European Mathematical Society

CONTENTS EDITORIAL TEAM EUROPEAN MATHEMATICAL SOCIETY

EDITOR-IN-CHIEF ROBIN WILSON Department of Pure Mathematics The Open University Milton Keynes MK7 6AA, UK e-mail: [email protected] ASSOCIATE EDITORS STEEN MARKVORSEN Department of Mathematics Technical University of Denmark Building 303 NEWSLETTER No. 35 DK-2800 Kgs. Lyngby, Denmark e-mail: [email protected] KRZYSZTOF CIESIELSKI March 2000 Mathematics Institute Jagiellonian University Reymonta 4 30-059 Kraków, Poland EMS News : Committee and Agenda ...... 2 e-mail: [email protected] KATHLEEN QUINN Message from the EMS President ...... 3 Open University [address as above] e-mail: [email protected] Editorial by Vagn Lundsgaard Hansen ...... 4 SPECIALIST EDITORS INTERVIEWS Introducing the WMY2000 team ...... 5 Steen Markvorsen [address as above] SOCIETIES Interview with Lars Gårding ...... 6 Krzysztof Ciesielski [address as above] 2000 Anniversaries : Sonya Kovalevskaya ...... 9 EDUCATION Vinicio Villani 2000 Anniversaries : Eugenio Beltrami ...... 11 Dipartimento di Matematica Via Bounarotti, 2 Societies Corner : Dutch Mathematical Society ...... 12 56127 Pisa, Italy e-mail: [email protected] Societies Corner : Danish Mathematical Society ...... 14 MATHEMATICAL PROBLEMS Paul Jainta Educational Section ...... 16 Werkvolkstr. 10 D-91126 Schwabach, Germany Oberwolfach Programme 2001 ...... 19 e-mail: [email protected] ANNIVERSARIES Problems Corner ...... 20 June Barrow-Green and Jeremy Gray Open University [address as above] Forthcoming Conferences ...... 25 e-mail: [email protected] and [email protected] and Recent Books ...... 32 CONFERENCES Kathleen Quinn [address as above] RECENT BOOKS Ivan Netuka and Vladimir Sou³ek Designed and printed by Armstrong Press Limited Mathematical Institute Unit 3 Crosshouse Road, Southampton, Hampshire SO14 5GZ, UK Charles University phone: (+44) 23 8033 3132; fax: (+44) 23 8033 3134 Sokolovská 83 Published by European Mathematical Society 18600 Prague, Czech Republic ISSN 1027 - 488X e-mail: [email protected] and [email protected] ADVERTISING OFFICER NOTICE FOR MATHEMATICAL SOCIETIES Vivette Girault Labels for the next issue will be prepared during the second half of May 2000. Laboratoire d’Analyse Numérique Please send your updated lists before then to Ms Tuulikki Mäkeläinen, Department of Mathematics, Boite Courrier 187, Université Pierre P.O. Box 4, FIN-00014 University of Helsinki, Finland; e-mail: [email protected] et Marie Curie, 4 Place Jussieu INSTITUTIONAL SUBSCRIPTIONS FOR THE EMS NEWSLETTER 75252 Paris Cedex 05, France Institutes and libraries can order the EMS Newsletter by mail from the EMS Secretariat, e-mail: [email protected] Department of Mathematics, P. O. Box 4, FI-00014 University of Helsinki, Finland, or by e-mail: OPEN UNIVERSITY Please include the name and full address (with postal code), telephone and fax number (with coun- PRODUCTION TEAM try code) and e-mail address. The annual subscription fee (including mailing) is 60 euros; an Liz Scarna, Toby O’Neil invoice will be sent with a sample copy of the Newsletter.

EMS March 2000 1 EMS NEWS EMS News: Committee and Agenda EXECUTIVE COMMITTEE EMS Agenda PRESIDENT (1999–2002) Prof. ROLF JELTSCH 2000 Seminar for Applied Mathematics 24-25 March ETH, CH-8092 Zürich, Switzerland Executive Committee Meeting, hosted by the Polish Mathematical Society and the Institute of e-mail: [email protected] Mathematics of the Polish Academy of Sciences, Bedlevo, near Poz•an (Poland). VICE-PRESIDENTS 15 May Prof. ANDRZEJ PELCZAR (1997–2000) Deadline for submission of material for the June issue of the EMS Newsletter Institute of Mathematics contact: Robin Wilson, e-mail: [email protected] Jagellonian University 13-20 June Raymonta 4 EMS Lectures by Prof. Dr. George Papanicolaou, (Stanford, USA). PL-30-059 Krakow, Poland 13-16 June: ETH, Zürich (Switzerland): Financial Mathematics e-mail: [email protected] 18-20 June: University of Crete, Heraklion, Crete (Greece): Time Reversed Acoustics Prof. LUC LEMAIRE (1999–2002) contact: David Brannan, e-mail: [email protected] Department of Mathematics 17-22 June Université Libre de Bruxelles EURESCO Conference in Mathematical Analysis at Castelvecchio Pascoli (Italy): C.P. 218 – Campus Plaine Partial Differential Equations and their Applications to Geometry and Physics Bld du Triomphe Organiser: J. Eichhorn, Greifswald (Germany), e-mail: [email protected] B-1050 Bruxelles, Belgium [This series of conferences is financed by the ESF.] e-mail: [email protected] 3-7 July SECRETARY (1999–2002) ALHAMBRA 2000: a joint mathematical European-Arabic conference in Granada (Spain), promot- Prof. DAVID BRANNAN ed by the EMS and the Spanish Royal Mathematical Society Department of Pure Mathematics contact: Ceferino Ruiz, e-mail: [email protected] website: www.ugr.es/~alhambra2000 The Open University 6 July Walton Hall Executive Committee Meeting in Barcelona (Spain) Milton Keynes MK7 6AA, UK 7-8 July e-mail: [email protected] EMS Council Meeting at Institut d’Estudis Catalans, Carrer del Carme 47, E-08001 Barcelona TREASURER (1999–2002) (Spain), starting at 10 a.m. (The agenda will be sent to delegates in April.) Prof. OLLI MARTIO contact: EMS Secretariat, e-mail: [email protected] Department of Mathematics 10-14 July P.O. Box 4 Third European Congress of Mathematics (3ecm) in Barcelona (Spain) FIN-00014 University of Helsinki e-mail: [email protected] website: www.iec.es/3ecm/ Finland 24 July-3 August e-mail: [email protected] EMS Summer School, in Edinburgh (Scotland): ORDINARY MEMBERS New analytic and geometric methods in inverse problems Prof. BODIL BRANNER (1997–2000) Organiser: Erkki Somersalo, Otaniemi (Finland), e-mail: [email protected] Department of Maathematics 15 August Technical University of Denmark Deadline for submission of material for the September issue of the EMS Newsletter Building 303 contact: Robin Wilson, e-mail: [email protected] DK-2800 Kgs. Lyngby, Denmark 17 August-2 September e-mail: [email protected] EMS Summer School at Saint-Flour, Cantal (France): Probability theory Prof. DOINA CIORANESCU (1999–2002) Organiser: Pierre Bernard, Clermont-Ferrand (France), e-mail: [email protected] Laboratoire d’Analyse Numérique Autumn Université Paris VI Fifth Diderot Mathematical Forum on ‘Mathematics and Telecommunications’ 4 Place Jussieu Date and programme to be announced. 75252 Paris Cedex 05, France contact: Jean-Pierre Bourguignon, e-mail: [email protected] e-mail: [email protected] 22-27 September Prof. RENZO PICCININI (1999–2002) EURESCO Conference at Obernai, near Strasbourg (France): Dipto di Matem. F. Enriques Universit à di Milano Number theory and Arithmetical Geometry: Motives and Arithmetic Via C. Saldini 50 Organiser: U. Jannsen, Regensburg (Germany), e-mail: [email protected] I-20133 Milano, Italy EURESCO Conference at San Feliu de Guixols (Spain): e-mail: [email protected] Geometry, Analysis and Mathematical Physics: Analysis and Spectral Theory Prof. MARTA SANZ-SOLÉ (1997–2000) Organiser: J. Sjöstrand, Palaiseau (France), e-mail: [email protected] Facultat de Matematiques 30 September Universitat de Barcelona Deadline for proposals for 2001 EMS Lectures Gran Via 585 contact: David Brannan, e-mail: [email protected] E-08007 Barcelona, Spain 30 September e-mail: [email protected] Deadline for proposals for 2002 EMS Summer Schools Prof. ANATOLY VERSHIK (1997–2000) contact: Renzo Piccinini, e-mail: [email protected] P.O.M.I., Fontanka 27 14-15 October 191011 St Petersburg, Russia Executive Committee Meeting in the UK, hosted by the London Mathematical Society. e-mail: [email protected] 15 November EMS SECRETARIAT Deadline for submission of material for the December issue of the EMS Newsletter Ms. T. MÄKELÄINEN contact: Robin Wilson, e-mail: [email protected] Department of Mathematics P.O. Box 4 2001 FIN-00014 University of Helsinki 9-20 July Finland EMS Summer School at St Petersburg, Russia: tel: (+358)-9-1912-2883 Asymptotic combinatorics with application to mathematical physics fax: (+358)-9-1912-3213 Organiser: Anatoly Vershik, e-mail: [email protected] telex: 124690 3-6 September e-mail: [email protected] 1st EMS-SIAM conference, Berlin website: http://www.emis.de Organiser: Peter Deuflhard, e-mail: [email protected] 2 EMS March 2000 EMS NEWS

Message from the EMSEMS PPrresidentesident Rolf Jeltsch (Zürich)

The following is adapted from a letter sent and difficult industrial problem. It will be to the presidents of all our corporate soci- awarded for the first time at the 3ecm in eties. Barcelona this year. The deadline for nom- We have entered an exciting new year, inations was 1 March 2000. the World Mathematical Year 2000. I hope Further the EMS has started a coopera- that this year we will all encounter a big tion with SIAM, the Society for Industrial boost for our science and that we will suc- and Applied Mathematics. We have agreed ceed in making the general public aware of to start with the first joint EMS-SIAM con- the benefits of mathematics. The EMS ference, to be held in Berlin from 3-6 already contacted a series of journals which poster competition has been very success- September 2001. Finally, the number of may also enter this discussion. ful; you can find the results at EMS lectures have been doubled, in the Last year I asked for volunteers to work http://lrc.dtv.dk/ernest/wmy2000/index. html sense that they will take place not biannu- for the EMS. I am happy to announce that and further information on the WMY2000 ally but every year. The idea is to try to dis- Prof. Volker Merhmann has taken over the server http://wmy2000.math. jussieu.fr/. tribute the topics equally between pure and position as Editor-in-Chief of the ‘left- The biggest event for the EMS this year applied mathematics. This year Prof. hand-side’ of EMIS. I am convinced that will be the third European Congress of George Papanicolaou will give two sets of he will do a great job, and if you have Mathematics in Barcelona, from 10-14 lectures, one at ETH in Zürich on information for (or concerning) EMIS, July, and of course our Council meeting on ‘Mathematics in Finance’ from 13-16 June, write directly to him (volker.mehrmann 7-8 July. You can now register electronical- and one at the University of Crete in @mathematik.tu-chemnitz.de). Prof. Jean- ly for the Congress (see our web page, Heraklion on ‘Time Reversed Acoustics’ Pierre Bourguignon has agreed to be the www.emis.de) and all information on the from 18-20 June. We are now seeking pro- Chair of the Special Events Committee, Council can be obtained directly from our posals for EMS lecturers for the year 2001; and I am looking forward to the fifth secretary, Prof. David Brannan. the deadline is 30 September 2000. The Diderot Mathematical Forum on Let me first recount briefly what hap- same deadline applies to proposals for ‘Mathematics and Telecommunications’, pened in 1999. Our journal JEMS had a EMS summer schools for the year 2002. to take place this coming September. Prof. successful start; the first volume is com- In 1999 the executive committee creat- Renzo Piccinini is now the chairman of the plete and the level of articles has been ed a working group to study the possibility Summer School Committee and Prof. extremely high. For a journal as young as of founding EMSph, a European Claude Lobry chairs the Committee on JEMS, the number of subscriptions is Publishing House run by the EMS. To cre- Developing Countries. I wish them all suc- already quite large. ate such a publishing house has many cess in their positions. Last year the 5th framework pro- aspects and is no way a simple enterprise. Finally, let me mention that last year was gramme started and this created a lot of We therefore would like to get input and very exciting and exhausting. I would like activity for the EMS since we submitted five support from our corporate societies. We to apologise to those to whom I was slow in proposals. I am happy to report that all plan to have a meeting at the 3ecm in responding. I hope however to meet all of proposals have been approved. The Barcelona to discuss these plans. We have you at the 3ecm in Barcelona in July. largest, LIMES, involves partners from seven countries and the EMS and is designed to improve Zentralblatt MATH even further. At this point I might add that EMS Council for 2000: the EMS has entered contracts with FIZ Karlsruhe and the Zentrum für Didaktik individual members der Mathematik at Karlsruhe University concerning the Zentralblatt für Didaktik der The Secretariat received 16 nominations for the 15 delegates of individual members Mathematik / International Reviews on on the EMS Council from 2000-2003. One nomination was invalid. As a conse- Mathematical Education. You can find the quence, no ballot was needed. The new delegates for the individual members for electronic version, called MATH-DI, on this four-year period are: our web page www.emis.de. Giuseppe Anichini (Italy), [email protected] When taking office I promised to Vasili Berinde (Romania), [email protected] strengthen the applied mathematics within Giorgio Bolondi (Italy), [email protected] the EMS. One change you may have Alberto Conte (Italy), [email protected] already noticed: the ‘Applications C. T. J. Dodson (UK), [email protected] Committee’ (now called the ‘Applied Jean-Pierre Françoise (France), [email protected] Mathematics Committee’) has made great Salvador S. Gomis (Spain), [email protected] efforts to supply more articles concerning Laurent Guillope (France), [email protected] applied mathematics to our Newsletter. Klaus Habetha (Germany), [email protected] In addition, the Felix Klein Prize has Willi Jäger (Germany), [email protected] been created. It will be awarded to a young Tapani Kuusalo (Finland), [email protected] scientist, or a small group of young scien- László Márki (Hungary), [email protected] tists (normally under the age of 38), who Andrzej Pelczar (Poland), [email protected] use sophisticated methods to give an out- Zeév Rudnick (Israel), [email protected] standing solution that meets with the com- Gerald Tronel (France), [email protected] plete satisfaction of industry to a concrete EMS March 2000 3 EDITORIAL (excuse me for being patriotic). All the For a long time, I have been eager to see major breakthroughs in physics are written these motifs which form such a beautiful in the language of mathematics. And what bridge between the Arab and European Editorial about astronomy, chemistry and, in more cultures. Editorial recent times, biology and geology. I was thrilled and thought proudly to myself: Mathematical posters WMY2000 mathematics stretches back in time to the dawn The other main project of the EMS com- of civilisation, but its impact is really felt in mittee of the WMY2000 is the campaign modern society, and mathematics will continue for suitable posters to be displayed in sub- Vagn Lundsgaard to be vital in the future. The only thing that ways and other public places. As one way of Hansen bothered me was that I found this declara- stimulating the creation of good ideas for tion stated nowhere in the Science posters, the EMS arranged a poster com- (Lyngby, Denmark) Museum. There was only a tiny little room petition in the Spring of 1999, the results with mathematics on display – or so it of which were announced in the December appeared at first sight. But with hindsight 1999 Newsletter. The poster designers have When asked what the World Mathematical I found the museum filled with realisations given permission for their proposals to be Year 2000 is about, I answer that it is to of mathematical ideas. included in a web gallery. The gallery is make visible to the general public that still under construction, but has recently mathematics bridges gaps in culture, science and Bring out the message been made public; it can be found at the technology. For technology this is true, even The purpose of World Mathematical Year web address http://www. mat.dtu.dk/ems- in the very literal sense. For culture and 2000 is to bring the power of mathematics gallery/. Comments and suggestions for the science it is more profound. How this idea to the attention of the general public, web gallery are welcome. In particular, the dawned upon me, I am about to tell you. pupils in schools, students at the universi- gallery has a supplementary section, where ties, teachers at all levels, politicians,... My suitable ideas for posters independent of Getting overwhelmed goodness, we have a much better case than the competition can be included. During holidays in London last summer, I we think. They cannot do without mathe- first visited the Science Museum and the matics. The sad thing is that they do not Working hard next day the British Museum. As many know about it. We must tell them in every The main work in connection with the other visitors, I was thoroughly impressed. continent of the world that mathematics is World Mathematical Year has by necessity It is, indeed, fantastic what homo sapiens a language common to all nations and that to be done by all the local committees set has created throughout history. its values are eternal and universal. up in many countries to organise mathe- Nevertheless, my reflections on the matical events. It is, indeed, hard work to impressions followed an unexpected road. Greek vases catch the interest of the public media, Suddenly during the visit to the Science The next day we visited the British newspapers and TV, and it is very difficult Museum, I had a feeling of being com- Museum. Same experience. Mathematics to make the local postal services issue spe- pletely overwhelmed. Is there really more behind many things in art, paintings, cial stamps with a mathematical theme. to be invented? Are the achievements in sculptures, pottery, ornamentation, and The various mathematical societies and the natural sciences and technology in the marvellous mathematical ideas exploited teachers’ associations will organise compe- last millennium, and particularly in the last in the construction of clocks and instru- titions and lecture series, which take much few centuries, so extraordinarily impres- ments from the natural sciences. Despite thought to be properly prepared. The sive that it can be demoralising for the this, there was only one small cabinet reward is that the sum of these activities young and ambitious who think about their devoted to mathematics. But also in the will be to the benefit of mathematics all future careers? What is there still to be British Museum we found mathematics over the world. done? What can they do to make a differ- providing major links throughout the vari- It is fortunate that there is so much to ence? Somehow I felt that here might be ous exhibitions. Again a very happy day on tell about mathematics. For it is indeed one of the reasons for the decline in enrol- the beaches of the mathematical ocean. true that mathematics bridges gaps in culture, ment at universities in the natural and the A slight disappointment was that the science and technology. May I wish you all a technological sciences. I got, admittedly, exhibition of Greek vases was closed on the very exciting World Mathematical Year slightly depressed. But wait, there are hope day of our visit. And my main purpose in 2000. and encouragement to be found. this particular visit to the British Museum was to see the Greek vases. I wanted to see Slowly recovering the planar reflection symmetry realised by Where is the red thread connecting all the ‘umklappung’ (a spatial 180-degree rota- things in the Science Museum? Where are tion) in the famous motif of Ajax and the great unifying principles? Where is the Achilles at a board game. common language? Suddenly it came to me in a Platonic flash: the principles of math- The tour goes to Spain ematics are behind every single construction in This summer the tour goes to Spain – and technology. My adrenalin got a kick. I start- first to Granada, where a satellite confer- ed talking and talking to my very tolerant ence to the Third European Congress of and patient wife about this wonderful Mathematics takes place from 3 to 7 July. power of mathematics: “Look at these The conference in Granada is one of the steam engines, look at these marvellous main projects which the EMS committee of clocks. And do you think that man would the WMY2000 has worked for. We are ever have gone to the moon without math- extremely grateful to the people in ematics?” Granada who have realised the project of bringing Europeans and Arabs together in Here we go the old city with the famous Alhambra, cas- We went all the way from the philosophy of tle of the Moorish kings. A main part of the Plato and Aristotle on the mathematical conference will discuss the historical per- principles behind the universe, past the spectives of both cultures to our present work of the ancient Greek mathematicians mathematical knowledge. A central role is and philosophers, to Galileo (who died in played by the fascinating ornamentation in 1642, the year of the birth of Newton), and Alhambra, in which the seventeen planar Greek vase in the British Museum, depicting then to Maxwell, Einstein and Bohr crystallographic groups are represented. Ajax and Achilles at a board game. 4 EMS March 2000 EMS NEWS Introducing the WMY2000 team

