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. 52 e-mail: [email protected] KRZYSZTOF CIESIELSKI Mathematics Institute June 2004 Jagiellonian University Reymonta 4, 30-059 Kraków, Poland EMS Agenda ...... 2 e-mail: [email protected] STEEN MARKVORSEN Editorial by Ari Laptev ...... 3 Department of Mathematics, Technical University of Denmark, Building 303 EMS Summer Schools...... 6 DK-2800 Kgs. Lyngby, Denmark EC Meeting in Helsinki ...... 6 e-mail: [email protected] ROBIN WILSON On powers of 2 by Pawel Strzelecki ...... 7 Department of Pure Mathematics The Open University A forgotten mathematician by Robert Fokkink ...... 9 Milton Keynes MK7 6AA, UK e-mail: [email protected] Quantum Cryptography by Nuno Crato ...... 15 COPY EDITOR: KELLY NEIL Household names in Swedish mathematics by Kjell-Ove Widman ...... 17 School of Mathematics University of Southampton "Sauvons la recherche" ...... 21 Highfield SO17 1BJ, UK e-mail: [email protected] Portugal in the 1st years of the IMU by E. Hernandez-Manfredini, A. Ramos ...... 23 SPECIALIST EDITORS INTERVIEWS Interview with Ari Laptev ...... 25 Steen Markvorsen [address as above] SOCIETIES ERCOM: EURANDOM (Eindhoven) ...... 28 Krzysztof Ciesielski [address as above] EDUCATION Abel Prize, ICME-10 and ICMI awards ...... 30 Tony Gardiner University of Birmingham Forthcoming Conferences ...... 36 Birmingham B15 2TT, UK e-mail: [email protected] Recent Books ...... 39 MATHEMATICAL PROBLEMS Paul Jainta Designed and printed by Armstrong Press Limited Werkvolkstr. 10 Crosshouse Road, Southampton, Hampshire SO14 5GZ, UK D-91126 Schwabach, Germany telephone: (+44) 23 8033 3132 fax: (+44) 23 8033 3134 e-mail: [email protected] e-mail: [email protected] ANNIVERSARIES June Barrow-Green and Jeremy Gray Published by European Mathematical Society Open University [address as above] ISSN 1027 - 488X e-mail: [email protected] and [email protected] The views expressed in this Newsletter are those of the authors and do not necessari- CONFERENCES Vasile Berinde [address as above] ly represent those of the EMS or the Editorial team. RECENT BOOKS Ivan Netuka and Vladimír Souèek NOTICE FOR MATHEMATICAL SOCIETIES Mathematical Institute Labels for the next issue will be prepared during the second half of August 2004. Charles University, Sokolovská 83 Please send your updated lists before then to Ms Tuulikki Mäkeläinen, Department of 18600 Prague, Czech Republic Mathematics and Statistics, P.O. Box 68 (Gustav Hällströmintie 2B), FI-00014 University of e-mail: [email protected] Helsinki, Finland; e-mail: [email protected] and [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 June 2004 1 EMS NEWS EMS Committee EMS Agenda EXECUTIVE COMMITTEE PRESIDENT 2004 Prof. Sir JOHN KINGMAN (2003-06) 19-24 June Isaac Newton Institute EURESCO Conference. 20 Clarkson Road Symmetries and Integrability of Difference Equations: EuroConference on Analytic Linear and Nonlinear Difference Equations and Special Functions Cambridge CB3 0EH, UK Helsinki (Finland). e-mail: [email protected] 25 June VICE-PRESIDENTS EMS Executive Committee meeting at Uppsala (Sweden). Prof. LUC LEMAIRE (2003-06) Contact: Helge Holden, email: [email protected] Department of Mathematics 26-27 June Université Libre de Bruxelles EMS Council meeting at Uppsala (Sweden). C.P. 218 – Campus Plaine Contact: Helge Holden, email: [email protected] Bld du Triomphe or Tuulikki Mäkeläinen email: [email protected] B-1050 Bruxelles, Belgium 27 June - 2 July e-mail: [email protected] 4th European Congress of Mathematics, Stockholm. Prof. BODIL BRANNER (2001–04) Web site: http://www.math.kth.se/4ecm Department of Mathematics 4-24 July EMS Summer School at Cortona (Italy). Technical University of Denmark Evolution equations and applications. Building 303 Contact: Graziano Gentili, email: [email protected] DK-2800 Kgs. Lyngby, Denmark 15-23 July e-mail: [email protected] EMS Conference at Bêdlewo (Poland). SECRETARY Analysis on metric measure spaces. Prof. HELGE HOLDEN (2003-06) Web site: http://impan.gov.pl/Bedlewo/ Department of Mathematical Sciences 15 August Norwegian University of Science and Technology Deadline for submission of material for the September issue of the EMS Newsletter. Alfred Getz vei 1 Contact: Martin Raussen, email: [email protected] NO-7491 Trondheim, Norway 30 August - 3 September e-mail: [email protected] EMS Summer School at Universidad de Cantabria, Santander (Spain). Empirical processes: Theory and Statistical Applications. TREASURER Web site: http://www.eio.uva.es/ems/ Prof. OLLI MARTIO (2003-06) 3-5 September Department of Mathematics EMS-Czech Union of Mathematicians and Physicists (Mathematics Research Section). P.O. Box 4, FIN-00014 Mathematical Weekend, Prague (Czech Republic). University of Helsinki, Finland Web site: http://mvs.jcmf.cz/emsweekend/ e-mail: [email protected] 5-6 September EMS Executive Committee Meeting in Prague. ORDINARY MEMBERS Prof. VICTOR BUCHSTABER (2001–04) Steklov Mathematical Institute Russian Academy of Sciences 2005 5-13 February Gubkina St. 8, Moscow 117966, Russia EMS Summer School at Eilat (Israel). e-mail: [email protected] Applications of braid groups and braid monodromy. Prof. DOINA CIORANESCU (2003–06) 25 June - 2 July Laboratoire d’Analyse Numérique EMS Summer School at Pontignano (Italy). Université Paris VI Subdivision schemes in geometric modelling, theory and applications. 4 Place Jussieu 17-23 July 75252 Paris Cedex 05, France EMS Summer School and European young statisticians' training camp at Oslo (Norway). e-mail: [email protected] 13-23 September Prof. PAVEL EXNER (2003-06) EMS Summer School at Barcelona (Catalunya, Spain). Department of Theoretical Physics, NPI Recent trends of Combinatorics in the mathematical context. Academy of Sciences 16-18 September 25068 Rez – Prague, Czech Republic EMS-SCM Joint Mathematical Weekend, Barcelona (Catalunya, Spain). e-mail: [email protected] 25 September - 2 October Prof. MARTA SANZ-SOLÉ (2001-04) EMS Summer School and Séminaire Européen de Statistique at Warwick (UK). Facultat de Matematiques Statistics in Genetics and Molecular Biology. Universitat de Barcelona 12-16 December Gran Via 585 EMS-SIAM-UMALCA joint meeting in applied mathematics. E-08007 Barcelona, Spain Venue: the CMM (Centre for Mathematical Modelling), Santiago de Chile. e-mail: [email protected] Prof. MINA TEICHER (2001–04) Department of Mathematics and 2007 Computer Science 16-20 July ICIAM 2007, Zurich (Switzerland). Bar-Ilan University Ramat-Gan 52900, Israel e-mail: [email protected] Cost of advertisements and inserts in the EMS Newsletter, 2004 EMS SECRETARIAT Ms. T. MÄKELÄINEN (all prices in British pounds) Department of Mathematics 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)-9-1912-2883 Intermediate rates: Full page: £176; half-page: £98; quarter-page: £59 fax: (+358)-9-1912-3213 telex: 124690 Inserts e-mail: [email protected] Postage cost: £14 per gram plus Insertion cost: £58 (e.g. a leaflet weighing 8.0 grams will website: http://www.emis.de cost 8x£14+£58 = £170) (weight to nearest 0.1 gram) 2 EMS June 2004 EDITORIAL EditorialEditorial Ari Laptev (Stockholm) Chairman, 4ECM Organizing Committee

The Fourth European Congress of stantial funds amazingly quickly. Mathematics will take place in Among our major sponsors are the Knut Stockholm between June 28 and July 2 and Alice Wallenberg Foundation, the and is organized by the Royal Institute Swedish Foundation for Strategic of Technology (KTH) in collaboration Research, the Swedish Research with Stockholm University (SU). Council, the Nobel Committees for My possible involvement in the orga- Physics and Chemistry of the Royal nization of this event became apparent Swedish Academy of Sciences and many in July 2000 at a Conference in Bristol, others. Ari Laptev when I received an e-mail from Anatoly The Swedish Ministry of Higher Vershik, who was a member of the EMS Education and the bank of Sweden the members of the Prize Committee for Executive Committee (EC) and was par- Tercentenary Foundation have also con- their hard work in selecting excellent ticipating in the 3ECM in Barcelona. He tributed to 4ECM funds by each provid- candidates for the EMS prizes. told me that there were some difficulties ing financial support of 1/2 million SEK. By now the Organizing Committee has in raising funds for organizing the next We are extremely grateful to all the considered two rounds of 4ECM poster 4ECM in 2004, which was planned to be sponsors. Without their support the applications. There were two deadlines - held Göteborg. On behalf of the EMS 4ECM could not have been organized. 15th of February and 20th of April. EC he asked me for some help since I The 4ECM Committees worked very Altogether 358 posters were accepted. was the Vice President of the Swedish well. Lennart Carleson as the President Unfortunately not all of them will be Mathematical Society (SMS) at that of the Scientific Committee was most presented at the 4ECM since some of the time. Anatoly found it hard to believe efficient in organizing two meetings of poster applicants depend on receiving that Sweden did not have the funds to his Committee at KTH. At the first grants in order to attend. organize the 4ECM. Naturally, I did not meeting the decision regarding the struc- There were more than 400 4ECM have much to say, but in the end of ture of the 4ECM was made and in June grant applications, mostly from Eastern August 2000 there was a meeting at 2003, at the Committee’s second meet- Europe and the Former Soviet Union. KTH, where it was preliminarily agreed ing, all the speakers were chosen, with Unfortunately our financial possibilities to try to raise funds for the 4ECM in the exception of those speakers for allow us to give only 209 grants, which Stockholm. Science Lectures. The Organizing cover travel, a double hotel room, lunch- At that meeting everybody assumed Committee chose the latter afterwards, es and the Conference fee. We are very that, as the future President of the SMS, in coordination with L. Carleson. grateful to the EMS EC for helping us to I would take the major responsibility for The task of finding a President of the obtain $25,000 from UNESCO-ROSTE, organizing 4ECM, although this was Prize Committee was rather difficult and which allowed us to support participants certainly not my original intention. the EMS Executive Committee was very from Central and Eastern European In November 2000, Anders Lindquist, happy when Nina Uraltseva finally countries. In total we have more than chairman of the Mathematics agreed to take on this important respon- 1,600,000 SEK for the 4ECM grants, Department, KTH, Ulf Persson and then sibility. The Prize Committee was which are approved mostly for young President of the SMS, joined me in quickly formed and its work culminated participants. London to attend the EMC Executive at the meeting in Stockholm at the end of Although organizing a Congress with Committee meeting. We were met with February, when 10 EMS Prize Winners more than 1000 participants is hard an element of friendly curiosity. Trying were chosen. work, I have found it very rewarding to play a “hard game” our delegation Being invited to the meeting of this too. It allowed me to start seeing wanted to have as much freedom in Committee I can confirm that the main Mathematics much more as a whole sub- organizing 4ECM as possible. To our decisions were made surprisingly quick- ject, with many people trying to create great surprise we found a lot of under- ly and in a very friendly way. It is not beautiful new structures and find bril- standing among the EMS EC members. easy to choose 10 prizewinners from liant methods with which to solve old In particular, when we suggested more than 50 young mathematicians and new exciting problems. Lennart Carleson as a possible President nominated for an EMS prize and who Having only 1½ months remaining I of the 4ECM Scientific Committee, the almost all deserve such a prize. am beginning to feel slightly apprehen- EC members said that they had the same On behalf of the 4ECM Organizing sive about how it will all go. Almost idea. So the meeting ended on a happy Committee I would like to express my four years ago I could only guess how note and in mutual agreement. gratitude to the members of the much work has to be done and even now Mostly due to Anders Lindquist’s con- Scientific Committee for choosing an I am still not quite sure I understand tacts we were able to raise some sub- extremely interesting programme and to fully! EMS June 2004 3 EEvvoolluuttiioonn EEqquuaattiioonnss aanndd AApppplliiccaattiioonnss

SUMMER SCHOOL of the European Mathematical Society Scuola Matematica Interuniversitaria Supported by the Istituto Nazionale di Alta Matematica and the European Commission

Cortona (Palazzone), Italy 4th to 24th July 2004

The SUMMER SCHOOL will be devoted to evolution problems, includ- ing: • semigroups and evolution equations in Banach spaces; • linear and non-linear parbolic type problems using functional analytical methods.

The School will provide an occasion to facilitate and stimulate discus- sion between all participants, in particular, young researchers and lecturers, for whom financial support is available.

Courses • Semigroups for linear evolution equations (Reiner Nagel, University of Tubingen, Germany) • Linear and non linear diffusion problems (Alessandra Lunardi, University of Padova, Italy)

Website: http://www.matapp.unimib.it/smi EMS NEWS EMSEMS SummerSummer SchoolsSchools Luc Lemaire (Brussels) and David Salinger (EMS Publicity Officer)

Some time ago, we wrote that the 17-23 July 2005 at Oslo, Norway: will help with advertisement and European Mathematical Society had European young statisticians’ training organisation, as well as the applications been unsuccessful in its application for camp; for financial support. Framework 6 support of its summer Proposals for future Summer Schools school programme. No sooner had the 13-23 September 2005 at Barcelona, can be sent at any time by e-mail to Luc article appeared, than Vice-President Spain: Recent trends of Combinatorics Lemaire (e-mail: llemaire @ulb.ac.be) Luc Lemaire was contacted and told in the mathematical context; The deadline for proposals for EMS that the project had been promoted Summer Schools for 2006, 2007 and from the reserve list. The EMS was 25 September - 2 October at Warwick, 2008 is tentatively fixed on December working in close collaboration with UK: Statistics in Genetics and 31, 2004. the European Regional Committee of Molecular Biology. the Bernoulli Society, the Centre de EMS mathematical weekends Recerca Matematica (CRM) of We also obtained support from The EMS has launched a new format of Barcelona, the Scuola Matematica Unesco-Roste for the Summer joint meetings with its corporate mem- Interuniversitaria (SMI) of Firenze, Schools: ber societies, following the model set the Mathematical Research and - Analysis on metric measure spaces, out by the Portuguese Mathematical Conference Centre in Bêdlewo Bêdlewo, Poland, 15 to 23 July Society in the meeting that took place (Poland), the Emmy Noether Institute 2004; in Lisbon from September 12 to 14, (German-Israeli research institute - EMS Course on Empirical 2003 (http:// www.math.ist.utl.pt/ems/). located in Israel) and various groups of Processes: Theory and Statistical These “EMS- joint mathematical mathematicians working in European Applications in Laredo, from 29 week-ends” will start on a Friday, and universities. We shall pass over the August to 3 September 2004. finish on the Sunday, both at traumatic hours of form filling and fur- lunchtime, so that they can be easily ther negotiation, familiar to anyone For information on participation and attended during term-time. Each would who gets involved in European grants, conditions for financial support, please cover around 4 subjects, chosen by the but in March the EMS obtained a check on the Society’s web site: local organisers to fit the research major grant of the European http://www.emis.de/etc/ems-summer- strengths of the local mathematicians, Commission (Marie Curie Series of schools.html or new subjects they would want to Conferences) for the following eight develop. For each subject, a plenary Summer Schools. In the list below, Summer schools for the following lecture and two half-days of parallel attentive readers will perceive that not years sessions will be organised. Past experi- all the events happen in summer, but On June 24, the EU will have decided ence shows that such an internationali- all bar the Bêdlewo meeting do count on some modifications of the rules of sation of the meetings of national soci- as EMS Summer Schools. applications for Summer Schools sup- eties helps to substantially increase port. We shall thus launch a new call participation. The EMS will help with 4-24 July 2004 at Cortona, Italy: for proposals. scientific organisation, publicity and Evolution equations and applications; funding applications. With more than Call for proposals fifty corporate members, the EMS 15-23 July 2004 at Bêdlewo, Poland: The EMS will pursue its programme of hopes to see regular meetings of this Analysis on metric measure spaces; Summer schools, aiming at running format. Note that mathematics depart- such schools in pure and applied math- ments or individual members can also 12-19 December 2004 at Munich, ematics. The guidelines for such events plan such meetings. Germany: The Statistics of Spatio- are that there must be a very strong Two more Week-Ends are planned, Temporal Systems; component of training of young one in Prague, Czech Republic, researchers (in the first 10 years of their September 3-5, 2004 (http://mvs.jcmf. 5-13 February 2005 at Eilat, Israel: career) by means of integrated courses cz/emsweekend), and one in Barcelona, Applications of braid groups and and lectures at advanced level. This can September 16-18, 2005. braid monodromy; be supplemented by conference type research lectures, but the training com- Call for proposals 25 June - 2 July 2005 at Pontignano, ponent is needed. The courses should Proposals for future Joint Italy: Subdivision schemes in geomet- aim at an international audience (no Mathematical Weekends can be sent at ric modelling, theory and applica- more than 30% of participants should any time by e-mail to Luc Lemaire tions; come from a single state). The EMS (e -mail: [email protected]). EMS June 2004 5 EMS NEWS

TheThe HelsinkiHelsinki MeetingMeeting ofof thethe ExecutiveExecutive CommitteeCommittee David Salinger, EMS Publicity Officer

It wasn't cold for Helsinki at the end of nations had been received for the Felix now that it was being published by our February, just a few degrees below Klein prize. Publishing House, it was no longer zero. The streets were slippery and the It was agreed to support a mathe- included in the Springer package. sea was frozen, but the Executive matical weekend in Prague in An additional 3000 euro was Committee met indoors and got on September 2004. The offer of a math- approved to be used for grants to sup- with its business. ematical weekend in Barcelona was port 4ecm. The best news was that, after all, we welcomed: this is now to take place on The French ministry had produced a were likely to get funding for the sum- 16-18 September 2005. report on Zentralblatt, which, while mer schools from the EU's sixth frame- The President reported that Gérard critical of a few aspects, gave overall work programme. Final confirmation Huet, Terry Lyons, Simon Tavare and support. It had recommended the set- arrived after the meeting and the full Martin Groetschel had agreed to speak ting up of an advisory board and the list is in this issue's agenda. This is the at the European Science Open Forum Committee endorsed this approach. culmination of very hard work, in meeting in Stockholm in August. Luc Unfortunately, no one on the negotiation and form-filling, in partic- Lemaire had been invited to give a ple- Committee was able to carry forward ular by Luc Lemaire and Tuulikki nary lecture. the Digital Maths Library initiative. Mäkeläinen. It means also that an A meeting had been held in Brussels The EMS publishing house had additional staff member, Miku about the proposed European Research started producing books and journals. Koskenoja, can be employed at the Council. If this went ahead, the It hoped to move into profit in 2005. Society's Helsinki office. Society would have to be active to The Newsletter now had a copy edi- Other good news is that the Society ensure that funding through the tor. Plans were afoot to have parts of admitted as Institutional members the Research Council would be mathemat- the Newsletter in an electronic data- Mathematisches Forschungsinstitut ics-friendly. base. Oberwolfach, Institut Henri Poincaré, The Committee discussed whether The Committee had received a pro- Centre Emile Borel, the International the Society's journal, JEMS, should posal asking it to support a centre of Centre for Mathematics, and the Abdus have a second series devoted to applied mathematical excellence in Palestine Salam International Centre for mathematics. The chair of the Applied (Bir-Zeit University). It was agreed to Mathematical Physics. The Committee Mathematicss committee would be ask the chair of the Developing recommended to Council the following asked to investigate the viability of Countries committee to look into this. societies for corporate membership: such a journal. It was important that The President thanked Olli Martio the Cyprus, Malta and Romanian JEMS should be recommended to and Tuulikki Mäkeläinen for organis- Mathematical Societies and the libraries by mathematicians, because, ing the meeting. Belgian Statistical Society. It was hoped that other statistical societies would follow that lead; three of the seven forthcoming summer schools were in statistics. The Society had been asked to sug- gest names to the European Science Foundation for a panel to consider applications for young investigators' fellowships. The Committee agreed to recom- mend to Council that the corporate membership fee should be increased slightly, but that the individual mem- bers' rate should remain the same. The Committee had received no pro- posals for Diderot forums or EMS lec- tures. It was recognised that the for- mer were difficult to organize and depended on local enthusiasm. Several different models were suggested to revive the EMS lectures: it was agreed to discuss this further at Council. There was a strong field of candi- dates for the EMS prizes, but no nomi- EC meeting 6 EMS June 2004 FEATURE

OnOn powerspowersOn powers of 2 ofof Paweł22 Strzelecki (Warsaw)

Pawe³ Strzelecki (Warsaw) Pawe³ Strzelecki (Photo "die arge lola", Stuttgart)

Let’s have a look at the following interesting little problem: ( ) Problem: Consider a sequence an n∈N consisting of the first digits of the successive powers of 2:

1, 2, 4, 8, 1, 3, 6, 1, 2, 5, 1, 2, 4, 8, ...

Will 7 ever appear in this sequence?

This is a translation from Polish of an article that appeared in the Polish journal Delta 7/1994.

This problem, in different versions, k we have log(2), a = log(77) − 1, b = log(78) − 1, is well known in mathematical litera- then because of the equalities 1 = k n k ture. It can be found in A walking mathe- 7 · 10 < 2 < 8 · 10 [log 77]=[log 78], we conclude that matician, by Zbigniew Marciniak (Delta the digit 7 can even appear twice, in 7/1991), or in the famous book Ordi- We can get a simpler description of this the first two places of the decimal no- nary differential equations, by V.I.Arnold. condition if we take the decimal loga- tation of a power of 2. Using an ana- It is usually accompanied by auxiliary rithms of both sides; this yields logous argument we readily discover facts or suggestions, intended to make + ( ) < ( ) < + ( ) that any finite sequence of digits can a solution more accessible. Neverthe- k log 7 n log 2 k log 8 . appear at the beginning of a decimal less, I do not know of any publication notation of a power of 2, like 1995 or Since decimal logarithms of 7 and 8 lie in which the problem is presented to- 1234 or 567890 etc. At the end of this between 0 and 1, we conclude that k is gether with a complete solution, inclu- article we exhibit a table of important simply the integer part of the number ding every detail. Let us try to re- historical dates compared with the cor- n log(2), which leads to the following medy this unfortunate situation. We responding powers of 2 - to satisfy the inequalitites: start with a rather desperate solution sceptics. We can even deduce more: requiring just a piece of paper and a log(7) < n log(2) − [n log(2)] < log(8). pencil, or perhaps some more advan- Corollary. If an integer p > 1 is not an ced versions of these tools (a good cal- And now it suffices to bring together integer power of 10, then any sequence culator or a computer?). An assiduous some known facts. of digits can appear at the beginning reader will easily verify that of the decimal notation of some nth po- ( ) 246 = 70, 368, 744, 177, 664. Lemma 1. The number log 2 is irra- wer of p for some n. tional. Going further with this experiment it To prove the corollary, just observe can be seen that 7 is the first digit of the Lemma 2. If a number x is irrational that log(p) is irrational and then repeat th th th th th 56 ,66 ,76 ,86 and 96 power of 2 and c(n) := nx − [nx], then for any a the same argument as above. th (but the first digit of the 106 power of and b in [0,1] infinitely many members But then, why does 7 not appear 2 is 8, not 7). This method, however, is of the sequence c(n) lie within the in- among the first members of the se- clearly far from being elegant. terval (a, b). quence we introduced at the very be- It is time then for a better solution, ginning? Why does this deceitful se- which might also lead to new conclu- Before we prove these two lemmas quence pretend to be periodical? The sions. First of all, we should try to let’s have a look at their consequences. reason is simple. The number log(2)= realize what the statement, that 7 is First, by applying Lemma 2 so that 0.3010299956... can be very well ap- n the first digit of a number 2 , actually x = log(2), a = log(7), b = log(8), proximated by the rational number 0.3, means. we get that 7 is the first digit of infi- and for all rational x the sequence The answer is simple: 7 is the first di- nitely many powers of 2. If we apply c(n)=nx − [nx] is periodical. This is git of 2n if and only if for some natural Lemma 2 again, to get the numbers x = why after seeing the first initial mem- EMS June 2004 7 FEATURE a(7,n) log(8)−log(7) bers of the sequence a(n) we come to can be rephrased as follows: If we walk →∞ = limn a(8,n) log(9)−log(8) an erroneous conclusion that the se- along a circumference of perimeter 1 = 1.1337... > 1. quence has period 10 and 7 is not a for long enough and make steps of an This means that looking at suffi- member of it, while 8 appears quite of- irrational length, then we shall step on ciently long initial fragments of the se- ten. each hole with a frequency which is quence a(n), we will see slightly more directly proportional to the size of the In 1910 Waclaw Sierpinski, Hermann sevens than eights. Weyl and P. Bohl proved indepen- hole. The result of Bohl, Sierpinski´ and dently of each other that for an irra- Let’s translate this fact into our lan- Weyl which we have previously men- tional x the sequence c(n)=nx − [nx] guage of powers of 2. Let a(7, n) tioned and our little fact on sevens is homogeneously distributed over the and a(8, n) be the number of sevens and eights are actually simple conse- interval [0,1]. More precisely, if we take and eights, correspondingly, among quences of a very general and deep arbitrary a and b (with a < b) in [0,1], the first n members of the sequence ergodic theorem due to G.D.Birkhoff and let k(n, a, b) denote the number of a(n). By the last formula, we have (1931), but that’s another story. elements of the set a(7, n) A problem for the reader: For what {c(i) :1≤ i ≤ n, c(i) ∈ (a, b)}, lim = log(8) − log(7), n does the number 2n have four conse- n→∞ n cutive sevens at the beginning? What then about five sevens? How do you esti- ( ) ( ) k n, a, b a 8, n = ( ) − ( ) mate from above, the least n such that lim = b − a lim→∞ log 9 log 8 , n→∞ n n n the decimal notation of 2n begins with In more illustrative terms, the theorem so consequently 1995 consecutive sevens?

