorial board of various journals inclu- colleagues from Austria, Belgium, Brazil ding “Annali di Matematica”, “Rendiconti and France. dell’ Accademia Nazionale dei Lincei”, Klaus Schmidt’s research is mainly in Olga has been a member of the “Ricerche di Matematica” and “Bollettino the area of ergodic theory and its connec- Executive Committee of the Real dell’Unione Matematica Italiana”, which tions with other branches of mathematics Sociedad Matemática Española (Royal he currently edits. (like operator algebras and arithmetic). In Spanish Mathematical Society, RSME) Carlo is a Fellow of the Accademia 1993, he was awarded the Ferran Sunyer i during the last four years, and she is now Nazionale dei Lincei, Vice-President of Balaguer Prize for the monograph one of the vice-presidents. Amongst other the Accademia Pontaniana and Secretary ‘Dynamical Systems of Algebraic Origin’. things, she has been the Editor-in-Chief of of the Società Nazionale di Scienze He became a full member of the Austrian the newsletter of the RSME from its Lettere e Arti in Naples. He is the Academy of Science in 1997. beginning. President of the Unione Matematica From 1995 - 2003 he was one of the two Italiana and a member of the Technical Scientific Directors of the Erwin Carlo Sbordone was born in 1948 in Secretariat of the Italian Minister of Schrödinger International Institute for Naples, Italy. He was a visiting researcher Education, University and Research. He is Mathematical Physics (ESI), Vienna, in the Scuola Normale Superiore of Pisa also a member of the Italian Committee Austria, and he is now President of the for two years, working with Ennio De for the OECD-PISA (Programme for ESI.

4 EMS December 2004 EMS NEWS 2nd2nd JointJoint MathematicalMathematical WWeekendeekend Prague, September 3-5, 2004

Jan Kratochvíl, Jiøí Rákosník (Prague)

The framework seminar, blending with all the advantages of In order to support the growing sense of modern technology. A short explanation European identity that follows the political about its history was given by Ivan Netuka, changes in Europe, the European the dean of the faculty, in his welcoming Mathematical Society organizes many regu- speech. The participants admired the skills lar events, the European Mathematical the architects had performed during the Congresses being the largest and most impor- recent renovation works, allowing the School tant. To maintain more frequent contact with of Computer Science to fit so well in a build- national societies and to help promote the ing with old stone doorframes and ceilings ideas of cross-European collaboration, the decorated with 300-year-old frescos. EMS has launched a new series of annual meetings called the Joint Mathematical Minisymposia and plenary lectures Weekends. The three-day meetings are finan- The meeting itself was opened by Jan cially supported by the EMS, under the Kratochvíl, the president of the Czech requirement that the organizing national soci- Mathematical Society. The European ety arranges additional funds so that a large Mathematical Society was represented by the number of local, as well as other European welcoming address of Luc Lemaire, a vice- mathematicians, may participate without president of the EMS. The Friday afternoon paying an expensive conference fee. The programme started with the plenary lecture meeting consists of several parallel minisym- “Some recent applications of the classifica- posia whose topics are chosen to represent tion of finite simple groups” by Jan Saxl from the strongest areas of the organizing society. the University of Cambridge. His plenary Each minisymposium is coordinated by a talk introduced the minisymposium local expert who chooses one plenary speak- “Discrete Mathematics and Combina- er to represent the area and to give a general torics”, organized by Jaroslav Nešetøil from lecture introducing the subject to all partici- Charles University in Prague and director of pants of the conference. Together they select ITI. As the title of the talk suggests, it pre- several leading European experts in the area sented interactions of algebra and combina- as further invited speakers. Contributed talks torics, presenting applications in various were also the subject of the Saturday after- are also accepted as the time schedule allows. areas of graph theory and discrete mathemat- noon plenary talk “Probabilistic and varia- The first meeting of this type was organized ics. In the programme of the following days, tional methods in problems of phase coexis- by the Portuguese Mathematical Society in the invited speakers in this minisymposium tence”, given by Errico Presutti from Lisbon in September 2003. were Peter Cameron (University of London), Universita degli Studi di Roma Tor Vergata. Gábor Tardos (Alfréd Rényi Institute of This talk introduced the minisymposium The genius loci Mathematics, Budapest), Oriol Serra “Mathematical Statistical Physics”, coordi- The 2004 Joint Mathematical Weekend was (Universitat Politècnica de Catalunya, nated by Roman Kotecký from Charles organized by the Czech Mathematical Barcelona) and Patrice Ossona de Mendez University in Prague. The pattern of this min- Society in Prague, with further financial and (CNRS, Paris). Five contributed talks were isymposium differed slightly from the other logistic help from the Faculty of Mathematics also included. three. It had only three invited speakers and Physics of the Charles University in The minisymposium “Mathematical Marek Biskup (University of California, Los Prague and from the Czech national research Fluid Mechanics” was coordinated by Angeles), Dmitry Ioffe (Technion centre Institute for Theoretical Computer Eduard Feireisl from the Mathematical University, Haifa), and Kostya Khanin (Isaac Science (ITI), which is a platform for the Institute of the Academy of Science of the Newton Institute, Cambridge), but the modus cooperation of research teams from several Czech Republic in Prague. Since his plenary operandi hinged on informal discussions fol- institutions. The meeting took place in a fac- speaker, H. Beirao da Veiga (Università di lowing each of the invited talks. ulty building at the Lesser Town Square in Pisa), could not attend the meeting due to an The last plenary talk “Some current trends the close vicinity of Prague Castle, the injury, E. Feireisl himself presented this min- in proof complexity” was given on Sunday Charles Bridge, the Astronomical Clock and isymposium’s plenary talk “On the mathe- morning by Alexander Razborov from other celebrated historical monuments. The matical theory of viscous compressible flu- Steklov Mathematical Institute in Moscow participants enjoyed the beautiful atmosphere ids” on Saturday morning. He introduced the and the Institute for Advanced Studies in of the historical building, a former Jesuit audience to methods used for modelling Princeton. This talk belonged to the min- dynamical aspects of liquid and gas media. isymposium “Complexity of Computations Invited speakers in this minisymposium were and Proofs”, organized by Jan Krajíèek from Josef Málek (Charles University, Prague), the Mathematical Institute of the Academy of Victor Starovoitov (CAESAR, Bonn), Science of the Czech Republic in Prague. Marius Tucsnak (Université Nancy), and The audience was entertained not only by his Antonín Novotný (Université du Sud very colourful presentation of methods and Toulon-Var). Moreover, there were three reasoning in contemporary theoretical com- more contributed talks. puter science, but also by a malicious coinci- The mathematical weekend (almost) got its Mathematical models of the real world own stamps! dence that some software-hardware commu- EMS December 2004 5 nication problems delayed the beginning of Mathematicians and Physicists, and they dean of the Faculty of Mathematics and this very computer science oriented talk. The have always enjoyed this coexistence and Physics of the Charles University Ivan invited talks of this minisymposium were cooperation of closely related sciences. Netuka and the director of the Mathematical given by Peter Buergisser (Universität Within the Union, they are grouped in pro- Institute AS CR Antonín Sochor. Paderborn), Oleg Verbitsky (Lviv fessional units, one of them having been Finally, there was a meeting of the Czech University), Peter Bro Miltersen (University called the Mathematics Research Section National Committee for Mathematics - the of Aarhus), Albert Atserias (Universitat since its foundation in 1970’s. This name body consisting of leading Czech mathemati- Politecnica de Catalunya, Barcelona), Stefan reflected the era of mainly Eastbound orient- cians, established under the auspices of the Dantchev (University of Leicester), and ed international links, governed by the politi- Academy of Sciences of the Czech Republic Pavel Pudlák (Mathematical Institute AS CR, cal situation in Europe in the second half of to deal with basic conceptual and organiza- Prague) and further six contributed talks were the 20th century, and always required tedious tional questions of mathematical sciences in included. explanations to foreign colleagues. Therefore the country and to represent the Czech math- After the last plenary session, Sir John the name “Czech Mathematical Society” has ematical community in the IMU. One of the Kingman, the president of the EMS, official- been frequently, informally and somewhat main topics was to discuss the somewhat ly addressed all participants. He appreciated illegally used in foreign correspondence in dangerous features of a new evaluation sys- the importance and significance of this new the last years. Now, after rigorous voting dur- tem for the R&D being prepared by the gov- series of meetings, emphasized the role of the ing the general assembly meeting, this name ernment. EMS, invited the participants to join the soci- has become official and legal (with its Czech We can proudly admit that the organizing ety, and thanked the Czech Mathematical equivalent Èeská matematická spoleènost). team, Jan Kratochvíl, Jiøí Rákosník, Jiøí Fiala, Society for hosting the Weekend. This does not, however, change the fact that Daniel Hlubinka and Pavel Exner (the Czech we are still organized within the frames of the Mathematical Society), Anna Kotìšovcová Conference banquet and photo Union together with our sister Czech (Conforg Agency), Hana Polišenská (secre- On Saturday evening all participants were Physical Society. The first formal act accom- tary of the ITI) and Jaroslav Nešetøil, Eduard invited to a conference banquet in the base- plished under the new name of the Czech Feireisl, Roman Kotecký and Jan Krajíèek ment of the building, to experience rooms Mathematical Society was signing the agree- (the minisymposia coordinators), did a good that during the academic year serve as a cafe- ment of collaboration and reciprocal mem- job and prepared pleasant and friendly work- teria at lunch time and as the faculty club in bership with the Catalan Mathematical ing conditions for the participants. As Luc the afternoons. The pleasant atmosphere was Society during the EMS weekend banquet, Lemaire mentioned in his welcoming speech, enhanced by toasts of Jaroslav Nešetøil, Bodil an act which fitted well the spirit of the EMS one meeting does not make a series, two Branner and David Salinger, and later also by activities and of the Joint Weekend in partic- meetings in a row do. We in the Czech live music performed by some participants ular. (The next agreement on cooperation Mathematical Society feel honoured by the and organizers of the meeting. was signed with the Royal Spanish commission to organize the second joint Earlier that day the official conference Mathematical Society shortly after the weekend and proud of being present at the photo was taken during the lunch break from Weekend.) birth of a new tradition of what we all believe the windows of the conference building. The The other event associated with the will prove to become a useful and important participants were lined up in front of a histor- Weekend, the meeting of the EMS Executive series of meetings of European mathemati- ical plague column, one of the baroque sculp- Committee, was perhaps less ceremonial but cians. ture jewels in Prague. more busy. Most members of the Committee Jan Kratochvil [[email protected]. cz], born in 1959, did his Ph. D. in discrete mathematics at Charles University in Prague. Since 1994, he has been at the Department of Applied Mathematics of this university, and since 2003 he is the head of the department and a full professor. He has also lectured several times as a visiting professor at Computer and Information Science Department of the University of Oregon in Eugene, OR. His main research interests are graph theory and computational complexity. He is currently president of the Czech Mathematical Society.

Jiøí Rákosník [[email protected]], born in 1950, graduated in mathematical analysis from Charles University in Prague. His research interest concentrates on theory of function spaces and theory of integral and Additional meetings enjoyed the mathematical programme of the differential operators. He is associated to the Three important work meetings took place entire weekend, while their working meeting Mathematical Institute of the Academy of behind the scene at the weekend. The Czech was scheduled on Sunday afternoon and Sciences of the Czech Republic. Since 2001, Mathematical Society made use of this Monday morning in hotel Vaníèek. In recog- he is also a member of the Academy Council opportunity and called its general assembly nition of the importance of the EMS, a festive of the AS CR, responsible for economic and meeting for Friday evening. One of the main dinner was given for the Executive financial matters of the Academy and for agenda items was the change of name for the Committee members by the president of the cooperation of the Academy in preparing society. For almost one and a half centuries Academy of Sciences of the Czech Republic legislation concerning research and develop- mathematicians and physicists of this region Helena Illnerová, the president of the Czech ment. He heads the Czech Editorial Unit of have been organized in the Union of Czech Mathematical Society Jan Kratochvíl, the Zentralblatt MATH.

6 EMS December 2004 EMS NEWS devolved to a subcommittee, but should rest with the Executive Committee itself. A new committee will, however, be set up EMSEMS executiveexecutive committeecommittee to look at applications of mathematics.

Meetings meeting at Prague The next ‘mathematical weekend’ will meeting at Prague be in Barcelona in September 2005, jointly organised by the Catalan Mathematical David Salinger, EMS Publicity Officer Society and the EMS. In 2006 there will be a conference organised by the EMS and the French and Italian mathematical soci- Meetings of the Executive would look Recognising the importance of contacts eties in Turin. We shall submit an appli- dull to a traditional newspaper. No with the EU and the burden it placed on cation for further Summer Schools to the punch-ups, a calm business-like atmos- those members who were involved, the EU and have already advertised a call for phere predominates, but underlying it all Executive Committee resolved to broaden proposals from members. is a sense that the committee is working the Group on Relations with Europe to the 4ecm had been a high quality meeting, for the good of mathematics and mathe- whole Executive Committee and to wel- but had attracted rather too few partici- maticians. It was a particular pleasure to come the help of others who were pre- pants. There was general satisfaction with welcome future members, Olga Gil- pared to lobby for mathematics in the selection of EMS prize-winners, but Medrano and Carlo Sbordone, to the Brussels. some regret that it had proved impossible Prague meeting. to award a Felix Klein Prize. Membership and Representation Mathematics and the EU Individual membership remains stubborn- Subcommittees Certain issues come round each time. It is ly on a plateau of about 2,300, but institu- Mina Teicher would become chair of the not just a routine complaint to observe tional and corporate membership has Education Committee. She submitted a that mathematics is under-represented on grown. Nearly all the mathematical insti- list of suggested members and an outline influential EU committees: members of tutes which are ERCOM members have plan of action. The committee on special the executive committee also do some- now joined. Statistical Societies have events would be abandoned and its work thing about it, from lobbying ministers to begun to join the Society, too and our taken over by the General Meetings sitting on extra committees, and generally, Summer School programme includes Committee, which would lose the word grabbing opportunities to do things when- events involving the Bernoulli Society. ‘general’ from its name. Andrzej Pelczar ever they arise. At Prague, Luc Lemaire The EMS, naturally, believes that it’s to was stepping down from the Committee on Eastern European mathematicians after many years of service to the Society. Jan Kratochvíl of the Czech Mathematical Society would take over the chair. The work of the Committee would take account of changing circumstances in Eastern Europe. Pavel Exner would become chair of the Committee on Electronic Publications: it was felt that an Executive member should take on that role because of the growing importance of the committee’s remit and, in particular, because it oversees the Electronic Library on EMIS. The Publications Committee was dis- banded: the Executive itself would take on the task of dealing with issues concerning the Publishing House. It agreed to trans- fer the publication of the Newsletter from the Society to the Publishing House, as part of the task of persuading other mathematicians to publish with EMSph.

Closing Matters Our stay in Prague had been most enjoy- able, thanks in no small part to Pavel was able to report on the panel session the advantage of mathematics to have one Exner and the Czech Mathematical organised by the EMS and on his plenary society representing it at the European Society, who had organised an exemplary lecture to the EuroScience Open Forum in level and is working hard to include all Mathematical Weekend beforehand. The Stockholm. Some good contacts were strands of mathematics, including particu- President also thanked the retiring mem- made, but attendance at sessions was larly applied mathematics and statistics, in bers of the Committee: Bodil Branner, poor. The Executive Committee resolved its work (and in the committee member- Marta Sanz-Solé and Mina Teicher. The to encourage local mathematicians to get ship). It has decided that, as applied next meeting would be in Capri (Italy) in involved, when the Forum, now intended mathematics is central to the Society’s April at the invitation of the Unione to be a two-yearly event, came their way. work, responsibility should no longer be Matematica Italiana.

EMS December 2004 7 EMS NEWS The new library of the Faculty of Mathematics and Physics, Charles University in Prague Ivan Netuka and Vladimir Souèek (Prague) The August floods in 2002 dramatically affected the mathematical and information science's part of the library that is situated in the Faculty's building in Karlin. The water reached a height of 2.90 meters above pavement level. 13000 books, 468 titles of journals, 6800 textbooks and 2000 diploma theses were damaged (see the information in EMS Newsletter, Issues 45 and 49). In the period after the floods we received great support from foreign and home organizations, institutions, societies, universities and individual donors. Financial donations reached almost 104,000 euros. The principal financial support was provided by The Ministry of Education, Youth and Sports, and Charles University. In addition, the Faculty received 5907 books and 156 journal titles from 165 donors. The donors The mathematical library after the floods ... and after reconstruction that were identified were being sent letters of thanks. Two press conferences were the new library was finished. After the pre- The Faculty of Mathematics and Physics held on the occasion of the presentation of vious unfortunate experience, it was decid- would like to thank all the people and insti- a large collection of books and journals ed to situate it on the first floor. The library tutions that contributed to the renewal of from French and German mathematicians. has just been opened for the beginning of the library with their understanding and In September 2004, the construction of the academic year 2004/2005. help. EUROPEANEUROPEAN MAMATHEMATHEMATICALTICAL SOCIETYSOCIETY ARARTICLETICLE COMPETITIONSCOMPETITIONS The Committee for Raising Public lished, or be about to be published, in an tion: January 1, 2006. Awareness of Mathematics of the European international magazine or a specialized Mathematical Society (acronym RPA) national magazine, bringing articles on Prizes and conditions for submissions believes that it is of vital importance for the mathematics to an educated public. Articles For each of the two competitions there will recognition of mathematics in society that for the competition shall be submitted both be prizes for the three best articles, to the articles popularising mathematics are writ- in the original language (the published ver- sum of 200, 150 and 100 Euros respective- ten. The experience gained from the first sion) and preferably also in an English ly, and many of the winning articles will be article competition organized by the RPA- translation. Articles (translations) may also published in the Newsletter of the EMS. committee of the EMS, completed in 2003, be submitted in French, German, Italian or Other articles from the competition may show that it is beneficial to distinguish Spanish. The English (or alternative lan- also be published if space permits. between articles addressing the educated guage) version should be submitted both By submitting an article for one of the layman and articles addressing a general electronically and in paper form. competitions it is assumed that the author audience; see http://www.mat.dtu. dk/peo- Deadline for submission to this competi- gives permission for the translation of the ple/V.L.Hansen/rpa/resultartcomp. html. tion: August 1, 2005. article into other languages and for its pos- Therefore the EMS now wishes to encour- sible inclusion on a web site. Translations age the submission of articles on mathe- Articles for the general public into other languages will be checked by matics to two competitions, one for articles To be considered an article must be pub- persons appointed by relevant local mathe- for the educated layman and one for articles lished, or be about to be published, in a matical societies and will be included on for a general audience. It is hoped that valu- daily newspaper or some other widely read the web site. able contributions will be collected, which general magazine, thereby providing some Articles should clearly indicate to which deserve translation into many languages. evidence that the article does catch the one of the two competitions the article is The EMS is convinced that such articles interest of a general audience. Articles for being submitted and should be sent before will contribute to raising public awareness the competition shall be submitted both in the given deadlines to the chairman of the of mathematics. the original language (the published ver- RPA-committee of the EMS: The RPA-committee of the EMS invites sion) and preferably also in an English professional mathematicians, or others, to translation. Articles (translations) may also Professor Vagn Lundsgaard Hansen, submit manuscripts for suitable articles on be submitted in French, German, Italian or Department of Mathematics, mathematics for one of two competitions. Spanish. The English (or alternative lan- Technical University of Denmark, guage) version should be submitted both Matematiktorvet, Building 303, Articles for the educated layman electronically and in paper form. DK-2800 Kongens Lyngby, Denmark. To be considered an article must be pub- Deadline for submission to this competi- e-mail: [email protected]

8 EMS December 2004 EMS NEWS Dear Colleagues,

AppliedApplied MathematicsMathematics andand I apologize that for personal reasons I cannot attend this Conference. ApplicationsApplications ofof MathematicsMathematics However, I would like to say a few words for the Opening Ceremony. (AMAM, Nice, February 10-15, 2003) It was a great pleasure for me to work jointly with my co-chair, More than 500 mathematicians from 25 Professor P.-L. Lions, Scientific countries took part into the conference. Secretary, Professor A. Damlamian Almost 20% of the total budget was and other members of the Committee, dedicated to 30 grants which supported in selecting the main speakers. I the participation of young researchers, remember it as a parade, demonstrating students or mathematicians from East a series of great achievements in many European countries, Asia and Africa. different areas of applied mathematics, The conference showed the unity of displaying many beautiful mathemati- mathematics and their role in various cal applications. It covers a broad fields of science and technology. It also spectrum of subjects including the emphasized the interest of the EMS in practical use of mathematics in busi- promoting Applied Mathematics in the ness and finance, practical cryptogra- community of mathematicians in gener- phy, and the impressive development In order to increase the place of Applied al. It should be mentioned that more of computational and theoretical meth- Mathematics in the EMS, its Council and more women are now choosing sci- ods in the natural sciences. Many proposed to organize a conference on entific careers. Among the participants, times I felt sorry that we were restrict- this subject, jointly with the French 27 % were women. The conference also ed in the number of invited talks. I mathematical societies SMF and SMAI. exposed to young European students, believe we made extremely good This conference Applied Mathematics various topics in mathematics. choices and left many excellent candi- and Applications of Mathematics took In conclusion, AMAM 2003 rein- dates for future meetings. place in February 2003 in Nice at the forced the links between Pure and The mathematicians who founded the Palais des Congrès. The co-presidents Applied Mathematics and presented new European Mathematical Society over of AMAM 2003 were Rolf Jeltsch challenges and issues for mathemati- 10 years ago always believed in the (EMS), Michel Théra (SMAI) and cians, arising from varied areas of sci- unity of mathematics, artificially Michel Waldschmidt (SMF). The ence and technology. It also showed the divided into pure and applied parts. It Scientific Committee was co-chaired by unity of mathematics and their role is the duty of mathematics to support Pierre Louis Lions (France) and Sergey within the modern world. (taken from its applied component. I always treat- Novikov (Russia), both Fields medal- the report to the EMS Council on ed the opposing point of view as some- ists. The Organizing Committee was http://www.math.ntnu.no/ems/council thing non-serious, a sort of philosophy co-chaired by Doina Cioranescu (SMAI) 04/uppsalapapers/Report_AMAM_type made from scientific weakness no mat- and Mireille Martin-Deschamps (SMF). set.pdf ). ter how broad it became distributed. Our unity is especially important The Scientific Committee established now. Mathematical education has the following list of topics: Inaugural Address reached a state of terrible crisis in all 1) Applications of number theory, civilized countries. What is going to including cryptography and coding. Sergey Novikov (University of happen to the most fundamental exact 2) Control theory, optimization, opera- Maryland, College Park, USA, and theoretical sciences of the past century tions research and system theory. Landau Institute for Theoretical like mathematics and theoretical 3) Applications of mathematics in biol- Physics, Moscow) physics? According to my observa- ogy, including genomics, medical tions, mathematics has a better chance imaging, models in immunology, at survival than theoretical physics, but modeling and simulation of biologi- our unity is necessary for that. cal systems. It was already obvious to everybody 4) Scientific computation, including ab in the 1990s that biology had become initio computation and molecular the main candidate for the position of dynamics. “Miss Science - XXIst Century”. 5) Meteorology and climate, including Unfortunately, we have also been wit- global change. nesses to the decay of theoretical 6) Financial engineering. physics during the last decade. What 7) Signal and image processing. does this mean for those of us who 8) Nonlinear dynamics. have dedicated their scientific activity 9) Probability and statistics, inverse to the interaction of mathematics with problems, fluid dynamics, material the natural sciences? Of course, we are sciences and other applications. happy to support all realistic and useful mathematically based investiga- The conference consisted of 9 plenary tions made for the needs of biology. speakers, 38 mini-symposia, two round Some of them are represented in this tables (Mathematics in developing conference. No doubt we should help countries, and Education) and two to increase their number. poster sessions (with 92 presentations). However, the connection of mathe-

EMS December 2004 9 EMS NEWS matics and physics was so deep that we should say a few words about today’s crisis. In a sense, theoretical physics COMMITTECOMMITTE FORFOR DEVELOPINGDEVELOPING has always been considered as the mathematics of the real world, a main source of mathematical ideas since the COUNTRIES:COUNTRIES: XVII century, the main driving force for the development of 90% of mathe- AA SUMMARSUMMARYY OFOF WORKWORK ANDAND matics, and the main road joining mathematics with other natural sciences. In the XXth Century, theoreti- PLANPLAN cal physics reached its highest level. It became a leading exact theoretical sci- (Chair: Herbert Fleishner; vice-chair: Tsou Sheung Tsun) ence. It was era of dinosaurs for physics. Its great leaders were capable of using and sometimes of creating The following is a slightly edited version of a have a wider geographical spread. paper presented by Tsou Sheung Tsun to the Lecture notes distribution: we have just start- very deep abstract mathematics when it IMU Developing Countries Strategy Group ed this scheme with the help of ICTP. The was needed for the study of the real meeting, 16-17 October 2004, at ICTP, idea is to make a collection of undergradu- world. New fundamental laws of Trieste, Italy. ate lecture notes and make them available nature were discovered and they to the developing world, in the form of a invented great new technology. It What are we doing? CD for example. I have already collected changed our world forever. We have active individual members working those available from Oxford. We hope to At the same time, there was a split- with their own programmes/organizations: extend this scheme to other languages, e.g. ting of the communities of physicists CIMPA, ISP, Vietnam, etc. I hesitate to French and Spanish. We shall be looking and mathematicians. As a corollary, add China, as her status of being a devel- for good universities in these language several important mathematical oping country is unsure. We exchange groups in the developed world. experience and liaise closely. Organize short-term placements (say 3 achievements made by physicists (such We have a well-supported book/journal dona- months) for visitors from the developed as quantum field theory, for example) tion scheme, the shipping costs of which world to give a lecture course in a develop- remain, until now, outside the mathe- are almost entirely funded by ICTP. There ing country. This may interest mathemati- matics community. Mathematical edu- is a vast source of books and journals, from cians who have recently retired, or fresh cation knew nothing about them even retirees and libraries going electronic in the PhDs. in the best times. Its language became developed world. Mobilize support for mathematicians in a incompatible with the mathematical We have a good network of distribution cen- developing country to attend conferences: I language of physics of the early stages. tres, particularly in Africa. These are suppose this is done by the IMU-CDE. Pure and applied mathematics have departments or institutes, mostly where Consolidate and expand existing MSc and both lost contact with high-level mod- there is some personal contact with mem- PhD programmes in countries regularly bers of our committee so that we can make visited by our members. ern physics in the past century. sure that the books and journals are used Closer liaison with IMU, ICTP, CIMPA, Mathematical language and the tech- properly, e.g. made available to all mathe- LMS, AMS, IMPA, AIMS, etc. nique of theoretical physics were espe- maticians. If material is donated that a par- cially designed as the best mathemati- ticular centre cannot use, or already have, What do we need? cal tools for the solution of real world they promise to send that to nearby institu- Continued support from ICTP for the book problems. In trying to replace them tions. donation scheme. with something absolutely formal and We have shipped about one and a half tonnes Help to start and develop the lecture notes rigorous, you make them useless. of such material, with a couple of 100kg distribution in English, French and Therefore, pure mathematics alone will being processed at the moment. Spanish. not be capable of adjusting itself to The recipients where we have contacts are Funds for sending lecturers to developing well monitored: sub-Saharan Africa, countries for short periods. Subsistence “physical mathematics”. Zimbabwe, and Vietnam. can usually be provided locally. We see now that the era of dinosaurs Referees’ honorarium donation: this is a Conference support for participants from is probably over and theoretical scheme by which referees donate their hon- developing countries. physics is going down. Maybe it hap- oraria from publishers, usually augmented pened as a consequence of overdevel- if exchanged for books. Princeton What can we offer? opment? Is it possible that some unex- University Press has donated some. Good field knowledge and contacts in Africa, pected great achievement will return Zentralblatt has donated twice, amounting South East Asia, and some parts of Latin its momentum? to some 1500 euros. We have talked with America, through personal experience and If not, let me ask the following ques- John Ewing of the American Mathematical the contacts of our members. tion: Who is going to preserve this Society and we are currently devising some Good network of book distribution, with variant of the above scheme with him. monitoring. great knowledge? Certainly it will A good point about this programme is that the Enthusiasm and some new ideas. remain important for many engineering recipient, typically a department with a applications and it would be dangerous small active group of researchers, can Finally, an appeal for humanity to forget it. choose the books they need, even though it We are in need of funds for our various activ- To my opinion, only the joint forces is a small number of books. ities and urge our colleagues in the developed of pure and applied mathematics may This could involve the whole mathematical world to help with donations, however small. help here. I do not know any other part community, and would raise awareness Donations can be made directly to the EMS- of science capable of preserving this about situations in the developing world. CDC account: great mathematical knowledge. I have no doubts that this conference What do we hope (in the short- and medi- Account number: 157230-381160 um-term)? Nordea Bank Finland Plc is going to be very successful. I wish Expand the book donation scheme: to include Swift: NDEAFIHH you great working days in Nice. more distribution centres, particularly to IBAN: FI78157230000381160 10 EMS December 2004 FEATURE

SySy mm mm ee tt Symmetriesrr ii ee ss Alain Connes (Paris) A l a i n C on n e s (Pa ri s)

This article appeared originially (in French) in the magazine Pour la Science, no. 292, Feb. 2002. It was submitted to the article competition of the EMS Raising Public Awareness Comittee by the journal’s director Philippe Boulanger and was given a runner-up award, cf. issue 50 of the Newsletter. The Newsletter thanks Alain Connes and Philippe Boulanger for the permission to republish the article. The concept of symmetry goes well beyond that of simple geometric symmetries. From the fair organisation of the ﬁnal stages of soccer competitions to the solving of equations, via the icosahedral game and Morley’s theorem, we’ll discover the multiple aspects of this concept.