Below we feature some of the international team for World Mathematical Year 2000.

Vagn Lundsgaard Hansen has been Professor of mathematics at the Technical University of Denmark, Lyngby (Copenhagen), since 1980. He earned a Masters degree in mathematics and physics from the University of Aarhus, Denmark, in 1966, and a PhD in mathematics from the University of Warwick, England, in 1972. He has held positions as Assistant professor, University of Aarhus, 1966-69; Research fellow, University of Warwick, 1969-72; associate professor, University of Copenhagen, Denmark, 1972-80, and Visiting professor at the University of Maryland, USA, in Fall 1986. He is fascinated by the interaction between the abstract and the concrete in mathematics, and by finding serious mathematics in the description and explanation of phenomena from the real world. He has written research papers in topology, geometry, and global analysis. Other research interests include mathematical education and the history of mathematics. He is the author of several books, including the general books Geometry in Nature and Shadows of the Circle. He is currently President of the Danish Academy of Natural Sciences and has been Vice-chairman of the Danish Natural Science Research Council. He is currently Chairman of the World Mathematical Year committee of the EMS.

Ronnie Brown was an undergraduate and postgraduate at Oxford University, and was supervised by Henry Whitehead (who died suddenly in 1960) and Michael Barratt. He taught at the Universities of Liverpool and Hull, before being appointed in 1970 to the Chair of pure mathematics at the University of Bangor, North Wales, where he has remained ever since. His first work was on the algebraic and general topology of function spaces, where he initiated the notion of a category adequate and convenient for topology. Later, while writing a book on topology, his attempts to explain the fundamental group led to an exposition that emphasised the use of groupoids as a flexible generalisation of groups, leading to the view that groupoids can be used to define higher-dimensional versions of groups which would reflect some old intuitions in topology and algebra. This has led to a substantial new theory with wide ramifications. Since 1983 he has become involved in the popularisation of mathematics, and led a team that produced an exhibition ‘Mathematics and knots’ for the UK ‘PopMaths Roadshow’ in 1989. This exhibition has since been shown widely and in 1997 was put on the web. The exhibition also led to a collaboration with the sculptor John Robinson (see the website: http://www.bangor.ac.uk/ma/CPM/).

Mireille Chaleyat-Maurel is Professor of mathematics at the Université René Descartes, Paris, France. Her main area of research is probability theory, particularly stochastic differential equations. She was a member of the Council of the Société Mathématique de France from 1987 to 1994 and in charge of communication and Publicity Officer of the EMS from 1994 to 1998. She is the Chair of the IMU Committee for WMY2000 and has been Editor-in-chief of the WMY2000 Newsletter since 1995. She is currently the coordinator of the project RPA-MATHS (Raising Public Awareness in Mathematics) which has been selected by the European Commission as a part of the European Science and Technology Week (6-12 November 2000).

José Francisco Rodrigues is Director of the Centro de Matemática e Aplicações Fundamentais of the University of Lisbon, Portugal. He is also Professor of mathematics in the Faculty of Science at the same university, where he received his PhD in 1982, after studying at the Université de Paris VI and lecturing at the ENSM/Université de Nantes. He has visited several universities and Research Institutes in Europe, USA and Asia. His mathematical interests lie mainly in non-linear partial differential equations and con- tinuuum mechanics, but he is also interested in the history and communication of mathematics, where he has written and edited a few books and articles. He chaired the European Science Foundation scientific programme ‘Mathematical treatment of free boundary problems’ (1993-97) and he is coordinating the edition of the new Oxford University Press mathematical journal Interfaces and free boundaries. He was Vice-President of the Sociedade Portuguesa de Matemática during the foundation of the European Mathematical Society, and coorganised the Fourth Diderot Mathematical Forum on ‘Mathematics and Music’. EMS March 2000 5 INTERVIEW acters of the classical groups and a later paper with Brauer on spinors in n dimensions. All this led to my 1944 disputation InterInterviewview withwith with Fenchel as the opponent of the faculty. In my now long-forgotten thesis I constructed maps from one representation space to another one on which the two- LLarsars GårGårdingding sided action by a group induces a linear representation of the same group. With interviewers: Karl Gustav Andersson and Anders Melin this I could construct a lot of wave equations which share with Dirac’s equation the property that every component satisfies You have just passed 80. How are you? the wave equation. This was the end of that I am well for my age and still able to per- subject for me, except for a lasting interest form simple tasks. in quantum physics.

Where did you go to school? And after the thesis? As a child I went to four schools, I believe. In my first published paper I deduced After that it was the local gymnasium in from its characteristic function the so- Motala, a small town in Sweden. I left it in called Wishart distribution for the second- 1937. order moments in multivariate normal dis- tributions. This gave me an interesting How did you and mathematics meet? Were analogue of Riesz’s fractional integration. there any special mathematical questions After my thesis I spent a year in that occupied your mind? Cambridge, England, as a British Council I was attracted to the subject very early. scholar. I spent my time adapting my frac- After reading a simple introduction to tional integration to solve Cauchy’s prob- infinitesimal analysis I got interested in lem for two interesting hyperbolic differ- series. As a special study at school I studied ential equations and discovered that the the chapter on series in de la Vallée supports of their fundamental solutions Poussin’s textbook and solved the simple had large codimensions. In other words exercises there. Mathematics was my first they had lacunas – open subsets between subject at Lund University when I arrived Lars Gårding the expected conical supports and the here in 1937. actual supports. Leonard Eugene Dickson’s Elementary In Cambridge I met Hardy, Littlewood, theory of equations. It was difficult, espe- How was academic life in your time as a Besicovich and Dirac and other interesting cially the proof by induction that symmet- student? persons, but the university had very little ric functions of the roots of an equation are Briefly, the situation in the 1930s and mathematical research in wartime. I and polynomials in its coefficients. 1940s may be described as follows. Lund the Polish mathematician Aronszajn start- was a small, classical university with four ed a mathematical seminar of our own. faculties and had around 2500 students. Who were your teachers? The year ended with a course for British Otto Frostman taught elementary analysis. All newcomers had passed a centrally Council Scholars at St Andrews with the Marcel Riesz had a seminar for semi- organised high school examination in title: The Ideals of the British Empire. Under advanced students. The problems included about six subjects. Every subject at the uni- this title the local faculty lectured on post- difficult elementary inequalities. This sem- versity was taught by just one professor war problems and their young listeners inar was my first encounter with the expe- who examined his students at most six had a wonderful time among themselves. rience and judgement of a first-class math- times a year. A regular curriculum had ematician. Later he became my mentor three grades each requiring in theory half And then? also outside mathematics. a year of study. After that there were the A year at home. The year 1946-47 I spent higher examinations with a doctor’s at Princeton University on a stipend from degree at the top. The university adminis- At the time Riesz was known for his work in the American-Scandinavian Foundation. tration employed less than ten people. summation theory and his convexity theo- Solomon Bochner helped me to rent a Students from all faculties lived in rented rem. Did he teach you classical function cheap room and got me a desk at Fuld rooms. For social life they were organised theory? Hall, then the mathematics department. No, he had acquired other interests. There in fraternities based on Swedish regions. My roommate was Raphael M. Robinson, were two of them: modification of They also had an academic society with a married to the logician Julia Robinson. He Hadamard’s partie finie by analytical con- house of its own. gave a course on analytic functions. I felt I tinuation with respect to a parameter, that Mathematics had two professors and had been promoted above my competence gave a kind of fractional integration, and every year received around 30 students. level. also the nature of Dirac’s spinors. These Most of them saw a future in teaching and were the constant subjects of conversation. aimed at a science exam with chemistry, What other mathematicians were there at I often met Riesz in a café, alone or togeth- physics and mathematics that took that time? er with some members of the regular between four and five years. Many. The mathematicians at the depart- mathematics seminar. During the war we Now the University is fifteen times larg- ment – for instance, Solomon Lefschetz, had very few visitors, but Harald Bohr er, teaching is organised in lessons rather Emil Artin and Claude Chevalley, and from Copenhagen lectured once and in than lectures, examinations are frequent many others. The main subject at the 1943 the flight of Jews from Denmark and many students refer to the university department was topology. Lefschetz was made Werner and Käthe Fenchel stay in as ‘the school’. I am not saying that things writing a colloquium treatise on the sub- Lund for two years. were better before. Student life has lost its ject. There were also guest lectures. I unity but the fraternities are still there and remember William Hodge talking about many things are as before. In fact, univer- What did your contacts with Riesz lead to? his harmonic integrals and explaining that To my interest in van der Waerden’s sity life for people in their twenties has the topology he got was independent of Moderne Algebra and in group representa- many time-invariant features. the metric he used. I was interested in tions. I got a kick out of Herman Weyl’s quantum mechanics and had the good for- three articles from the 1920s on the char- What did you start with in mathematics? tune to meet a graduate student in physics, 6 EMS March 2000 INTERVIEW Arthur Wightman. We became friends and There are plenty of lacunas.’ I found work with lacunas. And, perhaps, my book a few years later we collaborated on the Petrovsky’s big article on the subject which on mathematics in Sweden before 1950. representations of the commutation and was an intriguing mixture of analysis, alge- anti-commutation relations. braic geometry and topology. I stayed up How was your time as a professor? The department was only a twenty- all night trying to read it, but I failed to On the whole it was a wonderful time. In minute walk away from the Institute for understand even the simplest topological the 1950s the department had only three Advanced study. There one could see the criterion for a lacuna. I tried out some permanent positions, two full professors permanent members Herman Weyl, John topologists at the department for the sim- and one assistant professor, Nils-Erik von Neumann, Marston Morse and Carl plest parts of the article, but did not get Fremberg, who taught elementary calculus Ludvig Siegel and, less frequently, Kurt very far. Lefschetz was quoted but I did not and analytical geometry. After the war it Gödel and James Waddell Alexander. All dare consult him. It was only twenty years was also possible to employ young people except Morse and Alexander had left later that I understood enough about fun- as assistants. This made the department a Europe to escape Nazism. damental solutions to be able to enlist alge- lively place. Plenty of gifted young people braic and topological help from Atiyah and wanted to study mathematics. I can only Did you meet them? Bott. mention a few, Carl Hyltén-Cavallius, I heard von Neumann lecture on function- Lennart Sandgren, Jan-Erik Roos, Jaak al analysis and Weyl on integral equations. Then what? Peetre. And, of course, Lars Hörmander. I also met with and talked to most of them. I returned to Lund for a year, married Eva, His 1955 thesis was a big event. Very soon There were plenty of opportunities at the and spent another year at the Institute and afterwards he became a professor in very democratic American afternoon teas applied for a professorship. Lund had then Stockholm. and cocktail parties. I can also boast that I two retiring professors, Marcel Riesz and I enjoyed teaching. My first thesis stu- was invited to Bochner’s dinner parties Nils Zeilon. In 1952 I and Åke Pleijel were dents, Lennart Sandgren and Gunnar with von Neumann as a featured guest. nominated professors of mathematics at Bergendal, left mathematics for successful Lund University. Pleijel had studied with careers in politics and administration. My But what did you do? Carleman and worked mostly with the other rather few thesis students stayed in I wanted to construct fundamental solu- asymptotics of vibration problems. the academic pen. I am the originator of a tions of hyperbolic differential operators, number of minor theses that were pub- but then I found that this required me to What about Gårding’s inequality? lished as papers, but I found it difficult to define such operators. In the end I found At the end of the 1940s I realised by read- answer for major theses. As a thesis super- an intrinsic definition: finite propagation ing Herman Weyl that the solution of visor one feels the responsibility to give velocity and continuity. But I was also try- Dirichlet’s problem for high-order elliptic problems or tasks with enough carrying ing to learn modern mathematics. At the equations just amounted to finding lower power to give a substantial result. This, by time this meant abstract harmonic analysis. bounds for the corresponding Dirichlet the way, was a relatively new situation. My most intimate friends belonged to a integrals. I found the proof one morning Before the war, one had to write a thesis group of younger mathematicians who vis- after a bad night following a disastrous entirely on one’s own with very little out- ited the Institute – Irving Segal, Fred intervention in a disputation on econom- side intervention. Mautner (an ardent admirer of Herman ics. I am known internationally mostly for During all my active time we lived Weyl) and Richard Arens. Their chief ide- this inequality. under the pressure of swelling crowds of ologue was Segal. The Gods were André beginners. More and more assistants were Weil, who had written a book about What do you consider are your main math- employed till the government decided to Fourier analysis on commutative locally ematical achievements? create new permanent teaching positions compact groups, and Israel Gelfand who The intrinsic definition of hyperbolicity, called lectorships. This reform came in the had invented normed rings a few years ear- Gårding’s inequality, my existence proof middle 1960s and coincided with the 1968 lier. The literature of the field was new but by functional analysis alone of Cauchy’s student movement. The tricentennial of scarce: photoprints of new articles in problem for hyperbolic equations, my Lund University that year had to be cele- Russian journals by Gelfand, Rauikov, Naimark and other Russians. Bochner called it the mathematics of photoprints. Both Segal and Mautner were great talk- ers. The most common phrase in the common room of the Institute used to be ‘take a locally compact group’ .