A calendar involving powers of 2

EVENT Year Power of 2 Foundation of the Polish State 966 2568 = 9.66... × 10170 The King of Poland Jan III beats the Turcs at Vienna 1683 2709 = 1.683... × 10212 The first edition of Newton’s Principia 1687 26143 = 1.687... × 101849 The Battle of Waterloo 1815 2931 = 1.815... × 10280 Foundation of the Warsaw University 1816 27339 = 1.816... × 102209 Birth of Stefan Banach 1892 28133 = 1.892... × 102448 Beginning of World War II 1939 25522 = 1.939... × 101662 End of World War II 1945 21931 = 1.945... × 10581 First published issue of ”Delta” 1974 21549 = 1.974... × 101081

Appendix

Proof of Lemma 1 ful representation of points. Imagine the the point 0 and follow along the circle T for If the equality log 2 = r held true for real axis is wound around a circle T with an infinitely long time in the same direction m ε some natural numbers r and m, then by radius 1 and a distinguished point 0. For making steps of length , then you step in- r ∈ [ ] ( ) the definition of logarithm, 10 m = 2 and each pair of numbers a, b 0, 1 the arc of T finitely many times over the arc a, b . In- [ ] − consequently 10r = 2m. This, however, is a that corresponds to the interval a, b on the deed, its length b a is bigger than the ( ) contradiction, since 5 is a factor of any po- real axis will be also denoted by a, b . Let length of your step. This completes the −→ wer of 10, whereas it is certainly not a factor f : T T represent the anticlockwise ro- proof. π of a power of 2. tation by an angle of 2 x radians. Now, ins- Paweł Strzelecki [[email protected]] tead of considering the numbers cn as ele- studied mathematics in Warsaw and graduated ments of the interval [0, 1], let’s think of the Proof of Lemma 2 in 1987. He finished his PhD-thesis under the images of the distinguished point 0 under supervision of Bogdan Bojarski in 1993. Since First note that all the members of the se- successive iterations of the function f on T. that time he has been working as an assistant quence (cn) ∈ are different. Indeed, if n N In fact, a moment of reflection should make professor at the Institute of Mathematics at c = cm for some k, m ∈ N with k > m, then k it clear that if Warsaw University where he now also serves (k − m)x = kx−mx. However, the pro- n as the deputy director for teaching affairs. In duct of a positive natural number and an
  = n( )= ◦ ◦ ( ) 1994-1999 he served as the mathematics editor irrational number cannot be an integer. bn f 0 f ... f 0 , for the Polish popular mathematical-physical- Now, let n be a natural number such that then the length of the arc (0, bn) is equal to 1 < − astronomical monthly ”Delta”. In 1993-1998 n b a. The numbers c1, c2, c3, ..., cn+1 cn. Thus, due to equation (1), the length of ∈ [ ] he presented mathematics in a Polish TV pro- being pairwise different with ci 0, 1 for the arc between the points bs and bs+t is less gram for young people. He was an Alexander = + th i 1, 2, 3, ..., n 1, we can use the pigeon- than b − a. This means that the t iteration von Humboldt fellow in Bonn (1999 – 2001 hole principle to infer that for some s and t, of f is a rotation by an angle of 2πε radians; + and 2002 – 2003). He likes nonlinear elliptic both s and s t are included between 1 and the direction of this rotation has no signifi- + PDEs, clever hard-analytic proofs, compensated n 1 and satisfy the inequality cance for our purposes. compactness and Sobolev spaces, bright moun- 1 Now, this clearly implies that there are tain mornings, bright people and good food (no (1) 0 < ε := |cs − cs+t|≤ < b − a. n infinitely many points bt, b2t, b3t, ... that be- matter from which corner of the world). He is From now on we will refer to a very use- long to the arc (a, b). Thus if you start at happily married to a non-mathematician. 8 EMS June 2004 FEATURE

AA forgottenforgottenA forgotten mathematician mathematicianmathematicianRobbert Fokkink (Delft, the Netherlands) Robbert Fokkink (Delft, the Netherlands)

The original Dutch version of this article was published in Nieuw Archief for Wiskunde 5 (1), March 2004. The Newsletter’s thanks go to Nieuw Archief for the premission to reproduce it and to Reinie Erne (Leiden) for the translation into English.

Not so long ago, I was listening to a lecture in which a fine result was ca- sually mentioned. It was a classical em- bedding theorem of Haefliger, which roughly says that you can embed a space X in Rn if the reflection (x, y) → (y, x) on X × X is a conjugate of the an- − tipodal map x →−x on Sn 1. Some research into the literature on this area quickly showed that E.R. van Kampen was the first to come up with such embedding theorems, sometime in the early thirties. Very little about his work can be found in mathematical history books and his collected works were not to be found in the library. This is re- markable, because E.R. van Kampen is a much cited mathematician, who even had a component of the AMS classifica- tion named after him (20F06). So who Egbert van Kampen (l) with his sister Elizabeth and brother Laurens. Like Egbert, Eli- was he? zabeth studied mathematics in Leiden. Laurens became a professor of economy at the Vrije Universiteit in Amsterdam. Both lived to the age of 96. Photo: Archief Mieke van der Veen The life of unique talent in Europe, who has ob- general varieties and expands Veblen’s tained his teaching qualifications for theory of intersections and intersection E.R. van Kampen mathematics, but nevertheless was not multiplicities. In 1929, he obtains his exempted from this subject. His study Ph.D. with the thesis Die kombinato- Egbertus Rudolf van Kampen, known of mathematics and physics continues rische Topologie und die Dualit¨atss¨atze. as Egbert, was born on 28 May 1908 in Leiden, where he probably did get The thesis, directed by Van der Woude, in Berchem as the youngest in a family the necessary exemptions as he com- is missing from the University Library of three children. Egbert’s parents had pletes the studies very quickly. In 1927, in Leiden. moved from the Netherlands to Bel- at the age of only nineteen, he travels In the summer of 1928, Van Kam- gium a couple of years before, when his to Gottingen.¨ There he meets Alexan- pen spends five months in Hamburg, father became an accountant at the Mi- droff and van der Waerden and is gi- where he meets Emil Artin. Around nerva car factory in Antwerp. ven the subject for his Ph.D. thesis: find that time, he is recruited by Johns Hop- The children spend the war years a computable, topological definition of kins University, but because he is still 1914-18 with family in Amsterdam. the notion “variety”. Egbert solves the too young to enter the United States When the war is over, the family move problem by showing that the local ho- without the accompaniment of his pa- to The Hague, where Egbert completes mology group is a topological inva- rents, he first becomes an assistant to high school, Eerste Christelijke H.B.S. riant. According to Van Kampen, a va- Schouten in Delft. He writes a num- At his final examination in 1924, he riety is a simplicial complex with the ber of papers with Schouten on ten- is mentioned in the national press. right local homology. Armed with this sor analysis, which is Schouten’s spe- The Telegraaf and the Amsterdammer definition, he generalizes the Alexan- cialty. [4, 6, 9] report of a young mathematician, a der duality theorem from spheres to EMS June 2004 9 FEATURE In 1931, Van Kampen departs to Johns Hopkins. In his autobiography, Mark Kac gives an anecdote about Van Kam- pen’s arrival in the USA: ”University professors, clergymen and those of se- veral other occupations could emigrate outside the national quotas and Van Kampen had one of these highly co- veted non-quota visas. There was a story that at the immigration desk he was asked a routine question and ins- tead of answering ’professor’ or ’tea- cher’ or even ’mathematician’, he ans- wered ’topologist’. The immigration of- ficial could not find ’topologist’ on the list of occupations qualifying for a non- quota visa and he naturally asked what a topologist does. As Van Kampen tried to explain, the immigration official gai- ned the impression that he was dealing with a nut. He detained Van Kampen for further examination and Johns Hop- kins authorities had to step in to extri- cate him. Van Kampen never confirmed Van Kampen (l) with Ehrenfest, Rotterdam circa 1927 or denied this story - but it was comple- tely in character”. In his first year in the U.S., Van Kam- The work of endpoints of α and β, which, for that pen writes a paper on the fundamental matter, is not strictly necessary. With group, with which he truly establishes E.R. van Kampen some effort one can then see that the a name for himself. In 1933, he spends Egbert van Kampen has worked on complement of α ∪ β ∪ γ in R3 has the a year at the Institute for Advanced many subjects: topology, group theory, same fundamental group as the com- Study, and in 1935 he gives a talk on tensor analysis, harmonic analysis and plement of α ∪ β in half-space. This topological groups at the International probability theory. His most well- group has three normal subgroups of Congress at Moscow. know articles are on topology and index 2 and Van Kampen notes that 3 In his mid-twenties, Egbert van group theory. He was thought of as one of those has the group Z ⊕ Z/3Z Kampen seems to be at the beginning somewhat of an oracle, as part of his as quotient. It follows from this that of a brilliant career, but in reality he work was only discovered when others the fundamental group can not be free. has already done his most important started realizing it again. The theory that Van Kampen is expan- work. He remains particularly produc- The first publication in 1928 ding is at that time brand new and not entirely worked out yet. The compu- tive and, together with Aurel Wintner, Van Kampen’s first article [1] appears tations that he needed to do were no fills the American Journal of Mathema- in the Hamburger Abhandlungen in small matter. Perhaps that is why he tics. 1928. It consists of ten sentences and left out all the details. Nowadays the At the end of the thirties, in a let- one illustration and is exactly what one computation is much simpler, not in ter home, Van Kampen begins to com- would expect from a brilliant young the least because of the work of Van plain of headaches. He sees a specia- person: original and illegible. Kampen himself. list who diagnoses a neck problem. As It concerns two curves α and β that a result he starts physiotherapy but it are attached to a surface V, of which it Van Kampen’s embedding obs- is later determined that the problem is is questioned whether they can be se- truction more serious, a melanoma that origina- parated from each other, without the In 1932, Van Kampen publishes an ar- ted from a birthmark near his left eye. endpoints leaving the surface. ticle concerning embeddings of com- In April 1941, it is operated on and in In 1925, in an article with the same plexes in Euclidean spaces [7]. Again, the autumn Egbert takes up his courses title, Emil Artin had claimed the follo- Artin’s article from 1925 provides again, full of good hope for his reco- wing: if α and β are linked, then either the inspiration. Van Kampen stu- very. This however is not to last and α or β has a knot. The illustration in dies an embedding problem that will he once more suffers from violent hea- Van Kampen’s article shows that this only be definitively solved much la- daches and becomes deaf in his left ear. statement is incorrect. It is clear that ter: under which conditions does an In December 1941 he is once more ad- α and β do not have a knot, but that n-dimensional complex fit in R2n?In mitted to the hospital where his health they are linked anyway. Van Kampen 1930, Kuratowski had solved this for deteriorates quickly. In January 1942 a shows this by computing that the fun- n = 1. Perhaps Van Kampen was not second operation follows in vain. On damental group of the complement of aware of that result, because he does February 10, Egbert falls into a coma. α ∪ β in half-space is not free. He first not mention Kuratowski and he dis- He dies the next day, only 33 years old. draws an auxiliary line γ through the poses of the case n = 1 as being simple.

10 EMS June 2004 FEATURE

Van Kampen’s proof only holds for n ≥ 3, but his criterion is best illus- trated using the “simple” case n = 1. In that case the complex is a graph G. For example, draw 5 points in the plane and join each of them to all the others. The result is a depiction of the complete graph with 5 nodes in the plane. Some connecting lines may in- tersect each other, so the map is not an embedding. You might try to make it into an embedding by trial and error, by removing intersecting edges and re- placing them by new edges. For the graph with 5 nodes, any such attempt ultimately ends unsuccessfully. In Fi- gure 1, for example, there is only one intersection point left, which can not be removed. Van Kampen shows that this trial and error takes place within a co- homology class and that the graph can be embedded precisely when that class is zero. At this time he uses other ter- minology, because in 1932, cohomology theory has not really been developed yet.

Figure 1 Thecompletegraphwith5 The first article (1928) of Van Kampen in the Hamburger Abhandlungen nodes For a map f : G → R2 with fini- dimensional cells. The definition of the Afterwards, Van Kampen sends a hasty tely many double points, Van Kam- vector v uses the “intersection num- rectification indicating a partial correc- pen defines a vector v with coordinates ber”, the theory of which Van Kampen tion. He writes that it is “unneces- in Z/2Z. The number of coordinates had set up in his thesis. He first notes sary to neglect orientation”, by which of v is equal to the number of pairs that the obstruction against an embed- he probably means that the coefficients {I, J} of disjunct edges of the graph. ding can only lie in the n-dimensional must be taken in Z instead of in Z/2Z. The {I, J}th coordinate of v determines cells. Next he deduces a criterion in After this Van Kampen no longer pu- the parity of f (I) ∩ f (J).VanKam-four steps. Lemma 1 states that double blishes about embeddings, even when penthendefinesvectorsw(p, E) in the points can be removed pairwise, so Whitney develops a complete theory N same space (Z/2Z) as v,wherep is a only the parity matters. Lemma 2 a couple of years later. Only in 1957 vertex and E is an edge that does not states that double points on adjacent does Arnold Shapiro show that Van contain p.The{I, J}th coordinate of edges are no problem, so only disjunct Kampen’s embedding criterion follows w(p, E) equals 1 if and only if p ∈ I and subgraphs matter. Lemma 3 states that from the results of Whitney. = + E J,orvice-versa.LetL be the linear for every vector in v L there is a map As noted before, Van Kampen’s ( ) → 2n space generated by the w p, E .Van f : K R giving that vector. Lemma proof only holds for dimension n ≥ 3. → Kampen’s embedding criterion is then 4 conversely states that every f : K It is therefore surprising when in 1991, 2n + that G can be embedded if and only if R gives a vector in v L.Thecom- Sarkaria shows that the criterion is also ∈ v L. For the map of the graph above, plete proof is barely one page long. As correct for n = 1. In 1994, Freedman, v has a single 1 and zeros everywhere an added bonus, Van Kampen shows Krushkal, and Teichner show that the else. For all vectors w(p, E) the sum of that every n-dimensional variety can = 2n criterion does not hold for n 2. Kru- the coordinates is even, so in this case be embedded in R , a result that is shkal and Teichner also introduce the ∈ v L. now known as Whitney’s embedding name “Van Kampen’s embedding obs- In the higher dimensional case n ≥ 3, theorem. truction”, some sixty years after the pu- the edges become n-dimensional cells It is an amazing result with only one blication of the original article. and the vertices become (n − 1)- flaw: the proof of Lemma 1 is wrong. EMS June 2004 11 FEATURE

Group photo of staff and post docs at the Johns Hopkins campus in 1933, with f.l.t.r.: Egbert van Kampen, George Schweigert, Beatrice Aitchison, William Morrill, Oscar Zariski, Charles Wheeler and John Williamson. Copyright: Johns Hopkins University Library

Three consecutive articles on The articles have been written in now known as the Zariski-van Kam- the fundamental group an uninvolved tone. The first article pen theorem. states: As the resulting proof seemed too The second article continues with the At the beginning of the thirties, Os- algebraic for this simple and nearly topo- question of how one computes the fun- car Zariski, arguably the most famous logical question, Dr. Zariski asked me to damental group of a reunion U ∪ V mathematician from Hopkins, works on publish a topological proof which is contai- from the groups of U and V. In its mo- the fundamental group of the comple- ned in this paper. Part two states: The dern form this result is formulated by ment of an algebraic curve. An alge- opportunity of simplifying the treatment of means of coverings, but Van Kampen braic curve can be knotted in the pro- a fundamental group by means of this theo- turns his attention to the cut that di- jective space, which, as in the case of rem has been overlooked several times [...] vides the space in two or more pieces. a usual curve in R3, can be read off the for this reason we do not think it super- He is not entirely satisfied with his re- fundamental group of the complement. fluous to devote a separate paper to it. Part sult and writes: a path is shown to a Zariski determined the generators of three states: It is this 2-dimensional me- general theorem, of which the general for- the fundamental group and found re- thod of the proof and not any original re- mulation would be more confusing than lations between them, but it was not sult which justifies the publication of this helpful, so that it is suppressed. This clear to him then whether or not there paper. suppressed general theorem probably were more relations. Zariski presented The first part of the three articles concerns a cut that is as general as pos- the problem to Van Kampen who then concerns Zariski’s problem, which Van sible, which divides the space into an solved it and wrote three consecutive Kampen solves by skillfully cutting the arbitrary number of pieces. It is Corol- papers [10, 11, 12] on the problem in space in two pieces U and V which lary 2 of this article, in which a sim- the American Journal of Mathematics, each have a free fundamental group. ply connected set cuts the space into the internal publication of Johns Hop- He then divides every homotopy into two pieces, that will later be called kins. All three articles have become small pieces that lie either completely the Seifert-van Kampen Theorem. This standard references, although they are in U or completely in V. As the fun- theorem was in fact first proved by Sei- rarely mentioned together. The alge- damental group of both U and V is fert, another Ph.D. student of Van der braic geometers refer to the first article, free, the only freedom of choice is Waerden. the topologists to the second, and the for the pieces of homotopy that lie in An annoying technical detail that group theorists to the third. This is re- the intersection U ∩ V. This freedom Van Kampen comes across is that when markable as in fact only the first article turns out to exactly correspond with he divides a homotopy, he must conti- stands alone, and even then only ba- the relations of Zariski. Therefore there nue to link everything to the base rely. are no other relations. This result is 12 EMS June 2004 FEATURE point. All small pieces of homotopy ten for the general public, in contrast to Word of thanks must be linked to it. In the algebraic Van Kampen’s previous work, which translation, the pieces of homotopy be- gave the impression of being merely a This overview would not have been come relations, and the links to the reminder for the author himself. possible without the photographs and base point become conjugations. In This marks the beginning of a new newspaper cut-outs of Mieke van der the third part of the trio of articles, period. After 1935, Van Kampen leaves Veen, daughter of Elizabeth van Kam- Van Kampen makes a virtue of neces- the area of algebraic topology for what pen. I would like to thank her for the sity by noting that the inverse transla- it is and mostly studies almost-periodic authorization to use this material and tion gives a graphic representation of functions and differential equations. It for the care with which she saved it. group relations. He translates words in is no coincidence that this is the area of Boudewijn van Kampen, also a family groups into bouquets of curves, which interest of Aurel Wintner, a colleague member, provided the photograph of each represent the conjugacy class of a at Johns Hopkins. Egbert van Kampen Egbert van Kampen and Paul Ehren- relation. With this manner of thinking has always allowed himself to be inspi- fest. Egbert van Kampen’s great-great- he simplifies the proof of a result of red by the people around him. At the grandfather was Jacobus van der Blij, Schreier. This third article is only dis- end of his life, when the young Mark and he was also the great-grandfather covered in 1966, when Roger Lyndon Kac comes to Baltimore, Van Kampen of the retired professor of mathematics and Carl Weinbaum use the diagrams even writes a number of articles on van der Blij from Utrecht. to study relations with “small cancella- probability theory. His last article, of The anecdote about the visum of Van tion”. The bouquets of curves are now considerable size, deals with product Kampen comes from the memoirs of called van Kampen diagrams. measures and convolutions. It appears Mark Kac, Enigmas of chance, London, in the middle of 1940 [49]. Harper and Row, 1985. More anecdotes can be found in these memoirs, around page 85. The autobiography of Kac was Conclusion brought to my attention by Cor Kraai- kamp. At the age of only sixteen, Egbert Van Slava Krushkal helped me with the Kampen appeared in the newspaper as details of the Van Kampen embedding an unprecedented mathematical talent. obstruction, Mark de Bock of the muni- He was a renowned mathematician but cipality of Antwerp and Albert Schilt- nevertheless, his work had to wait a meijer from Amsterdam sent me data long time for recognition. Perhaps this concerning the Van Kampen family. was due to his inaccessible manner of Jim Stimpert of Johns Hopkins Library The typical form of a Van Kampen diagram: writing, a proof is frequently over be- sent information concerning Van Kam- relations on stalks. Rewriting a word cor- fore you notice it. Perhaps it was pen’s publication. responds to gluing the bouquet together. due to his discretion. In an Ameri- can newspaper a colleague noted, “He never spoke very much about his ac- References Pontrjagin-van Kampen dua- complishments”. While Van Kampen’s lity and later work fame is based on the work of his begin- [1] Zur Isotopie zweidimensionaler Flache im R4, ning years, after 1933 he wrote a fur- Abh. Math. Sem. Hamburg 6 (1928), 216. In 1933, Van Kampen spends a year ther fourty articles. Who knows what [2] Eine Verallgemeinerung des Alexanderschen in Princeton, at which time Pontrjagin beautiful things still await us. Dualitatsatzes, Proc. Kon. Ak. Weten. Amster- proves that compact Abelian groups dam 31 (1928), 899. are dual to discrete Abelian groups. [3] Die kombinatorische Topologie und die Dua- Pontrjagin assumes the groups to be litatss¨ atze,¨ Diss. Leiden, (Van Stockum, Den Haag), 1929. separable, because in 1933, the Haar measure has only been constructed for [4] with J.A. Schouten, Zur Einbettung und Krum-¨ mungstheorie nichtholonomer Gebilde, Math. such groups. Van Neumann remarks Ann. 103, 752-783, 1930. that duality also holds without separa- [5] Komplexen in Euclidische Ruimten, Handelin- bility, if instead of the Haar measure, gen XXIIIe Nederl. Natuur- en Geneeskundig one uses Von Neumann’s theory for Congres (1931), 123-124. almost-periodic functions. Van Kam- [6] with J.A. Schouten, Uber die Krummung¨ einer pen takes this up and writes a sur- Vm in Vn; eine revision der Krummungstheorie,¨ vey article on the duality of groups[16]. Math. Ann 105 (1931), 144-159. Since then this duality is also called the [7] Komplexe in euklidischen Raumen,¨ Abh. Math. Sem. Hamburg 9 (1932), 72–78, Berichtigung Pontrjagin-van Kampen duality. dazu, (1933), 152–153. On this topic, Van Kampen is quite [8] Some remarks on the join of two complexes and lavish and writes sixteen pages, begin- on invariant subcomplexes, Amer. J. Math. 44 ning with the definition of a topolo- (1932), 545–550. gical group and ending with the dua- [9] with J.A. Schouten, Beitrage¨ zur Theorie der De- lity of locally compact Abelian groups formation, Prace Mat.-Fiz. 41 (1933), 1–19. Egbert van Kampen, 1940. without the restriction of separability. [10] On the fundamental group of an algebraic curve, The article is well readable and writ- Photo: Archief Mieke van den Veen Amer. J. Math. 55 (1933), 255–260. EMS June 2004 13 FEATURE

[11] On the connection between the fundamental [26] On the argument functions of simple closed [41] with M. Kac and A. Wintner, On Buffon’s pro- groups of some related spaces, Amer. J. Math. curves and simple arcs, Compositio math. 4 blem and its generalizations. Amer. J. Math. 61 55 (1933), 261–267. Het artikel staat integraal in: (1937), 271-275. (1939), 672–676. Two decades of mathematics in the Netherlands, 1920-1940, A retrospection on the occasion of the [27] Remarks on systems of ordinary differential [42] with M. Kac, Circular equidistributions and sta- bicentennial of the Wiskundig Genootschap, E. equations, Amer. J. Math. 59 (1937), 144-152. tistical independence. Amer. J. Math. 61 (1939), M.J.Bertin,H.J.M.BosandA.W.Grootendorst, 677–682. Amsterdam, 1978. [28] with A. Wintner, On a symmetrical canonical re- duction of the problem of three bodies, Amer. J. [43] On the asymptotic distribution of a uniformly al- [12] On some lemmas in the theory of groups, Amer. of Math. 59 (1937), 153–166. most periodic function. Amer. J. Math. 61 (1939), J. Math. 55 (1933), 268–273 729–732. [29] with A. Wintner, Convolutions of distributions [13] Locally compact abelian groups, Proc. Nat. Acad. on convex curves and the Riemann zeta function, [44] with A. Wintner, A limit theorem for probability Sc. (1934), 434–436. Amer. J. Math. 59 (1937), 175-204. distributions on lattices. Amer. J. Math. 61 (1939), 965–973. [14] The structure of a compact connected group, [30] with P. Hartman and A. Wintner, Mean motions Amer. J. Math. 57 (1935), 301-308. and distribution functions, Amer. J. Math. 59 [45] with M. Kac and A. Wintner, On the distribution (1937), 261-269. of the values of real almost periodic functions. Amer. J. Math. 61 (1939), 985–991. [15] On some characterizations of 2-dimensional ma- nifolds, Duke Math. Journ. 1 (1935), 74-93. [31] with A. Wintner, On an absolute constant in the theory of variational stability, Amer. J. Math. 59 [46] A remark on asymptotic curves, Amer. J. Math. 61 (1939), 992–994. [16] Locally bicompact abelian groups and their cha- (1937) 270-274. racter groups, Annals Math. 36 (1935), 448–463. [32] with A. Wintner, On divergent infinite convolu- [47] with M. Kac and A. Wintner, Ramanujan sums tions, Amer. J. Math. 59 (1937), 635-654. and almost periodic functions, Amer. J. Math. 62, [17] The topological transformation of a simple clo- (1940), 107–114. sed curve into itself, Amer. Journ. of Math. 57 [33] On the addition of convex curves and the den- (1935), 142-152. [48] with P. Erd¨os, M. Kac, and A. Wintner, Ramanu- sities of certain infinite convolutions, Amer. J. jan sums and almost periodic functions. Studia Math. 59 (1937), 679-695. [18] On the structure of a compact group, Rec. math. Math. 9, (1940), 43–53. Moscou, 1 (1936), 699-700. [34] with A. Wintner, On a singular monotone func- [49] Infinite product measures and infinite convolu- tion, J. London math. Soc. 12 (1937), 243-244. [19] Almost periodic functions and compact groups, tions. Amer. J. Math. 62, (1940), 417–448. Ann. Math. (2) 37, 78–91. [35] with A. Wintner, On the reduction of dynamical [50] with A. Wintner, On the almost periodic beha- systems by means of parametrized invariant re- [20] A unicity theorem for ordinary differential equa- vior of multiplicative number-theoretical func- lations. Trans. Amer. Math. Soc. 44 (1938), no. 2, tions, Amer. J. Math. 62, (1940), 613–626. tions and a postulate system for Lie groups, Bull. 168–195. Amer. Math. Soc. 42 (1936), 330. [51] On uniformly almost periodic multiplicative and [36] The theorems of Gauss-Bonnet and Stokes, additive functions, Amer. J. Math. 62, (1940), [21] Note on a theorem of Pontrjagin, Amer. J. Math. Amer. J. Math 60 (1938), 129–138. 58 (1936), 177-180. 627–634. [37] with P. Hartman and A. Wintner, On the dis- [52] Elementary proof of a theorem on Lorentz ma- [22] The fundamental theorem for Riemann integrals, tribution functions of almost periodic functions, trices, Bull. Am. Math. Soc. 47 (1941), 288-290. Amer. J. Math. 58 (1936), 847-850. Amer. J. Math. 60 (1938), 491-500. [53] Remark on the address of S. S. Cairns. Lectures [23] with A. Wintner, On the canonical transforma- [38] Invariants derived from looping coefficients, in Topology, pp. 311–313. University of Michigan tions of Hamiltonian systems, Amer. J. Math. 58 Amer. J. Math. 60 (1938), 595–610. Press, Ann Arbor, Mich., 1941. (1936), 851–863. [39] with A. Wintner, On Liouville systems, Nieuw [54] Notes on systems of ordinary differential equa- [24] On almost periodic functions of constant abso- Arch. Wiskd. 19 (1938), 235–240. tions. Amer. J. Math. 63, (1941), 371–376. lute value, J. London math. Soc. 12 (1937), 3-6. [40] with P. Hartman and A. Wintner, Asympto- [55] with A. Wintner, On the asymptotic distribution [25] with A. Wintner, On bounded convolutions, Bull. tic distributions and statistical independence, of geodesics on surfaces of revolution, Cas. Mat. Amer. Math. Soc. 43 (1937), 564-566. Amer. J. Math. 61 (1939) 477-486. Fys. 72 (1947), 1-6.