The purpose of this article is to in- As we move to equations of higher secutive ‘trisectors’ (the two straight troduce the reader to the mathematical degree, we describe the icosahedral lines which divide an angle into three notion of symmetry, by way of a few game and the ‘icosions’, defined in the equal parts). In 1988, I gave an alge- illustrative examples. nineteenth century by the Irish mathe- braic formulation of this result and a To demonstrate the ubiquity of this matician, William Hamilton. proof which we shall see below. concept, as a mathematician under- We finish with a commentary on a stands it, we’ll start by evoking the theorem in geometry, proved by Frank connection between the final stages of Morley around 1899, where the sym- The final stages of a soccer cup soccer competitions and the way we metry of an equilateral triangle arises solve quartic equations. miraculously from an arbitrary trian- Let’s start with the organisation of a gle, by taking the intersection of con- soccer competition, for example, the

1. THE SYMMETRIES OF THE FINAL STAGES OF A SOCCER COMPETITION

F D FFC H The three days, 1,2,3 remain globally unchanged by the permutations of F, D, C, H. Thus: CCH DDH F D C H 1 2 3 FIRST DAY SECOND DAY THIRD DAY

FIRST DAY F D C H 1 2 3 interchanging F and D interchanges days 2 and 3,

F D C H 1 2 3

F C SECOND F D C H 1 2 3 DAY interchanging D and C interchanges days 1 and 2, THIRD DAY D F D C H 1 2 3

H F D C H 1 2 3 interchanging F with D and C with H leaves each day the same.

EMS December 2004 11 FEATURE millennial European Cup. During the point of intersection of the lines repre- Resolution of quartic equations final stage, teams were placed in pools senting the matches on that day. Thus by radicals of four, and arrived at an order of merit the matches FD and CH are associated within each pool. For example, one with the intersection of the lines FD This surprising symmetry underlies group consisted of Denmark, France, and CH. Continuing for the two other the general method for solving quar- Holland, and the Czech Republic (here pairs, the second day is at the intersec- tic equations ‘by radicals’. The solu- abbreviated to D, F, H,andC). tion of the lines FC and DH and the tion amounts to expressing the zeros 4 To arrive at an order of merit fairly, third is at the intersection of the lines a, b, c,andd of the polynomial x + 3 3 each team had to play each of the three FH and DC. nx + px + qx + r =(x − a)(x − b)(x − others, which meant that games had The figure thus constructed, formed c)(x − d) as a function of the coeffi- to played on three days. For instance, by four points and six lines, is called a cients n, p, q,andr using the extraction when D and F met, then H and C could complete quadrilateral. It is perfectly of roots. meet on the same day and so three days symmetrical (in an abstract sense, even To understand this assertion, we were enough for all possible configura- if none of the usual geometric symme- must go back in time and look at the tions of matches. tries – symmetries with respect to a solution of equations of degree lower In that European Cup, the matches point or a line – are present), because than four. were, FD and CH on the first day, FC each of the four points F, D, C,andH Though the technique of solving and DH on the second, and FH and playsexactlythesameroleastheoth- quadratic equations goes back to fur- DC on the third. Intuitively we can ers, and the same is true for the points thest antiquity (Babylonians, Egyp- see that this is a fair procedure, be- of intersection representing the three tians...), it wasn’t extended to cubic cause none of the teams has an advan- days. equations until much later and was tage. One checks indeed that, if we ar- Having visualised this complete published only in 1545 by Girolamo bitrarily permute certain teams, for in- quadrilateral, we can also give an alge- CardanoinChapters11to23ofhis stance, if we interchange D and H,this braic formulation of the symmetry un- book Ars magna sive de regulis alge- amounts to a simple permutation of the der discussion. In the following way: braicis. In fact, it wasn’t realised until first and third days. the quantity α, determined from the the 18th century that the key to the so- We can visualise the symmetry four numbers a, bcand d by the for- lution by radicals of the cubic equation 3 2 which is at work by putting the letters mula α = ab + cd only takes three val- x + nx + px + q = 0, with zeros a, b, D, F, C, H (representing the teams) at ues altogether when we permute a, b, c and c (the ‘roots’), lay in the existence four points of the plane. The line join- and d. The other values are β = ac + bd of a polynomial function f (a, b, c) of a, ing two points represents a match be- and γ = ad + bc. b,andc, which takes only two different tween the corresponding teams. Each values under the action of the six pos- of the three days corresponds to the sible permutations of a, b,andc.

2. SYMMETRY AND THE SOLUTION OF CUBIC AND QUARTIC EQUATIONS BY RADICALS

Solving polynomial equations by 1) The cubic equation: radicals requires the construction of x3 + 3px + 2 q = (x – a)(x – b)(x – c) = 0. auxiliary functions of the roots, which Reduced equation: x2+ 2qx – p3 = (x – α)(x – β). display symmetries when the Three Two auxiliary functions: Symmetry: Solutions: roots are permuted. 3 roots: α = [(a + bj + cj2) /3]3 The symmetry given Let u = √α β = [( 2 3 √β3 ROOTS UX. FCT a, b, c. a + bj + cj) /3] by any permutation of and v = A . where j = (–1+i √3)/2, a, b, and c leaves the such that uv = – p. j2 = (–1– i √3)/2, set of auxiliary Then a b c d α β γ with i being a square root functions {α,β} globally a = u + v of –1, so that invariant. b = j2u + j v j3 = 1 and j2+ j +1=0. c = ju + j2v.

a b c d α β γ 2) The quartic equation: x4 + px2 + qx + r = (x – a)(x – b)(x – c) (x – d) = 0 Reduced equation: x3 – px2 – 4rx + (4pr – q2) = (x – α)(x – β)(x – γ) = 0 The set of these auxiliary functions is globally unchanged by Four Three auxiliary Symmetry: Solutions: permutations of the roots of the roots: functions: The symmetry given by 1) Knowing α=ab+cd and r=(ab)(cd), equations. For cubic and quartic a, b, c, d α = ab + cd any permutation of a, b, gives the products ab and cd. equations, the auxiliary functions enable β = ac + bd c, and d leaves the set 2) If ab=cd, the system (a+b)+(c+d)= 0 us to reduce the solution to that of a γ = ad + bc of auxiliary functions {α, and cd(a+b) + ab(c+d) = – q, "reduced" equation of lower degree. β, γ} globally invariant. gives a+b and c+d. 3) Knowing ab and a+b gives a and b. Similarly for c and d.

12 EMS December 2004 FEATURE Cardano’s method amounts to writ- ab + cd. Here again, we can start with c + d and hence, finally, a, b, c,andd). 2 3 3 ing α =[(1/3)(a + bj + cj )] ,where a polynomial with no term in x (using The fundamental role of the permu- the number j is the first√ cube root of the same technique as before), namely tations of the roots a, b, c ... and of 4 2 unity, namely (−1 + i 3)/2, with i de- x + px + qx + r =(x − a)(x − b)(x − the auxiliary quantities α, β ..., was noting the square root of −1. The cyclic c)(x − d). The set of three numbers α = brought to light by Joseph Louis La- permutation taking a to b, b to c,and ab + cd, β = ac + bd,andγ = ad + bc, grange in 1770 and 1771 (published c to a simultaneously, clearly leaves α is invariant under each of the 24 per- in 1772) and, to a lesser degree, by invariant and the only other value ob- mutations acting on a, b, c,andd.The Alexandre Vandermonde in a memoir tained from the six possible permuta- numbers α, β and γ are thus the roots published in 1774, but written around tions of a, b and c is β =[1/3(a + cj + of a cubic equation whose coefficients 1770, as well as by Edward Waring 2 3 bj )] , where, for example, b and c have are easily expressed as a function of in his Meditationes algebraicae of 1770 been transposed. As the set consisting p, q,andr. A calculation shows that and by Francesco Malfatti. Today we of the two numbers α and β is invari- the polynomial (x − α)(x − β)(x − γ) rightly call those auxiliary quantities 3 2 2 ant under all the permutations of a, b, equals x − px − 4rx +(4pr − q ).It ‘Lagrange resolvents’. and c, it is easy to express the quadratic can thus be decomposed, as we have The resolvents are not unique (we α β α β γ equation whose roots are and ,in seen above, to yield , and .In- could equally well have put terms of the coefficients of the initial deed, it’s enough to find one of these α =(a + b − c − d)2 inthecaseofthe α equation roots, say, to determine a, b, c,and quartic equation, which corresponds to 3 + 2 + + 2 + − 3 = α x nx px q:itisx 2qx p d (because we then know the sum Descartes’ method), but they are the ( + + )( + − ) x q s x q s ,wheres is one of and the product r of the two numbers key to all general solutions by radicals. the square roots of p3 + q2 and where ab and cd, thus giving these numbers we have also rewritten the initial equa- via a quadratic equation; all we then tion in the equivalent form x3 + 3px + have to do is to exploit the equations Abel and Galois 2q (without the square term) by an ap- (a + b)+(c + d)=0andab(c + d)+ propriate translation of the roots. We cd(a + b)=−q to determine a + b and Of course, mathematicians wanted to introduced the factors 2 and 3 to sim- go further: Descartes certainly tried, plify the formulas. and he wasn’t alone. The next step 3. THE ORDER OF A PERMUTATION A simple calculation then shows that would clearly be that of the quintic equation. This was found to pose each of the roots a, b,andc of the ini- The order of a permutation is the tial equation can be expressed as the least integer n such that if the apparently insuperable obstacles, and permutation is applied n times, we sum of one of the three cube roots of since the time of Abel and Galois (who get the identity permutation. Thus obtained their results around 1830) we α and one of the three cube roots of β. the permutation u has order 2 and have known why the search was in These two choices are connected by the the permutation v has order 3 : − vain. fact that their product must equal p a b c d e a b c d e (so there are only three pairs of choices In all the previous cases, we were u = v = − to account for, which is reassuring, in- able to find a family of n 1numbers α β γ stead of the nine possibilities which we , , . . ., determined as polynomials v = might have thought of a priori). u = of the n roots a, b, c,andd ... (with n less than or equal to 4). This family These formulas logically required a b c d e v = the use of complex numbers. Indeed, was globally invariant under the per- Checking that mutations of the roots. More precisely, even in the case where the three roots a b c d e 3 + 2 the permutation if we let Sn denote the group of bijec- are real numbers, p q can be nega- { } tive, and then α and β are necessarily uv has order 5. tions of the set a, b, c, d ... with itself, whatcanbedoneforn strictly less than complex. a b c d e 5istodefineamappingofSn onto Sn−1 The solution of cubic equations, de- a b c d e which preserves compositions of per- scribed above, took a long time to uv = = mutations. reach its final form (we know that at Since the beginning of the nineteenth least one of the special cases was be- century, we have known that this is im- ing worked out by Scipione del Ferro between 1500 and 1515). On the other a b c d e possible for n larger than 4. The same is true for a (non-constant) composition- hand, the solution of quartic equations followed quickly, because it is preserving mapping of the group An of also in the Ars magna (Chapter 39), permutations of even order (the products of an even number of transposi- where Cardano attributes it to his secretary Ludovico Ferrari, who appar- tions) onto the group Sm when m is less (uv)5 = than n and n is greater than 4. This ently found it between 1540 and 1545 shows that Lagrange’s method can’t (René Descartes would publish an- = other solution in 1637). And it is this be extended to the case n 5orto higher values of n,butis,ofcourse,not solution which brings us back to the first symmetry we met, that of the or- enough to show that a solution by rad- ganisation of soccer finals, the com- icals is impossible for the general equa- a b c d e tion of degree 5 or higher: other, more plete quadrilateral, and the expression general, methods might succeed where EMS December 2004 13 FEATURE

4. THE GROUPS A4 AND A5 A) THE GROUP A 4

1) The group A4 is the group of the even permutations of

the four letters (a,b,c,d). This group is generated by the 5 4 1 permutations s = a b c d , which maps a to b, b to a, c 6. 11 [ b a d c [

a b c d 2 u

to d and d to c, and t = , which maps a to a, b 6.

[a c d b [

1 3 to c, c to d and d to b. They obey the rules: s2 =1, t3 =1, and (st)3 =1. 10 2) There's a geometric representation of the group A4 : it's the group of rotations preserving the regular tetrahedron a,b,c,d .

a 1

s 4 5 4 6. 6.

2 v

1 5 110 b d 11 3 a c t C) PRESENTATION OF A 5 The patient reader may check that the simplifying rules u2 =1, v3 =1, (uv)5 =1 together with the group law, are enough to show that there are only 60 distinct words formed out of the letters u and v. We start by putting s = u and t = k–2uk b d (where k = uv), and then show that s and t are generators 2 3 of A4, that is, they obey the simplifying rules s =1, t =1, c (st)3 =1. We then show, using the simplifying rules above, s is represented by the symmetry with respect to the line that every word formed from the letters u and v can be joining the midpoints of ab and cd. written in the form km h, where m equals 0,1,2,3,4 and h is t is represented by a rotation through an angle 2π/3 about a word formed from the letters s and t. the axis of the tetrahedron passing through a. As there are precisely 12 distinct words h, we see that the st = a b c d is the rotation through 2π/3 about the axis group A5 is given by the above relations. [ b d c a [ through c. D) A : A GROUP OF MATRICES The patient reader may check that the rules of 5 2 3 3 simplification s =1, t =1, (st) =1 form a presentation of 1) Let F5 denote Z/5Z, the field of remainders modulo 5. In the group, that is, together with the group law, they are this field, 4 + 2 =1, 3 + 2 =0, 4 x 2 = 3, 3 x 2 =1, etc. enough to show that there are only twelve distinct "words" formed out of the letters s and t. 2) We represent u and v as the following mappings of the projective space P1(F5). This projective space contains the B) THE GROUP A five points of F , together with a point "at infinity", denoted 5 5 by 1/0. Let us put u(z) = –1/z, for a point z in P (F ). 1) The group A5 is the group of the even permutations of 1 5 the five letters (a,b,c,d,e). This group is generated by the Clearly, u2(z)=z, that is u2=1. Now let us put v(z) = –1/(z+1) : we can check that v3=1. We see that we have a permutations u = a b c d e , which maps a to b, b to [b a d c e [ representation of A because k=uv is given by k(z) = z +1 a b c d e 5 a, c to d, d to c, and e to e, and v = and km(z) = z+m, so that k5=1 since 5 equals 0 in F . [e b a d c ,[ which 5 maps a to e, b to b, c to a, d to d, and e to c. These obey 3) We give a matrix representation of the elements u and v. the rules: u2 =1, v3 =1, (uv)5 =1 (of course, u and v do not If a,b,c,d are elements of F , with ad – bc =1, we associate commute). 5 the matrix a b , with the mapping f of P (F ), given by: 2) This group has 60 elements and is isomorphic to the [c d [ 1 5 group of rotations of a regular dodecahedron. f(z) a b z = . Thus u is one of the 15 symmetries with respect to a line [ 1 [ [ c d [ 1 [ joining the midpoints of two edges which are themselves The group of mappings obtained in this way is called symmetric about the centre. PSL(2,F5), for "Projective Special Linear" group of F5. Similarly, v is one of the 20 rotations through 2π/3 about a Then u, v, k, and t, y are represented by the matrices: line joining two vertices which are symmetrically placed u = 0 –1 v = 0 1 km = 1 m and t = –2 –3 about the centre. [1 0 [ [ –1 –1 [ [0 1 [ [ 1 1 [

14 EMS December 2004 FEATURE

Figure 5: Lagrange’s text on 3rd degree equations Figure 6: Icosahedral game and Hamiltonian circuit (1772) according to Sainte Lag¨ue

Lagarange had failed. Nowadays, be- the roots which left the function invari- Let an equation be given which has roots cause of Abel and Galois, we know ant. However, he assumed incorrectly a, b, c ...m. It will always have a group that even such a generalisation is im- that the radicals involved in solving the of permutations of the letters a, b, c ...m possible.Many of the most celebrated equation necessarily had to be rational with the following properties: mathematicians were interested in this functions of the roots. fundamental and complex problem, In the event, it was 1824 before Niels 1. every function of the roots which among them Leonhard Euler, who re- Abel, in his Mémoire sur lesequations ´ is invariant by substitutions of this turned to it several times and, above algébriques, justified Ruffini’s intuition. group, will be rationally determined; all, Karl Friederich Gauss (1801) and Abel, having at first thought, on the 2. conversely, every rationally deter- Louis-Augustin Cauchy (1813). contrary, that he had found a general mined function of the roots will be We’ll stay with the case of the quintic method of solution, proved the impos- invariant under these substitutions. equation. Descartes, for one, was con- sibility of solving the general quintic vinced that no formula like that of Car- by radicals in his 1826 Mémoire sur une Galois then studied how this group of dano could be found. In 1637 Descartes classe particulière d’équations résolubles ‘ambiguities’ could be modified by ad- suggested a graphical solution – us- algébriquement, in which he sketched a joining auxiliary quantities thenceforth ing the intersection of circles and cubic general theory which would only be considered ‘rational’. Solving an equa- curves – which he had invented for the fully worked out by Galois, towards tion by radicals was then reduced to purpose. From 1799 to 1813 (the date 1830. Galois’ work inaugurated a new solving its Galois group. of publication of his Riflessioni intorno era in mathematics where calculations The impossibility of reducing the alla solutione delle equazioni algebraiche gave way to the consideration of their quintic equation to equations of lower generali), Paolo Ruffini published di- potential, and concepts, such as those degree comes from the ‘simplicity’ of verse attempts at proofs, each of them of abstract group or of algebraic exten- the group A5 of the sixty even permu- more refined than the last, attempt- sion, occupied the foreground. tations of the five roots a, b, c, d,ande ing to demonstrate the impossibility of Galois’ great insight was to associate of the quintic. We say that an abstract solving the general quintic equation by to an arbitrary equation a group of per- group is ‘simple’ if there is no non- radicals.Hehadthecorrectideaof mutations which he defined in these constant composition-preserving map- assigning, to each rational function of terms: ping of the group into a smaller group. the roots, that group of permutations of EMS December 2004 15 FEATURE

The group A5 is the smallest non- commutative group which is simple, and it arises very frequently in math- 7. THE UNIVERSAL COVERING AND NON-EUCLIDEAN GEOMETRY ematics. This group can be described very economically: it is generated by two elements u and v satisfying the re- 2 3 5 lations u = 1, v = 1and(uv) = 1, vuv which gives us an excuse to move to 2 Hamilton’s icosions. vuv v uv2 d abO c Hamilton’s icosions J J' v2 uv After discovering quaternions, William vuv Hamilton tried, in 1857, to construct 2 vuv v uv2 a new algebra of generalised numbers, d abO c which he called icosions. Two of them, J J' A KLEIN'S MODEL denoted by u and v, which Hamil- v2 uv ton termed ‘non-commutative roots of unity’, were to satisfy u2 = 1, v3 = 1 ( )5 = Length of segment and uv 1. A childishly simple cal- [a,b] = Log ac x bd culation shows that, if uv = vu,then ad x bc we have v = 1v = u5v5v = u5v6 = B POINCARÉ'S MODEL (u2)2u(v3)2 = u,sou = v = vu2 = v3 = 1. So we can’t represent u and v as ele- To get a geometric understanding of the group generated by two elements u and v and presented by the relations u2=1, v3=1, and (uv)5=1, we start by ments of the groups S for n less than n considering the two first relations (u2=1, v3=1). The group thus generated or equal to 4. To represent u and v as el- is the group PSL(2,Z) which can be understood by looking at its action on ements of the group A5 of even permu- an infinite tree T in which three edges exit from each vertex. The third tations of the five letters a, b, c, d,and relation ((uv)5=1) can then be understood by identifying the tree T with the e, it is enough to put u =(b, a, d, c, e), universal covering of the graph in box 8 below. The universal covering, in the permutation which fixes e but ex- the sense of Poincaré , is obtained by considering all the paths which follow changes a with b and c with d,andto the edges of the regular dodecahedron. put v =(e, b, a, d, c),thepermutation The infinite tree T is represented by two models of non-Euclidean geometry: which fixes b and d but changes a to Klein's model (A) and Poincaréé 's model (B). In each model, the set of points e = v(a), c to a = v(c),ande to c = v(e). of plane geometry lie in the interior of a disk. In Klein's model, the straight The product uv is then the cyclic per- line joining two points is the same straight line as in Euclidean geometry, only the length of the line is changed. In this model, the length of a straight line is mutation (eabcd) which is indeed of or- given by the logarithm of the cross-ratio (ab,cd) of ab with the points c and d der 5. In fact, we could similarly repre- where the line ab intersects the circle C. Thus, [a,b] = log ac x bd . The ad x bc sent u and v in altogether 120 separate edges of the tree T are line segments of equal length. (but pairwise isomorphic) ways as ele- In the Poincaré model, the "straight lines" are arcs of circles otrhogonal to the ments of A5. circle C, and the idea of angle remains the same as in Euclidean geometry. The group A5 is isomorphic to the The distances are given by 2log(ab, cd) where the cross-ratio is calculated group of rotations preserving a regu- on the circle through (a,b,c,d), and where the factor 2 arises in comparing (A) lar icosahedron or, which amounts to with (B). the same thing, a regular dodecahe- The group PSL(2, Z) is represented by isometries of non-Euclidean 2 dron (these are the two most interest- geometry. A presentation of this group is given by the relations u =1 and 3 ing of the five platonic solids, whose v =1. The element u is given by symmetry about the origin O, and the π other members are the regular tetrahe- element v, by the non-Euclidean rotation with centre J and angle 2 /3. We don, the cube and the regular octahe- operate on the edge JJ' by the transformations represented by the words (such as uvvuuuvuv...) whose letters are the elements u and v. This gives dron, formed by the six centre of the exactly the tree T of the universal covering of the graph of Hamilton's icosion faces of the cube. These are the only game. In particular, each Hamiltonian path is a point of the universal regular convex polyhedra which exist covering. in our usual three-dimensional space). We get the dodecahedron by identifying the edges of the tree T which are To construct the isomorphism referred congruent modulo 5. This congruence means that we pass from one edge to to above, it is enough to associate u a b another by a non-Euclidean isometry given by an element [] c d of the group with one of the 15 rotations of order two PSL(2, Z) satisfying a = 1 (modulo 5), b = 0 (modulo 5), c = 0 (modulo 5), (a symmetry whose axis is one of the 15 and d = 1 (modulo 5). In this way, the group A5 whose presentation requires lines joining the midpoints of parallel the extra relation (uv)5 = 1, arises from quotienting PSL(2, Z) by the normal edges) and to associate v with one of the subgroup G generated by (uv)5.The quotient of T by G is precisely the graph 20 rotations of order three (whose axis formed by the edges of the dodecahedron. The quotient group PSL(2, Z)/G of symmetry connects one of ten pairs is the group PSL(2, F5) of box 4. of two diametrically opposite vertices of the dodecahedron, or the centres of two parallel faces of the icosahedron) in such a way that the product uv is one of 16 EMS December 2004 FEATURE the 24 rotations of order five (whose Morley’s triangle If G is the affine group of a commu- axis of symmetry connects one of the tative field k (that is, the group of map- six pairs of centres of parallel faces of pings g of k into k which can be writ- It is wrong to oppose the ’angel of ( )= + the dodecahedron, or pairs of diamet- geometry’ with the ’devil of alge- ten in the form g x ax b,where rically opposed vertices of the icosa- a = a(g) is non-zero), then for each bra’. What takes place is a fruitful co- ( ) hedron). The 60 rotations preserving operation between the visual parts of triple f , g, h of elements of G such either solid can be expressed simply that j = a( fgh) is not the identity and the brain, which can perceive the har- as products of the generators u and v. such that fg, gh,andhf are not transla- mony of a configuration at a glance, Even though the two icosions u and v and those parts of the brain which pro- tions, the following two assertions are generate the group A5 and satisfy the equivalent: 2 = 3 = ( )5 = cess language and distil that harmony relations u 1, v 1and uv 1, into algebraic formulae. We shall end it’s not immediate that these relations this introduction to the idea of sym- a) f 3g3h3 = 1 (the identity mapping constitute a presentation of the group, metry with a nice example of that co- 1(x)=x); that is, it’s not immediate that every operation by way of Morley’s theorem. relation between u and v follows from This is a subject where concrete sym- these. There are two ways, algebraic or 3 2 metries arising from geometry become b) j = 1andα + jβ + j γ = 0, geometric, of showing this (boxes 4 and abstract and algebraic when we look where α istheuniquefixedpoint 7). at them from a different angle. Their of fg, β that of gh,andγ that of The graph of the edges of the dodec- forceful interplay gives a genuine sense hf. ahedron, which has the same symme- of beauty. tries as that of the icosahedron, gave The British mathematician Frank rise to Hamilton’s ‘icosahedral game’ We need to show how this very ab- Morley was one of the first university which he also called the ‘game of non- stract property helps us better under- teachers in America. At the end of the stand (and, at the same time, prove) commutative roots of unity’. This nineteenth century, while pursuing re- game is the first example of what is Morley’s theorem. We shall take k to search into families of cardioids tan- be the (field of) complex numbers. Its now called the search for a Hamilto- gent to the three sides of a given trian- affine group is that of the direct simi- nian circuit, which is a very important gle, he discovered the following prop- concept in modern graph theory (box larities and has the rotations as a sub- erty: the three pairs of trisectors of the group (precisely when a has modulus 8). The challenge is to pass precisely angles of a triangle (that is, the straight once through each vertex of the dodec- 1). We let f , g and h be the rotations lines which divide the interior angles ahedron, using only the edges, and to about the three vertices of the triangle into three equal parts) intersect in six with each angle of rotation being two- finish at a vertex which is joined by points, of which three are vertices of an an edge to the starting point. A re- thirds of the relevant angle of the tri- equilateral triangle. angle. Thus f is the rotation with cen- markable account of this is given in The original proof is quite difficult tre A and angle 2a (the internal angle AndréSainte-Laguë’s 1937 essay Avec des nombres et des lignes, reissued by and depends on ingenious calculations at A being denoted by 3a), g is that based on a profound mastery of ana- with centre B and angle 2b and h that Vuibert in 1994. lytic geometry. There are now many with centre C and angle 2c.Theprod- 3 3 3 8. THE GAME OF ICOSIONS proofs of this result, as well as gener- uct of the cubes f g h is 1, because, for 3 14 15 alisations which produce 18, or 27 (or instance, f is the product of the two even more) equilateral triangles which symmetries with respect to the sides of 13 12 11 can be constructed from the 108 points the angle at A, so that these symme- of intersection of the 18 trisectors ob- tries simplify pairwise in the product 3 3 3 20 2 10 tained from the original ones by rota- f g h . 1 3 tions of π/3. These proofs include ones The equivalence above shows us that 5 4 by trigonometric calculation as well as α + jβ + j2γ = 0, where α, β,andγ 19 9 purely geometrical ones, such as that are the fixed points of fg, gh,andhf, 18 16 6 8 given by Raoul Bricard in 1922. and where the number j = a( fgh) is There is a proof which is entirely the first cube root of unity, which we 7 different, which illuminates the result have already met in the course of this from an interesting angle, because it article. The relation α + jβ + j2γ = 0is 17 lets us extend the result (a priori en- a well-known characterisation of equi- In his book Avec des nombres et des lignes Sainte-Lagu brought the tirely Euclidean) to the geometry of lateral triangles (it can be written in the (α − β) (γ − β)=− 2 game of icosions, invented by the affine lines over an arbitrary field k. form / j ,which βγ Irish mathematician Hamilton (1805- The purely algebraic result, which in- shows that we pass from the vector 1865), back to life. The game cludes and extends the trisector prop- to the vector βα by a rotation through consists of completing the circuit erty, is so general that its proof be- π/3). going through all the vertices of an comes a simple verification (a very An old recipe, well-known to those icosahedron once and once only: to general result is often easier to prove thoroughly trained in the rigours of start, the first six vertices are given. than a particular case, because the gen- classical geometry, shows that the Here's and example: (1, 2, 3, 4, 5, 6, erality can reduce the number of hy- point α,definedbyf (g(α)) = α,is 19, 18, 14, 15, 16, 17, 7, 8, 9, 10, potheses). none other than than the intersection of 11, 12, 13, 20). The statement is as follows: the trisectors of the angles A and B ly- EMS December 2004 17 FEATURE