Did you write in this field? I proved that the infinitesimal generators of unitary representations of a Lie group have a common dense domain. This was partly a result of lectures at the department by Jean Delsarte in which he introduced Laurent Schwartz’s theory of distrib- utions. In fact, there I learned about infi- nitely differentiable functions with compact support. Delsarte had only two people in his audience and I was one of them. The students at the department were too busy with topology or homotopy to be interested. I tried topology but found the introductory triangulations uninspiring.

Did you try other things? My own examples of lacunas in wave propagation made me curious about the lacuna phenomenon. When Irving Segal heard about my interest he said ‘Why don’t you Lars Gårding with Michael Atiyah at the Nordic Summer School of Mathematics, Tjörn, in 1969. look at the latest issue of the Sbornik. (Photograph taken by Christer Bennewitz) EMS March 2000 7 INTERVIEW brated under the cover of mounted police. Petrovsky, then Rector of Moscow University and receiver of an honorary degree, was tactfully unimpressed by the Journal Surveys on police. The minister of education at the time, Olof Palme, decided to pacify the students by letting them into practically of the Mathematics every university committee. With time this had a calming effect. European Since the 1960s we have experienced a for Industry constant expansion in terms of students and professors. Now with the influx of Mathematical Managing Editor: H. Engl, Linz mathematicians from the east we are get- Editorial Board: T. Beth, C. ting more and more like an American Society Cercignani, M. Deistler, R. E. mathematics department. Ewing, D. Ferguson, A. Friedman, A. Gilg, R. Glowinski, M. Can you tell us, finally, about some impor- (JEMS) Groetschel, H. Hagen, R. Janssen, tant events in your professional life? K.-H. Keil, U. Langer, T.-T. Li, B. The Contents lists of the fifth, sixth My many visits to the States and the many Lindorfer, A. Louis, P. Markowich, and seventh issues of the JEMS are as lifelong friendships that I acquired there – R. Mattheij, H. Neunzert, J. follows: for instance with Fritz John, Peter Lax and Priaux, P. Rentrop, A. A. Louis Nirenberg. Hörmander’s 1968 Samarskii, W. Toernig Volume 2, Number 1: return to Lund from the Institute for Surveys on Mathematics for Charles Pugh and Michael Shub, Advanced Study was very important. It has Industry aims to bridge the gap Stable ergodicity and julienne quasi-con- been fascinating to see the birth and devel- between university and industry by formality opment of microlocal analysis of distribu- presenting mathematical methods Alan Weinstein, Almost invariant sub- tions. Hörmander dominated mathematics relevant for industry, and by manifolds for compact group actions at the department both by his papers and exposing industrial problems that Fang-Hua Lin and Tristan Rivière, books and his excellent teaching material are of interest to mathematicians. Erratum to Complex Ginzburg-Landau on various branches of analysis. His four To achieve this goal, the journal equations in high dimensions and codimen- volumes on the analysis of partial differen- publishes surveys on new mathe- sion two area minimizing currents tial operators form an impressive master- matical techniques; on established piece. mathematical techniques with a Volume 2, Number 2: Petrovsky’s lacuna paper and the fact new range of applications; on Alain-Sol Sznitman, Slowdown estimates that Russian mathematicians were forced industrial problems for which and central limit theorem for random to write in Russian made me try to learn appropriate mathematical models walks in random environment this language at the end of the 1940s. I or methods are not yet available; Bruno Kahn and R. Sujatha, Motivic came so far that I could review mathemat- and broad historical surveys. cohomology and unramified cohomology of ical papers in Russian for both the Furthermore, coverage includes quadrics Mathematical Reviews and Zentralblatt. comparisons of mathematical mod- Birkett Huber, Jörg Rambau and Actually, all of my professional life was els or methods for particular Francisco Santos, The Cayley trick, lift- marked by the cold war and the restricted industrial problems, and descrip- ing subdivisions and the Bohne-Dress the- access to Russia. The first opening in tions of mathematical modelling orem on zonotopal tilings mathematics came in 1956 when a number techniques. Articles of general of Western mathematicians were invited to interest about the use of mathe- Volume 2, Number 3: a congress in Moscow for mathematicians matics in industry will also be con- J. Lindenstrauss and D. Preiss, A new from the entire Soviet Union. We were the sidered. Papers should be submit- proof of Fréchet differentiability of first foreign guests in twenty years. For me ted to a member of the Editorial Lipschitz functions this visit was an unforgettable mixture of Board or to the Managing Editor J. Filo and S. Luckhaus, mathematics, confidences about repression who also welcomes suggestions for Homogenization of a boundary condition and prisons and my first encounter with possible topics by prospective for the heat equation some important mathematicians. Olga authors. (Abstracted/indexed in G. van der Geer and T. Katsura, On a Ladyzhenskaya, Olga Oleinik, Ivan Mathematical Reviews, Zentralblatt stratification of the moduli of K surfaces Petrovsky, Israel Gelfand and many others 3 für Mathematik and Database became my friends. One day Pravda car- MATH.) ried an article on the cult of the personali- Special offer for members of ty. This was part of the speech that EMS: Khruschev had given at the 20th party con- DM 102, including carriage gress and really dealt with the crimes of the Correction to charges for a subscription to Stalin era. Our interpreters were dumb- Surveys on Mathematics for Industry founded. Newsletter 34 (regular price: DM 1151). The My collaborations with Arthur subscription is for one volume On page 28 of EMS Newsletter 34 Wightman and with Michael Atiyah and (four issues) and can be started (December 1999), the article on the Raoul Bott were both important and with either Volume 9 or 10; EMS-WiR Summer School, Numerical delightful. My collaboration with Jean Volume 9/1 and 9/2 have already Simulation of Flows was written by Leray in the 1960s petered out, but was been published. The table of con- Jürgen Geiser and Torsten Fischer, important for me since I learned a lot from tents appears on the journal’s members of the research team of Prof. him about Petrovsky’s paper. homepage: http://www. G. Wittum (University of Heidelberg), I have now talked too long, but let me springer.at/smi Orders must be and not by Rolf Jeltsch. Also, it was say that my marriage to Eva, my many placed directly with Springer- not mentioned that support for this friendships with mathematicians in Verlag, Subscription Department, summer school was given by Scandinavia and abroad, and the kindness Sachsenplatz 4-6, A-1201 Wien, UNESCE-ROSTE, the Venice office of and understanding I have met from math- Austria. Fax: 0043 1 330 24 26 62 UNESCO. We apologise for these ematicians in Lund – for instance, my two or email: [email protected] errors. interviewers – have given me a happy life. 8 EMS March 2000 ANNIVERSARIES 2000 Anniversaries

