European Mathematical Society

Home , Henri Cartan

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. 53 e-mail: [email protected] KRZYSZTOF CIESIELSKI Mathematics Institute September 2004 Jagiellonian University EMS Agenda ...... 2 Reymonta 4, 30-059 Kraków, Poland e-mail: [email protected] Editorial - Manuel Castellet ...... 3 STEEN MARKVORSEN Department of Mathematics, Technical EMS meetings at Uppsala ...... 4 University of Denmark, Building 303 DK-2800 Kgs. Lyngby, Denmark EMS Summer Schools ...... 6 e-mail: [email protected] Report on the 4th ECM - Ulf Persson ...... 7 ROBIN WILSON Department of Pure Mathematics Visions of Young Mathematicians - Jan Larsson and Mikael Johansson...... 12 The Open University Milton Keynes MK7 6AA, UK Reflections on ICME-10 - Vagn Lundsgaard Hansen ...... 14 e-mail: [email protected] COPY EDITOR: 25 years proof of the Kneser conjecture - Mark de Longueville ...... 16 KELLY NEIL 100th Birthday of Henri Cartan ...... 20 School of Mathematics University of Southampton Interview with Nuno Crato ...... 22 Highfield SO17 1BJ, UK e-mail: [email protected] Interview with Michael Atiyah and Isadore Singer ...... 24 SPECIALIST EDITORS Abel Prize Celebration 2004 ...... 31 INTERVIEWS Steen Markvorsen [address as above] Greek Mathematical Society ...... 34 SOCIETIES Krzysztof Ciesielski [address as above] French-Polish cooperation in Approximation Theory ...... 36 EDUCATION Tony Gardiner ERCOM: CRM Barcelona ...... 37 University of Birmingham Forthcoming Conferences ...... 40 Birmingham B15 2TT, UK e-mail: [email protected] Recent Books ...... 42 MATHEMATICAL PROBLEMS Paul Jainta Werkvolkstr. 10 Designed and printed by Armstrong Press Limited D-91126 Schwabach, Germany Crosshouse Road, Southampton, Hampshire SO14 5GZ, UK e-mail: [email protected] telephone: (+44) 23 8033 3132 fax: (+44) 23 8033 3134 ANNIVERSARIES e-mail: [email protected] June Barrow-Green and Jeremy Gray Open University [address as above] Published by European Mathematical Society e-mail: [email protected] ISSN 1027 - 488X and [email protected] CONFERENCES The views expressed in this Newsletter are those of the authors and do not necessari- Vasile Berinde [address as above] ly represent those of the EMS or the Editorial team. RECENT BOOKS Ivan Netuka and Vladimír Souèek Mathematical Institute NOTICE FOR MATHEMATICAL SOCIETIES Charles University, Sokolovská 83 Labels for the next issue will be prepared during the second half of November 2004. 18600 Prague, Czech Republic Please send your updated lists before then to Ms Tuulikki Mäkeläinen, Department of e-mail: [email protected] Mathematics and Statistics, P.O. Box 68 (Gustav Hällströmintie 2B), FI-00014 University of and [email protected] Helsinki, Finland; e-mail: [email protected] ADVERTISING OFFICER INSTITUTIONAL SUBSCRIPTIONS FOR THE EMS NEWSLETTER Vivette Girault Institutes and libraries can order the EMS Newsletter by mail from the EMS Secretariat, Laboratoire d’Analyse Numérique Department of Mathematics and Statistics, P.O. Box 68 (Gustav Hällströmintie 2B), FI-00014 Boite Courrier 187, Université Pierre University of Helsinki, Finland, or by e-mail: ([email protected] ). Please include et Marie Curie, 4 Place Jussieu the name and full address (with postal code), telephone and fax number (with country code) and 75252 Paris Cedex 05, France e-mail address. The annual subscription fee (including mailing) is 80 euros; an invoice will be e-mail: [email protected] sent with a sample copy of the Newsletter.

EMS September 2004 1 EMS NEWS EMS Committee EMS Agenda EXECUTIVE COMMITTEE 2004 PRESIDENT 15 November Prof. Sir JOHN KINGMAN (2003-06) Isaac Newton Institute Deadline for submission of material for the December issue of the EMS Newsletter 20 Clarkson Road Contact: Martin Raussen, e-mail: [email protected] Cambridge CB3 0EH, UK e-mail: [email protected] 12-19 December VICE-PRESIDENTS EMS Summer School and Séminaire Européen de Statistique at Munich (Germany) Prof. LUC LEMAIRE (2003-06) The Statistics of Spatio-Temporal Systems Department of Mathematics Web site: http://www.stat.uni-muenchen.de/semstat2004/ Université Libre de Bruxelles C.P. 218 – Campus Plaine Bld du Triomphe 2005 B-1050 Bruxelles, Belgium 5-13 February e-mail: [email protected] EMS Summer School at Eilat (Israel) Prof. BODIL BRANNER (2001–04) Applications of braid groups and braid monodromy Department of Mathematics Technical University of Denmark 16-17 April Building 303 EMS Executive Committee meeting at Capri (Italy) DK-2800 Kgs. Lyngby, Denmark e-mail: [email protected] 25 June – 2 July SECRETARY EMS Summer School at Pontignano (Italy) Prof. HELGE HOLDEN (2003-06) Department of Mathematical Sciences Subdivision schemes in geometric modelling, theory and applications Norwegian University of Science and Technology Alfred Getz vei 1 17-23 July NO-7491 Trondheim, Norway EMS Summer School and European young statisticians’ training camp at Oslo e-mail: [email protected] (Norway) TREASURER Prof. OLLI MARTIO (2003-06) 13-23 September Department of Mathematics EMS Summer School at Barcelona (Catalunya, Spain) P.O. Box 4, FIN-00014 Recent trends of Combinatorics in the mathematical context University of Helsinki, Finland e-mail: [email protected] 16-18 September ORDINARY MEMBERS Prof. VICTOR BUCHSTABER (2001–04) EMS-SCM Joint Mathematical Weekend at Barcelona (Catalunya, Spain) Steklov Mathematical Institute Russian Academy of Sciences 25 September – 2 October Gubkina St. 8, Moscow 117966, Russia EMS Summer School and Séminaire Européen de Statistique at Warwick (UK) e-mail: [email protected] Statistics in Genetics and Molecular Biology Prof. DOINA CIORANESCU (2003–06) Laboratoire d’Analyse Numérique 12-16 December Université Paris VI 4 Place Jussieu EMS-SIAM-UMALCA joint meeting in applied mathematics 75252 Paris Cedex 05, France Venue: the CMM (Centre for Mathematical Modelling), Santiago de Chile e-mail: [email protected] Prof. PAVEL EXNER (2003-06) 2006 Department of Theoretical Physics, NPI 22-30 August Academy of Sciences International Congress of Mathematicians in Madrid (Spain) 25068 Rez – Prague, Czech Republic e-mail: [email protected] Web site: www.icm2006.org Prof. MARTA SANZ-SOLÉ (2001-04) Facultat de Matematiques 2007 Universitat de Barcelona Gran Via 585 16-20 July E-08007 Barcelona, Spain ICIAM 2007, Zurich (Switzerland) e-mail: [email protected] Prof. MINA TEICHER (2001–04) Department of Mathematics and 2008 Computer Science July Bar-Ilan University 5th European Mathematical Congress, Amsterdam (The Netherlands) Ramat-Gan 52900, Israel e-mail: [email protected] Cost of advertisements and inserts in the EMS Newsletter, 2004 EMS SECRETARIAT Ms. T. MÄKELÄINEN (all prices in British pounds) Department of Mathematics 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 September 2004 EDITORIAL EditorialEditorial Manuel Castellet (Barcelona) Chairman of ERCOM

ERCOM is a committee of the European at the same time: it is a building and it is an Mathematical Society created in 1996 con- organisation of people working together and sisting of Scientific Directors of European dedicated to the pursuit and support of Research Centres in the Mathematical research. But at its best it is more. It starts life Sciences, or their chosen representatives. as an idea in the mind of one or more per- Only centres in which the number of visiting sons, of insight and imagination, and then staff substantially exceeds the number of per- lives and grows by spreading the spirit, manent and long-term staff and that broadly imbued by their founders, through the hearts cover Mathematical Sciences are eligible for and minds of all those benefiting from its representation in ERCOM. The eligibility of presence and contributing to its future”. centres is decided by the EMS Executive In mathematical research, exchange of In particular, ERCOM has established an Committee. ideas plays a central role and needs a deep exchange programme for short-term visits of The aims of ERCOM are to contribute to human contact. A contact which is the true post-doctoral fellows from one ERCOM the unity of Mathematics, from fundamental mathematical laboratory, and, as Friedrich centre to another, flexible and capable of to applications, with the purpose of constitut- von Siemens said at the end of the nineteenth adapting to different centres and situations, ing a forum for communication and century, “Laboratories are the fundamental the main goal being to stimulate the exchange of information and fostering col- basis of knowledge and power”. The high exchange and the increase of knowledge and laboration and co-ordination between the degree of abstraction of mathematics and the to create synergies enriching research. centres themselves and the EMS, to foster compact way it is presented need direct per- The Administrators have prepared a report advanced research training on a European sonal communication, since mathematics is on the percentage of women doing research level, to advise the Executive Committee of an attempt to develop tools that can be used in the ERCOM centres. There are about 14% the EMS on matters relating to activities of to achieve a better understanding of the of female researchers conducting long term the centres, to contribute to the visibility of world’s measurable aspects and to identify visits and an average of 16% female the EMS and to cultivate contacts with simi- processes that appear in very different natur- researchers conducting short term visits in lar research centres within and outside al situations but that are essentially analo- the ERCOM centres, while, on the other Europe. gous. hand, the administrative staff is mostly The 24 centres at present represented in But mathematics, apart from this, is a truly female and the directorate is mostly male. ERCOM have different working and organi- international science, perhaps the most inter- Last March in the Aarhus meeting the fol- sational structures, they are also different in national of sciences, since, compared with lowing resolution was adopted: “ERCOM size and scope, but they can be grouped in other disciplines, it is based less on the use of encourages its members to take actions to three different classes, as centres mainly for instruments and more on a strong human facilitate the presence of women in their sci- meetings, centres without any permanent contact. This is where the research institutes entific activities, and to collect data regarding researchers, only visitors and post-docs, and play a crucial role, allowing not only the the number of applications from female centres with a small permanent staff and a exchange of ideas between specialists in the researchers received and approved”. large number of visitors. They broadly cover same field, but also profound and sometimes ERCOM, as a committee of the European the European Area, from Russia and Sweden surprising links between different lines of Mathematical Society, wishes and has the to Portugal and Israel. A Chairman and a research. capacity to act as one of the EMS means of Vice-chairman, at present myself, as Director In order to foster collaboration and co- scientific outreach, fostering the presence of of the Centre de Recerca Matemàtica, and ordination between the European research mathematicians in the new emerging multi- Kjell-Ove Widman, Director of the Institut institutes and to provide communication and disciplinary research domains. With this aim Mittag-Leffler, co-ordinate the annual exchange of information, ERCOM meets we are preparing a proposal to be submitted ERCOM meeting and the other initiatives yearly, usually by March, in one of the mem- to the European Commission as a NEST proposed by the ERCOM members. Sir John ber institutes. Since the 1999 meeting in Support Action through a Co-ordination Kingman, Director of the Isaac Newton Cambridge the Administrators have been Action instrument, with the main objective Institute, acts as link between ERCOM and invited to discuss separately and jointly with of designing curricula and finding ways to the EMS Executive Committee. the Directors matters of their competence. train researchers into emerging specialities of The Editor-in-Chief of the EMS Several topics are considered for discus- Mathematics, linked especially with System Newsletter, Martin Raussen, recently offered sion in the meetings: The European Research Biology, Neuroscience, Risk Assessment to the ERCOM institutions the opportunity to Area and the presence of Mathematics in the and Data Security, and identifying future present themselves in order to give the read- Framework Programmes of the European research opportunities on the interface ers an idea of how they might use the possi- Union, the annual call for post-doctoral between Mathematics and suitable areas of bilities offered by the centres. EURAN- grants by the European Post-doctoral Medicine, Industry and Social Sciences. DOM, Eindhoven, presented itself in the Institute for the Mathematical Sciences A home page (http://www.ercom.org), June issue of the Newsletter and the Centre (EPDI), the situation of Mathematics and the from which you can reach that of each de Recerca Matemàtica (CRM), Barcelona, mathematicians in Eastern Europe, the ERCOM centre, provides information on the is doing so in the present issue. INTAS programme, the Digital Math ERCOM initiatives and, in particular on As Peter Hilton already wrote twenty years Library, issues arisen from the EMS open positions offered by the centres and ago “A research institute is at least two things Executive Committee, etc. scheduled conferences and training courses. EMS September 2004 3 EMS NEWS EMSEMS CouncilCouncil MeetingMeeting Uppsala, Sweden, 26th-27th June 2004 David Salinger, EMS Publicity Officer

The Council of the European Mathematical Society (its “parliament”) met in the weekend before the European Congress in Stockholm on the premises of Uppsala University, about 60 kilome- tres north of the Swedish capital. There were 65 delegates representing 43 Mathematical Societies from all over Europe, 14 delegates representing individual members, and institutions were represented by 9 delegates. The EMS executive committee and several invitees made up the rest of the assembly.

Membership issues, budget and elec- tions Council had its share of routine business, but finished with several lively discussions. The routine business was, however, important, starting with the approval of the membership applications from the Mathematical Societies of Malta, Cyprus and Romania and the Statistical Societies of Belgium, Sweden, Norway Host Sten Kaijser, Uppsala Univ., and EMS-President Sir John Kingman and Spain. The President underlined two impor- Committee. The President thanked the some work on the Pisa report and work tant facets of the Society’s many activi- outgoing members of the Committee: closely with the education committees of ties: its lobbying for mathematics at a Bodil Branner, Marta Sanz-Solé, and national societies. European level, in which Luc Lemaire Mina Teicher. Vagn Lundsgaard Hansen talked about played an invaluable role; and the suc- the activities of the Committee for cessful establishment of its own publish- Reports and Debates Raising Public Awareness and the neces- ing house, for which Rolf Jeltsch had Martin Raussen, editor of the Newsletter sity for mathematicians to engage with been the driving force. All activities reported on a smooth handover from politicians, journalists and the general depended on a small group of dedicated Robin Wilson. Much of the Newsletter public. This stimulated several sugges- people, in particular he warmly thanked would be made available on EMIS, and tions and examples from delegates, such Secretary Tuulikki Mäkeläinen, but the it was hoped that the Book Reviews and as the forthcoming public activities in Society needed more volunteers as its Conferences could be made available in Copenhagen during the ICME meeting, work expanded. The President com- a searchable form. Council thanked links on EMIS and a website for teachers mended the willingness of mathemati- Robin Wilson and Martin Raussen for (like that of the DMV). cians to serve on the recently created their work for the Newsletter. There was a report from the Special Scientific Advisory Panel. Council There had been some excellent work Events Committee and from the Eastern agreed to send congratulations to Henri on behalf of the applied mathematics Europe committee, whose principal Cartan on the occasion of his 100th community, including the AMAM con- function remains the allocating of travel birthday. ference in Nice and the focus on applica- grants. The financial statements, presented by tions of mathematics in the forthcoming Luc Lemaire reported on the success Olli Martio, were accepted. The finances 4ecm, but the applied mathematics com- of the General Meetings Committee in were in good health, but a slight rise in mittee had been unable to decide attracting a grant of half a million euros the corporate membership fee was need- whether to recommend a separate series for EMS Summer Schools. A new round ed, which Council approved. of JEMS for applied mathematics and of applications was due. The Pavel Exner was elected as Vice- there had been a loss of momentum in its Mathematical Weekends had started President; Victor Buchstaber, Olga Gil- activities. with a successful meeting in Lisbon in Medrano, Carlo Sbordone, and Klaus Mina Teicher was the new chair of the 2003, with another due in Prague this Schmidt as ordinary members of the Education Committee. She would do September and a further one planned for 4 EMS September 2004 EMS NEWS a strategy, but should not be allowed to distort or dominate. Laurent Guillopé spoke of the French government’s review of Zentralblatt-MATH. Reporting on the work of the Committee on Women and Mathematics, Emilia Mezzetti spoke of the limited success of the questionnaire, so that it was not possible to base action on it. But the Committee was collecting data and this did show some improvement in the position of women, particularly at starting levels. The Committee was also involved with mentoring schemes. Council concluded with a discussion on the position of mathematics within European research programmes. The themed networks were not very good for mathematics. In the Marie Curie and other schemes, the success rate, though Delegates and members of the EC during a break low, was fixed to be same in each subject. A delegate spoke passionately Barcelona in 2005. On the other hand, 2008. about the time-consuming and demean- there had been no suggestions received The President spoke of the importance ing bureaucratic hurdles encountered if for EMS lecturers. of having a viable Zentralblatt-MATH, an application was successful. Though a A short report on the state of the as a counterweight to Mathematical report had come out in favour of a Bologna process was followed by a stim- Reviews-MathSciNet. There were cur- European Research Council with signifi- ulating, and occasionally agonised, dis- rently some difficulties, in that the finan- cant funding, the political will did not cussion. The President summed up by cial base (mainly German and French) appear to be there; nor did this appear to saying that the Society should be willing needed to be broadened and that the pro- be a high priority for the accession coun- to back the mathematical community in ject rested heavily on the shoulders of tries. particular countries rather than formulat- too few people. There was a need for a In his closing remarks, the President ing a general policy. scientific advisory board, not to manage, appealed to delegates to bore people Thomas Hintermann, managing direc- but to provide scientific input. An EU about the importance of mathematics, tor of the EMS Publishing House, spoke funding application (two had failed) particularly the development of abstract of its progress. From this year, EMSph should be a secondary activity to support mathematics as ‘tomorrow’s tools’. was publishing JEMS, Interfaces and Free Boundaries, Oberwolfach Reports and, from 2005, Commentarii Helveticae Council agreed that the search for a new EMS-president should and Elemente der Mathematik would begin now. The terms of office is four years starting with January join the list. Book series had also started. A delegate suggested reduced prices 2007. Members of the Society are invited to send suggestions to for developing countries. President Sir John Kingman at [email protected] The Committee on Developing Countries continued to be very active. Herbert Fleischner said the books project was continuing, and the Committee was trying to increase the geographical spread of its work. It had a working fund with contributions from EMS and Zentralblatt-MATH. John Ball spoke of the work the IMU was doing. Helge Holden talked about the Digital Mathematical Library project. This was being pursued at the national level; the EMS might have a co-ordinating role, but it could not fill it without additional volunteers. John Ball spoke of the importance with which the IMU regards this project, which was moving well. Council unanimously agreed to hold the fifth European Congress of The Newsletter's Editorial Board held a meeting at Uppsala, too. From left to right: Martin Mathematics in Amsterdam in July Raussen, Steen Markvorsen, Krzysztof Ciesielski, Robin Wilson. EMS September 2004 5 EMS NEWS all expenses of key speakers which are non eligible. CALL FOR PROPOSALS The proposers should be aware of the CALL FOR PROPOSALS rules of funding by the EU: the travel and living expenses of all eligible researchers EMS SUMMER SCHOOLS IN FUNDAMENTAL AND can be covered, up to a rather generous INTERDISCIPLINARY MATHEMATICS maximum determined by the EU. If a fee is charged to all participants, the EU will reimburse the fees of eligible ones but then Following the success of its first E.U. ences, contact making opportunities…) subtract this amount from the expenses application (which allows the funding of - Management and feasibility (organisa- below. All other expenses, including eight Summer Schools or Conferences in tion and management, plans for public- expenses of non-eligible key speakers, will 2004-2005, see the EMS agenda page 2 or ity, dissemination of the results by pub- be covered at a proportion equal to the ratio http://www.emis.de/etc/ems-summerschools. lications, web sites, any other means…) eligible participants / all participants. html), the European Mathematical Society - Relevance to the objectives of the EU The EU will pay a maximum of 80 % in is now launching a new call for proposals action and added value to the EU (rele- advance, the last 20 % only when all for such Schools and Conferences for 2006, vance of the themes in relation with reports are in, and possibly a year later. 2007 and 2008. The deadline for this call is European interests and achievements, Thus a statement from an organisation January 12, 2005, by e-mail to the measures taken in favour of gender bal- (university, conference centre…) must address: llemaire@ ulb.ac.be. ance, use for integration of less devel- accompany proposals, agreeing that it will The deadline will allow the EMS to pre- oped regions, plans to favour public cover the complement of organisational sent a coherent proposal of activities for understanding of science…). expenses, and also be ready to advance 20 EU funding, thereby allowing organisers of % of the funding until payment from the single meetings to be part of a series of Concerning the last two criteria, part of the EU has arrived. events. EMS direct support being limited, value comes from the EMS organisation, To give an idea, for the eight schools in the result of this application will make a and this will be added in the final applica- the present project, we had a total of 140 major difference to the funding for the tion. However, any little point will count, participants of category 1, 155 of category meetings selected by EMS. There will be and each organiser can help by stressing 2, 20 of category 3. Their funding amounts similar calls every two or three years in the qualities. to 366 000 euros, the organisational future. Proposers are asked to present a project expenses of each event varies between This call for proposals concerns all as detailed as possible, with (provisional!) 12000 and 17 000 euros, of which the EU Summer schools or Conferences that any lists of speakers, subjects, and addressing covers the prescribed percentage. group of mathematicians – pure or applied all questions above. They should also Note that by a new rule, EU will not fund - would like to run in 2006, 07 or 08 in the include an estimate of the number of eligi- series of events in a single specific subject. EU or associated states. ble participants for each of the three cate- In case that several proposals submit a par- The EU guidelines for these events must gories (EU or associated states and less ticular theme, the EMS will have to make a be followed strictly in order to have a rea- than 4 years research experience; less than choice or have to determine if a merger sonable hope for funding (the success rate 10 years; no time limit but a researcher between these proposals makes sense. for EU-applications being extremely low). from EU or associated states living outside Thus, there must be a very strong com- that zone). For these, no financial estimates Luc LEMAIRE ponent of training of young researchers (in should be given. An estimate of “organisa- Phone: 00-32-2-6505837 the first 10 years of their career) by means tional” expenses is also needed, including e-mail: [email protected] of integrated courses and lectures at an advanced level. These can be supplement- ed by conference type research lectures, but EURYI Call for Proposals the training component is needed. Each course must aim at an international European Young Investigators Awards audience (no more than 30% of participants The European Heads of Research Councils (EUROHORCS) in collaboration should come from a single state; otherwise with the European Science Foundation (ESF) have created the European Young the event in not eligible for funding). Investigator (EURYI) Awards, launched for the first time in 2003. Awards will The EMS will make a selection amongst last for a 5 year period with a total value normally no less than 750 kEuro. In the proposals received, taking into consid- July 2004, 25 awards were granted after the first call. eration the criteria applied by the EU, i.e.: Young researchers from anywhere in the world with between 2 and 10 years - Scientific quality of the project (high postdoctoral experience – taking into account career breaks – are eligible. The quality, cutting edge subject, speakers selection criteria for EURYI Awards are the research quality and potential of the of high calibre…) applicant and the originality as well as the groundbreaking character of the - Quality of the research training activity research proposal and its feasibility. (training provided to the young Applications must be submitted through a national EUROHORCS organisa- researchers, level of interaction tion no later than November 30, 2004. They will then be assessed first at the between the main speakers and the national level and later at the European level. Awards are expected to be young ones…) announced at the end of July 2005. Detailed information can be obtained from - Quality of the host (quality of infra- www.esf.org/euryi . structure, experience in running confer- 6 EMS September 2004 EMS NEWS SStockholmtockholm 20042004 TheThe FourFourthth EurEuropeanopean CongrCongressess ofof MathematicsMathematics Ulf Persson (Göteborg, Sweden)

How it came about together with the recent acceleration in the In spite of a difficult start, the fourth number of specialized conferences, often European Congress in Mathematics (ECM) referred to as workshops to emphasize their took place after all. As its President Ari focused and businesslike character, has Laptev revealed in an earlier issue of this sown doubts in the minds of many as to the newsletter, its eventual location in scientific relevance of such gargantuan Stockholm was not initially planned but meetings, which, cynics protest, are more was the result of a crisis perceived in the the occasion for tourism than serious scien- summer of 2000. Despite what many out- tific interchange. In such a perspective, the conveying of an idea (usually one is siders may have assumed, finding sufficient establishment of a European Congress may enough) and some motivation (which funding from Swedish sources was not a seem redundant. One may point out though should not be confused with justification straightforward matter (as the first bid that the situation of mathematics is not often of the type: ‘this has applications to made painfully clear). It was only due to the unique, but is shared by most academic dis- physics’) and placing the subject matter in daring and ‘savoire faire’ of Ari Laptev and ciplines, and that it is very important, a wider mathematical context. Apart from his colleague at Kungliga Tekniska among other things, to counter the frag- that, a good lecture can contain highly tech- Högskolan (KTH), Anders Lindquist, that mentation of a discipline with opportunities nical material, and there is no reason why sufficiently promising leads were extracted for the contemplation of it as a whole. For the audience should understand most of it from some of the major Swedish funding this to be fully successful there is both a as long as they have gotten something to institutions, thus enabling the Swedish need for the lectures to be directed to a gen- take home with them. It would be very dan- Mathematical Society to send a small dele- eral mathematical audience and for that gerous if mathematicians were to abandon gation to the executive meeting in London audience to seek out lectures not primarily their tradition of honesty and aim for mere- later that fall for negotiations. However, a within their own speciality. Popular sur- ly the ‘flashy’. To reflect on why your work final commitment to hold the congress was veys do not rank highly among the priori- is interesting and to try to truly motivate it, not made until just a week before Christmas ties of research mathematicians. They are should not be seen as a concession to igno- - after a rather tense encounter with the then notoriously difficult to do and the rewards rance, but rather as an additional source of President of the European Mathematical are marginal. A scientific committee selects inspiration for your own research. Society (EMS), Rolf Jeltsch. The meeting their choices primarily on the basis of sci- only came to a desired conclusion after a entific excellence and topicality of the sub- New features final short interview with the President of ject matter, not on expository skills; and Obviously this is not the place to com- the KTH, Anders Flodström, in which the notwithstanding the desirability of the latter ment upon how successfully the various latter agreed to the necessary financial this is basically a sound principle, tamper- speakers performed their task of communi- underwriting. The upshot was that the plan- ing with which would court disaster and cating to a general mathematical audience. ning and organization of the congress was seriously undercut the legitimacy of the Fortunately, although few lectures can be shortened by at least a year from what has whole enterprise. Thus selection as a speak- expected to please everyone, the lecture is been customary for this kind of event. er is seen primarily as recognition of scien- rare indeed that does not bring rewards to at tific merit and not as an opportunity, as well least a token number in the audience, and Why big congresses? Was it worth it? as an obligation, to communicate effective- one may argue that this is indeed all that is This is obviously not the place to delve ly. needed to make it worthwhile. However, deeper into that philosophical question, yet What is needed is a change of culture, there was a feeling among the organizers a few reflections may not be amiss. In the something that cannot be decreed but has to that something extra was needed to justify good old days, which meant up into the evolve slowly. On what constitutes a good yet another general conference in mathe- fifties, conferences were few and small, and lecture, one can of course argue, there obvi- matics, as well as serving as a rejuvenation in particular the International Congresses of ously being no specific rules on which of the concept. Two new features were Mathematicians (ICM) were rather select everyone can agree. And besides, in all added. One was to invite scientists in natur- and as a consequence not bigger than kinds of human interaction rules are simply al science, not only in chemistry and allowing group portraits to be taken. But there to be broken. It suffices to point out physics, but also mathematical biologists, when the ICM was held in Stockholm back that a scientific lecture, and especially a to give lectures. The other was to latch on in 1962, it was the largest scientific con- mathematical one, is not necessarily made to the pre-existing structure of the gress that had ever been held in Sweden. more accessible by the simple process of European Networks and thus give to the Since then the ICM have turned from rather ‘watering down’ (i.e. removing technicali- ECM a natural European anchoring. The exclusive meetings to large affairs attended ties and making unwarranted simplifica- first may be seen as somewhat controver- by the ‘mathematical masses’. This fact, tions). What is required of a lecture is the sial, wedding mathematics and its ultimate EMS September 2004 7 EMS NEWS justification too tightly to applications. Yet, (physically) distant from both. It is of fairly lunchtime, there was the temptation of the whether we like it or not, practical mathe- recent vintage, located on the main campus official cafeteria situated halfway between matical applications are what brings in of the University of Stockholm, easily the two locations (actually accessible from material resources, and for public relations accessible by the Stockholm Underground the Aula Magna remaining indoors the the importance of the willingness of mathe- (‘T-banan’, ‘T’ as in tunnel). Shaped like entire way, a godsend in the case of maticians to interact externally, and in the an amphitheatre, with options of subdivi- inclement weather). It provided the stan- process maybe also acquiring greater pub- sion, the Aula actually boasts a capacity dard Swedish lunch fare to be expected lic visibility, should not be underestimated. well in excess of the number of actual par- from that kind of self-serve establishment, From a less opportunistic standpoint, ticipants (around 900), which resulted in confidently assured of a captive audience, applications should not merely be seen as the somewhat unfortunate impression of not only by its isolated existence but also necessary justifications, but also as sources not only the lectures but also the opening due to pre-paid lunch coupons. of inspiration. The second new feature may hopefully ensure that the new tradition of ECM’s continues. The funding and organization of a big international congress is indeed a major undertaking and the difficulties involved may be expected to increase with time rather than decrease. The European Union has already invested a formidable apparatus of networks replete with their own special conferences. What would be more natural than a unifying one, which should bring with it a large body of active participants, and hopefully also channels for necessary funding? At Stockholm, admittedly the network lectures played a rather marginal role, but if the idea is accepted and developed in future ECM’s they may provide the core activity onto which various extras may be attached as embellishments.