9. MORLEY'S THEOREM

C rotation g takes the point α to α', which is the reflection of α α∋ α α c c c through AB. The rotation f takes the point to : so is a fixed point of the product of rotations fg. Similarly, β is a γ β fixed point of the product of rotations gh, and γ is a fixed point of the product of rotations hf.

α a b C a b a b A B x" AC α, Morley's theorem states that the three meeting points eflection β, γ of the trisectors of an arbitrary triangle ABC, as R indicated in the diagram, form an equilateral triangle. (n x' red). f 3 α

A Reflection AB B a g f b v a b A B x Now we consider the product f3g3h3. The rotation f3 through an angle 6a about A is the product s(AC) s(AB) α' and the reflection s(AB) in the side AB, and the reflection s(AC) in the side AC. Similarly, g3 is the product s(AB) f, g, h are the three rotations about the vertices of the s(BC) and h3 is the product s(BC) s(AC). So f3g3h3 = given triangle, through angles which are two thirds of the s(AC) s(AB) s(AB) s(BC) s(BC) s(AC). As the square of a angles at the vertices. reflection is equal to 1 this, together with the algebraic So f is the rotation through 2a about A, g is the rotation theorem, shows that the triangle is equilateral. through 2b about B, and h is the rotation through 2c about C. We shall look at the properties of these rotations. The ing closest to the side AB.Thereader lows us to expand the boundaries of objects;...symmetricmeanssomething can be convinced of this by checking our knowledge of geometry, already like well-proportioned, well-balanced, that the rotation g, centred on B,ofan- liberated from its Euclidean chains by and symmetry denotes that sort of con- gle 2b, transforms the point of intersec- the arrival of non-Euclidean geome- cordance of several parts by which tion into its symmetric image with re- tries (box 7). they integrate into a whole.’ spect to the line AB and that the rota- The discovery of quantum mechan- Alain Connes [[email protected]] is a tion f ,centreA, of angle 2a,returns ics and of the non-commutativity of winner of the Fields Medal and the the point to its original place. The the coordinates of phase space for an Crafoord Prize. Recently, he received same applies to the points β and γ.We atomic system has given rise, during the Gold Medal 2004 of the French have thereby shown that the triangle the past twenty years, to what can be CNRS. He is a professor at the Collège ABC is equilateral. As a bonus, we can seen as an equally radical evolution of de France and the Institut des Hautes see that, in this order, the triangle is geometric concepts, freeing the idea of Etides Scientifiques. He would like described in a positive direction (anti- space from the commutativity of co- to thank André Warusfel for his in- clockwise). This proof applies equally ordinates. to the other Morley triangles: the 18 tri- valuable help in preparing this arti- In non-commutative geometry, the cle, originally a lecture organised by sectors obtained from the interior tri- idea of symmetry becomes more sub- sectors by rotations through π/3 en- Jean-Pierre Bourgignon at the Centre tle, the groups sketched in this arti- Georges Pompidou in September 2000. able us to modify f , g and h without cle being replaced by the algebras in- The Newsletter would like to thank changing the product of their cubes, vented by the mathematician Heinz David Salinger (Leeds, UK) for the thus giving new solutions of equation Hopf, exemplifying Hermann Weyl’s ) translation of the article into English, a , and new equilateral triangles. fine definition, taken from his book and also Martin Qvist (Aalborg, Den- The evident duality of algebra and Symmetries ‘. . . the idea [of symmetry] mark) for help with the management of geometry in the examples above al- is by no means restricted to spatial the figures.

Apology

Unfortunately, a very disturbing in the formulas and also the line break- article can be read and downloaded mistake occurred during the printing signs disappeared. The Newsletter as a PDF-ﬁle from the following URL process of the recent issue 53 of the wishes to apologise sincerely for the http://www.emis.de/newsletter Newsletter. In the feature article 25 inconvenience that this has caused for /current/current9.pdf. years of Kneser’s conjecture by Marc de the readership; its apologies go also to Longueville, p.16 – 19, all minus-signs the author. A corrected version of this 18 EMS December 2004 FEATURE

MathematicsMathematicsMathematics isis is alivealivealive and andand well andwellwell thriving in Europe andand thrivingthriving inin EuropeEurope Luc Lemaire (Universit´e Libre de Bruxelles)

Luc Lemaire (Universit´e Libre de Bruxelles)

This is the text of a lecture given at the ﬁrst Euroscience Open Forum in Stockholm in August 2004. The text will keep unashamedly the characteristic of the lecture: no previous mathematical knowledge is assumed.

Numbers And now we consider the infinite Let us push a bit further. In the product above formulas, could we replace the 1 1 1 1 1 squares by cubes? In fact we have This being about mathematics, I may 1 − 1 − 1 − 1 − 1 − ... as well start with numbers, and shall 22 32 52 72 112 1 1 1 1 write three short formulas. Like the infinite sum above, this yields 1 + + + + ...= 23 33 43 (1 − 1 )(1 − 1 )(1 − 1 ) ... The first formula will have essen- a (finite) number and we write : 23 33 53 tially no ingredients, except the posi- 1 B = tive integers : 1,2,3,4,... Let us write : 1 1 1 1 butthisisnolongerafractiontimes 1 − 1 − 1 − 1 − ... 3 22 32 52 72 π . 1 1 1 Likewise, provided x is a real num- 1 + + + + ...= A. 22 32 42 The third formula will use an in- ber greater than 1, we have gredient from a totally different area, Whenever I stop summing after a finite namely classical geometry. As we all 1 1 1 number of terms, I just have a sum of 1 + + + ...= know, the length of a circle of radius R 2x 3x ( − 1 )( − 1 )( − 1 ) 1 2x 1 3x 1 5x ... fractions that can be computed by hand is given by 2πR,whereπ has approxi- or with a pocket calculator. mate value This has now become a function of the But the ... means that I want to number x, called the zeta function of continue summing forever, writing the π = 3.14159265358979323846... Riemann : ζ(x). sum of an infinite number of terms. Let us write This should not frighten anybody, we In 1859, in a prodigious paper, Rie- know of other examples of such infinite π2 mann analyses this function and shows C = sums yielding a finite number, like 6 its deep relation with number theory, and the distribution of prime numbers 3 3 3 1 So we have written three numbers, + + + ...= 0, 3333... = . in particular. 10 100 1000 3 coming from different areas, with no ζ apparent relation. Yet, one can show For that, he shows that can be ex- In the first formula, the sum also gives that A = B = C,theyarethesame tended as a function defined also for a number, which we’ll call A. < ζ( ) number. x 1, and even√ to the case of z , The second formula will have more where z = x + −1y is a complex It you pause to think about it (do it!) complicated ingredients, namely the number. He states his belief that the this is unbelievable. It means in partic- complete list of prime numbers. Re- zeros of zeta, i.e. the values of z for ular that some facetious god of mathe- call that a prime number is an inte- which ζ(z)=0, beyond the “easy” matics has encoded the length of a cir- ger ≥ 2 which has no divisor except ones −2, −4, −6, ... are all situated on cle in the list of prime numbers, totally 1 and itself. So 6 is not prime be- a line, namely are all of the form z = = × unrelated a priori. 1 √ cause 6 2 3, and 7 is prime be- + −1y. cause such a decomposition does not We draw three first conclusions: 2 exist. The first prime numbers are 1) Mathematics has beauty and Unable to prove it, he assumes it as 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31... Prime magic. a hypothesis, leaving to others the task numbers have always fascinated math- 2) Mathematics is not about specific of proving it. ematicians and Euclid, 24 centuries fields like algebra, geometry, analy- The “Riemann hypothesis”, still un- ago, proved that there is an infinite sis,... but is about the relations be- proven, is considered by many as the supply of primes. tween different areas, allowing us to most important single open question in Note that his proof is so perfect, use methods of one to solve problems mathematics today. shortandelegantthatitisstillthebest of the other. Why is this obscure looking question proof of this result, in fact it can be used 3) Ancient notions and proofs are as about the zeros of a specific compli- as a first illustration of what a proof is. fresh today as 24 centuries ago. cated function so important? From EMS December 2004 19 FEATURE

Figure 1: Graphs of the Riemann zeta function the first formulas, we get a glimpse the problem. Today, the obscure ques- Anybody who wants to receive en- that it is related to questions of number tion of Riemann appears to be at the coded messages will choose two very theory. In fact, hundreds of theorems centre of a spider web of mathematical large prime numbers - say of one hun- about prime numbers, or about the- fields and theories. dred digits. He will not reveal to any- oretical computer science, have been Physics has also joined the web: one the two chosen numbers, the “se- proven under the assumption that Rie- Michael Berry related the zeros of zeta cret key”. mann’shypothesisistrue,andwere to quantum chaos, and Alain Connes to But he will compute the product of not proven otherwise. So when - or if the eigenvalues of an operator, similar the two numbers, obtaining a two hun- - the hypothesis is proven, hundreds of to the absorption spectrum of a star. dred digit number that he will adver- other results will be proven at the same But is all this confined to the realm tise freely (the public key). time. of “pure” or fundamental mathemat- Using some clever mathematics In 1900, the great German math- ics, highly amusing to mathematicians (available as a software), anybody who ematician David Hilbert drew a vi- but unrelated to real life? wants to send him a message will en- sionary list of 23 open problems that G. H. Hardy, who contributed to the crypt it using the 200 digit numbers shaped part of the development of Riemann hypothesis in 1914, claimed and can send it without much care. mathematics in the twentieth century. that number theory’s beauty was re- Indeed, the only possibility to decode Riemann’s hypothesis is one of lated to its uselessness, so that it was it is to know the two prime numbers, them, and the only specific question of not related to dull realities or applica- known only to one person. the list unsolved after one century. tions. This is the principle of RSA cryp- In 2000, the Clay foundation (estab- Well, Hardy was wrong, and all of tography, created in 1977 by Rivest, lished by philanthropist Landon Clay) you use prime numbers regularly, but Shamir and Adleman (and slightly ear- announced a list of 7 Millennium prob- without knowing it. lier by Ellis, Cocks and Williamson of lems and, times having changed, of- Indeed, whenever you use a bank the British secret service, but they kept fered a reward of one million dollars machine, or order anything through it secret). for each. Riemann’s hypothesis is - of the internet, your bank data is of course But, you might say, the two hun- course - one of them. encrypted. And the only totally safe dred digit number is decomposable in The main effect of trying to solve encrypting code today is based on a unique way as the product of the two such a difficult problem is to relate it to prime numbers. original prime numbers, and it would new results and theories - a very posi- Basically, it works as follows: be sufficient to find that factorisation to tive result even when it does not solve decode all messages that use this num-

20 EMS December 2004 FEATURE ber. limit of 40 for the award of a medal, To sum up, mathematics in Europe is The point is that it would take cen- illustrating the fact that in most cases at top level, quite cheap to run and ex- turies for the larger computer to find mathematical genius can be detected at tremely efficient. the decomposition, because of the size ayoungage. It must be acknowledged that the of the numbers. For example, Jean-Pierre Serre was usefulness of fundamental mathemat- Could one maybe find by chance one awarded the medal in 1954 at the age ics will not always appear quickly of the prime numbers, hence the other? of 27 - because he was already an ac- (even if 24 centuries from Euclid to in- The odds would be no better than try- knowledged master with impressive ternet is an extreme example). ing to pick up at random a specific par- discoveries. But as stated by Timothy Gowers ticle of the known universe. On the other hand, Andrew Wiles, in his millennium address at the Clay So we get two more conclusions who achieved eternal (and even news- foundation: “If you were to work out about mathematics paper) fame by proving Fermat’s last what mathematical research has cost 4) Old notions still give rise to the theorem (a question open since 1637), the world in the last hundred years, most pressing open problems of today did not get a Fields medal because he then work out what the world has and tomorrow. was 41 when his proof was completed. gained in crude economic terms, you will discover that the world has re- 5)“Pure” mathematics, studied only Now, many political and economi- ceived an extraordinary return on a for the sake of elegance and beauty, cal studies comparing the effectiveness very small investment”. suddenly finds crucial applications to of scientific research in various conti- science or economic development. To nents refer to Nobel prizes as an indi- Since return is not immediate, it falls quote the physicist Eugene Wiegner, cator and draw a negative conclusion outside the scope of an industrial com- this is the “unreasonable effectiveness about Europe. pany that needs to meet its objectives of mathematics applied to natural sci- Indeed, between 1980 and 2003, the within a few years. Needless to say, it ences”. Nobel prizes in biology and medicine, is outside the aims of a financial group physics and chemistry give : buying a company with the sole aim to 68 for Europe sell it after five years, after raising its stock exchange value and nothing else. Nobel prizes, Fields 154 for the USA Thus, funding for fundamental cu- medals, Abel prizes and with the gap growing with time. riosity driven mathematics must come economic development For mathematics, the Fields medals from public money, for everybody’s give a much better picture for Europe. long term interest. In the same period, Fields medals There are no Nobel prizes in mathe- This should be managed by pro- were awarded to : matics. The mathematicians managed grammes better suitedformathematics to keep alive for decades the legend 9 Europeans (one working in the at the level of the European Commis- that Alfred Nobel’s girlfriend eloped USA) sion and the national policies. with the Swedish mathematician Gosta 5 U.S. citizens To include this better in the overall Mittag-Leffler, so that Nobel would not 4 Russians (one working in Paris, scientific planning of the E.U., a mini- create a prize that might go to his ri- two in the US) mum step would be to include a mathematician in EURAB, the 45 member val. However, Nobel’s sex life is not so 1 Japanese well documented, and the fun went out advisory committee of the European 1 New Zealander (working in the Commission on science (so far, 67 peo- of the story when geologists and oth- US) ers tried to spread similar stories about ple have been members of EURAB - All together, including Russia in Eu- their science. none of them mathematicians). rope where it should be, we get The point is much more probably It is sometimes said that mathemati- 10 working in Europe that Nobel - as an industrialist - was ciansworkbythemselvesintheirof- interested in “inventions and discover- 9 working in the USA fice, with a pen and a piece of paper. First one should not forget the waste ies” and didn’t see mathematics fitting a success story for Europe. there. paper basket, much used in mathemat- Fields medals are awarded for recent ical research. Anyway, the mathematicians created developments, so we see no growing And then there are the serious needs a prize recognised today as the “No- gap at all in mathematics. for mathematical work. bel of Mathematics”: the Fields medal, More recently, to celebrate the two First, of course, we need positions, first awarded in 1936. hundredth anniversary of the birth either in universities or research insti- There are three main differences be- of Niels Henrik Abel, the Norwegian tutes. Mathematics is done by people - tween Nobel prizes and Fields medals. government created a prize for math- and this is the priority. Compared to Nobel prizes, the ematics similar to the Nobel prize, fit- medal comes only with a minimal fi- tingly called the Abel Prize. In Europe, we face the same para- dox as in other sciences. Indeed, we nancial prize. After two awards, the laureates are educate high level mathematicians up Every four years, 2, 3 or 4 medals are Jean-Pierre Serre (49 years after his to the doctoral level, then have some awarded and they are never shared be- Fields medal), Michael Atiyah and post-doctoral positions, but usually do tween mathematicians for a joint dis- Isadore Singer - two Europeans and not provide the bridge between these covery. one US citizen. positions and the tenured ones. At that The major difference is an upper age EMS December 2004 21 FEATURE stage, the USA step in and offer tenure- and this again requires public funding way it is folded. track jobs to scientists fully educated in now. Europe. Note that this database could be We should strongly promote the cre- made available at minimal cost in less ation of timely tenure-track positions developed countries, where high level in our universities. mathematicsispresentbutfacesmajor In this respect, I believe the Euro- economic difficulties. pean Commission is making a mistake Finally, note that mathematical rein putting the accent on “training” of search is accomplished both in univer- doctoral students (up to four years re- sities and in a string of high level research experience) and not more ex- search centres, with regular movement perienced ones (up to 10 years) in all of researchers from one to the other, so its Marie Curie activities. Doesn’t this that all are necessary. simplyeducatemorescientiststobe swallowed by the American system ? Figure 2. Airbag simulation A special consideration should be The most efficient (and cost efficient) given to Central and Eastern Europe. Differential equations way to find the best shape is by com- The extremely high level of the math- puter modelling. The researchers in ematical tradition there explains why Since the invention of calculus by New- Kaiserslautern developed a computer for instance we have four Russian ton and Leibnitz, differential and par- software that allows them to model Fields medallists out of twenty. But the tial differential equations have been any shape and gas speed, and look for dramatic economic situation explains the central tool in mathematics appli- an efficient one. They can vary the that only one remained in Russia. cable to science - first astronomy and shape and folding many times and ob- A smallish investment to preserve physics, then chemistry, biology and serve the blowing up of the bag. that tradition would be in everyone’s now economy and finance. Comparing the “films” of their interest. It is a huge subject, both in funda- mathematical bags with the film of con- Secondly, mathematicians need reg- mental mathematics and its applica- crete realisation of the bags built af- ular contacts with other researchers, tions. terwards shows that they obtain a re- the world over. This happens during To take but a specific example, I’ll markable precision. conferences, and by short or long term consider the Navier-Stokes equations. But this achievement required the visits. They are a system of partial differential development of new mathematics. These provide an immense accelera- equations that model the movements The standard way to model partial tion of research. In one or two hours, in fluid dynamics. They were written differential equations on a computer is a specialist can explain the basics and by Navier in 1822, then justified more to choose a mesh on the domain of recent trends in his field, whereas it precisely by Stokes a few years later. the equation, then approximate deriva- would take months to get that informa- They are both extremely useful in ap- tives by differences of the values at the tion from the literature. Also mathe- plications and extremely hard to study different vertices. maticians are not working together in with mathematical rigour. Here the domain (the airbag) varies very large centres, so they regularly After almost two centuries, we have quickly, because of the varying gas need to see specialists in their domain, no formula giving the solutions (this is pressure. Therefore, “mesh free” meth- often rather thinly spread. usual for partial differential equations), ods have to be developed. Thirdly, easy access to the literature and we don’t even have decent results We see here the same scheme ap- is necessary. We have seen that older on their existence and properties. pearinginanendlessseriesofexam- articles keep all their value, and access In fact, their mathematical study is ples. There is first a mathematical the- is needed to all good level literature the object of one of the seven Millen- ory, maybe fairly old. It is still the ob- present and past. nium problems of the Clay foundation. ject of difficult theoretical questions. It No university library today can keep But still they are used in applications also gives rise to unexpected applica- up with the rising cost of journals, but like shaping cars and planes, mod- tions which in turn give rise to new electronic access offers new unprece- elling the flow of blood in the cardio- mathematical questions. Their study dented opportunities. vascular system and many others. will provide new theories which again Most new papers are typed in the I shall briefly describe an applica- will motivate new problems and give software TEX, and can be made acces- tion developed at the Fraunhofer Insti- rise to new unexpected applications. sible to all. tute of Applied Mathematics in Kaiser- Talking in Stockholm on partial dif- Another objective is to digitise the lautern, Germany: the conception of ferential equations, I should mention whole literature - an operation es- airbags for cars. a grand master of the subject: Lars timated at around 50 million euros. For many years, it was an inaccessi- Hörmander. Good co-ordination is needed, so that ble idea. Indeed, in case of an accident, His early work was so exceptional all digitised papers are accessible with the bag has to be inflated fully in 1/20 that he was appointed full professor in the same standard. Also, the owner- of a second or it is too late. Stockholm at the age of 25. ship of the database should not be left Everything will count: the way the At the age of 31 he got a Fields to purely profit-making organisations, gas is injected, the shape of the bag, the medal, but moved to the Institute for 22 EMS December 2004 FEATURE Advanced Study in Princeton. The obvious example is the Greek The first mathematical ideas were This brain drain provoked a strong civilisation, which left us as heritage practical: counting objects (like cattle), reaction in Sweden, and the parliament (with contributions of the Arabic civil- computing the area of a field, the vol- voted the “Lex Hörmander”, a law al- isation) art, mathematics, philosophy ume of a pyramid. lowing the creation of personal chairs and the beginnings of science. It is But then why did Euclid care about in exceptional case. these four aspects, and not the value of prime numbers, or abstract proofs Hörmander came back to Lund and their stock exchange or whatever they of geometric theorems? Why did the law also allowed them to attract had, that founded the rebirth of our the Babylonians, 1000 years before back Lennard Carlesson, another major civilisation after the Renaissance. Pythagoras, engrave his theorem on figure. Civilisation as we live it is not de- their clay tablets? Why did the Egyp- Together, they had 36 PhD students fined in economic terms, and is our tians develop a sophisticated and quite in Sweden, who in turn had PhD stu- most precious asset. useless system of fractions? dents, so that the total number of their The princes of Medicis, wealthy as On a bone, found by anthropologist mathematical descendants in Sweden they were, will be remembered forever Jean de Heinzelin in Ishango, Africa, is now over 180. for triggering that Renaissance. a series of scratches provides the be- Thus, their presence in Sweden Likewise, how should we remem- ginning of a multiplication table and a helped not only the general reputation ber Carl Wilhelm Ferdinant, Duke of short list of prime numbers. The bone of the country, but more concretely the Brunswick? is dated between 7000 and 20000 years local development of science and in- In my dictionary, he is described as BC and the scratches could be a coinci- dustry. a duke soldier who was beaten by the dence - or one of the first appearances of abstract mathematics. Let us have a Lex Hörmander at the French in Valmy, then again in Iena. European level! But I must say I looked only in a I believe that the human mind, early French dictionary. Still, an uninspiring in its evolution, has the need to think notice. in abstract terms, and that mathematics unavoidably appears at this stage. Pour l’honneur de l’esprit But one day, he got a report from humain a school teacher that a young boy Finally, why do mathematicians do seemed remarkably gifted in mathe- mathematics? Why did Michelangelo paint and So far, this lecture has concentrated matics. The boy was the son of a poor sculpt, why did Beethoven compose? on the far reaching economic benefits gardener and bricklayer, so his future following the development of funda- should have been rather bleak. All for the same reason: because they mental mathematics. I described only But the Duke liked mathematics, must. As David Hilbert put it in 1930: two examples, for lack of space, but saw the boy and was convinced by his Wir mussen ¨ wissen, wir werden wis- couldgoonendlesslyinmostifnot obvious talent (if not by his good man- sen (we must know, we shall know). all fields of human activity. Fast devel- ners). Thus he supported his studies Asked why he insisted in try- oping examples include medical imag- and career throughout his life. ing to climb Mount Everest, the fa- ing, image compression for storage or The boy’s name was Carl Friedrich mous mountaineer George Mallory an- transmission, epidemiology, biostatis- Gauss, and we owe to him (and the swered: “Because it is there”. tic, mathematical genomics, finance, Duke) the Gauss law of prime num- Likewise, mathematicians attack control theory for plane safety or en- bers, the Gauss distribution in prob- their own Everest (like Riemann’s hy- ergy saving ... ability, the Gauss laws of electromag- pothesis) because it is there. In this presentation, I followed the netism, most of non-Euclidean geom- But when they reach the top, they present day trend of having to prove etry, and the Gauss approximation in have changed the scenery by their the economic value of all activities, in- optics. achievement. cluding art, mathematics, and culture. Obviously, we need more Dukes of Looking backward, they see the hard But this is a sad evolution of our so- Brunswick in our governments. path that they have followed, but also ciety, and mathematics should also be More to the point, we need a Euro- some much easier and simplified paths pursued “for the honour of the human pean Research Council to cater for fun- now open to others. mind”, to coin the phrase of Carl Gus- damental research. Looking forward, they see higher tav Jacobi. Going back to the idea of civilisa- mountains which were invisible or So I want to point out now that tion, it seems that abstract mathemat- maybe did not exist before. Mountains mathematics, like art, philosophy and ics - disjoint from practical applications that must now be climbed. science, are essential parts of civilisa- - appears early in the development of Because they are there. tion. the human mind.