to me as an old friend. [3, p.123]. (in absentia). She was the first woman in Sonya Kovalevskaya As Sonya began to develop mathemati- modern Europe to receive a doctorate in cally she also matured politically, becom- mathematics and one of the first women to (b. 1850) ing increasingly interested in the nihilist receive a doctorate in any field. June Barrow-Green philosophy. The nihilists wanted to change The three dissertations were on partial differential equations, Abelian integrals, Sophia (Sonya) Vasilievna Korvin- and Saturn’s rings, but it was the one on Krukovsky was born in Moscow on 15 partial differential equations that had real- January 1850, the daughter of a general in ly excited Weierstrass. In it was contained the Russian artillery. In 1858 her father what is today known as the Cauchy- retired and took his family to live at their Kovalevskaya theorem, an important tool large estate in Palibino. As she related in in establishing the existence or non-exis- her autobiography, the move had an unex- tence of analytic solutions of partial differ- pected consequence with regard to her ential equations. The paper was published mathematical development: in Crelle’s Journal in 1875 [4] and her work When we moved permanently to the country, the was greatly admired by other mathemati- whole house had to be redecorated and all the cians, including Charles Hermite and rooms had to be freshly wallpapered. But since Henri Poincaré, both of whom spoke of the there were many rooms, there wasn’t enough result in glowing terms [2, p.241; 9, p.26]. wallpaper for one of the nursery rooms. Because Meanwhile, in 1872, Vladimir had ordering wallpaper involved sending to St gained a doctorate in geology at Jena. The Petersburg it was a very complicated business, couple had reconciled in 1873 and they and it really wasn’t worthwhile to go through all returned to Russia expecting to be of that for just one room. It was all waiting for appointed immediately to prestigious a propitious occasion, and in expectation of this teaching posts in St. Petersburg. However, the maltreated room just stood there for many Kovalevskaya soon discovered that as a years with one of its walls covered with ordinary the traditional tsarist society and believed woman she was unable to obtain a position paper. But by happy chance, the paper for this in the power of education to implement in the Russian system of higher education preparatory covering consisted of the litho- social change. They had a strong faith in and the only opening for her in mathe- graphed lectures of Professor Ostrogradsky on the natural sciences and unequivocally sup- matics was to teach in the lower grades of a differential and integral calculus, which my ported the equality of women. It was a girls’ school. But since, as she sarcastically father had acquired as a young man. combination of ideas that Sonya found observed, “she was not strong in the multi- These sheets speckled all over with strange particularly appealing and in September plication tables” [2, p.127] she did not seri- unintelligible formulas, soon attracted my atten- 1868 she married fellow nihilist Vladimir ously consider such a position. Instead she tion. I remember as a child standing for hours Kovalevsky. The marriage which was ini- turned to other intellectual pursuits – writ- on end in front of this mysterious wall, trying to tially ‘fictitious’ – it had been contracted in ing fiction, theatre reviews and popular sci- figure out at least some isolated sentences and to order to give Sonya the possibility of study- ence reports for newspapers. Initially she find the sequence in which the sheets should fol- ing at a university abroad – proved a diffi- tried to combine mathematics with her low one another. From this protracted daily con- cult one. other interests and kept up a mathematical templation, the outer appearances of many of Initially the couple lived in St correspondence with Weierstrass. these formulas imprinted themselves in my mem- Petersburg, but as a woman Kovalevskaya However, in September 1875 her father ory; indeed their very text left a deep trace in my was unable to gain admission to university, died. His death came as a shock to her and brain, although they were incomprehensible to and early in 1869 they travelled to Vienna. for reasons possibly associated with his me when I was reading them.” [3, pp.122- Unable to find any sympathetic mathe- death, her health and the change in her 123]. maticians in Vienna, they moved on to relationship with Vladimir (they had final- Sonya was fortunate that her father, who Heidelberg. In Heidelberg Kovalevskaya, ly consummated their marriage in early was well educated and fond of mathematics the first woman student at the University, 1875), she turned completely away from and science, encouraged her study of studied physics with Gustav Kirchhoff, mathematics. From October that year and mathematics, at least at an elementary physiology with Hermann Helmholtz, and for the next three years Weierstrass heard level. Her ability was recognised by a fami- mathematics with Leo Königsberger and nothing from her. The Kovalevskys ly friend, Nikanorovich Tyrtov, professor Paul DuBois-Raymond. immersed themselves in the salon life in of physics at the St Petersburg Naval In the autumn of 1870, with recommen- the capital, engaging in what was to Academy, and in 1867, persuaded by dations from Königsberger and DuBois- become a catalogue of ruinous financial Tyrtov, her father consented to her taking Raymond, Kovalevskaya journeyed to speculation. lessons from Alexander Strannolyubsky in Berlin to work with Karl Weierstrass. The In early 1876 Kovalevskaya was visited St Petersburg. From Strannolyubsky she University was closed to women but by another of Weierstrass’s students, the studied differential and integral calculus, Weierstrass agreed to tutor her privately. young Swedish mathematician Gösta and, as she recalled, her hours of perusing As Kovalevskaya herself was later to say: Mittag-Leffler. On this occasion he had the wallpaper at Palibino finally paid off: These studies had the deepest possible influence been sent by their common adviser to try Many years later ... I took my first lesson in dif- on my mathematical career. They determined to reawaken her interest in mathematics. ferential calculus from the eminent Petersburg finally and irrevocably the direction I was to fol- He did not succeed, but he left with a glow- professor Alexander Strannolyubsky. He was low in my later scientific work: all my work has ing impression: amazed at the speed at which I grasped the con- been done precisely in the spirit of Weierstrass. More than anything else in St. Petersburg what cepts of limit and of derivatives, “exactly as if [3, p.218] I found most interesting was getting to know you knew them in advance”. I recall that he Under Weierstrass’s supervision Kovalevskaya ... As a woman, she is fascinating. expressed himself in just those words. And, as a Kovalevskaya completed three disserta- She is beautiful and when she speaks, her face matter of fact, at the moment when he was tions and in 1874, with recommendations lights up with such an expression of feminine explaining these concepts I suddenly had a vivid from Lazarus Fuchs and Heinrich Weber, kindness and highest intelligence, that it is sim- memory of all this, written on the sheets of she was granted her doctoral degree summa ply dazzling. Her manner is simple and natur- Ostrogradsky; and the concept of limit appeared cum laude from the University of Göttingen al, without the slightest trace of pedantry or pre- EMS March 2000 9 ANNIVERSARIES tension. She is in all respects a complete “woman the “mathematical mermaid”. be a pernicious and useless monstrosity [1, of the world”. As a scholar she is characterised by Meanwhile she continued to correspond p.109]. In January 1884 she gave her first her unusual clarity and precision of expression with Mittag-Leffler who had been battling lecture, the beginning of a course on par- ... I understand fully why Weierstrass considers with the university in Helsinki to obtain a tial differential equations. The lectures her the most gifted of his students. [8, p.172]. position for her. Ironically his efforts had were well received, she completed her pro- In the spring of 1878 Kovalevskaya’s life come to grief, not because she was a bationary term successfully, and in the turned again. She became pregnant. woman but because she was a known summer Mittag-Leffler secured her During the course of what was to be a diffi- nihilist and the university administrators appointment as an assistant professor. cult pregnancy she reflected on the direc- had feared that her appointment would In addition to joining the staff of the tion of her life, and in August she draw the anger of the tsarist government University, Kovalevskaya became an editor reopened her correspondence with that occupied Finland at the time. In 1881 of Acta Mathematica, the journal founded by Weierstrass, writing to him for advice on Mittag-Leffler became head of the mathe- Mittag-Leffler in 1882 – the first woman to resuming her mathematics. However her matics department at the newly founded join the board of a scientific journal. On return to mathematics was delayed by the university in Stockholm and continued his behalf of Acta she liaised with the mathe- birth of her daughter in October and the efforts on her behalf. maticians of Paris, Berlin and her native Later in 1881 she moved to Paris to Russia, providing an important link work alone on her mathematics. In May, between Russian mathematicians and their Mittag-Leffler arrived and, surprised to western European counterparts. Acta also discover that she had not met any French proved a good vehicle for her own publica- mathematicians, immediately took her to tions. Her doctoral paper on Abelian inte- meet Hermite. In July 1882 she became a grals was published in 1884 [5] and her member of the French Mathematical paper on the propagation of light in 1885 Society and soon got know several of the [6]. (Her paper on the shape of Saturn’s best mathematicians, including Hermite, rings was also published in 1885, but in Poincaré, Picard and Darboux. However, Astronomische Nachrichten [7].) while Kovalevskaya was making an impres- 1885 was a year that was to prove one of sion on mathematicians in Paris, Vladimir, the most productive of her mathematical who had completely abandoned palaeon- career. She had been working hard at the tology in favour of financial speculation, rotation problem and after four years had was facing complete financial ruin. In April finally made a breakthrough. By the spring 1883 he committed suicide. When of 1886 she had solved in principle as Kovalevskaya heard the news she locked much of the problem as she could hope to herself into her room, refused to eat or do, and the rest was a matter of working allow a doctor near her, blaming herself out the details, although a considerable death of her mother in February 1879, fol- for his death. Five days later she lapsed amount remained to be done. Aware of the lowed by the failure of the Kovalevsky into a coma and the doctor was able to importance of her work, she left for Paris investments. force feed her. She recovered conscious- in order to communicate her ideas to the The financial disaster, which culminat- ness the following day. She began working mathematicians there. Her visit was to ed in the sale of most of the family posses- intensively and soon completed her have important consequences. sions, affected Vladimir deeply and he research on the light diffraction problem. In the autumn of 1886 the announce- withdrew from society. Kovalevskaya on In the early summer she travelled to ment for the Prix Bordin of the French the other hand responded more positively. Moscow and on the way stopped off in Academy of Sciences to be awarded in She contacted the Russian mathematician Berlin to show Weierstrass her work. 1888 was made. The topic: “Improve, in Chebyshev who invited her to give a paper Weierstrass was satisfied and encouraged some important point, the theory of the at the 6th Congress of Natural Scientists to her to write up her results for presentation. movement of a rigid body” had been cho- be held in St Petersburg in January 1880. At the end of August she delivered her sen specifically with Kovalevskaya in mind. She dug out her unpublished dissertation paper at the 7th Congress of Natural Due to competing concerns in her person- on Abelian integrals, translated it from Scientists in Odessa. al life she was unable to complete her work German to Russian in one night, and pre- Meanwhile, Vladimir’s death had by the deadline of 1 June 1888, but she sented it to the conference. It met with an caused relief in mathematical circles. As a sent in a half-finished version and, with the enthusiastic reception, and Mittag-Leffler, widow Kovalevskaya would encounter Prize Committee’s permission, submitted a who was in the audience and now a profes- fewer social obstacles to her mathematical revised, although still incomplete, version sor at the University of Helsinki, left deter- career than she would as a single or mar- in the late summer. mined to find her a university position. ried woman, and Mittag-Leffler seized the In December she went to Paris to hear In the spring of 1880 the Kovalevskys chance provided by her new status. the result. As she herself later wrote: moved to Moscow, Kovalevskaya intent on Together with a group of other professors Some fifteen papers were presented, but it was participating fully in scientific life. In he made a concerted assault on the admin- mine that was found deserving of the prize. And October she travelled to Berlin to visit istration of the new Stockholm University that was not all: in view of the fact that the same Weierstrass. On her return in January and by skilful manoeuvring managed to topic had been assigned three times running and 1881 she found that Vladimir was again obtain for her a temporary post. remained unsolved each time, and also in view badly in debt. She was completely exasper- In September 1883 Sonya heard the of the significance of the results achieved, the ated and to all intents and purposes their news. The offer – as a privat-docent – was a Academy voted to increase the previously marriage finally ended. The household common one given to people on the com- announced award of 3,000 francs to 5,000. [3, was dissolved in March 1881 and she and pletion of their doctorate. She was to be on p.227]. her daughter left for Berlin where she probation for a year, during which time Kovalevskaya’s memoir was a triumph threw herself totally into mathematics. She she would receive no salary and no official and earned lavish praise from her contem- saw Weierstrass frequently and devoted status. Her pupils would pay by private poraries. She had discovered a special (and her research to the study of two topics: the arrangement and her situation would be complicated) case of the problem that was propagation of light in a crystalline medi- reviewed at the end of the academic year. capable of a closed solution. It was a case in um – a subject to which she had been led In November she arrived in Stockholm. which the body is asymmetric. The partic- by studying the work of the French physi- Her reception was mixed – one of the pro- ular novelty of her solution lay in her cist Lamé – and the rotation of a solid body gressive Stockholm newspapers hailing her application of the recently developed theo- about a fixed point, a particularly attrac- as a “princess of science” [2, p.179] while ry of theta functions to solve hyperelliptic tive but elusive problem which had become the Swedish dramatist Strindberg consid- integrals. Prior to her work the problem known by the German mathematicians as ered a female professor of mathematics to had only been completely solved for two 10 EMS March 2000 ANNIVERSARIES cases, in both of which the body is symmet- Differentialgleichen, Journal für die reine und angewandte an Italian translation of Gauss’s celebrated rical. In the first, solved by Euler, the cen- Mathematik 80 (1875), 1-32. work on conformal representation, and tre of gravity of the moving body coincides 5. S. Kovalevskaya, Über die Reduction einer bes- took up the question of when the geodesics with the moving point, and in the second, timmten Klasse abel’scher Integrale dritten Ranges auf on a surface can be represented as straight solved by Lagrange, the centre of gravity elliptische Integrale, Acta Mathematica 4 (1884), 393- lines on the plane. When this can be done and the fixed point lie on the same axis. In 414. the equation for the geodesics takes a spe- the words of the Prize Committee (who, in 6. S. Kovalevskaya, Über die Brechung des Lichtes in cial form. Beltrami deduced that geodesics theory, were not supposed to know the crystallinischen Mitteln, Acta Mathematica 6 (1885), 249- cannot always be represented as straight identity of the entrants to the competi- 304. lines, and then investigated those surfaces tion): 7. S. Kovalevskaya, Zusätze und Bemerkungen zu for which such a representation can be The author has not merely added a result of very Laplace’s Untersuchung über die Gestalt der found, and showed that they are precisely high interest to those that were bequeathed to us Saturnringe, Astronomische Nachrichten 111 (1885), 37- the surfaces of constant curvature. by Euler and Lagrange; he has made a pro- 48. His celebrated discovery of the repre- found study of the result due to him, in which all 8. G. Mittag-Leffler, Weierstrass et Sonja Kowalewsky, sentation of a surface of constant negative the resources of the modern theory of theta func- Acta Mathematica 39 (1923), 133-198. curvature on the interior of the unit disc tions of two independent variables allow the 9. H. Poincaré, Sur le problème des trois corps et les followed in 1868. Beltrami, who knew complete solution to be given in the most precise équations de la dynamique, Acta Mathematica 13 (1890), Hoüel’s French translation of some of and elegant form. One has thereby a new and 1-270. Lobachevskii’s work in 1866 (but probably memorable example of a problem of mechanics in 10. Comptes Rendus Hebdomadaires des Séances de not his translation of Bolyai’s the next which these transcendental functions figure, l’Académie des Sciences, CV (1888), 1042. year) explained clearly how the disc car- whose applications had previously been limited ried non-Euclidean geometry. But publica- to pure analysis and geometry. [10] June Barrow-Green [j.e.barrow-green@open. tion of his Saggio (Essay) was delayed, In May 1889, with recommendations ac.uk] is a research fellow in the history of math- because Cremona was worried that the from Bjerknes, Hermite and Beltrami, the ematics at the Open University, UK. paper rested on a vicious circle. It was a University of Stockholm made thorough-going piece of differential geom- Kovalevskaya a full professor – the first etry, but the calculus rested on Euclidean woman to achieve such a position. When Eugenio Beltrami geometry: it was not clear that it could be she returned to Stockholm later in the year (d. 1900) used to describe an alternative geometry. she published two variations on the prize- Beltrami laid the paper aside for some winning memoir. These papers clarified Jeremy Gray time as a result, until he decided that it was points that had been left obscure in her ‘substantially in agreement with some ideas haste to meet the deadline for the Bordin The Italian mathematician Eugenio of Riemann’. Oddly enough, he seems not competition. She had also, on the recom- Beltrami (16 November 1835 – 18 to have discussed geometry with Riemann, mendation of Chebyshev and in opposi- February 1900) is best remembered for his although Riemann had spent a good deal tion to the prevailing academic attitudes work on non-Euclidean geometry, but he of time in Italy talking to Betti and to have towards women, been elected a corre- worked for most of his life on different top- learned of his ideas only through their sponding member of the Russian Academy ics in applied mathematics. He first stud- posthumous publication. of Sciences – the first woman to be so hon- ied mathematics at the University of Pavia, The accounts of non-Euclidean geome- oured. where Brioschi was Professor, but then for try by Bolyai and Lobachevskii were in the In May 1890 she travelled to St a few years financial problems forced him last analysis accounts of the consequences Petersburg to discuss with Chebyshev the to work as the secretary to a railway engi- of making a new definition of parallels. possibility of filling a recent vacancy in the neer, in Verona and Milan after the They did not show that the alternative def- Russian Academy of Sciences – that is, to Austrians had been driven out during the inition did not lead to a contradiction. So become a full member rather than a corre- unification of Italy. His fortunes now Beltrami’s was the first account that was spondent. Chebyshev was encouraging but rooted in sound mathematics, and as such the prejudices were too great and the pro- it drew its share of criticisms from those to ject fell through. Her case was not helped whom a novel geometry was a palpable by the attacks made on her by A. A. absurdity. But Beltrami went on to elabo- Markov, who claimed that her rotation rate an n-dimensional version the next papers were not only incorrect but also year, and among mathematicians at least incompetent – there was a gap in one of the battle for non-Euclidean geometry was her arguments, but nothing irreparable. then steadily won. After the autumn term in 1890 In the 1870s Beltrami’s interests turned Kovalevskaya travelled to Genoa. On the towards physics. He investigated how the return journey she arrived in Copenhagen theory of the Newton gravitational poten- in early February without any Danish tial would have to be modified in spaces of money and had to carry her bags in pour- negative curvature, and formulated the ing rain. When she arrived in Stockholm appropriate generalisation of the Laplace she was ill but managed to teach her first operator (nowadays called the Laplace- class of the new term before taking to her Beltrami operator in his honour). His work bed. Three days later she seemed better on differential parameters led to a theory and discussed her work plans with Mittag- of intrinsic functions and properties of sur- Leffler. But she had contracted pneumonia faces, later taken up by Ricci and Levi- and quite suddenly, early the next morn- Civita. He also contributed significantly to ing, 10 February 1891, she died. the history of mathematics, rescuing Girolamo Saccheri’s prescient study of the Bibliography parallel postulate (published in 1733) from 1. R. Cooke, The Mathematics of Sonya Kovalevskaya, New improved, and he occupied a succession of obscurity. York, 1984. professorships in Bologna, Pisa (where he In 1898 he became President of the 2. A. H. Koblitz, A Convergence of Lives. Sofia came to know Enrico Betti well), and then Accademia dei Lincei and in 1899 a sena- Kovalevskaia: Scientist, Writer, Revolutionary, Boston, Rome, Pavia, and finally Rome again. tor of the kingdom of Italy. 1983. His interest in geometry was sharpened 3. S. Kovalevskaya, A Russian Childhood, New York, by his acquaintance with Cremona, and in Jeremy Gray [[email protected]] is a senior 1978; translated by B. Stillman. the 1860s he studied the representation of lecturer in mathematics at the Open University, 4. S. Kovalevskaya, Zur Theorie der partiellen curved surfaces on a plane. He published UK. EMS March 2000 11 SOCIETIES Societies Corner

Societies corner is a column concerning the society still carries the motto he chose: Een reviving the glorious seventeenth century. mathematical societies in European countries. onvermoeide arbeid komt alles te boven The Napoleonic era transformed the The articles in this column could describe the (Untiring labour overcomes all). It Dutch republic into a kingdom. A new history of a particular society or discuss some expressed the attitude of a membership class of scientists and mathematicians event connected with the society. If you feel that that, in addition to the amateurs just men- emerged: those who performed research your society would interest others, please contact tioned, comprised schoolteachers, survey- on a professional basis. The Wiskundig the column editor, Krzysztof Ciesielski (e-mail: ors, bookkeepers, engineers, instrument Genootschap became more tightly organ- [email protected]) in the first instance. makers, and other practically minded ised. Their yearly meetings were replaced mathematicians. The frontispiece of the by monthly ones, with lectures that, Genootschap’s earliest publication speaks instead of addressing the utility of mathe- volumes. One senses the spirit of Stevin matics and its role in the pursuit of happi- Dutch Mathematical and van Ceulen and, maybe, the hope of ness, now had actual mathematical con- Society Wiskundig Genootschap

In 1600 the Netherlands were at war. Prince Maurits van Nassau (1567-1625), son of William the Silent (1533-84), led the young republic in its fight for inde- pendence from Spain. A military genius who believed in scientific warfare, he counted among his advisers Simon Stevin (1548-1620), a versatile mathematician whose accomplishments ranged from the design of fortifications to the introduction of decimal fractions. At the instigation of Stevin, the Prince attached an engineering school to the newly founded university at Leiden. Its first professor was Ludolph van Ceulen (1540-1610), famous to this day for having computed 35 decimals of π. He taught his courses in the vernacular, using the extensive Dutch mathematical vocabu- lary that Stevin had zealously devised. Dutch is still the only western language having a word of its own for mathematics: wiskunde, which literally translates into ‘knowledge’. The Dutch succeeded in breaking the power of Spain, and their seaborne empire developed into the wealthiest nation of the seventeenth century. Arts and sciences flourished during the Dutch Golden Age. Rembrandt van Rijn (1606- 69) and Baruch de Spinoza (1632-77) achieved world fame, as did Christiaan Huygens (1629-95), Europe’s greatest mathematician in the period before Isaac Newton (1642-1727). The Wiskundig Genootschap has the dis- tinction of being the oldest of all present- day national mathematical societies. It was founded in 1778 by Arnoldus Bastiaan Strabbe (1741-1805), preceptor of mathematics and astronomy and gauger of wine casks of the City of Amsterdam. In 1770, he had started the Oeffenschool der Mathematische Wetenschappen (Training school of mathematical sciences), a periodical that sought to enlighten the many intellectuals who in the Age of Reason solved mathematical problems and puz- zles as a pastime. Commercially, the enterprise was a failure, and Strabbe originally founded the Genootschap in order to finance his numerous publications. The The frontispiece of the first issue of the Wiskundig Genootschap in 1782. 12 EMS March 2000 SOCIETIES tent. Order was brought into the Publications Mathématiques, which the chair at Amsterdam to his brilliant advisee Genootschap’s library, and during the nine- Genootschap published from 1893 to 1934. Luitzen Egbertus Jan Brouwer teenth century the emphasis in their pub- He was a professor at Leiden, whose schol- (1881–1966), whose theorem on the lications gradually shifted from problems arly reputation rested on voluminous invariance of dimension (1910) and fixed- to original work. A prominent and active tables of definite integrals (1858). His bib- point theorem (1911) heralded the advent member was Rehuel Lobatto (1797–1866), liography of early Dutch scientific publica- of algebraic topology. Brouwer created an expert in weights and measures and in tions (1883) is still widely used. intuitionism, and was involved in a famous actuarial sciences, and professor at Delft. The reader may know several Dutch struggle with David Hilbert (1861–1943) Generations of Dutch mathematicians mathematicians who were active around on foundational issues. The Genootschap learnt higher algebra from Lobatto’s the turn of the century. Thomas Joannes published his Collected Works in 1975–76, Lessen over Hoogere Algebra, which Stieltjes (1856–94) was too great for the and instituted in 1970 the Brouwer medal, appeared in 1845 and reached its ninth Netherlands. On the recommendation of awarded once every three years to a math- edition in 1921. Charles Hermite (1822–1901) he was ematician of the highest calibre. Its first During the second half of the nine- appointed to a professorship at Toulouse. recipient was René Thom (b. 1923). teenth century, the Wiskundig Genootschap The Wiskundig Genootschap published his Since 1875, all members of the established contacts with newly founded Oeuvres Complètes in 1914–18. A central Wiskundig Genootschap have received the national mathematical societies in other figure in Dutch mathematics was Diederik Nieuw Archief voor Wiskunde. Itself a European countries. The compilation of Johannes Korteweg (1848–1941), who renewed version of the Archief, which had bibliographic reference works, necessitat- served on the board of the Genootschap for started in 1856, the Nieuw Archief keeps ed by the growing body of literature, 58 years. The Korteweg-de Vries equation renewing itself, a fifth series commencing required international cooperation. David first appeared in the thesis (1894) of his in 2000. The Problem section of this quar- Bierens de Haan (1822–95) was the dri- student Gustav de Vries (1866–1934). In terly is as alive as ever. The Mededelingen ving force behind the Revue Semestrielle des 1913, Korteweg generously ceded his (Notices) van het Wiskundig Genootschap are now largely distributed electronically; the printed edition is expected to be discon- tinued soon. A few years ago, the society assumed responsibility for publishing Pythagoras, a magazine for high school students. The Wiskundig Genootschap never did much government advising. In 1918, they were instrumental in increasing the number of mathematicians at the Universiteit van Amsterdam. After World War II, they were indirectly involved in founding the Mathematisch Centrum, a government- funded institution for research in applied mathematics that serves as a meeting point between industry and academia. In 1954 the Genootschap hosted the International Congress of Mathematicians