3 Robbert Fokkink [[email protected]] studied mathematics in Amsterdam and Delft. His PhD thesis from 1991 is on topological dynamics. From 1992 to 1999 he worked at Delft Hydraulics. He is currently a member of the probability group at Delft University.

14 EMS June 2004 FEATURE

QuantumQuantum CrCryptographyyptography Nuno Crato (Lisbon)

"Quantum cryptography" is the final of a does not mean "impossible to break". three-article sequence written by Nuno The safety of one of the most reliable Crato and published at the Portuguese modern cryptography systems - the so newspaper Expresso in September 2001. called RSA, which we have been report- These three articles were submitted to ing on in this Revista - is based on the the Public Awareness of Mathematics difficulty of finding the prime factors of 2003 competition and won its First very large numbers. There are no algo- Prize. rithms known to date that enable their Crato's articles were published in the factorization in a reasonable amount of for a new cryptography system that will widely read magazine Revista, which is time, even if we use the most powerful be absolutely secure. Before RSA can included in the weekly newspaper computers available. However, if a math- fail us, mathematicians, physicists, and Expresso. Readership for this newspaper ematician discovers a process that computer scientists hope to develop a is about half a million people, which is a enables a speedy factorization, or if a new procedure that will be virtually large number for Portugal. In fact, new generation of computers such as unbreakable. This might well be possi- Expresso is the most sold newspaper in quantum computers are commercialized, ble, because the new system scientists the country. the world of communication as we know envision to create is based on the most The first and second articles referred it can be easily jeopardized. In the event profound laws of matter, those that gov- briefly to the history of cryptography, that one of these revolutionary techno- ern the uncertainty of the quantum discussed the asymmetric key solution logic advances becomes widespread, world. The impossibility of getting to and the RSA method. They stressed the electronic commerce would cease to be know a priori the behavior of elementary importance of mathematical concepts secure, the military would have to rein- particles is what will guarantee commu- for safe communication and showed vent its communication systems, and nication security. today's relevance of mathematical cryp- bank institutions would have to take a The idea has been brewing in the tography. These two articles discussed step back and make transactions at a very minds of scientists for some time now. concepts that were novel for the general slow pace. It would be the collapse of Charles Bennet, a computer scientist at reader, but that are sufficiently well informational technology in our present the IBM Watson Research Center has known for most mathematicians. The day society. been one of those looking for a solution third article discussed the emergent Not surprisingly, people are looking to this problem. Finally, in the 80's he research on quantum cryptography, and his colleague Gilles Brassard man- explaining one way by which the uncer- aged to conceptualize a quantum cryp- tainty of quantum world can be used to tography system. For a long time their provide safe communications. It is this ideas were only dreams. During the last third article we reproduce here. two years, however, technological and In the next issue of this Newsletter, scientific advances have made it possible we will publish an interview with Nuno to build prototypes of quantum systems Crato by José Francisco Rodrigues. This that seem to be absolutely unbreakable. conversation between the two To discuss issues related to ciphered Portuguese mathematicians addresses messages, experts usually refer to three subjects such as the importance of math- different fictitious characters: Alice, ematics popularization, the difficulties Bob, and Eve. The first represents the and caveats of this work, and its rele- sender, the second the receiver, and the vance for the public and the public status third the intruder who attempts to break of Mathematics. the secret communication. At the core of the process proposed by Charles Bennet QUANTUM CRYPTOGRAPHY and his collaborators, there is a random It sounds like science fiction, but it is a code key as long as the message itself. reality: the most bizarre properties of This key is simply a binary number, a subatomic particles allow us to create sequence of zeros and ones led by a one. unbreakable ciphers Alice begins by transforming the text she The security of bank transactions, elec- wants to send Bob by translating it into tronic commerce, and email communica- A new book explaining how quantum another binary number. Then, she adds tion is based on the safest cryptography mechanics can be used to conceive an the key number to her message number. systems created by man. But "safest" innovative computing system She sends the result to Bob, who has the EMS June 2004 15 FEATURE ton passed through a horizontal filter, Alice must have polarized it diagonally and it can be assigned the value one. Those photons that were retained remain a mystery to Bob. But now Alice needs to know which photons went through Bob's filters. To know this, Alice and Bob can communi- cate through a less secure system and they can even be heard by Eve. Even if Eve finds out which particles reached Bob, she still cannot figure out which fil- ter they passed through. Thus, the key is established solely on the basis of the photons that completed the voyage. Now, Alice and Bob can communicate with complete security. The uncertainty of the quantum world gives them the cer- key that Alice used. By subtracting the vertically polarized photon and makes tainty that their code cannot be broken. key from the message received, he gets one correspond to a 45º polarized pho- Maybe we are close to implementing the original message. To read it, he only ton. Bob has two other polarizers, one of this cryptography system. Only a few has to transform the sequence of zeros them horizontally oriented and another years ago it all sounded like science fic- and ones into a sequence of letters, but one at –45º. When he receives Alice's tion, but recent technological break- this is a routine procedure any computer photons, he makes them pass through throughs made it possible to create pro- is capable of doing. one of his polarizers, alternating the two totype computation models with viable For this system to be unbreakable, it is in a completely random fashion. applications of these ideas. As we can important that the key is a random The photons that Alice sends and Bob anticipate, the technological challenges sequence of zeros and ones and that it is receives may or may not pass through his are huge. How do we transmit light a used only once. This usually means that device, depending on how the polarizers photon at a time? How do we make sure those numbers have to be generated in are oriented. If Alice sends a photon these particles reach their destination? advance and that Alice has to send them polarized vertically and Bob attempts to Nonetheless, these problems are being to Bob. This is where problems usually make it pass through his horizontal solved little by little. We are already able arise. If Alice and Bob never meet face to polarizer, the particle is stopped and does to use quantum cryptography through face, as it happens in e-commerce, they not pass. If Alice sends Bob a photon fiber cables and through air for a few have to exchange the key through some polarized at 45º and Bob receives it with kilometers. Thus, we may not be far from communication channel. How can they the polarizer at –45º, the particle is also being able to protect our secrets by mak- do that? Well, they could agree on anoth- stopped and does not go through. ing them travel a particle at a time - at the er key, but this again does not solve the Polarizers perpendicular to the photon speed of light. problem, as to do this Alice and Bob polarization do not let the particles pass. would have to meet or trust a messen- Surprises arise when Alice sends a Nuno Crato [[email protected]] is ger... Since Eve is always peering over vertically polarized photon and Bob Associate Professor with habilitation at them, absolute security seems impossi- receives it with the diagonal polarizer or the Mathematics Department of Instituto ble. when Alice sends a diagonally polarized Superior de Economia e Gestão, Lisbon This is where the quantum world photon and Bob receives it with the hor- Technical University. He received his comes into play by the hands of Charles izontal polarizer - that is, when Alice and Ph.D. from the University of Delaware Bennett and other computer scientists. It Bob's polarizers make a 45º angle. In this and worked previously in the U.S. for a is a strange world, with rules difficult to case, the uncertainty principle of quan- number of years. His research interests understand on the basis of our everyday tum mechanics is set in motion: half of lie in Stochastic Processes and Time experiences. One of those rules is uncer- the particles go through Bob's polarizer Series Analysis, with various applica- tainty. And this uncertainty is not based and the other half are stopped. And this tions, from finance and economics to cli- on our insufficient knowledge; rather it happens without any prior knowledge of mate models. He is a member of the is intrinsic to the very life of particles. which particles are going to go through board of the Portuguese Mathematical How could this be used for safe commu- and which particles are going to be Society (SPM). In parallel with his nication? stopped. research and teaching activity, he is a In order to create the random code At this stage, only Alice knows the science writer. He has a column at the key, Alice begins by sending Bob a polarization of the photons she sent. And Portuguese newspaper Expresso and has sequence of light particles, that is, a only Bob knows which ones finished worked for a few TV science documen- sequence of photons. She has two polar- their trip. This way, Bob discovers the taries. He is co-author of Trânsitos de izers in her system, one vertically orient- polarization established by Alice for the Vénus (2004), Eclipses! (1999) and other ed and another tilted at 45º, as we can see photons that passed through. If a photon science books. The European in the figure. To set up the key, she ran- passed through a diagonal filter, Alice Mathematical Society awarded him the domly alternates the polarizers and, for must have polarized it vertically and it First Prize in mathematics populariza- example, makes zero correspond to a can be assigned the value zero. If a pho- tion writing in 2003. 16 EMS June 2004 SOCIETIES

HouseholdHousehold namesnames inin SwedishSwedish mathematicsmathematics Some who should be and some others Kjell-Ove Widman (Institut Mittag-Leffler, Stockholm)

For natural reasons, this causerie on were well up to the standard of the day. also for his discovery of the famous error Swedish mathematicians – just in time for Not wanting to stay in Germany, and find- in Poincaré’s first paper in the Acta. the 4th European Congress of ing no suitable job in Sweden, he applied Having succeeded Sonya Kovalevsky as a Mathematics in Stockholm - contains very for, and was appointed to, a professorship professor, Phragmén left mathematics for little in the way of flesh on its bones. The in Helsinki, where he had one student with insurance and politics after about ten canonical, meaty source is Gårding, 1998, a name that has remained in the books: years. from which I have used much material. To Hjalmar Mellin (1854-1933), inventor of Decidedly less known is Helge von avoid delicate questions concerning my the transform bearing his name. As a pro- Koch (1870-1924, professor in colleagues, I have decided not to mention fessor in Stockholm from 1881 to 1911, he Stockholm), a precocious mathematician the names of any mathematicians under made a successful career as a researcher, whose work on infinite determinants and the traditional Swedish retirement age of teacher, thesis adviser, and founder of the on continuous non-differentiable functions 65. This of course, for the most part, Acta Mathematica, while at the same have receded into the darkness of history. leaves contemporary mathematics out of time managing to amass a substantial for- Two of Mittag-Leffler’s students gave the picture, but that is the way it has to be. tune as a businessman. He willed his up pure mathematics for other fields of house and fortune to the Royal Swedish endeavour: Gustav Cassel (1866-1945) The Mittag-Leffler Era Academy of Sciences, making the future decided mathematics was too difficult and On an international scale, Swedish mathe- Institute Mittag-Leffler possible. A last- went on to become the father of the matics begins with Gösta Mittag-Leffler ing contribution to history was his suc- Stockholm school of economics, with (1846-1927). He learned analysis, includ- cessful effort to bring Sonya Kovalevsky Nobel Prize winners Bertil Ohlin and to Sweden. Gunnar Myrdal as prime exponents. Several of Mittag-Leffler’s students Number theory, more specifically ana- became well known and respected mathe- lytic number theory, had been taken up maticians in their day. Ivar Fredholm both by Phragmén and von Koch in the (1866-1927, professor in Stockholm) is early years of the century, inspired by the best known for his work on integral equa- successes of Hadamard and de la Vallée tions and spectral theory, with a theorem Poussin. Harald Cramér (1893-1985, whose generalization to abstract spaces professor in Stockholm), probably goes under the name of “Fredholm’s alter- inspired and tutored by Marcel Riesz, native”. His main interest was mathemat- wrote a thesis on the subject in 1917, and ical physics and partial differential equa- continued to produce good papers for sev- tions, where he constructed fundamental eral years. According to legend however, solutions for various classes of equations he fell out with Mittag-Leffler, who in two and three variables. These studies promised that as long as he, Mittag- were continued by Fredholm’s student, Leffler, had any influence, Cramér would Nils Zeilon (1886-1957, professor in never get a professorship. This can Lund), who managed to considerably arguably be considered a stroke of luck, enlarge the classes of equations for which both for Swedish mathematics and for a fundamental solution could be shown to Cramér himself, since Cramér then turned Gösta Mittag-Leffler exist. This line of investigation was later to probability and mathematical statistics carried to a logical end in the thesis of and founded a thriving Swedish school in ing complex analysis, in Uppsala in the Hörmander. the area. His magnificent monograph 1860’s, from his teachers H. Daug and G. Ivar Bendixson (1861-1935, professor from 1946, Methods of Mathematical Dillner. However, the knowledge gained in Stockholm), learned set theory from the Statistics, has rightly been considered a would hardly have sufficed for a research papers by Cantor, solicited by Mittag- standard reference in the area. career had he not gone to Hermite, and in Leffler for the Acta, and dynamical sys- Marcel Riesz (1886-1969, brother of particular to Weierstrass, to learn more. In tems from the papers of Poincaré. The Fredrik Riesz), came to Sweden to assist fact a glance at Mittag-Leffler’s thesis result was the Cantor-Bendixson theorem Mittag-Leffler in the editing of the Acta, does not convince one that he had mas- in set theory and the Poincaré-Bendixson but stayed on at the University of tered the definition of the derivative of a theorem on the existence of closed cycles Stockholm as a docent until 1927, when he complex-valued function of a complex of ODEs in two dimensions. became professor in Lund. Riesz intro- variable. “Die Strenge des Weierstrass” Edvard Phragmén (1863-1937, profes- duced new areas of analysis in Sweden, was not lost on him, however, and he start- sor in Stockholm), is of course best known and had a profound influence on Swedish ed producing mathematics papers which for the Phragmén-Lindelöf theorem, but mathematics. One of Riesz’s most famous

EMS June 2004 17 SOCIETIES theorems, generalized by one of his stu- had several students, the best known of dents and known as the Riesz-Thorin inter- which was Björn Dahlberg (1950-1998, polation theorem, gave rise to a whole professor in Göteborg), an imaginative new subject. Other claims to Riesz’s fame analyst who produced a result on the are his theorem on conjugate functions and mutual singularity of harmonic and his work on Riesz potentials. The latter boundary surface measure in Lipschitz work was used in potential theory and in domains. connection with the wave operator and the Anders Wiman had a colleague in propagation of singularities. Riesz’s per- Uppsala, Erik Holmgren (1873-1943, sonality was decidedly un-Swedish. He professor in Uppsala), whose name would liked to talk at length, and was a continu- live on because of an oft-cited theorem. ous source of inspiration to his students His interest was partial differential equa- through his indefatigable enthusiasm for tions, and he proved a uniqueness theorem mathematics. which plays an important part in the mod- ern theory of the subject. Björling and Holmgren descendants Torsten Carleman (1892-1949, profes- Before continuing with the students of sor in Stockholm), was easily Holmgren’s Marcel Riesz, we must go back to another Arne Beurling germ of modern Swedish mathematics, namely to Carl Björling (1839-1910, pro- Finnish mathematician Lars Ahlfors, fessor in Lund). Inspired by the Danish requiring Beurling to extend his investiga- mathematician Zeuthen, and in competi- tions. Beurling had a brilliant career, first tion with his colleague Anders Bäcklund, as professor in Uppsala, then as a member he took up algebraic geometry. This was of the Institute for Advanced Study. He is a subject much in vogue at the time due probably unique in Swedish mathematics mainly to the work of Italian mathemati- in that he might have influenced the course cians11 Among them was Pasquale del of history through his ingenious cryptana- Pezzo, Duca di Cajanello and brother-in- lytic efforts during World War II. The law of Gösta Mittag-Leffler. most dramatic event was the breaking of Björling had a student by the name of the German “Geheimschreiber”, T-52, Anders Wiman (1865-1957), who later which made it possible for the Swedish became professor in Uppsala and had two military command to read top level famous students, Fritz Carlsson and Arne German communications for two or three Beurling. Wiman had started out in the years during the most critical part of the footsteps of his teacher Björling, but in war (the story is described in detail in Uppsala he also worked in analysis, in par- Beckman, 2003). There were other ticular in analytic functions. Swedish mathematicians involved as well: Torsten Carleman Björling’s colleague, Anders Bäcklund Gunnar Blom and C-O Segerdahl, both (1845-1922), also dabbled in algebraic students of Cramér, and Beurling’s student most famous student. He wrote about geometry. By a quirk of history, Bäcklund Bo Kjellberg. integral equations, partial differential found a new field of interest when Sophus In his own country, Beurling is consid- equations, analytic functions, quasi-ana- Lie was among the applicants for a mathe- ered to be one of the handful of really bril- lytic functions, harmonic analysis, and matics chair in Lund, in 1871. A friend of liant mathematicians that Sweden has pro- kinetic gas theory, and made fundamental Lie was supposed to hand in the applica- duced, but unfortunately, except for a contributions in all of these areas. Like tion, but was negligent in his duties and small group of cognoscienti, he is little Beurling however, he did not enjoy the the application arrived one day late. The known internationally. Part of the reason international reputation that we the act stayed in the office of the registrar, might be that his most productive time Swedes think he deserves. Again, part of however, and Bäcklund was allowed to coincided with a trend away from con- the reason was the concurrent develop- study Lie’s papers, among which he found crete, down-to-earth analysis. Also, he ment of more abstract theories, which a description of Lie’s contact transforma- seldom published, and had a tendency to attracted more attention. Other causes tions. The Bäcklund transformation, a keep his ideas to himself, guarding them may have been that he was actually ahead generalization of Lie’s concept, ensued. with suspicion and jealousy. He had a of his time, as for example with his gener- Bäcklund later used his considerable pow- lasting influence on Swedish mathematics alization of Holmgren’s uniqueness theo- ers of analysis on problems in mechanics through his many students, one of which rem, which only drew attention through and electricity. None of Bäcklund’s stu- was to join him on the Swedish mathemat- later developments in the area. dents continued with mathematics. We ical Olympus. Carleman’s personality was complicated. will now return to Wiman’s students, The second of Wiman’s students, Fritz To quote Gårding: he was neither at ease Beurling and Fritz Carlson. Carlson (1888-1952, professor in with the world nor with himself. Arne Beurling (1905-1986), was a stu- Stockholm), also continued in the path of Carleman was director of the Mittag- dent of Holmgren as well as of Wiman, but his teacher’s interest in analytic functions, Leffler Institute for 20 years, but there was took up Wiman’s interest in analytic func- and can indeed be seen as continuing hardly any activity, due a great deal to the tions. His thesis, in which he started the Mittag-Leffler’s program in the area. lack of funds. development of the concept of extremal Carlson had two students, with an interna- Carleman did not have many students. length, was delayed for a year because tional reputation in different fields, Hans One of them however, Åke Pleijel (1913- Beurling’s father wanted him to join him Rådström (1918-1969, professor in 1989, professor at KTH, in Lund and in for an alligator safari in Panama. An addi- Linköping) and Tord Ganelius (1925-, Uppsala), continued and extended tional reason was that some of the results professor in Göteborg). Ganelius is best Carleman’s work in spectral theory. in the thesis had been anticipated by the known for his Tauberian theorems. He Pleijel is perhaps best known for his

18 EMS June 2004 SOCIETIES results on the asymptotic distribution of KTH), took up Beurling’s interest in eigenvalues for the Laplacian on Nevanlinna theory, studying the relations Riemannian manifolds and for ordinary between maximum and minimum moduli differential equations. of analytic functions in the plane. To finish the line of Holmgren’s descen- The most famous of Beurling’s students, dants, we will mention Jaak Peetre and the only one to reach - and surpass - (1935-, professor in Lund and in his level, is Lennart Carleson (1928-, Stockholm). Peetre started out in PDEs as professor in Stockholm and Uppsala). a student of Pleijel, but then branched out Carleson started out doing bone-hard into interpolation theory and other fields, analysis, studying exceptional sets of vari- broadening his areas of interest substan- tially.

The Riesz Legacy The first Riesz student to reach fame was Einar Hille (1894-1980, professor at Yale). A couple of years after his Ph.D. he left for a postdoc position at Harvard and was lost to Sweden for ever, although the Lars Hörmander author took a graduate course in algebra lishing widely on PDEs, several complex from him in Uppsala in the early 1960’s. variables, Fourier integral operators, and Hille published widely, but is of course convexity. His early monograph on linear best known for his book on semigroup the- PDEs grew with time into a formidable ory. four-volume oeuvre. After some political Otto Frostman (1907-1977, professor skirmishes, Hörmander left Stockholm for in Stockholm) wrote a beautiful thesis on the Institute of Advanced Study, causing potential theory. Using modern measure the Swedish parliament to institute a new theory he could complete and considerably law, Lex Hörmander, aiming at curtailing Lennart Carleson extend Gauss’s ideas from the 1820’s to the brain-drain. After a few years howev- kernels of a very general type. Several ous classes of harmonic and analytic func- er, Hörmander returned to his Alma Mater. good papers followed the thesis, but after tions. Interpolation problems for holo- Through his dozen or so Swedish PhD stu- his time as docent in Lund, Frostman had morphic functions led to the solution of dents, he has over 80 descendants (MGP to support himself as a high school teacher the corona problem. For the general does not list all of them). It would proba- until a chair in Stockholm became avail- mathematical public, Carleson is best bly not be too much of an exaggeration to able. For a 15-year period, he was also known for proving the i.e. convergence of say that Hörmander is the Swedish mathe- director of the Institute Mittag-Leffler. the Fourier series of L2 functions, a prob- matician who has had the largest impact Lars Gårding (1918-, professor in lem which had eluded analysts for more ever. Lund) soon developed into one of the pil- than half a century. Outside harmonic Two of his early students, Germund lars of Swedish mathematics in the last analysis he has, in particular in his later Dahlquist and Vidar Thomée, moved half of the century. He worked initially in years, been very active in dynamical sys- away from their master’s voice to contin- group representation theory - albeit with tems. Carleson seems to be the only one to ue their careers in different parts of numer- strong connections to PDEs - but soon have profited by Lex Hörmander, having ical mathematics, more about which will made a name for himself in PDEs, where been awarded a personal professorship by be explained later. Other students include he published widely on elliptic and hyper- the Government in the 1960s. As a thesis Jan Boman (1931-, professor in bolic equations and systems of equations. adviser he has been prolific, with over two Stockholm) and Christer Kiselman Presumably, his name will live on forever dozen students, and his descendants, close (1939-, professor in Uppsala). Kiselman in connection with the Gårding inequality. to 100 in number, have populated the has carried the torch of several - indeed, Later work on hyperbolic equations mathematics departments of Swedish uni- sometimes infinitely many - complex vari- includes studies on lacunas, partly togeth- versities and university colleges. He was ables, and has in turn had over a dozen stu- er with Atiyah and Bott. Gårding also also in practice the lone editor of the Acta dents. used his considerable literary skills to Mathematica for close to a 30-year period. advantage; one example is the already In addition to traditional mathematical The Beurling Legacy cited book on Swedish mathematics activities, Carleson contributed greatly by Beurling’s students were given problems before 1950. Less known abroad are prob- infusing life into the slumbering Institut in very different areas, attesting to their ably his plays and essays, with mathemat- Mittag-Leffler. By finding funds to master’s wide interests. ical and philosophical themes. His colour- refurbish the Mittag-Leffler villa and Carl-Gustav Esseen (1918-2001, pro- ful and outspoken personality, combined building apartments for guests, and by fessor at KTH and in Uppsala), wrote a with a solid interest in music and culture, conceiving the format of the institute’s sci- thesis on the best remainder in the law of has never failed to make a lasting impres- entific programs, he managed to create a large numbers, and continued in mathe- sion on anybody who has come his way. centre for mathematical research of a matical statistics. Sweden’s only Fields medallist so far, unique character. Göran Borg (1913-1997, professor at Lars Hörmander (1931-, professor in Carleson’s most productive student has KTH), was among the first to successfully Stockholm and Lund), received his PhD in been Hans Wallin (1936-, professor in study the inverse spectral problem for Lund in 1955, and was a full professor in Umeå), both when it comes to number of ODEs. Due to the isolation caused by the Stockholm in 1957, at the age of 25. His papers and number of students (MGP lists war, his results did perhaps not have an thesis on linear partial differential equa- 32 descendants, of which 24 were stu- immediate following, receiving their tions was a tour-de-force, and he has con- dents). The rest of us, including Gunnar recognition much later. tinued in the same vain for decades, pub- Aronsson (1939-, professor in Bo Kjellberg (1920-1999, professor at EMS June 2004 19 SOCIETIES Linköping), Lars Inge Hedberg (1935-, came back and founded a school in the These however, failed to materialize, and professor in Linköping), Ingemar Wik subject in Stockholm, which is very active the female gender has been sadly under- (1934-, Umeå), and the author, have been to this day. represented in the professorial ranks. A more modest in these respects. Bernt Lindström (1933-, professor at few have received PhDs though, and some The last one of Beurling’s students, KTH) was formally a student of Otto have become lecturers. One can mention Yngve Domar (1928-, professor in Frostman, but wrote a thesis on combina- Sonja Lyttkens (student of Carleson), Uppsala), wrote a beautiful thesis that did torial methods in number theory. He had Anna-Lisa Wold-Arrhenius (Domar), not receive the attention it probably several students, thus creating a Swedish and Kersti Haliste (Carleson). (There are deserved. Domar had a productive career school of combinatorics, represented some unmentionables also.) in Uppsala, doing harmonic analysis and today by at least three professors. When in the 1990s, there were finally synthesis. He had 12 PhD students, one of one or two women appointed to mathe- whom, Matts Essén (1932-2003, profes- Foreign Influx matics and applied mathematics chairs, sor in Uppsala), came under the influence With the fall of the iron curtain, a number they were imports from Germany and of Bo Kjellberg and continued his career of Russian mathematicians immigrated to Finland. in majorization problems for harmonic and Sweden, changing the face of mathematics analytic functions. here inalterably. Also, professors have Swedish mathematics of late been recruited from Germany, Holland, For several decades after 1950, the num- Numerical Mathematics Italy, Norway, Poland, Romania, the UK, ber of professors of mathematics in Numerical analysis took off as an indepen- and the United States (I have probably for- Sweden was very small, on the order of dent subject in the 1960s, and three more gotten one or two). Due to my self- 10-15, and most of them worked in analy- or less independent schools were created. imposed age limit, the names will not be sis. Life was dominated, and the standard My impression is that Sweden has had a mentioned here, with the exception of set, by the three giants, Carleson, Gårding, very good reputation in this area. A result Vladimir Maz’ya (1937- ), who was and Hörmander. This meant that you sel- of these activities was the creation of BIT, drafted to Linköping by Lars Inge dom published, and preferably only your a Nordic journal for numerical mathemat- Hedberg. Maz’ya, working mostly in best and polished results. In retrospect, ics with Carl-Erik Fröberg (1918-, pro- PDEs, and enormously productive by this time seems idyllic, although I do not fessor in Lund) as one of the driving Swedish standards (several books by his think we, the lesser souls, thought so at the forces. hand have appeared in the last few years), time, the worries of finding a bread-win- Although a student of Hörmander, has given analysis in Sweden and in ning position occupied our minds too Germund Dahlquist (1925-, professor at Linköping new impetus and new direc- much. Since then, Swedish mathematics KTH) worked in numerical methods of tions. has taken on a completely different, inter- ODEs and associated numerical algebra national flavour, for good and for bad (on problems from the beginning. Among his Women Mathematicians balance, surely for good). The number of students, in excess of 20, was Åke Björck As mentioned above, Mittag-Leffler man- professors is more than 75, although due (1934-, professor in Linköping). aged to convince the board of the then pri- to a strange law, some of these are teach- Heinz-Otto Kreiss (1930-, professor in vate Stockholm University (known as ing positions, and professorships in name Uppsala and at KTH), came from Stockholm’s högskola in those days) to only. The breadth of areas of mathematics Germany but took a PhD at the KTH and appoint Sonya Kovalevsky (1850-1892) being studied has increased enormously, went on to found a thriving school of while some of the traditional subjects have numerical analysis of PDEs in Uppsala. receded almost into oblivion. The tradi- He has also had long stints abroad. tion of publishing has altered, the pressure Among his students are Bertil Gustafsson to publish has increased, and so has the (1939-, professor in Uppsala) and Olof competition for positions. It is not uncom- Widlund (1939- ), who left Sweden early mon that a lectureship draws 50 appli- on for a career at the Courant Institute. cants. In short, we have joined the world. Another of Hörmander’s students was Vidar Thomée (1933-, professor in References Göteborg), who started doing more tradi- Gårding, Lars: Mathematics and tional Hörmander mathematics, but then Mathematicians. Mathematics in branched out and founded a very success- Sweden before 1950. American ful school of PDE numerical analysis in Mathematical Society and London Göteborg, with an international reputation Mathematical Society 1998. and a number of good students to his cred- it. Beckman, Bengt: Codebreakers: Arne Beurling and the Swedish Crypto pro- New Areas gram during World War II. American In the author’s youth, Swedish mathemat- Mathematical Society 2003. ics was dominated by analysis. True, there was Trygve Nagell (1895-198) in Uppsala Kjell-Ove Widman [[email protected]] doing number theory, and Olof Hanner received his PhD under Lennart Carleson Sonya Kovalevsky (1924?-) in Göteborg doing general topol- in Uppsala in 1966. Professorships in ogy, but both represented rather modest- Uppsala and Linköping were followed by scale activities. to a chair of mathematics. Unfortunately, a longish stint abroad. He then returned This started to change when Jan-Erik she died at a relatively young age, before to Sweden to become Director of Institut Roos (1935-, professor in Stockholm) really making an imprint here. Mittag-Leffler in 1995. His scientific went away from Lund to study in Paris, With Kovalevsky as a model, one would interests include partial differential equa- and found out that there was something have expected locally groomed women tions, calculus of variations, and cryptog- called algebra and algebraic geometry. He mathematicians to appear soon after. raphy.