Luc Lemaire [[email protected]] received a Doctorate from the Université Libre de Bruxelles in 1975 and a Ph.D. from the University of Warwick in 1977. From 1971 to 1982 he held a research position at the Belgian F.N.R.S., and has been a professor at the Université Libre de Bruxelles since then. His research interests lie in differential geometry and the calculus of variations, with a particular interest in the theory of harmonic maps. A former chairman of the Belgian Mathematical Society, he has been associated with the European Mathematical Society since its creation in 1990, being a member of the Council from 1990 to 1997, a member of the group on relations with European Institutions since 1990, Liaison Officer with the European Union since 1993, and Vice President since 1999. EMS December 2004 23 SOCIETIES TheThe PolishPolish MathematicalMathematical SocietySociety (PTM)(PTM) Janusz Kowalski (Warsaw)

PTM in the early years headquarters had been moved to Warsaw The Polish Mathematical Society was and its statute changed). established in 1919. The Reader can find The number of members of the Polish information regarding the circumstances Mathematical Society equaled 49 persons of its rise, as well as a description of its in 1921 and 155 persons in 1939. The activity during the first year of its exis- Society was of a scientific character, as tence, in an article by Józef Piórek in the stated in the statute, where active and European Mathematical Society passive voting powers were given exclu- Newsletter ([4]). Stefan Banach, sively to authors of mathematical publi- Franciszek Leja, Otto Nikodym, Stanis³aw cations. A new statute, resolved in Lvov Zaremba and Kazimierz ¯orawski were in 1936, established a federal organization called into being by the government, the among its founder members, and of five sections with its headquarters in Polish Mathematical Society had been the Stanis³aw Zaremba was elected as the Warsaw. The General Meeting of PTM, only central institution representing Polish President of the Society. In 1921, the which included delegates from all sec- mathematics at home and abroad. Apart Mathematical Society in Cracow tions and the President of PTM, elected from assemblies of Polish mathematicians (Towarzystwo Matematyczne w Krakowie) a president, a secretary and a treasurer - organized by this right, PTM was in con- was transformed into a national Polish all of them constituting the General tact with public institutions, voiced opin- Mathematical Society (Polskie Management - as well as a Board of ions on subjects related to science and Towarzystwo Matematyczne; PTM) with Control. The General Management also education and issued its own periodical - its headquarters in Kraków (Cracow). automatically included the presidents of “Annales de la Société Polonaise de According to its first statute, the the sections, who held titles of “vice- Mathématique“ - distributed nationally presidents of PTM“. Sections held local and internationally. meetings, where local General Management members were elected as The post war years well as those of local boards of control In the years 1919-39, the time when the and delegates to the General Meeting of famous Polish mathematical school was PTM. The aforementioned statute of 1936 established, Polish mathematics was a substantially extended the aims of PTM. great success and met with a high esteem on the international forum. During the Second World War, when Poland fell under occupation, all official activity of the Polish Mathematical Society came to a standstill; only clandestine scientific sessions were held in Cracow and in Warsaw. The second period of PTM’s activity involves the years 1945-53, when Kazimierz Kuratowski acted as President. In 1945 the Cracow section was reacti- Stanis³aw Zaremba, vated and in 1946 sections in Poznañ and PTM’s first president Warsaw started to operate again. Within the period 1946-53, six new sections Society’s aim was “a comprehensive cul- were established. The number of mem- tivation of pure and applied mathematics bers of PTM grew from 144 persons in by means of scientific sessions combined Stefan Banach, elected president in 1939 1946 to 339 in 1953. with lectures“. The first change to the During the first years of the post-war statute - still in 1921 - resulted in the Apart from those adopted before, new period, the Polish Mathematical Society following insertion: “publication of a ones were added, among others to orga- was, similarly to the situation before periodical and maintenance of contacts nize competitions, to gather collections of 1936, the only institution actively cover- with the mathematical scientific move- publications, to improve work conditions ing the whole range of issues related to ment“. for mathematicians, to maintain contacts Polish mathematics. As such, it cooperat- Members not residing in Cracow were with scientific institutions both within the ed with Polish authorities on the recon- admitted to PTM through local sections. country and abroad and to invite mathe- struction of the 3rd level education sys- Within the period 1921-1939, the follow- maticians from abroad to give lectures. tem (among others, it prepared a reform ing sections existed: one in Lwów The basic rules under this statute have regarding mathematical studies and M.Sc. (Lvov), since 1921, and three others since been in force until present times. degrees in mathematics). PTM also coop- 1923 in Warszawa (Warsaw), Poznañ and Until 1936, when a Council for Exact erated on a regular basis with the Wilno (Vilnius). Since 1937, mathemati- and Applied Science (Rada Nauk Ministry of Education as well as other cians from Cracow have constituted part Œcis³ych i Stosowanych) and its organ educational authorities, being a founding of the Cracow section (after the Society’s called the Mathematical Committee were body of the first Olympic Games for sec-

24 EMS December 2004 SOCIETIES ondary school students in the country, i.e. the sections in Toruñ, Katowice and file to a Society involved in large-scale the Mathematical Olympic Games. The Szczecin - established in the years 1952- scientific and social activity, became Society was also an initiator of research 55 - and those in Bia³ystok, Rzeszów, reflected in the development of various work and systematically convened scien- S³upsk, Czêstochowa, Kielce, Olsztyn, organizational structures, created within Opole, Nowy S¹cz and Zielona Góra - the Society’s frame to fulfill defined established in the years 1970-75. The sat- assignments. By the end of 1975, six uration of big national scientific centres committees operated under the supervi- with specialist seminars made it necessary sion of PTM’s General Management: the to do research work within a much wider Committee for Popularization of scope that might be of interest to all Mathematics and Higher Education, the mathematicians. There was also a need to Committee for Mathematics at hold a council dedicated to social and Universities, the Committee for School organizational issues. This type of activ- Handbooks, the Committee for ity was conducted by successive sections, Application of Mathematics, the as well as by the General Management Committee for Publication and the of PTM, during section meetings, coun- Committee for an Information and cils, conferences and assemblies of Polish Service Centre. Apart from these, there mathematicians. The Polish Mathematical was a Main Committee for Mathematical Society also fulfilled important tasks Olympiads, which operated together with Kazimierz Kuratowski within the scope of cooperation with the its local committees, seven editorial com- educational authorities, and introduced mittees, one editorial board and four tific sessions. Within the years 1946-49, new forms of teaching young people with competition jury boards. Analogous teams four assemblies of Polish mathematicians a talent for mathematics. It also cooper- operated under the supervision of local took place, the last one together with ated with foreign mathematical centers General Managements. Czechoslovak counterparts. At the same and initiated new publications. An aspect Roman Sikorski, who in the years time, annual competitions for the best of significant value was PTM’s help to 1953-75 impacted PTM’s activity and works within the field of mathematics newly established sections, effectuated development the most, was the Society’s were organized. Editorial activities were mainly through delegating lecturers. President between 1965 and 1977. launched by PTM as early as in 1945, At the end of the 1960s, the Polish Another important character in those when the 18th volume of “Annales de la Mathematical Society got involved in a years was Tadeusz Iwiñski, Secretary in Société Polonaise de Mathématique“ was campaign to establish a new profession the years 1960-1981. published. for mathematicians working within vari- In 1948, the National Mathematical ous branches of science, economy and Conferences and Assemblies Institute (Pañstwowy Instytut Matema- state administration. This problem was Since the very beginning of PTM’s exis- tyczny) was established. That meant that the focus subject for the 10th Assembly tence, scientific lectures and discussions PTM ceased to be the only central math- of Polish Mathematicians - organized in during local meetings, conferences and ematical institution; therefore its activity Katowice in 1970, together with the assemblies have been the basic form of started to be gradually limited to the National Mathematical Committee of the activity. In the years 1919-39, as many profit of the Institute. After the Polish Polish Academy of Sciences. This Academy of Sciences (Polska Akademia Assembly, being the largest one in Nauk) had been called into being in 1952 PTM’s history as far as the number of (the National Mathematical Institute participants is concerned (547), formulat- becoming a part of it after being renamed ed new assignments for the Polish to: Mathematical Institute of the Polish Mathematical Society and desiderata Academy of Sciences - Instytut addressed to other institutions as well as Matematyczny PAN), a third national a definite activity program. In this way, mathematical institution came into being: PTM managed to be a focal point for the National Mathematical Committee of the majority of mathematicians working the Polish Academy of Sciences (Komitet in various branches of economy. Nauk Matematycznych PAN). Statutory Members of the Polish Mathematical tasks of the Mathematical Institute cov- Society are employees of institutes of the ered many fields, previously being with- Polish Academy of Sciences as well as in the scope of activity of the Polish educational centers such as various types Mathematical Society. In bigger scientific of 3rd level education schools (including centres, numerous specialist seminars technical, pedagogic, economic, medical were established and, as a consequence, and agricultural ones), various teaching scientific sessions of PTM lost much of colleges and secondary schools. their appeal. Financial resources are derived from However, there still existed important membership fees, subsidies granted by the Roman Sikorski, scientific and social projects, which could Ministry of Scientific Research and PTM’s president 1965 – 1977 only be carried out within the frame of Information Technology and by the a national scientific society that would Ministry of National Education and Sport. as 1143 lectures were given, while in the assemble all scientific and didactic According to the provisions of the first period of 1949-75, the corresponding employees from within one field. A clear statute, PTM was supposed to “compre- number was 5998 (there are no data cov- indication of this was a spontaneous hensively cultivate pure and applied ering the period of 1945-48). During the establishment of PTM’s new sections in mathematics“, but in practice was con- period between 1976 and 2003, as many cities and towns where new scientific and fined to “scientific sessions combined as 3860 lectures were given. It is worth academic centres emerged. These were with lectures“. A change in PTM’s pro- mentioning that members of other sec- EMS December 2004 25 SOCIETIES tions and mathematicians from abroad debates of the General Meeting of PTM, over all these publications, including constituted a significant part of all lec- constitute the key elements of PTM’s PTM’s official organ “Annales de la turers. Foreign lecturers represented over assembly. Forty sessions were held before Société Polonaise de Mathematique“, which was then renamed to “Annales Polonici Mathematici“. The Polish Mathematical Society started a new phase of publishing activity in

PTM’s Assembly in Wroc³aw, 1946

40 countries from all over the world. 2004, which were dedicated to a general The Polish Mathematical Society has so overview of selected issues related to “Functions of a Complex Variable” far organized 15 assemblies of Polish contemporary mathematics, being of mathematicians (including 3 in the inter- interest to all mathematicians. The the- 1955, when the first of each of two war period), which usually take place matic content of scientific sessions held series of “PTM’s Annals“ were issued. every 5 years; in the remaining years, in the years 1999-2002 embraced an Series 1: “Mathematical Papers“ (“Prace regular PTM meetings have been held. In overview of the most significant achieve- Matematyczne“), which became 1929, PTM organized the First Congress ments in mathematics throughout the 20th “Commentationes Mathematicae“ in 1967, of Mathematicians of the Slavic century. published 43 volumes up to the year Countries. The assemblies are of scientif- 2003. Series 2: “Mathematical News“ ic character. Their programmes involve Publications (“Wiadomoœci Matematyczne“) published lectures and reports of mathematical sub- The publication of periodicals has been stance: cross-thematically during plenary the second basic form of activity of the meetings and those of a more specialized Polish Mathematical Society. In 1921, an character during meetings in sections. organ of PTM entitled “Dissertations of the Polish Mathematical Society“ was called into being, and one year later it was changed into a periodical named “Annales de la Société Polonaise de Mathematique“. 25 volumes of this periodical were issued during the period 1922-52. In the years 1948-53, PTM started publishing other periodicals: a bi- monthly for teachers entitled “Mathe- matics“ (“Matematyka“), launched by PTM in 1948 but taken over by the Ministry of Education in 1953, and a series called “Mathematical Library“ (“Biblioteka Matematyczna“), 1953. Apart Tadeusz Iwiñski, from the ones mentioned above, the fol- PTM’s secretary 1960 – 1981 lowing titles were also published: “Fundamenta Mathematicae“, “Studia These assemblies create an opportunity to Mathematicae“, “Colloquium Mathemati- Wiadomósci Matematyczne ponder on more general issues regarding cum“, “Mathematical Monographs“ and science, the system of education and “Mathematical Dissertations“ (since 39 volumes up to the year 2003. In social matters. These issues could even 1952). In the years 1948-53, PTM super- 1973, Series 3 was launched entitled be a motive to convene a meeting (e.g. vised - by the order of the Ministry of “Applied Mathematics“, which became in the years 1969, 1970 and 1972). Education - all Polish mathematical pub- “Applied Mathematics. Mathematics for Since 1962 the Polish Mathematical lications. the Society“ in 2000, with 45 volumes Society has been organizing 2- to 4-day- In 1953 the Mathematical Institute of published up to the year 2003. In 1977, long scientific sessions that, together with the Polish Academy of Sciences took Series 4 was launched, “Fundamenta 26 EMS December 2004 SOCIETIES Informaticae“, publishing 56 volumes up field of applied mathematics and practi- papers, organized by the editorial board to the year 2003. Finally, in 1982 Series cal elaborations (named after Hugo of “Delta“ monthly. This competition’s 5 was launched, “Didactics of Steinhaus and Wac³aw Pogorzelski); final- finals take place during the annual Mathematics“ (“Dydaktyka Matematyki“); Scientific Session of the Polish 25 volumes were published up to the Mathematical Society. In the period 1976- 2004, nearly one hundred schoolchildren were awarded with prizes and distinc- tions. First of all, they were from comprehensive secondary schools in big cities; however, among them there were also some pupils from secondary voca- tional schools in smaller towns.

Committees In order to carry out its statutory tasks, PTM brings into being specialized committees. In 1953 a Committee for Popularization of Mathematics and in Hugo Steinhaus 1958 a Committee for Secondary Education were called into being by the ly, for achievements for the benefit of General Management of the Polish mathematical culture (named after Samuel Mathematical Society. In the period 1959- Dickstein). PTM Prizes for young math- 61, a subcommittee for a reform of pro- ematicians have also been set up. The grammes and teaching methods worked Toruñ Section organizes the “Józef out projects and drafts regarding some Marcinkiewicz Competition“ for the best school handbooks. This programme was Delta introduced to schools with minor amend- ments and PTM cooperated with the Ministry of Education in its being carried year 2003. out. In 1968 a Committee for School Since 1971, PTM has been distributing Handbooks was called into being to orga- among its members an “Information nize debates in working teams (in the Bulletin of the Polish Mathematical period 1968-71), publish articles, deliver Society“, which includes current news relevant materials to authors, and prepare regarding scientific and organizational reviews of handbooks (including orga- issues. At first it was published several nized debates over them). The times per year, but later this frequency Committees for Secondary and Primary diminished. After a two-year break (2000- Education and for Popularization of 2001) its publication was relaunched and Mathematics worked within three spheres: continued on a more regular basis. Since scientific activity (focusing on the most 1974, further to an initiative of the Polish recent results of scientific research relat- Mathematical and Physical Societies, a ed to didactics of mathematics), analysis popular mathematical/physical monthly, of documents and cooperation in their entitled “Delta“, has been published. The Wac³aw Sierpiñski being elaborated (legal acts and instruc- thematic scope of this publication was tions concerning educational policy), and extended in 1979 by issues regarding student’s paper, the Wroc³aw Section working out methods of modernizing the astronomy. For more information about does similarly for the best student’s paper process of teaching mathematics as well “Delta“, see [4]. Two other periodicals in the field of probability theory and as preparing the mathematics teachers for are published with PTM’s cooperation: mathematical applications, and the edito- new tasks. “Mathematics“ (a periodical for teachers) rial board of “Didactics of Mathematics“ The Committees for Secondary and and “Gradient“ (a periodical for teachers, organizes the “Anna Zofia Krygowska Primary Education and for Popularization pupils and their parents). Competition“ for the best student’s paper of Mathematics, in cooperation with the in the field of didactics of mathematics. Association of Teachers of Mathematics, Competitions and prizes The Polish Mathematical Society is also organized, among others, a national sci- The Polish Mathematical Society is also involved in organizing and supervising entific seminar devoted to didactics of busy with organizing competitions and competitions for prizes named after mathematics. During this seminar, some awarding prizes. There are two competi- Kazimierz Kuratowski, Stanis³aw Mazur proposals of methodical and didactic solutions open for entry to all Polish mathe- and W³adys³aw Orlicz. It also takes part tions were put forward which concerned maticians, one for young mathematicians in carrying into effect the idea of lectures various levels of education in relation to under 28 years of age and three for uni- which are awarded with the “Wac³aw mathematics. Special attention was paid versity students. Special PTM prizes have Sierpiñski medal“. On the whole, 836 by the Committee to the role of mathe- been set up, named after outstanding prizes and 25 medals have been award- matics at the “matura“ examination (i.e. Polish mathematicians: Grand Prizes for ed so far. the final high school examination). This scientific achievements (named after The Polish Mathematical Society is a was exemplified by the “Open letter by Stefan Banach, Stefan Mazurkiewicz, patron of activity aimed at bringing to the Polish Mathematical Society“, accept- Stanis³aw Zaremba, Wac³aw Sierpiñski, light pupils with a talent in mathematics. ed by PTM’s General Meeting in Lublin Tadeusz Wa¿ewski, and Zygmunt This activity takes the form of a com- in 1992 (published in numerous dailies Janiszewski); for achievements in the petition for schoolchildren’s mathematical and weeklies). EMS December 2004 27 SOCIETIES In 1962 the Polish Mathematical Conferences of Application of degree competitions, 20718 in 2nd degree Society called into being a Committee for Mathematics, where mathematical models competitions and 3846 in 3rd degree University Education which worked in applicable to specified practical issues competitions. The total number of laure- two teams: one dedicated to non-univer- were acquired and presented. 32 such ates amounted to 817, and of those sitary schools and one dedicated to uni- conferences took place until the year awarded with a distinction title, to 444. versities and pedagogic schools. These 2003. Since 1959, a 6-8 person delegation has teams embraced among other activities: a In 1972, an Information and Service been chosen to participate in internation- reform of the course and the programmes Centre was set up. Up until the end of al mathematical Olympiads. Three of of studies (1964-67), educating mathe- the 1980s, it was responsible for solving such Olympiads were organized in Poland maticians at technical schools (1962-65), problems submitted by various scientific in 1963, 1972 and 1986. Since 1977, the inter-school exchange of students and and economic institutions, acting as advi- candidates for a doctor’s degree, methods sor and conducting training for groups of of educating future mathematicians, employees in their workplace. During assessment of experimental programs, PTM’s scientific sessions, assemblies and modernization of teaching methods, and conferences, problems from applied math- education by the media. In the following ematics have been the subject of a much years, smaller specialised committees more detailed scrutiny now than in the emerged from this Committee. past. In 1993, the Committee of Mathematics at Universities, Pedagogical Academies Popularization and Teaching Colleges, together with its The Polish Mathematical Society under- counterparts in the Polish Physical, takes various types of actions to com- Chemical and Biological Societies, hand- memorate Polish mathematicians. Many ed to the Ministry of Education a streets in Cracow, Warsaw and Wroc³aw “Memorial regarding education within the have been named, on PTM’s initiative, scope of non-major basic subjects at ter- after outstanding mathematicians: tiary-level schools“. The Committee ana- Stanis³aw Go³¹b, Bronis³aw Knaster, lyzed the phenomenon of intensified Miros³aw Krzy¿añski, Kazimierz diversification of programmes related to Kuratowski, Franciszek Leja, Edward Stefan Straszewicz, organiser of teaching mathematical subjects in mathe- Marczewski, Zdzis³aw Opial, Witold Mathematical Olympic Games matical faculties at individual universities. Pogorzelski, Marian Rejewski, Wac³aw On the initiative of the Committee of Sierpiñski, Stefan Straszewicz, Jacek there has also been a 6-person delegation Mathematics at Technical Universities, Szarski, Tadeusz Wa¿ewski and Stanis³aw chosen to participate in the Polish- tests were conducted in 1995 to verify Zaremba. Austrian mathematical competitions, orga- the level of mathematical knowledge of Further to PTM’s application, on 23rd nized each year alternately in Poland and 1st year students at several technical ter- November 1982 the Polish Post-Office in Austria. Since 1992, a 5-person dele- tiary schools. The results, which turned issued four stamps in a “Polish mathe- gation has been chosen to participate in out to be somewhat alarming, were maticians“ series with portraits of Stefan mathematical competitions of the Baltic passed on to educational authorities and Banach, Zygmunt Janiszewski, Wac³aw States, organized by Poland in 1998. made public. Following this, the Sierpiñski and Stanis³aw Zaremba. On the Several mathematicians have been Committee insisted that an entry exami- initiative of PTM’s Cracow Section, a working on the organization of the nation in mathematics be compulsory at monument dedicated to Stefan Banach Olympic Games. The first Chairman of all technical tertiary schools until a com- was erected, to be unveiled with due cer- the Main Committee was Stefan pulsory “matura“ examination in mathe- emony on 30th August 1999 during the Straszewicz, one of the creators of the matics was introduced. XV Assembly of Polish Mathematicians. Mathematical Olympic Games, who held The Committee for Mathematics in The Polish Mathematical Society carries this position for 20 years (1949-1969). Economic Studies was busy with prob- on its activities in the school environ- The Toruñ, Wroc³aw and Nowy S¹cz lems related to “the New Matura“ exam- ment. It organizes lectures for teachers sections, popularized among pupils and ination in mathematics and a “programme and schoolchildren, as well as the gener- students, as well as among adults, the base“ for mathematics in economic stud- al public. 3637 such lectures were given idea of participation in three internation- ies. in the years 1952-2003. Since 1954, PTM al mathematical competitions: Since 1987, the Committee for the has been managing inter-school mathe- “Kangaroo“, “International French History of Mathematics, under the aus- matical circles and competitions for tal- Championship in Mathematical and pices of PTM’s General Management and ented pupils. 1100 such forms of work Logical Games“ and “Mathématiques sans in cooperation with mathematical insti- have been recorded so far. Frontières“. tutes of schools of higher education, has Occasionally, other forms of popular- been organizing annual Schools of ization of mathematical knowledge are International Cooperation History of Mathematics. 18 such schools used: scientific camps for young people, The Polish Mathematical Society has were organized until the year 2004. distance learning studies (delivery of been actively cooperating with the math- During the PTM’s assembly in 1970, a source materials and assignments to stu- ematical community abroad. Since the Committee for Application of dents and sending back corrected assign- first years of PTM’s existence, lectures Mathematics was established in response ments), and guidance units for schools’ have been organized to be given by for- to the fact that many “non-academic“ mathematical circles. eign mathematicians at various occasions (working in various branches of econo- The Mathematical Olympic Games or at a special invitation to give a lec- my) mathematicians had joined the Polish have been operating under the auspices ture in Poland. These lectures are given Mathematical Society. This obliged the of the Polish Mathematical Society. 55 during scientific sessions in PTM’s local Committee to organize (in cooperation Olympiads were held in the years 1949- sections, during PTM’s conferences or with other institutions) annual 2004: 81972 pupils took part in 1st assemblies. Nearly 2000 such lectures 28 EMS December 2004 SOCIETIES were held in the years 1949-2003. The Polish Mathematical Honorary Members Society made agreements for an exchange with the following In recognition for contributions to the development of math- societies: Czechoslovak (1962), Bulgarian (1968), Hungarian ematics, its being taught, applied and popularized, as well as (1973) and Greek (1980). Within this framework, in the years in acknowledgement of devoted participation in PTM’s activ- 1976-1990, 390 persons profited from the exchange program on ities, the Polish Mathematical Society confers the dignity of both sides, spending both in Poland and in the other above Honorary Member of PTM. The following mathematicians mentioned countries a total of 2641 so-called “exchange days“. have been given this dignity so far: Pawe³ S. Aleksandrow, In the case of Greece, such cooperation took place in the years Donald W. Bushaw, Karol Borsuk, Zygmunt Butlewski, 1980-1982. Zbigniew Ciesielski, Mieczys³aw Czy¿ykowski, Samuel Stanis³aw Zaremba, the President of the Mathematical Society Eilenberg, Pàl Erdõs, Eugeniusz Fidelis, Stanis³aw Go³¹b, with its seat in Cracow, represented Polish mathematics during Tadeusz W. Iwiñski, Wiktor Jankowski, Leon Jeœmanowicz, the International Mathematical Congress, held in 1920 in Bronis³aw Knaster, Andriej N. Ko³mogorow, Jan Kozicki, Strasbourg. The International Mathematical Union (IMU) was Anna Krygowska, W³odzimierz Krysicki, Kazimierz established there by representatives from 11 countries: Belgium, Kuratowski, Andrzej Lasota, Jean Leray, Franciszek Leja, Czechoslovakia, France, Greece, Japan, Poland, Portugal, Serbia, Stanis³aw £ojasiewicz, Edward Marczewski, Stanis³aw Mazur, United States of America, Great Britain and Italy. Among the Jan Mikusiñski, Julian Musielak, Jerzy Sp³awa-Neyman, members of the Executive Committee of the International Witold Nowacki, W³adys³aw Orlicz, Franciszek Otto, Mathematical Union were Kazimierz Kuratowski (1959-1962), a Aleksander Pe³czyñski, Helena Rasiowa, Marian Rejewski, patron of one of PTM’s prizes, and Czes³aw Olech (1979-1982, Edward S¹siada, Wac³aw Sierpiñski, Hugo Steinhaus, Roman 1983-1986). In the years 1963-66, K. Kuratowski acted as Vice- Sikorski, Stefan Straszewicz, W³adys³aw Œlebodziñski, Andrzej President of the Union. Turowicz, Eustachy Tarnawski, Kazimierz Urbanik, Antoni The Polish Mathematical Society sent its own delegations to Wakulicz, Tadeusz Wa¿ewski, Lech W³odarski, Zygmunt 4 international congresses organized by the International Zahorski, Antoni Zygmund. Mathematical Union (Stockholm in 1962, Moscow in 1966, Nice in 1970 and Helsinki in 1978), as well as to 16 scientific conferences in the years 1950-75. In 1983, Poland was the organizer of the International Mathematical Congress in Warsaw. Due to the Martial Law imposed in 1981, the Congress, initially planned to take place in 1982, could only take place as late as 1983. The Organizing Committee was presided by Czes³aw Olech, who was also elected President of the Congress. The Polish Mathematical Society has also been cooperating with the American Mathematical Society (AMS) and the Canadian Mathematical Society (CMS) on the basis of a reci- PTM’s presidents procity agreement. (1946 – 1977) The Polish Mathematical Society is also one of the founder members of the European Mathematical Society (EMS), called Presidents into being in December 1990 during a meeting held in the The function of President of the Polish Mathematical Society Polish Academy of Sciences conference centre in M¹dralin near was performed by: Stanis³aw Zaremba (1919-21, 1936-37), Warsaw. EMS was established thanks to the initiative of about Wiktor Staniewicz (1921-23), Samuel Dickstein (1923-26), 30 mathematical societies representing nearly all European coun- Zdzis³aw Krygowski (1926-28), Wac³aw Sierpiñski (1928-30), tries. Kazimierz Bartel (1930-32), Stefan Mazurkiewicz (1932-36, During the founding meeting, the Polish Mathematical Society 1937-39), Stefan Banach (1939-45), Karol Borsuk (1946), was represented by Bogdan Bojarski (then director of the Kazimierz Kuratowski (1946-53), Stefan Straszewicz (1953- Mathematical Institute of the Polish Academy of Sciences) and 57), Edward Marczewski (1957-59), Tadeusz Wa¿ewski (1959- Andrzej Pelczar, PTM’s President in the years 1987-1991. One 61), W³adys³aw Œlebodziñski (1961-63), Franciszek Leja of EMS’s first Vice-Presidents, elected for the term ending in (1963-65), Roman Sikorski (1965-77), W³adys³aw Orlicz 1992, was Czes³aw Olech. During the term 1992-96, Andrzej (1977-79), Jacek Szarski (1979-81), Zbigniew Ciesielski (1981- Pelczar acted as a member of the Executive Committee of EMS 83), Wies³aw ¯elazko (1983-85), Stanis³aw Balcerzyk (1985- and later, in the years 1997-2000, as EMS’s Vice-President. 87), Andrzej Pelczar (1987-91), Julian Musielak (1991-93), Another organ of EMS’s authority is the Society’s Council, Kazimierz Goebel (1993-99), Boles³aw Szafirski (1999-2003), where the Polish Mathematical Society holds a two-person rep- Zbigniew Palka (2003- ). resentation. During the term 1999-2002, its representatives were Julian Musielak and Andrzej Pelczar. Kazimierz Goebel and skich [More than 50 years of activities of Polish Zbigniew Palka have been elected for the term 2002-2006. Mathematicians], Pañstwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warszawa 1975. This elaboration has been written on the basis of an article by [4] J. Piórek, From the minutes of the Mathematical Society Tadeusz Iwiñski [2] in Cracow, Newsletter of the European Mathematical Society 32 (1999). References [1] K.Ciesielski, Z.Pogoda, Interview with Marek Kordos, Janusz Kowalski [[email protected]] graduated in math- Newsletter of the European Mathematical Society 41 (2001). ematics at Warsaw University in 1969. He currently works at [2] T.Iwiñski, Polskie Towarzystwo Matematyczne [The Polish the £azarski Law and Trade University in Warsaw. For many Mathematical Society] in: S³ownik polskich towarzystw years he taught at secondary schools in Warsaw. Since 1988 naukowych [Dictionary of Polish scientific societies], he has been the Vice-Secretary of the Polish Mathematical Warszawa 1975. Society. He is also a member of the Polish Mathematical [3] T.Iwiñski, Ponad pó³ wieku dzia³alnoœci matematyków pol- Society Commission of Teaching and Popularization. EMS December 2004 29 ERCOM