Ludolph van Ceulen in 1596, with 20 decimal places of π. in Amsterdam. Queen Juliana (born 1909), another member of the House of Nassau, received the Fields Medallists in her garden, and the Mathematisch Centrum produced table mats, now collec- tors’ items, displaying the Gaussian primes. Soon, the Wiskundig Genootschap will coordinate a committee to advise the Cover design of the Nieuw Archief voor Wiskunde – the first issue of the fifth series (2000). government on educational issues. The EMS March 2000 13 SOCIETIES Mathematisch Centrum celebrates the oldest mathematical societies. It was founded in 1907, the library and the run- computer age with a new name: CWI. founded in 1873, eight years after the ning of literature meetings were moved In 1965, the Genootschap’s monthly London Mathematical Society, five years there. meetings, attendance of which had been after the Finnish Mathematical Society But the responsibility of organising lec- declining, were replaced by the annual and one year after Société Mathématique tures remained within the society for a Nederlands Mathematisch Congres. This two- de France. long time, with a special obligation to day conference, which takes place in April, In the nineteenth century, the possibil- establish and maintain international con- draws a large part of the Dutch mathemat- ities for mathematical studies improved tacts; the first foreign speaker was G. ical community, including high school and the mathematical community was Mittag-Leffler in 1900. Besides the many teachers and industrial and applied math- growing. The Technical University individual lectures given since then by for- ematicians. The annual Winter Symposium (Polyteknisk Læreanstalt) had been found- eigners, it is of interest to mention the is specifically aimed at high school teach- ed in 1829 and a military school in 1830, special invitation to prominent mathe- ers. after the French model. Moreover, the fac- maticians to deliver a series of lectures, This summer, the Wiskundig ulty of mathematics and the natural sci- approximately every other year; this tradi- Genootschap will organise Pi in de ences had been established at tion started in 1921 with David Hilbert. Pieterskerk. The 35 decimals of π that Copenhagen University in 1850, making it The Second World War put a stop to it, Ludolph van Ceulen computed four cen- possible to obtain a degree in mathemat- but the series was taken up again soon turies ago were first published as an ics, so there was a need for a common after the war, with the invitation of Hopf inscription on his tombstone in the meeting place. in 1949. Pieterskerk (Peter’s church) in Leiden. The society was founded on the initia- The international contacts were built Sometime during the early nineteenth tive of Th. N. Thiele. The original rules on personal relationships. There was a dis- century the stone disappeared. On 6 July, stated that the goal was to ‘promote a vivid cussion in 1921 as to whether the society 2000, a reproduction will be unveiled in a cooperation for the benefit of the science should become a member of L’Union ceremony that honours the roots of Dutch of mathematics and its practical applica- Internationale Mathématique. There were mathematics. tions’; later ‘the benefit of teaching in divided opinions, due to the Union’s dis- The Wiskundig Genootschap gratefully mathematics’ was added. The first board crimination against mathematicians from acknowledges the assistance of D. J. Beckers of the society consisted of Thiele, H. G. countries that had been on the losing side and H. W. Lenstra jr. Consult http://www. Zeuthen and Julius Petersen. At that time in the First World War. Harald Bohr was wiskgenoot.nl/ for more information on the there were two professorships in mathe- strongly against membership, and no Wiskundig Genootschap. matics, soon after held by Zeuthen and action was taken. After the Second World Petersen, while Thiele was professor in War, the situation was quite different; the astronomy and later mathematical direc- international mathematical community tor of the life-insurance company Hafnia. was not to repeat the failures from the Danish Mathematical The early history of the society is inti- inter-war period. Børge Jessen served as mately connected with the mathematical secretary of the Interim Executive Society life in Denmark at the time. The society Committee of the IMU from 1950–52, and was responsible for arranging lectures by in September 1951 he could declare the Dansk Matematisk Forening members, who talked about their own official rebirth of the Union. Bodil Branner work or introduced new mathematical As described above, the activities of the topics and concepts. The society also sub- society have often originated in, or been scribed to some journals and established a supplements to, activities within the math- The Danish Mathematical Society – Dansk small library. When the Mathematical ematical community in the Copenhagen Matematisk Forening (DMF) – is one of the Laboratory at Copenhagen University was area, a situation which has only changed

H. G. Zeuthen (1839–1920), known for his Julius Petersen (1839–1910), known in partic- Th. N. Thiele (1838–1910), known for his work on enumerative geometry and the history ular for his contributions to graph theory. work in astronomy, actuarial science and of mathematics. statistics.

14 EMS March 2000 SOCIETIES slowly and is still under transformation ferent mathematics departments, with lec- Mathematica Scandinavica and Nordisk although the mathematical landscape tures and discussions of common interests. Matematisk Tidsskrift (NORMAT). Every started to change half a century ago. In The society has published the Collected fourth year one of the Scandinavian math- 1954, the Department of Mathematics at Mathematical Works of Harald Bohr, edited ematical societies hosts a conference. This Aarhus University was founded as the first by E. Følner and B. Jessen in 1952. (They year the DMF is the host of the First AMS- outside Copenhagen. From the very are still sold through the DMF.) Moreover, Scandinavian International Mathematics beginning it had strong international links the society initiated the publication of the Meeting, the XXIII Scandinavian and many foreign teachers. A colloquium Collected Mathematical Papers of Jakob Congress of Mathematicians. It will take tradition was also started, a tradition Nielsen, edited by Vagn Lundsgaard place in Odense in June: see www.imada. which soon spread to Copenhagen, and Hansen and published by Birkhäuser in sdu.dk/~hjm/AMS.Scand.html/ the other mathematics departments to be. 1986. The DMF (web address: www.dmf. Thus the former important role of the For the last 25 years, a newsletter MAT- mathematics.dk) has currently about 340 society has diminished. Other changes are NYT has been published and distributed members, of which one-fifth are also indi- due to the creation of societies such as the to all members almost weekly. It consisted vidual members of the EMS. A general Danish Society of Theoretical Statistics, mainly of advertisements for lectures and assembly is held once a year, and in even the Danish Operations Research Society, meetings. This service was changed last years members of the Board are elected. the Danish Society of Computer Science year by splitting it into an electronic cal- Re-election may occur, although the pres- and the Danish Center of Applied endar (see www.matnyt.mathematics.dk/) and ident can be re-elected only once and Mathematics and Mechanics. a newsletter with articles (mainly in other members at most three times. The society still arranges lectures in the Danish) printed four times a year and dis- Individuals can apply for membership to Copenhagen area, but less frequently than tributed to all members. the Board, which has the power to accept before. During the last ten years regular Together with other Scandinavian members. meetings have been organised once or mathematical societies, the DMF is Bodil Branner is currently president of the twice a year, alternating between the dif- responsible for the publication of DMF.

Mathematicians on their way to congratulate Zeuthen on his 80th birthday in 1919. From left to right: T. Bonnesen, J. Mollerup, Chr. Juel, Harald Bohr, Niels Nielsen, C. Crone, J. Hjelmslev and J. L. W. V. Jensen. They were all closely involved in the life of the Society.