Aula Magna After those general preambles it may be 4ecm staff studying new books at the booth of the EMS publishing house appropriate to describe the Fourth European Congress of Mathematicians as ceremony being sparsely attended. As The opening ceremony an actual event tied to a physical location at expected, the Aula along with its adjacent The conference got a head start on Sunday a given slot in time. The basic problem of hallways became the locus of the meeting. afternoon, on June 27, by providing regis- organizing a conference for one thousand Walking along its perimeter you had imme- tration outside the Aula. This involved get- odd expected participants is to find a lec- diate access to the young assistants donning ting a handy black briefcase, doubling as a ture hall big enough. Unlike the case of the yellow T-shirts, as well as to the various rucksack, with the logo of the 4ECM suffi- Olympics, the erection of new buildings is bookstalls providing not only opportunities ciently discreet to encourage post-confer- not an option. Of course big conferences for browsing but also seducing discounts. ence use. It would be tedious to list its con- are legion these days, but commercially The Springer stall also supplied piles of tents of ‘goodies’, but I am sure that most available space often comes with price-tags copies of the Stockholm Intelligencer (still people appreciated the free public transport not within the capabilities of mathematical smelling of fresh print), featuring short arti- passes intended to cover the entire period. meetings. The Royal Swedish Institute of cles on Sweden and mathematics, including One of its more trivial items was a coupon Technology (Kungliga Tekniska a short but morbid list of distinguished to be exchanged for a single glass of wine Högskolan - KTH) was not able to provide mathematicians who died here. Climbing a (of optional colour) to be served at the such a hall on its premises, making a direct few stairs you could also inspect the vari- hour-long reception starting at six o’clock. collaboration with the University of ous poster-sessions. Additional smaller lec- The next day, Monday June 28, involved Stockholm a necessity on this basis alone. ture halls were available at the proverbial the official opening of the meeting at nine The fact that there are two separate depart- stone-throws distance, as well as what in thirty in the morning, preceded by an ments of mathematics in Stockholm has recent years has become an absolute neces- opportunity for last-minute registrations. incidentally been a bone of contention for sity for wayward mathematicians - access Back at the ICM of 1962, the old Swedish at least fifty years, and may continue to be to e-mail. A fairly spacious room, accessi- King Gustavus VI attended, giving out the so for another fifty years, but this is of ble only by pressing a code at the entrance, Fields Medals. Such a spectacle of royalty course only a matter of local interest. The was filled with a sufficient number of com- at a mathematical meeting was, alas, not Aula Magna is the official grand lecture puters to keep waiting lines rather than tem- repeated this time. The presence of the hall of the University of Stockholm, and pers short. Furthermore, a small staff of Swedish Majesty back then has been attrib- has of course no direct connection, physical knowledgeable yellow-shirts was always at uted to the above mentioned fact that at the or not, to its department of mathematics, hand during opening hours. For those able time, it was the largest scientific meeting nor to that of the KTH, being about equally to resist the allure of the screen at ever to have taken place in Sweden. No 8 EMS September 2004 EMS NEWS such distinction could be given to the The week in review the customary juggle of balancing your 4ECM, so neither the old King’s grandson, The conference was kicked off by the first wineglass, your plate, and your knife and the present King Charles XVI, nor the plenary lecturer, Oded Shramm, associated fork with just two hands, while standing great-granddaughter, the crown princess with Microsoft Research. An appropriate firmly on your feet and trying to make Victoria, were able to squeeze the events beginning in view of the fact that President coherent conversation. Afterwards into their busy schedules. (One should not Laptev, earlier in the ceremony, got stuck Kingman rose to the occasion thanking the overestimate the love the general public, on his power-point presentation, cursing hosts referring to the great success of the including that of royalty, feels for mathe- modern computer technology. Schramm, conference (so far). He concluded by deliv- matics) Instead two academic notables, however, did not address such wordly ering a splendid celebration of the impor- Bremer and Franke, provided the required issues of practicality, but expounded on tance of mathematics, reminding everyone official lustre. The latter, Sigbrit Franke, conformally invariant random processes that while back in 1900, some of the scien- the chancellor of the university system in instead, albeit with many a computer visu- tific accomplishments honoured at this very Sweden, also served the function of award- alization. And then there was time for place involved no mathematics, nowadays ing the EMS prizes to the young mathe- lunch, and in spite of the customary denial this would be almost impossible, and that maticians, to which we will return shortly. of the existence of such entities, free to all all scientists should acknowledge this fact. Both Bremer and Franke, not surprisingly, participants. The afternoon was devoted to In fact, he reminded us all that the entire made a special point in referring to Sofia parallel sessions, four in fact, involving human race will benefit from the develop- Kovalevskaya in their short speeches. twelve lectures in total. The day was ment of mathematics. In order to gently Kovalevskaya, as can have escaped the capped off by an EMS reception at the old usher out the mathematical crowd from the notice of few mathematicians, was the first location for the department of mathematics premises, a guided tour was offered at the ever woman professor in mathematics, at KTH, a building commonly referred to as end providing a natural conclusion. For holding her position at the precursor of Sing-sing, due to its intimidating lay-out. those of us who afterwards lingered on by Stockholm University at the end of the 19th The reception wisely took place on the the waterfront, we may late forget the glo- century, and ever since then being a mathe- ground floor, the limited space of which rious view made sublime (as one used to matical role model for half the population quickly got extremely crowded. One sur- say) by the slowly setting sun. The tradi- of the world. Kingman, in his capacity as mises that afterwards there were only tional association of Stockholm with the President of the EMS, gave the mathe- empty wine-bottles among the left-overs. Venice, made particularly palpable by the matical speech with commendable aplomb, The next day started out with presentations Town Hall, is not plucked out of thin air, stressing the importance of mathematics, of the prize winners and their work (it but rests on water. Wednesday was taken and urging young mathematicians to go for should be noted that some of them had also, over by plenary talks in the morning and the hard problems, irrespective of whether independently one assumes, been invited as three additional science lectures in the pure or applied, because you can never tell. regular speakers as well.) Those were fol- afternoon. The Austrian Nowak expounded He also encouraged the new generation by lowed by invited lectures and then in the on mathematical biology with special reminding them that they are as great and afternoon there were three Science lectures. emphasis on evolutionary processes, and imaginative as the great ones of the past, The last one was that of R.Ernst, a Nobel Berry presented a cascade of computer gen- because what puzzled our ancestors does Prize winner in Chemistry, giving a survey erated pictures relating to optics and con- not necessarily puzzle us. The chair of the from Fourier to Medical imaging, constitut- comitant singularities. Thursday was only Prize Committee, Nina Uraltseva from St. ing no doubt a very instructive lecture to half a day, with invited lectures in the Petersburg, who complained that she was a the mathematicians, giving them, among morning and scheduled excursions in the bit too short for the microphone, explained other things, a sense of the somewhat alien afternoon. In the evening, the French that the work of the prize committee had culture of big science. Tuesday was capped ambassador gave a reception for the nota- been very hard. There were fifty nominees, off by a visit to the Town Hall of bles of the EMS, the organizers and last but out of which ten had to be selected. To Stockholm replete with a buffet courtesy of not least the young prize-winners. France those who did not make it, there is only one the city of Stockholm. The Town Hall, as a country and culture should be com- thing to say - work harder and try to outdo designed by the Swedish architect Ragnar mended for the official respect it accords those who were selected. This is not the Östberg and built in the twenties, is one of mathematics and for always recognising place to give a list of the winners, along the most commonly pictorially mathematical achievement. I fear that the with short descriptions of their accomplish- reproduced landmarks of the city, located at 4ECM may very well have received more ments. It suffices to add that in addition to the edge of an island, and commanding a publicity in France than in Sweden. Friday the traditional EMS prizes, a so called BIT presence on the main waterway. Its design was the closing up, with parallel network prize was awarded for work in numerical was inspired by the palace of the dodges in lectures in the morning and a series of ple- analysis. No official ceremony is complete Venice and its main feature is the great ban- nary talks in the afternoon, the last fittingly without some kind of entertainment. The quet hall inside, somewhat puzzlingly delivered by a local speaker, Johan Håstad musical interlude in this case was provided known as the Blue Hall (guides of the at KTH, talking on the difficulty of proving by a small Swedish ensemble specializing building are just delighted to explain to you the generally believed NP≠P. in Elizabethan music but also performing, the historical reason why), which every fittingly, some classical Swedish songs. December is host to the Nobel banquet fol- Wrapping up Dressed in period costumes, they did their lowing the prize awards at the Concert The concluding ceremony was, as such thing, singing with their own instrumental Hall. This time however, the setting was things tend to be, rather short. Kingman support, concluding their act by strutting somewhat less sumptuous. Two grand buf- expressed the pride of the EMS to be asso- around the stage pilfering the belongings of fets (presumably identical) were displayed ciated with the ECM and thanked the orga- those unfortunate enough to have the privi- on two long tables on one side. Food was nizers, and especially its President. There lege of sitting on the first row. plentiful, but to savour it you needed to do was a lot of applause. Then there was a ref- EMS September 2004 9 EMS NEWS erence to the upcoming centenary birthday of Henri Cartan, who had an honorary role in the first ECM which took place in Paris EMSEMS PrizesPrizes 1992, and to whom the entire congress relayed its congratulations. Laptev then referred to the statistics of the event, with EMS Prizes 2004 with citations the final tally of 930 members, over three hundred posters, and sixty three scheduled Franck Barthe, Institut de talks, of which only one had to be can- Mathématiques: Laboratoire de celled. As expected he could not refrain Statistique et Probabilités, Toulouse, from congratulating himself on having France arranged such nice weather for the entire The EMS prizes are awarded by the Barthe pioneered the use of measure- duration. No mean feet indeed, and a defi- European Mathematical Society in recog- transportation techniques (due to nite contribution to the general pleasure. nition of distinguished contributions in Kantorovich, Brenier, Caffarelli, Mc Finally the congress was treated to an invi- Mathematics by young researchers not Cann and others) in geometric inequali- tation to the fifth ECM to take place in older than 35 years. The prizes are pre- ties of harmonic and functional analysis Amsterdam in 2008. After the usual prob- sented every four years at the European with striking applications to geometry of lems with an unresponsive machine, the Congresses of Mathematics. convex bodies. His major achievement is audience was eventually exposed to a The Prize Committee is appointed by an inverse form of classical Brascamp- power-point show by the representative of the EMS and consists of a number of rec- Lieb inequalities. Further contributions the next ECM, promising future delights. ognized mathematicians from a wide vari- include discovery of a functional form of By the time the audience emerged out of ety of fields. The prizes were first award- isoperimetric inequalities and a recent the Aula Magna for the last time, bookstalls ed in Paris in 1992 and then in Budapest in solution (with Artstein, Ball and Naor) of were being dismantled, posters taken down, 1996 and in Barcelona in 2000. a long-standing Shannon’s problem on and the last remaining conference briefcas- Each prize winner 2004 received 5,000 entropy production in random systems. es sold off at very reasonable prices. Euro. Clearly within hours, no physical traces Stefano Bianchini, Instituto per le would be left of the congress. On the other Prize committee Applicazioni del Calcolo “M. Picone”, hand, one would hope that mental traces, Enrico Arbarello, - Rome Rome, Italy preferably of the positive and edifying Victor Buchstaber, - Moscow Stefano Bianchini has introduced an kind, will remain for a very long time. John Coates,- Cambridge, UK entirely new perspective to the theory of Jacek Graczyk, - Orsay discontinuous solutions of one-dimen- Ulf Persson [[email protected]] has Bertil Gustafsson, - Uppsala sional hyperbolic conservation laws, rep- been professor at Chalmers University of Stefan Hildebrandt, - Bonn resenting solutions as local superposition Technology in Gothenburg (Göteborg) Jean-François Le Gall, - Paris of travelling waves and introducing inno- since 1989. He earned his Ph.D at Harvard Vladimir Lin, - Haifa vative Glimm functionals. His ideas have 1975 as a student of David Mumford. Leonid Polterovich, - Tel Aviv led to the solution of the long standing Works in Algebraic Geometry, especially Domokos Szasz, - Budapest problem of stability and convergence of compact complex surfaces. Hobbies Dimitri Yafaev, - Rennes vanishing viscosity approximations. In include mathematical pictures pro- Eduard Zehnder, - Zürich his best individual achievement, pub- grammed directly in PostScript. Has served lished in 2003 in Arch.Ration. Mech. as the President of the Swedish Anal., he shows convergence of semidis- Mathematical Society and has been active- crete upwind schemes for general hyper- ly engaged in public debate on mathemati- bolic systems. In the technically demand- cal education. Publishes regularly reviews ing proof the travelling waves are con- on scientific-philosophical matters in the structed as solutions of a functional equa- Swedish press and is the editor of the tion, appling center manifold theory in an newsletter of the aforementioned Swedish infinite dimensional space. Mathematical Society. Paul Biràn, School of Mathematical Sciences, Tel-Aviv University, Israel Paul Biràn has made fundamental and EMS Prize winners 2004 with the presi- influential contributions to symplectic dent of the Prize committee, Nina topology as well as to algebraic geometry Uraltseva. Sitting from the left: Xavier and Hamiltonian systems. His work is Tolsa, Paul Biràn, Sylvia Serfaty, characterised by new depths in the inter- Stefano Bianchini, Otmar Venjakob. actions between complex algebraic geom- Standing from the left: Franck Barthe, etry and symplectic topology. One of the Warwick Tucker, Nina Uraltseva earlier contributions is his surprising Starting the race towards 5ecm at (President of the Prize committee), Elon solution of the symplectic packing prob- Amsterdam in 2008. From left to right: Lindenstrauss, Andrei Oukonkov, lem, completing work of Gromov, H.J.J. te Riele,J.O.O. Wiegerinck, C.M. Ran Stanislav Smirnov. McDuff and Polterovich, showing that (http://sta.science.uva.nl/ brandts/5ECM/) 10 EMS September 2004 EMS NEWS compact symplectic manifolds can be important and influential result of affirmatively Melnikov’s conjecture, packed by symplectic images of equally Okounkov is a formula he found in joint Tolsa provides a solution of the Painlevé sized Euclidean balls without wasting work with Borodin, which expresses a problem in terms of the Menger curva- volume if the number of balls is not too general Toeplitz determinant as the ture. Xavier Tolsa has also published small. Among the corollaries of his proof, Fredholm determinant of the product of many important and influential resultsre- Biràn obtains new estimates in the Nagata two associated Hankel operators. The lated to Calderón-Zygmund theory and problem. A powerful tool in symplectic new techniques of working with random rational approximation in the plane. topology is Biràn’s decomposition of partitions invented and successfully symplectic manifolds into a disc bundle developed by Okounkov lead to a striking Warwick Tucker, Uppsala University, over a symplectic submanifold and a array of applications in a wide variety of Sweden Lagrangian skeleton. Applications of this fields: topology of module spaces, ergod- Warwick Tucker has given a rigorous discovery range from the phenomenon of ic theory, the theory of random surfaces proof that the Lorenz attractor exists for Lagrangian barriers to surprising novel and algebraic geometry. the parameter values provided by Lorenz. results on topology of Lagrangian sub- This was a long standing challenge to the manifolds. Paul Biràn not only proves Sylvia Serfaty, Courant Institute of dynamical system community, and was deep results, he also discovers new phe- Mathematical Sciences, New York, USA included by Smale in his list of problems nomena and invents powerful techniques Sylvia Serfaty was the first to make a sys- for the new millennium. The proof uses important for the future development of tematic and impressive asymptotic analy- computer estimates with rigorous bounds the field of symplectic geometry. sis for the case of large parameters in based on higher dimensional interval Theory of Ginzburg-Landau equation. arithmetics. In later work, Warwick Elon Lindenstrauss, Clay Mathematics She established precisely the values of the Tucker has made further significant con- Institute, Massachusetts and Courant first, second and third (with E.Sandier) tributions to the development and appli- Institute of Mathematical Sciences, New critical fields for nucleation of one stable cation of this area. York, USA vortex, vortex fluids and surface super- Elon Lindestrauss has done deep and conductivity. In micromagnetics, her Otmar Venjakob, Mathematisches highly original work at the interface of work with F. Alouges and T. Rivière Institut: Universität Heidelberg, Germany ergodic theory and number theory. breaks new ground on singularly per- Otmar Venjakob has made a number of Although he has worked widely in ergod- turbed variational problems and provides important discoveries in both the algebraic theory, his recent proof of the quantum the first explanation for the internal struc- ic and arithmetic aspects of non-commu- unique ergodicity conjecture for arith- ture of cross-tie walls. tative Iwasawa theory, especially on metic hyperbolic surfaces breaks fertile problems which appeared intractable new ground, with great promise for future Stanislav Smirnov, KTH, Sweden and from the point of view of the classical applications to number theory. Geneva University, Switzerland commutative theory. In arithmetic geom- Already, in joint work with Katok and Stanislav Smirnov’s most striking result etry, Iwasawa theory is the only general Einsiedler, he has used some of the ideas is the proof of existence and conformal technique known for studying the myste- in this work to prove the celebrated con- invariance of the scaling limit of crossing rious relations between exact arithmetic jecture of Littlewood on simultaneous probabilities for critical percolation on formulae and special values of L-func- diophantine approximation for all pairs of the triangular lattice. This gives a formu- tions, as typified by the conjecture of real numbers lying outside a set of la for the limiting values of crossing prob- Birch and Swinnerton-Dyer. Venjakob’s Hausdorff dimension zero. This goes far abilities, breakthrough in the field, which work applies quite generally to towers of beyond what was known earlier about has allowed for the verification of many number fields whose Galois group is an Littlewood’s conjecture, and spectacular- conjectures of physicists, concerning arbitrary compact p-adic Lie group ly confirms the high promise of themeth- power laws and critical values of expo- (which is not, in general, commutative), ods of ergodic theory in studying previ- nents. Stanislav Smirnov also made sev- and has done much to show that a rich ously intractable problems of diophantine eral essential contributions to complex theory is waiting to be developed. His approximation. dynamics, around the geometry of Julia most important results include the proof sets and the thermodynamic formalism. of a good dimension theory for modules Andrei Okounkov, Princeton over Iwasawa algebras, and the proof of University, USA Xavier Tolsa, ICREA and Universitat the first case of a structure theory for Andrei Okounkov contributed greatly to Autònoma de Barcelona, Spain modules over these algebras. With the field of asymptotic combinatorics. An Xavier Tolsa has made fundamental con- Hachimori he discovered the first exam- extremely versatile mathematician, he tributions to Harmonic and Complex ples of arithmetic Iwasawa modules found a wide array of applications of his Analysis. His most outstanding work which are completely faithful, as well as methods. His early results include a proof solves Vitushkin’s problem about semi- proving a remarkable asymptotic upper of a conjecture of Olshanski on the repre- additivity of analytic capacity. The prob- bound for the rank of the Mordell Weil sentations theory of groups with infinite- lem was raised in 1967 by Vitushkin in group of elliptic curves in certain towers dimensional duals. Okounkov gave the his famous paper on rational approxima- of number fields over Q whose Galois first proof of the celebrated Baik-Deift- tion in the plane. Tolsa’s result has impor- group is a p-adic Lie group of dimension Johansson conjecture, which states that tant consequences for a classical (100 2. Very recently, he found the key to the the asymptotics of random partitions dis- years old) problem of Painlevé about a problem of defining, in non-commutative tributed according to the Plancherel mea- geometric characterization of planar com- Iwasawa theory, the analogue of the char- sure coincides with that of the eigenval- pact sets are removable in the class of acteristic series of modules over Iwasawa ues of large Hermitian matrices. An bounded analytic functions. Answering algebras. EMS September 2004 11 NEWS

VVisionsisions ofof YYoungoung MathematicsMathematics inin EurEuropeope Jon Larsson and Mikael Johansson (Stockholm, Sweden)

Jon Larsson Mikael Johansson

During the week immediately prior to the organisers realised that a periodicity of Thrilled and excited after the experience, European Congress of Mathematics, one four years is much too sparse for the we were determined to seek out similar of the congress’s many satellites took intended age group, as a student would events back in Sweden, or - if need be - place at the Royal Institute of be able to both start and finish secondary bring them there. Technology (KTH) in Stockholm. This studies - or for that matter higher studies particular one differed slightly from most – between congresses and without any Stockholm 2004 satellite events in that the participants of contact with the congress activities. When news that the 4ECM was to be held this conference were aged 14-20 and not Thus, the third JMC was held in in Stockholm reached us three years ago, yet active research mathematicians. Potsdam, in conjunction with the ICM in it was clear that the JMC of 2004 must Berlin. The fourth was supposed to be also take place here - on our home turf. The Junior Mathematical Congress held in Spain, together with the With the help and guidance mainly of Dr The first Junior Mathematical Congress Barcelona ECM, but the intensity of Péter Körtesi of the University of was held in Paris, France, in conjunction arranging different activities was too Miskolc, and of the Swedish National with the first ECM. Several hundred high and thus in 2000, the JMC was held Centre for Mathematics Education, the young secondary school students, mainly in Miskolc. 2002 again saw a small JMC Stockholm district of the Swedish from France but also from several other in Miskolc, this time with mostly Federation of Young Scientists through European countries, gathered and lis- Hungarian participants. us started the preparations for the con- tened to seminars from participants and In Sweden, home of the authors of this gress. from invited lecturers. As the time came article, the JMC was unheard of until By the end of June 2004, after having for the second ECM in Budapest, the 1998, when Mikael got involved in the raised EUR 30,000 in funding and gath- JMC was this time held in Miskolc preparations for the Potsdam congress ered 40 participants from seven European (Northeast Hungary), again with a large during an exchange year in Germany. countries, it was finally time. A staff of international contingent present. Two years later, the two of us both twelve, all just a little older than the par- After the Miskolc event, some of the attended the fourth JMC in Miskolc. ticipants, got everyone through the week-

Congress participants at work ... and at dinner

12 EMS September 2004 NEWS long event which, according to the ques- Here in Stockholm, the work has now [email protected]. We believe the EJMS tionnaire at the end, was a great success. begun. The formation of a Swedish soci- has great potential and hope that you will The sixth JMC was indeed a success, ety for young mathematicians is under- share that belief. but it was also a lot of hard work, uncer- way in close cooperation with the tainty and confusion. During the course Swedish Federation of Young Scientists. Jon Larsson [[email protected]] is a stu- of preparations and the congress itself, With a national organisation in place, we dent of theoretical computer science at we experienced in a very profound way will have a starting point from which we the Royal Institute of Technology (KTH) the lack of continuity, established coop- can gather the contacts, support and in Stockholm, Sweden, and Chairman of eration and well-functioning communica- funding necessary to launch the EJMS. the Stockholm District of the Swedish tion and information networks in the In closing, we would like to make a call Federation of Young Scientists. young mathematics movements in for support to you, the mathematicians of Mikael Johansson Europe. It had been a difficult task at best Europe. If you feel that you can help us [[email protected]] has recently fin- to reach out to like-minded youngsters in any way, be it by founding a Junior ished his Master thesis at the Department throughout the continent and inform Mathematical Society in your country, of Mathematics of Stockholm University, them about the JMC’s existence and the providing us with funds or simply shar- Sweden. fact that we wanted them to attend it. ing valuable experience with us, we urge Both authors were members of the Something needed to be done. you to contact us by e-mail at 4ECM organising committee.

Founding a Mathematical Society In 2000, Péter Körtesi had already formulated the idea of cooperation across national borders on a more general level, Max-Planck-Institut für Mathematik in den Naturwissenschaften and as the work progressed, this idea Max Planck Institute for Mathematics in the Sciences matured in our discussions. A European Junior Mathematical Society would be The Max Planck Institute for Mathematics in the Sciences in Leipzig, Germany, is seeking a well fitted to supply the much needed Coordinating Scientist continuity between congresses by acting for the as formal principal of the JMC’s, keeping International Max Planck Research School records from earlier events and facilitat- Mathematics in the Sciences at the University of Leipzig ing personal contact between organisers The Max Planck Research School Mathematics in the Sciences is a cooperation of the Max Planck Institute for in different countries. On a more general Mathematics in the Sciences and the Department of Mathematics and Computer Sciences and the Department level, the EJMS can act as a common of Physics of the University of Leipzig. forum and a coordinating body for sever- It will lead interested graduate students towards research problems in the physical and life sciences. This al national or regional organisations involves a broad range of mathematical fields, including geometry, partial differential equations and functional within the field of young mathematics. analysis, stochastics, and discrete mathematics. Specific subject areas are:

The EJMS would not only be an asset - Partial Differential Equations and Material Sciences for congress organisers and assorted - Numerical Analysis and Scientific Computing NGO’s, but also, perhaps even more - Riemannian and Symplectic Geometry and Hamiltonian Systems - Quantum Field Theory, Particle Physics importantly, for the individual young - Algebraic Geometry, String Theory mathematicians on the continent. For the - Geometric and Functional Analytic Methods in Mathematical Physics teenager interested in mathematics, two - Stochastic Processes, Many Particle Systems - Complex Systems in Evolutionary Processes and Neurobiology of the main problems are often finding others who share their interest, and find- It is expected that candidates have a Ph.D. in one of the areas covered by the research programm and they are ing further mathematical stimulation and encouraged to pursue their research interest in parallel to the duties as coordinator of the research school. Ideally this research interest is in nonlinear pde or many particle systems. The coordinator will work closely guidance. These are both problems that with the speaker of the research school, Prof. Dr. Stephan Luckhaus. could hopefully be alleviated by the exis- His/her duties include the organization of the selection procedure of candidates, organization of lecture series tence of a Pan-European network of and summer schools, and the presentation of the IMPRS. He/she is responsible for the general management of the IMPRS and the support of its graduate students. young mathematicians. That network would also have a natural place in the We offer a position with diverse activities in an excellent research environment. The financial conditions greater quest to further and increase depend on the qualifications with a maximum salary base of BAT IIa/Ib (including Social Security benefits) interest in mathematics among young according to German public regulations. The position is temporary, with an initial appointment of three years and possible extension for another three years. The Max Planck Society is an equal opportunity employer and Europeans in general. encourages disabled persons to apply. It also aims at increasing the number of female staff members in fields These discussions were brought to a where they are underrepresented, and therefore encourages women to apply. larger audience at the JMC this summer. The successful applicant should, if possible, start the offered position not later than January 1, 2005. In order to Many good ideas were brought forth, by ensure full consideration, a curriculum vitae, publication list, statement of professional goals, and the name of youngsters and adults alike. The culmi- three references should be directed by October 15, 2004 to: nation was reached during the closing Prof. Dr. Stephan Luckhaus Max Planck Institute for Mathematics in the Sciences ceremony of the congress, when the par- Inselstraße 22 ticipants - young mathematicians from D-04103 Leipzig seven countries - unanimously declared Germany e-mail: [email protected] that it was the wish of the sixth JMC that a European Junior Mathematical Society Later applications will be considered as long as the evaluation process is going on. be founded as soon as possible. EMS September 2004 13 ICME-10 SOMESOME PERSONALPERSONAL REFLECTIONSREFLECTIONS ONON ICME-10ICME-10 Vagn Lundsgaard Hansen (Lyngby, Denmark)

In the second week of July 2004, the big easily brought them to the site of the con- quadrennial event in mathematical educa- gress. tion, or didactics of mathematics as we say in Europe, took place on the campus of the Researchers and Practitioners Technical University of Denmark situated The international congress on mathemati- at the outskirts of Copenhagen. We are cal education is the event that brings talking about the “10th International together active research mathematicians, Congress on Mathematical Education”, deeply devoted to develop and dissemi- abbreviated as ICME-10. nate their subject, and practitioners, About 2300 researchers in mathematics, strongly devoted to create a well-function- researchers in mathematics education and ing and rich teaching environment in teachers from all levels in the educational mathematics in their countries of origin, system from primary school to university, under the guidance of researchers in math- ods, the presentation and the assessment discussed new developments in relation to ematical education, who have contributed of mathematics in schools. quality in mathematical education. There significantly to structuring thinking about were participants from 119 different coun- problems related to the teaching and learn- Ceremonies and Awards tries. Registration took place on Sunday ing of mathematics. During the congress, I Many participants will remember the July 4 in the magnificent celebration hall came to realise how important it is that opening ceremony since two politicians of the University of Copenhagen, close to researchers in mathematics and made very good speeches and stayed for the hotels of the participants. The registra- researchers in mathematics education do the whole ceremony. The Danish minister tion day was a beautiful sunny day. not loose contact with each other to the of education, Ulla Tørnæs, faced the prob- During the conference, participants confusion of the practitioners in the class- lems for mathematics in schools without enjoyed rich variations in the Danish room, who face concrete problems with getting into general political polemic, and weather! At registration, the participants specific topics from mathematics. More the Mayor of Lyngby Municipality, Rolf received cards for the public transporta- than ever, I now believe that the voices of Aagaard-Svendsen, who holds a Ph.D. in tion system in the Copenhagen area, both groups of researchers are necessary mathematical statistics, made a witty which in the following days quickly and in the debate about the contents, the meth- speech where he even showed a page from his thesis. Other speeches were made by professor Hyman Bass, the president of ICMI (International Commission on Mathematical Instruction), professor Mogens Niss, the chair of the international programme committee (IPC), professor Morten Blomhøj, the chair of the local organizing committee (LOC), professor Christian Stubkjær, the dean of research at the Technical University of Denmark, and professor Bernard Hodgson, the general secretary of ICMI. The ceremony was further enlightened by a musical performance by Royal Danish Brass. The opening ceremony also included the awarding of the first Felix Klein Medal to professor Guy Brousseau, France, for his lifetime contributions to didactics of mathematics, and the first Hans Freudenthal Medal to professor Celia Hoyles, England, for outstanding contributions to research in the domain of technology and mathematics education1. With From the left; Morten Blomhøj (chair of the LOC), Elin Emborg (administrative the choices of the first recipients, a high secretary for LOC and IPC), Mogens Niss (chair of the IPC), all three IMFUFA, Roskilde University, Denmark and Gerd Brandell (chair of the Nordic Contact 1 The citations can be found in issue 52 of Committee for ICME-10), Lund University, Sweden. the Newsletter. 14 EMS September 2004 ICME-10 puzzles, boomerangs, juggling, origami, wood carving, beautiful Christmas deco- rations, various rope tricks and construction of kites. The mathematical circus was a great success and helped attract the attention of the news media to the congress in general, including leading Danish newspapers, TV and radio. Also a very interesting exhibition developed in cooperation between ICMI and UNESCO, “Why mathematics”, was shown during ICME-10. Being one of the local organisers of the congress, I must be careful with appraisal. But I do feel that the congress was successful in fulfilling its aim, namely stimulating the continual process towards good teaching of mathematics. As professor Jeremy Kilpatrick assured us under a very interesting plenary interview session with The president of the International Commission on Mathematical Instruction, four distinguished mathematical educa- Hyman Bass, presents the first Felix Klein Medal to Professor Guy Brousseau (on tors: “The strive for good teaching will the right-hand side). never stop. There will always be new chal- lenges.” standard has been set for these awards, ematics and mathematics education that The closing ceremony took place on making them especially prestigious for caused a heated debate. Although the tone Sunday, July 11, and the organisers were research in mathematics education. was not exactly polite, it was very direct relieved after more than four interesting and explicit, and it may actually lead to years of planning in collaboration with our The Programme reflections on the various roles of extremely effective and helpful congress And then we were ready for the confer- researchers in mathematics and bureau: Congress Consultants. ence. The programme was huge: 8 plenary researchers in mathematics education and ICME-11 takes place in Mexico in activities, some 80 regular lectures, 29 their possibilities for the shaping of good 2008. You should join it. An ICME con- topic study groups, 24 discussion groups mathematics teaching in the classroom. gress is worth the effort. and - as a new thing - a thematic afternoon with 5 choices. There were 5 national pre- Circus Mathematicus Vagn Lundsgaard Hansen sentations during one full afternoon: During the congress, a mathematical cir- [[email protected]] is a Professor Korea, Mexico, Romania, Russia and the cus “Circus Mathematicus” was arranged of Mathematics at the Technical Nordic countries (Denmark, Finland, at the congress site for the general public. University of Denmark, Lyngby, Iceland, Norway, and Sweden). Each The creativity shown was enormous, Denmark. He serves as chairman of the national presentation had a detailed and encompassing among other things: bowl- EMS-committee on Raising Public comprehensive programme. Obviously, ing with fractions, a labyrinth, life size Awareness of Mathematics. most activities were in parallel sessions with many options. All in all, there were more than half a million possibilities for composing your own programme. To assist the individual planning, participants got an extra copy of the one page sched- ule, where they could fill in which activities they wanted to attend and where to find them. For a programme of this size, no single participant can capture it all. But everyone felt the spirit and the energy and compas- sion with which the many contributors had prepared their contributions. In all fairness it is not possible to single out specific contributions in a general overview. But again you realise that the listeners and the eager- ly engaged contributors to discussions are really what make up a good congress. I do think that ICME-10 was a success in that respect. Children and congress participants engaged in mathematical activities in Circus I attended a thematic afternoon on math- Mathematicus. EMS September 2004 15 25 years proof of the Kneser conjecture The advent of topological combinatorics

Mark de Longueville (Berlin)

Figure 1: The origin of the Kneser conjecture in the hand writing of Martin Kneser.