20 EMS June 2004 SOCIETIES

SOCIETIES CORNER The Societies Corner is a column concern- France:France: "Sauvons"Sauvons lala rrecherecherche"che" ing mathematical societies in European countries. Over the last five years, this col- Colette Moeglin (Paris, Jussieu) umn has presented numerous articles that and described the history of a particular society Marie-Francoise Roy (Rennes, France) or the society's present day work. Many European Mathematical Societies have been portrayed, and we wait for the contri- butions from those societies that have yet to present. However, until now some important aspects of the societies' lives have not appeared in the column. The life of a society does not only consist of its historical development and its current active work. A number of events happen in the every day life of a society. Do you remember the general assembly of your society three years ago and a fasci- nating discussion that took place there? How about a farewell party and an exciting toast by the President, or a controversial decision about something undertaken by the council last year? Several anecdotes Colette Moeglin Marie-Francoise Roy concerning famous mathematicians or for- mer presidents of the society, connected During the first semester of 2004, a very sig- ing young scientists in Universities. with their work in the society? And what nificant initiative entitled 'Sauvons la On March 9, given the lack of satisfactory about your society competition for the Recherche' ('Let's save scientific research') has response from the government, over 2000 sci- paper written by school pupils, and some taken place in France, inducing an unexpected entists in charge of research responsibilities papers presented there? What about anoth- and powerful social movement of popular sup- (heads of research laboratories or research pro- er activity? A host of funny or important port for scientists' requests. Several dozen grams) collectively resigned from their stories, so many interesting events hap- thousand scientists, as well as several hundred responsibilities. This was considered as a pened... thousand non-scientists, have signed a peti- major event by the media. A few weeks later, Don't keep them to yourself! Please, tion, with opinion polls indicating that more the government lost the regional elections in a write about them and submit to the Society than 80% of the population are in support of spectacular way. A change of policy concern- column. Let others enjoy your stories! Let the scientists' requests. These requests were ing research was one of the three main changes others know about the everyday life of the extremely simple. The first one was that the recommended by President Chirac to take into society! If it is not written down now, after government cancels its recent funding restric- account the electoral regional failure (the two some years it may be forgotten. tions and gives the funding that was initially others being the funding of young artists, and Of course, we will not stop publishing promised back to the research institutions. The the support of unemployed people, two other articles of the same style as before, we just second one was that the 550 research positions topics for which important demonstrations and want to have something more. proposed by the government as non perma- actions had taken place in the preceding If you feel that anything like that would nent positions are turned into civil servants months). The ministry in charge of research interest others, please do not hesitate and positions as was the case in pre- was changed and the requests ceding years. Finally, the third of the scientists accepted with- was that a national discussion is in a few days. organized under the name The current situation, 'Etats généraux de la recherche'. beyond all initial hopes, is that In the French political and all the requests motivating the social tradition, Etats généraux 'Let's save scientific research' is thought of as a large bottom- initiative have been accepted. up discussion converging to The Etats Généraux de la national proposals for further actions and Recherche are currently being prepared in changes, following the model of the famous local research groups, universities and initial Etats Generaux that was prelude to the research institutions and should define propo- Krzysztof Ciesielski French revolution. When the influence of the sitions for the fall. A 'programmation law' initiative grew bigger, the opening requests, making scientific research a national priority contact the column editor, Krzysztof initiated in research institutions such as INRA should be voted before the end of the year and Ciesielski (e-mail: [email protected]) and CNRS, were rapidly added to by a request various changes to the organization of scientif- in the first instance. for the creation of 1000 new positions for hir- ic research in France should follow.

EMS June 2004 21 SOCIETIES The French mathematical community is choosing scientific studies in universities is ber of scientific and technological research naturally active in this context. The French dropping. areas is widely acknowledged. Industry and mathematical societies SMF (Société In this general context, French mathemati- services need more and more people with a Mathématique de France) and SMAI (Société cal data can quickly be summarized. The com- good mathematical background and an ability de Mathématiques Appliquées & munity of professional mathematicians (uni- to adapt to the rapid changes in science and Industrielles) have supported the 'Let's save versity teachers and full time researchers) has technology. French mathematicians have to scientific research' initiative from its beginning about 3500 members, only 10% of them being open their eyes and ears and redefine new cur- and encouraged their members to be active in full time CNRS researchers, and about half of ricula for students and new scientific collabo- the various demonstrations, contacts with the these are concentrated in Paris or the Ile de rations with the outer world. population and preparation of the 'Etats France region. In spite of the small number of A forum discussion 'Let's save scientific généraux'. full time researchers in mathematics, CNRS research' has been proposed by SMF and The organization of research and of scien- plays a very important role in the organization approved by SMAI, cf. http://smf.emath.fr tific curricula has some very specific features of French mathematical research: about 85% /Forum/. Concrete suggestions following this in France. Firstly in many areas of science, like of the mathematicians belong to one of about discussion will be proposed before the sum- physics, biology and chemistry, a majority of 60 joint CNRS-Universities research laborato- mer. On the European scale, a petition 'For a scientists are not university professors but ries, and the important equipments of the European Research' in support of a European rather full time researchers, and moreover, French mathematical community are support- Research Council (ERC) for basic research civil servants paid by the state. The main ed by CNRS (CIRM, IHP, the network of has been launched at http://fer.apinc.org/ . research institution is CNRS, which covers all math libraries, the project Numdam for European mathematical societies and individ- aspects of scientific research from archaeology numeration of old mathematical publication, ual European mathematicians are very wel- and linguistics to biology, computer science the network of computer engineers for mathe- come to participate in our French discussions and mathematics. However smaller and more matics). 25% of INRIA (around 100 people) and bring us some fresh air and new ideas! specialized research institutes like INRA for are mathematicians. The loss of students, Please visit and send your contribution (by e- biology and medicine, and INRIA for com- although less spectacular than in physics for mail to [email protected]. jussieu.fr). puter science, are also present, and the relative example, is high in mathematics, reaching roles of these institutes and CNRS is not clear. about 25% in the last four years. The number Colette Moeglin [[email protected]. fr] Important research programs are defined and of university positions in mathematics started graduated at Univ. Paris 6 under Jacques directed by the Ministry of Research, indepen- to decrease in 2003 and about 100 positions Dixmier with a thesis on primitive ideals of dently of research institutes. The role of have disappeared. The mechanism is the fol- enveloping algebras. Her current research is Universities, as well as regional institutions, in lowing: mathematicians retire or are promoted concerned with automorphic forms and repre- the orientation of research has to be redefined, and their position is given to another depart- sentations of p-adic groups. She is employed and the duality between full time permanent ment where student numbers are greater (sport by CNRS as Directrice de Recherche at the research positions and university positions is and theatre are currently popular among stu- Institute of Mathematics at Paris-Jussieu, being questioned. dents). where she also acts as Directrice Adjointe Another specific French feature is the exis- A vicious circle is thus currently at work: amd as editor-in-chief of JIMJ, the Journal of tence of 'classes prépatoires'. Rather than less students, less positions for mathemati- the institute. going to universities, the best scientific stu- cians, less motivation to study mathematics.... dents prepare competitive exams in specific and in a few years the French mathematical Marie-Francoise Roy [marie.francoise. classes, taught by very competent full time atmosphere, recently active and full of [email protected]] works on algorithmics professors in the secondary school administra- promise, could switch to long-term depres- within real algebraic geometry. She is a pro- tive system, having no contact with research. sion. fessor of mathematics at the University of The engineering schools they enter after this This situation is paradoxical since the Rennes 1 and a member of the council of the preparation are mostly independent of the uni- importance of mathematics in a growing num- SMF. versities. The very best of these students, if they enter into major 'grandes écoles' such as Ecoles Normales Supérieures, Ecole Polytechnique and a few others, will be in con- tact with scientific research in the remainder of their studies. However, an important propor- tion of these good students will stay apart from the university and scientific research system Monsieur François Fillon Ministre de l’Education Nationale during their whole curriculum. On the other de l’Enseignement Supérieur et de la Recherche hand, students deciding to choose universities 110 rue de Grenelle 75357 Paris 07 SP are often poorly motivated and not adequately prepared and do not meet the expectations of the university professors who are active scien- Paris, le 13 avril 2004 tists.

Monsieur le Ministre, Status and Perspectives L'organisation future de la recherche scientifique en France va faire l'objet de discussions et de The present context is moreover characterized décisions dans les semaines qui viennent. Nous nous félicitons de ces nouvelles perspectives. by two important facts: the number of scien- tists retiring from research or teaching posi- Par la présente, nous avons l’honneur de solliciter un entretien avec vous pour vous présenter quelques particularités de l'organisation actuelle des sciences mathématiques qu'il nous semble important tions is growing, and the number of students de voir prises en considération dans le cadre des adaptations nécessaires. Il nous revient aussi de vous 22 EMS June 2004 SOCIETIES PORPORTUGALTUGAL ININ THETHE FIRSTFIRST YEARSYEARS OFOF THETHE INTERNAINTERNATIONALTIONAL MAMATHEMATHEMATICALTICAL UNIONUNION (Enrique Hernández-Manfredini and Anabela Ramos, Universidade de Aveiro, Portugal)

In [5], Graciano de Oliveira describes the essentials of the life, aims and activities of the Portuguese Mathematical Society, together with the adverse historical context that seriously hindered its normal develop- ment and obstructed Portuguese mathemat- ics during the dictatorship years. De Oliveira pays special attention to the histo- ry of mathematics in Portugal in the 1940’s. Digging a little further back in time, and following Costa & Malonek [2], and Ramos & Malonek [6], [7], these notes aim to attract more attention to the efforts made by Portugal to establish contacts with the international scientific community in the post-Great War period; an aspect of Portuguese scientific history which has often not received due attention, if not been completely forgotten. Enrique Hernández-Manfredini Anabela Ramos We also intend these notes to be a contri- bution to recover- Congress of Mathematicians (ICM) in International Mathematical Union (IMU) ing the memory of Toronto, the Instituto’s Assembly named and his involvement in several Congresses Francisco Miranda as Corresponding Members the following of the International Research Council da Costa Lobo, renowned mathematicians: (IRC), which provided him with the oppor- (1864-1945), Élie Joseph Cartan, from France; tunity to become acquainted with several whose dedication John Charles Fields, from Canada; foreign mathematicians, with whom he and efforts to make Karl Rudolf Fueter, from Switzerland; developed strong ties of friendship. This is these contacts pos- Tullio Levi-Civita, from Italy; especially true of his participation in the sible and to stimu- Salvatore Pincherle, also from Italy; Toronto Congress in 1924. The long trip by late mathematical John Lighton Synge, from Ireland; sea provided an environment that was pro- life in Portugal in Grace Chisolm Young, from England; pitious for such developments. William Henry Young, from England, and Costa Lobo’s activities and contacts the 1920's seem to Francisco Miranda have faded from Clément Servais, from Belgium. enabled him to make the state of mathe- da Costa Lobo. matical research abroad known in his coun- memory as well. (1864-1945) Costa Lobo was Within the next couple of months, the try, as well as to report about the way not only a first-class scientist, but also an Assembly would also add to the list, the Mathematical Education was conducted in important ambassador for Portugal in the names of Nicolau Kryloff, from Kiev foreign universities. He was quite international scientific field, without whom University, who gave a two-month course impressed with several aspects of the an important chapter in the history of on Advanced Analysis at the University of University of Toronto, (its organization, Portuguese academic life would, no doubt, Coimbra, and Jacques Hadamard, from academic life, interaction with other uni- be missing. Costa Lobo, an astronomer, France. Finally, in 1925, the file was com- versities, the subjects offered in the differ- was Director of the Observatório pleted with the name of the mathematical ent courses, in particular those in mathe- Astronómico de Coimbra, and was also physicist Sir John Larmor, from England. matics), and describes them in detail in president of the Instituto de Coimbra from Some of the above were to become Lobo [4]. This can also be seen in Costa & 1915. The Instituto, recognised by the Presidents of different ICMs: Pincherle in Malonek [2]. International Association of Scientific 1928 in Bologna; Fueter in 1932, in Zurich; Mathematical activities in Portugal in the Academies as an academy of sciences, had and Hadamard, Honorary President of the 1920’s, and especially, Portugal’s early an interest in international intellectual life. 1950 Congress in Cambridge, USA. The association with the IMU are rather forgot- Under the influence of its director, the presence of these personalities in Portugal ten matters. Yet, it is to be noted that Instituto focused its interests in the area of allowed a much needed interchange of Portugal, was involved in the very creation mathematics. information and widened Portugal’s inter- of the IMU. First, taking part in the At the time, the Instituto made a consid- national scientific horizon. In particular, Constitutive Assembly of the International erable contribution to Portuguese mathe- one can see works by de la Vallé, G. C. Research Council in Brussels, in 1919, matical life, and opened the country to the Young, Kryloff, and Servais, published by where preliminary moves towards the cre- international mathematical world: In its the Instituto’s magazine. This impressive ation of the IMU were made, and later, in 1920 General Assembly, the Instituto list of collaborators is a token of Costa 1920, as a participant in the ICM in named Emile Picard, Charles de la Vallé Lobo’s dedication and commitment to Strasbourg, where the IMU was definitive- Poussin, and Gabriel Koenigs, as Honorary developing mathematics in Portugal, as ly founded. On this occasion, the foreign members. well as to forging relations between his Portuguese representative was Costa Lobo, Later, after the 1924 International country and the mathematical world. who presented a communication Sur la It was Costa Lobo’s connection with the courve déscrite par le pôle sur la surface EMS June 2004 23 SOCIETIES de la terre. Later, in 1924, Portugal attend- ed the Congress that took place in Toronto, represented by Costa Lobo and Fernando de Vasconcelos. The former presented a communication on Nouvelles théories physiques: application à l’astronomie; the latter, one entitled Sur quelques points de l’histoire des mathématiques des Egipciens et aussi sur le Sidhantas des Indiens. It will be remembered that political issues and sensitivities proper of the post- war period pervaded the life of the IMU and the IRC’s meetings. They were haunt- ed by animosities towards the countries that had fought against the allied powers. As a consequence, these countries were banned from membership of the IMU and from participation in the 1924 Toronto Congress. Regarding these political issues within the context of the IMU and the IRC, University of Coimbra it may be of interest to note that Costa Lobo did not directly address or comment was living in the USA. The others, exiled the EMS 46, December 2002. on them. However, his views about war in Brazil, limited themselves to sending [6] Ramos, Anabela & Malonek, related topics were the ones that could be their contributions. This was an indication Helmuth. Portugal e a fundação da expected from a citizen of one of the allied that political pressure exerted on the União Matemática Internacional. countries. (cf. Lobo [3]). Portuguese universities by the Salazar (Portugal and the foundation of the In the year that followed the Toronto regime had began to show its effects. It is International Mathematical Union). Congress, it has to be mentioned the rôle well known that during the Salazar regime, To appear. played by Costa Lobo in the realization in intellectual life and scientific research [7] Ramos, Anabela & Malonek, Coimbra, in 1925, of the Second Joint activities in Portugal were severely ham- Helmuth. Um capítulo esquecido – Congress of the Associação Portuguesa pered. For more details of how this situa- Sócios Matemáticos extrangeiros.do para o Avanço das Ciências1, together with tion arose and subsequently evolved, see de Instituto de Coimbra na terceira déca- its Spanish counterpart, the Associación Oliveira [5], Bebiano [1], and Ramos & da século XX. (A forgotten chapter - Española para el Avance de las Ciencias.2. Malonek [6]. foreign Mathematicians, as members The congress took place from 14th.to In these notes we have focussed our of the Coimbra Institute in the third 19th of July and embraced a wide spectrum attention on a specific part of the history of decade of the 20th century.) Boletim of topics: mathematics, astronomy and mathematics in Portugal in the 1920’s. de la Sociedade Portuguesa de physics of the Earth, physico-chemistry, Details of ensuing events, especially the Matemática, No. 49, 2003. natural sciences, philological, philosophi- foundation and life of the Portuguese cal and historical sciences, and medical sci- Mathematical Society can, again, be found Enrique Hernández Manfredini lectures in ences and applications. Apart from the in de Oliveira [5]. Mathematical Logic and Foundations of peninsular representatives, the congress Mathematics at the University of Aveiro- was attended, amongst others, by represen- References. Portugal. He graduated in mathematics tatives of The French Institute, The Royal education at Chile University and obtained Academy of Belgium and the French and [1] Bebiano, Natália. Bento Caraça his Ph. D. in mathematics from (the for- English Associations for the Advancement (1901-48): A beautiful mind.. News- mer) Bedford College, London University. of Science. letter of the EMS 44, June 2002. He has worked at the Chile University, the On this occasion, Costa Lobo shared [2] Costa, Teresa & Malonek, Helmuth. (former) Technical University of Santiago- responsibilities with an important and Uma iniciativa notável de Costa Lobo. Chile, the Cuban Academy of Sciences, the internationally known Portuguese mathe- (A remarkable initiative of Costa University of Havana, British Telecom, matician: Francisco Gomes Teixeira. The Lobo). Helios, http/www.mat.uc.pt/~ and at a variety of secondary and further latter took part as President of the obsv/obsv. education colleges. His main interest lies Associação Portuguesa, while Costa Lobo [3] Lobo, Francisco Miranda da Costa. in the realm of the foundations of mathe- acted as President of the Congress’ Congresso Internacional de Matem- matics. Executive Committee. áticas. (International Mathematical Anabela Ramos is an assistant at the Although we lose track of Portuguese Congress). O Instituto, Jornal Mathematics Department of Aveiro participation in the 1928, 1932 and 1936 Scientífico e Literário, No. 67, University, where she tutors and lectures in ICMs, we note the presence of Portuguese Coimbra, 1920. mathematical analysis. She completed her mathematicians at the 1950 Cambridge- [4] Lobo, Francisco Miranda da Costa. first degree in mathematics, and later her USA Congress. Communications were Congresso da União Matemática master’s degree, at the University of submitted by Hugo Ribeiro, A. Gião, Internacional. (Congress of the Coimbra. Her MSc thesis dealt with one of A. Monteiro, M. Peixito, and R. L. Gomes. International Mathematical Union). O Pedro Nunes’ works: O De Erratis Orontii However, these scientists took part in the Instituto, Jornal Scientífico e Literário, Finaei. She has previously worked as a Congress as individuals and not as repre- No. 71, Coimbra, 1924. secondary school mathematics teacher. sentatives of Portugal. Moreover, only [5] de Oliveira, Graciano. The Portuguese Her main interest focuses on Portuguese Hugo Ribeiro was personally present, as he Mathematical Society. Newsletter of mathematical history.

1 Portuguese Association for the Advancement of Science. 2 Spanish Association for the Advancement of Science 24 EMS June 2004 INTERVIEW

AnAn interviewinterview withwith AriAri LaptevLaptev Interviewer: Marta Sanz-Solé (Universitat de Barcelona)