EMS March 2000 15 EDUCATION large number of ‘examination boards’ – One of the main goals of the (right- most of which had historical connections wing) government of the 1980s was to with groups of universities. The main restrict the power of (mostly left-wing) MathematicsMathematics exam boards originally existed to provide urban LEAs. To this end they exploited the ‘matriculation tests’ for university language of ‘accountability’, and used this inin entrance, but in the 1960s they began to to justify mechanisms for central control. coordinate locally based exams at age 16 In education, this led naturally to the idea for non-academic pupils. of a centrally controlled national curricu- EnglishEnglish The move away from grammar schools lum, with centrally administered national coincided with an increasing concern to assessment. As indicated above, some schools provide for the (previously neglected) ‘bot- change was needed; but the motivation for schools tom 75%’. The consequences of these what happened was more political and changes were often driven as much by local bureaucratic than educational. A. D. Gardiner politics and administrative concerns as by England has no tradition of central con- educational policy. For example: trol in education. Unlike certain other History of the school system (a) The potential market for the examina- European countries, we had no cadre of Until 1988 education in England was a tion boards more than doubled, and the officials who understood the conditions local responsibility – with control vested in desire to generate income from the larger under which central control in education around 100 ‘Local Education Authorities’ part of this market was sometimes stronger can work, who appreciated its limitations, (LEAs). Each LEA received most of its than the demand for educational princi- or who knew how to administer a centrally funding from central government; LEAs ples to guide these new developments. controlled system. were responsible for compulsory education (b) Grammar schools had contained the The changes that were eventually made (ages 5-16) and for further education (ages bulk of highly qualified teachers, and so in the late 1980s were strongly influenced 16-19 and adult vocational training), and were the natural focus for academic educa- by the position of ‘mathematics’ in the raised a percentage of their budget tional provision at ages 16-19. Abandoning school curriculum. In response to wide- through local taxes. Each LEA provided grammar schools created a vacuum at spread complaints about the mathematical advice and support to schools and colleges. senior high school level. In the political abilities of school leavers, the government However, the details of what was taught euphoria following ‘comprehensivisation’, established in 1978 an inquiry – which and how it was taught remained the almost all schools were allowed to offer came to be known as the ‘Cockcroft responsibility of the individual school: academic courses for pupils aged 16-19. It Committee’, after its Chair, the late Sir there were no national or local curricula. was clear from the outset that this was Wilfred Cockcroft, a professor of mathe- This apparent recipe for anarchy was unworkable; however, to have allowed a matics. This committee was instructed ‘to constrained by the facts that what was minority of schools to retain academic pro- consider the teaching of mathematics ... taught at secondary level was strongly vision for ages 16-19 would have looked with particular regard to the mathematics influenced by ‘public’ examinations at ages like reinventing grammar schools by the required in further and higher education, 16 and 18, and that until the early 1970s back door. So many LEAs moved towards a employment and adult life generally’. what was taught at primary level was influ- new pattern of provision – with secondary They interpreted this instruction in a enced by selection tests for entry to ‘gram- schools restricted to ages 11-16, while sep- strongly utilitarian way: the very limited mar schools’ at ages 10-11. During the late arate ‘colleges’ provided academic and mathematics which adults actually use was 1960s and 1970s selection at ages 10-11 vocational courses for those aged 16-19. accepted as indicating what really needed was largely abandoned, and most of the This led to an outflow of qualified mathe- to be taught at school level. This had pro- existing grammar schools were reconstitut- matics teachers from 11-16 schools into found consequences. In particular, though ed as ‘comprehensive’ (all-ability) schools. 16-19 colleges. Official statistics [3] show the report itself was mostly moderate in Any such change had to be decided at a that in 1992 in maintained (publicly fund- tone, it was used by others to persuade a local level, and so had electoral implica- ed) secondary schools only one third of the whole generation of English educators and tions for local politicians; hence some mathematics lessons for pupils aged 11-14 administrators to undervalue the mental LEAs adopted compromise schemes with were taught by teachers whose pre-service ‘mathematical world’, and to overvalue the result that we still have 166 selective training included a degree with mathemat- practical work, ‘generic skills’ (such as secondary schools (out of 4000 or so) in ics as a main component, while almost two- teamwork and problem solving), ‘technolo- England. However, the present govern- thirds of mathematics lessons for pupils gy’, and ‘applications’ with little mathe- ment has introduced proposals which are aged 16-18 were taught by teachers with matical content. intended to make it easier for LEAs to such a qualification. This was the era of plausible slogans. change the status of the remaining gram- Technology, we were told, had changed mar schools. The present completely what schoolchildren needed to At neither primary nor secondary level Like the British constitution, many English learn. They no longer needed to learn facts was there any tradition of devising detailed institutions reflect what outsiders must see or algorithms, but should rather be taught curricula or teaching plans: some schools as an incoherent muddle. However, given ‘how to learn’. Multiplication tables and devised their own teaching plans, but most a measure of stability (and, in the case of written algorithms were too narrow; mathematics departments selected and fol- education, a sufficiently large, critical mass instead children needed a ‘sense of num- lowed a scheme or textbook series which, of competent teachers), this incoherence ber’. No-one explained how one could in their judgement, matched the abilities has often proved historically valuable by achieve such a ‘number sense’ without mas- of their pupils and which prepared them leaving room for innovation, flexibility and tering traditional arithmetic, or how one for the next exam they faced. There was compromise. could ‘solve problems’ without the requi- (and is) no official procedure for vetting or The dramatic changes in England from site facts and algorithms. Having woken up ‘approving’ textbooks: schools are not only 1965 to 1985 made educational change to the fact that ‘the mathematical world’ on free to choose, but are given no official essential, but they also undermined social its own is not ‘sufficient’ to ensure that information to assist in that choice. and political stability. Government intro- school leavers can use their mathematics, The examinations which effectively duced a new style of ‘radical politics’ in English mathematics educators appeared determined the curriculum were not which change was imposed through ‘per- to conclude that this formal universe was designed or controlled by government manent revolution’ – an endless string of also ‘unnecessary’ – and even unhelpful. agencies. The entry tests for grammar initiatives and ‘reforms’ which achieved Teachers were encouraged to believe that schools were administered by LEAs – often central control by undermining the tradi- one could teach pupils to use mathematics using commercially available tests in tional stability of English life. This style of directly, without first grappling with the English, Mathematics and Reasoning. The administration has since been adopted awkward abstractions of traditional school ‘public’ examinations were provided by a much more widely. mathematics. 16 EMS March 2000 EDUCATION The Cockcroft committee reported in observer it would appear that modest tional shortcomings were denied until rel- 1982. Its analysis and recommendations changes were insufficient for the then atively recently (and are still not acknowl- were concerned solely with mathematics, Prime Minister, so a new Minister of edged openly), but one could not escape but they stemmed from tensions and con- Education was appointed to introduce the consequences of the contradictions flicting demands which affected all sub- wholesale change. The resulting Education inherent in the structure of this curricu- jects. In particular, the report had to Reform Act (1988) promised a compulsory lum. However, these consequences could address the kind of changes which might national curriculum in ten subjects, and not be effectively addressed as long as their provide a better deal for ‘the bottom 75%’. provided for the subsequent introduction origin was denied. Thus there were repeat- The Cockcroft Report made some of centrally controlled national testing at ed attempts (in 1991, 1993 and 1995) to important and valuable recommendations, ages 7, 11, 14 and 16. revise the mathematics curriculum – but and achieved a remarkable balance The administration underestimated the only ‘to make it easier to administer’, not between ‘progressive’ and ‘traditional’ difficulties of what was being attempted. to correct its genuine weaknesses. views of mathematics education, even if To start the process of drafting and imple- Notwithstanding the inherent flaws in those who subsequently used the report to menting an agreed curriculum (in ten sub- the curriculum, by the mid-1990s teachers further their own agendas were often less jects) it was thought obvious to start with had got used to it and were tired of contin- sensitive to this need for balance. Mathematics! ual change. Politicians and bureaucrats However, the report had two main The abolition of the grammar schools used this weariness to argue that no further weaknesses: had created a national taboo: it was no changes were needed. (i) It was too concerned to resolve tensions longer possible to discuss the idea of ‘selec- The subsequent very public struggle – within the mathematics education commu- tion’, or to consider devising different cur- fuelled in late 1995 by the London nity, and so failed to notice more telling ricula for identifiably different groups Mathematical Society report Tackling the administrative and political trends, such as before the end of compulsory education at mathematics problem and by the very poor the increasing calls for ‘accountability’ and age 16. Hence the National Curriculum TIMSS (Population 2) results – led eventu- for central control. had to appear to provide equal opportuni- ally to the most recent revision, which is to (ii) The report was strongly influenced by ties for all. be implemented from September 2000. emerging evidence of the gulf between the However, the need to make some provi- Officially this revision was only allowed to ‘intended’ and the ‘achieved’ curriculum. sion for different abilities had been high- make minor changes of content, so most of Unfortunately it failed to distinguish lighted by the Cockcroft report, which had the changes made had to be presented as between universal facts and local patholo- revealed the dramatic gulf between the attempts to clarify the ‘intention’ of the gy. For example, data from the late 1970s achievements of the best and the worst existing (rather vague) curriculum by spec- which showed that certain topics appeared English pupils at age 11, a phenomenon ifying content in a more detailed and to be ‘too hard’ for most English 14 year- which was summarised in the phrase ‘the structured way, while still trying to encour- olds was interpreted at face value without seven-year gap’ – meaning that, after age desirable teaching styles. looking across the Channel to see whether removing outliers, the best 11-year-olds In parallel with, and perhaps more this was a universal fact or a temporary were as good as the average 14-year-old, important than, this recent curriculum local aberration – that is, the report did while the weakest were no better than the review has been the National Numeracy not take account of what was routinely average 7-year-old. Instead of proposing Strategy. This scheme constitutes a radical achieved in other European countries. measures to reduce this ‘gap’ (as in most shift in the way mathematics is taught at In the years immediately following the countries), it was assumed to be a fact of primary level. The approach has been Cockcroft Report, successive Ministers life. Thus we rejected the idea of a com- piloted over the last 2-3 years and was (with orders from the then Prime Minister, mon ‘year-by-year’ curriculum for all implemented in all primary schools in Margaret Thatcher) grappled with the pupils up to an age where divergence September 1999. In contrast to the vague problem of how to regain control of an made it natural for pupils to attend differ- and contradictory guidance of the last fif- educational system which had lost its tradi- ent kinds of schools – with each type of teen years, the Numeracy Strategy pro- tional contraints – but with limited success. school having a slightly different curricu- vides a highly structured model for the Some partial improvements were imple- lum. Rather, we tried to devise the maxi- teaching of elementary mathematics. The mented. For example, until 1985 public mal curriculum for ages 5-16 in the form of approach is pragmatic, has been formulat- examinations at age 16 were offered on a single ladder – with ten rungs, which all ed in great haste, and has many shortcom- two levels: the traditional ‘O level’ exams pupils would climb ‘at their own pace’. ings, but is generally accepted as a serious for the top 25-30%, and an unsatisfactory Schools were expected to accommodate – and on the whole sensible – attempt to watered-down version of these exams for those who progressed very slowly (perhaps improve the achievement of ‘the bottom the next 30-35%. Both exams officially reaching only the third or fourth rung by 75%’ of 11-year-old primary school excluded the bottom 40%. the age of 16) as well as those who could leavers. The strategy offers schools a very In line with the purely mathematical reach the top much more quickly (perhaps detailed year-by-year curriculum, and recommendations of the Cockcroft Report, by the age of 13). This created an unman- incorporates a requirement that each pri- the year 1988 saw the introduction of a ageable diversity within each classroom. mary school class should devote 45-60 uniform system of examinations at age 16 A problem had been recognised, but the minutes each morning to mathematics. for all subjects. The ‘system’ was uniform, curriculum structure which was adopted Though the strategy has many good fea- but there were dozens of different exams reflected a refusal to tackle it centrally: tures – its emphasis on ‘numeracy’ rather offered by different agencies – though they instead the buck was passed to individual than ‘mathematics’ – the absence of any all had to observe certain common criteria teachers and schools. similar programme for secondary schools – which led to the introduction of vetting Worse, this ‘bottom-up’ model of the is worrying. Also, the year-by-year struc- procedures which marked the beginning of curriculum as a single ladder meant that ture of the Numeracy Strategy is logically central control! the curriculum specification of important incompatible with the ‘level-by-level’ lad- The new system of exams served a larg- topics was determined by the needs and der structure of the National Curriculum. er fraction of the age group, and the syl- limitations of the slowest pupils. Thus, the Rather than confront this issue and make a labus for weaker students was designed details of what was needed at the level of rational choice, English pragmatism specifically to cater for their needs and introductory algebra were specified in the appears content to embrace both – for the abilities. However, since this requirement same way for those who might proceed to time being. was interpreted in line with the comments study science and engineering as for those in (ii) above, the resulting syllabuses were who would struggle to understand the sim- Main mathematical objectives partly responsible for institutionalising low plest formula. The 1989, 1991, 1993 and 1995 versions of expectations. The resulting mathematics curriculum the curriculum all played down the role of The detailed history of this period which was forced through in 1989 was ‘the mathematical world’. The motivation remains to be written. To the casual unworkable. Its mathematical and educa- was understandable, but superficial. EMS March 2000 17 EDUCATION England has no tradition of pedagogy being confident in the use of basic tech- cates something of the tension which lies and didactics. There is therefore no niques. scarcely beneath the surface. The reluc- accepted formal way of analysing the chal- The most recent revision of the curricu- tance to use the word ‘geometry’ in the title lenges which confront the mathematics lum has made a nominal attempt to incor- reflects the earlier distaste for any hint of teacher, or of communicating intended porate ‘using and applying’ as part of each formal methods. The presentation of con- modifications to existing or intending content strand. Previous versions of the tent is made more difficult by the need to teachers. The only vehicles are therefore curriculum emphasised ‘general abilities’ accommodate the persistent belief that pragmatic ones: from textbooks, syllabuses and ‘applications’, while failing to specify ‘transformations’ offer a viable approach and examinations, to personal example clearly either which aspects of ‘the mathe- to ‘useful geometry for the majority’. and encouragement to ‘reflect on one’s matical world’ were most important, or Nevertheless, the strand includes most of experience’, though without a theoretical how they should be taught in order to the basic material one would expect under framework. derive the most benefit from pupils’ will- the heading ‘elementary geometry’ – This tradition proved unable to handle ingness to ‘use what they know’ with confi- geometry of lines and triangles, approxi- the shift in the 1970s from a ‘top-down’, dence. The most recent revision of the cur- mate and exact constructions, congruence, university-driven agenda to ‘education for riculum spells out in much greater detail Pythagoras, and similarity. all’. The new majority were taught and those aspects of ‘the mathematical world’ The treatment varies from informal (for examined in (a watered-down version of) that are felt to be important, and encour- the majority) to semi-formal (for the top the old minority tradition, with unsatisfac- ages teachers to bring ‘using and applying’ 25% or so, who also meet the basic circle tory results. There was also increasing evi- back into the mainstream, but gives little theorems). ‘Space’ plays a limited role, dence that even the very best pupils under- indication of how this should be done. exploring 3D-shapes, working with stood much less than had been assumed. cuboids and prisms, with the top 25% This led to a reaction against ‘formal meth- Basic contents required to use Pythagoras in 3D. ods’. This reaction was strengthened by In the revised version of the mathematics ‘Measures’ are included in this strand only those who (ignoring the lessons of the last curriculum, the outline of content covers because there seemed to be nowhere else 400 years) claimed that traditional disci- more than 50 A4 pages, with an additional to put them. plines, each with its separate range of tech- 25 pages of general requirements. Handling data focuses on ‘collecting niques and methods, were now less impor- The content is presented under three data, processing and representing data, tant than ‘generic skills’ (or general abili- main headings: Number and algebra, Shape, interpreting and discussing results’, while ties), utility (or applications), and technol- space and measures, and Handling data. The also trying to convey the unpredictable ogy. additional theme Using and applying mathe- nature of random processes, the kind of Instead of reassessing the kind of ‘for- matics, extends across all three content questions that can be addressed using sta- mal mathematics’ which had to be taught, headings. tistical methods, and sources of bias in sta- and the way it should be taught, to serve Since the curriculum is still specified in tistical data. There is an obligation on the new majority better, there sprang up a terms of ‘levels’, rather than ‘grades’, it is teachers to involve pupils in designing belief that informal methods would suffice. difficult to infer from what is written what experiments and surveys so that they have The National Curriculum (and the associ- is expected of the average pupil, or of the to decide what data to collect in order to ated assessment) encouraged teachers: majority of pupils. answer a simple question. (1) to see school mathematics as being Number and algebra means essentially The only official indication of the rela- motivated and justified by its uses. (‘We ‘Number’ until age 11, after which algebra tive weighting of the three strands is that believe it should be a fundamental princi- begins to play an increasing role. The cur- Handling data does not appear explicitly ple that no topic should be included unless riculum for ages 11-16 contains most of during the first 2-3 years, but is viewed at it can be developed sufficiently for it to be what one would expect if the top 25% or so that level as a natural part of ‘Number’. applied in a way which pupils can under- are to solve quadratic equations, to handle Unofficially it would seem that during stand.’ ‘Pupils should be given opportuni- simple quadratic functions graphically, these early years, ‘Number’ may occupy as ties to use and apply mathematics in prac- and perhaps to find the points of intersec- much as 80% of the available time, with tical tasks [and] in real-life problems.’) tion of a straight line and a circle. Shape, space and measures occupying 20%. (2) to concentrate on encouraging pupils However, the curriculum for the bottom In later years (ages 12-15) Number and alge- to ‘use what they know’ with confidence, 70% or so suggests that they should not be bra may occupy 55% of the time, Shape, rather than to try to use more formal expected to work with quadratic expres- space and measures 30%, and Handling data methods which they do not understand. sions. 15%. (‘Very many pupils in secondary schools The title Shape, space and measures indi- are at present being required to follow mathematics syllabuses whose content is too great and which are not suited to their World Mathematical Year Stamp level of attainment.’) (3) to ensure that pupils are given opportunities to explore, and to investigate, simple situations with some mathematical con- This year several countries are issu- tent, to ‘find ways of overcoming difficul- ing postage stamps to commemo- ties that arise, develop and use their own strategies’. rate World Mathematical Year In this spirit, the 1995 curriculum 2000. The first country was began – not with the three ‘content strands’ (Number and algebra, Shape, space Belgium who issued this stamp in and measures and Handling data) – but with February. It features Stokes’ theo- a strand called Using and applying mathe- rem and Fermat’s last theorem matics, which emphasised ‘making and monitoring decisions to solve problems, inside a circle, with the normal communicating mathematically, and curve and a representation of the developing skills of mathematical reasoning’. While some teachers managed to use Ishango bone, the earliest known this emphasis to improve their teaching, mathematical artefact. the approach suffered from a failure to analyse the relationship between achieving fluency in mathematical techniques, and 18 EMS March 2000 NEWS (Warwick), Michael Newman (Canberra), Herbert Pahlings (Aachen)] OberOberwolfachwolfach PPrrogrammeogramme 20012001 5-11 August: Partial Differential Equations [Craig Evans (Berkeley), Ernst Kuwert (Freiburg), Stefan Müller (Leipzig)] Mathematisches Forschungsinstitut Oberwolfach 12-18 August: Relativistic Quantum Systems and Quantum Electrodynamics [Volker Bach (Mainz), Heinz Siedentop Lorenzenhof, D-77709 Oberwolfach-Walke, Germany (Regensburg), Jan Philip Solovej (Copenhagen)] 19-25 August: Mini-Workshops Names of organisers are in square brackets. 15-21 April: Asymptotic and Numerical (Hints for applications: see above) Participants of the meetings at Oberwolfach are Methods for Kinetic Equations 26 August-1 September: Complex invited personally by the director of the institute. [Pierre Degond (Toulouse), Axel Klar Geometry: Interactions between Algebraic, Participation is subject to such an invitation. (Darmstadt), Reinhard Illner (Victoria)] Differential and Symplectic Geometry Interested researchers, in particular young mathe- 22-28 April: Konvexgeometrie [Arnaud Beauville (Paris), Fabrizio Catanese maticians, can contact the administration of the [Paul Goodey (Norman), Peter M Gruber (Göttingen), Eduard Looijenga (Utrecht), institute. Since the number of participants is (Wien)] Christian Okonek (Zürich)] restricted, not all enquiries can be considered. 29 April-5 May: Phasenübergänge 2-8 September: Singularitäten Information is also available on our website [Hans Wilhelm Alt (Bonn), Stephan Luckhaus [Gert-Martin Greuel (Kaiserslautern), Joseph http://www.mfo.de. (Leipzig), Errico Presutti (Roma), Ekhard Steenbrink (Nijmegen), Victor Vassiliev Salje (Cambridge)] (Moskau)] 7-13 January: Finite Fields: Theory and 6-12 May: Aperiodic Order 9-15 September: Topologie Applications [Michael Baake (Tübingen), Jean Bellissard [Wolfgang Lück (Münster), Cameron Gordon [Joachim von zur Gathen (Paderborn), Igor (Toulouse), Robert Moody (Edmonton)] (Austin), Robert Oliver (Villentaneuse)] Shparlinski (NSW)] 13-19 May: Schrödinger Operators 16-22 September: Theory of the Riemann 14-20 January: Combinatorial Convexity [Volker Enß (Aachen), Christian Gerard Zeta and Allied Functions and Algebraic Geometry (Palaiseau)] [Martin Huxley (Cardiff), Matti Jutila [Victor Batyrev (Tübingen), Peter McMullen 20-26 May: Nonlinear Evolution Problems (Turku), Yoichi Motohashi (Tokyo)] (London), Tadao Oda (Sendai), Bernard [Michael Struwe (Zürich), Sergiu Klainerman 23-29 September: Combinatorics, Teissier (Paris)] (Princeton)] Probability and Computing 21-27 January: Berechenbarkeitstheorie 27 May-2 June: Schnelle Löser für partielle [Bela Bollobas (Memphis), Ingo Wegener [Klaus Ambos-Spies (Heidelberg), Steffen Differentialgleichungen (Dortmund)] Lempp (Madison), Ted Slaman (Berkeley)] [Randolph Bank (San Diego) Wolfgang 30 September-6 October: Stochastic 28 January-3 February: Topologische Hackbusch (Leipzig), Gabriel Wittum Evolution Equations and Applications Methoden in der Gruppentheorie (Heidelberg)] [Guiseppe Da Prato (Pisa), Michael Röckner [Herbert Abels (Bielefeld), Peter Kropholler 3-9 June: Oberwolfach-Seminars (Bielefeld), J. Zabczyk (Warszawa)] (London), Karen Vogtmann (Ithaca)] 10-16 June: Differentialgeometrie im 7-13 October: Arbeitsgemeinschaft mit 4-10 February: Mixed Finite Element Großen aktuellem Thema (wird in Heft 3/2001 der Methods and Applications [Werner Ballmann (Bonn), Jean-Pierre DMV-Mitteilungen bekanntgegeben) [Douglas Arnold (University Park), Carsten Bourguignon (Bures-sur-Yvette), Wolfgang 14-20 October: Oberwolfach Seminars Carstensen (Kiel), Ronald Hoppe (Augsburg)] Ziller (Philadelphia)] 21-27 October: Theoretische und 11-17 February: Funktionentheorie 17-23 June: Numerik von Mikrostrukturen Mathematische Biologie [Kari Astala (Jyväsklyä), Walter Bergeweiler [Carsten Carstensen (Kiel), Wolfgang [Andreas Dress (Bielefeld)] (Kiel), Reiner Kühnau (Halle)] Hackbusch (Leipzig), Thomas Hou 28 October-3 November: Stable Laws, 18-24 February: Geometric Rigidity and (Pasadena)] Processes and Applications Hyperbolic Dynamics 17-23 June: Two Hundred Years of Number [Werner Linde (Jena), Jan Rosinski [Werner Ballmann (Bonn), Anatole Katok Theory after Carl-Friedrich Gauß’s (Knoxville), Gennady Samorodnitsky (Ithaca)] (University Park), Gerhard Knieper Disquisitiones Arithmeticae 4-10 November: Mini-Workshops (Bochum)] [Catherine Goldstein (Paris), Norbert (Hints for applications: see above) 25 February-3 March: Mini-Workshops Schappacher (Strasbourg), Joachim 11-17 November: Oberwolfach Seminars (Hints for applications: see above) Schwermer (Düsseldorf)] 18-24 November: Numerical Integration 4-10 March: Algebraische Gruppen 24-30 June: Algebraische Zahlentheorie and its Complexity [Michel Brion (Grenoble), Jens Carsten [Christopher Deninger (Münster), Peter [Harald Niederreiter (Wien), Knut Petras Jantzen (Aarhus), Peter Slodowy (Hamburg)] Schneider (Münster), Anthony Scholl (Braunschweig), Henryk Wozniakowski 11-17 March: Stochastics in the Sciences (Durham)] (Warszawa/New York)] [Anton Bovier (Berlin), Richard Gill 1-7 July: 4-dimensional Manifolds 25 November-1 December: Modellierung, (Utrecht), Willem van Zwet (Leiden)] [Stefan Alois Bauer (Bielefeld), Peter Simulation und Optimierung integrierter 18-24 March: Gewöhnliche Kronheimer (Harvard), Ronald Stern Schaltkreise Differentialgleichungen (Irvine)] [Kurt Antreich (München), Roland Bulirsch [Jean Mawhin (Louvin-la-Neuve), Klaus 8-14 July: Vision and Related Subjects (München), Albert Gilg (München/Perlach), Schmitt (Salt Lake City), Hans-Otto Walther [Jean-Michel Morel (Paris), Christoph von der Peter Rentrop (Karlsruhe)] (Gießen)] Malsburg (Bochum/Los Angeles), David 2-8 December: Finite Geometries 25-31 March: Representations of Finite Mumford (Providence)] [Aart Blokhuis (Eindhoven), James Hirschfeld Groups 15-21 July: Dynamische Systeme (Sussex), Dieter Jungnickel (Augsburg), [Michel Broue (Paris), Richard Dipper [Helmut Hofer (New York), Jean-Christophe Joseph Thas (Gent)] (Stuttgart), Burkhard Külshammer (Jena), Yoccoz (Orsay), Eduard Zehnder (Zürich)] 9-15 December: C*-Algebren Geoffrey Robinson (Birmingham)] 22-28 July: Explicit Methods in Number [Dietmar Bisch (Santa Barbara), Eberhard 1-7 April: Arbeitsgemeinschaft mit Theory Kirchberg (Berlin), Georges Skandalis (Paris)] aktuellem Thema (wird in Heft 1/2001 der [Henri Cohen (Talence), Hendrik Lenstra Jr 16-22 December: Mathematical Methods in DMV-Mitteilungen bekanntgegeben) (Berkeley/Leiden), Don Zagier Manufacturing and Logistics 8-14 April: Numerical Methods for Singular (Bonn/Utrecht)] [Rainer Burkard (Graz), Horst Hamacher Perturbation Problems 29 July-4 August: Computational Group (Kaiserslautern), Hartmut Noltemeier [Pieter Hemker (Amsterdam), Hans-Görg Theory (Würzburg)] Roos (Dresden), Martin Stynes (Cork)] [Gerhard Hiß (Aachen), Derek Holt EMS March 2000 19 PROBLEM CORNER PPrroblemoblem CornerCorner :: ContestsContests frfromom RRomaniaomania