ERCOM: ICMS Edinburgh mathematics that are relevant to other The inaugural event of Edinburgh’s sciences , industry and commerce International Centre for Mathematical • Promote international cooperation with Sciences (ICMS) was a meeting on particular reference to mathematicians Geometry and Physics held at Newbattle working in developing countries Abbey near Edinburgh in March 1991. Alain Connes, Roger Penrose and Simon As a joint venture of its two founding Donaldson were among the speakers and universities, ICMS has, under the leader- Jacques-Louis Lions (Collège de France) ship of Angus MacIntyre, its first was chair of the ICMS Programme Scientific Director, established a strong Committee. reputation for running high quality inter- ICMS had been founded by a partner- national workshops across the full range ship that involved Edinburgh and Heriot- of mathematical sciences and for bring- Watt universities, Edinburgh City ing the best mathematicians in the world Council, and the International Centre for to Edinburgh and the UK. This record is Theoretical Physics in Trieste, with sup- all the more remarkable because it has, port from some of Edinburgh’s financial until now, been achieved in the absence institutions. An important figure during of systematic core funding from govern- those early days of ICMS was John Ball, ment agencies, using only funds raised currently president of the International through the initiative of organisers who Mathematical Union. The Nobel Prize brought their projects to Edinburgh. winner Abdus Salam also took a keen interest in its development. Editorial services, workshops and In 1994 ICMS moved to its current meetings premises, 14 India Street, which is the To compensate for the lack of core fund- house in Edinburgh’s New Town where ing for its activities, ICMS has been the celebrated physicist James Clerk developing a modest commercial arm by Natural and Environmental, and the Maxwell was born on 13 June 1831. offering editorial services to journals and Engineering and Physical Sciences, management services to research agen- Research Councils. Purpose of ICMS cies. Currently, editorial work for the Determined that ICMS should succeed From the outset the purpose of ICMS Proceedings of the Royal Society of with its mission despite limited means, was to Edinburgh and Proceedings of the over the past decade a small but highly • Create an environment in which the Edinburgh Mathematical Society are committed and resourceful staff has pro- mathematical sciences will develop in done at ICMS headquarters, and ICMS duced a range of meetings of outstanding new directions manages the Environmental Mathematics quality across the entire range of the • Encourage and exploit those areas of and Statistics Workshops on behalf of the Mathematical Sciences, from New Mathematical Developments in Fluid Mechanics (1995) organised by Constantin, P-L Lions and Majda to the recent celebration of the Centenary of Sir William Hodge who was born in Edinburgh in 1903. This meeting, which was organised by Atiyah, Bloch, Donaldson, Griffiths and Witten, with members of Hodge’s immediate family attending the historical lectures and the banquet, was described as ‘stellar’ by one distinguished participant. Details of meetings held since 1995 are available at http://www.icms.org.uk/previous/index. html. In addition to research workshop activity ICMS has played a full part in the mathematical life of Scotland and of the UK. On the one hand it had a pivotal role in the organisation of ICIAM which Delegates at Geometry and Physics, Newbattle Abbey, 1991 brought over 2000 delegates to 30 EMS December 2004 ERCOM Toland (University of Bath), and the Programme Committee are now solicit- ing high quality adventurous proposals in all branches of the mathematical sciences and scientific areas with significant mathematical content and, to be fair to inter-disciplinary proposals, will find ways to counter the inherent conser- vatism of the peer review process. They will also continue to look for ways to support participants from developing countries, an important part of the ICMS mission since it was founded, but still difficult because of a lack of available funding. Within the United Kingdom the role of ICMS in promoting and supporting short workshop activity in Edinburgh comple- ments that of the Isaac Newton Institute in Cambridge where support for researchers in residence in Cambridge is the key to the success of its long-term research programmes. In fact the two institutes are developing a good understanding and a close working relation- ship. New Mathematical Developments in Fluid Mechanics, Edinburgh, 1995 - part of the UNESCO 50th anniversary celebrations. Institutional Structure ICMS is a joint venture of Heriot-Watt Edinburgh in 1999 and, on a totally dif- the lecture-room facilities and the intro- and Edinburgh universities, is an institu- ferent scale, it runs workshops and sup- duction of new break-out rooms and tional member of the European port meetings for research students in computer equipment. Workshop organis- Mathematical Society and is represented Scottish universities. It also encourages ers and participants will still benefit from on European Research Centres on workshops on industrial problems from the experience of the Centre Manager, Mathematics (ERCOM). The scientific the local region. Most recently, in 2004 Tracey Dart, who will now be able to programme is controlled by the ICMS joined with the Faculty of designate one of the ICMS staff as a Programme Committee with membership Actuaries to set up a major meeting for Conference Coordinator to each work- drawn from across the UK and abroad. senior professional actuaries on the prob- shop. ICMS will continue to offer edito- Resources from EPSRC, SHEFC, Heriot- lems currently facing the pensions indus- rial and managerial services to the acad- Watt and Edinburgh Universities, and try in the UK and worldwide. emic community. elsewhere are allocated to individual With the new arrangements in place workshops by a Management Committee The Future the ICMS Scientific Director, John of the two universities, following advice With its strong record of past achievement on a limited budget, in November 2004 ICMS and its two founding universities entered a partnership with the Scottish Higher Education Funding Council (SHEFC) and the UK’s Engineering and Physical Sciences Research Council (EPSRC) with the aim of having guaranteed funds for workshops from now until 2008/9. During that period the ICMS Programme Committee, under the chairmanship of Jerry Bona (University of Illinois at Chicago), will have at its disposal funds for about eight international workshops per year, each with about 40 participants, and at the same time will continue to solicit proposals for workshops of various sizes from organisers who can make other funding arrangements. The new arrangement will enable ICMS to improve its facilities and very soon past Distinguished participants at the Hodge Centenary Conference, Edinburgh, 2003, including visitors to India Street will see changes, past and current scientific directors Angus MacIntyre (bottom row, second left) and John with new decoration, modernisation of Toland (middle row, third left). EMS December 2004 31 ERCOM

Call for Proposals Proposals are invited for workshops to be held at ICMS in Edinburgh in 2005/6. The International Centre for Mathematical Sciences (ICMS) is based in central Edinburgh, in the birthplace of James Clerk Maxwell. Following new funding arrangements with EPSRC for the period 2005 to 2008 ICMS is able to offer support to run workshops and symposia on all aspects of the mathematical sciences in new or traditional subjects and interdisciplinary areas with significant mathematical content. The core of ICMS activity will be the rapid-reaction research workshop programme (R3WP). ICMS therefore particularly welcomes proposals for workshops in rapidly-developing and newly-emerging areas where there is a need to eval- uate new developments quickly. ICMS will respond quickly to such proposals. Organisers can expect preliminary comments from reviewers in 8 weeks. Decisions will be made by the Programme Committee four times a year (December, March, May and September). Small meetings can be organised in 6-8 months from acceptance. Potential organisers should contact ICMS as early as possible to discuss ideas, before submitting a firm proposal. The proposal document should not normally exceed five pages and should be submitted electronically (PDF, PS, Word or DVI). Proposals may be submitted at any time. Full instructions on how to submit a proposal, together with details of the refereeing process and criteria for selection can be found on the webpages: http://www.icms.org.uk/call/index.html Anyone unable to read these pages or download documents can order print versions from ICMS. If your application is successful, you will be offered a funding package to contribute to the travel and subsistence of a proportion of the participants. ICMS staff will undertake all non-scientific administration connected with the workshop (such as issuing invitations, processing registrations, organising accommodation, preparing material, financial administration). One of the Scientific Organisers (often an author of the initial proposal) will be appointed Principal Organiser and be the main point of contact. For all enquiries about ICMS or the procedures for submitting a proposal, please contact Tracey Dart, Centre Manager, ICMS, E-mail [email protected] (14 India Street, Edinburgh EH3 6EZ, Tel +44 (0)131 220 1777; Fax +44 (0)131 220 1053). from the Programme Committee. ICMS is advised by the ICMS Board, whose membership is drawn from organisations with an interest in ICMS, such as the Ramanujan Prize Edinburgh and London Mathematical Societies, the Royal Society of for Young Mathematicians Edinburgh, ICTP, and the City of from Developing Countries Edinburgh. The Abdus Salam International Centre for Theoretical Physics (ICTP) is International Centre for Mathematical pleased to announce the creation of the Ramanujan Prize for young mathe- Sciences 14 India Street, Edinburgh EH3 6EZ maticians from developing countries. The Prize is funded by the Niels Henrik Abel Memorial Fund. Scientific Director & Chair of the Management Committee: The Prize will be awarded annually to a researcher from a developing country John Toland (University of Bath) less than 45 years of age at the time of the award, who has conducted outstanding research in a developing country. Researchers working in any branch Programme Committee Chair: of the mathematical sciences are eligible. The Prize carries a $10,000 cash Jerry Bona (University of Illinois at award and travel and subsistence allowance to visit ICTP for a meeting where Chicago) the Prize winner will be required to deliver a lecture. The Prize will usually be Management Committee Deputy Chairs: awarded to one person, but may be shared equally among recipients who have Chris Eilbeck (Heriot-Watt University) contributed to the same body of work. Tony Carbery (Edinburgh University) The Prize will be awarded by ICTP through a selection committee of five emi- Centre Manager: nent mathematicians appointed in conjunction with the International Tracey Dart (ICMS) Mathematical Union (IMU). The first winner will be announced in 2005. The The classical Georgian house in central deadline for receipt of nominations is July 31, 2005. Edinburgh that is the headquarters of ICMS has proved an attractive venue for Please send nominations to [email protected] describing the work of the nomi- workshops and with the new funding nee in adequate detail. Two supporting letters should also be arranged. arrangements will, it is hoped, continue to be so far in the future. 32 EMS December 2004 PROBLEM CORNER PPrroblemoblem CornerCorner ContestsContests frfromom BulgariaBulgaria PPartart IVIV Paul Jainta

Unquestionably, the peak of most sports solve three questions within four hours. the International Olympiad afterwards. events culminates in a grand finale. And All problems are compiled and worked out Bulgaria is one of the founder countries it’s often been this way with intellectual by the Regional Inspectorates of of the International Mathematics challenges, say, maths contests. Finalists Education while the present teachers are Olympiad (IMO) and has participated in in these meetings have to clear the hurdles asked to correct the examination papers of all its 42 editions. Only two other coun- of divers selection criteria and/or come their own charges. About 30 per cent of all tries can look back to a similar constance: through several qualifyings, i.e. local and participants in this round will pass to the Romania and the Czech Republic. regional qualifying rounds bearing partic- second stage, the regional round. Students Bulgarian students have won 32 gold, 73 ularly exotic names, and observe specific of different grades are gathered in region- silver and 81 bronze medals altogether. rules on the whole. For example, in al centres and have to solve three problems During the last 10 years, Bulgaria has Bulgaria they are trading under unusual within four hours again. This time, the ranked among the top ten countries with names such as the ‘Virgil Krumov’ contest responsibility for the set of questions lies best performance, and regarding the last 4 or the ‘Chernorizets Hrabar Tournament’ in the hands of the National Olympiad years, Bulgaria has even moved into the and so on . Commission and the examination papers best five. France, Germany, England, Italy The last three issues of Problem Corner are marked under the supervision of and other countries that are world centres have dealt with the wealth of different regional inspectors. The third round, or the of Mathematics, have much lower rank- competitions. Here, all roads lead to the National round, is reserved for students ings. In 1998 in Taiwan, Bulgaria was sec- national highlight for mathematically able from grades 8 to 12. In former times there ond, in 1999 and in 2000 - fifth, in 2001 youngsters, the National Mathematics was an additional round for 7-graders, and with the participation of exactly 83 coun- Olympiad (NMO). But it would go beyond these results were partially used as a per- tries - third. The International Jury awards the scope of this Corner to enumerate the mit to enter a Mathematics-, Foreign exceptional prizes to students with extra- whole plethora of contests that Bulgaria Language-, or Technical School. Alas, this ordinary achievements. Since 1987, only offers to its adolescents in this field. For, tradition does not exist any more because two prizes have been awarded. Both of in spite of this diversity, the separate of a complete restructuring of the veteran these were addressed to Bulgarians: in maths trials are distinguishing from one educational system, which is still going Australia in 1988 and in Canada in 1985. another only by nuances. The National on. The third round is a two-day event like The recent winner of a special prize was Bulgarian Mathematics Olympiad repre- the International Olympiad. Students are Nikolay Valeriev, who represents a real sents a country-wide showdown for native asked to solve three problems each day phenomenon of the IMO: from four partic- pupils to decorate themselves with the within 4 or 5 hours. The National ipations he has won three gold and one sil- unofficial title of champion of maths, as Commission in Mathematics creates the ver medal. During the long history of the reported by Prof. Sava Grozdev, Institute problems and is also responsible for mark- IMO, the mother of all maths contests of Mechanics, Bulgarian Academy of ing the examination papers. The coordina- worldwide, there have been only five other Sciences, Sofia. Here comes his final part tion of the results is carried out in the pres- participants with comparable achieve- of a long story that describes the efforts ence of both teachers and students. This ments. Four of them were before 1974, made in his country towards getting young procedure is a fully objective, fair and thus when the number of participating coun- people really interested in mathematics. democratic element of the construct, tries was limited. The fifth was Ride called BNMO. Usually, the number of par- Burton from the USA team, who obtained The National Mathematics Olympiad ticipants in the third round is about 100, four gold medals out of four and maintains The Bulgarian National Mathematics with a growing tendency to drop further. a world record in this branch until today. Olympiad (BNMO) originates in 1949 and For instance, in May 2002 the total num- Nowadays, the Bulgarian student was actually the first competition orga- ber of students in the third round of the Alexander Lishkov has the possibility to nized in Bulgaria. Because of a two year National Olympiad reached a minimum of improve this mark, since he has one silver interruption during the period 1957-59, the 38 ‘survivors’. The champions of the final medal and four more participations to 53rd National Olympiad took place in round are awarded the possibility to study come through. May, 2004. Initially, only students from Mathematics at a Bulgarian University of grades 8 to 11 were permitted to partici- their choice, without passing entrance What is expected from those who have pate in the Olympiad. At present the con- examinations. The universities in Bulgaria worked their way from a large field of test is open to grades 4 to 12. The BNMO (about 40) follow an autonomous policy, starters up to the finale can best be learnt is run in 3 rounds. Naturally, the first which includes such individual tests, but by an examination of the problems posed round, better known as ‘school’ round, is they respect the results achieved at the in my recent Corner. The new set is a conquered by participants. About ten years National Olympiad, and thus are acknowl- bundle of questions which were used ago, the number of starters evened out at edging high level talents. Besides this, the while preparing the Bulgarian team for 150 000 individuals, almost 10 per cent of 12 students with the highest scores in the participating in the International all participating students. Currently, the third round are invited for further selection Mathematical Olympiad, held in Glasgow, initial number is reduced to about 20 000. each year. The selection includes two two- Scotland, July 17 - 30, 2002. All problems One of the reasons for this decrease is a day tests consisting of six problems in all. are original and are worked out by mem- reduction in the total number of students When the six constituents of the National bers of the Team for Extra Curricula due to a strongly abating birth-rate. team have finally been identified, they Research in cooperation with the Union of During the first round students have to have to undergo a fortnight preparation for Bulgarian mathematicians. EMS December 2004 33 PROBLEM CORNER

164 Strange animals live in a building with n t 3 floors. The roof is considered the (n+1)st floor. Exactly one animal is living on the first floor, while one animal at most is living on each of the other floors. Within a month curious things will happen: Exactly once a month each animal gives birth to a new animal. Immedi- ately after his birth the new animal moves to the nearest upper floor and remains there if the floor is unoccu- pied by another animal. If not, the newborn animal eats the previous inhabitant and walks on to the next upper floor repeating the same procedure. But a giant is dwelling on the roof and if an animal enters the roof, it will be eaten by the giant. The same things will happen the next month and so on. It is known that all the animals give birth to new animals simultaneously. What is the smallest number of months, after which the building will be settled the same way as it was at the beginning?

165 Consider a regular polygon with 2002 vertices and all its sides and diagonals. How many different ways can you choose some of them, such that they form the longest continuous (= connected) route?

a 166 Given the sequence: xxxxn ,1,(1) 2 t, where a is a positive integer. Prove that x 112 nnn n is irrational for every n t 3 .

167 All points on the sides of an acute triangle ABC are coloured white, green or red. Prove that there exists 3 points of the same colour, which form the vertices of a right triangle, or rather there exists 3 points of different colours, which are vertices of a triangle similar to the given one.

3 2 168 Given is a sequence of polynomials: P1(x) = x, P2(x) = 4x +3x, …, Pn+2(x)= (4x +2)·Pn+1(x) - Pn(x), n 1. Prove that there are no positive integers k, l and m, such that Pk(m) = Pl(m+4).

169 Find all continuous functions f: RĺR, such that f satisfies the functional equation: fx(())() fx fx for all real x.

It remains for me to present solutions to questions 152 to 157, published in Issue 49 of the Corner. All problems come from Hungarian sources, which stand for high quality of course.

152 The first four terms of an arithmetic progression of integers are a1, a2, a3, a4. 2 2 2 2 Show that 1ǜa1 + 2ǜ a2 + 3ǜ a3 + 4ǜ a4 can be expressed as the sum of two perfect squares.

Solution by J.N. Lillington, Wareham, UK .

2 2 2 2 Let a1 = a, a2 = a+d, a3 = a+2d, a4 = a+3d, where a,d are integers. Then 1ǜa1 + 2ǜ(a+d) + 3ǜ(a+2d) + 4ǜ(a+3d) = a2 +2a2 +4ad+2d2+3a2+12ad+12d2+4a2+24ad+36d2 = 10a2+40ad+50d2 = a2+(3a)2+40ad+d2+(7d)2 = (3a+7d)2 + (a-d)2 . Also solved by Niels Bejlegaard, Copenhagen, Denmark; Pierre Bornsztein, Maisons-Laffitte, France; Erich N. Gulliver, Schwäbisch-Hall, Germany; Gerald A. Heuer, Concordia College, Moorhead, MN, USA, and Dr Z Reut, London, UK.

153 Is it possible to get equal results if 102 nn 10 and 102 nn 10 1 are rounded to the nearest interger? (n is a positive interger).

Solution by Gerald A. Heuer, Concordia College, Moorhead, MN, USA .

1 No. From the fact that 1022nn 10 10 nn 10 10 2 nn 10 1 it follows that 4 1 1022nn 10 10 n 10 nn 10 1 . Therefore the nearest integer to 102nn 10 is at most 10n-1, 2 while the nearest integer to 102nn 10 1 is 10n. Also solved by Niels Bejlegaard, Pierre Bornsztein, E.N. Gulliver, J.N. Lillington, and Dr Z Reut, London.

34 EMS December 2004 PROBLEM CORNER

154 An Aztec pyramid is a square-based right truncated pyramid. The length of the base edges is 81 m, the top edges are 16 m and the lateral edges 65 m long. A tourist access is designed to start at a base vertex and to rise at a uniform rate along all four lateral faces, ending at a corner of the top square. At what points should the path cross the lateral edges?

Solution by J.N.Lillington. K (Ed. We refer to the opposite figure. It can easily be seen that the lateral faces of the truncated 0 pyramid form trapezoids with 60 base angles. Line segments CD, EF, GH, IJ are drawn parallel I 16 J to base AB.) G First, we show that triangle ABK is equilateral. Triangles ABK and IJK are similar according H KJ KJ 65 E F to the preface, thus (because the lateral edges measure 65 m) or KJ = 16, 16 81 C D which gives the result. 0 Let the tourist walk be the path ADCFEHGJI crossing the right lateral edge at D,F,H. K 60 A 81 B 81 AD Applying the sine rule on triangles ABD, and ACD gives: and sin 12000 G sin 60 AD CD or (because of sin1200 = sin600 and sin(1200-G ) = sin[1800-(1200-G ) = sin(600+G )) sin12000 sin(60 G )

0 81 CD sin 60 G this leads to or 81 CD . sin 6000GG sin 60 sin 600 G

4 §·sin 600 G Then applying successively to triangles CDF, CEF, EFH, EGH, GHJ and GIJ we get 81 16 ¨¸0 ©¹sin 60 G 33tan G or which finally simplifies to 5tanG 3. 2 3tan G DB 81 81 Applying the sine rule on triangle ABD gives or DB . Hence DB = 27 sinGG sin 600 31 cot G 22 and applying successively we yield DF = 18, FH = 12, and HJ = 8. Also solved by Niels Bejlegaard.

155 The billposters of the Mathematician’s Party observed that people read the posters standing 3 meters away from the centres of the cylindrical advertising pillars that have a 1.5 m diameter. The Party wants to achieve that, after sticking the posters around a pillar, a whole poster will be visible from any direction. How wide should the posters be?

Solution by Niels Bejlegaard, Copenhagen, slightly revised by the editor. (Ed. A person standing at M, a distance of 3 meters away from the centre, is able to see the part of the cylindrical advertising pillar that is bounded by two tangent planes drawn to it. Clearly, it will be sufficient to view a circle as a cross-section of the pillar). According to the figure a simple calculation shows: 3.7522 tanM 15 or M | 75.52o . The length of a bill along the cylinder is s = .75 2 75.52o arctan 15 § 1.98 m (or s |.75 2S ). 3600 Obviously two equal bills are not sufficient to be viewed completely because the visual angle would be 0o then. s If we denote the width of the posters by w, then we must have w . Since we are looking for the maximal 2 width, we may assume the posters to tile the pillar, i.e. the circumference of the pillar is an integer multiple of .75 2S s the poster width. If we have, say, n posters, then d must hold, from which we get n > 4.767 … . So 52 .75 2S the smallest possible value of n is 5, and the poster width must be w = | .942 m. 5 Also solved by J.N. Lillington.

EMS December 2004 35 PROBLEM CORNER

156 Find a simpler expression for the sum S = 3 + 2·32 + 3·33 + … + 100·3100.

Solution by Dr Z Reut, London. 100 100 100 100 The sum can be written as Sn ¦¦3nnnn ( n 1 1)3 ¦ ( n 1)3 ¦ 3. The first term can now nn 11 n 1 n 1 100 100 99 100 be written as follows: ¦¦(nnnn 1) 3nnnn 3 ( 1) 31100 3 ¦¦ 3 3 ( 3 100 3 ) nn 11 nn 11 = 3·(S - 100·3100). The second term is a geometric series, which is reduced as follows: 100 313100 3 ¦33n 3100 1. The first equation becomes SS 3 100 3100 3 100 1 . Solv- n 1 31 2 2

Also solved by Pierre Bornsztein, Niels Bejlegaard, Erich N. Gulliver, Gerald A. Heuer, and J.N. Lillington.

157 Given are two non-negative numbers x,y, that satisfy the inequality x3 + y4 x2 + y3. Prove that x3 + y3 2.

Solution by Pierre Bornsztein. First, the result is trivially true for x = y = 0. Thus, we assume (x,y) (0,0). Suppose, for a contradiction, that x3 + y3 > 2.

22 33 xy x y The inequality between means of order 2 and order 3 gives d 3 . 2 2 222121 1 Thus, x2+y2 x33yxyxyxyxy33332233 33 33 33 33. It follows that x2 – x3 < y3 – y2 . On the other hand we have y3-y2 y4-y3 0 y2·(y-1)2, and this inequality is generally true. It follows that x2-x3 < y3-y2 y4-y3, which contradicts the statement of the problem. Thus, x3 + y3 2, as desired. Also solved by Niels Bejlegaard and J.N. Lillington.