Ari Laptev (Stockholm) is the Chairman of the 4ECM Organizing Committee. Ari Laptev Marta Sanz-Solé Role of the European Congress throughs were achieved by com- decided to involve the European of Mathematics (ECM) bining techniques from different Networks by giving members an areas of Mathematics. Therefore I opportunity to speak at the You are chairing the Organizing think that IMU and EMS Congress. Committee of the 4ECM, one of Congresses will play an even In order to receive a grant from the major mathematical events in more important role than before, Brussels or ESF, one has to gather Europe in 2004. What is in your allowing mathematicians to focus a very representative bunch of opinion the role and the necessi- on problems of general interest. teams from different European ty of big, interdisciplinary meet- countries working in the same ings like ECMs? Since 1897, mathematicians have area of Mathematics. The compe- I used to be quite sceptical about met periodically at International tition is extremely high but once large-scale meetings in general Congresses (ICMs), with pro- the project is awarded, the and EMS Congresses in particular. grammes including all disci- Network obtains quite substantial However, my involvement in plines. Mathematics is the only funding for developing a specific organizing the 4ECM has radical- scientific field keeping this tradi- area of research. However, they ly changed my attitude. I see now tion. The much younger ECMs have very little interaction with how useful such meetings can be. might be seen as ICMs on a mathematicians who do not Compared with specialized scien- smaller scale. What makes ECMs belong to their Network. tific conferences, where one meets so special? Therefore, European Congresses close colleagues, big meetings At the time when it became clear can play a coordinating role for have quite a different function. that Stockholm might be the next these programmes. They give an overview of new host for the European Congress of By inviting the speakers from dif- tendencies in Mathematics and Mathematics, the opinion of some ferent European Networks to the play an important role in develop- of my colleagues was that EMS 4ECM, we want to give European ing its different areas. There is an Congresses are not needed, pre- Networks the possibility of com- opinion that Mathematics has cisely for this reason. They municating with each other. We become too large and that it is dif- claimed that such meetings imi- also hoped that by inviting such ficult for the majority of partici- tate IMU Congresses on a smaller speakers we could make the pants to follow most of the talks at scale, are not of the highest quali- 4ECM more attractive to the these meetings. During the last ty and therefore unnecessary. I members of the corresponding century many new fields have disagree. Looking back at the research groups. Indeed, as we see been created, and established three EMS Congresses organized now, many members of these branches of Mathematics have in Paris, Budapest and Barcelona, groups will be participating in the split into sub fields. However, it is noticeable that they were all 4ECM and will have their own development during the last twen- different from the IMU satellite meetings before or after ty years has to a large extent been Congresses in character as well. the Congress. the opposite. There is a clear ten- When organizing the 4ECM we This time we only have half a dency towards unification of dif- also hoped to introduce new fea- day for the Network presenta- ferent branches of mathematical tures. For example, already at the tions. I personally think that it research. Many new break- very early organizational stage we would have been even better if we EMS June 2004 25 INTERVIEW could have devoted a whole day to Mathematics in Science and organizing the 4ECM we had the the European Network activities. Technology idea of using the fact that Sweden For example, half a day for gener- is the country of Nobel Prizes and al Invited Lectures and another The theme of the 4ECM - invited representatives from other half for networks mini-symposia, Mathematics in Science and Sciences to give Plenary talks. We where the members of the corre- Technology - might be a state- believe that a partnership with sponding Network could choose ment addressed to those who specialists from other scientific the speakers themselves. One of think that Mathematics can areas could be extremely fruitful. the advantages of such mini-sym- develop without any cross con- We have made an effort to estab- posia compared with “usual” nections and interactions with lish better contact with Applied Network meetings could be that scientists and engineers. At the Mathematics by inviting Plenary such a meeting would gather a same time, it is carrying a clear and Invited Speakers covering a broader audience, allowing people message about the new opportu- variety of applications. to both invite speakers from paral- nities and developments for lel groups and participate them- Mathematics in the 21st century. Visibility of Mathematics selves in other Network mini- Could you comment on this? symposia. In his article written for the The celebration of a large scien- We also kept in mind that Barcelona Intelligencer, Jean- tific meeting like the 4ECM is an including European Network pre- Pierre Bourguignon pointed out excellent opportunity to increase sentations might be important for the importance of interactions the visibility of Mathematics. future EMS relations with the between different areas within What aspects of the scientific authorities from Brussels and with Mathematics and also the interac- programme help the most to show the ESF in Strasbourg. We are tion of Mathematics with other why Mathematics is useful and expecting some representatives subjects. He writes: why a knowledge-based society from Brussels to attend the 4ECM “Mathematicians are already and needs mathematically well-edu- Opening Ceremony. as I see it, will be more and more, cated people? Another new feature of the confronted with the need to broad- It is a very good opportunity 4ECM is the involvement of sci- en the conception they developed indeed. Looking at the list of reg- entists from areas outside of their discipline and how they istered participants and at some of Mathematics. Nobel Institutes in practice it.” the applications to Poster ses- Physics and Chemistry of the With the title “Mathematics in sions, I can see many scientists Swedish Royal Academy of Science and Technology” we who cannot be classified as Pure Sciences are supporting the wanted to emphasize the impor- Mathematicians. Obviously, this Congress. This enables us to tance of developing Mathematics means that people from areas of invite prominent specialists in for other Sciences and research related to Mathematics Physics, Chemistry and Biology, Engineering, and try to identify are interested in the 4ECM and in who will share their views on the the place of Mathematics within developing contacts with mathe- significance of Mathematics in the modern scientific community. maticians. This is a promising ten- their subject in a lecture of their The attitude that Mathematics can dency and once more shows the own choice. be good only for mathematicians importance of big general meet- Finally, I would also like to should be changed and we all have ings of Mathematics where people mention that there are 10 prizes to try to be more open minded from different areas can meet each awarded to young mathematicians towards a large number of new other. at EMS Congresses. They are dif- areas in Science and Engineering ferent in character compared with developed during the last 20 Fields Prizes and allow us to draw years. There is a danger that math- attention to a group of young ematicians will lose respect from European mathematicians who are other scientists if they are not pre- at the beginning of their scientific pared to collaborate. career and have already made sub- There is also a danger that stantial contributions in their Mathematics as a subject could be When trying to the predict areas of research. put aside and become insignifi- future, we sometimes have discus- I am sure that the organizers of cant in the main stream of the sions in our Department about the 5ECM in Amsterdam will also development of Science. We how much one will need have new ideas, which will con- should not take the importance of Mathematics in perhaps 50 years tribute to the future style of the Mathematics for granted; we have time. Will future engineers need EMS Congresses. to continue to prove it. to know the multiplication table? As I already mentioned, when For example, this skill is hardly a 26 EMS June 2004 INTERVIEW requirement for a supermarket European countries, will support that we shall be able to involve the cashier nowadays, but it definitely the popularity of Mathematics in media. The possibility of getting was very important in the past. countries where in general, condi- extra publicity is also an added Will the teaching of Mathematics tions for mathematicians are still advantage of inviting Nobel Prize survive or will it die out in a sim- very poor. winners to give lectures at the ilar way to the teaching of Latin? Congress. The Swedish Academy It is difficult to say. However, it is You live in a country with inten- of Science has already shown its clear that the training of logical sive activity and influence in the interest in publicizing the 4ECM. thinking will always be necessary. development of Science and However, it is difficult to compete Moreover, progress of Science Technology. Do education policy with the European Football Cup in and Engineering relies more and makers put enough effort into Portugal, the last week of which more on complicated computa- ensuring good mathematical coincides with the 4ECM. You can tions, which always have to be education at Swedish schools? guess which event will be the evaluated. It is difficult to imagine Are scholars aware of the attrac- most attractive to journalists. that this can be done without deep tive, exciting side of knowledge of Mathematics. Mathematics? And the outcome? It is clear that we have a big prob- Young mathematicians lem with mathematical education You and your colleagues on the in Swedish schools and it looks Organizing Committee are devot- One cannot think of the future of like it is a common phenomenon ing enormous efforts into the science without the incoming for many other European coun- preparation of this important flow of the younger generations. tries too. One of the main reasons event, and many of the country’s Why do young people feel attract- behind such development is that Institutions are sponsoring the ed to do research in nowadays the status of teachers is Congress. What kind of tangible Mathematics? What is the impor- rather low. As a result, this profes- benefits could you expect from tance of Congresses showing sion is not very attractive to tal- all this? (For instance, increas- both the unity and the diversity of ented people. Our government ing student’s interest for Mathematics and its most recent tries to improve this situation but Mathematics, gaining govern- breakthroughs? it is not an easy problem, especial- ment support to improve mathe- It is a mystery how society contin- ly if one is not prepared to invest matical education and research, uously produces a number of peo- a substantial amount of money in strengthening scientific coopera- ple who are gifted in Mathematics it. A relatively “cheap” solution - tion between different depart- and who have the necessary pas- investing into research in mathe- ments across the country, etc.) sion to pursue it. At the beginning matical education - has not had I do not think that one should try of May we had a meeting at our much of an effect. to measure how profitable the Department at KTH evaluating Being a good teacher in any sub- 4ECM will be for Mathematics in applications for PhD positions. ject, in particular Mathematics, is Europe, Sweden or for our There were 30 applications. a talent that is based on a solid Department at KTH. However, I Although our financial possibili- knowledge of the subject. The believe that a meeting of this ties allowed us to accept only four resource of talented teachers is dimension, as well as everything applications, we were very happy limited and it is vital for any soci- important which is happening in to see that Mathematics is an ety not to lose such people and to Mathematics, is definitely helpful attractive subject for young peo- make the teaching profession for its visibility. I believe that ulti- ple in Sweden. However, such an more attractive. At University mately it will play an important attraction is based on many fac- level, we try to do our best to put role in the future funding of tors, which are often insignificant future Engineers on the “right Mathematics at every level. when taking separately, but track”, but it becomes more and become very encouraging when more difficult because students We expect that the 4ECM will be viewed together. I believe that the have not been sufficiently pre- very good for Swedish PhD stu- visibility of the EMS Congresses pared in Mathematics at school. dents. Many of them are partici- with their awards to young pating in the Congress, and meet- researchers plays an important Stockholm is known as a “City of ing prominent mathematicians part in making Mathematics Science”. Do you plan to present personally will definitely inspire attractive to young people. the 4ECM to journalists, to let their commitment to Mathematics. Needless to say that more than people know that it will be the 200 grants, which were distrib- “City of Mathematics” around Thank you very much for this uted to mostly young participants the dates of the congress? interview. I am sure that the from Central and Eastern We shall try our best and hope 4ECM will be a success. EMS June 2004 27 EURANDOM ERCOM: European Research Centres on Mathematics A lot of mathematical activity is hosted at the 24 European research centres - quite different in size and scope - represented in the EMS committee ERCOM. The Newsletter wishes to give these institutions the opportu- nity to present themselves in the coming issues and years. This will hopefully give the readers an idea of how they might use the possibilities offered by these centres: as organiser or participant of a workshop or a sem- inar or as a visitor. Quick information about the ERCOM institutions and about the ERCOM committee of their directors is found on the web site http://www.crm.es/ERCOM.

EURANDOM, a European research institute for the stochastic sciences “The feeling of sparkling Champagne”

History and people University of Technology (TU/e). Together - Disordered Systems The stochastic sciences have been develop- - Combinatorial Probability ing so fast in the past decades that an increasing need was felt for international Queuing and Performance Analysis (QPA): cooperation and bundling of research - Performance Analysis of Production resources. This led to the establishment of Systems EURANDOM, in July 1997. The institute - Performance Analysis of opened its doors in September 1998. The Communication Systems first scientific director was Willem van - Queuing Theory Zwet. Since October 2000, Frank den Hollander has been the scientific director, Statistical Information and Modelling with Henry Wynn (London, United (SIM): Kingdom) as scientific co-director. - Statistical Learning they provide 60% of the budget. Both par- - Biomedical and Bio-molecular ties have a representative on the Board of Statistics EURANDOM and they jointly appoint a - Industrial Statistics. third external member. Other funding sources are: European science organisa- tions, industrial contracts, EU-funds, ESF- networks, etc.

Mission The mission of EURANDOM is to foster research in the stochastic sciences and their applications. It achieves this mission by helping talented young researchers find their way to tenured positions in academia and industry. It does this by carrying out Frank den Hollander, scientific director and facilitating research through post-doc- toral and graduate appointments, visitor Peter Bickel (Berkeley, USA) chairs the exchange and workshops, and by taking Harry Kesten (Cornell University, USA), scientific council, which consists of initiatives for collaborative research at the EURANDOM Chair in 2002 renowned European scientists in the areas European level. of probability, statistics and stochastic RSS focuses on systems consisting of a operations research. In July 2004, Don Focus large number of interacting random com- Dawson (Ottawa, Canada) will take over From the summer of 2004 onwards, the ponents, possibly with disorder. It aims to this position. research at EURANDOM will be re- capture their large space-time behaviour aligned into three parallel programmes, through a combination of probabilistic, Financial resources each with three themes: combinatorial and ergodic techniques, with EURANDOM has two main financiers: the special emphasis on critical phenomena Netherlands Organisation for Scientific Random Spatial Structures (RSS): and universality. Research (NWO) and Eindhoven - Critical Phenomena 28 EMS June 2004 ERCOM QPA is concerned with the performance and senior visitors, and through various close ties with MaPhySto (Aarhus, analysis of stochastic networks, in particu- European networks and programmes in Denmark), the Stochastic Centre lar, production and communication sys- which EURANDOM participates. (Göteborg, Sweden), and the Norwegian tems. Queuing theory and stochastic analy- Computing Centre (Oslo), which share a sis provide the means to deal with control Links to research in The Netherlands strong focus on the stochastic sciences. and optimization issues. EURANDOM is located on the TU/e cam- pus in Eindhoven. EURANDOM is an Several national research organizations SIM combines analytic and algebraic independent foundation. The location on (e.g. DFG, FWO, NSF) have supported methods to address a variety of statistical the university campus allows for special EURANDOM through grants for post-doc- questions, including statistical classifica- ties with the university, in particular with torates and visitors to carry out their tion, bootstrap, micro-arrays, gene expres- the Department of Mathematics and research at EURANDOM. sion, design of experiments, predictive Computer Science. Through the appoint- maintenance and quantum statistics. ment of scientific advisors, the institute Location and facilities maintains intensive relations with other EURANDOM is situated in a spacious Each programme hosts six to eight post- mathematics departments in The building, full of light, glass and colours, on doctorates and graduate students, is super- Netherlands. the TU/e campus in Eindhoven. vised by senior scientific advisors, and is guided by an international steering com- mittee. Through the choice of themes, spe- cial attention is given to the interfaces with physics, biology, telecommunication and industry.

In addition to the above three research pro- grammes, EURANDOM runs two research projects, one on reinsurance and one on battery management.

Post-doctorates get together on a daily basis, run a bi-weekly post-doctorate semi- nar, and participate in the busy schedule of Links to research in Europe seminars and mini-courses the institute European cooperation runs via the scientif- offers. Social events are organised at regu- ic council and the steering committees of lar intervals, ranging from hiking on bike the research programmes, via the appoint- (with or without umbrella!), to admiring ment of post-doctorates, graduate students, tulip exhibitions, ice-skating, canoeing, and through the workshop and visitors pro- etc. gramme. Each year EURANDOM organis- Staff and research es some 10 workshops on a range of topics, EURANDOM offers an attractive environ- with on average 40-45 participants from ment for carrying out research on random around the world. Each year some 40 phenomena. Since its start, some 60 post- junior and senior researchers from outside doctorates and graduate students have been The Netherlands visit EURANDOM for working at EURANDOM. About 80% of periods of one week up to several months the junior staff leaves EURANDOM for (on average 60-70 visitor weeks per year). tenured positions in academia or industry EURANDOM thus provides a stimulating (11 so far in The Netherlands). At any environment to enhance collaboration time, about 25 junior researchers are work- within Europe. ing at EURANDOM. EURANDOM participates in an EC Vacancies at EURANDOM may occur at Thematic Network (Pro-Enbis), two EC any time during the year. The opening of Networks of Excellence (PASCAL and positions is not restricted by deadlines, and EURO-NGI), a Dutch-German Bilateral Junior staff and long-term visitors have an depends on the number of junior staff per Research Group (BRG), and chairs an ESF office with computer facilities and are programme at a given moment of time. Network (RDSES). EURANDOM has assisted at various levels, such as finding Candidates who wish to apply for an exter- accommodation and arranging a visa and nal research grant are welcome. The insti- work permit. There is a small in-house tute offers assistance in the application library, carrying most international sto- procedure. Post-doctoral appointments are chastic science journals. Extensive library typically for two years, though shorter facilities are available on campus, much of periods are possible as well. Graduate stu- it on-line. dents are appointed for three to four years. The TU/e acts formally as employer of The EURANDOM visitor programme is EURANDOM personnel; therefore all staff another important element that makes the can use the facilities on campus (library, institute into a lively place. Senior and sporting, restaurant, etc.). junior researchers come from all over the world, either on invitation or upon request, EURANDOM looks forward to welcoming and participate in the various activities. you at the institute. Come and share with Workshop topics are chosen by the scien- us the “feeling of sparkling Champagne”. tific advisors and by the steering commit- For more detailed information on the insti- tees of the research programmes. Ideas for tute and its current activities, visit our topics are collected from post-doctorates website: http://www.euran dom.tue.nl. EMS June 2004 29 NEWS AbelAbel PrizePrize 20042004 toto AtiyahAtiyah andand SingerSinger

The Norwegian Academy of Science and Letters has decided to award the theorem has had innumerable applica- Abel Prize for 2004 jointly to tions, first within mathematics and Sir Michael Francis Atiyah, University of Edinburgh then, beginning in the late 70's, in the- and oretical physics: gauge theory, instan- Isadore M. Singer, Massachusetts Institute of Technology tons, monopoles, string theory, the the- ory of anomalies, etc. "for their discovery and proof of the index theorem, bringing together topol- At first, the applications in physics ogy, geometry and analysis, and their outstanding role in building new came as a complete surprise to both bridges between mathematics and theoretical physics." the mathematics and physics commu- of the surrounding space. nities. Now the index theorem has A simple case is illustrated by a become an integral part of their cul- famous paradoxical etching of M.C. tures. Atiyah and Singer, together and Escher, "Ascending and Descending", individually, have been tireless in their where the people, going uphill all the attempts to explain the insights of time, still manage to circle the castle physicists to mathematicians. At the courtyard. The index theorem would same time, they brought modern dif- have told them this was impossible! ferential geometry and analysis as it The Atiyah-Singer index theorem applies to quantum field theory to the was the culmination and crowning attention of physicists and suggested I.Singer, M. Atiyah and King Harald achievement of a development of ideas new directions within physics itself. during the Prize Ceremony more than 100-years old, from This cross-fertilization continues to be (Photo: Knut Falch/Scanpix) Stokes's theorem, which students learn fruitful for both sciences. in calculus classes, to sophisticated Michael Francis Atiyah and Isadore The second Abel Prize has been modern theories like Hodge's theory of M. Singer are among the most influen- awarded jointly to Michael Francis harmonic integrals and Hirzebruch's tial mathematicians of the last century, Atiyah and Isadore M. Singer. The signature theorem. and are still working. Over a period of Atiyah-Singer index theorem is one of The problem solved by the Atiyah- 20 years they worked together on the the great landmarks of twentieth cen- Singer theorem is truly ubiquitous. In index theorem and its ramifications, tury mathematics, profoundly influ- the 40 years since its discovery, the and with their theorem they changed encing many of the most important later developments in topology, differ- ential geometry and quantum field the- ory. Its authors, both jointly and indi- vidually, have been instrumental in repairing a rift between the worlds of pure mathematics and theoretical par- ticle physics, initiating a cross-fertil- ization that has been one of the most exciting developments of the last decades. We describe the world by measuring quantities and forces that vary over time and space. The rules of nature are often expressed by formulas involving their rates of change, so-called differ- ential equations. Such formulas may have an "index", the number of solu- tions of the formulas minus the num- ber of restrictions that they impose on the values of the quantities being com- puted. The index theorem calculated Wreath ceremony at the this number in terms of the geometry Abel monument, Oslo 30 EMS June 2004 NEWS the landscape of mathematics. Academy of Art and Sciences, the nent positions, among them the highly Atiyah and Singer came originally American Philosophical Society and prestigious Savilian Chair of from different fields of mathematics - the National Academy of Sciences Geometry at Oxford and that of Master Atiyah from algebraic geometry and (NAS). He served on the Council of of Trinity College, Cambridge. Atiyah topology, Singer from analysis. Their NAS, the Governing Board of the has also been professor of mathemat- main contributions in their respective National Research Council, and the ics at the Institute for Advanced Study areas are also highly recognized. White House Science Council. Singer in Princeton. Atiyah's early work on meromorphic was vice president of the American Atiyah rejuvenated British mathe- forms on algebraic varieties, and his Mathematical Society from 1970 to matics during his years at Oxford and important 1961 paper on Thom com- 1972. Cambridge. He was also the driving plexes, are such examples. Atiyah's In 1992 Singer received the force behind the creation of the Isaac pioneering work together with American Mathematical Society's Newton Institute for Mathematical Friedrich Hirzebruch on the develop- Award for Distinguished Public Sciences in Cambridge and became its ment of the topological analogue of Service. The citation recognized his first director. Atiyah is now retired Grothendieck's K-theory, had numer- "outstanding contribution to his pro- and an honorary professor at the ous applications to classical problems fession, to science more broadly and to University of Edinburgh. of topology and turned out later to be the public good." Michael Francis Atiyah has deeply connected with the index theo- Among the other awards he has received many honours during his rem. received are the Bôcher Prize (1969) career, including the Fields Medal Singer, jointly with Richard V. and the Steele Prize for Lifetime (1966). He was elected a Fellow of the Kadison, initiated the subject of trian- Achievement (2000), both from the Royal Society in 1962, at the age of gular operator algebras. Singer's name American Mathematical Society, the 32. He was awarded the Royal Medal is also associated with the Ambrose- Eugene Wigner Medal (1988), and the of the Society in 1968 and its Copley Singer holonomy theorem and the National Medal of Science (1983). Medal in 1988. Atiyah was president Ray-Singer torsion invariant. Singer, When Singer was awarded the Steele of the Royal Society from 1990 to together with Henry P. McKean, point- Prize, his response, published in the 1995. Atiyah has also served as presi- ed out the deep geometrical informa- Notices of the AMS, was: "For me the dent of the London Mathematical tion hidden in heat kernels, a discov- classroom is an important counterpart Society (1974 - 1976) and has played ery that had great impact. to research. I enjoy teaching under- an important role in the shaping of graduates at all levels, and I have a today's European Mathematical Isadore M. Singer was born in Detroit host of graduate students, many of Society (EMS). in 1924, and received his undergradu- whom have ended up teaching me Atiyah was responsible for the ate degree from the University of more than I have taught them." Singer founding of the Inter-Academy Panel has also written influential textbooks that brought together many of the that have inspired generations of worlds academies of science. The mathematicians. Inter-Academy Panel has now been permanently established and will play Michael Francis Atiyah was born in a major role in the integration of sci- London in 1929. Atiyah got his B.A. entific policy throughout the world. and his doctorate from Trinity College, Atiyah also instigated the formation of the Association of European Academies (ALLEA). Atiyah has been president of Pugwash Conferences on Science and World Affairs. Among the prizes he has received are the Feltrinelli Prize from the Accademia Nazionale dei Lincei (1981) and the King Faisal International Prize for Science (1987). Michigan in 1944. After obtaining his Michael Francis Atiyah was knighted Ph.D. from the University of Chicago in 1983 and made a member of the in 1950, he joined the faculty at the Order of Merit in 1992. Massachusetts Institute of Technology (MIT). Singer has spent most of his professional life at MIT, where he is Cambridge. He spent the greatest part The Newsletter plans to publish an currently an Institute Professor. of his academic career in Cambridge interview with the laureates in its next Singer is a member of the American and Oxford. He has held many promi- issue. EMS June 2004 31 NEWS ICME-10ICME-10 andand ICMIICMI awardsawards

budget, but not that bad: 484 from Artigue of the University Paris 7. the Nordic Countries, 590 from ICMI is proud to announce the the rest of Europe and 1032 from first awardees of the Klein and the rest of the world including 355 Freudenthal Medals. from the USA. If you want the The Felix Klein Medal for 2003 exact number of participants from is awarded to Guy Brousseau, your country check at www.icme- Professor Emeritus of the 10.dk. University Institute for Teacher The programme will be as Education of Aquitaine in planned in the second announce- Bordeaux, for his lifetime devel- ment. opment of the theory of didactic situations, and its applications to The ICMI Felix Klein and Hans the teaching and learning of math- Freudenthal medals for 2003 ematics. The International Commission on The Hans Freudenthal Medal Mathematical Instruction (ICMI), for 2003 is awarded to Celia The 10th International Congress founded in Rome in 1908, has, for Hoyles, Professor at the Institute on Mathematics Education, the first time in its history, estab- of Education of the University of ICME-10, in Denmark this sum- lished prizes recognizing out- London, for her seminal research mer, has got the following visions: standing achievement in mathe- on instructional uses of technology matics education research. The in mathematics education. • A well organised congress cele- Felix Klein Medal, named for the Citations of the work of these brating the ICME tradition. first president of ICMI (1908- medalists can be found below. • A congress with a strong scien- 1920), honours a lifetime achieve- Presentation of the medals, and tific programme representing ment. The Hans Freudenthal invited addresses of the medalists, the state-of-the-art in both Medal, named for the eighth pres- will occur at ICME-10 in mathematics education research ident of ICMI (1967-1970), recog- Copenhagen, July 4-11, 2004. and teaching practices. nizes a major cumulative program • A strong Nordic representation of research. These awards are to Citation for the 2003 ICMI amongst participants. be made in each odd numbered Felix Klein Medal to Guy • A large representation of math- year, with presentation of the Brousseau ematics teachers from all edu- medals, and invited addresses by The first Felix Klein Award of the cational levels. the medallists at the following Internal Commission on • A congress that makes a differ- International Congress on Mathematical Instruction (ICMI) ence to the practice of teaching Mathematical Education (ICME). is awarded to Professor Guy mathematics. These awards, which pay tribute Brousseau. This distinction recog- • A gender balanced congress. to outstanding scholarship in nises the essential contribution • A congress that gives space and mathematics education, serve not Guy Brousseau has given to the time for informal discussions only to encourage the efforts of development of mathematics edu- among the participants. others, but also to contribute to the cation as a scientific field of • A financially balanced con- development, through the public research, through his theoretical gress. recognition of exemplars, of high and experimental work over four standards for the field. The decades, and to the sustained The last vision might need some awards represent the judgment of effort he has made throughout his further development. Two months an (anonymous) jury of distin- professional life to apply the fruits before the congress the number of guished scholars of international of his research to the mathematics registrations is a little below the stature, chaired by Prof. Michèle education of both students and 32 EMS June 2004 NEWS teachers. when the dominant vision was COREM provided an original Born in 1933, Guy Brousseau cognitive, strongly influenced by organisation of the relationships began his career as an elementary the Piagetian epistemology, he between theoretical and experi- teacher in 1953. In the late sixties, stressed that what the field needed mental work. after graduating in mathematics, for its development was not a Guy Brousseau is not only an he entered the University of purely cognitive theory but one exceptional and inspired Bordeaux. In 1986 he earned a allowing us also to understand the researcher in the field, he is also a ‘doctorat d’état,’ and in 1991 social interactions between stu- scholar who has dedicated his life became a full professor at the dents, teachers and knowledge that to mathematics education, tireless- newly created University Institute take place in the classroom and ly supporting the development of for Teacher Education (IUFM) in condition what is learned by stu- the field, not only in France but in Bordeaux, where he worked until dents and how it can be learned. many countries, supporting new 1998. He is now Professor This is the aim of the theory of doctoral programs, helping and Emeritus at the IUFM of didactic situations, which has pro- supervising young international Aquitaine. He is also Doctor gressively matured, becoming the researchers (he supervised more Honoris Causa of the University of impressive and complex theory than 50 doctoral theses), contribut- Montréal. that it is today. To be sure, this was ing in a vital way to the develop- a collective work, but each time ment of mathematical and didactic there were substantial advances, knowledge of students and teach- the critical source was Guy ers. He has been until the nineties Brousseau. intensely involved in the activities This theory, visionary in its inte- of the CIEAEM (Commission gration of epistemological, cogni- Internationale pour l’Etude et tive and social dimensions, has l’Amélioration de l’Enseignement been a constant source of inspira- des Mathématiques) and he was its tion for many researchers through- secretary from 1981 to 1984. At a out the world. Its main constructs, national level, he was deeply such as the concepts of adidactic involved in the experience of the and didactic situations, of didactic IREMs (Research Institutes in contract, of devolution and institu- Mathematics Education), from tionalization have been made their foundation in the late sixties. widely accessible through the He had a decisive influence on the translation of Guy Brousseau’s activities and resources these insti- principal texts into many different tutes have developed for promot- Guy Brousseau languages and, more recently, the ing high quality mathematics publication by Kluwer in 1997 of training of elementary teachers for From the early seventies, Guy the book, ‘Theory of didactical sit- more than 30 years. Brousseau emerged as one of the uations in mathematics - 1970- leading and most original 1990’. Citation for the 2003 ICMI researchers in the new field of Although the research Guy Freudenthal Medal to Celia mathematics education, convinced Brousseau has inspired currently Hoyles on the one hand that this field must embraces the entire range of math- The first Hans Freudenthal Award be developed as a genuine field of ematics education from elemen- of the International Commission research, with both fundamental tary to post-secondary, his major on Mathematical Instruction and applied dimensions, and on contributions deal with the ele- (ICMI) is awarded to Professor the other hand that it must remain mentary level, where they cover Celia Hoyles. This distinction close to the discipline of mathe- all mathematical domains from recognises the outstanding contri- matics. His notable theoretical numbers and geometry to proba- bution that Celia Hoyles has made achievement was the elaboration bility. Their production owes to research in the domain of tech- of the theory of didactic situa- much to a specific structure – the nology and mathematics educa- tions, a theory he initiated in the COREM (Center for Observation tion, both in terms of theoretical early seventies, and which he has and Research in Mathematics advances and through the develop- continued to develop with unfail- Education) – that he created in ment and piloting of national and ing energy and creativity. At a time 1972 and directed until 1997. international projects in this field, EMS June 2004 33 NEWS aimed at improving through tech- major theoretical advances in the designed workplace training using nology the mathematics education field, such as the notions of web- a range of new media. of the general population, from bing and situated abstraction, In recent years Celia Hoyles has young children to adults in the ideas that are well known to become increasingly involved in workplace. researchers irrespective of the spe- working alongside mathematicians cific technologies they are study- and teachers in policy-making. ing. She was elected Chair of the Joint From the mid nineties, her Mathematical Council of the U.K. research on technology integrated in October 1999 and she is a mem- the new possibilities offered by ber of the Advisory Committee on information and communication Mathematics Education (ACME) technologies as well as the new that speaks for the whole of the relationships children develop mathematics community to the with technology. She has recently Government on policy matters co-directed successively two pro- related to mathematics, from pri- jects funded by the European mary to higher education. In Union: the Playground project in 2002, she played a major role in which children from different ACME’s first report to the countries designed, built and Government on the Continuing shared their own video games, and Professional Development of the current WebLabs project, Teachers of Mathematics, and con- which aims at designing and eval- tributed to the comprehensive Celia Hoyles uating virtual laboratories where review of 14-19 mathematics in children in different countries the UK. In recognition of her con- Celia Hoyles studied mathemat- build and explore mathematical tributions, Celia has recently been ics at the University of and scientific ideas collaboratively awarded the Order of the British Manchester, winning the Dalton at a distance. As an international Empire for “Services to prize for the best first-class degree leader in the area of technology Mathematics Education”. in Mathematics. She began her and mathematics education, she Celia Hoyles belongs to that career as a secondary teacher, and was recently appointed by the special breed of mathematics edu- then became a lecturer at the ICMI Executive Committee as co- cators who, even while engaging Polytechnic of North London. She chair of a new ICMI Study on this with theoretical questions, do not entered the field of mathematics theme. lose sight of practice; and recipro- education research, earning a However, Celia Hoyles’ contri- cally, while engaged in advanc- Masters and Doctorate, and bution to research in mathematics ing practice, do not forget the became Professor of Mathematics education is considerably broader lessons they have learned from Education at the Institute of than this focus on technology. theory and from empirical Education, University of London Since the mid nineties, she has research. Celia Hoyles’ commit- in 1984. been involved in two further major ment to the improvement of math- Her early research in the area of areas of research. The first, a ematics education, in her country technology and mathematics edu- series of studies on children’s and beyond, can be felt in every cation, like that of many understanding of proof, has pio- detail of her multi-faceted, diverse researchers, began by exploring neered some novel methodological professional activity. Her enthusi- the potential offered by Logo, and strategies linking quantitative and asm and vision are universally she soon became an international qualitative approaches that include admired by those who have been leader in this area. Two books pub- longitudinal analyses of develop- in direct contact with her. It is lished in 1986 and 1992 (edited) ment. The second area has thanks to people like Celia Hoyles, attested to the productivity of her involved researching the mathe- with a clear sense of mission and research with Logo. This was fol- matics used at work and she now the ability to build bridges lowed, in 1996, by the publication co-directs a new project, Techno- between research and practice of Windows on Mathematical Mathematical Literacies in the while contributing to both, that the Meanings: Learning Cultures and Workplace, which aims to develop community of mathematics educa- Computers, co-authored with this research by implementing and tion has acquired, over the years, a Richard Noss, which inspired evaluating some theoretically- better-defined identity. 34 EMS June 2004 PERSONAL COLUMN PersonalPersonal columncolumn Please send information on mathematical awards and deaths to the editor.