Paul Jainta

Mathematical competitions have a country. in 1894) has appeared without a long tradition in Romania. The first In the native country of mathemat- break, and played a crucial role in event goes back to 1898, when the ics competitions, the authors do not organising the first mathematical com- Ministry of Public Education adver- understand why this way of matching petitions in Romania, as well as in dis- tised a national contest for students at one’s strength against others has lost covering talented young students in secondary school. Part of this compe- none of its appeal. They declare mathematics, encouraging interest in tition was designed to rate the mathe- openly: ‘One extremely difficult ques- mathematics, and directing mathematical capability of the participants. tion faced us when we began to design matical education. Many years later, the International the details of this note: how to explain, The Romanian Mathematics Competition Mathematics Olympiad (IMO), the pin- after almost a century of tradition, the Organisation System The fact that nacle of competitions among individu- current high rate of interest in mathe- Romania organised the first two als, was the brainchild of Romania’s matics competitions still existing International Mathematics Olympiads Tiberiu Roman, an educator of monu- amongst both students and teachers in (IMO) and instigated the idea arises mental vision, and the first two IMOs Romania’. But our initial perplexity from the above-mentioned enthusiasm were held in Romania, in 1959 and soon evaporates. Their review of the for mathematical bouts in this country. 1960. Roman’s pioneering efforts in contest scene in Romania presents an So Romania should be appreciated mathematics were followed by other impressive improvement in the math- even more nowadays, when similar disciplines, leading to yearly ematical abilities of young Romanians. Olympiads for physics, chemistry, International Olympiads in physics, Three ingredients have caused this computer science, and so on, have chemistry and informatics. success. become popular throughout the Nevertheless, there is a striking dis- The Romanian mathematics education sys- world. parity between Romania’s role in iden- tem This was established in 1898 under These international competitions tifying and nourishing mathematically the overall control of the Minister of are generally accepted as successful talented youngsters and the way in Education, Spiru Haret, a mathemati- attempts to identify, encourage and which these efforts were publicised. I cian. The basic principles of Haret’s challenge gifted youngsters in all knew nothing of the Romanian law proved to be ahead of their time countries, and to create opportunities groundwork in this field, except for an and endured to a great extent in the for the exchange of information on article that I stumbled over last year in Romanian education system through- school syllabuses and practice the Journal of the World Federation of out the 20th century. throughout the world. The 40th IMO National Mathematics Competitions. The centennial journal Gazeta of 1999 was hosted by the Romanian The author of that account, Vasile Mathematica This periodical (founded Society for Mathematical Sciences in Berinde, is Dean of the Faculty of Science and head of the Department of Mathematics and Computer Science at the Northern University of Baia Mare. He is heavily involved in the organisation of local and regional mathematical fixtures in his country, and is assisted by Mircea Becheanu, Associate Professor in the Faculty for Mathematics at the University of Bucharest, a Vice-president of the Romanian Society for Mathematical Sciences who has coached Romanian IMO teams. On inquiry Berinde allowed me to examine data on the rapidly improving mathematical competence among Romanian students, and on the basis of his report I now spotlight the competitions in that 20 EMS March 2000 PROBLEM CORNER Bucharest, yielding excellent results was the degree of excellence of sub- decade 1980-90 it achieved a monthly for the Romanian team which came mitted solutions. From 1905 the con- circulation of 130000-140000 copies; third, after Russia and the People’s testants were examined in certain cen- it now has 120000 subscribers. Republic of China. Incidentally, the tres, after preliminary registration, The astonishing career of mathe- only student in the 39-year history of and in 1909 written and oral exams matics as a refurbished and sporting the IMO who achieved three perfect were introduced. From 1910 the com- discipline in Romania will employ us scores in succession was a young petition, now called The Gazeta in forthcoming issues of this Corner. Romanian, Ciprian Manolescu (1996- Matematica Annual Contest, was consid- The following problems come from 98). These arguments show impres- ered almost a national event, due to its various Romanian competitions, and sively how organising mathematics large field of entrants. illustrate the hundred-year-old tradi- competitions open to talented stu- Over the years dents in Romania arises from a com- Gazeta mendable educational system. Matematica and its competitions The beginnings of the National have been Mathematics Olympiad helped financial- A century ago Haret’s law introduced ly and scientifi- three categories of secondary schools cally by respect- into the educational scene: scientific, ed Romanian modern and classical. The main tar- mathematicians. get was to create new opportunities for The awards were young people to develop their natural endowed by the skills and abilities in an up-and-com- journal itself, ing education system. But, due to the and later (1909) antiquated schooling, the level of sci- funded by a soci- entific education was very poor in both ety of the same the high-school and academic areas. name. After the Further, progress in getting teaching Second World shipshape quickly succeeded to uni- War, the Annual versities because of the younger lectur- Gazeta ers who, after getting a PhD in Matematical Western European countries, returned Contest became home in droves. Soon, an alarmingly the National substandard level of mathematical Mathematical knowledge was detected amongst the Olympiad entrance candidates to the National (NMO), a large- Technical School (now the Polytechnic scale competi- University of Bucharest). A group of five tion with finan- young engineers involved in the cial backing by entrance procedure decided to the state. In the increase the mathematical level of con- meantime, the testants, instead of decreasing the education system demands. Their plans gradually itself had been matured, and they launched a mathe- given a shake-up through the ‘great tion that is inherent in the promotion matical magazine intended for high- education reform’ of 1948, when of mathematically talented Romanian school students, whose main aim was mathematics became a centrepiece in youngsters. to identify mathematically-able people both secondary and high-school cur- Please send me your solutions as and encourage them to study mathe- ricula. well as contest materials, and propose matics. The spearhead of this mathe- These changes have generated a problems for readers to solve. matical fostering was the Gazeta second flourishing period in the histo- Wherever possible, proposals should Mathematica. ry of mathematics in Romania, which be accompanied by a solution, refer- The role of this notable journal is culminated in the first IMO in ences and other insights, to help the unanimously recognised in Romania. Bucharest (1959). Two words are editor. The problems can range from It turned out that a most efficient way apposite to the exceptional contribu- elementary to advanced, from easy to to entice further high-school students tion of Gazeta Matematica during this difficult, and original ones are particu- to solve problems posed in the Gazeta time: tradition and continuity. larly sought. So, please submit any Mathematica was to create the first Through its prestige, Gazeta interesting problems you come across, mathematics competition in 1902, Matematica quickly won more and especially from (problem) books and during the spring holidays. The con- more readers. The number of sub- contests that are not easily accessible - testants were selected from the best scriptions increased yearly from 144 in but other interesting problems may be correspondents of the monthly and 1895 to 410 in 1912, and more signif- acceptable provided that they are not handpicked from public and military icantly after the Second World War well known and references give their schools. One criterion for their choice (1964: 25000, 1974: 75000). In the provenance. EMS March 2000 21 110 The quadrilateral ABCD has two parallel sides. Let M and N be the midpoints of DC and BC, and let P be the common point of the lines AM and DN. If PM/AP = ¼, prove that ABCD is a parallelogram.

111 Let k be an integer and P(X) be the polynomial P(X) = X1997 – X1995 + X2 – 3kX + 3k + 1. Prove that: (i) the polynomial has no integer roots; (ii) the numbers P(n) and P(n) + 3 are relatively prime, for every integer n.

112 Find the image of the function f: R → R, defined by f (x) = (3 + 2sinx)/[√(1 + cos x) + √(1 – cos x)].

113 Let a, b, c, d ∈ R and f: R → R, f (x) = ax3 + bx2 + cx + d , such that f (2) + f (5) < 7 < f (3) + f (4). Prove that there exist u, v ∈ R such that u + v = 7 and f (u) + f (v) = 7.

114 Let f: N × N → N be a function which satisfies the following conditions: (i) f (0, y) = y + 1, for all y ∈ N (ii) f (x + 1, 0) = f (x, 1) for all x ∈ N (iii) f (x + 1, y + 1) = f (x, f (x + 1, y)) Compute f (3, 1997).

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

Solutions to some earlier problems

91 Prove that for any positive number n ≥ 4 and any positive real numbers a1,a2, ... ,an, the following inequality holds:

1/(a1 + a2)+1/(a2 + a3) + ⋅⋅⋅ + 1/(an-1 + an) + 1/(an + a1)

> 4/3[1/(a1 + a2 + a3) + 1/(a2 + a3 + a4) + ⋅⋅⋅ + 1/(an-1 + an + a1) + 1/(an + a1 + a2)]

Solution by Ronald van Lujk (Utrecht); also solved by Aldric L. Brown (Chandigarh) For x, y > 0 we have 4/(x + y) ≤ 1/x + 1/y.

Let x = aI + (ai+1/2) and y = (ai+1/2) + ai+2 .

We get 4/(aI + ai+1 + ai+2) ≤ 2/(2aI + ai+1) + 2/(ai+1 + 2ai+2). But 2/(2x + y) + 2/(x + 2y) < 3/(x + y). Thus, summation over n yields n 4 n 2 2 n 3 ≤ ()+ < . ∑ aa++ a ∑ 2aa+ a+ 2 a ∑ aa+ 1 ii++12 i 1 ii++11 i i + 2 1 ii+1

97 Suppose a, b and c are the lengths of the sides of a triangle with a ≤ b ≤ c. Let S and T be real numbers. Find the minimum value S and the maximum value T, satisfying the inequality S ≤ (a + b + c)2/ b.c ≤ T.

Determine when there is equality.

Solution by Niels Bejlegaard (Stavanger); also solved by Dr. J. N. Lillington (Dorchester) and Claude Lamoureux (Paris).

Since a ≤ b ≤ c we conclude that (a + b + c)2/ bc > (b + c)2/ bc = b/c + c/b + 2 ≥ 2 + 2√(b/c).(c/b) = 4, by the arithmetic-geometric mean inequality. Conversely, the following reduction is valid: (a + b + c)2/ bc ≤ (2b + c)2/ bc = 4 + 4b/c + c/b = 4 + k (where k = 4b/c + c/b). Now we minimise the value of k: we have 4b/c + c/b ≤ k, because b ≤ c. We arrive at 4b2 – kcb + c2 ≤ 0, giving b ≤ c{k ± √(k2 – 16)}/8. Now {k + √(k2 – 16)}/8 = 1 implies that k = 5; we deduce that 4 < (a + b + c)2/ bc ≤ 9. Equality occurs only when the triangle is equilateral.

98 Find all sets of four points in the plane so that the sum of the distances from each of the points to the other three is constant.

Solution by Pietro Fanciulli (Porto S. Stefano, Italy) and the proposer.

Let the points be Pi (i = 1, 2, 3, 4), and consider Xi,j the distances PiPj of the point Pi from Pj (where i ≠ j and ij = ji). We obtain the following equations (where k = constant)

X12 + X13 + X14 = k; X12 + X23 + X24 = k, X13 + X23 + X34 = k, X14 + X24 + X34 = k. Subtract each of the last three from the first:

X13 + X14 = X23 + X24; X12 + X14 = X23 + X34; X12 + X13 = X24 + X34 . This leads (by adding two of the equations and subtracting the third) to:

X12 = X34 ; X14 = X23 ; X13 = X24 ; that is, the distance between any two of the points equals the distance between the other two points. This holds

only for the vertices of a (possibly degenerate) rectangle. [For example, assuming that P1P2P3P4 is a simple (non- crossing) quadrilateral, the first two equations imply that the quadrilateral is a parallelogram, and then the third equation forces it to be a rectangle.]