Laszl´ o´ Lovasz’´ s proof of the Kneser conjecture about 25 years ago marked the beginning of the history of topological combinatorics: applications of topological methods and theorems to problems of discrete mathematics which did not (seem to) have any connection to topology.

With this article I want to take the op- torics can be characterized by a sche- pological properties. portunity to sketch the development of me that many proofs in this field pur- topological proofs in discrete mathe- sue. If we want to solve a combinato- Thus we are concerned with a (func- matics beginning with the proof of the rial problem by topological means we torial) procedure, which is common- Kneser conjecture about 25 years ago, carry out the following steps. ly used in mathematics. However, the which eventually led to a new discipli- difference to other fields is that the ne: topological combinatorics. (1) Associate a topological constructions have to be invented and space/continuous map to the gi- In the beginning of the twentieth tailored anew for almost each indivi- ven discrete structure such as a dual problem. This calls for a certain century the discipline of combinatorial graph/graph homomorphism. amount of ingenuity and deeper in- topology already made use of combina- sight, in particular in steps (1) and (2) torial concepts in topology, finally lea- (2) Establish a relationship bet- of the above procedure. ding to the emergence of algebraic to- ween suitable topological inva- pology. Meanwhile, discrete mathema- riants of the space, e.g. dimensi- The first proof of this kind is the tics did not make much use of tech- on, k-connectedness, homology proof of the Kneser conjecture by niques from (algebraic) topology until groups, etc. and the desired com- Laszl´ o´ Lovasz.´ Because of its relevance the seminal proof of the Kneser con- binatorial features of the original for the emergence of topological com- jecture. This situation was going to structure. binatorics, its elegance, and its 25th an- change in an unexpected and fascina- niversary, I will sketch this proof and ting way. (3) Show that the associated space report on the development of topologi- The essence of topological combina- resp. the map has the desired to- cal combinatorics.

1 Figure 2: From “Jahresbericht der DMV” 1955. Figure 3: Martin Kneser. Figure 4: Laszl´ o´ Lovasz.´

The Kneser conjecture and its The afore mentioned partitions of will make use of some topological no- proof the k-subsets of an n-set now corre- tions that will not be further explained, spond to partitions of the vertex set of but which can be found in almost any The occupation with an article by Ir- the graph in so called colour classes. The textbook on topology, such as [11]. ving Kaplansky on quadratic forms, property that no partition set contains Lovasz’´ proof of the Kneser conjec- from 1953, led Martin Kneser to que- a pair of disjoint k-sets now translates ture pursues the scheme that we men- stion the behaviour of partitions of the into the property that no colour class tioned in the introduction. Step (1) is family of k-subsets of an n-set: contains two vertices that are adjacent based on the invention of a simplicial via an edge in the graph. Such a partiti- complex associated with any graph G, Consider the family of all on is called a graph colouring. The chro- the so called neighbourhood complex k-subsets of an n-set. It matic number of a graph is the smallest N (G). Simplices in the neighbourhood is easy to partition this number of colour classes in a graph co- complex are defined by sets of vertices family into n − 2k + 2 louring. As mentioned above it is easy that have a common neighbour in the classes C1 ∪ · · · ∪ Cn−2k+2, to define a graph colouring of KGn,k graph. such that no pair of k-sets with n−2k+2 colour classes. The follo- within one class is disjoint. As an example we consider the wing example shows 5−2·2+2 = 3 pos- G Is it possible to partition the graph in Figure 6. It defines the sible colour classes C1, C2, C3, which G n − 2k + 1 neighbourhood complex N ( ): family into clas- define a colouring of KG5,2. ses with the same property? C1 = {{1, 2}, {1, 3}, {1, 4}, {1, 5}} N (G) = {∅, {1}, {2}, {3}, {4}, {5}, Kneser conjectured that this is not C2 = {{2, 3}, {2, 4}, {2, 5}} {1, 2}, {1, 3}, {1, 4}, {1, 5}, possible! He presented his conjecture C3 = {{3, 4}, {3, 5}, {4, 5}} {2, 3}, {2, 5}, {3, 4}, in the “Jahresbericht der Deutschen Mathematiker–Vereinigung” in 1955 in (The example already suggests the ge- {1, 2, 5}, {1, 3, 4}} the form of an exercise [6] (cf. Figure 2). neral principle for a partition into n − We will translate Kneser’s conjecture 2k + 2 colour classes!) The Kneser con- The inclusion maximal simplices in into graph theory language. For that jecture now states that it is impossible N (G) are given by {1, 2, 5}, {1, 3, 4} purpose we define the Kneser graph to colour the graph with fewer colour and {2, 3}. In the graph G they cor- KGn,k in a suggestive manner: the ver- classes, i.e., a lower bound of n−2k+2 for respond to the neighbourhoods of the tices are the k-subsets of an n-set and the chromatic number of the graph KGn,k. vertices 3, 2, 1 respectively. Figure 7 the edges are given by pairs of disjoint Twenty three years after Kneser po- shows the topological space that reali- k-sets. Figure 5 shows this graph for sed his problem Laszl´ o´ Lovasz´ initia- zes the simplicial complex geometrical- the parameters n = 5 and k = 2. For ted the development of topological ly. The simplices {1, 2, 5} and {1, 3, 4} example, there is an edge between the combinatorics with the publication of correspond to the shaded triangles, the sets {1, 2} and {3, 5} because they have his proof of the Kneser conjecture [7]. simplex {2, 3} corresponds to the line empty intersection. In the following sketch of his proof we segment from 2 to 3. {1, 2}

{3, 5} {4, 5} {1, 3} {3, 4} , {2 5} 4 2 3 2 {1, 4} 1 {2, 4} 1 {2, 3} {1, 5} 5 3 4 5

Figure 5: The familiar Petersen graph Figure 6: A graph G. Figure 7: The neighbourhood complex in the guise of the Kneser graph K5,2. N (G) associated to G.

In step (2) Lovasz´ applies the Borsuk– to the complete graph Km+2 on m + 2 of Borsuk–Ulam’s theorem and its ge- Ulam theorem from topology, about vertices in which all pairs of verti- neralizations. Maybe one of the most which Gian–Carlo Rota, in an essay ces are connected by an edge. Such important generalizations is a theorem about Stan Ulam, once wrote: a graph homomorphism induces a presented by Albrecht Dold [3] in 1983: continuous map N (G) → N (Km+2) G “While chatting at the Scot- of topological spaces. It is easy Let be a non-trivial finite group which acts free- tish Cafe´ with Borsuk, an to verify that N (Km+2) is an m- outstanding Warsaw topo- dimensional sphere. Now, if N (G) ly on (well behaved) spaces X Y X logist, he [Ulam] saw in is n-connected, then with the help and . Suppose is (n − 1)-connected and Y a flash the truth of what of the map N (G) → N (Km+2), one m is now called the Borsuk– can construct an antipodal continuous has dimension . If there +1 G Ulam theorem. Borsuk had map f : Sn → Sm. Hence by the exists a -equivariant map X Y m n to commandeer all his tech- Borsuk–Ulam theorem m ≥ n + 1, resp. from to , then ≥ . nical resources to prove it.” n ≤ m − 1 as desired. For X = Sn, Y = Sm, and G the In step (3) Lovasz´ completes his two–element group acting via the an- One of the commonly used versions proof by verifying that N (KGn,k) is in- tipodal map, we re-obtain the Borsuk– of this theorem reads: deed (n − 2k − 1)-connected. Ulam theorem. As a matter of fact, If there exists an anti- With his proof Lovasz´ had identified there is such a variety of applications podal continuous map the topological core of the problem: of this theorem and its generalizations f : Sn → Sm from the n- the Borsuk–Ulam theorem. In the sa- that Jirˇ´ı Matousekˇ dedicated an entire sphere to the m-sphere, i.e. me year, 1978, a much shorter proof by (and entirely wonderful) book to them a continuous map that sa- Imre Bar´ any´ followed, which employs with the title “Using the Borsuk–Ulam tisfies f(−x) = −f(x) for the Borsuk–Ulam theorem in a mo- Theorem” [8]. all x ∈ Sn, then m ≥ n. re direct fashion. As recently as 2002, While Borsuk–Ulam’s theorem so far another substantial simplification has plays the most prominent role in topo- Lovasz´ now shows: if N (G) is to- been found by the American mathe- logical combinatorics, most standard pologically m-connected, then G is not matics student Joshua Greene [4]. For tools from algebraic topology have (m + 2)-colourable, i.e., there is no co- his proof, which can be considered as now found their applications in combi- louring of G with m + 2 colour classes. “Proof from the Book”, Greene has be- natorics, from homology- and cohomo- In the example of Figure 7, the neigh- en awarded the 2003 AMS-MAA-SIAM logy computations, characteristic clas- bourhood complex is 0-connected, i.e. Morgan prize. ses up to spectral sequences. A recent connected, but not 1-connected since example is the article “Complexes of for example the loop 1→2→3→1 can- graph homomorphisms” by Eric Bab- not be contracted to a point. This re- Topological combinatorics son and Dmitry Kozlov [1]. Even some flects the fact that G is not 2-colourable methods from differential topology ha- but 3-colourable. The role of the Borsuk–Ulam theorem ve found a combinatorial analog, e.g. in We want to sketch this essential part is not restricted to the proof of the Kne- the invention of discrete Morse theory by of the proof from a slightly more mo- ser conjecture. General bounds for the Robin Forman. dern point of view (see e.g. [2] and chromatic number of a graph, partiti- I want to mention a few applications also quite current is [10]). An (m + 2)- on results, complexity bounds for algo- of these methods. Most notably there colouring of a graph G induces a graph rithmic problems, and much more, ha- are graph colouring problems. By now a homomorphism G → Km+2 from G ve all been established with the help multitude of bounds for the chromatic number of graphs and hypergraphs homology groups and other invariants [2] Anders Bjorner¨ . Topologi- have been established with topological of simplicial complexes which led to ef- cal methods. In R. Graham, methods. Partition results of different ficient programs for the computation of M. Grotschel,¨ and L. Lovasz,´ kinds were solved, such as the neck- these invariants. Editors, Handbook of Combinato- lace problem, which was solved in its rics, 1819–1872. North Holland, full generality in 1987 by Noga Alon. As mentioned in the introduction, Amsterdam, 1994. Moreover, complexity problems, such as combinatorial topology led to the for- the complexity of linear decision tree mation of algebraic topology. The term [3] Albrecht Dold. Simple proofs of algorithms and the complexity of mo- “algebraic topology” was apparently some Borsuk–Ulam results. Con- notone graph properties in connection first used in a public lecture by Solo- temp. Math., 19:65–69, 1983. with the evasiveness conjecture have be- mon Lefschetz at Duke University in en addressed with techniques from to- 1936 (cf. [5]): [4] Joshua Greene. A new short pological combinatorics. Another hu- proof of Kneser’s conjecture. ge topic is the topology of partially or- “The assertion is often ma- Amer.Math.Monthly, 109:918–920, dered sets. In the early 80s the Swe- de of late that all mathe- 2002. dish mathematician Anders Bjorner¨ in- matics is composed of alge- troduced the concept of shellability of bra and topology. It is not [5] Ioan M. James. From Combinato- a partially ordered set. With a partial- so widely realized that the rial Topology to Algebraic Topolo- ly ordered set one can associate a sim- two subjects interpenetrate gy. In I. M. James, Editor, History of plicial complex and thus a topologi- so that we have an alge- Topology, 561–573. North Holland, cal space. Shellability of a partial or- braic topology as well as a Amsterdam, 1999. der as defined by Bjorner¨ implies that topological algebra.” [6] Martin Kneser. Aufgabe 360. Jah- the associated topological space is a The latter has now also become true resbericht der DMV, 58(2):27, 1955. bouquet of spheres. This combinatori- for combinatorics and topology. al concept along with its topological consequences found numerous appli- [7] Laszl´ o´ Lovasz.´ Kneser’s conjec- Addendum: Prof. Dr. Martin Kneser ture, chromatic number and ho- cations, such as in the theory of Bru- died on February 16, 2004 in Gottingen.¨ hat orders and questions in the area motopy. J. Combinatorial Theory, of algebraic combinatorics. It should Ser. A, 25:319–324, 1978. The EMS-Newsletter is grateful to the be remarked that using the concept of editors of the Newsletter of Deutsche [8] Jirˇ´ı Matousek.ˇ Using the Borsuk– shellability, it is easy to see that the Mathematiker–Vereinigung for their kind Ulam Theorem. Lectures on topolo- neighbourhood complex N (KGn,k), as permission to publish the author’s English gical methods in combinatorics and it appears in Lovasz’´ proof of the Kne- translation of this article which original- geometry. Written in cooperation ser conjecture, is indeed (n − 2k − 1)- ly appeared in Mitteilungen der DMV, with Anders Bjorner¨ and Gunter¨ M. connected. 4/2003 under the title “25 Jahre Beweis Ziegler. Universitext, Springer, der Kneservermutung”. Heidelberg, 2003. Back to combinatorics Mark de Longueville [[email protected] [9] Jirˇ´ı Matousek.ˇ A combinatorial Ever since combinatorial theorems berlin.de] studied mathematics in Berlin proof of Kneser’s conjecture. Pre- were proved with topological me- and New Orleans. Supported by a grant print 2000, Combinatorica, to appe- thods, the natural question followed as from German Research Foundation (DFG), ar. to whether the topological argument he received his Ph.D. from Technische Uni- could be replaced by a combinatorial versitat¨ Berlin. He has been assistant pro- [10] Jirˇ´ı Matousekˇ and Gunter¨ M. one. A first breakthrough in this direc- fessor at University of Minnesota in Min- Ziegler. Topological lower bounds tion was made by Jirˇ´ı Matousekˇ in the neapolis and is now assistant professor at for the chromatic number: A hier- year 2000 with a combinatorial proof Freie Universitat¨ Berlin. He is mainly in- archy. Jahresbericht der DMV, Pre- of the Kneser conjecture [9]. His proof terested in combinatorial problems that po- print 2003, to appear. relies on a special case of a combina- tentially have connections to other fields in torial lemma by A.W. Tucker which is mathematics. [11] James R. Munkres, Elements of al- essentially “equivalent” to the Borsuk– gebraic topology, Addison Wesley, Ulam theorem. Combinatorial relatives 1984. of the Borsuk–Ulam theorem have also Literature been applied in the construction of ap- [12] Rade T. Zivaljeviˇ c.´ Topological proximation algorithms in connection [1] Eric Babson and Dmitry N. methods. In Jacob E. Goodman with fair division problems . Moreover, Kozlov. Complexes of et al., Editoren, Handbook of discrete the concrete questions in discrete ma- graph homomorphisms. and computational geometry, 305– thematics pointed out the need for ex- http://arXiv.org/abs/ 329. CRC Press, Second edition, plicit methods for the computation of math/0310056, Preprint 2003. 2004. ANNIVERSARY 100th100th BirBirthdaythday ofof HenriHenri CarCartantan

On July 8, 2004, the outstanding French mathematician Henri Cartan turned one hundred years old. The French Mathematical Society held a seminar in honour of Henri Cartan on June 28. Its programme, some of the contributions, a biography, several photos and documents can be found on the web site http://smf.emath.fr/VieSociete/Rencontres/JourneeCartan. The 100th issue of Gazette des Mathématiciens, the news publication of Société Mathématique de France, carried some tributes to Cartan, two of which are reproduced in this issue of the Newsletter along with a resolution from the International Mathematical Union (IMU). At the closing ceremony of the 4th European congress of mathematicians, EMS-president Sir John Kingman sent his cordial congratulations to Prof. Cartan in the name of all the participants.

Three-quarters of a century with Henri one more child. Then as research advisor, Cartan who did not propose a thesis topic, but who, Jean Cerf (CNRS, Université Paris- one day, pointed out to me the article of Sud, Orsay, France) Feldbau concerning homeomorphisms of spheres, which became the starting point of This article appeared (in French) in Gazette my thesis. des Mathématiciens 100, April 2004, p. 7-8. Even before this last period, I was suffi- The Newsletter’s thanks go to the author and ciently close to him to occasionally glean to the editor of Gazette for the permission to confidential remarks such as the following reproduce it and to Larry Siebenmann (Univ. that I remember concerning René Thom, Paris-Sud) for the translation into English. whom he ‘discovered’ before anyone else in spite of their quite different ways of thinking: In a special section in Le Monde of 3 March “Thom is a boy brimming with ideas, but 2004 entitled “Matière grise: la bataille mon- how hard it is to make him write them diale” (Grey Matter: the worldwide contest), down!” Then, over a long period: his battles, one of the first observations by the biologist mostly victorious, to, as he formulated it, E.E. Baulieu reads: “French mathematicians “raise the level of the Sorbonne in are among the very best in the world”. Above Mathematics”; his vain efforts to have a chair and beyond reservations concerning certain created at the Collège de France for André excesses perpetrated under the Bourbaki concern! I saw him again in 1939 in Weil. His precocious commitment in favour banner, it is my conviction (hopefully, one Clermont-Ferrand, where the University of of Franco-German reconciliation – in a peri- widely shared) that this high reputation is in Strasbourg had been moved following the od when that demanded lofty vision, a utopi- large measure the heritage of that group of outbreak of war, and later in La Bourboule, an vision whose realization is today one of young men who, in the thirties of the last after the defeat of June 1940, of which he had our grounds for hope in the face of seeming- century, conceived the Bourbaki project to been one of the few to voice premonitions. ly insoluble conflicts. His militance in the write a treatise that would develop mathe- Then again in Strasbourg in 1945 after the defence of Human Rights everywhere in the matics from the ground up. In this group, I abyss of war from which my own family was world, and for the construction of Europe as see, in the front rank, Henri Cartan flanked rising safe and sound, but not his. Next in a federal state, an aim to which he attaches by his friends André Weil and Jean Paris at the École Normale (rue d’Ulm), such high value. Dieudonné. when he was the professor and I a student: Jean-Pierre Serre has written: “I believe I have had the privilege to frequently cross the course for second year students, in which that the Cartan style is the finest to be found the path of Henri Cartan. The first occasion he taught us what a differentiable manifold in mathematics”. (by his own account) was in Strasbourg in was, and where, from the back of the lecture I would add that the Cartan style is the 1930; admittedly, I have forgotten this room, an unknown person (it was Alexander finest to be found in life in general. Thank encounter – I was just two years old – but I Grothendieck) ventured to dialog with him you, Monsieur Cartan, for showing, by your have retained clear memories from succeed- as an equal; the first Cartan Seminar on example, that it is possible to become with ing years. Cartan was a colleague of my Topology (1948), for which, under his guid- age ever more humane. father’s and a friend of the family, a friend ance, I presented “exposé no. 3” (thereafter who was admired, but somewhat feared for entirely written up by him alone); and fre- Jean Cerf has been Professeur and Directeur his caustic turn of mind and tongue; at the quently in his family gathering, Boulevard de Recherche (CNRS) at Université Paris- same time, his fragile health was a cause for Jourdan, on Sunday, where I was a little like Sud (Orsay), France.

20 EMS September 2004 ANNIVERSARY Personal souvenirs of Henri Cartan made us share his enthusiasm for recent Pierre Samuel (Université de Paris- discoveries or trends, e.g. the notion of a Sud, Orsay, France) functor and various topics from S. Eilenberg and his book “Homological This article appeared (in French) in algebra”. Gazette des Mathématiciens 100, April In spite of the fact that our domains of 2004, p. 13-14. The Newsletter thanks the research were quite distinct, our friendship editor of gazette for the permission to and our common concerns intensified. reproduce it and the author for providing Gradually, and especially when I became the translation into English. an environmental activist, he made me share his enthusiasm for the unification of My first meeting with Henri Cartan took Europe. place in August 1940. I was a candidate at I also admired his ceaseless actions for the Ecole Normale Supérieure and he was persecuted mathematicians. During the one of the examiners. Classmates, who May 1968 movement, we were both open had already taken the examination, told to some demands of the students, e.g. the me that he was a “nice” examiner as inclusion of personal studies in the cur- opposed to some examiners at the Ecole riculum. In 1970, we both successfully Polytechnique who tried to destabilize the requested to be transferred to the new candidates. In fact, Henri Cartan wanted to of a topic which has been the subject of Université de Paris-Sud (Orsay). There he make the candidates reveal as much as recent research, but in a field distinct from worked very hard to embody statutes pro- possible in order to find out which ones the field of the thesis itself). Cartan chose viding a good balance between teaching were the most promising. I noticed the the relations between homology and and research. same quality when, much later, we sat homotopy and spent a lot of time pointing Other mathematicians, more competent together in doctorates juries. The good out the articles to be read and making sure than me, will surely describe his results on questions he put to the candidates enabled that I was mastering this topic that was functions of several complex variables, him to predict from their answers which new to me. sheaves, algebraic topology, homological candidates would become first-rate mathe- In the meantime, I became full-fledged algebra, etc. I just know that they are fun- maticians. He was never wrong. member of Bourbaki. During the damental and that the “Séminaires I met Henri Cartan again in 1944-1945 Congresses, I admired the acuteness of Cartan”, the topics of which were chosen at the Ecole Normale Supérieure, when I Cartan’s comments and his mastery of by him each year with a remarkable intu- was preparing for the Agrégation de most branches of Mathematics. If one of ition of what would be important, awak- Mathématiques. At that time, the oral part his proposals was not accepted immediate- ened the vocations of many first-rate of this competitive examination consisted ly, he tried hard to convince other mem- mathematicians. in preparing (without documents) and giv- bers during the recesses; either they were ing lectures on classical subjects (like or the proposals got improved. When trav- Pierre Samuel is Professeur émérité at Euclidian Geometry, Analytic Geometry, elling by train to the Congresses, he also Université Paris-Sud (Orsay), France. Calculus) in front of the jury – just like in front of a high school class. We practiced lecturing in front of Henri Cartan. His crit- On the Occasion of the 100th Birthday of icisms were incisive and to the point and Henri Cartan his suggestions were very valuable. Even It is a great honour for the International Mathematical Union to associate itself with at this elementary level, we learned a lot of the hundredth birthday of Henri Cartan, on July 8, 2004. deep Mathematics from him. The son of the great mathematician Elie Cartan, his contributions to mathematics I had written to Bourbaki, pointing out have been fundamental, from several complex variables to algebraic topology and mistakes in the exercises of the published homological algebra. A member of the Bourbaki group, his participation in the reju- volumes, so he took an interest in me and venation of the French mathematical school was essential, in particular through his invited René Thom and myself as “guinea seminar held at the École Normale Supérieure. His roles as teacher and mentor were pigs” to a Bourbaki congress, which took also exceptional, and were felt well beyond national boundaries. place in July 1945 at the Ecole Normale During the critical years after the second world war, Cartan's enduring friendship Supérieure. This was one of the great with the German mathematician Heinrich Benhke, and his own personal generosity, experiences in my life. contributed greatly to the rebirth of German mathematics. He was made an honorary From 1945 to 1947, I got a research fel- member of the German Mathematical Society (DMV) in 1994. lowship at Princeton University and I His natural preoccupation with international cooperation led to his active involve- returned with a research project on inter- ment with the International Mathematical Union, of which he was President from section multiplicities in Algebraic 1967 to 1970. As such he chaired the Fields Medal Committee for the Nice Geometry, mostly inspired by Claude International Congress of Mathematicians in 1970. Chevalley. When this work was completed He has been actively involved in the defense of mathematicians who were jailed or in 1949, I presented it as a thesis to the discriminated against in their countries, and is an ardent defender of European unity. Université de Paris. I asked Henri Cartan Mathematicians worldwide unite with respect and admiration in warmly congratu- to be in the jury by giving me the subject lating Henri Cartan, great mathematician and great man, on this happy occasion. of the so-called “seconde thèse” (in which the candidate studies and gives an outline International Mathematical Union EMS September 2004 21 INTERVIEW

BRINGING MATH TO THE PUBLIC FRONTLINE Interview with Nuno Crato, winner of the 2003 ‘Raising Public Awareness of Mathematics’ article competition of the EMS, conducted by José Francisco Rodrigues (Lisbon, Portugal)