(Edinburgh, UK) and to Isadore M. Singer (MIT, USA) for their AWARDS discovery and proof of the index theorem. More information can be found in this issue of the Newsletter.

In November 2003, the African Mathematics Union (AMU) and The 2004 Gödel Prize for outstanding papers in the area of theo- the International Conference of Mathematical Sciences presented retical computer science, sponsored jointly by the European medals to a group of promising young African Mathematicians. Association for Theoretical Computer Science (EATCS) and the The winners are: Special Interest Group on Algorithms and Computing Theory of Olatunde Akiulade (Nigeria; mathematical physics) the Association of Computing Machinery (ACM-SIGACT), has Bamidele Awojoyogbe (Nigeria; mathematical physics) been awarded to Maurice Herlihy (Brown Univ., RI, USA), Nir Abba Gumel (Nigeria, now Canada; applied mathematics) Shavit (Sun Labs, USA), Michael Saks (Rutgers, NJ, USA) and Oluwole D. Makinde (Nigeria, now South Africa; applied math- Fotios Zaharoglou (Univ. CA, San Diego, USA) for important ematics) papers on the topological structure of asynchronous computability. Guy M. Kniet (Gabon; statistics) James A. Oguntuase (Nigeria, now Italy; pure mathematics) Former Newsletter Editor Robin Wilson (Open University, UK) has been appointed Gresham Professor of Geometry for the years The French Academy of Sciences has awarded several prizes in 2004-07. Congratulations! mathematics for 2003: Claire Voisin (Paris) has been awarded the Prix Sophie Germain; Louis Boutet de Monvel (Paris) received Erratum: the Prix Fondée par l’État; The Prix Jacques Hadamard was award- The editor wishes to apologize for an incorrect statement in issue ed to Wendelin Werner (Paris-Sud); Claude Bardos was award- 50 about awards given in Poland in 2003. The corrected ed the Prix Marcel Dassault; Gilles Lebeau (Nice) received the announcements are: Prix Ampère de l’Électricité de France; The Prix Gabrielle Sand et Micha³ Szurek (Warsaw) has received the Dickstein Great Prize M. Guido Triossi was awarded to Damien Gaboriau (CNRS) and of the Polish Mathematical Society for his achievements in popu- Jean-Marc Delort (Paris-Nord) received the Prix Langevin. larizing mathematics. Pawe³ Wilczyñski (Kraków) has been awarded the first The 2003 Fermat Prize for Mathematical Research has been Marcinkiewicz Prize of the Polish Mathematical Society for stu- awarded to Luigi Ambrosio (Scuola Normale Superiore) for his dents’ research papers. contributions to the calculus of variations and geometric measure theory and their relations with partial differential equations. DEATHS Two scientists were awarded the Max Planck Research Award 2003 in Mathematics from the Max Planck Society: Stephan Luckhaus (Leipzig, Germany) for his trail blazing model We regret to announce the deaths of: showing clear cut solutions for the transport of water and other substances in soil. These solutions can also describe the growth of Martine Babillot (July 2003) tumour cells in healthy tissue, which, up until now, was only described theoretically; and Andrey Andreevich Bolibruch (11 November 2003) Wolfgang Lück (Münster, Germany) for his landmark research on algebraic topology. Armand Borel (11 August 2003)

In 2003 the first John von Neumann Awards were presented to David Fowler (13 April 2004) Marina von Neumann Whitman, Charles Simony and to George Dyson. Miguel de Guzman (14 April 2004)

Mikhael L.Gromov (IHES, Bures-sur-Yvettes, France) and Jay John Hammersley (2 May 2004) Gould (Courant Institute, NYU, NY, USA) have been named recipients of the 2003-2004 Frederic Esser Nemmers Prize in Hans Hermes (10 November 2003) Mathematics. Eckehart Hotzel (28 September 2003) The Institut de Recherche Mathématique Avancée (IRMA) has created an annual prize in memory of the mathematician Paul Martin Kneser (16 February 2004) André Meyer (1934-2003). The prize honours an outstanding young probabilist, working in the field of stochastic processes. John Lewis (21 January 2004) The first IRMA prize in Memory of Paul André Meyer was award- ed in February 2004 to Thomas Duquesne (Paris-Sud). Hans-Joachim Nastold (26 January 2004)

The 2004 Emil Artin Junior Prize in Mathematics has been award- Peter Owens (30 December 2003) ed to Gurgin R. Asatryan (Yerevan, Armenia). Paul Wauters (26 October 2003) The Norwegian Academy of Science and Letters has awarded the second Abel Prize (2004) jointly to Sir Michael Atiyah Heiner Zieschang (5 April 2004) EMS June 2004 35 CONFERENCES fundamental groups, equivariant families, non-archimedean uniformization, and group theoretic aspects Keynote participants include: A. Abbes; ForthcomingForthcoming conferencesconferences P. Bradley; T. Chinburg; R. Guralnick; D. Harbater; F. Kato; S. Maugeais; R. Pries; M. Raynaud; N. Stalder; compiled by Vasile Berinde (Baia Mare, Romania) S. Wewers; M. Zieve Information: e-mail: [email protected] or [email protected] Web site: http://pascal.lc.leidenuniv.nl/ Please e-mail announcements of European J.-I. Itoh (Kumamoto Univ., Japan); conferences, workshops and mathematical S. Marcus (Univ. of Bucuresti, Romania); 16-21: Bunyakovsky International meetings of interest to EMS members, to J. Myjak (Univ. dell’Aquila , Italy) Conference, Kyiv, Ukraine one of the following addresses Deadline: for preliminary registration Theme: Honouring the bicentenary of [email protected] or vasile_berinde (express of interest) is April 10, 2004. B.Ya.Bunyakovsky (1804-1889), eminent @yahoo.com. Announcements should be Information: Eva Vasarhelyi, e-mail: mathematician of Tsarist Russia, born in written in a style similar to those here, and [email protected] Ukraine. The scope of the conference will sent as Microsoft Word files or as text files include the range of the problems (but not as TeX input files). Space permit- 1-15: Moonshine - the First Quarter Bunyakovsky was interested in and made ting, each announcement will appear in Century and Beyond. A Workshop on his own contribution detail in the next issue of the Newsletter to the Moonshine Conjectures and Vertex Topics: (1) mathematical analysis; (2) go to press, and thereafter will be briefly Algebras, International Centre for number theory; (3) theory of probability noted in each new issue until the meeting Mathematical Sciences, Edinburgh UK and its applications, actuarial mathematics; takes place, with a reference to the issue in Information: (4) mathematics education and mathemati- which the detailed announcement e-mail: [email protected] cal terminology appeared. Web site: http://www.ma.hw.ac.uk/icms/ Languages: English, Ukrainian, Russian meetings/2004/moonshine Organizers: Institute of Mathematics of July 2004 [For details, see EMS Newsletter 51] the National Academy of Sciences of Ukraine, Ministry of Science and 3-10: Conference on Symplectic Education of Ukraine, National Technical June 26 - July 1: 7th International Topology ECM Satellite Conference University of Ukraine “KPI”, Vinnytsya Conference of The Mathematics Stare Jab³onki, Poland National Technical University, Education into the 21st Century Project, Information: T. Shevchenko Kyiv National University, Ciechocinek, Poland. e-mail: [email protected] Ukrainian Mathematical Society Information: Web site: http://symp.univ.szczecin.pl Programme committee: Y. Berezansky; e-mail: [email protected] [For details, see EMS Newsletter 51] O. Boholyubov; V. Vyshensky; Web site (for previous conferences only): O. Hannyushkin; M. Gorbachuk; http://math.unipa.it/~grim/21project.htm 15-23: EMS Conference at Bêdlewo M. Zgurovsky; V. Klochko; V. Korolyuk; [For details, see EMS Newsletter 50] (Poland). Analysis on metric measure V. Melnyk; B. Mokin; M. Portenko; spaces. A. Yurachkivsky; M. Yadrenko (all from 27 June - 2 July: 4th European Information: Ukraine); L. Brylevskaya (St-Petersburg, Congress of Mathematics, Stockholm, Web site: http://impan.gov.pl/Bedlewo/ Russia); E. Seneta (Sydney, Australia), Sweden. R. Andrushkiw (New York, USA) Information: 26-31: 6th World Congress of the Organising committee: A. Samoilenko Web site: http://www.math.kth.se/4ecm Bernoulli Society and the 67th Annual (Chairman); M. Gorbachuk and [For details, see also EMS Newsletter 50, Meeting of the Institute of Mathematical M. Swirzhevsky (Vice-Chairmen); H. Syta 51] Statistics, Barcelona (Spain) (Secretary); V. Boyko; Y. Vynnyshyn; Information: I. Yegorchenko; T. Karataeva; O. Magda; June 30-July 2: Convexity and Discrete e-mail: [email protected] V. Ostrovskyi; N. Ryabova; S. Spichak - Geometry. A conference on the occasion Web site: http://www.imub.ub.es/events/ Institute of Mathematics of the NASU; of the sixtieth birthday of Tudor wc2004/ N. Pankratova - National Technical Zamfirescu [For details, see EMS Newsletter 49] University of Ukraine “KPI”; V. Grabko - Theme: This conference will reflect Tudor Vinnytsya National Technical University Zamfirescu’s lifelong passion for geome- August 2004 Location: Kyiv, Institute of Mathematics try, his broad range of mathematical inter- of National Academy of Sciences of ests, and his commitment to students and Ukraine Tereshchenkivska Str.,3 01601, young mathematicians 16-20 Workshop “Automorphisms of Kyiv-4, Ukraine Location: Eötvös Loránd University, Curves”, Lorentz Center, Leiden, The Note: From August 20 to 21, conference Faculty of Science, Pázmány Péter sétány Netherlands. participants have the opportunity to take 1/a H-1117 Budapest, Hungary Organizers: Gunther Cornelissen and part in a two days’ after-conference excur- Organizers: K. Bezdek (Univ. of Calgary Frans Oort (Universiteit Utrecht) sion to Bar, the native place of Viktor & Eötvös Univ.); K. Böröczky (Eötvös Aim: We will discuss the latest develop- Bunyakovsky, and to Vinnytsya, staying Univ.); A. Heppes (Eötvös Univ.); Éva ments in the theory of group actions on the night in the City Hotel in Vinnytsya Vásárhelyi (Eötvös Univ.) algebraic curves, a field that has been lift- Deadline: for abstracts May 15, 2004 Confirmed Speakers: K. Bezdek (Univ. ed to a new level of sophistication in recent Information: of Calgary, Canada); P. Gritzmann (Tech. years Web site: http://www.imath.kiev.ua/~syta/ Univ. of Munich, Germany); P.M. Gruber Topics: deformation theory, lifting to char- bunyak (Tech. Univ. of Vienna, Austria); acteristic zero, relation to moduli of A. Heppes (Eötvös Univ., Hungary); curves, patching method, techniques from 18-21: The Thirteenth International 36 EMS June 2004 Workshop on Matrices and Statistics, in (Limoges) Tel. +39-55-434975 / +39-55-4237111 Celebration of Ingram Olkin’s 80th Local Organizing Committee: FAX +39-55-434975 / +39-55-4222695 Birthday, Bêdlewo, near Poznañ, Poland V. Radulescu (chairman), D. Busneag, [For details, see EMS Newsletter 51] Information: N. Constantinescu, M. Cosulschi, e-mail: [email protected] D. Ebanca, M. Gabroveanu, M. Ghergu, 20-24: 12th French-German-Spanish Web site: http://matrix04.amu.edu.pl/ A. Matei, M. Preda, N. Tandareanu, Conference on Optimization Avignon, [For details, see EMS Newsletter 50] C. Vladimirescu, I. Vladimirescu. France Invited Sessions: Information: 23-September 2: International 1) Contrôle des systemes gouvernés par e-mail: [email protected]; Conference-School on Geometry and des équations aux dérivées partielles, orga- Web site: http://www.fgs2004.univ-avig Analysis dedicated to the 75th anniver- nized by Marius Tucsnak (Nancy I) non.fr sary of Academician Yu. G. Reshetnyak, 2) Homogénéisation et applications aux [For details, see EMS Newsletter 50] Novosibirsk, Russia sciences des matériaux, organized by Horia Information: contact Sergei Vodopyanov Ene (Bucharest) 23-26: 4th International Conference on (Chairman of the Organizing 3) Mathématiques financieres, organized Applied Mathematics (ICAM-4), Baia Committee), e-mail: [email protected] by Radu Tunaru (London) Mare, Romania (previous editions in Web site: http://www.math.nsc.ru/confer 4) Biomécanique, organized by Marc 1998, 2000 and 2002) ence/ag 2004/indengl.htm Thiriet (Paris) Information: e-mail: [email protected]; [For details, see EMS Newsletter 51] 5) Inégalités et applications, organized by [email protected] Constantin Niculescu (Craiova) Web site: http://www.ubm.ro/site-ro/facul 24-27: International Conference on Information: To participate in the tati/departament/manifestari/icam4/index. Nonlinear Operators, Differential Colloquium it is necessary to fill out the html Equations and Applications (ICN- Registration Form and send it to the Local [For details, see EMS Newsletter 49] ODEA-2004), Cluj-Napoca, Romania Organizing Committee. Information: e-mail: [email protected] Abstracts of conference lectures and com- 26-October 1: Potential theory and cluj.ro munications will have been published by related topics, Hejnice, Czech Republic Web site: http://www.math.ubbcluj.ro/ the start of the Colloquium. The Aim: to introduce progress, prospects and ~mserban/confan.htm Organizing Committee plans to publish the perspectives of the field [For details, see EMS Newsletter 49] Proceedings of the Colloquium in the local Scope: various aspects of potential theory mathematical journal, Annals of the and related fields 30 - September 3: 7th French-Romanian University of Craiova, Series Mathematics Main speakers: L. Beznea (Romania); Colloquium in Applied Mathematics, and Informatics. Additional information A. O’Farrell (Northern Ireland); Craiova (Romania) about the Colloquium may be obtained at: S. Gardiner (Northern Ireland); W. Hansen The previous editions of this Colloquium http://www.inf.ucv.ro/colloque2004 (Germany); O. Martio Finland); were organized in Iasi (1992), ENS-Paris Contacts: M. Basarab, Laboratoire M. Rockner (Germany) (1994), Cluj (1996), Metz (1998), Jacques-Louis Lions, Université Paris 6, Note: During the conference there will be Constantza (2000), and Perpignan (2002). 175 rue du Chevaleret, 75013 Paris, a chance to congratulate our colleague Ivan The present edition is organized by the France. Netuka on the occasion of his birthday. His Department of Mathematics of the Faculty e-mail: [email protected] works will briefly be described by Jurgen of Mathematics and Informatics, V. Radulescu, Department of Mathematics, Bliedtner (Germany) University of Craiova (Romania), under University of Craiova, 200 585 Craiova, Format: invited and contributed talks the patronage of the Institute of Statistics Romania, e-mail: [email protected] Sessions: only plenary section of the Romanian Academy, Université Organizers: Technical University Liberec Pierre et Marie Curie (Paris 6), and the September 2004 and other institutions Société de Mathématiques Appliquées et Organizing committee: M. Brzezina Industrielles (SMAI). (Technical Univ. Liberec); Scientific Committee: D. Cioranescu 3-5: EMS-Czech Union of mathemati- M. Dont (Czech Technical Univ., Prague); (chairman, Paris 6), M. Iosifescu (chair- cians and Physicists (Mathematics K. Janssen (Heinrich-Heine-Univ., man, Romanian Academy), V. Radulescu Research Section). Mathematical Dusseldorf); J. Lukes (Charles Univ., (chairman, Craiova), V. Barbu (Iasi), Weekend, Prague (Czech Republic). Praha); D. Medkova (Czech Academy of S. Basarab (Bucharest), F. Bethuel (Paris Information: Web site: Sciences, Prague); Karl-Theodor Sturm 6), H. Brezis (Paris 6), P. Deheuvels (Paris http://mvs.jcmf.cz/emsweekend/ (Rheinische Friedrich-Wilhelms-Univ., 6), J.-M. Deshouillers (Bordeaux), Bonn); Jiri Vesely (Charles Univ., Praha) C. Duhamel (Paris), D. Gaspar 8-11: Dixiemes journees montoises d’in- Proceedings: will not be published (Timisoara), U. Herkenrath (Duisburg), formatique theorique, a Liege (Tenth Location: Hejnice, Czech Republic I. Ionescu (Chambéry), Y. Maday (Paris 6), Mons theoretical computer science days, Information: C. Niculescu (Craiova), A. Rascanu (Iasi), in Liege) e-mail: [email protected] I.A. Rus (Cluj), G. Salagean (Cluj) Information: e-mail: [email protected]; Web site: http://www.karlin.mff.cuni.cz/ Invited Speakers: V. Bally (Paris), Web site: www.jm2004.ulg.ac.be PTRT04 C. Bandle (Basel), J.-Y. Chemin (Paris), [For details, see EMS Newsletter 50] Ph. G. Ciarlet (Paris and Hong Kong), 29-October 2: XII Annual Congress of A. Damlamian (Paris), G. Dinca 13 – 18: 3º CIME Course - Stochastic the Portuguese Statistical Society, (Bucharest), C. Faciu (Bucharest), Geometry, Martina Franca, Taranto, Evora, Portugal O. Goubet (Amiens), G. Haiman (Lille), Italy Theme: all subjects in Probability and D. Iftimie (Lyon), P. Jebelean (Timisoara), Information: e-mail: [email protected] Statistics and their applications C. Le Bris (Paris), B. Miara (Noisy-le- Web site: http: //www.math.unifi.it/CIME Aim: to promote the exchange of ideas and Grand), S. Micu (Craiova), G. Nenciu Address: Fondazione C.I.M.E. c/o the spread of recent scientific develop- (Bucharest), D. Polisevschi (Bucharest), Dipart.di Matematica “U. Dini” ments R. Precup (Cluj), T. Ratiu (Lausanne), Viale Morgagni, 67/A - 50134 FIRENZE Main speakers: L. Centeno (Portugal); M. Sofonea (Perpignan), M. Théra (ITALY) D. Cox (UK); B. Efron (USA); J. Mexia EMS June 2004 37 (Portugal); D. Peña (Spain); J. Branco Normale Supérieure, Paris); D. Kroener Catania, Italy); O. Sánchez (Univ. de (Portugal; tutorial on Cluster Analysis) (Univ. Freiburg); S. Noelle (RWTH Granada, Spain); A. Santos (Universidad Format: invited conferences, contributed Aachen); L. Pareschi (Univ. di Ferrara); de Extremadura, Spain); H.J. Schroll talks, tutorial M. Pulvirenti (Univ. di Roma); J. Soler (Lund Univ., Sweden); S. Serna (Univ. de Sessions: plenary conferences, 3 or 4 par- (Univ. de Granada); C. Villani (École Valencia, Spain); B. Sjogreen (KTH allel sessions, poster sessions, tutorial Normale Supérieure, Lyon) Stockholm, Sweden); M. Torrillhon Languages: Portuguese and English Speakers: E. Caglioti (Univ. di Roma I, (ETHZ, Switzerland); G. Toscani (Univ. di Call for papers: if you wish to present a Italy); B. Despres (Univ. Paris VI, France); Pavia, Italy); J.J.L. Velázquez (Univ. contributed presentation, please send an L. Desvilletes (ENS Cachan, France); Complutense de Madrid, Spain) abstract before 15 May 2004 to the e-mail: F. Filbet (Univ. d'Orléans, France); Oral Presentations and Posters: There [email protected], indicating keywords J.A. Font (Univ. de Valencia, Spain); are 10 free time slots of 25' for oral pre- and the preferred form of presentation (talk A. Goldshtein (Technion Haifa, Israel); sentations. Also, there will be a poster ses- or poster). Maximum one A4 page in Word L. Gosse (IAC Bari, Italy); T. Goudon sion throughout the whole conference. We format with a 12 font and 1.5 line spacing (Univ. des Sciences et Technologies Lille would appreciate your registration as soon Organizers: Portuguese Statistical Society 1, France); C. Helzel (IAM Bonn, as possible and that a title of your presen- (SPE), University of Evora Germany); J.M. Ibánez (Univ. de Valencia, tation is sent no later than June 6th. We Executive and Scientific Committee: Spain); P.E. Jabin (ENS Paris, France); will send notification of acceptance and the F. Rosado (Lisboa, Congress President); K.H. Karlsen (Univ. of Bergen, Norway); type of your contribution, before June C. Braumann (Evora); M. E. Graça Martins D. Levermore (Univ. of Maryland, USA); 15th. (Lisboa); P. Oliveira (Coimbra); A. Mangeney (Institute de Physique du Deadlines: For application for financial C. D. Paulino (Lisboa) Globe de Paris, France); J.M. Marti (Univ. support, June 5, 2004. For registration and Local Organising Committee: de Valencia, Spain); C. Mouhot (ENS payment, June 30, 2004 C. Braumann; R. Alpizar-Jara; F. Mendes; Lyons, France); S. Osher (UCLA, USA); Information: M. Oliveira T. Poeschel (Humboldt-Universität - e-mail: [email protected] Proceedings: refereed, to be published Charité, Germany); S. Rjasanow (Univ. Web site: http://www.crm.es/Complex after the Congress (assuming, as it hap- Saarland, Germany); G. Russo (Univ. di Flows pened in all the previous 11 Congresses, that adequate financial support will be available) Location: Evora Hotel (Evora), special accommodation fares at the Hotel for par- ticipants and accompanying persons Social program: dinner, excursion, other events Reduced registration fees: for students and members of the Society Notes: registration and accommodation forms can be downloaded from the web site Deadlines: for abstracts, 15 May 2004; registration with reduced fee if sent before 15 May 2004 Information: e-mail: [email protected] Web site: www.eventos.uevora.pt/spe2004 Address: Comissao Organizadora XII Congresso SPE, Departamento de Matematica, Universidade de Evora, Rua Romao Ramalho 59, 7000-671 EVORA, PORTUGAL Fax: +351-266745393; Phone: +351-266745370

October 2004

6-9: HYKE Conference on Complex Flows, Centre de Recerca Matematica, Bellaterra, Spain Aim: The main objective of the conference is to highlight new developments of either numerical or analytical nature in kinetic and hydrodynamic equations. We would like to foster the interaction with applica- tions, making an emphasis on two applica- tions: granular media and astrophysical flows, with special sessions devoted to them. Co-ordinators: J.A. Carrillo (ICREA- UAB); A. Marquina (Univ. de Valencia) Programme Committee: A.V. Bolylev (Karlstadt Univ.); F. Bouchut (École