(Ed. Marcel G. de Bruin, Faculty of Electrical Engineering, Mathematics and Computer Science, Department of Applied Mathematics, Delft, The Netherlands, has proposed an improvement of the outcome of problem 147, given in the NewsletterNo. 51. The question reads as follows: 11 1 Given 100 positive integers x1, x2, …, x100, such that ... 20 , prove that at least two of xx12 x 100 the integers must be equal. A simple sharpening of the integral-test allows one to establish a better estimation, for the following inequality 1 n 1 2 dx can be easily shown: 1 , n \0.^ ` Assuming x1, … ,xm to be distinct positive integers, we find ³n nx2

mm 1 m 1 112 dx 1 d 1 22.m The ‘best’ integer m follows easily from 22m ¦¦³ 2 nk 11xnxn 2 2 20, i.e. m = 114.)

That completes this issue of the Corner. Send me your nice solutions and generalizations.

36 EMS December 2004 (Warwick), Richard Thomas (London) and Ulrike Tillmann (Oxford).

PersonalPersonal columncolumn Amongst those elected to Fellowship Please send information on mathematical awards and deaths to the editor. of the Royal Society in May 2004 were: Samson Abramsky (Oxford), Julian Besag (Seattle), David Epstein (Warwick) and David Preiss Gabriele Veneziano (CERN) has been (London). Awards awarded the Dannie Heineman Prize for Mathematical Physics for his pio- Björn Sandstede (Surrey) has been neering work in dual resonance mod- awarded a five-year Royal Society Alain Connes (IHES) was awarded els. Wolfson Research Merit Award for the Gold Medal 2004 of the French research in coherent structures in sci- CNRS for the creation of non-com- Frank Allgöwer (Stuttgart) has been ence and biology: dynamics, stability mutative geometry, revolutionizing the awarded a 2004 Leibniz Prize by the and interaction. theory of operator algebras, and for Deutsche Forschungsgemeinschaft for his contributions to the mathematical his work in nonlinear systems and David Acheson (Oxford) has been problems arising from quantum control theory. awarded a National Teaching physics and relativity theory. Fellowship. Jerzy Jezierski (Warsaw) and Pierre Deligne (Princeton) has been Wac³aw Marzantowicz (Poznañ) were Ben Green (Cambridge) was present- awarded the 2004 Balzan Prize in awarded the Banach Great Prize of ed a 2004 Clay Research Award in Mathematics for major contributions the Polish Mathematical Society for recognition of his work on arithmetic to several important domains of math- their research papers on topology and progressions of primes. ematics. nonlinear analysis. Jens Marklof (Bristol) has received Mikhael L. Gromov (IHES) and Jay Witold Wiês³aw (Wroc³aw) was one of the five 2004 Marie Curie Gould (Courant NYU) received the awarded the Dickstein Great Prize of Excellence Awards of the European 2003-2004 Frederic Essers Nemmers the Polish Mathematical Society for Commission for his work in quantum Prize in Mathematics. his achievements in the history of chaos and number theory (Semi- mathematics and popularization. Classical Correlations in Quantum Gérard Laumont and Bao-Châu Ngô Spectra). (Paris-Sud) were presented a 2004 Dariusz Buraczewski (Wroc³aw) was Clay Research Award in recognition awarded the Kuratowski Prize. of their work on the Fundamental Lemma for unitary groups. Jakub Onufry Wojtaszczyk (Warsaw) Deaths was awarded the first Marcinkiewicz Guy David (Paris-Sud) received the Prize of the Polish Mathematical Ferran Sunyer i Balaguer Prize of Society for students' research papers. We regret to announce the deaths of: the Institut d’ Estudis Catalans, in recognition of his monograph Sir Roger Penrose OM FRS (Oxford) John Chalk (28.6.2004) Singular Sets of Minimizers for the was awarded the De Morgan Medal Mumford-Shah Functional. He was of the London Mathematical Society Janusz Jerzy Charatonik (11.7.2004) also awarded the Prix Servant of the for his wide and original contribu- French Academy of Sciences. tions to mathematical physics. Klaus Doerk (2.7.2004)

Henri Berestycki (Paris) was award- Boris Zilber (Oxford) was awarded Anatoolii Ya. Dorogovtsev ed the Grand Prix Sophie Germain of the Senior Berrick Prize for his paper (22.4.2004) the French Academy of Sciences for “Exponential sums equations and the his fundamental contributions to the Schanuel conjecture”, J. London Günther Hellwig (17.6.2004) analysis of nonlinear partial differen- Math. Soc. (2), 65, (2002). tial equations. Arno Jaeger (24.2.2004) Richard Jozsa (Bristol) was awarded Colette Moeglin received the Prix the Naylor Prize of the LMS for his Olga Alexandrovna Ladyzhenskaya Jaffé of the French Academy of fundamental contributions to quantum (12.1.2004) Sciences. information science. Thomas Maxsein (10.5.2004) Laurent Stolovitch was awarded the Ian Grojnowski (Cambridge) is the Prix Paul Doistau-Emile Bluter of the first recipient of the LMS Fröhlich Jerzy Norwa (14.6.2004) French Academy of Sciences. Prize for his work on problems in representation theory and algebraic Flemming Damhus Pedersen Paul Biran (Tel Aviv) has received geometry. 23.2004) the 2003 Oberwolfach Prize for outstanding research in geometry and The Whitehead Prizes of the LMS Gert Kjærgaard Pedersen (13.4.2004) topology, particularly symplectic and were awarded to Mark Ainsworth algebraic geometry. (Strathclyde), Vladimir Markovic Helmut Pfeiffer (2.7.2004) EMS December 2004 37 CONFERENCES of Mathematics International Programme Committee: L.D. Faddeev, chair; A.B. Zhizhchenko, deputy chair; S.K. Godunov; V.V. Kozlov; A.P. Klemeshev; S.P. ForthcomingForthcoming conferencesconferences Novikov; V.A. Sadovnichii; V.M. Buchstaber (Russia); O. Pironneau (France); F. Hirzebruch; W. Jäger; C. Zenger (Germany); M. Atiyah (UK); L. compiled by Vasile Berinde (Baia Mare, Romania) Karleson (Sweden) Organising Committee: A.P. Klemeshev, chair; S.A. Ishanov, deputy chair; B.N. Chetverushkin, deputy chair; G.M. Fedorov; L.V. Zinin; A.I. Maslov; S.I. Aleshnikov; K.S. Latyshev Please e-mail announcements of European confer- Troitsky, V. Zoller Format: Plenary lectures and working groups. ences, workshops and mathematical meetings of Information: Apart from plenary sessions and working groups, interest to EMS members, to one of the following e-mail: [email protected], the scholars are welcome to make presentations and addresses [email protected] or vasile_berinde@ [email protected] give lectures for students and post-graduates at yahoo.com. Announcements should be written in a Web site: dgfm.math.msu/conf/ KSU departments. style similar to those here, and sent as Microsoft Kaliningrad State University covers the lecturers’ Word files or as text files (but not as TeX input files). 17-21: CERME 4 (Fourth Congress of the Euro- royalties, accommodation costs and per diem. Space permitting, each announcement will appear pean Society for Research in Mathematics Edu- Abstracts: are to be published before the confer- in detail in the next issue of the Newsletter to go to cation), Sant Feliu de Guíxols, Spain ence. press, and thereafter will be briefly noted in each Aim: CERME is a congress designed to foster a Conference fee: 200 €; for students and post-grad- new issue until the meeting takes place, with a ref- communicative spirit. It deliberately and distincti- uates: 100 € erence to the issue in which the detailed announce- vely moves away from research presentations by Deadline: January 10, 2005 ment appeared. individuals towards collaborative group work. Its Information: main feature is a number of thematic groups whose e-mail: [email protected] January 2005 members will work together in a common research area. Researchers wishing to present a paper at the congress should submit the paper to one of these May 2005 6–9: 24th Nordic and 1st Franco-Nordic groups. In addition to the thematic working groups, Congress of Mathematicians, Reykjavik, there will be plenary sessions, including plenary 9-20: The Third Annual Spring Institute on Iceland speakers and debates, poster sessions, and ERME Noncommutative Geometry and Operator alge- Information: e-mail: policy sessions. bras & 20th Shanks Lecture, Vanderbilt [email protected] Programme Committee: A. Arcavi (IL); C. Berg- University, Nashville, Tennessee, USA Web site: http://www.raunvis.hi.is/1Franco sten (S); R. Borromeo Ferri (G); M. Bosch (E); J.- Main speakers: A. Connes (College de France, NordicCongress/ L. Dorier (F); N. Gorgorió (E); B. Jaworski (UK- IHES & Vanderbilt University), V. Jones (UC [For details, see EMS Newsletter 53] N); J. Pedro Ponte (P); H. Steinbring (G); N. Steh- Berkeley), U. Haagerup (Univ. of Southern likova (CH); E. Swoboda (PL); R. Zan (G) Denmark), S. Popa (UCLA) and D. Voiculescu 28–30: 4th Mediterranean Conference on Speakers: Y. Chevallard (F); M. Brown (UK); J.D. (UC Berkeley). Mathematics Education, Palermo-Italy Godino (E); M. Artigue (F); P. Ernest (UK); F. Organisers: A. Connes (director), D. Bisch, B. Information: e-mail: Furinghetti (I); J. Matos (P); C. Tzanakis (GR) Hughes, G. Kasparov and G. Yu. [email protected] Working Groups: The congress is organised aro- Information: Web sites: http://math.unipa.it/~grim/ mediter- und some thematic groups working in parallel. e-mail: [email protected] ranean_05.htm; Over 4 days, groups will have at least 12 hours to Web site: http://www.math.vanderbilt.edu www.cms.org.cy meet and progress their work. Participants are /~ncgoa05 [For details, see EMS Newsletter 53] encouraged to attend and contribute to the work of just one group. 16-18: Conference “Algorithmic Information February 2005 Information: Theory”, University of Vaasa, Finland Web site: http://cerme4.crm.es Theme: Algorithmic Information Theory is an 5–13: Applications of braid groups and braid March 2005 international meeting bringing together mathemati- monodromy, EMS Summer School, Eilat, Israel cians, computer theorists, logicians, mathematical physicists and scientists in related fields to assess Information: 9–April 1: 14th INTERNATIONAL WORK- Web site: www.emis.de/etc/ems-summer-scho- recent development in the Foundations of SHOP ON MATRICES AND STATISTICS ols.html Mathematics and the limits to Computability and (IWMS-2005), Massey University, Albany Computation and to stimulate further research in Campus, Auckland, New Zealand 14-19: Topology, analysis and applications to these and related fields. Information: mathematical physics, Moscow, Russia Topics: Algorithmic Information Theory; Formal Web site: http://iwms2005.massey.ac.nz/ Dedicated to the memory of Prof. Yu.P. Solovyov Systems; Symbolic Computation; Cellular and [For details, see EMS Newsletter 53] (8.10.1944-11.9.2003) Quantum Automata; Operator and Spectral Theory Topics: Geometry and topology, analysis, applica- Main speakers: B. Buchberger (RISC Institute in tions to mathematical physics including the follo- April 2005 Linz); G. Chaitin (IBM Thomas Watson Research wing topics of particular interest of Yu.P. Solovyov: Center, New York); G. Dasgupta (Columbia Univ., algebraic and differential topology, noncommutati- New York); J. Kari (Univ. of Turku); H. Langer 4-8: International conference on Selected ve geometry, elliptic theory, operator algebras, path (Technische Univ. Wien); H. de Snoo (Univ. of integration Problems of Modern Mathematics, dedicated to Groningen); K. Sutner (Carnegie-Mellon Format: Plenary lectures (1.5 hour) and session the 200th anniversary of K.G. Jacobi, and the University) lectures (45 min) 750th anniversary of the Koenigsberg founda- Organising committee: M. Laaksonen (Univ. of Organizing and Programme Committee: V.A. tion Vaasa); M. Linna (Univ. of Vaasa); S. Mäkinen Sadovnichij (chairman), A. Bak, P. Baum, V.V. Aim: Kaliningrad State University hosts an interna- (Vaasa Polytechnic); Jari Töyli (Univ. of Vaasa); Belokurov (vice-chairman), V.M. Buchstaber, D. tional conference, organised under the aegis of the M. Wanne (Univ. of Vaasa), chairman Burghelea, S.M. Gussein-Zade, A.A. Egorov, J. Russian Academy of Sciences (RAS). Leading Program Committee: S. Hassi (Univ. of Vaasa); Eichhorn, A.T. Fomenko (vice-chairman), M. mathematicians from Russia and other countries are V. Keränen (Rovaniemi Polytechnic); C.-G. Karoubi, G.G. Kasparov, R. Krasauskas, A. Lipko- expected to participate. Källman (Vaasa Polytechnic); chairman, M. Linna vski, A.S. Mishchenko (vice-chairman), Th.Yu. Topics: Algebra and Geometry; Mathematical (Univ. of Vaasa); H. Niemi (Vaasa Polytechnic); T. Popelensky, V.V. Sharko, V.M. Tikhomirov, E.V. Physics; Applied Mathematics, Mathematical Vanhanen (Stadia) Modelling and IT; Theoretical Mechanics; History Deadline: for extended abstracts January 31, 2005 38 EMS December 2004 CONFERENCES Information: e-mail: [email protected] (Technion Univ.); S. Galbraith (Royal Holloway School Director: T. Thompson, Dalhousie Univ. Web site: http://www.uwasa.fi/ait05 Univ. of London) Information: Program Co-chairs: R. Cramer; T. Okamoto e-mail: [email protected]; 31-June 3: 4th Kortrijk Conference on Discrete Coordinators: J.L. Villar and C. Padró (Univ. [email protected]; Groups and Geometric Structures, with Politècnica de Catalunya) [email protected] Applidations, Oostende, Belgium Information: Topics: Recent developments concerning all inter- e-mail: [email protected] 17-23: European young statisticians training actions between group theory and geometry includ- Web site: http://www.crm.es/jornadas_ criptologia camp, EMS summer school, Oslo Norway ing geometric group theory, group actions on man- associated with the European Meeting of Statistics ifolds, crystallographic groups and all possible 25–July 2 : Subdivision schemes in geometric (see below). generalizations (affine, polynomial, projective crys- modelling, theory and applications, EMS Information: tallographic groups, almost-crystallographic Summer School, Pontignano, Italy Web site: www.emis.de/etc/ems-summer-scho- groups), discrete subgroups of Lie groups, graphs of Information: ols.html#2005 groups Web site: www.emis.de/etc/ems-summer-scho- Scientific Committee: Y. Félix (Univ. Catholique ols.html 24-28: 25th European Meeting of Statistics, de Louvain), W. Goldman (Maryland, College [For details, see EMS Newsletter 53] Oslo, Norway Park), F. Grunewald (Univ. Düsseldorf), P. Igodt Organizer: University of Oslo and the Norwegian (K.U.Leuven/Kortrijk), K.B. Lee (Oklahoma) 28-July 2: Barcelona Conference on Geometric Computing Center Main Speakers: O. Baues (Univ. Karlsruhe), Y. Group Theory, Centre de Recerca Matemàtica, Format: 8 invited lectures, 24 invited sessions and Benoist (ENS, Paris), M. Bridson (Imperial Campus of the Universitat Autònoma de contributed sessions College, London), B. Farb (Univ. of Chicago), O. Barcelona Invited lecturers: D. Donoho (Stanford), Y. Peres Garcia-Prada (Univ. Autonoma de Madrid), E. Speakers: M. Bestvina (Univ. of Utah); T. Delzant (Berkeley), O. Aalen (Oslo), J. Bertoin (Paris), W. Ghys (Ecole normale Supérieure, Lyon), D. Toledo (Univ. de Strasbourg); G. Levitt (Univ. of Caen); K. Kendall (Warwick), N. Shephard (Oxford), L. (University of Utah) Vogtmann (Cornell Univ.) Davies (Essen), K. Worsley (Mc Gill) Proceedings: will be published as a part of a spe- Programme Committee: N. Brady; J. Burillo; E. Programme Committee: A. van der Vaart (chair, cial issue in Geometriae Dedicata Ventura; J. Burillo, coordinator Amsterdam), Ø. Borgan (Oslo), U. Gather (Dort- Location: Hotel Royal Astrid, Oostende Information: mund), S. Richardson (London), G. Roberts (Lan- Conference fee: 145€ e-mail: [email protected] caster), T. Rolski (Wroc³aw), J. Steif (Göteborg) Deadline: with abstract: April 12, 2005; without: Web site: http://www.crm.es/Geometric Deadlines: for grant application: March 1, 2005; May 1, 2005 GroupTheory for submission of contributed papers and posters: Information: March 31, 2005; for early registration: May 1, e-mail: [email protected] 2005; for registration: June 30, 2005. Web site: http://www.kulak.ac.be/ July 2005 Conference fee: 310€, 280€ (Bernoulli Society of workshop ISI members), 200€ (students) 5-15: Advanced Course on the Geometry of the Information: e-mail: [email protected] June 2005 Word Problem for finitely generated groups, Web site: www.ems2005.no; Centre de Recerca Matemàtica, Campus of the Universitat Autònoma de Barcelona 30-August 6: Groups St. Andrews 2005, 12–24: Foliations 2005, £odz, Poland Speakers: H. Short (Univ. d’Aix-Marseille I); N. University of St. Andrews, St. Andrews, Information: Brady (Oklahoma Univ.); T. Riley (Yale Univ.) Scotland e-mail: [email protected] Coordinator: Josep Burillo Aim: This conference, the seventh in the series of Web site: http://fol2005.math.uni.lodz.pl Registration Fee: 160 EUR Groups St Andrews Conferences, will be organised [For details, see EMS Newsletter 53] Deadline: May 15, 2005 along similar lines to previous events in this series. Grants: The CRM offers a limited number of Speakers: P.J. Cameron (Queen Mary, London); 13–17: Computational Methods and Function grants for registration and accommodation R.I. Grigorchuk (Texas A&M); J.C. Meakin Theory CMFT 2005, Joensuu, Finland addressed to young researchers. The deadline for (Nebraska-Lincoln); A. Seress (Ohio State) Aim: The general theme of the meeting concerns application is April 5, 2005. Programme: The speakers above have kindly various aspects of interaction of complex variables Information: agreed to give short courses of lectures. In addition and scientific computation, including related topics e-mail: [email protected] there will be a programme of one hour invited lec- from function theory, approximation theory and Web site: http://www.crm.es/Word Problem tures and short research presentations. numerical analysis. Another important aspect is to Topics: The conference aims to cover all aspects of promote the creation and maintenance of contacts 17-August 14: Summer School of Atlantic group theory. The short lecture courses are intend- with scientists from diverse cultures. Association for Research in the Mathematical ed to be accessible to postgraduate students, post- Format: Invited one-hour plenary lectures, invited Sciences, Campus of Dalhousie University in doctoral fellows, and researchers in all areas of and contributed session talks, and poster sessions. Halifax, Nova Scotia, Canada group theory. Plenary speakers: L.A. Beardon (Oxford), P. Aim: The Summer School is intended for graduate Location: Mathematical Institute, St. Andrews. Clarkson (Canterbury), H. Farkas (Jerusalem), A. students and promising undergraduate students in Scientific Organising Committee: C. Campbell Fokas (Cambridge), H. Hedenmalm (Stockholm), Mathematics from all parts of the world. Each par- (St. Andrews); N. Gilbert (Heriot-Watt); S. Linton D. Khavinson (Fayetteville), A. Martinez- ticipant is expected to register for two courses. The (St. Andrews); J. O’Connor (St. Andrews); E. Finkelshtein (Almeria), M. Seppälä (Helsinki), N. courses will be: 1. Convexity and Fixed Point Robertson (St. Andrews); N. Ruskuc (St. Trefethen (Oxford) and R. Varga (Kent) Algorithms in Hilbert Space (H. Bauschke, Univ. of Andrews); G. Smith (Bath) Scientific committee: St. Ruscheweyh Guelph); 2. Integral Geometry of Convex Bodies Information: (Würzburg), E. B. Saff (Nashville), O. Martio and Polyhedra (D. Klain, Univ. of Massachusetts at e-mail: [email protected] (Helsinki) and I. Laine (Joensuu) Lowell); 3. The Mathematics of Finance (W. Web site: http://groupsstandrews.org Grants: Limited amount of support available for Runggaldier, Univ. di Padova); 4. Mathematical participants coming from developing countries. Statistics (B. Smith, Dalhousie Univ.) August 2005 Information: e-mail: [email protected] Format: Each course consists of four sixty-minute Web site: http://www.joensuu.fi/cmft lectures and two ninety-minute problem sessions each week, for four weeks; 21-26: SEMT ‘05 (International Symposium on 20-22: Workshop on Mathematical Problems Organizer: The Atlantic Association for Research Elementary Mathematics Teaching), Czech and Techniques in Cryptology, Centre de in the Mathematical Sciences” (Canada) Republic, Charles University in Prague, Faculty Recerca Matemàtica, Campus of the Universitat Location: Campus of Dalhousie University in of Education Autònoma de Barcelona Halifax, Nova Scotia, Canada. Accommodation is Theme: Understanding the environment of the Speakers: A. Lenstra (Eindhoven Univ. of available on Campus mathematics classroom. Technology); R. Cramer (Leiden Univ.); T. Grants: Some financial support in Halifax is possi- Format: Plenary lectures, presentation of papers, Okamoto (NTT Laboratories); H. Krawkzyc ble. workshops and discussion groups. EMS December 2004 39 Information: e-mail: [email protected] Web site: http://www.pedf.cuni.cz/kmdm/ index.htm

September 2005

13-23: Advanced Course: Recent Trends on Combinatorics in the Mathematical Context; Centre de Recerca Matemática, Campus of the Universitat Autònoma de Barcelona Speakers: B. Bollobas (Trinity College Cambridge and Univ. of Memphis); J. Nesetril (Charles Univ. and ITI, Prague) Coordinator: O. Serra Registration Fee: 160 EUR Deadline: July 10, 2005 Grants: The CRM can offer a limited number of grants to young researchers covering the registration fee and/or accommodation. The deadline for application is June 1, 2005. Information: e-mail : [email protected] Web site: http://www.crm.es/Recent Trends

October 2005

11-18: EMS Summer School and Séminaire Européen de Statistique, Statistics in Genetics and Molecular Biology, Warwick, UK Grants: 40 Marie Curie Actions grants available Local Organization: B. Finkenstadt (Warwick) Information: e-mail: [email protected] Web site: http://www2.warwick.ac.uk/fac/ sci/statistics/news/semstat/

17-21: Nonlinear Parabolic Problems, Helsinki Aim: An international program on nonlinear parabolic partial differential equations will be organized in Helsinki, Finland, during Sept-Nov 2005. The program will run at the University of Helsinki and at the Helsinki University of Technology (HUT) and is sponsored by the governmental agency The Academy of Finland. Within the framework of this programme, a conference on nonlinear parabolic problems will be also held in Helsinki. Organizers: H. Amann (amann@math. unizh.ch); J. Taskinen (jari.taskinen@ helsinki.fi); S.-O. Londen ([email protected]) Main topics: Qualitative theory of parabolic equations; Reaction-diffusion systems; Fully nonlinear problems; Free boundary problems; Navier-Stokes equations; Maximal regularity; Degenerate parabolic problems Speakers: M. Chipot (Zurich); Ph. Clement (Delft); J. Escher (Hannover); M. Fila (Bratislava); M. Hieber (Darmstadt); G. Karch (Wroclaw); H. Kozono (Sendai); Ph. Laurencot (Toulouse); J. Lopez-Gomez (Madrid); S. Nazarov (St.Peters- burg); W.-M. Ni (Minnesota); M. Pierre (Rennes); J. Pruess (Halle); P. Quittner (Bratislava); J. Rehberg (Berlin); G. Simonett (Nashville); H.Sohr (Paderborn); V. Solonnikov (St. Petersburg); Ph. Souplet (Versailles); J.L.Vazquez (Madrid); D. Wrzosek (Warszawa); L. Weis (Karlsruhe); E. Yanagida (Sendai) Location: The conference will take place at HUT, located northwest of downtown Helsinki. Information: Web site: http://www.math.helsinki.fi/ research/FMSvisitor0506