Nuno Crato is a mathematician and a well-known science writer in his native Portugal. He is frequently present in public discussions related to research, education, and science popularization. He regularly writes for the press and often N. Crato (left) and J.F. Rodrigues (right) appears in radio and television programs. read an article in, say, Science, or Annals er. If he or she is hooked with the first sen- He was recently elected President of the of Statistics, or even Scientific American, tences, there is a chance the article will be Portuguese Mathematical Society (SPM), faster than most interested readers without read. If not… and took charge this September. a scientific background. As I read journals Last year, he won the First Prize of the and magazines very often, just out of my So you try to bring people into science EMS competition Raising Public simple curiosity, I very often indeed come using ordinary subjects as a pretext. Do Awareness (RPA) on Mathematics. The across simple ideas on topics. Then I think you consider this work as science popu- last EMS Newsletter, issue 52, reproduced “I’d like to explain this to someone”. It’s a larization or science vulgarization? part of the three-article series on cryptog- natural thing for a teacher. Unfortunately, Maybe it’s both. But I can’t give general raphy he submitted to the competition. I can’t go to class and say “today, instead rules. Sometimes it works the way I English and Portuguese versions of these of power series, let me explain you how described. Sometimes it is different. Some articles are now available online at these guys from Scotland are assessing the people explain mathematics and science in http://pascal.iseg.utl.pt/~ncrato/EMS. Hun- number of stars in our galaxy using a a straight forward manner; other people garian and Italian versions will soon be novel simulation algorithm”… use pretexts, as I usually try to do; other available as well. people write only chronicles, that is, com- Chaired by Vagn Lundsgaard Hansen Then, you decide to write an article on mentaries on the public scene that are and formed by eight mathematicians from it… made from a special point of view (in our an equal number of countries, the RPA More or less... But I have to do some case, science or math). Everything is use- committee selected Crato’s articles among research on the topic, think about the way ful, when it is done seriously and serves to 26 proposals from 14 different countries. to explain it, restrain myself to a limited expand people awareness of science, I José Francisco Rodrigues, who has been space, and wait for the right moment… think. collaborating with the EMS since its foundation (at which he represented the Couldn’t you write immediately? Why do you think popularization of Portuguese Mathematical Society), con- Not usually. It seems to be better to wait mathematics is useful? Some people say ducted the conversation that follows. for a moment when the topic gains actual- it is useless, since the public cannot be ity and becomes interesting for the large educated with light articles that only Let me first congratulate you, Nuno, on public. For example, going back to the scratch the surface of things. the well-deserved prize. I have been articles on the competition, I have been I’m sorry: I completely disagree with this reading your articles in the Portuguese seduced by the marvels of modern cryp- idea. We can’t confuse formal education press and one thing that surprises me is tography for a while. I have read David with journalism, and science writing is the way you chose the topics. They are Kahn’s and Simon Singh’s books and a akin to journalism. Popularization of very diverse and very often are not number of expository papers and articles. mathematics brings our discipline to pub- related to your research. Is this true? But I waited. Suddenly, there was a public lic attention. That’s it! If some people are Thanks, José Francisco. My research is in debate in Portugal on the use of credit enticed by an article and read further, Stochastic Processes and applications, cards and electronic purchasing. Some that’s fantastic. But if most people only mainly now on long-memory time series, a banks even created a special debit card read it and get a general idea, that’s also kind of generalization of fractal Brownian that could be used as a credit card, but only fine. I have nothing against it. motion. This is necessarily a restricted for electronic transactions. They were try- Now, a completely different problem is area and only once I wrote about it to the ing to dispel the fears on the public. It was the rigor we put in the writing. We can’t general public. But I am usually interested the ideal moment to talk about cryptogra- say mistakes and we are always treading in many other subjects, as you are and as phy. If you read my first article, you will dangerous grounds. We have to write everybody who likes mathematics and sci- notice it starts precisely discussing these things in such a way people are not bored ence usually is. What I find is that I can fears. I wrote it this way to entice the read- and, at the same time, avoid making 22 EMS September 2004 INTERVIEW errors. How do we do it? I can’t give gen- this benefit? nalists have a basic scientific background eral rules. We just have to be careful and It’s always difficult to do ratings, but this and the professionalism necessary to do imaginative. benefit is certainly important. I would add science popularization well. Ideally, we Let me just give an example. If you are a couple of others, putting in first place the would have plenty of good science jour- talking about something that grows fast, appreciation of the essential ethic of math nalists. But we do not have so. And it is you can’t say: “it’s an exponential and science: the intellectual honesty, the also refreshing that different worlds and growth”, unless it really is. A politician, critical rationalism, the respect for the different people communicate and, occa- though, could say it, since “exponential reality, the international cooperation, the sionally and for a while, even trade places. growth” is now a common phrase. But if effort to avoid prejudices. you are a mathematician or a scientist— In general, I would say mathematics and people read you in a different way. If you science are an essential part of our culture don’t want to go into details and explain and they deserve to be on the forefront of the difference between polynomial and public life. This is important for the sup- exponential growth, you can just say: “it is port of our efforts. And, more importantly, a fast growth”. for the creation of a general culture that respects mathematics and science and tries Let me insist: do you think it is possible to educate the citizens accordingly. to popularize mathematics and science without vulgarizing it? You have recently been elected Many mathematicians and scientists are President of SPM—Sociedade afraid of vulgarization, I know, because Portuguesa de Matemática. Do you they identify it with oversimplification. I think our societies should give more don’t like to discuss words, but I have attention to scientific popularization? Illustration from the prize-winning articles nothing against simplification, that’s what I can’t speak in general. I believe scientif- we are always doing in life. I condemn ic societies should promote research, edu- What do you mean? errors and try to avoid mistakes, which is cation, and popularization. All these goals Think about research and teaching: people another question. are important, but not all societies can pro- who do research can bring to teaching an mote them equally. insider’s view and an insight on math problems that other people usually can’t. … as we all should? The same way, professional mathemati- I believe scientific organizations should cians can bring to math popularization a not forget their role in society, which rigor and a point of view no good journal- includes popularization of science. But ist can. So, I believe it’s useful for science this is a goal for organizations, not for journalism that some of us, once a while, individuals. I think it would be a terrible practice science popularization. The more mistake to try to involve everybody in people talk about math and sciences on the popularization. Some people may like to newspapers and the better they do it, the do it and may have some aptitude for it. better math and science are appreciated by Other people not. The basic thing is society. And the better society appreciates research. Teaching comes next. science, the better education is. Then, Popularization is at the end of the list. more resources and more people come to mathematics and sciences. And this is Illustration from the prize-winning articles Isn’t it possible to do all three? good for everybody. It’s very difficult, and one activity always But don’t you think that, simplifying harms the other. Very few people are like José Francisco Rodrigues [rodrigue matters, the popularization of science Ian Stewart, who apparently can write a @ptmat.fc.ul.pt] is professor of favors a wrong idea about scientists great book a year, a research paper a Mathematics at the Faculty of Science of work? month, and still be a dedicated teacher. For the University of Lisbon, where he direct- It may and may not. It all depends on the most of us, to do science writing necessar- ed the Centro de Matemática e Aplicações way things are done. You can say “on a ily harms research and teaching. It’s true Fundamentais for several years until dark night, he suddenly had the idea…” or that these activities are also complemen- 2002. His research interests are in the you can say “after many discussions and tary. Sometimes, research helps us with field of nonlinear PDEs and their applica- many days thinking about the problem, he ideas for writing to the public, and teach- tions, in particular, in free boundary prob- came to the idea…” My impression, ing can give us ideas on ways to explain lems. He is currently collaborating with though, is that the public has already a ter- things. But I never noticed a good research the Centro de Matemática of the ribly wrong idea about the way science idea coming out a good effort on popular- University of Coimbra, also in Portugal. works. Everything you can do to clarify ization. He co-organized the 1999 Diderot things is positive. Mathematical Forum on “Mathematics So you think popularization should be and Music” and is a member of the EMS You didn’t talk yet about one often cited left to journalists? Raising Public Awareness of Mathematics benefit of mathematics popularization, Popularization should be done by people Committee. He is a main editor of which is the appreciation of mathemat- who know what they are talking about and “Interface and Free Boundaries”, an EMS ics by the public. How would you rate who know how to talk about it. Some jour- journal. EMS September 2004 23 INTERVIEW InterviewInterview withwith MichaelMichael AtiyahAtiyah andand IsadoreIsadore SingerSinger Interviewers: Martin Raussen and Christian Skau

The interview took place in Oslo, on the 24th of May 2004, prior to the Abel prize celebrations.

The Index Theorem

First, we congratulate both of you for having been awarded the Abel Prize 2004. This prize has been given to you for “the discovery and the proof of the Index Theorem connecting geometry and analysis in a surprising way and your outstanding role in building new bridges between mathematics and theoretical physics”. Both of you have an impressive list of fine achievements in mathematics. Is the Index Theorem your most important from left to right: I. Singer, M. Atiyah, M. Raussen, C. Skau result and the result you are most pleased with in your entire careers? cal, but actually it happens quite different- there are generalizations of the theorem. ATIYAH First, I would like to say that I ly. One is the families index theorem using K- prefer to call it a theory, not a theorem. SINGER At the time we proved the Index theory; another is the heat equation proof Actually, we have worked on it for 25 years Theorem we saw how important it was in which makes the formulas that are topolog- and if I include all the related topics, I have mathematics, but we had no inkling that it ical, more geometric and explicit. Each probably spent 30 years of my life working would have such an effect on physics some theorem and proof has merit and has differ- on the area. So it is rather obvious that it is years down the road. That came as a com- ent applications. the best thing I have done. plete surprise to us. Perhaps it should not SINGER I too, feel that the index theorem have been a surprise because it used a lot of Collaboration was but the beginning of a high point that geometry and also quantum mechanics in a has lasted to this very day. It’s as if we way, à la Dirac. Both of you contributed to the index theo- climbed a mountain and found a plateau rem with different expertise and visions – we’ve been on ever since. You worked out at least three different and other people had a share as well, I proofs with different strategies for the suppose. Could you describe this collabo- We would like you to give us some com- Index Theorem. Why did you keep on ration and the establishment of the result ments on the history of the discovery of after the first proof? What different a little closer? the Index Theorem.1 Were there precur- insights did the proofs give? SINGER Well, I came with a background sors, conjectures in this direction already ATIYAH I think it is said that Gauss had in analysis and differential geometry, and before you started? Were there only ten different proofs for the law of quadrat- Sir Michael’s expertise was in algebraic mathematical motivations or also physical ic reciprocity. Any good theorem should geometry and topology. For the purposes ones? have several proofs, the more the better. of the Index Theorem, our areas of exper- ATIYAH Mathematics is always a contin- For two reasons: usually, different proofs tise fit together hand in glove. Moreover, uum, linked to its history, the past - nothing have different strengths and weaknesses, in a way, our personalities fit together, in comes out of zero. And certainly the Index and they generalize in different directions - that “anything goes”: Make a suggestion - Theorem is simply a continuation of work they are not just repetitions of each other. and whatever it was, we would just put it that, I would like to say, began with Abel. And that is certainly the case with the on the blackboard and work with it; we So of course there are precursors. A theo- proofs that we came up with. There are dif- would both enthusiastically explore it; if it rem is never arrived at in the way that log- ferent reasons for the proofs, they have dif- didn’t work, it didn’t work. But often ical thought would lead you to believe or ferent histories and backgrounds. Some of enough, some idea that seemed far-fetched that posterity thinks. It is usually much them are good for this application, some did work. We both had the freedom to con- more accidental, some chance discovery in are good for that application. They all shed tinue without worrying about where it answer to some kind of question. light on the area. If you cannot look at a came from or where it would lead. It was Eventually you can rationalize it and say problem from different directions, it is exciting to work with Sir Michael all these that this is how it fits. Discoveries never probably not very interesting; the more per- years. And it is as true today as it was happen as neatly as that. You can rewrite spectives, the better! when we first met in ’55 - that sense of history and make it look much more logi- SINGER There isn’t just one theorem; excitement and “anything goes” and “let’s 24 EMS September 2004 INTERVIEW see what happens”. SINGER Certainly computers have made sis. And I think the same goes for geome- ATIYAH No doubt: Singer had a strong collaboration much easier. Many mathe- try and analysis for people like Newton. expertise and background in analysis and maticians collaborate by computer instant- It is artificial to divide mathematics into differential geometry. And he knew cer- ly; it’s as if they were talking to each other. separate chunks, and then to say that you tainly more physics than I did; it turned out I am unable to do that. A sobering coun- bring them together as though this is a sur- to be very useful later on. My background terexample to this whole trend is prise. On the contrary, they are all part of was in algebraic geometry and topology, so Perelman’s results on the Poincaré conjec- the puzzle of mathematics. Sometimes you it all came together. But of course there are ture: He worked alone for ten to twelve would develop some things for their own a lot of people who contributed in the back- years, I think, before putting his preprints sake for a while e.g. if you develop group ground to the build-up of the Index on the net. theory by itself. But that is just a sort of Theorem – going back to Abel, Riemann, ATIYAH Fortunately, there are many dif- temporary convenient division of labour. much more recently Serre, who got the ferent kinds of mathematicians, they work Fundamentally, mathematics should be Abel prize last year, Hirzebruch, on different subjects, they have different used as a unity. I think the more examples Grothendieck and Bott. There was lots of approaches and different personalities – we have of people showing that you can work from the algebraic geometry side and and that is a good thing. We do not want usefully apply analysis to geometry, the from topology that prepared the ground. all mathematicians to be isomorphic, we better. And not just analysis, I think that And of course there are also a lot of people want variety: different mountains need dif- some physics came into it as well: Many of who did fundamental work in analysis and ferent kinds of techniques to climb. the ideas in geometry use physical insight the study of differential equations: SINGER I support that. Flexibility is as well – take the example of Riemann! Hörmander, Nirenberg… In my lecture I absolutely essential in our society of math- This is all part of the broad mathematical will give a long list of names2; even that ematicians. tradition, which sometimes is in danger of one will be partial. It is an example of being overlooked by modern, younger peo- international collaboration; you do not Perelman’s work on the Poincaré conjec- ple who say “we have separate divisions”. work in isolation, neither in terms of time ture seems to be another instance where We do not want to have any of that kind, nor in terms of space – especially in these analysis and geometry apparently get really. days. Mathematicians are linked so much, linked very much together. It seems that SINGER The Index Theorem was in fact people travel around much more. We two geometry is profiting a lot from analytic instrumental in breaking barriers between met at the Institute at Princeton. It was nice perspectives. Is this linkage between dif- fields. When it first appeared, many old- to go to the Arbeitstagung in Bonn every ferent disciplines a general trend – is it timers in special fields were upset that new year, which Hirzebruch organised and true, that important results rely on this techniques were entering their fields and where many of these other people came. I interrelation between different disci- achieving things they could not do in the did not realize that at the time, but looking plines? And a much more specific ques- field by old methods. A younger genera- back, I am very surprised how quickly tion: What do you know about the status tion immediately felt freed from the barri- these ideas moved… of the proof of the Poincaré conjecture? ers that we both view as artificial. SINGER To date, everything is working ATIYAH Let me tell you a little story Collaboration seems to play a bigger role out as Perelman says. So I learn from about Henry Whitehead, the topologist. I in mathematics than earlier. There are a Lott’s seminar at the University of remember that he told me that he enjoyed lot of conferences, we see more papers Michigan and Tian’s seminar at Princeton. very much being a topologist: He had so that are written by two, three or even more Although no one vouches for the final many friends within topology, and it was authors – is that a necessary and com- details, it appears that Perelman’s proof such a great community. “It would be a mendable development or has it draw- will be validated. tragedy if one day I would have a brilliant backs as well? As to your first question: When any two idea within functional analysis and would ATIYAH It is not like in physics or chem- subjects use each other’s techniques in a have to leave all my topology friends and istry where you have 15 authors because new way, frequently, something special to go out and work with a different group they need an enormous big machine. It is happens. In geometry, analysis is very of people.” He regarded it to be his duty to not absolutely necessary or fundamental. important; for existence theorems, the do so, but he would be very reluctant. But particularly if you are dealing with more the better. It is not surprising that Somehow, we have been very fortunate. areas which have rather mixed and inter- some new [at least to me] analysis implies Things have moved in such a way that we disciplinary backgrounds, with people who something interesting about the Poincaré got involved with functional analysts with- have different expertise, it is much easier conjecture. out losing our old friends; we could bring and faster. It is also much more interesting ATIYAH I prefer to go even further – I them all with us. Alain Connes was in for the participants. To be a mathematician really do not believe in the division of functional analysis, and now we interact on your own in your office can be a little bit mathematics into specialities; already if closely. So we have been fortunate to dull, so interaction is stimulating, both psy- you go back into the past, to Newton and maintain our old links and move into new chologically and mathematically. It has to Gauss… Although there have been times, ones – it has been great fun. be admitted that there are times when you particularly post-Hilbert, with the axiomat- go solitary in your office, but not all the ic approach to mathematics in the first half Mathematics and physics time! It can also be a social activity with of the twentieth century, when people lots of interaction. You need a good mix of began to specialize, to divide up. The We would like to have your comments on both, you can’t be talking all the time. But Bourbaki trend had its use for a particular the interplay between physics and mathe- talking some of the time is very stimulat- time. But this is not part of the general atti- matics. There is Galilei’s famous dictum ing. Summing up, I think that it is a good tude to mathematics: Abel would not have from the beginning of the scientific revo- development – I do not see any drawbacks. distinguished between algebra and analy- EMS September 2004 25 INTERVIEW lution, which says that the Laws of Nature of mathematics having evolved out of the There are times when this goes out of fash- are written in the language of mathemat- physical world; it therefore is not a big sur- ion or when parts of mathematics evolve ics. Why is it that the objects of mathe- prise that it has a feedback into it. purely internally. Lots of abstract mathematical creation, satisfying the criteria of More fundamentally: to understand the matics does not directly relate to the out- beauty and simplicity, are precisely the outside world as a human being is an side world. ones that time and time again are found to attempt to reduce complexity to simplicity. It is one of the strengths of mathematics be essential for a correct description of the What is a theory? A lot of things are hap- that it has these two and not a single external world? Examples abound, let me pening in the outside world, and the aim of lifeblood: one external and one internal, just mention group theory and, yes, your scientific inquiry is to reduce this to as sim- one arising as response to external events, Index Theorem! ple a number of principles as possible. the other to internal reflection on what we SINGER There are several approaches in That is the way the human mind works, the are doing. answer to your questions; I will discuss way the human mind wants to see the SINGER Your statement is too strong. I two. First, some parts of mathematics were answer. agree with Michael that mathematics is created in order to describe the world If we were computers, which could tabu- blessed with both an external and internal around us. Calculus began by explaining late vast amounts of all sorts of informa- source of inspiration. In the past several the motion of planets and other moving tion, we would never develop theory – we decades, high energy theoretical physics objects. Calculus, differential equations, would say, just press the button to get the has had a marked influence on mathemat- and integral equations are a natural part of answer. We want to reduce this complexi- ics. Many mathematicians have been physics because they were developed for ty to a form that the human mind can shocked at this unexpected development: physics. Other parts of mathematics are understand, to a few simple principles. new ideas from outside mathematics so also natural for physics. I remember lec- That’s the nature of scientific inquiry, and effective in mathematics. We are delighted turing in Feynman’s seminar, trying to mathematics is a part of that. Mathematics with these new inputs, but the “shock” explain anomalies. His postdocs kept is an evolution from the human brain, exaggerates their overall effect on mathe- wanting to pick coordinates in order to which is responding to outside influences, matics. compute; he stopped them saying: “The creating the machinery with which it then Laws of Physics are independent of a coor- attacks the outside world. It is our way of Newer developments dinate system. Listen to what Singer has to trying to reduce complexity into simplicity, say, because he is describing the situation beauty and elegance. It is really very fun- Can we move to newer developments with without coordinates.” Coordinate-free damental, simplicity is in the nature of sci- impact from the Atiyah-Singer Index means geometry. It is natural that geome- entific inquiry – we do not look for com- Theorem? I.e., String Theory and try appears in physics, whose laws are plicated things. Edward Witten on the one hand and on independent of a coordinate system. I tend to think that science and mathe- the other hand Non-commutative Symmetries are useful in physics for matics are ways the human mind looks and Geometry represented by Alain Connes. much the same reason they’re useful in experiences – you cannot divorce the Could you describe the approaches to mathematics. Beauty aside, symmetries human mind from it. Mathematics is part mathematical physics epitomized by these simplify equations, in physics and in math- of the human mind. The question whether two protagonists? ematics. So physics and math have in com- there is a reality independent of the human ATIYAH I tried once in a talk to describe mon geometry and group theory, creating a mind, has no meaning – at least, we cannot the different approaches to progress in close connection between parts of both answer it. physics like different religions. You have subjects. prophets, you have followers – each Secondly, there is a deeper reason if your Is it too strong to say that the mathemati- prophet and his followers think that they question is interpreted as in the title of cal problems solved and the techniques have the sole possession of the truth. If you Eugene Wigner’s essay “The that arose from physics have been the take the strict point of view that there are Unreasonable Effectiveness of lifeblood of mathematics in the past; or at several different religions, and that the Mathematics in the Natural Sciences”3. least for the last 25 years? intersection of all these theories is empty, Mathematics studies coherent systems ATIYAH I think you could turn that into then they are all talking nonsense. Or you which I will not try to define. But it stud- an even stronger statement. Almost all can take the view of the mystic, who thinks ies coherent systems, the connections mathematics originally arose from external that they are all talking of different aspects between such systems and the structure of reality, even numbers and counting. At of reality, and so all of them are correct. I such systems. We should not be too sur- some point, mathematics then turned to ask tend to take the second point of view. The prised that mathematics has coherent sys- internal questions, e.g. the theory of prime main “orthodox” view among physicists is tems applicable to physics. It remains to be numbers, which is not directly related to certainly represented by a very large group seen whether there is an already developed experience but evolved out of it. of people working with string theory like coherent system in mathematics that will There are parts of mathematics where the Edward Witten. There are a small number describe the structure of string theory. [At human mind asks internal questions just of people who have different philosophies, present, we do not even know what the out of curiosity. Originally it may be phys- one of them is Alain Connes, and the other symmetry group of string field theory is.] ical, but eventually it becomes something is Roger Penrose. Each of them has a very Witten has said that 21st century mathe- independent. There are other parts that specific point of view; each of them has matics has to develop new mathematics, relate much closer to the outside world very interesting ideas. Within the last few perhaps in conjunction with physics intu- with much more interaction backwards and years, there has been non-trivial interaction ition, to describe the structure of string the- forward. In that part of it, physics has for a between all of these. ory. long time been the lifeblood of mathemat- They may all represent different aspects ATIYAH I agree with Singer’s description ics and inspiration for mathematical work. of reality and eventually, when we under 26 EMS September 2004 INTERVIEW impact on number theory as the ideas flow across mathematics – on one extreme number theory, on the other physics, and in the middle geometry: the wind is blowing, and it will eventually reach to the farthest extremities of number theory and give us a new point of view. Many problems that are worked upon today with old-fashioned ideas will be done with new ideas. I would like to see this happen: it could be the Riemann hypothesis, it could be the Langlands program or a lot of other related things. I had an argument with Andrew Wiles where I claimed that physics will have an impact on his kind of number theory; he thinks this is nonsense but we had a good argument. I would also like to make another prediction, namely that fundamental progress on the physics/mathematics front, String Isadore Singer and Sir Michael Atiyah receive the Theory questions etc., will emerge from a Abelprize from King Harald. (Photo: Scanpix) much more thorough understanding of stand it all, we may say “Ah, yes, they are in String Theory that will give us greater classical four-dimensional geometry, of all part of the truth”. I think that that will insight into the structure of the subject, and Einstein’s Equations etc. The hard part of happen. It is difficult to say which will be along with that will come new uses of physics in some sense is the non-linearity dominant, when we finally understand the mathematics. of Einstein’s Equations. Everything that picture – we don’t know. But I tend to be Alain Connes’ program is very natural – has been done at the moment is circum- open-minded. The problem with a lot of if you want to combine geometry with venting this problem in lots of ways. They physicists is that they have a tendency to quantum mechanics, then you really want haven’t really got to grips with the hardest “follow the leader”: as soon as a new idea to quantize geometry, and that is what non- part. Big progress will come when people comes up, ten people write ten or more commutative geometry means. Non-com- by some new techniques or new ideas real- papers on it and the effect is that everything mutative Geometry has been used effec- ly settle that. Whether you call that geom- can move very fast in a technical direction. tively in various parts of String Theory etry, differential equations or physics But big progress may come from a differ- explaining what happens at certain singu- depends on what is going to happen, but it ent direction; you do need people who are larities, for example. I think it may be an could be one of the big breakthroughs. exploring different avenues. And it is very interesting way of trying to describe black These are of course just my speculations. good that we have people like Connes and holes and to explain the Big Bang. I would SINGER I will be speculative in a slightly Penrose with their own independent line encourage young physicists to understand different way, though I do agree with the from different origins. I am in favour of non-commutative geometry more deeply number theory comments that Sir Michael diversity. I prefer not to close the door or than they presently do. Physicists use only mentioned, particularly theta functions to say “they are just talking nonsense”. parts of non-commutative geometry; the entering from physics in new ways. I think SINGER String Theory is in a very special theory has much more to offer. I do not other fields of physics will affect mathe- situation at the present time. Physicists know whether it is going to lead anywhere matics - like statistical mechanics and con- have found new solutions on their land- or not. But one of my projects is to try and densed matter physics. For example, I pre- scape - so many that you cannot expect to redo some known results using non-com- dict a new subject of statistical topology. make predictions from String Theory. Its mutative geometry more fully. Rather than count the number of holes, original promise has not been fulfilled. Betti-numbers, etc., one will be more inter- Nevertheless, I am an enthusiastic support- If you should venture a guess, which ested in the distribution of such objects on er of Super String Theory, not just because mathematical areas do you think are noncompact manifolds as one goes out to of what it has done in mathematics, but also going to witness the most important devel- infinity. We already have precursors in the because as a coherent whole, it is a marvel- opments in the coming years? number of zeros and poles for holomorphic lous subject. Every few years new devel- ATIYAH One quick answer is that the functions. The theory that we have for opments in the theory give additional most exciting developments are the ones holomorphic functions will be generalized, insight. When that happens, you realize which you cannot predict. If you can pre- and insights will come from condensed how little one understood about String dict them, they are not so exciting. So, by matter physics as to what, statistically, the Theory previously. The theory of D-branes definition, your question has no answer. topology might look like as one approach- is a recent example. Often there is mathe- Ideas from physics, e.g. Quantum es infinity. matics closely associated with these new Theory, have had an enormous impact so insights. Through D-branes, K-theory far, in geometry, some parts of algebra, and Continuity of mathematics entered String Theory naturally and in topology. The impact on number theory reshaped it. We just have to wait and see has still been quite small, but there are Mathematics has become so specialized, it what will happen. I am quite confident that some examples. I would like to make a seems, that one may fear that the subject physics will come up with some new ideas rash prediction that it will have a big will break up into separate areas. Is there EMS September 2004 27 INTERVIEW a core holding things together? like biology these days where there is lots ATIYAH I like to think there is a core to be discovered. Communication of mathematics holding things together, and that the core is When I was young the job market was rather what I look at myself; but we tend to good. It was important to be at a major uni- Next topic: Communication of mathe- be rather egocentric. The traditional parts versity but you could still prosper at a matics: Hilbert, in his famous speech at of mathematics, which evolved - geometry, smaller one. I am distressed by the coer- the International Congress in 1900, in calculus and algebra - all centre on certain cive effect of today’s job market. Young order to make a point about mathemati- notions. As mathematics develops, there mathematicians should have the freedom cal communication, cited a French are new ideas, which appear to be far from of choice we had when we were young. mathematician who said: “A mathemati- the centre going off in different directions, cal theory is not to be considered com- which I perhaps do not know much about. The next question concerns the continuity plete until you have made it so clear that Sometimes they become rather important of mathematics. Rephrasing slightly a you can explain it to the first man whom for the whole nature of the mathematical question that you, Prof. Atiyah are the ori- you meet on the street”. In order to pass enterprise. It is a bit dangerous to restrict gin of, let us make the following on to new generations of mathemati- the definition to just whatever you happen gedanken experiment: If, say, Newton or cians the collective knowledge of the pre- to understand yourself or think about. For Gauss or Abel were to reappear in our vious generation, how important is it example, there are parts of mathematics midst, do you think they would under- that the results have simple and elegant that are very combinatorial. Sometimes stand the problems being tackled by the proofs? they are very closely related to the continu- present generation of mathematicians – ATIYAH The passing of mathematics on ous setting, and that is very good: we have after they had been given a short refresh- to subsequent generations is essential for interesting links between combinatorics er course? Or is present day mathematics the future, and this is only possible if and algebraic geometry and so on. They too far removed from traditional mathe- every generation of mathematicians may also be related to e.g. statistics. I think matics? understands what they are doing and dis- that mathematics is very difficult to con- ATIYAH The point that I was trying to tils it out in such a form that it is easily strain; there are also all sorts of new appli- make there was that really important understood by the next generation. Many cations in different directions. progress in mathematics is somewhat inde- complicated things get simple when you It is nice to think of mathematics having pendent of technical jargon. Important have the right point of view. The first a unity; however, you do not want it to be ideas can be explained to a really good proof of something may be very compli- a straitjacket. The centre of gravity may mathematician, like Newton or Gauss or cated, but when you understand it well, change with time. It is not necessarily a Abel, in conceptual terms. They are in fact you readdress it, and eventually you can fixed rigid object in that sense, I think it coordinate-free, more than that, technolo- present it in a way that makes it look should develop and grow. I like to think of gy-free and in a sense jargon-free. You much more understandable – and that’s mathematics having a core, but I do not don’t have to talk of ideals, modules or the way you pass it on to the next genera- want it to be rigidly defined so that it whatever – you can talk in the common tion! Without that, we could never make excludes things which might be interesting. language of scientists and mathematicians. progress - we would have all this messy You do not want to exclude somebody who The really important progress mathematics stuff. Mathematics does depend on a suf- has made a discovery saying: “You are out- has made within 200 years could easily be ficiently good grasp, on understanding of side, you are not doing mathematics, you understood by people like Gauss and the fundamentals so that we can pass it on are playing around”. You never know! Newton and Abel. Only a small refresher in as simple a way as possible to our suc- That particular discovery might be the course where they were told a few terms – cessors. That has been done remarkably mathematics of the next century; you have and then they would immediately under- successfully for centuries. Otherwise, got to be careful. Very often, when new stand. how could we possibly be where we are? ideas come in, they are regarded as being a Actually, my pet aversion is that many In the 19th century, people said: “There is bit odd, not really central, because they mathematicians use too many technical so much mathematics, how could anyone look too abstract. terms when they write and talk. They were make any progress?” Well, we have - we SINGER Countries differ in their attitudes trained in a way that if you do not say it 100 do it by various devices, we generalize, about the degree of specialization in math- percent correctly, like lawyers, you will be we put all things together, we unify by ematics and how to treat the problem of too taken to court. Every statement has to be new ideas, we simplify lots of the con- much specialization. In the United States I fully precise and correct. When talking to structions – we are very successful in observe a trend towards early specializa- other people or scientists, I like to use mathematics and have been so for several tion driven by economic considerations. words that are common to the scientific hundred years. There is no evidence that You must show early promise to get good community, not necessarily just to mathe- this has stopped: in every new generation, letters of recommendations to get good first maticians. And that is very often possible. there are mathematicians who make enor- jobs. You can’t afford to branch out until If you explain ideas without a vast amount mous progress. How do they learn it all? you have established yourself and have a of technical jargon and formalism, I am It must be because we have been success- secure position. The realities of life force a sure it would not take Newton, Gauss and ful communicating it. narrowness in perspective that is not inher- Abel long – they were bright guys, actual- SINGER I find it disconcerting speaking ent to mathematics. We can counter too ly! to some of my young colleagues, because much specialization with new resources SINGER One of my teachers at Chicago they have absorbed, reorganized, and sim- that would give young people more free- was André Weil, and I remember his say- plified a great deal of known material into dom than they presently have, freedom to ing: “If Riemann were here, I would put a new language, much of which I don’t explore mathematics more broadly, or to him in the library for a week, and when he understand. Often I’ll finally say, “Oh; is explore connections with other subjects, came out he would tell us what to do next.” that all you meant?” Their new conceptu- 28 EMS September 2004 INTERVIEW al framework allows them to encompass out my ideas, I’m wrong 99% of the time. I succinctly considerably more than I can Individual work style learn from that and from studying the ideas, express with mine. Though impressed techniques, and procedures of successful with the progress, I must confess impa- I heard you, Prof. Atiyah, mention that one methods. My stubbornness wastes lots of tience because it takes me so long to reason for your choice of mathematics for time and energy. But on the rare occasion understand what is really being said. your career was that it is not necessary to when my internal sense of mathematics is remember a lot of facts by heart. right, I’ve done something different. Has the time passed when deep and Nevertheless, a lot of threads have to be important theorems in mathematics can woven together when new ideas are devel- Both of you have passed ordinary retire- be given short proofs? In the past, there oped. Could you tell us how you work best, ment age several years ago. But you are are many such examples, e.g., Abel’s how do new ideas arrive? still very active mathematicians, and you one-page proof of the addition theorem ATIYAH My fundamental approach to have even chosen retirement or visiting of algebraic differentials or Goursat’s doing research is always to ask questions. positions remote from your original work proof of Cauchy’s integral theorem. You ask “Why is this true?” when there is places. What are the driving forces for ATIYAH I do not think that at all! Of something mysterious or if a proof seems keeping up your work? Is it wrong that course, that depends on what foundations very complicated. I used to say – as a kind mathematics is a “young man’s game” as you are allowed to start from. If we have of joke – that the best ideas come to you dur- Hardy put it? to start from the axioms of mathematics, ing a bad lecture. If somebody gives a terri- ATIYAH It is no doubt true that mathemat- then every proof will be very long. The ble lecture, it may be a beautiful result but ics is a young man’s game in the sense that common framework at any given time is with terrible proofs, you spend your time you peak in your twenties or thirties in terms constantly advancing; we are already at a trying to find better ones, you do not listen of intellectual concentration and in original- high platform. If we are allowed to start to the lecture. It is all about asking ques- ity. But later you compensate that by expe- within that framework, then at every stage tions – you simply have to have an inquisi- rience and other factors. It is also true that there are short proofs. tive mind! Out of ten questions, nine will if you haven’t done anything significant by One example from my own life is this lead nowhere, and one leads to something the time you are forty, you will not do so famous problem about vector fields on productive. You constantly have to be suddenly. But it is wrong that you have to spheres solved by Frank Adams where the inquisitive and be prepared to go in any decline, you can carry on, and if you man- proof took many hundreds of pages. One direction. If you go in new directions, then age to diversify in different fields this gives day I discovered how to write a proof on you have to learn new material. you a broad coverage. The kind of mathe- a postcard. I sent it over to Frank Adams Usually, if you ask a question or decide to matician who has difficulty maintaining the and we wrote a little paper which then solve a problem, it has a background. If you momentum all his life is a person who would fit on a bigger postcard. But of understand where a problem comes from decides to work in a very narrow field with course that used some K-theory; not that then it makes it easy for you to understand great depths, who e.g. spends all his life try- complicated in itself. You are always the tools that have to be used on it. You ing to solve the Poincaré conjecture – building on a higher platform; you have immediately interpret them in terms of your whether you succeed or not, after 10-15 always got more tools at your disposal own context. When I was a student, I years in this field you exhaust your mind; that are part of the lingua franca which learned things by going to lectures and read- and then, it may be too late to diversify. If you can use. In the old days you had a ing books – after that I read very few books. you are the sort of person that chooses to smaller base: If you make a simple proof I would talk with people; I would learn the make restrictions to yourself, to specialize in nowadays, then you are allowed to essence of analysis by talking to Hörmander a field, you will find it harder and harder – assume that people know what group the- or other people. I would be asking ques- because the only things that are left are hard- ory is, you are allowed to talk about tions because I was interested in a particular er and harder technical problems in your Hilbert space. Hilbert space took a long problem. So you learn new things because own area, and then the younger people are time to develop, so we have got a much you connect them and relate them to old better than you. bigger vocabulary, and with that we can ones, and in that way you can start to spread You need a broad base, from which you write more poetry. around. can evolve. When this area dries out, then SINGER Often enough one can distil the If you come with a problem, and you need you go to that area – or when the field as a ideas in a complicated proof and make to move to a new area for its solution, then whole, internationally, changes gear, you that part of a new language. The new you have an introduction – you have already can change too. The length of the time you proof becomes simpler and more illumi- a point of view. Interacting with other peo- can go on being active within mathematics nating. For clarity and logic, parts of the ple is of course essential: if you move into a very much depends on the width of your original proof have been set aside and dis- new field, you have to learn the language, coverage. You might have contributions to cussed separately. you talk with experts; they will distil the make in terms of perspective, breadth, inter- ATIYAH Take your example of Abel’s essentials out of their experience. I did not actions. A broad coverage is the secret of a Paris memoir: His contemporaries did not learn all the things from the bottom happy and successful long life in mathemat- find it at all easy. It laid the foundation of upwards; I went to the top and got the ical terms. I cannot think of any counter the theory. Only later on, in the light of insight into how you think about analysis or example. that theory, we can all say: “Ah, what a whatever. SINGER I became a graduate student at the beautifully simple proof!” At the time, all SINGER I seem to have some built-in sense University of Chicago after three years in the ideas had to be developed, and they of how things should be in mathematics. At the US army during World War II. I was were hidden, and most people could not a lecture, or reading a paper, or during a dis- older and far behind in mathematics. So I read that paper. It was very, very far from cussion, I frequently think, “that’s not the was shocked when my fellow graduate stu- appearing easy for his contemporaries. way it is supposed to be.” But when I try dents said, “If you haven’t proved the EMS September 2004 29 INTERVIEW to make sure that Georgia is part of Europe. Now, the politicians have to decide where the borders of Europe are. It is good that the EMS exists; but you should think rather broadly about how it is evolving as Europe evolves, as the world evolves, as mathematics evolves. What should its function be? How should it relate to national societies? How should it relate to the AMS? How should it relate to the governmental bodies? It is an opportunity! It has a role to play!