38 EMS June 2004 BOOKS have also included the theory of characteristic classes of vector bundles. The book has two appendices (A: Proof of the Dold-Thom theo- RecentRecent booksbooks rem, B: Proof of the Bott periodicity theorem), references consisting of 83 items and a long edited by Ivan Netuka and Vladimír Souèek (Prague) list of symbols. There are many remarks and comments making the orientation of the read- Books submitted for review should be sent to and Physics, Graduate Studies in er in the field of algebraic topology easier, and the following address: Mathematics, vol. 52, American Mathematical suggesting directions for further studies. The Ivan Netuka, MÚUK, Sokolovská 83, 186 75 Society, Providence, 2002, 343 pp., US$59, reading requires rather modest prerequisites; Praha 8, Czech Republic ISBN 0-8218-2951-3 some basic point set topology and some basic information about groups. Let us mention also Y. A. Abramovich, C. D. Aliprantis: An The book is based on courses taught at the that the book had a predecessor, namely a pre- Invitation to Operator Theory, Graduate Humboldt University for students of mathe- liminary version in Spanish. (jiva) Studies in Mathematics, vol. 50, American matics and physics. Topics treated in the book Mathematical Society, Providence, 2002, 530 cover analysis, differential geometry, Lie J.P. Allouche, J. Shallit: Automatic pp., US$69, ISBN 0-8218-2146-6 groups and Lie algebras and main applications Sequences: Theory, Applications, The book is an immense graduate text devot- in mathematical physics. The main tools used Generalizations, Cambridge University ed to the basic notions of linear operator theo- throughout are differential forms and their cal- Press, Cambridge, 2003, 571 pp., £37,50 ry in Banach spaces and Banach lattices. Basic culus. They are introduced first (together with ISBN 0-521-82332-3 concepts of the theory (Banach spaces, opera- the Stokes theorem) on open subsets in Rn, The book can serve both as an introduction tors and functionals, Banach lattices, bases, then on submanifolds in Rn. The next essential into the study of automatic sequences, and as fundamentals of measure theory and basic chapter addresses Pfaffian systems. a systematic survey of known results and operator theory) are contained in preliminary Fundamental theorems on curves and surfaces applications. The topic mentioned in the title chapters. The chapters that follow these are on in R3 are discussed carefully using all the is far from being the only interesting theme of operators on AL- and AM-spaces and special advantages offered by the language of differ- the volume. Related topics range from combi- classes of operators (finite-rank operators, ential forms. To allow the reader to deal with natorics on words and formal languages multiplication operators, the Fredholm and symmetry questions, a brief introduction to through number theory and formal power strictly singular operators) and on integral Lie groups and homogeneous spaces is given. series to physics. The basic concept is defined operators. The next part of the book is devot- The last three chapters use material prepared in the fifth chapter using finite automata for- ed to spectral theory (including holomorphic earlier in the book for a discussion of basic malism. In the following chapter an equivalent functional calculus and the Riesz-Schauder applications in mathematical physics – sym- characterization in terms of fixed points of theory), positive matrices and irreducible plectic geometry and its relations to mechan- morphisms is given. Properties of automatic operators. The last two chapters also contain ics, statistical mechanics and thermodynam- sequences bring together in a natural way some recent results, published for the first ics, and finally electrodynamics. The book is results from number theory and theoretical time in the book. The first of these chapters nice to read. It is very clearly written, under- computer science, for example, questions of deals with the famous invariant subspace standable and contains a lot of pictures. Each transcendence of numbers and power series. problem, while the last one presents a com- chapter ends with many carefully chosen exer- Relations between these areas are often prehensive study of operators satisfying the cises. The book can be useful for students as neglected due to different conventions in lan- Daugavet equation. Every chapter ends with a well as for teachers preparing courses on the guage and notation. The book presents a lot of large number of exercises and problems of topics covered. (vs) results from both fields for the first time in a various levels of difficulty. There are more unified framework. Material is presented in a than 600 exercises offering useful practise to M. Aguilar, S. Gitler, C. Prieto: Algebraic clear and attractive way with motivation, exer- the reader, supplementing examples and coun- Topology from a Homotopical Viewpoint, cises, open questions, and a very rich bibliog- terexamples, or extending the theory present- Universitext, Springer, New York, 2002, 478 raphy. The presentation is self-contained in a ed in the text. Complete solutions of all exer- pp., €79,95, ISBN 0-387-95450-3 remarkable degree. The book contains a cises are given in a separate companion vol- This is a basic course on algebraic topology, review of various areas needed such as peri- ume “Problems in operator theory” (cf. the and I would like to stress at the beginning that odicity in words, subword complexity, alge- next review). The book is written in an under- it is excellently written and composed and can braic and transcendental numbers, numeration standable way and it requires only a basic be strongly recommended to anybody wishing systems, finite automata, Turing machines or knowledge of functional analysis, measure to learn the field. It is already clear from the continued fractions. Famous sequences like theory and topology, together with a suitable title that the authors adopted the homotopical Thue-Morse, Fibonacci or Rudin-Shapiro are background in real analysis. It can be warmly approach to algebraic topology. This means also studied. The book will be of use for recommended to graduate students and every- that the book is more topological and less beginners as well as for advanced students and body who wishes to become acquainted with algebraic. Important technical tools here are researchers. (šh) basic elements and deeper properties of linear infinite symmetric products and Moore operators on Banach spaces and Banach lat- spaces. For example, homology groups are F. J. Almgren, Jr.: Plateau’s Problem: An tices. (jl) defined as homotopy groups of infinite sym- Invitation to Varifold Geometry, revised edi- metric product of the space under considera- tion, Student Mathematical Library, vol. 13, Y. A. Abramovich, C. D. Aliprantis: tion. Eilenberg-Mac Lane spaces are infinite American Mathematical Society, Providence, Problems in Operator Theory, Graduate symmetric products of Moore spaces, and via 2001, 77 pp., US$19, ISBN 0-8218-2747-2 Studies in Mathematics, vol. 51, American Eilenberg-Mac Lane spaces, cohomology This is a small booklet describing germs of Mathematical Society, Providence, 2002, 386 groups are defined. Taking the approach cho- geometric measure theory and their applica- pp., US$49, ISBN 0-8218-2147-4 sen in the book into account, the authors were tions to the solution of the Plateau problem in This book contains complete solutions to well aware of the fact that they must pay spe- calculus of variation. It means that we are more than 600 exercises, given in the com- cial attention to computations. The reader can looking for a “surface” of smallest area having panion volume “An invitation to operator the- find many examples, calculations, and also a a given closed curve as its boundary. The ory” (cf. previous review) by the same number of exercises. (There is a whole chap- book starts with many examples of soap films authors. It can also be viewed as a supplemen- ter devoted to cellular homology.) However, illustrating an unexpected behaviour of solu- tary text to courses in operator theory and their we do not want to create an impression that tions. The author then introduces differential relatives. (jl) the book covers only homology and cohomol- forms and their integration over rectifiable ogy. The book deals also with K-theory and its sets. It leads in a natural way to a notion of a I. Agricola, T. Friedrich: Global Analysis: many applications, and later on, generalized current (i.e., an element of the dual to the Differential Forms in Analysis, Geometry cohomology theories are treated. The authors space of k-forms in a suitable topology) and it EMS June 2004 39 BOOKS is an inspiration for the definition of a vari- list of his publications, a long list of his doc- ters of the book contain part of a theory need- fold. The last part describes a solution of the toral students (40 altogether), together with ed in a study of Lie groups and Lie algebras. Plateau problem in the context of varifolds. the program of the conference organized in Chapter 4 is devoted to Coxeter groups and The first version of the book appeared in the Göttingen on this occasion in April 2000. Tits systems. Chapter 5 contains a discussion 60’s. The new revised edition contains an There are 17 research contributions on many of groups generated by reflections. Finally, expanded bibliography, reflecting the evolu- topics from complex geometry (often related Chapter 6 treats root systems, ending with tion of the topic. After all these years, the to work of H. Grauert) by W. Barth; T. Bauer their classification. (vs) book is still a beautiful introduction to the and his coworkers; I. Bauer, F. Catanese and Plateau problem and it is a pleasure to read. R. Pignatelli; A. Bonifant and J. Fornaess; C. O. Bratteli, P. Jorgensen: Wavelets Through (vs) Ciliberto and K. Hulek; J.P. Demailly; H. a Looking Glass, Applied and Numerical Flenner and M. Lübke; A. Huckleberry and J. Harmonic Analysis, Birkhäuser, Boston, K.T. Arasu, A. Seress, Eds.: Codes and Wolf; Y. Kawamata; S. Kebekus; T. Peternell 2002, 398 pp., €85,98, ISBN 0-8176-4280-3 Designs, Ohio State University Mathematical and A. Sommese; K. Oguiso and De-Qi The book gives a general presentation of some Research Institute Publications 10, Walter de Zhang; T. Oshawa, S. Schröer and B. Siebert; recent developments in wavelet theory, with Gruyter, Berlin, New York, 2002, 322 pp., Y.-T. Siu; E. Viehweg and K. Zuo, and by J. an emphasis on techniques that are both fun- €128, ISBN 3-11-017396-4 Wisniewski. (vs) damental and relatively timeless, having a This volume contains the proceedings of a geometric and spectral-theoretic flavour. The conference honouring Dijen K. Ray- M. C. Beltrametti, F. Catanese, C. exposition is clearly motivated and unfolds Chaudhuri on the occasion of his 65th birth- Ciliberto, A. Lanteri, C. Pedrini (Eds.): systematically, aided by numerous graphics. day held at Ohio State University in 2002. Algebraic Geometry, Walter de Gruyter, Excellent graphics show how wavelets depend Dijen Ray-Chaudhuri is a well-known combi- Berlin, 2002, 355 pp., €148, ISBN 3-11- on the spectra of the transfer operators. Some natorist who is most renowned for BCH- 017180-5 new results are presented for the first time codes, which appear in every textbook on cod- The book is based on talks given at the con- (e.g., results on homotopy of multiresolutions, ing theory. He is also an influential teacher. ference in memory of Paolo Francia, held in on approximation theory, and the spectrum of The volume contains 22 contributions, written Genova in September 2001, and it contains associated transfer and subdivision operators). mostly by his former students, and relates to invited papers dedicated to Paolo Francia. The The book is divided into six chapters. Each various aspects of algebraic combinatorics, contents cover a wide spectrum of algebraic chapter, and some sections within chapters, coding theory, tilings, block designs and geometry. The life and work of Paolo Francia open with tutorials or primer of varying graph theory. Highlights of Dijen Ray- is briefly described in the introduction, togeth- length. The tutorials are written in a style that Chaudhuri’s research are reviewed in the first er with the list of his publications. The book is much more informal. They are in fact meant contribution by A. Seress, which also includes contains 18 contributions, by L. Badescu and as friendly invitations to the topics to follow, a list of his papers. It is a nice volume hon- M. Schneider (Formal functions, connectivity with the emphasis on friendliness. Key topics ouring a very fine mathematician. (jneš) and homogeneous spaces), L. Barbieri-Viale of wavelet theory are examined: connected (On algebraic 1-motives related to Hodge components in the variety of wavelets, the S. Ariki: Representations of Quantum cycles), A. Beauville (The Szapiro inequality geometry of winding numbers, the Galerkin Algebras and Combinatorics of Young for higher genus fibrations), G. Borelli (On projection method, classical functions of Tableaux, University Lecture Series, vol. 26, regular surfaces of general type with pg = 2 Weierstrass and Hurwitz and their role in American Mathematical Society, Providence, and non-birational bicanonical map), F. describing the eigenvalue-spectrum of the 2002, 158 pp., US$31, ISBN 0-8218-3232-8 Catanese and F.O. Schreyer (Canonical pro- transfer operator, Perron-Frobenius theory, The main part of the book is devoted to a jections on irregular algebraic surfaces),C. and quadrature mirror filters. Concise back- study of representations of quantum algebras. Ciliberto and M. M.Lopes (On surfaces with ground material for each chapter, open prob- The author uses the quantum deformation of pg =2, q=1 and non-birational bicanonical lems, exercises, bibliography, and comprehen- (1) the Kac-Moody Lie algebra Ak as the main map), A. Conte, M. Marchisio and J. Murre sive index, make this work a fine pedagogical working example. The first three chapters (On unirrationality of double covers of fixed and reference source. The book also describes contain a brief review of concepts needed for degree and large dimension), A. Corti and M. important applications to signal processing, (1) a description of Ak by generators and rela- Reid (Weighted Grasmanians), T. de Fernex, communications engineering, computer tions. The next three chapters contain a dis- L. Ein (Resolution of indeterminacy of pairs) graphics algorithms, qubit algorithms and cussion of crystal bases for integrable repre- V. Guletskii and C. Pedrini (The Chow motive chaos theory, and is aimed at a broad reader- (1) sentations of Ak and a description of behav- and the Gedeaux surface), Y. Kawamata ship of graduate students, practitioners, and ior of crystal bases under tensor products. The (Francia’s flip and derived categories), K. researchers in applied mathematics and engi- existence of crystal bases is proved in chapters Konno (On the quadric hull of a canonical sur- neering. The book is also useful for other 7, 8 and 9. The proof is based on relations face) A. Langer (A note on Bogomolov’s mathematicians with an interest in the inter- between crystal bases and the Lusztig canoni- instability and Higgs sheaves), A. Lanteri and face between mathematics and communica- cal bases. The next two chapters contain a dis- R. Mallavibarrena (Jets of antimulticanonical tion theory. (knaj) cussion of the third possible approach to crys- bundle on Del Pezzo surfaces of degree < 2), tal bases, which uses a combinatorial con- M. M. Lopes and R. Pardini (A survey on the E. Bujalance, F.-J. Cirre, J.-M. Gamboa, G. 2 struction based on (multi)-Young diagrams, bicanonical map of surfaces with pg = 0 and K Gromadzki: Symmetry Types of due to Misra and Miwa. The final part of the >2), F. Russo (The antibirational involutions Hyperelliptic Riemann Surfaces, Mémoires book contains the proof of a conjecture (due to of the plane and the classifications of real Del de la Sociéte Mathématique de France, Vol Leclerc, Lascoux and Thibon) concerning the Pezzo surfaces), V. V. Shokurov (Letters of a 91, Sociéte Mathématique de France, Paris, representation theory of cyclotomic Hecke bi-rationalist: IV.Geometry of log flips), A. J. 2001, 122 pp., FRF 150, ISBN 2-85629-112-0 algebras of classical type. The book offers a Sommese, J. Verschelde and Ch. Wampler (A The aim of the publication is to present classi- nice introduction to combinatorics related rep- method for tracking singular paths with appli- fication of all real structures of hyper-elliptic resentation theories and their applications. cation to the numerical irreducible decompo- Riemann surfaces, together with their full (vs) sition). In addition the book contains informa- group of analytic and antianalytic automor- tion on the conference consisting of a list of phisms, their topological invariants, their I. Bauer, F. Catanese, Y. Kawamata, T. lectures and a list of participants. (jbu) description in terms of polynomials equations and explicit formulae for the corresponding Peternell, Y.-T. Siu, (Eds.): Complex τ Geometry, Springer, Berlin, 2002, 340 pp., N. Bourbaki: Elements of Mathematics. Lie real structures. A real structure on a €79,95, ISBN 3-540-43259-0 Groups and Lie Algebras, Chapters 4-6, Riemann surface X is an antianalytic involu- Springer, Berlin, 2002, 300 pp., €99,95, ISBN tion on X. A real form on X is a class of equiv- The book contains a collection of papers ded- τ icated to Hans Grauert on the occasion of his 3-540-42650-7 alence of modulo conjugation by elements of The book is one of the Bourbaki volumes, the full group of analytic and antianalytic 70th birthday. The reader can also find here a translated into English. The first three chap- authomorphisms. For example, if the curve X 40 EMS June 2004 BOOKS is defined by a set of polynomials with real the book treats Clifford algebras of a given ful for undergraduate students of mathematics coefficients, then the complex conjugation vector space with a non-degenerate scalar and computer science. The text is also acces- gives real structure on X. Most complex product. Embedded manifolds, corresponding sible to a more general audience interested in curves have no real forms but some have more covariant derivatives and spinor fields are computer algebra and its applications. (mer) than one. A number of connected components introduced in the second chapter. Various ver- of the fixed point set of the involution τ and sions of Dirac operators are introduced and E. von Collani (Ed.): Defining the Science connectivity of its complement in X together their properties are discussed in the third chap- of Stochastics, Sigma Series in Stochastics, form an invariant, which characterizes the ter. A description of conformal mappings Vol. 1, Heldermann Verlag, Lemgo, 2004, 260 conjugacy class of τ. The symmetry type of X using Clifford algebra 2 x 2 matrices and their pp., €30, ISBN 3-88538-301-2 is the (finite) set of these invariants for all real relation to Dirac operators are given in Although the words “stochastic” and “sto- forms of X. In the first chapter, the authors Chapter 4. The next two chapters contain a chastics” are frequently used, their exact introduce combinatorial methods used in the discussion of the unique continuation property meaning may not be generally known. The classification. The second chapter is devoted of solutions of a Dirac equation and their Greek noun stochos means target, aim, or to a description of the full group of automor- boundary values. (vs) guess. Stochastikos is a person who is skilful phisms and computation of the number of real in predicting. From this point of view, the title forms. The third chapter contains a computa- P. M. Cohn: Basic Algebra. Groups, Rings ‘Ars conjectandi’ of the famous book by Jacob tion of symmetry types of hyper-elliptic and Fields, Springer, London, 2003, 465 pp., Bernoulli has the Greek equivalent ‘στοχστικ Riemann surfaces, divided into ten different €64,95, ISBN 1-85233-587-4 ητεχνγ’ (stochastic techniques). Formerly, the subcases. (vs) This book, together with its sequel, “Further word “stochastic” was used mainly in the Algebra and Applications” (to appear), is a phrase “stochastic process”. Today one feels C.H. Clemens: A Scrapbook of Complex new and revised version of the author’s that “stochastic” means more than only “ran- Curve Theory, Graduate Studies in famous text “Algebra (Vols. 2 and 3)”, which dom”. The book contains contributions devot- Mathematics, vol. 55, American Mathematical is now out of print. Besides basics on groups, ed to the problem of how to define the science Society, Providence, 2003, 188 pp., US$39, rings, modules and fields, the book contains of stochastics. The articles are revised and ISBN 0-8218-3307-3 classical Galois theory of equations, an intro- updated versions of lectures delivered at the This is very nice book by C.H. Clemens and duction to quadratic forms and ordered fields, symposium “Defining the Science of would make an indispensable tool for anyone valuation theory, and classical commutative Stochastics”, which took place in Germany in willing to enter the realm of algebraic geome- ring theory (including primary decomposition October 2000. The publication is divided into try. Although not written in the tight mathe- and Hilbert Nullstellensatz). The final chapter three parts, each one having three papers. The matical form Definition-Theorem-Proof, it deals with infinite field extensions. The book first part describes the present state of statis- covers and explains motivation for the intro- contains numerous exercises, and moreover, tics and probability theory in comparison to duction of algebraic language to problems in motivation and many illuminating comments the science of stochastics. The second part geometry using many suitable examples. The on the subject. There is no doubt that the book deals with randomness, contingency, and first chapter recalls all notions of classical will take the position of its predecessor in white noise. The third part is devoted to sto- geometry related to conic curves - projective being one of the most outstanding introducto- chastics in public education. The authors of space, linear system of conics, cross ratio, ry algebra textbooks. (jtrl). individual contributions are S. Kotz, U. hyperbolic space and rational points on Herkenrath, V. Kalashnikov, J. Kohlas, C. S. quadrics. The second chapter describes cubic J. S. Cohen: Computer Algebra and Calude, T. Hida, M. Dumitrescu, and E. von (i.e., elliptic) curves. A unifying picture Symbolic Computation: Mathematical Collani (2×). The authors emphasize history emerges by comparison of elliptic curves over Methods, A. K. Peters, Natick, 2003, 448 pp., and nature of statistical science and propose to complex numbers (i.e., the variation of Hodge US$59, ISBN 1-56881-159-4 establish “Stochastics” as an independent, structure) and elliptic curves over finite fields The book is a readable introduction to com- mathematically based exact science, which (i.e., generating function for the number of puter algebra. It presents a theoretical basis of investigates the inherent variability exhibited rational points over successive extensions of recently published Computer Algebra and by any real phenomenon. A great variety of finite fields). The third and fourth chapters Symbolic Computation: Elementary topics in the published papers, reflects the fast offer an introduction to elliptic, modular and Algorithms, of the same author. The book growing diversification of statistical sciences. theta function theory, moduli spaces and basic explores applications of algorithms to auto- It might be that the reader finds some of them facts of cohomology theory - Jacobians, dual- matic simplification, greatest common divisor to be too far from his/her specialization, but ity in sheaf cohomology, Abel-Jacobi map, calculation, resultant computation, polynomi- everybody will find some contributions inter- etc. The last two chapters contain topics from al decomposition, and factorisation. After a esting and stimulating. In Biographical Notes, geometry of higher genus Riemannian sur- review of basic background concepts, algo- on p. 215, one reads that Thomas Bayes lived faces, e.g. Prym varieties, Schottky groups rithms for manipulation and evaluation of 1702-1761. According to the inscription on and hyperelliptic curves of higher genus. (pso) numerical objects are given. Automatic alge- his tomb in Bunhill Field cemetery in London, braic and trigonometric simplification is he died on April 7, 1761, at age of 59. Some J. Cnops: An Introduction to Dirac worked out thoroughly and it gives the reader authors introduce that he was probably born in Operators on Manifolds, Progress in rather detailed information about one of the 1701. (ja) Mathematical Physics, vol. 24, Birkhäuser, essential algorithms implemented in computer Boston, 2002, 211 pp., €72,90, ISBN 0-8176- algebra software. Another of the important J.-M. Delort: Global Solutions for Small 4298-6 numerical objects - single variable polynomi- Nonlinear Long Range Perturbations of Two A Dirac operator on a Riemannian manifold M als - is discussed in two chapters. Attention is Dimensional Schrödinger Equations, acts on sections of a bundle S of left modules paid in particular to the algorithm for polyno- Mémoires de la Sociéte Mathématique de over the canonical Clifford bundle Cl(M). mial division, which is the basis to the algo- France 91, Sociéte Mathématique de France, This construction allows for many special rithm for polynomial expansion, Euclidean Paris, 2002, 94 pp., €23, ISBN 2-85629-125-2 cases, a choice of the Dirac operator depends algorithm, algorithm for partial fraction In the book, a global existence of solutions to on a choice of the bundle S. The book is devot- expansion, and algorithm for performing the two dimensional Schrödinger equations is ed to such cases, where the manifold M inher- arithmetic operations for expressions in sim- proved. The equation allows for a quadratic its its Riemannian structure from an embed- ple algebraic number fields. This is followed nonlinearity containing the space derivatives. ding to a (pseudo)-Euclidean space Rp,q. The by a generalization of these concepts and algo- The initial data are small. The setting is ‘criti- special case of embedded manifolds makes it rithms to multivariate polynomials. Two cal’ in the sense that the nonlinearity becomes possible to study various special constructions important concepts of computer algebra, the non-integrable when applied to the fundamen- of the bundle S. The embedding of M into Rp,q resultant and Gröbner basis, are introduced. tal solution of the Schrödinger operator. To also allows the use of the Clifford algebra The book culminates in the last chapter, overcome this difficulty, a suitably chosen p,q Clp,q of R for various computations related to devoted to polynomial factorisation and a function is subtracted from the solution to can- the normal bundle of M. The first chapter of modern algorithm for it. The book will be use- cel out the ‘worst’ terms. The main technique EMS June 2004 41 BOOKS of the book is classical harmonic analysis. Yee, presents an example of an analytical tra- (racking, hammering, or amputation). Under Symbolic calculus, dyadic decomposition, jectory in a chaotic system. The last article, by classical conditions, minimization of qX1T-X2q carefully developed linear theory, together J. Della Dora and M. Mirica-Ruse, presents is the multivariate multiple regression prob- -1 with some nonlinear estimates (e.g., for prod- formalization for the concept of a hybrid sys- lem with the solution T=(X1′X1) X1′X2. The ucts) are the keystones of the final success, tem. (pkur) two-sided form of the Procrustes problem is to and the solution is found in a suitable weight- minimize qX1T1-X2T2q under some condi- ed space. The book is rather technical but is T. Forster: Logic, Induction and Sets, tions. A further variant consists of the double mostly self-contained. (dpr) London Mathematical Society Student Texts Procrustes problem concerning minimization 56, Cambridge University Press, Cambridge, of qT2X1T1-X2q. The solutions of the men- Y. Eliashberg, N. Mishachev: Introduction 2003, 234 pp., £18,95, ISBN 0-521-53361-9, tioned problems depend on the set of matrices to the h-Principle, Graduate Studies in ISBN 0-521-82621-7 T, which are allowed to be used. In some Mathematics, vol. 48, American Mathematical The book is an introduction to mathematical cases T is constrained to be an orthogonal Society, Providence, 2002, 206 pp., US$30, logic and set theory. It contains everything matrix, which represents a generalized rota- ISBN 0-8218-3227-1 that is expected from such a book - language, tion, in other problems T must be a permuta- The Gromov “h-principle” was introduced in syntax, semantics, model, deducibility, com- tion matrix and so on. Generally we can say the 80’s, in Gromov’s book on partial differ- pleteness theorem, normal form theorem, that Procrustean methods are used to trans- ential relations. This is really a principle, not a recursion and axioms of ZFC. The style of the form one set of data to represent another set of theorem, and it should be adapted to every book differs from similar textbooks. The cen- data as closely as possible. The book also con- new application. It describes the behaviour of tral notion in the book is a recursive data type, tains oblique Procrustes problems, weighting, a system of partial differential equations or which means a set defined by a recursion. This scaling, and missing values. Further topics are inequalities having a big space of solutions notion is introduced early, hence some knowl- accuracy and stability, and some applications. (e.g., dense in the space of functions or fields edge of set theory is tacitly assumed at the The authors inform that the process of writing considered). It was inspired by work on the very beginning (and it also induces some cir- the book was similar to the machinations of C1-isometric embedding theory (Nash and cular arguments at the end). The whole pre- Procrustes, the material seemed to be continu- Kuiper) and the immersion theory in differen- sentation is not as rigorous as could be expect- ously stretching and so it was necessary to tial topology (Smale and Hirsch). The ed in textbooks of this type. On the other hand, chop things. However, this is probably the fate Gromov book is not easy to understand. The the book contains a lot of philosophical expla- of many mathematical papers and books. (ja) book under review offers the reader a nice and nations pointing to aspects usually ignored. understandable treatment of two methods Another interesting point is that the set of F. von Haeseler: Automatic Sequences, de based on the “h-principle” – the holonomic exercises contains some, which are rare to Gruyter Expositions in Mathematics 36, approximation methods and convex integra- meet, for example: “Dress up the traditional Walter de Gruyter, Berlin, 2003, 191 pp., tion theory (which was treated in a more gen- proof that √2 is irrational into a proof by well- €78,50, ISBN 3-11-015629-6 eral situation in the book by D. Spring, pub- founded induction on N x N.” Once you are Automatic sequences can be equivalently lished in 1998). These two methods cover familiar with a classical textbook on the top- defined in terms of finite automata (they are many interesting applications. The book con- ics, reading this one as a second choice is fun. produced by them), substitutions (they are tains a discussion of applications of the holo- (psi) their fixed points), or by their kernel (which is nomic approximation method in symplectic finite). The book studies a generalized concept and contact geometry, including a review of R. Garnier, J. Taylor: Discrete Mathematics of automatic sequences, where the sequence is basic notions needed for the applications. The for New Technology, second edition, Institute indexed by elements of a finitely generated book is very interesting, readable and should of Physics Publishing, Bristol, 2002, 749 pp., group, instead of natural numbers. be of interest to more than just geometers. (vs) £25, ISBN 0-7503-0652-1 Accordingly, ordinary substitutions are The book is designed for students of comput- replaced by mappings based on expanding F. Fauvet, C. Mitschi, Eds.: From er science. It contains main mathematical top- group endomorphisms, and a corresponding Combinatorics to Dynamical Systems, IRMA ics needed in their undergraduate study. The notion analogous to finite automata is devel- Lectures in Mathematics and Theoretical main core of mathematics contained in the oped. These concepts are introduced and dis- Physics 3, Walter de Gruyter, Berlin, 2003, book belongs to standard material needed for cussed in the first three chapters. Chapter 4 241 pp., €36,95, ISBN 3-11-017875-3 any student of mathematics. The choice of studies automatic subsets and automatic maps. This is the proceedings of the conference topics was inspired by a first-year mathemat- The final chapter combines automatic “Journées de calcul formel en l’honneur de ics course for undergraduate students of com- sequences with some additional algebraic Jean Thomann”, which took place at IRMA puter science at the University of Nottingham. structures, and deals especially with Mahler (Institut de Recherche Mathématique The reader can find here sections on logic, equations over the ring of formal power series. Avancée) in Strasbourg in March 2003. The mathematical proofs, sets, relations, relational The book is written quite succinctly, and is not meeting was devoted to dynamical systems, databases, functions and functional depen- very easy to read. A certain degree of mathe- computer algebra, and theoretical physics. dence, matrix algebra and systems of linear matical maturity is required just to grasp the The first paper, by M. Espie, J.