40 EMS December 2004 BOOKS but to graduate students as well. (jmil)

H.-J. Baues: The Homotopy Category of Simply RecentRecent booksbooks Connected 4-Manifolds, London Mathematical Society Lecture Note Series 297, Cambridge University Press, Cambridge, 2003, 184 pp., £24,95, ISBN 0-521-53103-9 edited by Ivan Netuka and Vladimír Souèek (Prague) The main aim of this book is to understand the category of simply connected closed topological 4-manifolds and the M. Anderson, V. Katz, R. Wilson: Sherlock Holmes in tains some applications to physics (homogeneous Einstein homotopy classes of mappings between them. It is neces- Babylon and Other Tales of Mathematical History, and Kähler-Einstein metrics, Hamiltonian systems on gen- sary to mention that the problem consisted especially in the Spectrum, The Mathematical Association of America, eralized flag manifolds and homogeneous geodesics). description of the homotopy classes of maps. As an impor- Washington, 2003, 420 pp., $49,95, ISBN 0-88385-546-1 Notions introduced in the book are nicely illustrated by a tant tool the author uses the category CW(2,4). Its objects This excellent book contains 44 articles on the history of lot of examples. The size of the book is very small but it are CW-complexes X with one 0-cell, and then with cells mathematics, which were published in journals of the contains a wealth of interesting material. (vs) only in dimension 2 and 4, and its morphisms are homo- Mathematical Association of America (American topy classes of mappings. Within this category we find the Mathematical Monthly, College Mathematics Journal, M. Audin, A.C. da Silva, E. Lerman: Symplectic above-mentioned manifolds in the sense that every simply Mathematics Magazine, National Mathematics Magazine) Geometry of Integrable Hamiltonian Systems, Advanced connected 4-manifold is homotopy equivalent with such a over the past 100 years. The articles were written by dis- Courses in Mathematics CRM Barcelona, Birkhäuser, CW-complex X with only one 4-cell. The main achieve- tinguished past historians of mathematics (e.g., F. Cajori, J. Basel, 2003, 225 pp., €28, ISBN 3-7643-2167-9 ment of the book is a construction of a purely algebraic cat- L. Coolidge, M. Dehn, D. E. Smith, C. Boyer) as well as The three contributions contained in the book are based on egory, which is equivalent to the category CW(2,4). This, some contemporary ones (including E. Robson, R. lectures given by the authors at the Euro Summer School of course, enables the transform from various topological Creighton Buck, V. Katz). The articles are divided into “Symplectic geometry of integrable Hamitonian systems”, considerations into an algebraic setting. The advantage of four sections (Ancient mathematics, Medieval and renais- held in Barcelona in July 2001. The first paper (by M. such a transformation is obvious. The subject of the book sance mathematics, The seventeenth century, The eigh- Audin) contains a discussion of special Lagrangian sub- is relatively special, and this naturally brings the necessity teenth century) and they cover almost 4000 years (from the manifolds. This notion was studied very intensively during of special procedures and special computations. Therefore ancient Babylonians to the development of mathematics in recent years in connection with integrable systems and, in the reader may have an impression that the text is rather the eighteenth century). The most interesting topics from particular, with string theory and the mirror symmetry. technical. This is not at all true. On the contrary, the read- Babylonian, Greek, Roman, Chinese, Indian as well as Special Lagrangian submanifolds are rare beings, hard to er will obtain very deep information on the structure of rel- European mathematics are included. Each section is pre- construct, and their moduli space is finite dimensional. In evant categories. The text is very clearly written but the ceded by a Foreword, containing comments on historical the paper, explicit examples of special Lagrangian sub- author substantially uses many previous results of his own context, and followed by an Afterword. In some cases, two manifolds are constructed and the moduli space of special as well as many other results. This means that the reading articles on the same topic are included showing the Lagrangian submanifolds of a Calabi-Yau manifold is dis- is not very easy. Nevertheless, the results are so excellent progress in the history of mathematics. The comparison cussed. The second paper (by A. C. da Silva) contains an that they deserve some patience and effort. (jiva) shows that although modern research brings new informa- introduction to toric manifolds using the moment map and tion and new interpretations, the older papers are neither the moment polytope as the main tools. The text has two M. A. Bennet, B. C. Berndt, N. Boston, H. G. Diamond, dated nor obsolete. The book is not a classical textbook of parts. The first part uses a classification of equivalence A. J. Hildebrand, W. Philips, (Eds.): Number Theory the history of mathematics; the topics included do not classes of symplectic toric manifolds, using their moment for the Millennium, I, A. K. Peters, Natick, 2002, 461 pp., cover the whole development of mathematics. The book polytopes, and a computation of homology of symplectic $50, ISBN 1-56881-126-8 can be recommended to everybody interested in the histo- toric manifolds using the Morse theory. The second part M. A. Bennet, B. C. Berndt, N. Boston, H. G. Diamond, ry of mathematics and to anybody who loves mathematics. considers toric manifolds in algebraic geometry and it has A. J. Hildebrand, W. Philips, (Eds.): Number Theory (mnem) a structure parallel to the first part. The third contribution for the Millennium, II, A. K. Peters, Natick, 2002, 447 pp., (by E. Lerman) is devoted to the following problem: Let M $50, ISBN 1-56881-146-2 R. Arratia, A.D. Barbour, S. Tavaré: Logarithmic be the cotangent bundle of an n-dimensional torus without M. A. Bennet, B. C. Berndt, N. Boston, H. G. Diamond, Combinatorial Structures: A Probabilistic Approach, the zero section. Suppose that there is an effective action A. J. Hildebrand, W. Philips, (Eds.): Number Theory EMS Monographs in Mathematics, vol. 1, European of the group G=Rn/Zn on M, preserving the standard sym- for the Millennium, III, A. K. Peters, Natick, 2002, 450 Mathematical Society, Zürich, 2003, 375 pp., €69, ISBN 3- plectic structure on M and commuting with dilations. Is pp., $50, ISBN 1-56881-152-7 03719-000-0 the action of G necessarily free? To answer the question, This three volume collection of papers contains more than Many combinatorial (and other) objects can be decom- contact toric manifolds are introduced, and contact 1300 pages with 72 talks given at the Millennial posed into connected components and one can be interest- moment maps are used together with the Morse theory on Conference on Number Theory, which was held at the ed in numbers and sizes of components in a random object. orbifolds. The three sets of lectures complement each campus of the University of Illinois at Urbana-Champaign In this monograph, strong results are derived for the com- other nicely and the book offers a very useful and system- in May 2000. The papers cover the wide range of contem- ponent frequency spectrum in a rather general probabilistic atic introduction to a modern and interdisciplinary field. porary number theory. The conference was one of the situation, which is determined by two conditions (axioms): (vs) most important international meetings devoted to number the conditioning relation and the logarithmic condition. In theory, framed by 175 talks. Consequently the interested chapter 1, the decomposition of permutations to cycles and C. Bardaro, J. Musielak, G. Vinti: Nonlinear Integral reader finds here not only surveys on the most important the decomposition of integers to primes are compared and Operators and Applications, de Gruyter Series in contributions to number theory and its applications, but in chapter 2, many more examples of combinatorial struc- Nonlinear Analysis and Applications 9, Walter de Gruyter, also surveys on methods and techniques used in this impor- tures and their decompositions are presented. In the Berlin, 2003, 201 pp., €88, ISBN 3-11-017551-7 tant branch of mathematics. The collection offers a remaining eleven chapters, the authors build an abstract The book is devoted to quite general approximation meth- respectable view on contemporary number theory given by probabilistic approach to the problem of estimation of the ods based on convolution operators (both linear and non- prominent number theorists, and therefore could be recom- component frequency spectrum. Techniques and notions linear) and to the study of such operators. In order to devel- mended not only to number theorists but generally to all used include the Wasserstein distance, the Stein method, op the right setting for this investigation, the first two chap- mathematicians interested in various aspects of number the Ewens sampling formula, size biasing, the scale invari- ters contain preliminary results on modular spaces, which theory. (špo) ant Poisson process, the GEM distribution and the Poisson- are a suitable generalization of the Orlicz spaces as well as Dirichlet distribution. In the final chapters, the treatise of spaces with bounded ϕ-variation. The following two S. Bezuglyi, S. Kolyada, Eds.: Topics in Dynamics and becomes technical but the reader is rewarded by strength chapters present some error estimates of approximations in Ergodic Theory, London Mathematical Society Lecture and generality of obtained theorems. (mkl) terms of moduli of continuity. Classical linear convolution Note Series 310, Cambridge University Press, Cambridge, operators are generalized here to nonlinear convolutions 2003, 270 pp., £30, ISBN 0-521-53365-1 A. Arvanitoyeorgos: An Introduction to Lie Groups and with kernels, which either satisfy Lipschitz-type conditions The volume collects some of the mini-courses presented at the Geometry of Homogeneous Spaces, Student or are homogeneous in the generalised sense. Applications the International Conference and US-Ukrainian Workshop Mathematical Library, vol. 22, American Mathematical to the summability problem of a family of functions are “Dynamical Systems and Ergodic Theory”, held in Society, Providence, 2003, 141 pp., $29, ISBN 0-8218- given in Chapter 5. The convergence of approximations in Katsiveli (Crimea, Ukraine) in August 2000. The intro- 2778-2 ϕ-variation is possible only in certain subspaces of func- ductory contribution by A. M. Stepin is devoted to the The main topics treated in this small book are semisimple tions of bounded ϕ-variation. These subspaces are studied memory of V. M. Alexeyev, one of the best lecturers of the Lie groups, homogeneous spaces and their geometry, in Chapter 6. Two basic fixed-point theorems are used in Katsiveli’s school, who died in 1980 at the age of 48. The invariant metrics, symmetric spaces and generalized flag Chapter 7 to obtain solutions of nonlinear convolution paper “Minimal idempotents and ergodic Ramsey theo- manifolds. The book starts with the definition of a Lie equations. There is an important application of interpola- rems”, by V. Bergelson, reviews the construction of the group and its associated Lie algebra, together with a simple tions to the so-called sampling theorem in signal analysis. Stone-Èech compactification β N of N as the Ellis envelop- σ version of Lie theorems. A study of the Killing form leads The classical linear interpolation methods can be replaced ing semigroup of the map = x+1 on N. Using minimal to the notion of a semisimple Lie algebra. To equip Lie by nonlinear ones. Results in this direction for several idempotents, the celebrated van der Waerden theorem on groups with a structure of a Riemannian manifold, bi- classes of signals are shown in the last two chapters. The arithmetic progressions is proved. In “Symbolic dynamics invariant metrics are introduced, together with the associ- book, based mainly on results from the authors, can be con- and topological models in dimensions 1 and 2”, A. de ated connections and expressions for their curvature. sidered as a nonlinear continuation of the classical book of Carvalho and T. Hall present the classical kneading theory Homogeneous spaces are introduced and their Riemannian P. L. Butzer and R. J. Nickel (Fourier Analysis and of unimodal systems and extend it to two-dimensional sys- structure is defined by means of G-invariant metrics. Two Approximation, Academic Press, 1971). The presentation tems like the horseshoe. In “Markov odometers”, A. H. basic classes of examples - symmetric spaces and general- is clear and self-contained so that the book can be recom- Dooley proves that every ergodic non-singular transforma- ized flag manifolds - are classified. The last chapter con- mended not only to researchers in approximation theory tion is orbit equivalent to a Markov odometer on a Bratteli- EMS December 2004 41 BOOKS Vershik system. In “Geometric proofs of Mather’s con- courses given at the special program on authomorphic ciated moduli space of p-divisible groups. As the authors necting and accelerating theorems”, V. Kaloshin treats the forms at the Fields Institute in the spring of 2003. The consider unitary groups of more general signature, they wandering trajectories of exact area preserving twist maps. course by Cogdell is an introduction to standard L-func- have to get around the fact that one can no longer use In “Structural stability in 1D dynamics”, O. Kozlovski tions of automorphic forms on GL(n) and the Rankin- Drinfeld bases to define nice integral models of the rele- treats the structural stability in spaces of smooth and ana- Selberg L-functions on GL(m) x GL(n). It covers the fol- vant moduli spaces. Instead, the authors work with the cor- lytic maps. In “Periodic points of nonexpansive maps: a lowing topics: Whittaker models, local and global func- responding rigid analytic spaces defined by M. Rapoport survey”, B. Lemmas shows that trajectories of nonexpan- tional equations, converse theorems and functorial lifts for and T. Zink. In the first article, L Fargues shows that one sive maps converge to periodic orbits. In “Arithmetic classical groups. The course by Kim is a survey of the can realize the local Langlands correspondence (in the dynamics”, N. Sidorov deals with explicit arithmetic Langlands–Shahidi approach to L-functions via constant supercuspidal case) in the étale cohomology of certain expansions of reals and vectors that have a dynamical terms of Eisenstein series. It culminates in the recent Rapoport–Zink spaces. In the second article, E. Mantovan sense. In “Actions of amenable groups”, B. Weiss gener- proofs of functoriality of the symmetric cube and fourth for series uses the Newton–polygon stratification of the special alizes much of the classical ergodic theory to general GL(2). The course by Ram Murty centers on analytic fibre of the Shimura variety to relate its cohomology to the amenable groups like Zn; in particular he proves the properties of automorphic L-functions and their applica- cohomology of the associated Rapoport–Zink spaces and Shannon-McMillan theorem. (pku) tions to estimates for Hecke (and Laplace) eigenvalues. generalized Igusa varieties. (jnek) This book is a wonderful introduction to the Langlands N. Bourbaki: Elements of Mathematics. Integration I. program. It is heartily recommended to students (and G. N. Frederickson: Dissections: Plane and Fancy, Chapters 1-6, Springer, Berlin, 2004, 472 pp., €99,95, researchers) specializing in number theory and related Cambridge University Press, Cambridge, 2003, 310 pp., ISBN 3-540-41129-1 areas. (jnek) £16,95, ISBN 0-521-57197-9, ISBN 0-521-52582-9 Bourbaki, a collective author in the sixties, wrote the series The book by Frederickson on recreational mathematics is of books ‘Elements of Mathematics’. This is an English J. K. Davidson, K. H. Hunt: Robots and Screw Theory. devoted to the problem of how to cut a square (or triangle translation of the well-known original French edition. It Application of Kinematics and Statistics to Robotics, or hexagon) into the smallest number of pieces and how to contains the first six chapters (of nine) from the part devot- Oxford University Press, Oxford, 2004, 476 pp., £85, ISBN rearrange them into two squares (or triangles or hexagons). ed to integration. As in all books in the series, presentation 0-19-856245-4 The book also deals with others figures, e.g. stars, Maltese of material is abstract, proceeding from general to particu- This monograph can be regarded as a comprehensive text- Crosses, and with solids (polyhedra). Martin Gardner, a lar. Therefore a good knowledge of an undergraduate book on applications of screw theory in a variety of situa- well-known expert in recreational mathematics, can be course seems to be almost necessary. The approach to the tions in mechanics and robotics. Special attention is paid considered the unofficial godfather of the book. Even great integration theory here is functional, based on the notion of to modern applications of screws in the robotics of both mathematicians were interested in dissections. In 1900, a measure as a continuous linear functional on the space of serial and parallel manipulators. A screw is defined either David Hilbert presented the famous wide-ranging list of real, or complex, continuous functions with compact sup- as a pair of a force and a couple (a wrench) or as an infini- twenty-three problems and a third of them dealt with dis- port in a locally compact topological space. The six chap- tesimal space motion (twist). Later on, the connection to sections of polyhedra (the negative result proposed by him ters cover a detailed exposition of results on extension of the vector field of velocities of a spatial motion and to the was proven by M. Dehn within a few months). The read- measures. The chapter on integration of measures includes instantaneous motion is described. The basic properties of er will find solutions to all problems contained at the end the Lebesgue-Fubini theorem, the Lebesgue-Nikodým the- screws are described in chapter 3, including screw of the book. The bibliography is very comprehensive. A orem, and results on disintegration of measures. An essen- (Plücker) coordinates. The next three chapters deal with basic knowledge of high school geometry is sufficient for tial part of the theory of Lp spaces is also covered. The pre- coordinate transformations, screw systems and reciprocity reading the book. Every puzzle fan will like this interest- sentation of vector-valued integration is based on the weak (duality) of twists and wrenches. Properties of serial robot ing and amusing book. (lboc) integral. The text contains full proofs of the stated results, manipulators are studied in chapters 6 and 7, including many exercises, and worthwhile historical notes. It is writ- many examples of commercially used robots. Chapters 7 F. Gesztesy, H. Holden: Soliton Equations and Their ten very carefully to prevent misunderstandings and to and 8 are devoted to parallel robot-manipulators and their Algebro-Geometric Solutions, vol. I: (1+1)-Dimensional make orientation in the text easy. (ph) basic geometric properties, including the Jacobian and its Continuous Models, Cambridge Studies in Advanced computation. Special attention is given to the 3-3 parallel Mathematics 79, Cambridge University Press, Cambridge, D. Cerveau, E. Ghys, N. Sibony, J.-Ch. Yoccoz: manipulator - the octahedral structure. The rest of the book 2003, 505 pp., £65, ISBN 0-521-75307-4 Complex Dynamics and Geometry, SMF/AMS Texts and concentrates on more advanced topics, for instance on The field of completely integrable systems has developed Monographs, vol. 10, American Mathematical Society, manipulators combining serial and parallel structures, enormously in the last decades. The book under review Société Mathématique de France, Providence, 2003, 197 redundant robotic systems and legged vehicles. covers a part of this broad landscape. Its aim is to discuss pp., $59, ISBN 0-8218-3228-X Appendices give some useful formulas from line geometry in detail algebro-geometric solutions of five hierarchies of The book contains four survey papers on different but and basic ideas of the projective representation of screws in integrable nonlinear equations. The presented class of closely related topics in the theory of holomorphic dynam- connection with the projective line geometry. The Study solutions form a natural extension of the classes of soliton ical systems. The first paper (by D. Cerveau) is devoted to representation is mentioned at the very end. The book con- and rational solutions, and can be used to approximate a study of codimension 1-holomorphic foliations. The tains useful historical remarks on the origins of screw the- more general solutions (e.g. almost periodic ones). Basic main tool used here is a reduction of singularities. ory and related topics, references contain not only histori- tools in the description are spectral analysis and basic the- Particular attention is devoted to foliations in dimensions cal sources on the subject but also recent publications rele- ory of compact Riemann surfaces and their theta functions. two and three. The paper ends with applications to the sin- vant to considered problems. The main point of the book The basic KdV hierarchy is the most famous case, it con- gular Frobenius theorem and a discussion of invariant lies in applications of screws and not much space is devot- tains the equation for solitary waves on channels, which hypersurfaces. The second paper (by E. Ghys) contains a ed to the theory. This means that no proofs are given, the were discovered by Scott Russell in 1834. The solutions of discussion of Riemann surface laminations, arising in the algebraic structure of the screw space is not emphasized the KdV hierarchy are discussed in the first chapter. The theory of holomorphic dynamical systems. The Riemann and the style of the book is traditional. It can be recom- discussion is presented in more detail for this first case than surface laminations are more general objects than ordinary mended to all engineers and postgraduate students doing in the other four cases. The second hierarchy treated in foliations, the ambient space need not have a structure of a research in complicated mechanical systems, in particular Chapter 2 is a combined sine-Gordon and modified manifold. Their leaves are (not necessarily compact) in the mechanics of serial and parallel robot manipulators. Korteweg-de Vries hierarchy. The third chapter contains a Riemann surfaces. Classical questions for Riemann sur- (ak) discussion of solutions of the AKNS (Ablowitz, Kaup, faces (uniformization, existence of meromorphic func- Newell, Segur) system and related classical Boussinesq tions) are studied in this more general setting. The paper M. Emerton, M. Kisin: The Riemann-Hilbert hierarchies. The classical massive Thirring system is treat- by N. Sibony describes an analogue of the Fatou-Julia the- Correspondence for Unit F-Crystals, Astérisque 293, ed in Chapter 4. The last chapter describes solutions of the ory for a rational map f from Pk(C) to itself. The dynamics Société Mathématique de France, Paris, 2004, 257 pp., Camassa-Holm hierarchy. Individual chapters are orga- of f are described using properties of a suitable closed pos- €57, ISBN 2-85629-154-6 nized in such a way that they can be read independently. itive current of bidegree (1,1). The second part is devoted If k is a perfect field of characteristic p > 0 and X a smooth To reach this goal, similar arguments in constructions are to a study of regular polynomial biholomorphisms of Ck, Wn(k)-scheme, a well known generalization of the repeated in individual cases. Each chapter ends with and the last part to holomorpic endomorphisms of Pk(C). Artin–Schreier theory (due to N. Katz) establishes an detailed notes (e.g. notes for the first chapter have 17 For the convenience of the reader, properties of currents equivalent of categories between locally free étale sheaves pages) with references to literature, comments and addi- and plurisubharmonic functions are summarized at the end of Z/pnZ –modules on X and vector bundles E on X satis- tional results. In the Appendix (140 pages), it is possible to of the paper. The fourth contribution (by J.-C. Yoccoz, fying F*R ≈ E (where F is a lift of the Frobenius to X). find a summary of many fields (e.g. algebraic curves, theta written by M. Flexor) describes properties of the simplest The main goal of the book is to generalize this result to a functions, the Lagrange interpolation, symmetric func- nontrivial case of holomorphic dynamics, given by a qua- Riemann-Hilbert type correspondence between the derived tions, trace formulae, elliptic functions, spectral measures), b dratic polynomial in one complex variable. The discussion category D ctf (Xet, Z/pnZ) and a certain triangulated cate- which are used in the main chapters. At the end, the read- is centred around hyperbolic aspects, the Jacobsen theorem gory of arithmetic D–modules equipped with an action of er can find an extensive bibliography (30 pages of refer- and quasiperiodic properties, related to problems of small Frobenius. In fact, the case n=1 is treated separately, as it ences). The book is very well organized and carefully writ- divisors. The whole book starts with an introductory paper does not require any differential operators (nor the perfect- ten. It could be particularly useful for analysts wanting to on holomorphic dynamics (written by E. Ghys). It contains ness of k). (jnek) learn new methods coming from algebraic geometry. (vs) many interesting comments on the historical evolution of the discussed topics and their mutual relations. (vs) L. Fargues, E. Mantovan: Variétés de Shimura espaces V. I. Gromak, I. Laine, S. Shimomura: Painlevé de Rapoport-Zink et correspondances de Langlands Differential Equations in the Complex Plane, de Gruyter J. W. Cogdell, H. H. Kim, M. R. Murty: Lectures on locales, Astérisque 291, Société Mathématique de France, Studies in Mathematics 28, Walter de Gruyter, Berlin, Automorphic L-functions, Fields Institute Monographs, Paris, 2004, 331 pp., €66, ISBN 2-85629-150-3 2002, 299 pp., €88, ISBN 3-11-017379-4 American Mathematical Society, Providence, 2004, 283 This volume contains two articles that generalize some At the beginning of the 20th century, Painlevé studied pp., $63, ISBN 0-8218-3516-5 aspects of the recent work of M. Harris and R. Taylor on properties of second order differential equations in the This book consists of lecture notes from three graduate cohomology of certain unitary Shimura varieties and asso- complex plane and isolated a certain number of equations 42 EMS December 2004 BOOKS with distinguished behaviour. Their solutions have the excellent introduction to the more advanced monograph ter is devoted to the basics of wavelet theory. It includes property that there are no movable singularities other than written by the same authors (Introduction to the Modern the continuous as well as the discrete wavelet transform poles. Later on, attention has concentrated to six equations Theory of Dynamical Systems, Encyclopaedia of mathe- both in one- and in two-dimensional cases. A special P1, …, P6 of Painlevé type with the most interesting prop- matics and its applications, vol. 54, Cambridge University emphasis is paid to orthogonal wavelets with finite support, erties. The book is devoted to them. In the first chapter, Press, 1995). (jmil) because they play an important role in many applications. the authors show that the equations P1, P2, P4 and modifica- In the last chapter, some applications of wavelets in geo- tions of equations P3 and P5 have meromorphic solutions R. M. Heiberger, B. Holland: Statistical Analysis and sciences are reviewed. The book has developed from a only (with proofs in the first three cases). The growth Data Display, Springer Texts in Statistics, Springer, New graduate course held at the University of Calgary and is properties of solutions of these 5 types of equations togeth- York, 2004, 729 pp., 200 fig., €79,95, ISBN 0-387-40270-5 directed to graduate students interested in digital signal er with the value distribution theory for them are studied in The book is written as a text for a yearlong course in sta- processing. The reader is assumed to have a mathematical the next two chapters. The next six chapters are devoted to tistics. The importance of such a textbook follows from the background on the graduate level. (knaj) a study of the properties of solutions of six Painlevé equa- fact that today there are more than 15,000 statisticians in tions. Behaviour of solutions in a neighbourhood of a sin- the United States only, over 100 U.S. universities offer B. M. Landman, A. Robertson: Ramsey Theory on the gularity is studied. Integrable (systems of) equations usu- graduate degrees in statistics and a shortage of qualified Integers, Student Mathematical Library, vol. 24, American ally come in whole hierarchies. Higher order analogues of statisticians is expected to persist for some time. The stu- Mathematical Society, Providence, 2003, 317 pp., $49, the Painlevé equations of the first two types are discussed dents should learn not only purely mathematical tools but ISBN 0-8218-3199-2 in the book. For specific values of parameters of the equa- also the use of computers for obtaining numerical results The topic of this book is Ramsey theory on sets of integers. tions, solutions can be constructed using the Bäcklund and their graphical presentation. The book describes statis- Chapter 1 introduces notation and basic results of the field, transformations. For these values of parameters, solutions tical analysis of data and shows how to communicate the van der Waerden’s theorem, Schur’s theorem and Rado’s can be expressed in terms of rational functions or classical results. The topics included in the book are standard ones: theorem. The largest portion of the book, chapters 2-7, is transcendental functions. Relations of the Painlevé equa- an introduction to statistical inference, one-way and two- devoted to the first result. In Chapter 2, a proof of van der tions to hierarchies of integrable systems (KdV, way analysis of variance, multiple comparisons, simple Waerden’s theorem is given and bounds on van der Boussinesq, sine-Gordon, nonlinear Schrödinger, Einstein and multiple linear regression, design of experiments, con- Waerden’s numbers are discussed, including the break- and Toda type equations) are discussed in the last chapter. tingency tables, nonparametrics, logistic regression, and through by W. Gowers. Chapters 3-7 deal with various The two appendices offer a summary of basic facts needed time series analysis in time domain. An “invisible” but modifications and variations on the theme of arithmetical about solutions of ordinary differential equations in the extremely important part of the book is a web page (or CD) progressions (subsets and supersets of arithmetical pro- complex plane and on the Nevanlinna theory. Interest in with the data for all examples and exercises, which also gressions, homothetic copies of sequences, and modular properties of solutions of Painlevé equations is steadily contains codes in S (i.e., S-PLUS and R) and in SAS. In arithmetical progressions). Chapter 8 is devoted to Schur’s growing and they appear in many questions in mathemat- online files, the reader also finds the code and the theorem, Chapter 9 to Rado’s theorem and Chapter 10 to ics and mathematical physics. Hence, the careful treatment PostScript file for every figure in the text. other topics (Folkman’s theorem, Brown’s lemma and oth- of them in the presented monograph is a valuable addition The statistical analysis in the book is taught on interest- ers). Each chapter is followed by exercises, a list of to the existing literature. (vs) ing real examples. Sometimes (e.g. in logistic regression) research problems, and commentary. The book concludes the analysis is quite deep. An extraordinary care is devot- with a list of notation and an extensive bibliography with A. Grothendieck, Ed.: Revêtements étales et groupe fon- ed to graphical presentation of results. Many of the graph- 275 items. (mkl) damental (SGA 1), Documents Mathématiques 3, Société ical formats are novel. Finally, appendices to the book Mathématique de France, Paris, 2003, 325 pp., ISBN 2- contain useful remarks on statistical software, typography, J.-L. Lions: Oeuvres choisies de Jacques-Louis Lions, 85629-141-4 and a review of fundamental mathematical concepts. A vol. I, EDP Sciences, Paris, 2003, 722 pp., €70, ISBN 2- This is a new edition of the first volume of A. reader who studies the book and repeats the authors’ cal- 86883-661-5 Grothendieck’s SGA (Séminaire de géométrie algébrique), culations gains basic statistical skills and knowledge of J.-L. Lions: Oeuvres choisies de Jacques-Louis Lions, which is devoted to the theory of the algebraic fundamen- software for solving similar problems in applications. The vol. II, EDP Sciences, Paris, 2003, 864 pp., €70, ISBN 2- tal group. The book incorporates a few minor corrections book contains no theorems and no proofs. The methods 86883-662-3 and several updating remarks by M. Raynaud. A TEX file are only explained and then demonstrated. However, if a J.-L. Lions: Oeuvres choisies de Jacques-Louis Lions, of this text (as well as of the uncorrected version), typeset statistical course contains a theoretical part, then the book vol. III, EDP Sciences, Paris, 2003, 813 pp., €70, ISBN 2- by a team of volunteers headed by S. Edixhoven, is avail- can serve as a source of interesting examples and problems. 