Apart from mathematics…

The Abel lectures were given by I. Singer, E. Witten, M. Atiyah and J.- M. Bismut Could you tell us in a few words about your Riemann Hypothesis by age thirty, you countries; on the other hand, mathemati- main interests besides mathematics? might as well commit suicide.” How infan- cians are much more argumentative. When SINGER I love to play tennis, and I try to tile! Age means little to me. What keeps me it comes to the fine details of a constitution, do so 2-3 times a week. That refreshes me going is the excitement of what I’m doing then they are terrible; they are worse than and I think that it has helped me work hard and its possibilities. I constantly check [and lawyers. But eventually – in principle – the in mathematics all these years. collaborate!] with younger colleagues to be good will was there for collaboration. ATIYAH Well, I do not have his energy! I sure that I’m not deluding myself – that Fortunately, the timing was right. In the like to walk in the hills, the Scottish hills – I what we are doing is interesting. So I’m meantime, Europe had solved some of its have retired partly to Scotland. In happily active in mathematics. Another rea- other problems: the Berlin Wall had come Cambridge, where I was before, the highest son is, in a way, a joke. String Theory needs down – so suddenly there was a new Europe hill was about this (gesture) big. Of course us! String Theory needs new ideas. Where to be involved in the EMS. This very fact you have got even bigger ones in Norway. I will they come from, if not from Sir Michael made it possible to get a lot more people spent a lot of my time outdoors and I like to and me? interested in it. It gave an opportunity for a plant trees, I like nature. I believe that if you ATIYAH Well, we have some students… broader base of the EMS with more oppor- do mathematics, you need a good relaxation SINGER Anyway, I am very excited about tunities and also relations to the European which is not intellectual – being outside in the interface of geometry and physics, and Commission and so on. the open air, climbing a mountain, working delighted to be able to work at that frontier. Having been involved with the set-up, I in your garden. But you actually do mathe- withdrew and left it to others to carry on. I matics meanwhile. While you go for a long History of the EMS have not followed in detail what has been walk in the hills or you work in your garden happening except that it seems to be active. – the ideas can still carry on. My wife com- You, Prof. Atiyah, have been very much I get my Newsletter, and I see what is going plains, because when I walk she knows I am involved in the establishment of the on. thinking of mathematics. European Mathematical Society around Roughly at the same time as the collapse SINGER I can assure you, tennis does not 1990. Are you satisfied with its develop- of the Berlin Wall, mathematicians in gen- allow that! ment since then? eral – both in Europe and in the USA – ATIYAH Let me just comment a little on began to be more aware of their need to be Thank you very much on behalf of the my involvement. It started an awful long socially involved and that mathematics had Norwegian, the Danish, and the European time ago, probably about 30 years ago. an important role to play in society. Instead Mathematical Societies! When I started trying to get people interest- of being shut up in their universities doing ed in forming a European Mathematical just their mathematics, they felt that there The interviewers were Martin Raussen, Society in the same spirit as the European was some pressure to get out and get Aalborg University, Denmark, and Physical Society, I thought it would be easy. involved in education, etc. The EMS took Christian Skau, Norwegian University of I got mathematicians from different coun- on this role at a European level, and the Science and Technology, Trondheim, tries together and it was like a mini-UN: the EMS congresses – I was involved in the one Norway. French and the Germans wouldn’t agree; we in Barcelona – definitely made an attempt to spent years arguing about differences, and – interact with the public. I think that these 1 More details were given in the laureates’ unlike in the real UN – where eventually at are additional opportunities over and above lectures. the end of the day you are dealing with real the old-fashioned role of learned societies. 2 Among those: Newton, Gauss, Cauchy, problems of the world and you have to come There are a lot of opportunities both in terms Laplace, Abel, Jacobi, Riemann, to an agreement sometime; in mathematics, of the geography of Europe and in terms of Weierstrass, Lie, Picard, Poincaré, it was not absolutely essential. We went on the broader reach. Castelnuovo, Enriques, Severi, Hilbert, for probably 15 years, before we founded Europe is getting ever larger: when we Lefschetz, Hodge, Todd, Leray, Cartan, the EMS. started we had discussions about where Serre, Kodaira, Spencer, Dirac, On the one hand, mathematicians have were the borders of Europe. We met people Pontrjagin, Chern, Weil, Borel, much more in common than politicians, we from Georgia, who told us very clearly, that Hirzebruch, Bott, Eilenberg, are international in our mathematical life, it the boundary of Europe is this river on the Grothendieck, Hörmander, Nirenberg. is easy to talk to colleagues from other other side of Georgia; they were very keen 3 Comm. Pure App. Math. 13(1), 1960. 30 EMS September 2004 NEWS OsloOslo 2004:2004: TheThe AbelAbel PrizePrize celebrationscelebrations Nils Voje Johansen and Yngvar Reichelt (Oslo, Norway)

On 25 March, the Norwegian Academy of Science and Letters announced that the Abel Prize for 2004 was to be awarded to Sir Michael F. Atiyah of the University of Edinburgh and Isadore M. Singer of MIT. This is the second Abel Prize awarded following the Norwegian Government’s decision in 2001 to allocate NOK 200 million to the creation of the Abel Foundation, with the intention of awarding an international prize for outstanding research in mathematics. The prize, amounting to NOK 6 million, was insti- tuted to make up for the fact that there is no Nobel Prize for mathematics. In addition to awarding the international prize, the Foundation shall contribute part of its earnings to measures for increasing interest in, and stimulating recruitment to, Nils Voje Johansen Yngvar Reichelt mathematical and scientific fields. The first Abel Prize was awarded in machine – the brain and the computer, break those rules creatively, just like an 2003 to the French mathematician Jean- with the subtitle “Will a computer ever be artist or a musical composer. Pierre Serre for playing a key role in shap- awarded the Abel Prize?” Quentin After a brief interval, Quentin Cooper ing the modern form of many parts of Cooper, one of the BBC’s most popular invited questions from the audience and a mathematics. In 2004, the Abel radio presenters, chaired the meeting, in number of points were brought up that Committee decided that Michael F. which Sir Michael spoke for an hour to an Atiyah addressed thoroughly and profes- Atiyah and Isadore M. Singer should audience of about 50 people. He pointed sionally. share the prize for: out that while computers are extremely After a highly successful meeting last- adept at following pre-determined rules ing almost two hours, Atiyah answered their discovery and proof of the index and that he himself is not surprised that, his own question, “Will a computer ever theorem, bringing together topology, for example, very good chess programs be awarded the Abel Prize?”: – Only if the geometry and analysis, and their out- have been developed, what is surprising is Abel Prize Committee is replaced by com- standing role in building new bridges that people are still able to play chess on puters. between mathematics and theoretical an equal footing with machines. In other physics. words, computers are good at following Youth and maths in the celebration of rules, but what they are not able to do is to Niels Henrik Abel This year’s committee consisted of Erling break the rules in a creative manner. As Monday 24 April was Youth Day, on Størmer (Oslo, Leader), David Mumford an example, he cited Niels Henrik Abel’s which the Abel Committee invited the (Brown University), Jacob Palis (IMPA, proof of the impossibility of solving the winners of various mathematical competi- Brazil), Gilbert Strang (MIT) and Don general quintic equation. A computer tions for young people to Oslo. In addi- Zagier (Max-Planck-Institut für would have continued to search for the tion to the winners of the Abel Mathematik, Germany). solution, and would never have been able Competition for Norwegian upper sec- The Abel Prize for 2004 was presented to break the rules, as Abel did, and look at ondary schools and of KappAbel, the on 25 May, the occasion being marked by the inverse of the problem. As a mathe- mathematics competition for schools, the a number of associated events in Oslo. matician you have to know the rules, but winners of mathematics competitions in to create something new you have to Berlin and France were also invited. The Café Scientifique main event took place at Oslo Cathedral On Sunday 23 May, Sir Michael Atiyah School – the school at which Abel himself participated in the first event in connec- was a pupil. An audience of 200 was tion with the award of this year’s Abel assembled, mostly consisting of the Prize. In collaboration with the school’s own mathematics pupils. After a Norwegian Association of Young brief introduction by Paul Jasper, head- Scientists, the British Council arranged a master of the school, one of the pupils, Café Scientifique at the Kafé Rust in Jon Strand, took over and led the proceedings with capable hands. First the Oslo, in which Atiyah gave an informal Café Scientique: Quentin Cooper lecture on his chosen subject: Man versus (BBC) and Sir Michael Atiyah French students presented their winning EMS September 2004 31 NEWS colourful Abel Prize banners. This year’s modern mathematician. His whole prize winners arrived at the packed hall to approach, with its generality, its insight the sound of Klaus Sanvik’s recently- and its elegance, set the tone for the next composed Abel Fanfare, performed by two centuries. If Abel had lived longer, he Sidsel Walstad on electric harp, followed would have been the natural successor to by the arrival of the King and Queen. the great Gauss: a statement with which I Lars Walløe, the President of the fully concur except for the qualification Norwegian Academy of Science and that Abel was a much nicer man, modest, Letters, welcomed those present to the friendly and likeable. I am proud to have Awards to the winners of ceremony. Before the official presenta- a prize that bears his name.” the Abel competition tion, the audience were given a surprise in After that it was the turn of Isadore entry to the competition, “Niels Henrik the form of a new arrangement of Michael Singer, who started with a confession. Abel in the French tradition”. The audi- Jackson’s Billy Jean, performed by Sidsel “Outside of the university environment it ence were then given an introduction to Walstad (electric harp), Mocci Ryen is difficult to be a pure mathematician. No the winning project in this year’s (vocals) and Børre Flyen (percussion). one in my family understands what I do. KappAbel competition, and Sir Michael The lively and youthful performance was At parties, when someone learns I am a Atiyah and Professor Isadore Singer appreciated, at least by Isadore Singer, mathematician, they frown and say; “Oh, awarded a book prize to the young partic- who tapped his foot enthusiastically in I never could understand calculus”, and ipants. After this, Per Manne, the leader time with the music. they turn away.” After this description, of the Norwegian Mathematics Council, The leader of the Abel Committee, all too familiar to many, Singer described took the floor and presented information Erling Størmer, briefly explained the rea- how mathematicians are fascinated by the about a recently established prize for sons for the selection of Atiyah and beauty, logic and power of mathematics. good mathematics teaching, in memory of Singer as this year’s prize winners: “The The index theorem itself had “provided Bernt Michael Holmboe. Holmboe was Atiyah-Singer index theorem is one of the new insight in such fields as Gauge theo- the teacher who discovered Niels Henrik most important mathematical results of ry and String theory. Breakthroughs in Abel’s exceptional talent and who was his the twentieth century. It has had an enor- physics needed new mathematics, and the early tutor. The Norwegian Mathematics mous impact on the further development index theory frequently supplied what was Council arranges the Holmboe Prize, with of topology, differential geometry and needed. Mathematicians and physicists financial support from the Abel theoretical physics. The theorem also pro- began talking to each other again. Now Foundation. The prize is to be awarded vides us with a glimpse of the beauty of we take for granted this new discipline of annually to one or more teachers who mathematical theory in that it explicitly mathematical physics.” Finally, Singer have distinguished themselves through demonstrates a deep connection between stated that the establishment of the Abel high quality and inspiring mathematics mathematical disciplines that appear to Prize attracts the attention of the world teaching. In conjunction with the award of be completely separate.” and emphasises the fundamental role the prize, a symposium on mathematics After the address, His Majesty King which mathematics plays in modern liv- and mathematics teaching will also be Harald presented the Abel Prize to the two ing. held. winners. The ceremony was concluded with Sir Michael Atiyah commenced his Edvard Grieg’s Halling before the King Wreath laying at the Abel monument acceptance speech by thanking colleagues and Queen and the Abel Prize winners left At 5 p.m. there was a wreath laying cere- who had made important contributions to the hall. mony at Gustav Vigeland’s monument to the work, mentioning in particular Fritz Abel, which stands in front of the Royal Press conference Palace in Oslo. The ceremony began with Following the prize ceremony, a press a display by a troop from His Majesty’s conference was held in the “Annen Etage” Corps of Signals. Then the leader of the restaurant at the Hotel Continental. Jens Abel Committee, Jens Erik Fenstad, held Erik Fenstad and Jacob Palis commenced a short speech in which he explained by providing information about the about the origin of the Abel monument. involvement of the Abel Foundation and The ceremony culminated in the wreath the IMU with regard to the developing laying by Atiyah and Singer. Afterwards, countries, after which the floor was open the younger contingent retired to a restau- Abel Fanfare (Photo: Anne Lise Flavik) to the Press to put their questions to the rant, while the two prize winners and the prize winners. There was also an opportu- members of the Norwegian Academy of Hirzebruch, Raoul Bott, Graeme Segal nity to taste something new. Morten Science and Letters were invited to dinner and Nigel Hitchin. He went on to explain Hallan, the hotel’s Chef, had created a at the Academy. that right from the time of Newton to that completely new Abel cake, which was on of Einstein there has been a close rela- sale in the legendary Theatercafeen dur- The prize ceremony tionship between mathematics and the ing “Abel Week”. The recipe is of course After an audience with Their Majesties exploration of the natural world. “One of secret, but the different layers of the cake King Harald and Queen Sonja earlier in the unexpected joys of my partnership are as follows: First a base of chocolate the day, it was time for the prize ceremo- with Is Singer has been that these links cake soaked in blackcurrant liqueur, fol- ny in the Great Hall of Oslo University. with physics have been reinforced during lowed by a layer of chocolate truffle and The main street of Oslo, Karl Johans gate, our time” In conclusion, Atiyah said that then a layer of blackcurrant preserve, cov- was decorated for the occasion with in his opinion, “Abel was really the first ered by nut meringue, and topped off with 32 EMS September 2004 NEWS Italian meringue. The cake is decorated standing of the beauty of mathematics to link is illustrated by the Gauss-Bonnet with white and dark chocolate and their students. In connection with this she theorem. By means of the Atiyah-Singer caramel cornets filled with blackcurrants. quoted Sir Michael: By exploring the index theorem, an overall unification of Bon appétit! The cake will also be served whole country of mathematics you get to the disciplines was acquired. In turn, this in connection with future prize cere- the top of Mount Everest and look around. new insight formed the origin of countless monies. It’s a long route, and when you get to the applications in the field of theoretical top, it’s a big scene you can see. The physics. The exhibition paid special atten- Minister went on to express her delight tion to applications in the fields of gauge with the international response to the theory and string theory. The exhibition Abel Prize: it had been extremely posi- was located in the foyer outside the audi- tive, and she noted that the selection of torium in which the Abel Lectures were winners had gained widespread support. held and was the result of collaboration She then gave the floor to the president between the Departments of Mathematics of the European Mathematical Society, and Physics at the University of Oslo. Sir John Kingman. In his speech, Sir John touched on the The mathematicians’ own party criteria for the successful establishment of On the evening of 26 May, it was time to the Abel Prize: “The great name of Niels drop the formalities. The Abel days in Henrik Abel is important. Of course the Oslo were wound up with a party at the The Abel cake with the logo of the Abel actual value of the prize is important. It is Norwegian Academy of Science and Prize (Photo: Knut Falch/Scanpix) also important that those who select the Letters. Mathematicians from near and far Banquet prize winners select people that future were invited, in addition to other people At 7 p.m. the same day, the Norwegian prize winners will be proud to follow.” who had worked with this year’s events. Government held an Abel banquet at He went on to point out that such a prize The atmosphere was pleasant and every- Oslo’s historic Akershus Fortress, hosted could have a positive effect on recruit- body had an opportunity to meet old by Kristin Clemet, the Minister of ment, which is of considerable impor- friends and make new acquaintances. As a Education and Research. The banquet was tance since we need “new mathematicians contribution to the convivial atmosphere, attended by the King and Queen, doing new mathematics. Whether it is refreshments were served and throughout Norwegian and foreign mathematicians, what we call pure mathematics, mathe- the venerable building musical treats of eminent politicians and members of matics for its own sake, or whether it is various types could be enjoyed. Norwegian society. Many people recog- pursuing interesting new applications, nised the mathematician John Donaldson, applying the techniques which the so- Photos in this article without a credit are father of Danish Crown Princess Mary called pure mathematicians have invent- © Dept of Mathematics: UoO Donaldson, who was specially invited to ed.” the Abel Prize ceremony by King Harald Nils Voje Johansen [[email protected]. at the wedding of the Crown Prince of Exhibition no] and Yngvar Reichelt [reichelt@math. Denmark some weeks before. After a wel- In connection with the award of this uio.no] are associated to the Department coming cocktail in Christian IV’s Hall, year’s Abel Prize, an exhibition was of Mathematics at the University of Oslo. the party proceeded to Skriverstuen (the staged with the objective of informing Both were members of a group of mathe- Scribes’ Hall), where all were welcomed laypersons about the work of the prize maticians that formulated the suggestion by the Minister of Education and winners. The exhibition was based on to the Norwegian government to install Research, Sir Michael and Professor simple concepts in the fields of topology, the Abel Prize in commemoration of Niels Singer. Dinner was then served in the geometry and analysis, and demonstrated Henrik Abel’s bicentenary, which was Romerike Hall. The menu consisted of how the links between the disciplines had finally adopted by the Norwegian parlia- Norwegian trout and turbot, veal fillet and been discovered. An example of such a ment. caramel mousse. In her address, the Minister, Kristin Clemet, mentioned that there was one The Abel Prize 2005 winner (Jean-Pierre Serre) at the first King Harald of Norway will present the Abel Prize for 2005 to the winner Abel Prize ceremony, while at the second on May 24th in the Aula of the University of Oslo. The deadline for nom- there were two winners. Based on this, one could perhaps draw the conclusion inating candidates is the 15th of November 2004. that the number of prize winners would be Nominations letters should contain a CV and a description of the can- determined by the equation x = n, where n didate's work, together with names of distinguished specialists in the field is the number of years the prize has exist- of the nominee who can be contacted for independent opinion. The letter ed. Having reminded the audience of the should be marked "Abel Prize Nomination" and addressed to: story of Descartes, who died of pneumo- The Norwegian Academy of Science and Letters nia when he visited Scandinavia, she touched on the effect of the Abel Prize on Drammensveien 78 the recruitment of future mathematicians. NO-0271 OSLO The Holmboe Prize has now been created Norway to honour the best mathematics teachers – Detailed information is obtainable from the web site www.abelprize.no . those with the ability to impart under- EMS September 2004 33 SOCIETIES THETHE GREEKGREEK MAMATHEMATHEMATICALTICAL SOCIETYSOCIETY Its predecessors, its founders, and some highlights from its life

(Themistocles M. Rassias, Athens, Greece)

The beginnings member of a number of Honorary The history of “modern” Greek mathemat- Committees of International Congresses. ics begins with Nicholas Nicolaidis (1826- J. Hadzidakis is known for his work in 1889). Born during the Greek Revolution Geometry with the notion “Hadzidakis in the very center of Arcadia, officer-engi- transformation”, that is cited in M. neer in the Greek Army, he went with a Spivak’s “Differential Geometry” and in J. scholarship to Paris and became a Docteur McCleary’s “Geometry from a lines and surfaces, to Greece. Remoundos contributed a lot in function theory and in particular with his generalization of Picard’s theorem to algebraic functions. In the year 1905, Jacques Hadamard (1865-1963) proposed to P. Zervos the study of Monge’s conjecture (sometimes called Monge’s problem) in the field of Partial Differential Equations, which was posed in 1784 and has not been solved since. In 1905, P. Zervos was the first mathematician to notice the falsity of Monge’s conjecture. In the year 1912, after some intensive work by P. Zervos, E. Goursat, and O. Botasso among others, D. Hilbert solved a major part of Monge’s problem. Elie Cartan’s seminal paper of the year 1914 on Pfaff’s problem, begins as follows: “In a recent paper (1913), P. Zervos generalizes a theorem of D. Hilbert…” (Bull. de la Soc. Math. de France, 1914). Cyparissos Stephanos John Hadzidakis Foundation of the G.M.S. d’Etat es Sc. Math., with O. Bonnet in his Differential Viewpoint”. He was a pupil of In the year 1918, N. Hadzidakis, Committee. He published several original the Paris school and of Karl Weierstrass G. Remoundos and P. Zervos founded the works in the Editions of Gauthier-Villars. (1815-1897) in Berlin. He is also well G.M.S. as well as the Research Seminars When he returned to Greece he became known in Greece for writing good books in Mathematics at the University of professor at the University of Athens. His for all levels of mathematical education. Athens. In addition, the year after, in 1919, research work continues to enjoy some J. Hadzidakis and C. Stéphanos estab- they started the international journal: interest even in the present time (he is lished the mathematical tradition of the Bulletin of the G.M.S. The first issue of the known, in particular, for the term University of Athens and were the teach- Bulletin of the G.M.S. published “Nicolaidis’ theorem” in Kinematics). ers of the founders of the Greek P. Zervos’ inaugural lesson entitled With him begins our story. Mathematical Society (G.M.S.). These “Relation of Mathematics to the other The next to appear in Greek mathemat- founders, all three professors at the Sciences and to Philosophy” as well as ics were the professors Cyparissos University of Athens, were Nicholas some “technical” articles in mathematics Stéphanos (1857-1917) from the island of Hadzidakis (1872-1942), a son of by the above three mathematicians and Kea, and John Hadzidakis (1844-1921) J. Hadzidakis, George Remoundos (1878- others. Zervos had been in the audience of from Crete. Stéphanos was a Docteur 1928) from Athens, and Panayotis Zervos Poincaré’s lectures on Celestial d’Etat es Sc. Math. from Paris, with (1878-1952) from Cephalonia. All of them Mechanics in Paris during the period Charles Hermite (1822-1901) as a member had been pupils of the Paris School, in par- 1903-1905, and he was also the translator of his Committee. He was a geometer and ticular N. Hadzidakis of David Hilbert. into Greek of Poincaré’s celebrated book algebraist, with references made to his Following the geometric approach of Jean “Science and Hypothesis”. In 1935, the work by a number of mathematicians Gaston Darboux (1842-1917), Bulletin of the G.M.S. published a paper including the young (at that time) David N. Hadzidakis introduced Kinetic Geome- by Nicholas Criticos (1894-1985) that was Hilbert (1862-1943). He also served as a try, as well as the study of complexes of devoted to a new proof of the Jordan 34 EMS September 2004 SOCIETIES Curve Theorem. In the context of two lec- F. Severi, and T. Takagi. Two medals scientific side of the subject and have not tures given by him in the G.M.S., a kind of were awarded, one to Lars V. Ahlfors covered the general activity of the G.M.S. first advanced text in “Point Set Topology” (Harvard University) and one to Jesse In this connection, we recall the was provided, written in Greek. Douglas (M.I.T). It was C. Carathéodory Panhellenic Competition in Mathematics who presented the work of both medalists for High School Students, inaugurated by Papakyriakopoulos and Carathéodory during the opening of the International the G.M.S. in the year 1931. The first win- A rigorous presentation of Mathematical Congress (see Constantin Carathéodory: ner was George Tsamis from Patras, at Analysis on the basis of Dedekind Cuts An International Tribute, Vols I & II, that time a schoolboy, who was to become and Cantor’s Set Theory had been made World Scientific Publ. Co., 1991, edited by a very influential High School Teacher in by P. Zervos in his masterpiece: Th. M. Rassias). mathematics. In the sixties, the “Infinitesimal Calculus”. Both of the Now let us come back to our story. Mathematical Competition got a new lease above mentioned texts, by N. Criticos and Remoundos’ pupil Theodore Varopoulos of life by Aristide Pallas, a man who was P. Zervos, contributed greatly to the orien- (1894-1957), professor of the University entirely devoted to the G.M.S and at that tation in mathematical studies of the well- of Thessaloniki (father of the well-known time its president. known topologist Christos D. Papakyria- Paris Analyst Nicholas Th. Varopoulos), In 1975, the new Council of the G.M.S., kopoulos (Athens, 1913 - Princeton, the closest pupil of Paul Montel, had a spe- under the presidency of Professor 1976). The doctoral thesis of cial interest in the Geometry of S.P.Zervos, enlarged the domain of the Papakyriakopoulos, entitled “A new proof Polynomials. From time to time one could activities and publications of the G.M.S. of the invariance of the homology groups find articles in the Bulletin of the G.M.S. In that year, S.P. Zervos conceived the of a complex”, was published in the year devoted to polynomials (from both an idea for the foundation of a “University of 1943 in the Bulletin of the G.M.S. The algebraic and a geometric approach) influ- Aegean”, with a Pythagorean Department 154-page thesis of Papakyriakopoulos that enced directly or indirectly by of Mathematics, on the island of Samos, was published in the Bulletin also consti- Th. Varopoulos. For instance, in the thir- and other Departments (Schools) in tutes the first text written in Greek on ties one can find articles by Constantine various subjects on other islands. This idea Algebraic Topology. The (unofficial) ref- Yannopoulos and by John Papademetriou. was materialized later on. Some distin- eree for the Thesis was C. P. Papaioannou Around 1950, one has to mention the the- guished mathematicians delivered lectures (1899-1979), who was Professor of sis on polynomials written in Athens by during the “Pythagorean Days”, in Athens Mechanics at the University of Athens. Th. Varopoulos’ research student and in Samos. Among the mathematicians Papaioannou had the ability to immediate- Dionysios Vythoulcas. Moving further who delivered such lectures are the follo- ly foresee Papakyriakopoulos’ brilliant and with more essential results in the con- wing: K. Borsuk (Warszawa), L. Iliev research career. Another prophetic vision text of polynomials and especially their (Sofia), A. Kawaguchi (Japan), of Papaioannou appears in his inaugural localization of roots (zeros), Spyros M. Krasner (Paris), K. Kuratowski speech at the Academy of Athens in the P. Zervos (a son of P. Zervos) wrote his (Warszawa), Dj. Kurepa (Beograd), year 1965, in which he sees the universe as Thèse d’Etat in Paris (Ann. Sc. de l’Ecole M. Loi (Paris), O. Onicescu (Bucharest). Platon’s dodecahedron. This is something Normale Sup., 1960) in the spirit of A subsequent article will deal with the that was later very beautifully presented in Modern Mathematics. Furthermore, some ‘discovery’ of the ‘Olympiads’ for High a modern mathematical language by interesting work in the classical version of Schools in Greece, as well as to activities William P. Thurston (Fields Medal in polynomials and related subjects was of the G.M.S. with emphasis to the last Topology, 1982) in the Scientific obtained by the late I. Ferentinou- three decades. American. This seems to be a major sub- Nicolacopoulou and was published in the ject of research by satellites (cf. Notices of Bulletin of the G.M.S. Themistocles M. Rassias [trassias@math. the A.M.S., June-July 2004). ntua.gr] studied mathematics at the The 1943 issue of the Bulletin of the Other activities University of California at Berkeley, G.M.S. was, naturally, dedicated to Until now, we have been devoted to the where he received his Ph.D with Steven Constantin Carathéodory at the occasion Smale. He has published more than 170 of his 70-th birthday. Carathéodory was papers and six research books and he has born in Berlin, of Greek parents, on 13 edited 24 volumes on different subjects in September 1873, and he died in Munich Mathematical Analysis, Geometry/ on 2 February 1950. His work covered Topology and their applications. His several subjects of Mathematics, including research work has found more than 2000 the calculus of variations, function theory, citations. His work is known in the field of measure and integration, as well as applied Mathematical Analysis with the term mathematics. Carathéodory has also done “Hyers-Ulam-Rassias Stability” and in very fundamental work in mechanics, Geometry with the term “Alexandrov - thermodynamics, geometrical optics and Rassias Problem”. He has lectured exten- relativity theory. An example of sively and conducted research in various Carathéodory’s wide ranging influence in academic institutions in Europe and North the international mathematical community America. More than ten mathematical was seen during the first Fields Medals journals count him as a member of their awards at the International Congress of Editorial Board. He is Professor at the Mathematicians, Oslo, 1936. The selection National Technical University of Athens, committee consisted of George Greece. He is married and has two chil- C.Carathéodory D. Birkhoff, Elie Cartan, C. Carathéodory, dren. EMS September 2004 35 SOCIETIES FRENCH-POLISH COOPERATION IN APPROXIMATION THEORY AND REAL AND COMPLEX ANALYSIS Wies³aw Pleœniak (Jagiellonian University, Kraków, Poland)