-C. Novelli and equations, basic algebraic structures, Boolean basic definition of the automatic sequence. G. Racinet, contains computations of certain algebra and graph theory and its applications. The volume will be useful mainly for graded Lie algebras related to multiple zeta The book also contains applications (relation- advanced researchers in the field, interested in values. The article by E. Corel gives an alge- al databases, normal form of databases). In the broadening their view of the subject. (šh) braic and algorithmic treatment of formal second edition, the authors added a lot of new exponents at an irregular singularity. The arti- exercises and examples, illustrating discussed I. Chajda, G. Eigenthaler, H. Länger: cle by J. L. Martins studies irregular mero- concepts. The book contains a lot of well- Congruence Classes in Universal Algebra, morphic connections in higher dimensions. ordered and nicely illustrated material. (vs) Research and Exposition in Mathematics, vol. The paper by M. A. Barkatou, F. Chyzak and 26, Heldermann, Lemgo, 2003, 217 pp., €28, M. Loday-Richaud describes various algo- J. C. Gower, G. B. Dijksterhuis: Procrustes ISBN 3-88538-226-1 rithms for the rank reduction of differential Problems, Oxford Statistical Science Series The book explores a number of different fea- systems. The article by M. Canalis-Durand 30, Oxford University Press, Oxford, 2004, tures of congruence on universal algebras. It presents a “pattern recognition algorithm” for 233 pp., £55, ISBN 0-19-851058-6, covers the subject from the early works of Gevrey power series. The article by D. Let X1 and X2 be given matrices. The simplest Maltsev to recent results. One can find a good Boucher and J.-A. Weil gives a new proof of Procrustes problem is to find a matrix T that amount of material concerning properties of the non-integrability of the planar three-body minimizes qX1T-X2q. In analogy with the blocks (in general and in varieties), relation- problem. The paper by L. Brenig uses the story of Procrustes, son of Poseidon, we can ship to quotients and subalgebras or local Quasi Polynomial formalism to obtain effec- imagine that X1 is the unfortunate traveller, X2 properties of congruence. The first two chap- tive integrability for some dynamical systems. is the Procrustean bed, and T is the treatment ters explain basic notions and give quite a few The article by R. Conte, M. Musete and T.-L. 42 EMS June 2004 BOOKS examples of algebras. The study of congru- Bruhat, using a modification of WKB meth- diffeomorphisms, horocycle flows and inter- ence starts with the classical notions of per- ods for solutions of linear partial differential val exchange transformations. (pkur) mutability, distributivity, modularity and equations. Meanwhile, justification for ideas direct decomposability. Further chapters deal coming from geometric optics was developed B. Lapeyre, É. Pardoux, R. Sentis: with the role of subalgebras, single blocks, for solutions of certain semi-linear or quasi- Introduction to Monte-Carlo Methods for constant terms or ideals. Special chapters are linear equations. The first part of the book Transport and Diffusion Equations, Oxford devoted to extension properties of congru- contains an exposition of a structure of the Texts in Applied and Engineering ence, regular and coherent algebras, local con- equations. The second part describes families Mathematics, Oxford University Press, ditions and one-block congruence. (eb) of approximate solutions (to any order), in the Oxford, 2003, 163 pp., £60, ISBN 0-19- shape of single-phase expansions, rapidly 852592-3, ISBN 0-19-852593-1 P. Imkeller, J.-S. von Storch, Eds.: oscillating at high frequency. The third part The book is devoted to the applications of Stochastic Climate Models, Progress in contains a description of exact solutions, Monte Carlo methods (i.e., simulation tech- Probability, vol. 149, Birkhäuser, Basel, asymptotic to previous expansions. (vs) niques based on random numbers), for solu- 2001, 398 pp., DM 196, ISBN 3-7643-6520-X tions of partial differential equations. A prob- The proceedings of the workshop on J.-M. Kantor (Ed.): Ou en sont les mathé- abilistic representation of solutions of these Stochastic Climate Models, held in Chorin in matiques?, Société Mathématique de France, equations is given in the following way: the 1999, present an exiting and stimulating sum- Paris, 2002, 440 pp., ISBN 2-7117-8994-2 transport or diffusion type equations are inter- mary of probabilistic developments in climate The Société Mathématique de France has pub- preted as the Fokker-Planck equations associ- physics. Stochastic processes, stochastic and lished the journal Gazette des mathématiciens ated with Markov processes. Transport equa- partial differential equations, random dynami- since the 70’s. Its aim is a diffusion of mathe- tions in particle physics, the nonlinear cal systems, local and large deviations asymp- matical knowledge. The book under review Boltzmann transport equation, and the links totic procedures, have comprised the most brings a sample of papers from the journal between second order partial differential dif- active mathematical areas in the field over the since its foundation. Papers are complemented fusion equations and Brownian motion, are all past 30 years, since Klaus Haselmann sug- by comments written by a contemporary spe- investigated. For each type of problem, limits gested a climate stochastic model where cialist, describing recent evolution in the field. of methods are discussed and specific tech- weather fluctuations randomly force the cli- As a whole, it is a fascinating read. Papers niques used in practice are described. An mate in the same way that fluid molecules (written in French) describe eighteen different essential point is that even if Monte-Carlo force Brownian pollen particles. Hence the topics of enormous interest, reviewed in a methods need not converge, it is always possi- Langevin stochastic equation and stochastic readable and understandable way, and ble to control the reliability of the result using calculus, enter the model in a very natural designed for a general audience. The topics an inexpensive additional calculation. The way. The structure of the book reflects its aim include classical themes (the third and the fif- reader should have a good background in to review and explain recent mathematical teenth Hilbert problems, theory of invariants, mathematical analysis. (jste) additions to the list of tools for climate model- arithmetic algebraic geometry), important ling, to help the climate physicists to under- mathematical topics (stochastic calculus, W. K. Li W.: Diagnostic Checks in Time stand what these tools are about. Chapter 1 by deformation of various mathematical struc- Series, Monographs on Statistics and Applied D. Olbers, K. Fraedrich, J. S. von Storch and tures, symplectic geometry and topology, the Probability 102, Chapman & Hall/CRC, Boca R. Temam exhibits the most important climate work of M. Gromov, convex polytopes, singu- Raton, London, 2004, 216 pp., US$69,95, models, from those simple ones to more com- larities of maps, error correcting codes), very ISBN 1-58488-337-5 plex ones. Chapter 2 by L. Arnols, M. Denker, modern themes (mathematical ideas coming There are many books on time series analysis M. Kesseböhmer, Y. Kiefer, Ch. Rödenbeck, from quantum field theory, quantum groups but this is the first monograph specialized to Ch. Beck and H. Kantz, searches for possible and invariants of nodes, Wiles’ proof of the diagnostic checking. Construction of a model sources of stochasticity in the models from the Taniyama-Weil conjecture), as well as some for time series data usually consists of three mathematical as well as physical point of applications (image recognition, mathematics steps. At the beginning, a preliminary model is view. Chapter 3 by P. Imkeller, J. Duan, P. E. in meteorology, topology in molecular biolo- chosen. Then the parameters are estimated. Kloeden, B. Schmalfuss, J. Zabzyk and P. gy). It is a very nice collection, which can con- The third stage is called model diagnostic Müller, is a very informative compendium of tribute in an essential way to the general math- checking. It involves techniques like residual probabilistic tools mentioned above that are ematical education of any reader. (vs) plots and procedures for testing if the residu- already at work in the climate dynamics. als are approximately uncorrelated. If it is Chapter 4 by J. Egger, J.A. Freund, A. A. Katok: Combinatorial Constructions in found that the model is not adequate, the Neiman, L. Schimanski-Geier, A.H. Ergodic Theory and Dynamics, University process starts with the first step again, this Monahan, L. Pandolfo, P. Imkeller, P. Lecture Series, vol. 30, American time with some new information from the pre- Sardeshmukh, P. Penland, M. Newman and Mathematical Society, Providence, 2003, 121 vious analysis. The book describes a rich vari- W.A. Woyczynski, completes the presentation pp., US$29, SBN 0-8218-3496-7 ety of diagnostic checks. It starts with the uni- of climate mathematics with a discussion of The book is an updated version of the unpub- variate and multivariate linear models. some of the more special models for the fluc- lished notes “Constructions in ergodic theo- Although attention is paid mainly to ARMA tuating eastward flows or planetary waves. ry”, written by the author in collaboration with models, the author also writes about periodic The book is strongly recommended as an E.A.Robinson Jr., in 1982-83. The first part autoregression and Granger causality tests. A excellent source of information and inspira- entitled “Approximations and genericness in chapter is devoted to robust modelling and tion, both to mathematicians and physicists ergodic theory”, deals with the approxima- diagnostic checking, from a robust portman- interested in the field. (jste) tions of measure-preserving transformations teau test to the trimmed portmanteau statistic. by periodic processes, which are permutations An important part of the book describes P.-Y. Jeanne: Optique géométrique pour des of partitions of the space. Several types of results on nonlinear models. It contains good- systemes semi-linéaires avec invariance de such periodic processes are considered and the ness-of-fit tests, tests for general nonlinear jauge, Mémoires de la Sociéte Mathématique speed of approximation is evaluated. The structure, tests for linear vs. specific nonlinear de France, Sociéte Mathématique de France, main theorem says that the set of measure-pre- time series, and methods for choosing two dif- Paris, 2002, 160 pp., €23, ISBN 2-85629-123- serving transformations of a Lebesgue space, ferent families of nonlinear models. Then the 6 which admit a periodic approximation of a models for conditional heteroscedasticity are The main topic of the book is a study of the given type and speed, is residual. The second presented. Finally, fractionally differenced properties of solutions of field equations for a part entitled “Cocycles, cohomology and com- processes are described and a few special top- scalar field or a spinor field coupled to a binatorial constructions”, deals with algebraic ics are mentioned like non-Gaussian time Yang-Mills field, and construction of approx- constructions in ergodic theory. The concepts series and power transformations. Unit root imate solutions of a special shape. First, in the of rigidity, stability and effectiveness are and co-integration tests are not included. The 60’s, solutions that rapidly oscillated in high investigated. Main applications include author is a known specialist in time series frequencies were found by Y. Choquet- Diophantine translations of the torus, Anosov modelling. His approach is a practical one and EMS June 2004 43 BOOKS each topic is presented from a model builder’s integral representation of the solution to basic material, with special attention paid to point of view. A long list of references is also Maxwell’s equations called the Stratton-Chu Bol loops and Bruck loops. The second part is appreciated as an important part of the publi- formula. In addition, the author derives classi- devoted to smooth loops on low dimensional cation. The monograph is very useful for sta- cal series representations of the solution of manifolds. (tk) tisticians working in time series analysis. (ja) Maxwell’s equations. These are used in Chapter 10 to derive a semi-discrete method V. Paulsen: Completely Bounded Maps and S. Markvorsen, M. Min-Oo: Global for the scattering problem utilizing the elec- Operator Algebras, Cambridge Studies in Riemannian Geometry: Curvature and tromagnetic equivalent of the Dirichlet to Advanced Mathematics 78, Cambridge Topology, Advanced Courses in Mathematics Neumann map. A fully discrete domain- University Press, Cambridge, 2003, 300 pp., CRM Barcelona, Birkhäuser, Basel, 2003, 87 decomposed version of this algorithm is pro- £47,50, ISBN 0-521-81669-6 pp., €24, ISBN 3-7643-2170-9 posed and analysed in Chapter 11. In Chapter This book is intended as an (advanced) intro- The small booklet contains lecture notes from 12, the author studies a coupled integral equa- ductory text on the main results and ideas in two advanced courses. The first one was writ- tion and finite element method due to Hazard, some of the major topics of modern operator ten by S. Markvorsen (Distance geometric Lenoir and Cutzach. Chapter 13 is devoted to theory. The first half of the book is devoted to analysis on manifolds). It contains a compari- a discussion of some issues related to practical the study of completely positive and com- son theory for distance functions on immersed aspects of solving Maxwell’s equations. The pletely bounded maps between C*-algebras submanifolds in Riemannian manifolds and its final Chapter gives a short introduction to and their connections with the dilation theory. relations to seemingly unrelated topics. These inverse problems in electromagnetism. The The adverb completely is related to a collec- include isoperimetric inequalities, diffusion book will be of interest to mathematicians, tion of matrix norms on the algebra. The sec- processes (mean exit time comparison), tran- physicists and researchers concerned with ond part explains connections with various sience, warped products, and all that related to Maxwell’s equations. The three chapters, 4, 5 types of similarity problems (e.g. the Kadison the Laplacian. The second lecture notes are and 7, could be useful in a graduate course on conjecture and Pisier’s theory), and properties written by M. Min-Oo (The Dirac operator in finite element methods. (knaj) of operator systems (e.g., characterization of geometry and physics). The first part existence of isometric representations of oper- describes traditional results on the Dirac oper- S. Montgomery, H.-J. Schneider: New ator algebras - Blecher, Ruan, Sinclair theory, ator, its index formula and the Lichnerowicz Directions in Hopf Algebras, Mathematical which is applied to new proofs of classical formula. The second part treats the Gromov Sciences Research Institute Publications 43, results on interpolation of analytic functions). notion of K-area and the corresponding funda- Cambridge University Press, Cambridge, Another application is Pisier’s theory of the mental K-area inequality, while the third part 2002, 485 pp., £55, ISBN 0-521-81512-6 universal operator algebra of an operator discusses various aspects of the famous posi- Hopf algebras are abstractions of algebras of space and his results on similarity and factor- tive mass theorem. Both reviews are very use- functions on a group. As such, they were ization degree. The book is carefully written, ful for the orientation of the reader in the field, introduced to topology in the first half of the proofs are often accompanied with notes help- details should be found in the corresponding last century. Roughly speaking, Hopf algebra ing to explain the situation. Several theorems literature. (vs) is an associative algebra that also carries a are proved by various methods, e.g., there are “comultiplication”, and these two structures five different proofs of the von Neumann P. Monk: Finite Element Methods for are compatible in an appropriate sense. The inequality. The text can be used either for a Maxwell’s Equations, Numerical second advent of Hopf algebras was marked graduate course in operator theory or for an Mathematics and Scientific Computation, by a seminal talk on quantum groups, present- independent study, since it contains 205 exer- Clarendon Press, Oxford, 2003, 450 pp., ed by V. Drinfel’d in Berkeley in 1986. Since cises. The author recommends that the reader £59,95, ISBN 0-19-850888-3 then, Hopf algebras have found a great num- be acquainted with R. G. Douglas’ book The aim of this book is to provide an up-to- ber of applications, not only in the theory of “Banach Algebra Techniques in Operator date and sound theoretical foundation for quantum groups (which are particular types of Theory”. (jmil) finite element methods in computational elec- Hopf algebras) but also in low-dimensional tromagnetism. The emphasis is on finite ele- topology, algebra and geometry. The book R. Plato: Concise Numerical Mathematics, ment methods for scattering problems that under review is a collection of expository arti- Graduate Studies in Mathematics, vol. 57, involve the solution of Maxwell’s equations cles that attempt to summarize the recent American Mathematical Society, Providence, on infinite domains. Suitable variational for- developments and offer a new understanding 2003, 453 pp., US$85, ISBN 0-8218-2953-X mulations are developed and justified mathe- of classical topics. The contributions were This excellent textbook is a concise introduc- matically. The main focus of the book is an written by N. Andruskiewitsch, S. Gelaki, G. tion to the fundamental concepts and methods error analysis of edge finite element methods Letzler, A. Masuoka, D. Nikshych, F. van of numerical mathematics. The author covers that are particularly well suited to Maxwell’s Oystaeyen, D. Radford, P. Schauenburg, H.-J. many important topics using well-chosen equations. The methods are justified for Schneider, M. Takeuchi, L. Vainerman and Y. examples and exercises. Many of the present- Lipschitz polyhedral domains that can cause Zhang. Most of the authors were participants ed methods are illustrated by figures. For strong singularities in the solution. The book of the Hopf Algebras Workshop held at MSRI numerous algorithms, practical and relevant is divided into 14 chapters. After the prepara- as a part of the 1999-2000 Year on work estimates are given and pseudo codes are tory work in Chapters 2 and 3 (functional Noncommutative Algebra. (mm) provided that can be used for implementa- analysis and abstract error estimates, Sobolev tions. The author presents approximately 120 spaces and vector function spaces), Chapter 4 P.T. Nagy, K. Strambach: Loops in Group exercises with different levels of difficulty. contains a discussion of a simple model prob- Theory and Lie Theory, de Gruyter Topics covered include interpolation, the fast lem for Maxwell’s equations. Chapters 5 and Expositions in Mathematics 35, Walter de Fourier transform, iterative methods for solv- 6 form a central part of the book. They present Gruyter, Berlin, 2002, 361 pp., US$148, ISBN ing systems of linear and nonlinear equations, a detailed description of the original Nédélec 3-11-017010-8 numerical methods for solving ODE’s, numer- finite element spaces. In Chapter 7, the author The book is a comprehensive treatment on ical methods for matrix eigenvalue problems, gives the finite element discretization of the various topological and geometrical aspects of approximation theory, and computer arith- cavity problem and presents two convergence the theory of loops (i.e., generally non-asso- metic. The book is suitable as a text for a first proofs for this method. Chapter 8 presents ciative binary structures with neutral elements course in numerical methods, for mathematics basic topics concerning finite elements and and unique division). The loops are represent- students or students in neighbouring fields, gives a brief introduction to hp finite element ed as stable transversals (alias sections) in such as engineering, physics, and computer methods for Maxwell’s equations. A central their one-sided multiplication groups, and science. In general, the author assumes knowl- task of computational electromagnetism is the these are equipped with additional structure. edge of calculus and linear algebra only. approximation of scattering problems. The Thus topological, differentiable, algebraic (knaj) presentation of these problems starts in loops are treated via Lie groups, algebraic Chapter 9, with classical scattering by a groups, symmetric spaces, etc. The book is I. Reiner: Maximal Orders, London sphere, where the author derives the famous divided into two parts. The first one presents Mathematical Society Monographs 28, 44 EMS June 2004 BOOKS Clarendon Press, Oxford, 2003, 395 pp., £70, let us mention the Fenstad contribution groupoids (M. Enock: Quantum groupoids ISBN 0-19-852675-3 “Computability theory: structure and algo- and pseudo-multiplicative unitaries; P. Let R be a noetherian integral domain with rithms”, Rathjen’s “Explicite mathematics Schauenberg: Morita base change in quantum quotient field K, and let A be a finite dimen- with monotone inductive definitions. A sur- groupoids; K. Szlachányi: Galois actions by sional K-algebra. Recall that an R-order in the vey”, and Mancosu’s “On the constructivity of finite quantum groupoids; J.-M. Vallin: K-algebra A, is a sub-ring Λ of A, having the proofs. A debate among Behmann, Bernays, Multiplicative partial isometries and finite same unit element as A, and such that K . Λ = Gödel, and Kaufmann.” (jmlc) quantum groupoids). (jiva) { Σαi mi (finite sum)| αi ∈ K, mi ∈Λ}= A. A maximal R-order in A is defined in the usual Y. Talpaert: Differential Geometry with S. H. Weintraub: Representation Theory of way as such an R-order that is not properly Applications to Mechanics and Physics, Finite Groups: Algebra and Arithmetic, contained in any other R-order in A. After Monographs and Textbooks in Pure and Graduate Studies in Mathematics, vol. 59, algebraic preliminaries, basic properties of Applied Mathematics, vol. 237, Marcel American Mathematical Society, Providence, orders are investigated in Chapter 2. Maximal Dekker, Inc., New York, 2001, 454 pp., 2003, 212 pp., US$45, ISBN 0-8218-3222-0 orders in skew-fields, over discrete valuation US$185, ISBN 0-8247-0385-5 The book is an introduction to the group rep- rings and over Dedekind domains, are studied The book contains a careful exposition of resentation theory with emphasis on finite in Chapters 3, 5 and 6, respectively. Chapter 3 basic facts concerning manifolds, differential groups. Since group representations are just is devoted to Morita equivalence and Chapter forms and their integration, Stokes theorem, modules over group algebras, the classical 7 deals with crossed-product algebras. The exterior differential systems, Lie derivatives results on semisimple group representations last two parts investigate simple algebras over and basic facts on Lie groups and Lie alge- are naturally obtained in Chapter 3, as partic- global fields and a local and global theory of bras, Riemannian geometry and their use in ular cases of more general results concerning hereditary orders. (lbi) physics. Comparing the new English version the structure of modules over semisimple with the original French edition, the last three rings. The latter results are presented in A. Shen, N.K. Vereshchagin: Computable chapters were substantially reworked and Chapter 2. Chapter 4 deals with induced rep- Functions, Student Mathematics Library, vol. extended (these are chapters on Riemannian resentations and culminates in the proof of 19, American Mathematical Society, geometry, Lagrange and Hamilton mechanics, Brauer's theorem (concerning irreducible Providence, 2002, 166 pp., US$29, ISBN 0- and symplectic geometry). The book is written complex representations of G of exponent n 8218-2732-4 in a very understandable and systematic way, being defined over Q (n√1)). The remaining The book is based on lectures for undergradu- with a lot of figures. A very good feature of three chapters provide an introduction to mod- ate students at the Moscow State University the book is a collection of more than 130 exer- ular representation theory, again by applying Mathematics Department and the content cov- cises and problems, which are completely the powerful module theoretic approach. The ers a majority of basic notions of general the- solved, resp. answered. The book can be rec- book is well written, and contains many exam- ory of computation, excluding computational ommended for a wide range of students as a ples worked out in detail (for example, deter- complexity. It begins with a definition of com- first book to read on the subject. It can be also mining all irreducible complex and modular putable functions, based on an informal defin- useful for the preparation of courses on the representations of A5). It is a suitable text for ition of algorithms. Then, the classic notions topic. (vs) a year long graduate course on the subject. of topics such as decidability, universal func- (jtrl) tions, fixed point theorem, enumerable sets, L. Vainerman, Ed.: Locally Compact oracle computations and arithmetical hierar- Quantum Groups and Groupoids, IRMA V. I. Zhukovskiy: Lyapunov Functions in chy, are presented, clarified and developed. A Lectures in Mathematics and Theoretical Differential Games, Stability and Control: precise definition of an algorithm is given in Physics 2, Walter de Gruyter, Berlin, 2003, Theory, Methods and Applications, vol. 19, the ninth chapter (Turing machines); the word 247 pp., €34,53, ISBN 3-11-017690-4 Taylor & Francis, London, 2003, 281 pp., problem is treated here as well. Chapter 10 These are the proceedings of the workshop £70, ISBN 0-415-27341-2 (Arithmeticity of computable function), devel- "Quantum Groups, Hopf Algebras and their The book is devoted to the linear theory of dif- ops the subject on the mentioned precise base; Applications", which was held in Strasbourg ferential games under uncertainty. It is sup- Tarski’s theorem and Gödels’ undecidability in February 2002. They contain an posed that nothing is known on their nature theorem are presented. The last chapter is "Introduction of the editor" and 7 original arti- (i.e., the stochastic approach is ruled out). The devoted to recursive functions. The book can cles. All of them represent substantial contri- presence of uncertainties moves the problem appropriately provide knowledge of the theme butions to the theory of quantum groups and of the choice of strategy to the field of "multi- to programmers and to students of mathemat- groupoids, and every specialist in the field criteria" dynamical problems. The results are ics and computer science. (jml?) should be familiar with them. The introduc- mainly given for quadratic payoff functions, tion describes the development of the theory two players and non-cooperative games. The W. Sieg, R. Sommer, C. Talcott, Eds.: of quantum groups and groupoids in a very book is divided into two parts. The first one Reflections on the Foundations of concise and deep way, so that even a mathe- describes the foundations of differential Mathematics. Essays in Honour of Solomon matician who is not a specialist on this topic games under uncertainties. The central role is Feferman, Lecture Notes in Logic 15, A.K. can adequately understand why the theory has played by the notion of a vector guarantee. Peters, Natick, 2002, 444 pp., US$45, ISBN 1- proceeded in this way or that. For the non-spe- Two approaches, based on the analogue of the 56881-170-5 cialist, let us also mention that the paper by S. vector saddle point and the vector maximin, The book consists of contributions submitted Vaes and L. Vainerman (On low dimensional are used for solving the multicriteria dynami- mostly by participants of the symposium on locally compact quantum groups) contains cal problems. The second part is devoted to the foundations of mathematics, organized to Preliminaries, where we can find basic infor- the concept of equilibrium of objections and honour Solomon Feferman, which was held at mation about these groups. This paper repre- counter-objections, as well on the active equi- Stanford University, in December 1998. The sents the continuation of the research of both librium. The reader should have a basic volume is organized into four parts: Proof-the- authors on extensions of locally compact knowledge of ordinary differential equations oretic analysis, Logic and computation, quantum groups. J. Kustermans and E. and optimization. For example, the dynamic Applicative and self-applicative theories, and Koelink (Quantum SUq~(1,1) and its programming approach (i.e., the Belman Philosophy of modern mathematical and logi- Pontryagin dual), present an overview of the equation) and the method of Lyapunov func- cal thought. For example, in the first part we quantum group SUq~(1,1) and study its tions are frequently used. Both parts end with can find contributions concerning ordinal Pontryagin dual. A. Van Daele, in his paper results, comments on the history of ideas, and analysis (such as Avigad’s and Buchholz’s (Multiplier Hopf*-algebras with positive inte- references (131 items). These are well orient- contributions), Simpson’s paper that treats grals: A laboratory for locally compact quan- ed in works of the Russian and Ukrainian predicative analysis, where Feferman’s sys- tum groups), gives a survey of the theory of schools. Each part is also accompanied by tem IR is compared with Friedman’s ATR0, algebraic quantum groups and their relations exercises, with their solutions given at the end and Friedman’s article on combinatorial finite with locally compact quantum groups. The of the book. (jmil) trees. To demonstrate the variety of themes, remaining four papers deal with quantum EMS June 2004 45