86883-663-1 able from the arXiv.org e-print server. This book is indis- By the way, the authors’ web page also contains errata, Jacques-Louis Lions, outstanding scientist and leading per- pensable to any serious student of algebraic geometry. which is quite comfortable for the readers. It is surely not sonality in partial differential equations and many related (jnek) a final list of misprints. For example, the median of the fields, published more than 20 monographs and more than binomial distribution with n=2 and p=0.5 is h=1 which 600 research papers. The presented three volumes repre- B. Hasselblatt, A. Katok: A First Course in Dynamics: does not satisfy condition (3.10) on p. 28. (ja) sent a high level and representative sample of papers and with a Panorama of Recent Developments, Cambridge parts of monographs carefully chosen by the scientific University Press, Cambridge, 2003, 424 pp., £25,95, ISBN T. A. Ivey, J. M. Landsberg: Cartan for Beginners: committee (A. Bensoussan, P. G. Sciarlet, R. Glowinski 0-521-58750-6, ISBN 0-521-58304-7 Differential Geometry via Moving Frames and Exterior and R. Temam, with the collaboration of coordinators F. The book has two parts. The first one (A Course in Differential Systems, Graduate Studies in Mathematics, Murat and J.-P.Puel). Dynamics: From Simple to Complicated Dynamics) can vol. 61, American Mathematical Society, Providence, The first volume starts with an introduction (written by serve as a textbook for a beginner course in dynamical sys- 2003,378 pp., $59, ISBN 0-8218-3375-8 R. Temam) and a commentary by E. Magenes about his tems for senior undergraduate students of mathematics, Moving frames and exterior differential systems belong to joint work with J.-L. Lions. The volume is devoted to par- physics and engineering. The explanation is slightly classical methods for studying geometry and partial differ- tial differential equations and interpolation theory. It cov- unusual, since it is based on a number of simple examples ential equations. These ideas emerged at the beginning of ers, roughly speaking, results published in the period that present basic behaviour of dynamical systems. The the 20th century, being introduced and developed by sever- between 1950 and 1960 and it contains almost 30 papers. outset of theory, including topological and probabilistic al mathematicians, in particular, by Élie Cartan. Over the Let us quote at least a few of the most interesting points. J.- methods, begins from the detailed study of these examples. years these techniques have been refined and extended. L. Lions and E. Magenes developed a theoretical frame- The exposition is accompanied by many valuable com- The book is a nice introduction into classical and recent work for solving non-homogeneous boundary value prob- ments and exercises of different complexity. The first part geometric applications of the techniques. It covers classi- lems in a series of joint papers “Problèmes des limites non ends with complicated orbit structures like recurrence and cal geometry of surfaces and basic Riemannian geometry homogénes I - VII”, whose first five parts are reproduced mixing for systems on a torus. Chaotic behaviour is pre- in the language of the method of moving frames; it also here. The Hilbert part of the theory in Hs(Ω) spaces is con- sented and applications to coding are also given. includes results from projective differential geometry. tained in the three volume monograph of J.-L. Lions and E. The second part (Panorama of Dynamical Systems) is There is an elementary introduction to exterior differential Magenes. The Wk,p(Ω) theory is very well described in the more advanced and is intended as an introduction to mod- systems, basic facts from G-structures and general theory (fully reproduced) course by J.-L. Lions at the University ern achievements of the theory. Hyperbolic dynamics is of connections. Every section begins with geometric of Montreal. The interpolation between Sobolev spaces studied, and results like the closing and shadowing lemma examples and problems. There are four appendices devot- plays a very important role here and thus these results are are proved. A lot of attention is paid to the classical exam- ed to linear algebra and representation theory, differential closely connected with J.-L. Lions’ contributions to inter- ple of the quadratic map. A mechanism that produces forms, complex structures and complex manifolds and ini- polation theory both within Hilbert spaces and Banach horseshoes and thus gives rise to chaotic dynamics is tial value problems. Interesting exercises are included, spaces. Significant achievements were obtained also in shown. The famous Lorentz attractor, as a prototype of a together with hints and answers to some of them. The nonlinear partial differential equations. J.-L. Lions togeth- strange attractor, is examined in details. Variational authors presented it as a textbook for a graduate level er with J. Leray used monotonocity methods and general- approach does not belong to usual tools in dynamical sys- course. It can be strongly recommended to anybody inter- ized the results of F. Browder and G. Minty to functionals tems. The authors present how this approach can be ested in classical and modern differential geometry. (jbu) of the calculus of variations that are convex in the highest applied to the study of twist maps and afterwards to derivatives only. They also presented a new and elegant Riemann manifolds. The existence of infinitely many W. Keller: Wavelets in Geodesy and Geodynamics, proof to the results of M. Vishik. An important part of non- closed geodesics on the two-dimensional sphere equipped Walter de Gruyter, Berlin, 2004, 279 pp., 130 fig., €84, linear PDEs is the classical Navier-Stokes system and its with a Riemannian metric is proved. The second part ends ISBN 3-11-017546-0 generalizations. J.-L. Lions also contributed to the modern with the chapter devoted to relations between number the- This excellent textbook gives an introduction to wavelet theory of the Navier-Stokes system. He presented a new ory and dynamical systems. The book is written in a pre- theory both in the continuous and the discrete case. After and simpler variant of Hopf’s proof of the existence of cise and very readable style, there are many useful remarks developing the theoretical fundament, typical examples of weak solutions in three space dimensions - the most inter- and figures throughout. It can be highly recommended to wavelet analysis in geosciences are presented. The book esting from the point of view of physics - and in a joint anybody who is interested in dynamical systems and who consists of three main chapters. Fourier and filter theory paper with G. Prodi, they proved uniqueness of weak solu- has a basic knowledge of calculus. The book is also an are shortly sketched in the first chapter. The second chap- tions in two space dimensions. Together with Q. EMS December 2004 43 BOOKS Stampacchia, he introduced the concept of a variational cations of the compensated compactness to relaxation very comprehensive and presents a lot of material on each inequality, which proved to be very useful in many appli- problems. The book is carefully written and will be appre- of the themes. Moreover, it offers various ways of how to cations. ciated both by PhD students and experts in the field as one study “spaces” or ”algebras”, selecting and accentuating The second volume of the series with subtitles of a few books gathering the knowledge until recently dis- some of their features. The knowledge required for the “Controle” and “Homogenization”, is introduced by A. persed among research papers. (mro) reading of the book varies between the chapters, but only a Bensoussan and contains results published from the end of modest knowledge of category theory is supposed at the the sixties up to the end of the nineties. J.-L. Lions’ inter- G. Mislin, A. Valette: Proper Group Actions and the beginning. The authors of each chapter develop the neces- est in optimal control theory started soon after the pio- Baum-Connes Conjecture, Advanced Courses in sary categorical techniques themselves. The book will be neering books of L. S. Pontryagin and R. Bellman, which Mathematics CRM Barcelona, Birkhäuser, Basel, 2003, very useful for graduate students and teachers, and inspir- appeared in 1957. Already 11 years later, Lions’ book 131 pp., €28, ISBN 3-7643-0408-1 ing for the researchers interested in the discussed topics as “Controle optimal des systèmes ...“ was published, and it The book has two parts. The first contribution, by G. well as in category theory itself. (vtr) became soon a standard reference book. It was well known Mislin, contains a discussion of the equivariant K-homolo- that the problem of optimal control for distributed systems gy KG*(EG) of the classifying space EG for proper actions G. Pisier: Introduction to Operator Space Theory, can lead to problems with free boundary of Stefan type. J.- of a group G. The Baum-Connes conjecture states that the London Mathematical Society Lecture Note Series 294, L. Lions used variational inequalities to make this relation K theory of the reduced C* algebra of a group G can be Cambridge University Press, Cambridge, 2003, 475 pp., straightforward. In the seventies, he generalized the notion computed by the equivariant K-homology KG*(EG). The £39,95, ISBN 0-521-81165-1 of variational inequalities so that it was well suited to tools used in the exposition contain the Bredon homology The monograph is devoted to the study of operator space describe the problems of optimal stopping time and togeth- for infinite groups. Relations of the Baum-Connes conjec- theory. The book has three parts. The first part contains a er with A. Bensoussan, they studied the stochastic control ture to many other famous conjectures in topology are basic exposition of the theory and various illustrative and variational inequalities and impulse control and quasi- described in the Appendix. The second part (written by A. examples. An operator space is just a (usually complex) variational inequalities. Later he returned to optimal con- Valette with Appendix by D. Kucerovsky) contains a dis- Banach space X, equipped with a given embedding into the trol theory from a new point of view - namely the possibil- cussion of the Baum-Connes conjecture for a countable space B(H) of all bounded linear operators on a Hilbert ity of using control theory for stabilization of unstable or ill discrete group Γ. Suitable index maps provide the link space H. Natural morphisms in this category are com- posed problems. New aspects of control theory appear also between both sides of the conjecture. The second part of pletely bounded maps (a linear map is completely bound- in J.-L. Lions later papers, especially his famous HUM - the book contains a careful discussion of these maps. Both ed if the naturally induced mappings between respective Hilbert uniqueness method. Together with R. Glowinsky, lecture notes clearly cover the area around a beautiful, spaces of matrices have uniformly bounded norms). In he introduced numerical methods suitable for solving these interdisciplinary field and could be very useful to anybody view of this, we can have many operator space structures questions. In the second half of the seventies, J.-L. Lions interested in the subject. (vs) on a given Banach space. One of the basic tools for the the- was interested in the homogenisation theory for a large ory is the Ruan theorem, which shows the correspondence scale of equations with periodically oscillating coefficients. V. Mûller: Spectral Theory of Linear Operators and between operator space structures on X and some kind of The method he used was far-reaching and applicable also Spectral Systems in Banach Algebras, Operator Theory norms on the tensor product of X with the space K(L2) of to problems with an oscillating boundary or problems Advances and Applications, vol. 139, Birkhäuser, Basel, compact operators on L2. Basic operations on Banach posed on perforated domains. Thus it made it possible in 2003, 381 pp., €158, ISBN 3-7643-6912-4 spaces (dual space, quotient space, direct sum, complex fluid mechanics to deduce Darcy’s equation from Stokes’ The monograph is written as an attempt to organize a huge interpolation, some tensor products, etc.) can be defined in system, or to study the homogenisation of Bingham fluids. amount of material on spectral theory, most of which was the category of operator spaces. Some of these definitions At the same time it allowed the study of mechanical prop- until now available only in research papers. The aim is to are elementary, some make use of the Ruan theorem. It is erties of composite materials. present a survey of results concerning various types of also worth mentioning the existence (and uniqueness) of The third volume, with an introduction by P. G. Ciarlet, spectra in a unified, axiomatic way. The book is organized the “operator Hilbert space” - a canonical operator space is devoted to numerical analysis and applications of PDEs in to five chapters. At the beginning, the author presents structure on the Hilbert space. The second part is devoted in a wide scale of problems - the mechanics of fluids and spectral theory in Banach algebras, which forms a natural to C*-algebras, which form a subclass of operator spaces solid bodies, Bingham fluids, viscoelastic or plastic mate- frame for spectral theory of operators. The second chapter (notice that the structure of a C*-algebra on a Banach space rials, problems with friction, and plates described by linear is devoted to applications to operators. Of particular inter- induces a unique operator space structure). The main as well as nonlinear elasticity theory. Since 1990, J.-L. est are regular functions: operator-valued functions whose themes in this part are C*-tensor products and various Lions was attracted by very complex and complicated sys- ranges behave continuously. A suitable choice of a regular classes of C*-algebras and von Neumann algebras. Some tems of PDEs, i.e., by climatology models. Even in this function gives rise to the important class of Kato operators properties of C*-algebras are extended to general operator exceptional case, he, together with R. Temam and S. and the corresponding Kato spectrum. The third chapter spaces, and local theory of operator spaces is investigated. Wand, proved existence and uniqueness of solutions, their gives a survey of results concerning various types of essen- The third part deals with non-self-adjoint operator alge- asymptotic behaviour and ways to their numerical solution. tial spectra, Fredholm and Browder operators, etc. The bras, their tensor products and free products. The theory of These two papers and more than 20 others dealing with fourth chapter contains an elementary presentation of the operator spaces is also used to reformulate some classical highly interesting problems are contained in Volume III. Taylor spectrum, which is by many experts considered to similarity problems. (okal) The papers in the collection are, as well as all works by J.- be the proper generalization of the ordinary spectrum of a L. Lions, written in a very clear and concise form and they single operator. The most important property of the Taylor A. Polishchuk: Abelian Varieties, Theta Functions and are indispensable for researchers in PDEs and in numerical spectrum is existence of a functional calculus for functions the Fourier Transform, Cambridge Tracts in analysis. (jsta) analytic on a neighbourhood of the Taylor spectrum. The Mathematics 153, Cambridge University Press, last chapter is concentrated on the study of orbits and weak Cambridge, 2003, 292 pp., £47,50, ISBN 0-521-80804-9 Y. Lu: Hyperbolic Conservation Laws and the orbits of operators, which are notions closely related to the The book gives a modern introduction to theory of Abelian Compensated Compactness Method, Monographs and invariant subspace problem. All results are presented in an varieties and their theta functions. The text is based on lec- Surveys in Pure and Applied Mathematics 128, Chapman elementary way. Only a basic knowledge of functional tures by the author delivered at Harvard University (1998) & Hall/CRC, Boca Raton, 2003, 241 pp., $84,95, ISBN 1- analysis, topology and complex analysis is assumed. and Boston University (2001) and it provides an up-to-date 58488-238-7 Moreover, basic notions and results from Banach spaces, introduction to the subject, oriented to the general mathe- The book is devoted to the theory of the compensated com- analytic and smooth vector-valued functions and semi-con- matical community. One of the main goals is to give the pactness method, which is a principal tool for studying tinuous set-valued functions are given in the Appendix. first introduction to algebraic theory of Abelian varieties properties of systems of hyperbolic conservation laws. The monograph should appeal both to students and to and theta functions, employing Mukai’s approach to the Quasilinear systems of hyperbolic conservation laws in one experts in the field. It can also serve as a reference book. Fourier transform in the context of Abelian varieties. This space dimension are considered. This setting (systems in (jsp) approach is also supported by recently discovered links to one space dimension) is more or less the only case for the mirror symmetry problem in algebraic geometry and which the existence and uniqueness results can be obtained M. C. Pedicchio, W. Tholen, Eds.: Categorical quantum field theory. The exposition of material present- also in a classical way, e.g. by the method of wave-front Foundations: Special Topics in Order, Topology, ed in the book was influenced by the category approach to tracking. In this book a different approach is used, namely Algebra, and Sheaf Theory, Encyclopaedia of problems under consideration. The book is divided into that of compensated compactness. The author introduces Mathematics and Its Applications 97, Cambridge three main parts, then each of them into seven or eight basic elements of the theory of compensated compactness, University Press, Cambridge, 2004, 440 pp., $90, ISBN 0- chapters. Part I (Analytic Theory) discusses classical and based on results of Tartar and Murat from the 80’s. The 521-83414-7 recent aspects of transcendental theory of Abelian vari- notion of a Young measure is introduced and discussed. The book is a result of a collaborative research project of eties. Part II (Algebraic Theory) is devoted to general After these preliminaries, the author studies the Cauchy mathematicians from four European and three Canadian Abelian varieties over an algebraically closed field of arbi- problem for a scalar equation with L∞ and Lp data. It is to universities. During the years 1998 – 2001, small teams trary characteristic. Part III (Jacobians) contains theory of be noted that the simplified proof of the existence of the were formed to work on a variety of themes of current Jacobian varieties of smooth irreducible projective curves solution presented here does not need to use a concept of interest and to develop the categorical approach to them. over arbitrary fields. The chapters’ text contains many the Young measure. In the system case, the author works This book presents the results of their work. The book con- exercises complementing the material covered. The bibli- with all the usual concepts, such as the strict hyperbolicity, tains 8 chapters devoted in turn to ordered set and adjunc- ography is up-to-date and comprehensive, consisting of genuine non-linearity, linear degeneracy, Riemann invari- tion (by R. J. Wood), locales (by J. Picado, A. Pultr and A. 138 titles. The book is primarily intended for anybody ants, entropy-entropy flux pair and theory of invariant Tozzi), general topology (by M. M. Clementino, E. Giuli interested in modern algebraic geometry and mathematical regions to obtain uniform L-infinity estimates, symmetric and W. Tholen), regular, protomodular and Abelian cate- physics, with a good background not only in complex and and symmetrizable systems. The author then studies dif- gories (by D. Bourn and M. Gran), monads (by M. C. differential geometry, classical Fourier analysis, or repre- ferent important systems of hyperbolic equations, namely Pedicchio and F. Rovatti), sheaf theory (by C. Centazzo sentation theory, but also in modern algebraic geometry the system of Le Roux type, the system of the polytropic and E. M. Vitale) and to effective descent morphisms (by and categorical algebra. The book is written by a leading gas dynamic, Euler equations of one-dimensional com- G. Janelidze, M. Sobral and W. Tholen). Every chapter is expert in the field and it will certainly be a valuable pressible fluid flow, systems of elasticity and some appli- self-contained with its own list of references. The book is enhancement to the existing literature. (špor) 44 EMS December 2004 BOOKS J. F. Simonoff: Analyzing Categorical Data, Springer Then the theory is developed and this is quite often done on N. Saveliev: Invariants for Homology 3-Spheres, Texts in Statistics, Springer, New York, 2003, 496 pp., 64 two levels. For example, the Cauchy theorem is first done Encyclopaedia of Mathematical Sciences, vol. 140, fig., €84,95, ISBN 0-387-00749-0 in the version with sufficiently smooth positively oriented Springer, Berlin, 2002, 223 pp., €84,95, ISBN 3-540- The book can be divided into several parts. It starts with an (firstly based on the intuition) curves bounding a bounded 43796-7 introduction to regression models, including regression domain G and with function f = u + iv holomorphic in G The book can be considered as a fundamental monograph diagnostics and model selection. Then the author deals and u,v in C1-class on the closure of G. Later on it is on invariants of homology 3-spheres. Let us mention that with discrete distributions and corresponding goodness-of- proved for a rectifiable boundary and f holomorphic in G while the topic may seem rather specialised, these investi- fit tests. The main topics here are binomial, multinomial, and continuous on its closure. Many things, which are gations have proved to be extremely useful in the manifold and Poisson distributions complemented by the zero-inflat- briefly described in other books, in remarks or exercises, topology. The text covers almost all invariants from the ed Poisson model, the negative-binomial model, and the are given in full details (at least 5 different proofs of the classical Rokhlin invariant, through the Casson invariant beta-binomial model. Then regression models for count fundamental theorem of algebra are given). The book con- and its various refinements and generalizations, to recent data are presented. They are based mainly on generalized tains an exposition of analytic continuation, homotopy, invariants of the gauge type. Quite naturally, it also con- linear models. The part on contingency tables describes conformal mappings, special functions and boundary value tains many results on topology of 4-manifolds. Obviously, log-linear models, conditional analyses, structural zeros, problems, to name the less frequently treated material. The the book is designed for specialists in the field who will outlier identification, models for tables with ordered cate- authors will please readers interested mainly in applica- find there a practically complete survey of results and gories, and models for square tables. The last part of the tions as well as those who want to know how things really methods. On the other hand, it is written in such a clear book introduces regression models for binary data and for work and prefer deeper and more detailed treatment of the style that I would like to recommend it strongly to post- multiple category response data. The chapters end with a material. The book also contains more than 200 examples graduate students starting to make themselves familiar with section that provides references to books or articles related and 150 exercises. A certain drawback is that the fonts the field. The book will help them to learn basic notions to the material in the chapters. used in the typesetting of the book are rather small. and will gradually introduce them into contemporary The author based this book on his notes for a class with Regardless of this fact the book is nice and I recommend it research. The large number of references (311 items going a very diverse pool of students. The material is presented for courses in complex function theory (even on an up to 2001) will enable them the further orientation. (jiva) in such a way that a very heterogeneous group of students advanced level) and also as a reference book. (jive) could grasp it. All methods are illustrated with analyses of L. Schneps, Ed.: Galois Groups and Fundamental real data examples. The author provides a detailed discus- C. Villani: Topics in Optimal Transportation, Graduate Groups, Mathematical Sciences Research Institute 41, sion of the context and background of the problem. For Studies in Mathematics, vol. 58, American Mathematical Cambridge University Press, Cambridge, 2003, 467 pp., example, it is known that incorrect statistical analysis of Society, Providence, 2003, 370 pp., $59, ISBN 0-8218- £50, ISBN 0-521-80831-6 data that were available at the time of the flight of 3312-X This volume is the outcome of the MSRI special semester Challenger on January 28, 1986, led to its explosion. It is The monograph is an exhaustive survey of the optimal on Galois groups and fundamental groups, held in the fall less known that one of the recommendations of the com- mass transportation problem. It gives an overview of the of 1999. The book contains scientific and survey articles mission was that a statistician must be part of the ground recent knowledge of the subject and it introduces all tools from the most important extensions and ramifications of control team from that time on. All statistical modeling convenient for its investigation. One of the principal tools Galois theory - geometric Galois theory, Lie Galois theory and figures in the text are based on S-PLUS. The author used in the book is the Kantorovich duality on bounded and differential Galois theory, all in various characteristics. has set up a web site related to the book, where data sets continuous functions. Subsequent chapters introduce rele- The main focus of the study of geometric Galois theory is and computer code are available as well as answers to vant geometric arguments needed to prove the duality the- the theory of curves and objects associated with them - selected exercises (there are more than 200 exercises in the orem for the quadratic cost. Furthermore, Brenier’s Polar curves with marked points, their fields of moduli and their book). On the other hand, there are no theorems and proofs factorisation theorem, the Monge-Ampère equation, dis- fundamental group, covers of curves with their ramifica- in the book. Before using it as a textbook, the instructor placement interpolation and the probabilistic metric theory, tion information, finite quotients of the fundamental group should consult original papers and books to prepare theo- are all discussed. Relations to physical theories are also that are Galois groups of the covers. The articles present- retical background. At the same time, the style is not a mentioned. The optimal mass transportation problem is ed in the book include fundamental groups in positive char- ”cookbook”. The book is very interesting and it can be reformulated in terms of fluid mechanics. Reformulation acteristic, anabelian theory and Galois group action on fun- warmly recommended to people working with categorical and explanation, by means of energy and entropy produc- damental groups. The subject of Lie Galois theory origi- data. (ja) tion optimisation under variational inequalities, are stated. nates in the geometric situation, whose linearised version The monograph grew from a graduate course taught by the leads to graded Lie algebras associated with profinite fun- J. Stopple: A Primer of Analytic Number Theory: From author. The book is well organized and written in a clear damental groups. Special attention is paid to special loci in Pythagoras to Riemann, Cambridge University Press, and precise style. The text includes a list of illustrative the moduli space with a particular group of automor- Cambridge, 2003, 383 pp., £ 22,95, ISBN 0-521-81309-3, problems helping to understand the theory. A certain level phisms. Instead of considering finite groups as Galois ISBN 0-521-01253-8 of mathematical skill is required from the reader. The groups of Galois extensions of arbitrary fields, differential The book constitutes an excellent undergraduate introduc- monograph can be recommended to researchers and scien- Galois theory treats linear algebraic groups as Galois tion to classical analytical number theory. The author tists working or interested in the field as well as an appro- groups of so called Picard-Vessiot extensions of D-fields, develops the subject from the very beginning in an priate textbook for graduate and postgraduate courses on which are fields equipped with derivation. (pso) extremely good and readable style. Although a wide vari- the subject. (pl) ety of topics are presented in the book, the author has suc- T. Sheil-Small: Complex Polynomials, Cambridge cessfully placed a rich historical background to each of the C. Voisin: Théorie de Hodge et géométrie algébrique Studies in Advanced Mathematics 75, Cambridge discussed themes, which makes the text very lively. The complexe, Cours Spécialisés 10, Société Mathématique de University Press, Cambridge, 2002, 428 pp., £65, ISBN 0- author covers topics with roots in ancient mathematics like France, Paris, 2002, 595 pp., €69, ISBN 2-85629-129-5 521-40068-6 polygonal numbers, perfect numbers, amicable pairs, basic Cambridge University Press published the English transla- The book studies geometric properties of polynomials and properties of prime numbers and all central themes of the tion of the book in a two-volume version in 2002, 2003 rational functions in the complex plane. The book starts basic analytic number theory. Assuming almost no knowl- resp. The review of both parts of the English translation with a description of foundations of complex variable the- edge from complex analysis, he develops tools needed to appeared in the EMS Newsletter No. 53, p. 48. (pso) ory from the point of view of algebra as well as analysis. show the significance of the Riemann hypothesis for the For example, the degree principle is studied in connection distribution of primes. Problems of additive number theo- List of reviewers for 2004 with the fundamental theorem of algebra, falling out into ry are not covered, as well as the prime number theorem. the mini-course of plane topology. To mention another In the last three chapters of the book, the reader finds a cou- The Editor would like to thank the following for their example, the Rouché theorem is studied together with its ple of specific examples of L-functions attached to reviews this year. topological analogues, including the Brouwer fixed-point Diophantine equations. The material covered in these theorem. After the preliminary chapter on the algebra of chapters includes solutions of the Pell equation, elliptic J. Andìl, R. Bashir, M. Beèváøová-Nìmcová, L. Beran, polynomials, the book is divided in to 11 chapters contain- curves and analytic aspects of algebraic number theory. L. Bican, L. Boèek, J. Bureš, J. Dolejší, P. Dostál, ing a study of the Jacobian problem, analytic and harmon- The text contains a rich supplement of exercises, brief J. Drahoš, M. Ernestová, D. Hlubinka, Š. Holub, P. ic functions in the unit disc, trigonometric polynomials, sketches of more advanced ideas and extensive graphical Holický, M. Hušková, O. John, O. Kalenda, A. Kar- critical points of rational functions, self-inversive polyno- support. The book can be recommended as a very good ger, T. Kepka, M. Klazar, J. Kopáèek, V. Koubek, O. mials and many other topics in connection with the central first introductory reading for all those who are seriously Kowalski, P. Kùrka, P. Lachout, J. Lukeš, J. Malý, P. notion of a complex polynomial. Real polynomials are dis- interested in analytical number theory. (špor) Mandl, M. Markl, J. Milota, J. Mlèek, E. Murtinová, cussed in a separate chapter, including the Descartes rule of K. Najzar, J. Nekováø, J. Nešetøil, I. Netuka, O. signs, distribution of critical points of real rational func- W. Tutschke, H. L. Vasudeva: An Introduction to Odvárko, D. Praák, Š. Porubský, J. Rataj, B. Rie- tions, and real entire and meromorphic functions. Complex Analysis, Modern Analysis Series, Chapman & èan, M. Rokyta, T. Roubíèek, P. Simon, P. Somberg, Blaschke products are also mentioned at the end of the Hall/CRC, Boca Raton, 2004, 460 pp., $89,95, ISBN 1- J. Souèek, V. Souèek, J. Spurný, J. Stará, Z. Šír, J. book, with connection to harmonic mappings, convex 584-88478-9 Štìpán, J. Trlifaj, V. Trnková, J. Tùma, J. Vanura, J. curves and polygons. In general, the book offers a whole The authors present two parallel approaches to complex Veselý, M. Zahradník, M. Zelený. variety of concepts, oscillating among analysis, algebra function theory. One follows the idea that what can be and geometry. This is clearly one of the advantages of this used from the real analysis must be applied to lay the foun- All of the above are on the staff of the Charles nice publication. The book, with (or despite) its more than dations of complex function theory, while the other one is University, Faculty of Mathematics and Physics, 400 pages, gives the reader a feeling that it is both a con- “purely complex”. The book starts with a review (86 p.) of Prague, except: M. Markl and J. Vanura cise and a comprehensive monograph on the topic. It will basic methods and notions from real analysis such as met- (Mathematical Institute, Czech Academy of Sciences), surely be appreciated by graduate and PhD students, as ric spaces, liminf and limsup of a sequence of real numbers Š. Porubský (Technical University, Prague), B. Rieèan well as by researchers working in the field. (mro) or the Gauss-Green formula, and basics on the field C of (University of B. Bystrica, Slovakia), J. Nekováø complex numbers and on elementary functions (in C). (University Paris VI, France). EMS December 2004 45