There has always been a great tradition of Pleœniak (Institute of Mathematics of the academic exchange and contact between Jagiellonian University). France and Kraków in the field of mathemat- Thanks to the financial support from the ics. The origin of the formal French-Polish French Embassy in Poland and the Institute cooperation between the Institute of of Mathematics of the Jagiellonian Mathematics at the Jagiellonian University University, the coordinators had succeeded in and French universities in the domain of organising a series of seven French-Polish shop on Differential Analysis at the approximation theory and real and complex meetings in Poland (Karniowice near International Banach Center of Warsaw analysis is almost anecdotic. During his stay Kraków, the Niedzica Castle, and then (September 2000), (organized by Prof. Anne- at the Paul Sabatier University (Toulouse) in Krynica) under the common theme: Le Marie Chollet, Prof. Krzysztof Kurdyka 1985, Prof. Wies³aw Pleœniak learned that français des mathématiciens. These meetings (Université de Savoie a Chambery), and Prof. Prof. Jacek Gilewicz (Université de Toulon served as the permanent forum of the presen- Wies³aw Pleœniak. Moreover, the workshop et du Var and Centre de Physique Théorique, tation of the progress of both Polish and on Approximation Theory (January 2002) Marseille-Luminy) and the physicists from French participants of the ATP Project in was organised at the same International the University of Warsaw had been intending research concerning the subject of coopera- Banach Center (organised by Wies³aw to organize an international conference on tion. The meetings were also opportunities of Pleœniak with the cooperation of Prof. Anne- rational approximation at the Castle of first contacts with French scientific language Marie Chollet, Vincent Thilliez (Université £añcut, the home town of Pleœniak. This and French culture for young Polish mathe- des Sciences et Technologies de Lille), and inspired Prof. Pleœniak to go to Marseille and maticians that were planning for invited stays Jacques Chaumat (Université de Paris-Sud). meet Prof. Gilewicz. As a consequence of in French universities. Since 1998, the cooperation continues that meeting, Prof. Jacek Gilewicz was invit- First, we should make special note of an under the “Polonium” Project “Real and ed to the Institute of Mathematics at the important French-Polish meeting at the Complex Analysis” coordinated by Anne- Centre of the Polish Academy of Sciences in Marie Chollet and Wies³aw Pleœniak which Paris in November of 1995 that was orga- has been accepted by the French-Polish nized by Prof. Jacek Gilewicz and Prof. Government Commission for implementa- Wies³aw Pleœniak, At this meeting, the pro- tion in 2004. gramme of future co-operation in the above mentioned subject was scheduled. The next Wies³aw Pleœniak [[email protected]. pl], important step was the common “Réseau- born in 1944, is a professor of the Formation-Recherche” project that was Jagiellonian University at Kraków, Poland, implemented 1995-1998 under the coordina- where he holds the Chair in Approximation tion of Prof. Anne-Marie Chollet (Université Theory at the Department of Mathematics. The participants of the 1992 conference in des Sciences et Technologies de Lille) and His main mathematical interest is approxi- Niedzica Castle. Prof. Wies³aw Pleœniak. Thanks to this pro- mation theory and complex analysis. He is a Jagiellonian University and spent two ject, Polish partners from the Jagiellonian member of Editorial Boards of three jour- months there in 1986. During this visit, Prof. University and the Adam Mickiewicz nals: Annales Polonici Mathematici, Gilewicz and Prof. Pleœniak had succeeded in University of Poznañ obtained 56 months of Commentationes Mathematicae and East attracting interest from the French Embassy French Government fellowships for Polish Journal on Approximation. For some years in Warsaw. It ended in the cooperation that doctorate students and globally, more than he served as the President of the Kraków yielded a formal cooperation project under five hundred thousand French Francs for the branch of the Polish Mathematical Society; the ATP Programme (Action Thématique exchange programme. moreover, he was the Vice-Dean of the Programmée) supported by the French Other common projects included the work- Faculty. Currently he is the Vice-President of Ministry of Foreign Affairs and the Polish the Mathematical Committee of the Polish State Committee for Scientific Research, The Academy of Sciences. French universities were represented by the Wies³aw Pleœniak has received a doctorate Universite de Toulon et du Var, Universite honoris causa from the University of Toulon Paul Sabatier (Toulouse), Université de Paris in France. He is a member of Société Royale XI (Orsay), Université des Sciences et des Sciences de Liège and of the Polish Technologies de Lille, and then the Centre Academy of Arts and Sciences. International des Rencontres Mathématiques Among other distinctions, he was awarded (Marseille-Luminy). The project was coordi- the Mazurkiewicz Great Prize and the nated by Professor Jacek Gilewicz and then Zaremba Great Prize of the Polish by Professor Jean-Paul Brasselet (Centre Ceremony at Toulon, 23 October 2003. Mathematical Society (1977, 1989), and he International des Rencontres Mathém- Wies³aw Pleœniak, Jacek Gilewicz, Józef has been awarded scientific prizes from the atiques) and the undersigned Wies³aw Siciak, Miros³aw Baran. Polish Academy of Science three times. 36 EMS September 2004 ERCOM CRMCRM Centre de Recerca Matemàtica, Barcelona

Institutional framework ters emerging research directions by inviting The CRM was created in 1984 by the Institut outstanding mathematicians and organising d’Estudis Catalans (Institute for Catalan specialised research programmes, conferen- Studies), an academic, scientific and cultural ces, workshops, and advanced courses. body whose aim it is to promote research in It hosts post-doctoral fellows (thirteen in science, technology and humanities, while 2003), including several Marie Curie grant supporting all aspects of Catalan culture. holders, and it has also hosted doctoral stu- Since 2001, the CRM has been a consortium, dents as a Marie Curie Training Site of the with its own legal status, integrated by the European Union from 2000 to 2004. competitive calls. Other sources of funding, Institut d’Estudis Catalans and the Catalan Between 2000 and 2003, the CRM was a through competitive applications, are the Government. It is a research institute asso- node of an EC Training Network entitled Spanish Ministry of Education and Science ciated with the Universitat Autònoma de Modern Homotopy Theory, jointly with and the European Commission. Barcelona and located on the Bellaterra cam- teams in Aarhus, Aberdeen, Lille, Louvain- An open call for research programmes is pus, about 15 km from Barcelona. la-Neuve, Paris, and Sheffield. made every year. Each programme has to be The Governing Board of the CRM has approved by the CRM Governing Board. eight members and is chaired by the Minister Research programmes Proposals are presented by the Director on of Universities, Research and the In 2002, the Governing Board of the CRM the grounds of evaluation reports prepared by Information Society of the Catalan approved a quadrennial plan that includes the Scientific Advisory Board. Government. In 2002, the Governing Board two research programmes every year, toget- During the academic year 2003-2004, a re-elected Manuel Castellet as Director and her with complementary activities. Each research programme on Set Theory was appointed a new Scientific Advisory Board, research programme offers the following carried out. The organisers were Joan with the following members: Joan Bagaria, positions during one academic year: one full- Bagaria (ICREA and Universitat de Àngel Calsina, Carles Casacuberta, Vicent time local researcher, one full-time visiting Barcelona) and Stevo Todorcevic (CNRS Caselles, Alberto Facchini, Evarist Giné, researcher, two post-doctoral fellows, and 24 and Université Paris 7). For the academic Joan Girbau, Antoni Huerta, Jaume Llibre, months of visiting researchers for periods of year 2004-2005, a research programme on Xavier Massaneda, M. Pilar Muñoz, Joan one to three months. Activities include a the Geometry of the Word Problem is plan- C. Naranjo, David Nualart, Pere Pascual, weekly seminar, a conference or a workshop ned, under the direction of Josep Burillo and Joan Porti, Jordi Quer, Oriol Serra, and Juan and an advanced course at a doctoral or post- Enric Ventura (Universitat Politècnica de L. Vázquez. Since 2004, Carles Casacuberta doctoral level. Partial funding for visitors and Catalunya), and Noel Brady (University of and Jordi Quer have acted as Vice-Directors. activities is provided by the Department of Oklahoma). Shorter specialised programmes Universities, Research and the Information will be held on Control, Geometry and Aims and scope Society (DURSI) of the Catalan Engineering, co-ordinated by Miguel The CRM gives support to local research Government, under a contract programme C. Muñoz (UPC), and on Contemporary groups in all areas of mathematics and fos- that is revised every year, and by means of Cryptology, co-ordinated by Jorge Villar and Carles Padró (UPC). The Scientific Advisory Board selected two programmes for the academic year 2005-2006: one on Arakelov Geometry and Shimura Varieties, co-ordinated by José I. Burgos (Universitat de Barcelona) and Jörg Wildeshaus (Université Paris 13), and another one on Hilbert’s 16th Problem, co-ordinated by Armengol Gasull and Jaume Llibre (Universitat Autònoma de Barcelona), together with Chengzhi Li and Jiazhong Yang (Beijing University).

CRM Research Programmes 2003-2004 Set Theory 2004-2005 Geometry of the Word Problem 2005-2006 Arakelov Geometry and Shimura Varieties Excursion to Cabrera 1991 On Hilbert’s 16th Problem

EMS September 2004 37 ERCOM Scientific meetings In a similar vein to other ERCOM centres, Mathematics), a network of nine European The following conferences and courses were the CRM undertakes actions to reinforce the research institutes that jointly offer post- scheduled for 2004 and 2005: Advanced role of mathematics in the thematic priori- doctoral fellowships in mathematics and Course on Ramsey Methods in Analysis ties of the 6th Framework Programme of the mathematical physics every year. (January 19 to 28, 2004); Workshop on the European Commission. Funding is offered Foundations of Set Theory (June 9 to 12, to young mathematicians in order to foster Publications 2004); Advanced Course on Contemporary the development of the following topics, Besides its twenty-year old preprint collec- Cryptology (February 2 to 13, 2004); which were selected on the basis of reports tion the CRM has published, since 2001, a Conference on Mathematical Foundations prepared by local teams: Life sciences, monograph series entitled Advanced of Learning Theory (June 18 to 23, 2004); Workshop on Non-Linear Differential Galois Theory (June 28 to July 2, 2004); Advanced Course on Automata Groups (July 5 to 16, 2004); HYKE Conference on Complex Flows (October 6 to 9, 2004); 4th Congress of the European Society for Research in Mathematics Education (February 17 to 21, 2005); Barcelona Conference on Geometric Group Theory (June 8 to July 2, 2005); Advanced Course on the Geometry of the Word Problem for Finitely Generated Groups (July 5 to 15, 2005); Advanced Course on Recent Trends on Combinatorics in the Mathematical Context (September 12 to 23, 2005). Around the International Congress of Mathematicians (ICM2006) in Madrid, the CRM will organise a three-month research programme on Analysis and an advanced course on Computational Geometry, both co-ordinated jointly by researchers from Barcelona and Madrid. An ICM satellite Lecture room conference is also planned. genomics, and biotechnology for health; Courses in Mathematics CRM Barcelona, Links with European entities nanotechnologies and nanosciences; infor- which is produced and distributed by Since 2003, the CRM has been an institutio- mation society technologies; sustainable Birkhäuser Verlag (Basel). The series is nal member of the EMS. It is also a member development, global change and ecosys- especially addressed to doctoral and post- of ERCOM (European Research Centres on tems. Several activities, including doctoral doctoral students. Volumes contain care- Mathematics), a committee of the EMS con- training and workshops, are planned on neu- fully edited notes written by lecturers at sisting of scientific directors of European roscience and cryptology for the years 2004, CRM advanced courses. The following research centres in mathematics. In fact, the 2005 and 2006. volumes were published in 2003 and 2004: current CRM Director, Manuel Castellet, The CRM is a member of the EPDI Symplectic Geometry and Integrable has been the chair of ERCOM since 2002. (European Post-Doctoral Institute for Hamiltonian Systems, by M. Audin,

Congress at Institut d'Estudis Catalans. Andrew Barron, Steve Smale, and José M. Matías

38 EMS September 2004 ERCOM

Facilities at CRM A. Cannas da Silva and E. Lerman; Global in a commemorative volume and a CD. In 20th Anniversary of the CRM Riemannian Geometry: Curvature and this period, the CRM hosted 969 visitors Jean-Pierre Serre, Collège de France Topology, by S. Markvorsen and M. Min- from 58 different countries, including 44 Groupes finis: Choix de théorèmes Oo; Proper Group Actions and the Baum- post-doctoral fellows. Twenty-four congres- November 9, 2004 Connes Conjecture, by G. Mislin and ses were organised, together with 19 works- Institut d’Estudis Catalans, Barcelona A. Valette; Polynomial Identity Rings, by hops and 23 advanced courses. These events V. Drensky and E. Formanek. were attended by a total number of 3,480 participants, coming from 72 countries. Web site Facilities and infrastructure Many of them, hopefully all, enjoyed Additional information about the CRM, The CRM occupies 940 square metres in the Barcelona, the Catalan countryside, or the including full lists of visitors and activities, Science Building of the Universitat hospitality of their Catalan colleagues, and can be found at the internet address Autònoma de Barcelona. Office space keep pleasant memories from their stay. www.crm.es. allows the allocation of up to fifteen guests. Two lecture rooms are used for seminars Centres at present represented in ERCOM, and meetings. All offices and rooms are ordered by decreasing geographical latitude fully equipped and air conditioned. The CRM has a LAN Ethernet with twenty-five Euler International Mathematical Institute, St. Petersburg (59º55') working stations and four printers. Access to Institut Mittag-Leffler, Stockholm (59º20') major bibliographic data bases is provided. Network in Mathematical Physics and Stochastics, Aarhus (56º10') International Centre for Mathematical Sciences, Edinburg (55º57') In addition, visitors have free access to the Isaac Newton Institute for Mathematical Sciences, Cambridge (52º21') scientific infrastructure of the Catalan uni- Centrum voor Wiskunde en Informatica, Amsterdam (52º21') versities, including the use of the UAB Stefan Banach International Mathematical Center, Warszawa (52º15') Science and Engineering Library. The CRM Thomas Stieltjes Institute for Mathematics, Leiden (52º10') has several furnished apartments at its dis- Lorentz Center, Leiden (52º10') Mathematical Research Institute, Nijmegen (51º50') posal in the Vila Universitària of the EURANDOM, Eindhoven (51º26') Bellaterra campus and in the nearby town of Max Planck Institute for Mathematics in the Sciences, Leipzig (51º20') Sant Cugat. Max-Planck Institut für Mathematik, Bonn (50º44') Institut Henri Poincaré, Centre Emile Borel, Paris (48º52') 20th anniversary Institut des Hautes Études Scientifiques, Bures-sur-Ivette (48º52') Mathematisches Forschungsinstitut Oberwolfach (48º18') In the Autumn of 2004, the CRM will cele- Erwin Schrödinger International Institute for Mathematical Physics, Wien (48º13') brate its 20th anniversary. On this occasion, The Abdus Salam International Center for Theoretical Physics, Trieste (45º39') Jean-Pierre Serre will deliver a lecture Centro di Recerca Matematica Ennio Di Giorgi, Pisa (43º43') during an academic event on November 9, Centre International de Rencontres Mathématiques, Luminy (43º18') which will be presided by the foremost aca- Instituto Nazionale di Alta Matemática Francesco Severi, Roma (41º53') Centre de Recerca Matemática, Barcelona (41º25') demic and political authorities of Catalonia. Centro Internacional de Matemática, Lisboa (38º44') The history of the CRM during its first Emmy Noether Research Institute for Mathematics, Ramat-Gan (32º35') twenty years of existence will be published EMS September 2004 39 CONFERENCES (St. Petersburg), A. Valli (Trento) Location: Grand Hotel Bellavista, Levi- co Terme, Trento, Italy Information: ForthcomingForthcoming conferencesconferences e-mail: [email protected] Web sites: www.science.unitn.it/cirm/ , compiled by Vasile Berinde (Baia Mare, Romania) http://www.science.unitn.it/cirm/Lady lecture.html

January 2005 Please e-mail announcements of Baia Mare, Romania (previous edi- European conferences, workshops and tions in 1998, 2000 and 2002) mathematical meetings of interest to Information: e-mail: 6–9: 24th Nordic and 1st Franco- EMS members, to one of the following [email protected]; [email protected] Nordic Congress of Mathematicians, addresses [email protected] or Web site: http://www.ubm.ro/site-ro/ Reykjavik, Iceland [email protected]. facultati/departament/manifestari/icam4 Aim: The main goal of the congress is to Announcements should be written in a /index.html bring together mathematicians to pre- style similar to those here, and sent as [For details, see EMS Newsletter 49] sent recent results in several important Microsoft Word files or as text files (but areas of research in the Nordic countries not as TeX input files). Space permit- 26–October 1: Potential theory and and France. Around half of the pro- ting, each announcement will appear in related topics, Hejnice, Czech gramme will be devoted to 3 main detail in the next issue of the Newsletter Republic themes: Algebraic Geometry, Geometric to go to press, and thereafter will be Information: Analysis, and Probability. Satellite briefly noted in each new issue until the e-mail: [email protected] meetings will be organized in the days meeting takes place, with a reference to Web site: before the congress. the issue in which the detailed http://www.karlin.mff.cuni.cz/PTRT04 Scientific committee: H. Thorisson announcement appeared. [For details, see EMS Newsletter 52] (chair) (Iceland), J. Kr. Arason (Iceland), G. Grubb (Denmark), Ch. Kiselman (Sweden), J.-F. Le Gall September 2004 29–October 2: XII Annual Congress of the Portuguese Statistical Society, (France), P. Koskela (Finland), Mireille Evora, Portugal Martin-Deschamps (France), R. Piene 8–11: Dixièmes journées montoises Information: (Norway), and M. Waldschmidt d’informatique théorique, à Liège e-mail: [email protected] (France) (Tenth Mons theoretical computer sci- Web site: Plenary speakers: S. Helgason ence days, in Liège) www.eventos.uevora.pt/spe2004 (Iceland/USA), M. Yor (France), Information: Address: Comissao Organizadora XII M. Audin (France), S. Neshveyev e-mail: [email protected] Congresso SPE, Departamento de (Norway), H. Schlichtkrull (Denmark), Web site : www.jm2004.ulg.ac.be Matematica, Universidade de Evora, C. Villani (France), F. Loeser (France), [For details, see EMS Newsletter 50] Rua Romao Ramalho 59, 7000-671 A. Buch (Denmark), M. Passare EVORA, PORTUGAL (Sweden), Xiao Zhong (Finland), Olav 13–18: 3rd CIME Course - Stochastic Phone: +351-266745370 Kallenberg (Sweden/USA) Geometry, Martina Franca, Taranto, Fax: +351-266745393 Sessions and their organizers: Italy [For details, see EMS Newsletter 52] 1. Arithmetic theory of differential Information: equations and q-difference equations, e-mail: [email protected] Y. André (France); 2. Combinatorics, Web site: October 2004 E. Steingrimsson (Iceland/Sweden); http://www.math.unifi.it/CIME 3. Geometric topology, J. Dupont Address: Fondazione C.I.M.E. c/o 6–9: HYKE Conference on Complex (Denmark); 4. Group theory, Dipart.di Matematica “U. Dini” Flows, Centre de Recerca R.G. Moller (Iceland); 5. Homological Viale Morgagni, 67/A - 50134 FIREN- Matematica, Bellaterra, Spain methods in algebra and topology, ZE (ITALY) Information: B. Dundas and I. Reiten (Norway); Phone: +39-55-434975 / +39-55- e-mail: [email protected] 6. Lie groups and harmonic analysis, 4237111 Web site: F. Rouviere (France); 7. Mathematical Fax: +39-55-434975 / +39-55-4222695 http://www.crm.es/ComplexFlows physics, A. Kupiainen (Finland); [For details, see EMS Newsletter 51] [For details, see EMS Newsletter 52] 8. Nonlinear partial differential equations, K. H. Karlsen (Norway); 9. Real 20–24: 12th French-German-Spanish algebraic geometry and applications Conference on Optimization, 24–30: Partial Differential Equations (theme: Algebraic Geometry), Marie- Avignon, France in Mathematical Physics (in memory Francoise Coste-Roy (France); Information: of Olga A. Ladyzhenskaya), Trento, 10. Singularities (theme: Algebraic e-mail: [email protected] Italy Geometry), B. Teissier (France); Web site: Organizer: CIRM, Centro Internazio- 11. Foliations in algebraic and arith- http://www.fgs2004.univ-avignon.fr nale per la Ricerca Matematica, Trento, metical geometry (theme: Algebraic [For details, see EMS Newsletter 50] Italy Geometry), T. Ekedahl (Sweden); Scientific Committee: H. Beirao da 12. Moduli spaces (theme: Algebraic 23–26: 4th International Conference Veiga (Pisa), G. Seregin (St. Petersburg), Geometry), C. Faber (Sweden); on Applied Mathematics (ICAM-4), V. Solonnikov (Ferrara), N. Uraltseva 13. Complex geometry (theme: 40 EMS September 2004 CONFERENCES Geometric Analysis), A. Tsikh Manno; Secretary B: Katerina Nicolaou; Format: The Workshop will include (Russia/Sweden); 14. Quasiconformal Treasurer A: Claudia Sortino; Treasurer invited and contributed talks. techniques in analysis (theme: B: A. Philippou; Members: MATHEU Proceedings: It is intended that refereed Geometric Analysis), M. Zinsmeister partners; S. Antoniou Conference Proceedings will be pub- (France); 15. Spectral geometry (theme: Scientific/Program Committee: lished. Geometric Analysis), R. Nest A. Gagatsis; F. Spagnolo; B. D’Amore; Local Organising Committee: Jeff (Denmark); 16. Analysis on metric N. Alexandris; S. Grozdev; G. Makrides; Hunter (Chair) Juha Kinnunen (Finland). Pitta; A. Brigaglia; U. Bottazzini International Organizing Committee: 17. Percolation and spatial interaction Information: George Styan (Chair) ; Hans Joachim Werner (Vice- (Sweden); 18. Applied probability and Web sites: http://math.unipa.it/~grim/ Chair) and Simo Puntanen Probability), F. Baccelli (France); www.cms.org.cy 19. Branching processes (theme: Information: Web site: http://iwms Probability), P. Jagers (Sweden); February 2005 2005.massey.ac.nz/ 20. Large random matrices (theme: Probability), Alice Guionnet (France). June 2005 Information: e-mail: FrancoNordic 5–13: Applications of braid groups [email protected] and braid monodromy, EMS Summer Web site: http://www.raunvis.hi.is/ School, Eilat, Israel 12–24: Foliations 2005, £odz, Poland 1FrancoNordic Congress/ Information: Web site: www.emis.de Description: The conference is the /etc/ems-summer-schools.html fourth in a series devoted to the theory 28–30: 4th Mediterranean Conference of foliations. The previous three took on Mathematics Education, Palermo- March 2005 place in 1990 (£odz), 1995 and 2000 (both Warszawa). The main purpose of Italy the conference is to interchange new Description: The Fourth Mediterranean 9–April 1, 2005: 14th INTERNA- ideas in all aspects of foliation theory Conference on Mathematics Education TIONAL WORKSHOP ON MATRI- and related topics: contact and symplec- with International participation follows CES AND STATISTICS (IWMS- tic structures, confoliations, Engel and a successful series of three such confer- 2005), Massey University, Albany Goursat structures, groups and ences since 1997. At this stage educators Campus, Auckland, New Zealand pseudogroups of action on manifolds, and researchers who may be interested Aim: to stimulate research and, in an holonomy groups and pseudogroups of in attending the conference with or with- informal setting, to foster the interaction foliations etc. out a paper should send a message with of researchers in the interface between Format: 2 mini-courses, invited lec- full address, fax and e-mail to the orga- statistics and matrix theory. The tures and contributed talks nizing committee. Workshop will provide a forum through Invited speakers: V. Kaimanovich Themes: Mathematics in the Modern which statisticians may be better (mini-course), S. Matsumoto (mini- World; Mathematics and Didactics; informed of the latest developments and course), J. Alvarez Lopez, M. Asaoka, Mathematics and Society; Mathematics newest techniques in linear algebra and S. Fenley, V. Grines, X. Gomes Mont, and Talent; Mathematics and Sciences; matrix theory and may exchange ideas J. Heitsch, Y. Kordyukov, H. Minakawa, Mathematics and Technology; with researchers from a wide variety of K. Richardson, E. Zhuzhoma. Mathematics and Motivation; countries. The International Workshops Confirmation of speakers is in progress, Mathematics and Statistics; History of in Matrices and Statistics are held annu- see the conference web site: http: Mathematics ally in different locations. //fol2005.math.uni.lodz.pl Deadline: for full papers is 15 October This Workshop will be the fourteenth Organizers: Katedra Geometrii 2004 in the series, with the previous Uniwersytetu £odzkiego (£odz), Organizers: Cyprus Mathematical Workshops held at the following loca- Banach Centre (Warszawa) Society and University of Palermo tions: 1) Tampere, Finland, August Organizing Committee: S. Hurder Format: The conference will consist of 1990; 2) Auckland, New Zealand, (Chicago), R. Langevin (Dijon), several sessions covering multiple December 1992; 3) Tartu, Estonia, May T. Tsuboi (Tokyo), P. Walczak (£odz) themes but all under the general aim of 1994; 4) Montreal, Quebec, Canada, and M. Czarnecki, secretary, (£odz) Mathematics Education. July 1995; 5) Shrewsbury, England, July Location: Uniwersytet £odzki, £odz, Grants: The conference will offer a two 1996; 6) Istanbul, Turkey, August 1997; Poland day free accommodation to one presen- 7) Fort Lauderdale, Florida, USA, Grants: Several EU grants for young ter of each paper provided that their December 1998; 8) Tampere, Finland, mathematicians will probably be avail- papers are accepted by the scientific August 1999; 9) Hyderabad, India, able committee. All papers will be reviewed December 2000; 10) Voorburg, The Information: by at least two referees. Netherlands, August 2001; 11) Lyngby, e-mail: [email protected] Local Organising Committee: Chair: Denmark, August 2002; 12) Dortmund, Web site: http://fol2005.math.uni.lodz.pl F. Spagnolo, University of Palermo; Germany, August 2003 (http://www.sta Gianna Manno; Claudia Sortino; tistik.uni-dortmund.de/IWMS/main.html); 25–July 2 : Subdivision schemes in A. Scimone; Teresa Marino (all GRIM, 13) Bedlêwo, Poland, August 2004 geometric modelling, theory and Department of Mathematics, University (http:// matrix04.amu.edu.pl/). applications, EMS Summer School, of Palermo) IWMS-2005 is a Satellite Conference to Pontignano, Italy International Organizing Committee: the 55th Biennial Session of the Information: Chair: G. Makrides, Vice-Chair: International Statistical Institute to be Web site: www.emis.de/etc/ems-sum F. Spagnolo; Secretary A: Gianna held in Sydney, April 5 - 12, 2005. mer-schools.html EMS September 2004 41 BOOKS Lecture Note Series 309, Cambridge University Press, Cambridge, 2003, 476 pp., £37,95, ISBN 0 521-53931-5 RecentRecent booksbooks This is a first comprehensive monograph on corings and comodules. Corings are generalizations edited by Ivan Netuka and Vladimír Souèek (Prague) of coalgebras: while a coalgebra is an R-module over a commutative associative unital ring R, Books submitted for review should be sent to the parts of modern mathematics. A certain familiari- equipped with an R-linear coproduct and a comul- following address: ty with computer algebra programs like Maple or tiplication, a coring is an (A,A)-bimodule over an Ivan Netuka, MÚUK, Sokolovská 83, 186 75 Mathematica is helpful for a reader willing to try associative, but not necessarily commutative, uni- Praha 8, Czech Republic these things in practice. The first volume of the tal R-algebra A, equipped with a (A,A)-bilinear work contains a gentle introduction to experimen- coproduct and comultiplication. Corings can also B. Beckman: Codebreakers: Arne Beurling and tal mathematics in its historical context and to its be viewed as coalgebras in a particular monoidal the Swedish Crypto Program during World War methodology, using a series of well-chosen exam- category (the ‘tensor category’). As observed by II, American Mathematical Society, Providence, ples. The first chapter shows it as a rapidly devel- Takeuchi, important examples of corings are the 2002, 259 pp., $39, ISBN 0-8218-2889-4 oping field of mathematics, the second contains so-called entwining structures connecting R-alge- All stories about cryptanalytical work prior and further numerous illustrative examples. These bras with R-coalgebras. Moreover, in the particu- during the World War II read like a thriller. The quite often provoke a reader to work or play with lar case of bialgebra entwinings, entwined mod- book under review is no exception. The author, the the examples on his own PC. The third chapter ules are exactly the classical Hopf modules. So - former head of the cryptanalytical section of FRA describes the progress in computing π, where both and this is the point of the book - the theory of cor- (The Radio Agency of the Swedish Defence authors made significant contributions (BBP for- ings and comodules is a natural common general- Forces), presents a well of information. Its high- mula for computing π). A chapter on normality of ization of several theories connecting algebra with light is the detailed description of Arne Beurling’s numbers deals with another fascinating problem in category theory, noncommutative geometry, and solution of the German Siemens machine, one of which experimental mathematics promises at least quantum physics. the most magnificent achievements of cryptanaly- some path to future discoveries. In contrast to the The book consists of six chapters and an sis of that time. During the previous century, in the fact that the book introduces to the reader a highly Appendix. After developing coalgebra and thirties and the first half of the forties, the cryptan- up-to-date part of mathematics, the next chapter comodule theory from the module-theoretic point alytical work of cryptographic “superpowers” offers another look at classical themes like the fun- of view in Chapter 1, the authors deal with bialge- (Poland, England, and USA) became dominated damental theorem of algebra, the Gamma function bras, classical Hopf algebras and their modules in by mathematicians. The transition of secret ser- or Stirling’s formula. After an exposition on basic Chapter 2. The module theoretic approach of vices to the search of cryptanalytical talents tools, the book is closed by a chapter with the title Chapter 1 pays back in Chapter 3, where basics of among graduates of mathematics departments ‘Making sense of experimental mathematics’. the coring and comodule theory are developed. probably started in Poland and people like Marian For those wanting to know more on the pro- Chapter 4 deals with an important class of corings Rejewski became the founders of modern mathe- gression from numerical experiments to hypothe- coming from ring extensions (together with the matically based cryptology. Well known are the ses and finally to deep mathematical theorems BOCS’s of Rojter and Kleiner, these were the first contributions of Alan Turing to the war efforts in proven within the frame of “classical mathemat- examples of corings truly generalizing coalge- Bletchley Park. Arne Beurling is another, though ics”, the authors prepared the second volume of bras). Chapters 5 and 6 deal with the relation to less known, example of a top class mathematician the work together with R. Girgensohn. In fact, entwinings and weak entwinings. The Appendix σ serving his country as a cryptanalytician during both volumes can be read independently. recalls basics on the module category [M], which the war times. The second volume starts with a chapter on is one of the main tools for the theory. The reason The book starts with an explanation of what a sequences, series, products and integrals, which is is that any right comodule over a coring C is a left * code and a cipher is. Then the author overviews followed by the chapter on Fourier series and inte- module over the left dual ring C, and, for exam- α the Swedish cryptanalytical history to the end of grals. These chapters form one third of the book ple, the so called ‘left -condition’ just says that the World War II. Most of the book is based on and will be of interest to any open-minded person the category of all right C-comodules coincides σ declassified Swedish documents and it traces the with a deeper interest in mathematics, equipped with [*C C]. The book provides a unified and work of Arne Beurling in solving various ciphers with a basic knowledge of undergraduate mathe- general treatment of a theory whose pieces were used by different countries during the war, includ- matics. The following chapters on zeta functions originally developed by people working in rather ing the top secret German ciphers. For the review- and multi-zeta functions, partition, powers, primes distinct areas of algebra, category theory, non- er it is particularly interesting to read about the and polynomials are more specialised. The final commutative geometry, and mathematical Beurling solution of the cipher used by a Czech two chapters invite a reader to explore deeper physics. The book is a welcome addition to the lit- resident in Sweden named Vanìk to communicate methods and tools of experimental mathematics. erature on this young and rapidly developing sub- with the Czechoslovak Government exiled in Each chapter in both volumes is closed by com- ject. (jtrl) London. The second part of the book is dedicated mentaries and additional examples. The amount of to the life and mathematical work of Arne collected material is tremendous. It is impossible V. M. Buchstaber, T. E. Panov: Torus Actions Beurling, a close friend and collaborator of Lars to describe the content of the whole work in detail and Their Applications in Topology and Ahlfors. The book can be highly recommended to in just a few lines. These are very nice and pro- Combinatorics, University Lecture Series, vol. 24, anyone interested in the role of mathematics in voking books showing that experiments both were American Mathematical Society, Providence, classical cryptology and in the history of the 20th and are an important part of the development of 2002, 144 pp., $29, ISBN 0-8218-3186-0 century. (jtu) mathematics. New computer-based tools are This book is an outgrowth of an extensive paper broadly used mainly “outside” mathematics and “Torus actions, Combinatorial topology and J. Borwein, D. Bailey: Mathematics by the book shows “… how today, the use of homological algebra”, written by the authors and Experiment. Plausible Reasoning in the 21st advanced computing technology provides mathe- published in Uspekhi Mat. Nauk (Russian Math. Century, A. K. Peters, Natick, 2003, 350 p., $45, maticians with an amazing, previously unimagin- Surveys) in 2000. It is yet another contribution to ISBN 1-56881-211-6 able ‘laboratory’, in which examples can be the recent rich interplay of combinatorics and con- J. Borwein, D. Bailey, R. Girgensohn: analysed, new ideas tested, and patterns discov- vex geometry with algebraic geometry and topol- Experimentation in Mathematics. ered.” I strongly recommend visiting e.g. web-site ogy. In Chapters 1-5, the authors review basic Computational Path to Discovery, A. K. Peters, on URL http://www.expmath.info, where some notions of both the combinatorial and topological Natick, 2004, 350 p., $49, ISBN 1-56881-136-5 parts of the first volume are also available. Also side with a slight preference for combinatorial Despite the slightly different titles, the books form the very good typesetting and nice graphical notions (which they consider less familiar to a two volumes of the same publication. appearance of the whole work are worth mention- prospective reader). The last three chapters are Mathematical experiments performed on computing. It can be recommended not only to libraries devoted to their own contribution, which is cen- ers are more and more important factors of further but to all members of the mathematical communi- tred around moment angle complexes, their coho- development in the mathematics. The text is care- ty. (jive) mology and to subspace arrangements. This is a fully written and the plentiful short remarks on useful book that will be of interest to the (algebra- related topics are pleasant to read. The material is T. Brzezinski, R. Wisbauer: Corings and ic) combinatorics community as well as to mostly accessible without knowledge of advanced Comodules, London Mathematical Society researchers in algebraic geometry and topology. It is a carefully written state of the art book. (jneš) 42 EMS September 2004 BOOKS B. Buffoni, J. Toland: Analytic Theory of Global part, the C*- algebras of foliated spaces are studied tation of a reductive algebraic group is proved. Bifurcation, Princeton Series in Applied and some of the classical notions from The next chapter is devoted to linear rational rep- Mathematics, Princeton University Press, Riemannian geometry (heat flow and Brownian resentations of a non-reductive algebraic group, Princeton, Oxford, 2003, 169 pp., £29,95, ISBN motion) are generalized to foliated spaces. including the Nagata counterexample to Hilbert’s 0 691-11298-3 Necessary analytic background can be found in 14th problem. Covariants of the action are treated In the book, bifurcation problems for non-linear three appendices. The second part is devoted to in Chapter 5. Categorical and geometric quotients operator equations in infinite dimensional spaces characteristic classes and foliations. Here the read- and linearization of the action, a notion of stabili- are studied. To read the book requires certain er can find constructions of exotic classes based on ty of an algebraic action together with numerical knowledge, hence the first three chapters are the Chern-Weil theorem, vanishing theorem for criterion of stability are described next. The last devoted to a review (without proofs) of basic Godbillon-Vey classes and a discussion on three chapters treat some special cases (hypersur- notions and facts from linear functional analysis, obstructions to existence of a foliation transverse faces in projective space, ordered sets of linear nonlinear functional analysis (e.g., the implicit to the fibres of circle bundles over surfaces. In the subspaces in projective space, and toric varieties). function theory) and from the theory of analytic third part, compact 3-manifolds foliated by sur- (vs) operators in Banach spaces. Main facts on holo- faces are studied. Special methods of 3-manifolds morphic functions of several complex variables, topology yield existence theorems and further R. M. Dudley: Real Analysis and Probability, real analytic functions of several real variables and results unique for dimension three. There is an Cambridge Studies in Advanced Mathematics 74, on (finite-dimensional) analytic varieties, can be appendix with a proof and further discussion of Cambridge University Press, Cambridge, 2002, found in the next three chapters respectively. Palmeiras theorem, which says that the only sim- 555 pp., £32,95, ISBN 0-521-80972-X Analytic sets (in the complex setting, or their real ply connected n-manifold foliated by leaves dif- This book serves as a clear, rigorous, and complete version) have a nice, distinguished structure, feomorphic to Rn-1 is Rn. The book contains a lot of introduction to modern probability theory using which can be used in a study of analytic operator interesting results and can be recommended to methods of mathematical analysis, and a descrip- equations in infinite dimension. A tool needed for anybody interested in the topic. (jbu) tion of relations between the two fields. The first such a relation is a suitable version of the implicit half of the book is devoted to an exposition of real function theorem, reducing infinite dimensional X. Chen, K. Guo: Analytic Hilbert Modules, analysis. Starting with basic facts of set theory, the questions to finite dimension. The last two chap- CRC Research Notes in Mathematics 433, book treats e.g. the real number system, transfinite ters contain applications of previous methods to Chapman & Hall/CRC, Boca Raton, 2003, induction, and problems of cardinality, touching steady periodic water waves. (vs) 201 pp., $99,95, ISBN 1-58488-399-5 both the continuum hypothesis and axiom of The main theme of the book is the structure of choice together with its equivalences. General H. Cabral, F. Diacu, Eds.: Classical and modules over function algebras of holomorphic topology is discussed, including compactness and Celestial Mechanics: The Recife Lectures, functions in several complex variables. The book compactification, completion and completeness, Princeton University Press, Princeton, 2002, concentrates mainly on topics developed by both and metric. Measure theory and integration is 385 pp., £35, ISBN 0-691-05022-8 authors. The book starts with a nice introduction treated carefully, because it serves as the most This book has its origin in a lecture series hosted summarizing main results described in the book. important tool for probability theory. Among by the Federal University of Pernambuco in Chapter 2 describes the technique of characteristic more advanced topics of real and functional analy- Recife, Brazil, between 1993 and 1999. space theory for analytic Hilbert modules, which sis, we can find here an introduction to functional Immediately after opening the book, one gets a was developed by K. Guo. Chapter 3 shows that analysis on Banach and Hilbert spaces, convex feeling of the great enthusiasm of its editors, orga- analytic Hilbert modules in several variables have functions, convex sets and dualities, and measures nizers of the corresponding series of lectures. It is a much more rigid structure than in one variable. on topological spaces. The second half of the book a really attractive book, presenting many impor- The equivalence problem for Hardy submodules contains a description of modern probability theo- tant topics from Hamiltonian dynamics and celes- in the cases of the polydisk or the unit ball is treat- ry, including convergence laws, central limit theo- tial mechanics in an accessible way. The editors ed in Chapter 4. The next chapter describes the rems and laws of large numbers. Ergodic theory, and the mathematical centre in the equator, colo- structure of the Fock space, or more generally, as well as martingales, is studied. More advanced nial city of Recife, deserve great admiration for reproducing function spaces on Cn. Chapter 6 con- topics include convergence laws on separable met- producing such an impressive output - a record of tains a discussion of modules over the Arveson ric spaces, stochastic processes and Brownian their long term seminar, devoted to classical space of square integrable functions on the unit motion. The book can be considered as a textbook. mechanics. (Several contributions to the seminar ball in Cn. In the last chapter, the authors describe It is a self-contained text and all relevant facts are were published independently, before the publica- the extension theory of Hilbert modules over func- proved. The appendices show that the author care- tion of this book.) tion algebras. The book is written in a clear and fully filled all gaps in mathematical background The book contains the following contributions: systematic way and it describes an interesting part needed later. The book contains a number of exer- Central configurations and relative equilibria for of the recent development in the field. Each chap- cises helping to understand the contents. It could the N-body problem (by D. Schmidt), ter ends with useful bibliographic comments. (vs) be very useful for students interested in learning Singularities of the N-body problem (by F. Diacu), both topics, it can also serve as complementary Lectures on the two-body problem (by I. Dolgachev: Lectures on Invariant Theory, reading to standard lectures. Teachers preparing A. Albouy), Normal forms of Hamilton systems London Mathematical Society Lecture Note Series their graduate level courses can use the book as an and stability of equilibria (by H. E. Cabral), The 296, Cambridge University Press, Cambridge, excellent, rigorously written and complete source. Poincaré compactification and applications to 2003, 220 pp., £29,95, ISBN 0-521-52548-9 (mrok) celestial mechanics (by E. Pérez-Chavela), The In many problems in mathematics, the structure of motion of the moon (by D. Schmidt), Lectures on the set of all orbits of a group acting on a suitable J. Faraut, F. Rouviére, M. Vergne: Analyse sur geometrical methods in mechanics (by M. Levi), space is described by means of invariants of the les groupes de Lie et théorie des représentations, Momentum maps and geometric phases: group action. In particular, the case of the action of Séminaires & Congrés 7, Société Mathématique Overview, classical adiabatic angles, holonomy a linear algebraic group G on an algebraic variety de France, Paris, 2003, 177 pp., €40, ISBN for gyrostats, microswimming (by J. Koiller et al.), X is the case studied in classical invariant theory. 2-85629-142-2 and Bifurcation from families of periodic solu- The present book is intended for beginners as a The book is based on lectures presented at the tions (J. K. Hale and P. Táboas). (mzah) first introduction to the theory. Knowledge of fun- summer school on analysis on Lie groups and rep- damental notions and basic facts from algebraic resentation theory, held in 1999 in Kénitra, A. Candel, L. Conlon: Foliations II, Graduate geometry is expected. The purpose of the book is Morocco. It contains a written form of three series Studies in Mathematics, vol. 60, American to offer a short description of main ideas of the of lectures. The first one (given by M. Vergne, Mathematical Society, Providence, 2003, 545 pp., classical invariant theory illustrated by many spe- recorded by S. Paycha) contains a description of $79, ISBN 0-8218-0809-5 cific examples. Every chapter ends with a set of the equivariant cohomology of a manifold, which This is the second volume of the two-volume exercises and with hints for further reading. was introduced independently by N. Berline and series with the title Foliations. It has three inde- The first two chapters treat the classical exam- M. Vergne, E. Witten and M. F. Atiyah and pendent parts, describing three special topics in the ple of the action of the group GL(V) on homoge- R. Bott in the 80’s. In the special case of S1-action theory of foliations: Analysis on foliated spaces, neous polynomials of degree m on the space E, with isolated fixed points, the Paradan formula is Characteristic classes of foliations and Foliated 3- where E itself is the space of homogeneous poly- used for a description of the equivariant cohomol- manifolds. Each part contains a description of a nomials of degree d on V. In the next chapter, the ogy in terms of fixed points of the action. As an topic in foliation theory and its relation to another Nagata theorem on the algebra of invariant poly- important application, it is shown how to obtain field of contemporary mathematics. In the first nomials on the space of a linear rational represen- the Duistermaat-Heckman stationary phase for- EMS September 2004 43 BOOKS mula for a U(1)-action on a symplectic manifold. the Riemann sphere and their use in non-euclidean braic geometry. The book consists of notes written The second series of lectures (given by geometry. The second chapter contains a detailed by lecturers of the corresponding three series of F. Rouviére) was devoted to the Damek-Ricci description of discontinuous groups of motions of lectures. The first contribution (by D. Joyce) is spaces. A manifold is harmonic if the mean value the unit discs with its hyperbolic metric. In the devoted to the introduction and study of theorem holds for harmonic functions. The third chapter, associated surfaces and invariants Riemannian properties. The last contribution (by Damek-Ricci spaces are harmonic manifolds but needed for a classification are discussed. D. Huybrechts) describes compact hyperkähler they are not symmetric spaces. Basic facts from Elementary groups and elementary surfaces, the manifolds. The class of compact hyperkähler man- geometry and harmonic analysis on the Damek- decompositions of the discontinuous groups and ifolds form an interesting class of Ricci-flat mani- Ricci spaces are described in the course. their normal forms, are all treated in the fourth folds, playing an important role in different The third series of lectures (given by J. Faraut) chapter. The final chapter is devoted to isomor- branches of mathematics and mathematical is devoted to Hilbert spaces of holomorphic func- phisms of discontinuous groups, homeomor- physics. In this part, many interesting topics are tions invariant under the action of a group of auto- phisms of the corresponding discs and their exten- presented (including theory of holomorphic sym- morphisms of a complex manifold. The set of such sions to the boundary. A specific feature of the plectic manifolds, deformation theory of complex Hilbert spaces forms a convex cone and it is pos- book is that it is based entirely on geometric argu- structures, cohomology and other properties of sible to use methods of the Choquet theory for a ments. The Fenchel-Nielsen manuscript has been compact hyperkähler manifolds, twistor space and description of its structure and for an integral rep- famous for a long time already and its final publi- moduli space of a hyperkähler manifold and a dis- resentation of their reproducing kernels. The last cation is a valuable edition to mathematical litera- cussion of the projectivity of hyperkähler metrics). parts are devoted to the case of invariant domains ture. (vs) The themes of all three contributions are interre- in the complexification of a compact symmetric lated and together they give a nice introduction space. A part of the school’s program was M. Georgiadou: Constantin Carathéodory, into a very interesting field of research on the bor- reserved for lectures on the research work of par- Mathematics and Politics in Turbulent times, der between mathematics and physics. I would ticipants, the list of lectures and their abstracts can Springer, Heidelberg, 2004, 651 pp., €89,95, like to strongly recommend the book to anybody be found in the book. The book brings nice ISBN 3-540-44258-8 interested in the topic. (jbu) reviews of very interesting areas of the field. (vs) This is an excellent biography of a man who belongs to the most renowned mathematicians of B. C. Hall: Lie groups, Lie algebras, and repre- M. Feistauer, J. Felcman, I. Straškraba: the first half of the 20th century. More than 50 sentations: An Elementary Introduction, Mathematical and Computational Methods for years after his death – much later than his contri- Springer, Heidelberg, 2003, 351 pp., €59,95, Compressible Flow, Numerical Mathematics and bution to the science deserves – this book brings ISBN 0-387-40122-9 Scientific Computation, Clarendon Press, Oxford, his personality and work to life again. It depicts To present a circle of ideas around Lie groups, Lie 2003, 535 pp., £59,95, ISBN 0-19-850588-4 them in full context with that period, so rich on far algebras and their representations, it is necessary The book deals with the numerical solution of reaching revolutionary events, in a very lively and to make a few principal choices. The first question equations describing motion of compressible flu- suggestive way. Being a German mathematician is how to describe a relation between Lie groups ids. Classical as well as modern numerical born in an outstanding Greek family of diplomats, and Lie algebras. To make the book accessible to schemes are summarized here. At the beginning, Constantin Carathéodory was attached to German a broader audience, the author does not suppose the governing equations are determined and the intellectual tradition by his German education, as knowledge of the theory of manifolds. He restricts mathematical problems are defined. Some theoret- well as to Greece by his emotions. Moreover, a the attention to matrix groups. Lie algebra of G is ical results, such as existence and uniqueness of man of his intellectual influence must have also then defined using simple properties of the expo- solutions, are mentioned. The main part of the been a man of political significance. Because of nential map. As for the correspondence between book is concerned with the numerical solution of these facts and in connection with the dramatic Lie group homomorphisms and Lie algebra homo- inviscid as well as viscous compressible flows. period in which he lived, he is very suitable sub- morphisms, the author is using the Baker- Applications of various numerical methods (finite ject for a biography. The author has gathered with Campbell-Hausdorff theorem for its description. volume method, finite element method, discontin- exceptional care almost all accessible material that In the main part of the book, finite dimensional uous Galerkin method and their variants) for the has a bearing to the life of this scientist, and has representations of classical semisimple Lie groups Euler and Navier-Stokes equations are discussed. built a narrative that will fascinate everyone who are classified by their highest weights. Their con- Convergence properties of numerical schemes are opens the book. (jdr) struction is given in three different ways (as quo- mostly derived for a scalar nonlinear convection- tients of Verma modules, or by the Peter-Weyl diffusion equation. Several types of mesh adapta- M. Gidea, C. P. Niculescu: Chaotic Dynamical theorem, or by the Borel-Weil realization). The tion techniques are presented as well. In the book Systems: An Introduction, Centre for Nonlinear proof of the complete reducibility is based on the reader can find a lot of numerical examples Analysis and its Applications 3, Universitaria properties of representations of compact groups. demonstrating the efficiency of described meth- Press, Craiova, 2003, 239 pp., €10, ISBN The Weyl character formula and the classification ods. The book is suitable for researchers as well as 973-8043159-9 of complex semisimple Lie algebras end the main for students dealing with the numerical solution of The book aims to be an introduction to the theory part of the book. To keep prerequisites minimal, compressible flows. (jdol) of dynamical systems on the graduate level. It cov- the author also offers a few appendices. The book ers all classical topics of the subject, including is written in a systematic and clear way, each W. Fenchel, J. Nielsen: Discontinuous Groups structural stability, Lyapunov exponents, the chapter ends with a set of exercises. The book of Isometries in the Hyperbolic Plane, de Gruyter horse-shoe dynamics, hyperbolic and symbolic could be valuable for students of mathematics and Studies in Mathematics 29, Walter de Gruyter, dynamics, chaotic attractors, entropy, ergodicity physics as well as for teachers, for the preparation Berlin, 2003, 364 pp., €78,50, ISBN 3-1-017526-6 and invariant measures. The particular emphasis is of courses. It is a nice addition to the existing lit- In 1920’s and 1930’s, J. Nielsen published three given to the concept of chaos and the various erature. (vs) long papers in Acta Mathematica on discontinu- mathematical tools for its understanding. The ous groups of isometries of the non-euclidean exposition is clear and concise, yet perfectly rigor- U. Hertrich-Jeromin: Introduction to Möbius plane. His further research in the field led him to ous. Many simple examples and a number of exer- Differential Geometry, London Mathematical the idea to describe the theory of discontinuous cises serve excellently to the purpose of the book Society Lecture Note Series 300, Cambridge groups of isometries systematically and in full to be an elementary introduction. The only minor University Press, Cambridge, 2003, 413 pp., generality. After the Second World War, he start- setback seems to be a somewhat lesser quality of £29,95, ISBN 0-521-53569-7 ed a project along these lines together with the print. (dpr) The book is an introduction to the geometry of W. Fenchel and they prepared the first version of submanifolds of the conformal n-sphere. A the manuscript. W. Fenchel was able to continue M. Gross, D. Huybrechts, D. Joyce: Calabi-Yau Möbius transformation is a conformal (i.e., angle (with his collaborators) the work on the project Manifolds and Related Geometries, Universitext, preserving) transformation of the sphere. The after J. Nielsen’s death and the typewritten version Springer, Berlin, 2003, 239 pp., €49,95, ISBN sphere can be considered as a homogeneous space of the manuscript was ready at the end of the 80’s. 3-540-44059-3 of the group of Möbius transforms, hence it is a The book under review is the final version of Summer schools in Nordfjordeid, Norway, have model of Möbius geometry from F. Klein’s point the manuscript, which was prepared for publica- been organized regularly since 1996. Topics vary of view. There are several other models of Möbius tion by A. L. Schmidt after W. Fenchel’s death in each year but there are always three series of lec- geometry (projective model, quaternionic model 1988. It offers a systematic geometric treatment of tures by invited experts with evening exercises. and Clifford algebra model). Classically, a confor- discontinuous groups of isometries. It starts with a The school held in June 2001 was devoted to mal structure is represented by a Riemannian met- careful description of Möbius transformations of recent interaction between differential and alge- ric (modulo a multiplication by a positive func- 44 EMS September 2004 BOOKS tion). The change of Riemannian properties under how to model deterministic (and mostly continu- last part contains a discussion of Maxwell equa- conformal change is described and the Weyl and ous) processes in biology, physiology and ecolo- tions for inhomogeneous media, the Dirac equa- Schouten tensors are introduced. Conformal flat- gy. The style of writing is subordinated to these tion with potentials, and a generalization of the ness is discussed in details. The projective model purposes. It is remarkable that without the classi- Riccati equation to a nonlinear quaternionic equa- and related objects (congruencies, etc.) are intro- cal scheme (definition, theorem and proof) it is tion. The book can be useful not only for mathe- duced in the first chapter. The Cartan method of possible to explain rather deep results like proper- maticians interested in the field but also for engi- moving frames is often used for computations. As ties of the Fitz-Hugh-Nagumo model of nerve neers (the book is based on a course given by the an application of projective model constructions, impulse transmission or the Turing model of pat- author to future engineers). (vs) conformally flat hypersurfaces, isothermic and tern formation. This feature makes the reading of Willmore surfaces are discussed. A quaternionic this text pleasant business for mathematicians T. Lawson: Topology: A Geometric Approach, model introduced in the next chapter is used for a also. There exists a similar book written by Oxford Graduate Texts in Mathematics 9, Oxford study of isothermic surfaces in the four-dimen- J. D. Murray (Mathematical Biology), which con- University Press, Oxford, 2003, 388 pp., £45, sional case, while in higher dimensions, the tains more biological models. In comparison with ISBN 0-19-851597-9 Clifford algebra model is used. At the end of the it, the book under review is also a textbook on dif- The book is a nice introduction to topology with book, the reader can find a discussion of triply ferential equations. It can be recommended for an emphasis on its geometric aspects. It is written orthogonal systems, their Ribaucour transforma- students of mathematics who like to see applica- for two purposes. In the first part, consisting of tions and isothermic surfaces of arbitrary codi- tions, because it introduces them to problems on three chapters, there is material suitable for a one mension. The book is a well-written survey of how to model processes in biology, and also for semester basic course of topology. It starts with classical results from a new point of view and a theoretically oriented students of biology, because basic point set topology with special attention to nice textbook for a study of the subject. (jbu) it presents constructions of mathematical models the topology of Rn, followed by a description of the and the steps needed for their investigations in a classification of surfaces. The last topic introduced J. Ize, A. Vignoli: Equivariant Degree Theory, clear way and without references to other books. and studied in this part is the fundamental group of de Gruyter Series in Nonlinear Analysis and (jmil) a space, including its application to surfaces and Applications 8, Walter de Gruyter, Berlin, 2003, the vector field problem in the plane and on sur- 361 pp., €98, ISBN 3-11-017550-9 J. W. Kammeyer, D. J. Rudolph: Restricted faces. To compute fundamental groups, the The aim of the book is the development and appli- Orbit Equivalence for Actions of Discrete Seifert-van Kampen theorem is introduced and cations of the degree theory in the context of equi- Amenable Groups, Cambridge Tracts in proved. variant maps. (Equivariant simply means that the Mathematics 146, Cambridge University Press, The second part contains an extension of the mapping has certain symmetries, e.g., being Cambridge, 2002, 201 pp., £35, ISBN material of the first part to a full-year course. It even/odd, periodic, rotational invariant, etc.). The 0-521-80795-6 starts with the description of covering spaces, cov- theory is developed both in finite and infinite The main topic of the book belongs to ergodic the- ering transformations and universal covering dimension. The first chapter gives necessary pre- ory and measurable dynamics. It studies suitable space, followed by a study of CW-complexes and liminaries. The second chapter brings the defini- notions of similarity of two dynamical systems their properties from a homotopy point of view. tion of the degree and studies its basic properties. using structure of orbits of corresponding systems. Simplicial complexes and ∆-complexes for As the definition is somewhat abstract (the degree The authors discuss free and ergodic actions of CW-complexes are described as well. The last is defined as an element of the group of equivari- countable discrete amenable groups. The key chapter is devoted to the homology theory with ant homotopy classes of maps between two notions in the book are an orbit relation special attention to its relations to homotopy. spheres), it is useful to compute the degree in var- O={x,Tg(x)}g∈G ⊂ X x X generated by the action, There are about 750 exercises of different levels, ious particular cases. This is accomplished in an orbit equivalence (a measure preserving map which can attract students to a more active study Chapter 3. The last and also the longest chapter, carrying one orbit relation to the other), and of the subject. Solutions of selected exercises are deals with applications to particular ODE’s and to arrangements and rearrangements of orbits. Basic included as an appendix (solutions to all exercises bifurcation theory. The aim of the authors was to definitions and examples can be found in Chapter are available to the instructor in electronic form on write a book that would be easily accessible even 2 (the vocabulary of arrangements and rearrange- application to Oxford University Press). The book to non-specialists, thus the exposition is accompa- ments, and m-equivalence classes of an arrange- is an excellent introduction to the subject and be nied by a number of examples and the use of ment). Fundamental results by Ornstein and Weiss can recommended to anybody interested in geom- abstract special tools is limited. It is also worth on ergodic theory of actions of amenable groups etry and topology as his/her first reading. (jbu) noting that each chapter is accompanied by are reviewed in Chapter 3. The next two chapters detailed bibliographical remarks. (dpr) include key technical lemmas and entropy theory J. Lewin: An Interactive Introduction to for restricted orbit equivalences. Chapter 6 con- Mathematical Analysis, Cambridge University C. U. Jensen, A. Ledet, N. Yui: Generic tains a construction of a topological model for Press, Cambridge, 2003, 492 pp., £27,95, ISBN Polynomials: Constructive Aspects of the Inverse arrangements and rearrangements using a notion 0-521-01718-1 Galois Problem, Mathematical Sciences Research of Polish spaces and Polish actions. The last chap- The book under review provides an introduction to Institute Publications 45, Cambridge University ter contains a formulation and a proof of the equiv- mathematical analysis using possibilities of com- Press, Cambridge, 2003, 258 pp., £45, ISBN alence theorem. Similar questions were studied puters. The book starts with some background 0-521-81998-9 carefully in the last decades for Z-actions as well material concerning quantifiers, sets, logic formu- The inverse Galois problem is to determine, for a as for Zd -actions (d≥ 1). Relations of the results las, and methods of proofs. The second part deals given field K and a given finite group G, whether described in the book to results obtained in these with the set of real numbers, limits of sequences there exists a Galois extension of K, whose Galois special cases can be found in the Appendix. (vs) and functions, differentiation of functions, the group is isomorphic to G. And if there is such an Riemann integral, and infinite series. The next extension, to find an explicit polynomial over K, V. V. Kravchenko: Applied Quaternionic chapters are devoted to improper integrals, whose Galois group is the prescribed group G. The Analysis, Research and Expositions in sequences and series of functions and integration authors present a family of “generic” polynomials Mathematics, vol. 28, Heldermann, Lemgo, 2003, of functions of two variables. The enclosed CD for certain finite groups, which give all Galois 127 pp., €24, ISBN 3-88538-228-8 forms an important on-screen part of the book. extensions having the required group as their Function theory for the Dirac equation has grown The CD contains a lot of exercises with solutions. Galois group. The existence of such generic poly- substantially during the last several decades. The Some parts of the book are explained here in a nomials is discussed and a detailed treatment of case of dimension four is special in two ways – it more detailed way and some chapters are added their construction is given in those cases, when is a physical dimension and spinor fields can be (e.g., calculus of several variables or complex they exist. (jtu) identified with quaternionic functions of a quater- variable calculus) or are treated in an alternative nionic variable. Obtained results were applied to a (advanced) way. This book together with the CD D. S. Jones, B. D. Sleeman: Differential wide spectrum of problems in mathematical will be useful to many students. (mzel) Equations and Mathematical Biology, Chapman physics. The first part contains a summary of basic & Hall/CRC Mathematical Biology and Medicine facts from quaternionic analysis, including inte- E. L. Lima: Fundamental Groups and Covering Series, Chapman & Hall/CRC, Boca Raton, 2003, gral formulae for unbounded domains. The selec- Spaces, A. K. Peters, Natick, 2003, 210 pp., $49, 390 pp., $71,96, ISBN 1-58488-296-4 tion of topics is guided by their possible applica- ISBN 1-56881-131-4 The book is written with two aims: firstly, to be an tions. Boundary problems for Maxwell equations The book is an introduction to a part of algebraic introduction both to ordinary and partial differen- in homogeneous media and the Dirac equation for topology. It concentrates on the circle of ideas tial equations; secondly, to present main ideas on a free particle are treated in the second part. The around homotopy theory, fundamental groups and EMS September 2004 45 BOOKS covering spaces. The first part of the book intro- stability theorems. Chapter 3 contains the and cotangent fibre bundles of a manifold, jet bun- duces a general concept of a homotopy, together Haefliger theorem (there are no analytic foliations dles and they introduce linear differential opera- with a particular case of path homotopy. The fun- of codimension 1 on S3) and the Novikov theorem tors in this algebraic setting. As explained damental group is defined and computed for many (concerning existence of compact leaves in a codi- throughout the book, and in particular in the examples, including real and complex projective mension 1 transversely oriented foliation of a appendix (written by A. M. Vinogradov), it is pos- spaces and classical matrix groups. The funda- compact three-dimensional manifold). The sible to give a motivation coming from classical mental group of the circle is related to winding Molino structure theorem for foliations defined by mechanics for basic notions treated in the book. numbers of plane curves. The second part of the nonsingular Maurer-Cartan forms is treated in The commutative algebra F is related to the labo- book is devoted to basic properties of (differen- Chapter 4. A holonomy groupoid of a foliation is ratory itself, elements of F to measuring devices tiable) covering spaces and their relations to fun- a basic example of so called Lie groupoids. The and points in the spectrum of F to states of an damental groups. Each chapter ends with exercis- last two chapters describe properties of Lie observed physical system. The book contains es. The book starts from the beginning and its groupoids, a notion of weak equivalence between quite a few exercises and many useful illustra- reading requires only a basic knowledge. The Lie groupoids, a special class of étale groupoids, tions. (vs) book is a pleasure to read, it contains a lot of illus- and a Lie algebroid as an infinitesimal version of a trations and presentation is clear and systematic. Lie groupoid. The book is based on course lecture W. K. Nicholson, M. F. Yousif: Quasi- (vs) notes and it still keeps its qualities and nice pre- Frobenius Rings, Cambridge Tracts in sentation. (vs) Mathematics 158, Cambridge University Press, M. Métivier: Semimartingales. A Course on Cambridge, 2003, 307 pp., £55, ISBN Stochastic Processes, de Gruyter Studies in T. Mora: Solving Polynomial Equation Systems 0-521-81593-2 Mathematics 2, Walter de Gruyter, Berlin, 1982, I: The Kronecker-Duval Philosophy, Quasi-Frobenius algebras provide the basic setting 287 pp., €68, ISBN 3-11-008674-3 Encyclopedia of Mathematics and its Applications for modular representation theory of finite groups. The presented monograph is an advanced book on 88, Cambridge University Press, Cambridge, Indeed, the group algebra of any finite group is general martingale theory. A reader with a good 2003, 423 pp., £60, ISBN 0-521-81154-6 quasi-Frobenius. The presented book is a very knowledge of probability and discrete time This is an excellent book for readers interested in accessible introduction to basic properties of processes will appreciate the deep results con- algebraic methods. A part of the content will not quasi-Frobenius rings and the modules over them. tained in the book. Theorems are formulated for be new to most of them; its usefulness lies in the Rather than dealing with classical representation quasimartingales, some parts of the book are fact that so much is brought together in one book. theory, the authors consider a more general setting devoted to Hilbert space valued processes, and the In the first part of the book, the author describes of mininjective rings and show that basics of the theory is not restricted to continuous square inte- the Kronecker-Duval approach to the solution of classical theory can be developed using only ele- grable martingales as is usually the case. Due to systems of polynomial equations. The second part mentary module theory in a more general setting. these facts, the book is a necessity for all contains a discussion of factorisation of polynomi- (A ring is right mininjective, if any isomorphism researchers working with general stochastic als. The author says: “It is my firm belief that the between minimal right ideals is induced by a left processes. best way of understanding a theory and an algo- multiplication. By a theorem of Ikeda, quasi- The book consists of two parts. In the first part rithm is to verify it through computation …” Frobenius rings are exactly the right and left min- the general theory of martingales is given. It starts Accordingly, the book contains 52 numerical injective, right and left artinian rings). with basic definitions and facts from stochastic examples provided with solutions and 27 pro- While basic notions and results on (weak) self- processes; filtration, measurability, adapted and grams. I enjoyed the author’s language enormous- injectivity, CS- and C2-conditions, AB5*, and predictable processes, stopping time, and decom- ly. The author’s words from the preface “the num- dualities are developed through Chapters 2-7, the position. The next chapters continue with the mar- ber of hidden mistakes in a draft is always larger reader is gradually introduced into three challeng- tingale property. We find many classical results in than the number of the found ones” are true. I spot- ing open problems: the Faith Conjecture (whether a quite general context like Doob’s inequalities or ted a few, e.g., in Theorem 5.2.3 as well as in every left or right perfect right self-injective ring is convergence theorems. The first part is concluded Theorem 5.5.6, one has to add the assumption that quasi-Frobenius), the FGF-Conjecture (whether by a chapter devoted to square integrable semi- the degree of considered polynomials is at least the condition that every finitely generated module martingales, quadratic variation and Meyer’s one. This criticism of the text is a little more than embeds in a free module implies that the ring is process. Here we also find the theory of Hilbert a quibble and, in any case, is greatly outweighed quasi-Frobenius), and the Faith-Menal Conjecture space valued martingales and stochastic integrals by its virtues. (asking whether any right strongly Johns ring is with respect to them. In the second part, stochastic (lber) quasi-Frobenius. A ring R is right Johns, if R is calculus is the focus. First, the stochastic integral right noetherian and each right ideal of R is an is built and a semimartingale version of Itô trans- J. Nestruev: Smooth Manifolds and annihilator; R is right strongly Johns, if all full formation theorem is proved. Then, as an applica- Observables, Graduate Texts in Mathematics, vol. matrix rings Mn(R), n ≥ 1 are Johns). The latter tion, the Brownian and Poisson processes are con- 220, Springer, New York, 2003, 222 pp., €64,95, conjecture is investigated in Chapter 8, where an sidered and changes of probability, as well as ISBN 0-387-95543-7 example is given that the conjecture fails if the Girsanov formula, are presented. The book culmi- Main themes of the book are manifolds, fibre bun- term ‘strongly’ is omitted. Chapter 9 deals with the nates with a chapter on SDE. This is not a book dles and differential operators acting on sections Faith Conjecture, providing a generic construction which I would recommend as an introduction to of vector bundles. A classical treatment of these of examples using particular 3x3-upper triangular stochastic processes and martingales. But for any topics starts with a coordinate description of a matrix rings with coefficients in bimodules over rigorous work in the theory of martingales, name- manifold M; the algebra of smooth functions on M division rings. ly non-continuous processes, semimartingales and is a derived object in this approach. The present The book concludes with three Appendices: on multidimensional martingales, it is the book that book is based on an alternative point of view, Morita theory of equivalence, on Bass’ theory of should be consulted first. It is not easy to write a where calculus on manifolds is treated as a part of perfect rings, and on the Camps-Dicks Theorem concise book on general martingale theory but in commutative algebra. In particular, the initial (proving that the endomorphism ring of any artin- my opinion Michel Métivier has done great job. object is a commutative associative unital algebra ian module is semilocal). The authors have (dh) F with certain additional properties. The corre- achieved two seemingly incompatible goals: to sponding smooth manifold is reconstructed as the provide an elementary introduction to the classical I. Moerdijk, J. Mrèun: Introduction to spectrum of F. (A generalization to the non-com- theory of quasi-Frobenius rings, and to bring the Foliations and Lie Groupoids, Cambridge mutative case, which is usually called non-com- reader up to the current research in the field. This Studies in Advanced Mathematics 91, Cambridge mutative geometry, is based on this point of view. makes the book interesting both for graduate stu- University Press, Cambridge, 2003, 173 pp., £30, This generalization, however, is not treated in the dents and researchers in contemporary module ISBN 0-521-83197-0 book.) theory. (jtrl) This is just a small book nicely covering principal The first few chapters describe properties of notions of the theory of foliations and its relations algebras that correspond to smooth manifolds, L. I. Nicolaescu: The Reidemeister Torsion of 3- to recently introduced notions of Lie groupoids introduce a notion of charts and atlases and define Manifolds, de Gruyter Studies in Mathematics, and Lie algebroids. After the first chapter, con- smooth maps between such manifolds. The vol. 30, Walter de Gruyter, Berlin, 2003, 249 pp., taining a definition of a foliation and main exam- authors (J. Nestruev is an invented name hiding a €84, ISBN 3-11-017383-2 ples and constructions, the authors introduce the group of authors) then show equivalence of the This is a book on the Reidemeister torsion and its key notion of holonomy of a leaf, a definition of an algebraic definition with the usual one. In the sec- (mostly Turaev’s) generalizations. When compar- orbifold and they prove the Reeb and the Thurston ond part of the book, the authors define tangent ing it with Turaev’s book (Introduction to 46 EMS September 2004 BOOKS Combinatorial Torsions, Lectures in Mathematics, after significant steps forward by M. Melnikov, Definitions and basic results (self-similar ETH Zürich, Birkhäuser, 2001), which appeared V. Verdera, P. Mattila, M. Christ and others. In Laplacian, density of states); Preliminaries recently, we can immediately see that it has a dif- this research the Menger curvature appeared to be (Grassmann algebra, trace of a Dirichlet form); ferent character. While Turaev’s book can also relevant. In 2003, X. Tolsa characterized remov- The renormalization map (construction, the main serve as a kind of introduction to the subject (as able sets in terms of the Menger curvature. In this theorem in the lattice case); Analysis of the psh well as an introduction to the contemporary volume, the author presents the above mentioned function G (the dichotomy theorem, asymptotic research in the field), the book under review is results and related developments. degree, regularity of the density of states, some devoted to a wide range of applications of the tor- The first two chapters are devoted to the related rational maps); Examples; Remarks, ques- sion, and its reading requires certain prerequisites. Hausdorff measure and rectifiability, including tions and conjecture; and Appendix (plurisubhar- In the first chapter, which makes the reader beta numbers and uniform rectifiability. The monic functions and positive currents, dynamics familiar with the necessary algebraic notions, the Menger curvature is explained in Chapter 3. In of rational maps on projective space, iteration of reader is supposed to have some topological (and Chapter 4, the theory of singular integrals is meromorphic maps on compact manifolds, G - also algebraic) background. The author modestly applied to the Cauchy operator. The Painlevé Lagrangian Grassmanian). (mz) states in the introduction that this is a computa- problem and the analytic capacity are discussed in tionally oriented little book. But let us note that the Chapter 5. In Chapter 6, the proofs of the Denjoy I. Stewart: Galois Theory, third edition, “computations” we find here are very clever com- conjecture and of the Vitushkin conjecture as well Chapman & Hall/CRC Mathematics, Chapman & putations, and the wide variety of applications pre- as related results are presented. Some very recent Hall/CRC, Boca Raton, 2003, 288 pp., $44,96, sented here do not support the description of a lit- results including the Tolsa theorem are given in ISBN 1-58488-393-6 tle book. The book will be indispensable for spe- Chapter 7. The book excellently explains this This is the third edition of the classic textbook of cialists in the field, and I think that it is very good beautiful theory, which is a subtle mix of complex Galois theory, first published in 1972. But it is not that it exists together with Turaev’s book. It is very analysis, harmonic analysis and geometric mea- just a reprint of earlier editions. Those who know well written, with many examples and also many sure theory. The text is almost self-contained. The the first two editions will be surprised by a radical exercises. The wide range of applications will be history of this development is nicely reviewed. At change of presentation. The author reversed the interesting not only for topologists but also for dif- the end, main open problems of the theory are list- original Bourbakiste approach expressed by a slo- ferential geometers. (jiva) ed. (jama) gan “from general to concrete” and now presents the theory in the direction “from concrete to gen- S. P. Novikov, V. A. Rohlin (Eds.): Topology II, J. Roe: Lectures on Coarse Geometry, University eral”. Homotopy and Homology, Classical Manifolds, Lecture Series, vol. 31, American Mathematical Thus after a historical chapter, he starts with Encyclopedia of Mathematical Sciences, Springer, Society, Providence, 2003, 175 pp., $39, ISBN solutions in radicals of polynomial equations of Berlin, 2004, 257 pp., ISBN 3-540-51996-3 0-8218-3332-4 degree 2, 3, 4, and presents a quintic equation The volume under review consists of three chap- The book is based on lectures given by the author solvable in radicals. Factorization of complex ters: Introduction to Homotopy Theory (by at Penn State University. The notion of ‘coarse polynomials is developed from theory of polyno- O. Ya. Viro and D. B. Fuchs), Homology and geometry’ was invented to keep track of large mials with complex coefficients and the funda- Cohomology (again by O. Ya. Viro and scale properties of metric spaces. The first part of mental theorem of algebra. Field extensions of D. B. Fuchs), and Classical Manifolds (by the book introduces an abstract notion of a coarse rational numbers follow including the definition of D. B. Fuchs). The original Russian text was trans- structure, a notion of a bounded geometry coarse rational expressions and the degree of an exten- lated by C. J. Shaddock. The presentation of space, its growth and a notion of an amenable met- sion. As a digression, the author proves non-exis- notions and results in all three chapters is really ric space, and discusses coarse algebraic topology. tence of ruler-and-compass solutions of classical very nice. The first two chapters contain quite The main topic in the middle part is the Mostow geometric problems of squaring the cube, trisect- detailed exposition, while the third chapter has a rigidity theorem saying that if two compact hyper- ing the angle and squaring the circle. Then the more encyclopedia like character. This means that bolic manifolds of dimensions at least 3 are homo- Galois theory starts. After a short explanation of the first two chapters can also serve very well as a topy equivalent, they are isometric. The last part of Galois groups according to Galois, he presents textbook. I would even recommend them for this the book contains a discussion of a notion of modern definitions of the Galois correspondence, purpose, because the presentation is on one hand a asymptotic dimensions and uniform embeddings splitting fields, normal and separable extensions, detailed one (as already mentioned), and on the into Hilbert spaces, together with relations to the and field automorphisms. The fundamental Galois other hand it is not too long. The choice of mater- Kazhdan property of discrete groups. The book correspondence between the subfields of a field ial is very good, the text is saturated with many offers a very readable description of a circle of extension and subgroups of the automorphism examples, and we find here information about fur- ideas around the notion of coarse geometry. (vs) group of the extension is proved. An example of ther developments and recommendations for fur- the correspondence resulting from a quartic equa- ther study. Naturally, because this is an encyclo- C. Sabot: Spectral Properties of Self-Similar tion is also given. Solvable and simple groups are pedia, we find no exercises. The last chapter con- Lattices and Iteration of Rational Maps, introduced and the Galois theorem about solvabil- tains information about the topology of classical Mémoires de la SMF 92, Société Mathématique de ity of equations in radicals is proved. manifolds, and I do not think that information of France, Paris, 2003, 104 pp., €25, ISBN After all this, abstract rings and fields are intro- this type, in such a compact form and to such an 2-85629-133-3 duced and the abstract theory of field extensions is extent, can be found elsewhere. Generally, the The method of a renormalization group, widely developed. The last part of the book contains some whole volume makes a very good impression, and used in physics, produces some considerable applications, e.g., the construction of finite fields, I would say that it is very clearly written. (jiva) mathematical difficulties, when applied to systems constructions of regular polygons, circle divisions on usual, regular lattices. Hierarchical, self-similar (including cyclotomic polynomials) and an algo- H. Pajot: Analytic Capacity, Rectifiability, lattices are much better suited for a detailed math- rithm on how to calculate the Galois group of a Menger Curvature and the Cauchy Integral, ematical development of this method. The present polynomial equation. The penultimate chapter is Lecture Notes in Mathematics 1799, Springer, book gives a detailed treatment of the problems, about algebraically closed fields and the last chap- Berlin, 2002, 118 pp., €22,95, ISBN which were first studied by Rammal and Toulouse ter, on transcendental numbers, contains “what- 3-540-00001-1 on the Sierpiñski gasket, and puts them in a more every-mathematician-should-see-at-least-once”, The question of characterization of removable sets general perspective. The goal of the book is to the proof of transcendence of π. The book is for bounded analytic functions is known as the study systematically, in a mathematically rigorous designed for the second and third year undergrad- Painlevé problem and has attracted the interest of way, the spectral properties of Laplace operators uate courses. I will certainly use it. (jtu) mathematicians for more than 115 years. In a on self-similar sets. A new renormalization map, reformulation due to L. Ahlfors, the point is to which is a rational map defined on a smooth pro- D. Stirzaker: Elementary probability, Second describe for which sets the analytic capacity van- jective variety, is introduced. Then, the character- edition, Cambridge University Press, Cambridge, ishes. In 1977, A. P. Calderón proved the Denjoy istics of the spectrum of these operators are relat- 2003, 536 pp., £65, ISBN 0-521-83344-2 Conjecture that one-dimensional rectifiable curves ed with the characteristics of the dynamics of iter- This book provides an introduction to elementary are removable if and only if their Hausdorff length ates of such a renormalization map. An explicit probability and some of its applications. The word is zero. The proof was based on estimates of sin- formula for the density of states is given, and it is elementary means that the book does not need the gular integrals, namely of the Cauchy operator. shown that the spectral properties of the operator abstract Lebesgue measure and integration theory, The version of the Vitushkin conjecture that a depend substantially on the asymptotic degree of only elementary calculus is used. The strong fea- purely unrectifiable set of finite Hausdorff length the renormalization map. The contents (a slightly ture of the textbook is a choice of good examples. is removable has been proved in 1998 by G.David, shortened list of the chapters of the book) are: Each theoretical explanation is accompanied by a EMS September 2004 47 BOOKS large number of examples and followed by are devoted to preliminaries. The main result of tions of a given variety. The main explicit result in worked examples incorporating a cluster of exer- the first chapter is the Riemann and Hartogs this part is Nori’s connectivity theorem. cises. The examples and exercises illustrate the Theorems on extension of holomorphic functions The third (and final) part of the volume is devot- treated topics and help the student solve the kind and existence of local solutions to the Cauchy- ed to relations between Hodge theory and algebra- of problems typical of examinations. Each chapter Riemann equations. The second chapter contains ic cycles. Using infinitesimal techniques from the concludes with problems. Solutions to many of an introduction to (holomorphic) vector bundles second part, certain cycle class maps and equiva- these appear in an appendix, together with solu- and the Dolbeault complex on differentiable man- lence relations (rational, homological, algebraic tions to most of exercises. ifolds with complex structure (determined from an and Abel-Jacobi equivalence) are established. In The second edition of the book contains some almost complex structure via Newlander- particular, variations on the theme of the relation new sections. A new section provides a first intro- Nirenberg theorem). The third chapter contains a of Chow groups and Hodge theory of smooth duction to elementary properties of martingales, self-contained introduction to Kähler geometry, complex varieties are reviewed. (pso) which is now occupying a central position in mod- Kähler metrics and Chern connections on a holo- ern probability. Another section provides an ele- morphic vector bundle equipped with a Kähler mentary introduction to Brownian motion, diffu- metric. The introductory part ends with the fourth V. A. Zorich: Mathematical analysis I, sion, and the Wiener process, which has under- chapter describing theory of sheaves, cohomology Universitext, Springer, Berlin, 2004, 574 p., pinned much of classical financial mathematics, of a topological space with values in a sheaf and €53,45, ISBN 3-540-40386-8 such as the Black-Scholes formula for pricing theory of acyclic resolutions leading to a proof of V. A. Zorich: Mathematical analysis II, options. Optional stopping and its applications are the de Rham theorem. Universitext, Springer, Berlin, 2004, 681 p., introduced in the context of these important sto- The second part of the exposition is devoted to €53,45 , ISBN 3-540-40633-6 chastic models, together with several associated a proof of the Hodge and Lefschetz decomposition This is a very nice textbook on mathematical new examples and exercises. The list of chapters: theorems of cohomology of a complex manifold. analysis, which will be useful to both the students Introduction, Probability, Conditional Probability The first two chapters of this part summarize a and the lecturers. The list of chapters is as follows: and Independence, Counting, Random Variables: necessary background from analysis on Hilbert Vol. I. 1) Some General Mathematical Concepts Distribution and Expectation, Random Vectors: spaces used to define (formal) adjoints of elliptic and Notation, 2) The Real Numbers, 3) Limits, Independence and Dependence, Generating operators, their Laplace operators and finally, as 4) Continuous Functions, 5) Differential Calculus, Functions and Their Applications, Continuous an application, harmonic forms and their relation 6) Integration, 7) Functions of Several Variables, Random Variables, Jointly Continuous Random to cohomology. The following two chapters give 8) Differential Calculus in Several Variables, Variables, and Markov Chains. Some sections can conceptual applications of these results. The Some problems from the Midterm Examinations be omitted at a first reading (e.g. Generating notion of a polarized Hodge structure is explained and Examination Topics. Vol. II. 9) Continuous Functions, Markov Chains). The book is suitable and applied to the Kodaira embedding theorem, Mappings (General Theory), 10) Differential for a first university course in probability and very which states that a complex manifold is projective Calculus from a General Viewpoint, 11) Multiple useful for self-study. (br) if it admits an integral polarization. The next chap- integrals, 12) Surfaces and Differential forms in ter is devoted to the holomorphic de Rham com- Rn, 13) Line and Surface Integrals, 14) Elements plex and the interpretation of the Hodge Theorem of Vector Analysis and Field Theory, E. B. Vinberg: A Course in Algebra, Graduate in terms of degeneracy of the Frölicher spectral 15) Integration of Differential Forms on Studies in Mathematics, vol. 56, American sequence. The chapter ends with an introduction to Manifolds, 16) Uniform Convergence and Basic Mathematical Society, Providence, 2003, 511 pp., holomorphic de Rham complex with logarithmic Operations of Analysis, 17) Integrals Depending $89, ISBN 0-8218-3318-9 singularities for quasi-projective smooth varieties on the Parameter, 18) Fourier Series and Fourier The book covers all topics, which are usually with mixed Hodge structure on their cohomology. Transform, 19) Asymptotic Expansions, Topics included in basic courses on linear algebra and The third part is devoted to a study of variations of and Questions for Midterm Examinations and algebra in the first two years of study. In addition, Hodge structures. The notion of a family of com- Examination Topics. it also includes a lot of non-standard and interest- plex differentiable manifolds leads to the con- About style of explanation one can say that the ing material going into several different directions. struction of the Kodaira-Spencer map and Gauss- definitions are motivated and precisely formulat- The book is based on the long time teaching expe- Manin connection associated to the local system ed. The proofs of theorems are in appropriate gen- rience of the author. At the beginning, main alge- of cohomology of fibers of this family. Using the erality, presented in detail and without logical braic structures are introduced (groups, rings, Kodaira-Spencer map and the cup product in gaps. This is illustrated in many examples (many fields, algebras, vector spaces). Polynomial alge- Dolbeault cohomology, the period map and its dif- of them arise in applications) and each section bra is studied in detail and basic facts of group the- ferential are described. In the last chapter, various ends with a list of problems and exercises, which ory are covered. Linear algebra is covered in “cycle classes” are studied. In particular, Hodge extend and supplement the basic text. Finally, one Chapter 2, Chapter 5 and Chapter 6, together with classes arising in the study of morphisms of can make several remarks on the approaches used. bilinear and quadratic functionals. Chapter 8 con- Hodge structures are described in more detail. The Real numbers are introduced axiomatically, the tains a description of general multilinear algebra. Deligne cohomology and the Abel-Jacobi map are general concept of limits with respect to (filter) The last four chapters are devoted to commutative introduced, although they are studied in more base is explained and used, e.g. in the definition of algebra (principal ideal domains, Noetherian detail in the next volume of the series. (pso) Riemann integral, and real powers of a positive rings, algebraic extensions, and affine algebraic number are introduced as limits of rational pow- varieties), groups (Sylow theorems, simple groups C. Voisin: Hodge Theory and Complex ers. Trigonometric functions are firstly introduced Galois extensions and Galois theory), linear repre- Algebraic Geometry II, Cambridge Studies in intuitively using the unit circle, then as sums of sentations of associative algebras (complete Advanced Mathematics 77, Cambridge University some power series and finally as inverse functions reducibility, invariants, division algebras) and lin- Press, Cambridge, 2003, 351 pp., £60, ISBN 0- to functions arcsine and arccosine, which are ear Lie groups (the exponential map, the adjoint 521-80283-0 defined after definition of the length of a curve. representation, basic facts on linear representa- The book is the second volume of a two-volume The multiple integral is firstly introduced as a tions). The book is beautifully written, the choice treatise on Hodge theory and its applications. The Riemann integral over an n-dimensional interval of topics and their order is excellent and the book volume consists of three parts. The first part illus- (analogous to the 1-dimensional case). Then for is very carefully produced. It contains a huge num- trates various aspects of the qualitative influence the bounded set, D is defined as the integral over ber of exercises and it appeals to geometric intu- of Hodge theory on the topology of algebraic vari- an interval containing D of the product of the ition whenever possible. It can be highly recom- eties. In particular, the Deligne’s theorem on the given function and the characteristic function of mended for independent reading or as material for degeneration of Leray spectral sequence of ratio- the set D. The Lebesgue necessary and sufficient preparation of courses. (vs) nal cohomology of a projective fibration at E2 and criterion for integrability is proved and frequently its consequences, e.g. surjectivity of the map from used. Line and surface integrals are introduced as C. Voisin: Hodge Theory and Complex rational cohomology of the total space to (the base integrals of differential forms over surfaces. Algebraic Geometry I, Cambridge Studies in generated) monodromy invariant subspace of Smooth and piecewise smooth surfaces are con- Advanced Mathematics 76, Cambridge University rational cohomology of the fiber, are discussed. sidered. Fundamental integral formulas (including Press, Cambridge, 2002, 322 pp., £55, ISBN The second part is devoted to the study of infin- the general Stokes formula) are proved. This mate- 0-521-80260-1 itesimal variations of Hodge structure for a family rial is explained in more detail in Chapter 15. In As the title clearly implies, the book deals with the of smooth projective varieties and its applications, Chapter 14, vector versions of fundamental inte- basics of Hodge theory and its relation with com- in particular those concerning the case of complete gral formulas are stated and vector fields having a plex algebraic geometry. The first four chapters families of hypersurfaces of complete intersec- potential are also studied. (jkop) 48 EMS September 2004