2 99 Let x1 > 0 be any real number. Define the infinite sequence x2 , x3 , … by the recurrence relation xn+1 = xn + xn . k 1 Prove that lim exists, and sum the infinite series. k→∞ ∑ 1+ x 1 i

Solution by Dr. J. N. Lillington and the proposer. 2 From x1 > 0 and xn+1 = xn + xn, we have 2 2 2 2 4 3 (xn+2 – xn+1) – (xn+1 – xn) = (xn + xn) + (xn + xn) – 2(xn + xn) + xn = xn + 2xn > 0.

Thus we have: x2 – x1 = δ (say) and x3 – x2 > δ, . . . , xk – xk-1 > δ.

Hence xk – x1 > (k – 1) δ , by adding, and so xk → ∞ as k → ∞.

Now 1/x1 – 1/(1 + x1) = 1/x2 ; 1/x2 – 1/(1 + x2) = 1/x3 ; . . . , 1/xk – 1/(1 + xk) = 1/xk+1 ; k 1 Adding, we have 1/x1 – = 1/xk+1 . ∑ 1+ x 1 i k 1 Hence lim exists and equals 1/x1. k→∞ ∑ 1+ x 1 i

100 Given an arbitrary triangle ABC with circumcircle γ (O, R).

Let AO intersect BC in point A1, and let points B1 and C1 be determined analogously.

Show that OA1 + OB1 + OC1 ≥ 3R/2 .

Solution by Dr. J. N. Lillington; also solved by Maurice Brémond (Avignon, France), Pietro Fanciulli and the proposer.

We may assume the angles A, B, C are all acute for otherwise if A (say) is obtuse, then BC is the diameter of the

circumcircle and OA1+ OB1+ OC1 ≥ 2.

Let ∆A denote the area of triangle BOC, define ∆B , ∆C, similarly.

Then 1 + (OA1/R) = (R + OA1) / (R + OA1 – OA1) = ∆/(∆ – ∆A) = ∆/(∆B + ∆C)

Similarly, 1 + (OB1/ R) = ∆/ (∆A + ∆C)

and 1 + (OC1/ R) = ∆/ (∆A + ∆B) Summing these three equations we get

3 + (1/R)(OA1 + OB1 + OC1) = (∆A + ∆B + ∆C)/[1/(∆B + ∆C) + 1/(∆A + ∆C) + 1/(∆A + ∆B)]

= [(∆A + ∆B) + (∆A+ ∆C) + (∆B+ ∆C)][1/(∆B+ ∆C)+ 1/(∆A+ ∆C) + 1/(∆A+ ∆B)]/2 ≥ 9/2 (by applying the arithmetic-geometric mean inequality to both brackets).

Hence OA1 + OB1 + OC1 ≥ 3R/2.

103 Three points X, Y, Z are given in the plane, where X is the circumcentre, Y is the midpoint of the side BC, and Z is the foot of the altitude from B to side AC in the triangle ABC. Show how one can construct this triangle.

Solution by Pietro Fanciulli; also solved by Dr. J. N. Lillington and the

proposer. B Assume that the triangle ABC has been constructed. ∠ o c2 c1 Since BZC = 90 , the circle c (Y, R = ZY) intersects the side BC, so Y ZY = BY = CY. This is the key to the problem; proceed as follows: 1. Join the points X, Y by a straight line, and draw the perpendicular X A r p to XY at Y. Z C p 2. Draw the circle c1 (Y, ZY), which intersects the perpendicular line p in B and C. 3. Join Z and C by the straight line r.

4. Draw the circle c2 (X, R = BX = CX) which intersects r in points A, C. 5. Join A and B, and the points A, B, C are then defined. 104 A real-valued function f is defined for positive integers, and a positive integer a satisfies f (a) = f (1995), f(a + 1) = f (1996), f (a + 2) = f (1997), and f (n + a) = [f (n) – 1]/[f (n) + 1] for any positive integer n. Prove that f (n + 4a) = f (n) for any positive integer n, and determine the smallest possible value of a.

Composite solution by Dr. J. N. Lillington and Pietro Fanciulli. We have: f (n + 4a) = f [(n + 3a) + a] = [f (n + 3a) – 1]/[f (n +3a) + 1] = {f [(n + 2a) + a] – 1} / {f [(n + 2a) + a] + 1} = {[f (n + 2a) – 1] / [f(n + 2a) + 1}/{[f (n + 2a) – 1] / [f(n + 2a] + 1]} = [f (n + a) + 1] / (f (n + a) – 1] = {[f (n) – 1] / [f (n) + 1] + 1} / {[f (n) – 1] / [f (n) + 1] – 1} = – [–2f (n)] / 2 = f (n) , as required. Now if a is the least value, we have f (a) = f (5a) = f (9a) = … = f [(4n – 3)a] , n ≥ 1. Thus we seek the maximum integer n and the minimum integer a satisfying (4n – 3)a = 1995, where a ≠ 1 or 2. But a = 3 gives 4n – 3 = 665 and n = 167. Thus a = 3 is the required solution.

CONFERENCES 20-22: Schloessmann Seminar on Mathematical Models in Biology, Chemistry and Physics, Bad Lausick, FForthcomingorthcoming conferconferencesences Germany Information: compiled by Kathleen Quinn e-mail: [email protected] 20-25: MaPhySto and StocLab Summer School on Stereology and Geometric Please e-mail announcements of European confer- Analysis: (Non)smooth Analysis in Banach Tomography, Sandbjerg Manor, Denmark ences, workshops and mathematical meetings of Spaces, Paseky nad Jizerou, Czech Information: interest to EMS members, to k.a.s.quinn@ Republic e-mail: [email protected] open.ac.uk. Announcements should be written in Information: contact Katedra matematickè Web site: http://www.maphysto.dk/events/S- a style similar to those here, and sent as Microsoft analýzy, Matematicko-fyzikàln fakulta UK, and-GT2000/ Word files or as text files (but not as TeX input Sokolovskà 83, 186 75 Praha 8, Czech [For details, see EMS Newsletter 34] files). Space permitting, each announcement will Republic, tel./fax: +420-2-232-3390 26-27: Groups in Galway 2000, Galway, appear in detail in the next issue of the Newsletter e-mail: [email protected] Ireland to go to press, and thereafter will be briefly noted Web site: http://www.karlin.mff.cuni.cz/ Information: in each new issue until the meeting takes place, katedry/kma/ss/ e-mail: [email protected] with a reference to the issue in which the detailed [For details, see EMS Newsletter 34] 28-3 June: Combinatorics 2000, Gaeta, announcement appeared 26-28: Mathematical Education of Italy Engineers, Loughborough, UK Information: April 2000 Information: e-mail: [email protected] Web site: http://www.ima.org.uk/ http://www.mat.uniroma1.it 3-7: Workshop on Ergodic Theory of Zd- mathematics/conferences.htm /combinatorics2000 actions, Coventry, UK [For details, see EMS Newsletter 33] 28-3 June: Spring School on Analysis: Information: 25-6 May: NATO Advanced Study Some Recent Techniques in Harmonic e-mail: [email protected] Institute, Nonlinear Dynamics in Life and Analysis, Paseky nad Jizerou, Czech 5-14: ICMS Instructional Conference: Social Sciences, Moscow, Russia Republic Operator Algebras and Operator Spaces, Information: Information: contact Katedra matematickè Edinburgh, UK Web site: http://www.cas.mcmaster.ca/ analýzy, Matematicko-fyzikàln fakulta UK, Information: contact Allan M. Sinclair, ~sulisw/asi.html Sokolovskà 83, 186 75 Praha 8, Czech Department of Mathematics and Statistics, Republic, tel./fax: +420 - 2 - 232 3390 JCMB, KB, Edinburgh EH9 2DE, UK May 2000 e-mail: [email protected] e-mail: [email protected] Web site: http://www.karlin.mff.cuni.cz/ Web site: http://www.ma.hw.ac.uk/icms/ 11-15: Theory of Partial Differential katedry/kma/ current Equations and Special Topics of Theory of [For details, see EMS Newsletter 34] [For details, see EMS Newsletter 34] Ordinary Differential Equations, St 29-2 June: Deuxieme Rencontre 10-14: Summer School and Workshop on Petersburg, Russia Internationale sur les Polynomes a valeurs Algebraic and Co-Algebraic Methods in [dedicated to the 150th anniversary of the entieres CIRM, Luminy, France the Mathematics of Program Construction, birthday of Sofia V. Kovalevskaya] Information: Oxford, UK Topics: partial differential equations and e-mail: [email protected] Information: their applications, and special topics of 29-9 June: Foliations: Geometry and e-mail: [email protected] ordinary differential equations, preferably Dynamics Revisited, Banach Centre, 10-20: NATO Advanced Study connected with Kovalevskaya Warsaw, Poland Institute/EC Summer School, New Programme: four days devoted to presenta- Information: Theoretical Approaches to Strongly tions of papers, an excursion and celebra- Web site: http://fol2000.math.uni.lodz.pl/ Correlated Systems, Cambridge, UK tion of the anniversary of S. V. Information: Kovalevskaya June 2000 Web site: http://www.newton.cam.ac.uk/ Programme committee: L. D. Faddeev programs/scew.html (Chairman, St.Petersburg), O. A. 5-9: Advances in Convex Analysis and 11-14: Workshop on Harmonic Maps and Ladyzhenskaya (St Petersburg), V. M. Global Optimization Honouring the Curvature Properties of Submanifolds 2, Babich (St Petersburg), E. F. Mischenko Memory of C. Caratheodory (1873-1950), Leeds, UK (Moscow), P. Van Moerbeke (Louvain), M. Samos, Greece Information: contact J. C. Wood, School of A. Semenov-Tyan-Shansky (Russia-France), Information: Mathematics, University of Leeds, Leeds N. N. Ural’tseva (St Petersburg) Web site: http://samos.aegean.gr/math/acago/ LS2 9JT, UK Organisers: St Petersburg Department of 5-9: 6th International Conference on e-mail: [email protected] Steklov Institute of Mathematics (POMI), Probability, Poraj (near Czestochowa), Web site: http://www.amsta.leeds.ac.uk/pure/ Euler International Mathematical Institute Poland geometry/leeds2000.html Organising committee: L. D. Faddeev [dedicated to Professor Kazimierz Urbanik] [For details, see EMS Newsletter 33] (Chairman), V. M. Babich (Vice-chairman), Information: 12-15: Workshop on Mathematical N. Ya. Kirpichnikova, P. V. Krauklis, L. A. e-mail: [email protected] Methods of Regular Dynamics, Leeds, UK Molotkov, M. A. Semenov-Tyan-Shansky, E. [For details, see EMS Newsletter 34] [satellite meeting of the British V. Novikova (Secretary) 6-10: 8th International Conference on Mathematical Colloquium, Leeds, UK, 17- Language: English Groups and Group Rings, Wisla, Poland 20 April] Site: Euler International Mathematical Theme: groups, group rings, rings and Information: Institute their connections Web page: http://www.amsta.leeds.ac.uk/ Support: possibly support from RFBR to Aim: to extend the knowledge of algebraists ~vadim/work.html cover part of the expenses of participants (not only experts) on groups and group 17-20: 52nd British Mathematical from Russia; possibly reduced fees for par- rings and to stimulate research contacts Colloquium, Leeds, UK ticipants from countries in difficult econom- between mathematicians in East, West and Information: ic situations, and for young mathematicians Central Europe e-mail: [email protected] Information: Main speakers: A. A. Bovdi (Hungary), C. Web site: http://www.amsta.leeds.ac.uk/ bmc/ e-mail: [email protected] K. Gupta (Manitoba), N. D. Gupta [For details, see EMS Newsletter 33] Web site: http://www.pdmi.ras.ru/EIMI (Manitoba), O. H. Kegel (Freiburg), E. I. 23-29: Spring School on Functional /2000/sofia Khukhro (Novosibirsk), A. N. Krasilnikov EMS March 2000 25 CONFERENCES (Moscow), F. Menegazzo (Padova), D. M. [For details, see EMS Newsletter 33] ated states Riley (Western Ontario), A. L. Shmelkin 14-17: International Workshop for Information: (Moscow), I. Ya. Subbotin (USA), N. A. Operator Theory and Applications Web site: http://www.esf.org/euresco/00/ Vavilov (St Petersburg), J. S. Wilson (IWOTA), Bordeaux, France pc00094a.htm (Birmingham) Information: 18-21: International Conference on Monte Programme: 1 hour lectures given by the e-mail: [email protected] Carlo Simulation, Monte Carlo, Monaco main speakers, 15-30 minute short talks Web site: http://www.math.u-bordeaux.fr Information: Programme committee: O. H.Kegel /~iwota/ Web site: http://www.uibk.ac.at/c/c8/c810/ (Freiburg), E. I. Khukhro (Novosibirsk), J. [For details, see EMS Newsletter 33] conf/mcs_2000.html Krempa (Warsaw), O. Macedonska 15-17: 2nd Croatian Mathematical [For details, see EMS Newsletter 33] (Gliwice), A. Yu. Olshanskii (Moscow), D. S. Congress, Zagreb, Croatia 18-24: Perspectives of Mathematics, Passman (Wisconsin), V. Sushchanski Topics: all areas of mathematics Goslar, Germany (Gliwice), J. S. Wilson (Birmingham) Plenary speakers: A. Grossmann Information: contact K. Hulek, Institut für Organising committee: C. Baginski (Marseille/Versailles), G. Henniart (Paris), Mathematik, Universität Hannover, (Bialystok), J. Galuszka (Gliwice), W. M. Primc (Zagreb), A. Scedrov Postfach 6009, D-30060 Hannover, Holubowski (Gliwice), J. Krempa (Warsaw), (Pennsylvania), J. G. Thompson (Florida), Germany O. Macedonska (Gliwice), V. Sushchanski Z. Vondracek (Zagreb) e-mail: [email protected] (Gliwice) Programme: plenary lectures, parallel ses- Web site: http://www-ifm.math.uni-hannover. Sponsors: the Institutes of Mathematics of sions including invited lectures and short de/info/perspectives.html the Silesian University of Technology at communications, CMS award lecture [For details, see EMS Newsletter 33] Gliwice, Warsaw University and Bialystok Languages: English and Croatian 20-25: Mathematical Physics in University Programme and organising committee: I. Mathematics and in Physics: Quantum Site: Hotel Beskidy, Wisla, Poland Aganovic, D. Bakic, D. Butkovic, V. Hari, H. and Operator Algebraic Aspects, Siena, Grants: there are restricted funds to sup- Kraljevic, S. Kurepa, R. Manger, S. Italy port participants from East and Central Mardesic, P. Pandzic (secretary), M. Information : contact Roberto Longo Europe and former Soviet republics. Please Polonijo, M. Primc, R. Scitovski, D. Svrtan, Dipartimento di Matematica, Università di contact us if you need financial assistance H. Sikic (president), Z. Sikic, M. Tadic, Z. Roma ‘Tor Vergata’, I-00133 Roma, Italy, Deadlines: for registration, 1 April; for sub- Tutek, N. Uglesic, S. Varosanec, D. Veljan, fax: +39.0672594699 mission of abstracts, 30 April V. Volenec, Z. Vondracek e-mail: [email protected] Information: Sponsors: CVS, Tau d.o.o., Ministry of Web site: http://mat.uniroma2.it/~mp/ e-mail: [email protected] Science and Technology of Croatia siena2000.html Web site: http://www.polsl.gliwice.pl Site: Department of Mathematics, Bijenicka [For details, see EMS Newsletter 34] /~groups 30, Zagreb, Croatia 24-30: Numerical Methods for Evolution 7-11: PhD Euroconference on Complex Information: Partial Differential Equations, Anogia, Analysis and Holomorphic Dynamics, e-mail: [email protected] Crete Catalonia, Spain Web site: http://www.math.hr/~congress/ [part of the series: Euroconferences in Information: 17-22: EURESCO Conference on Mathematics on Crete] e-mail: [email protected] Mathematical Analysis: Partial Differential Main speakers: T. Gallouet (Marseille), R. Web site: http://crm.es/cad2000 (from 1 Equations and their Applications to Nochetto (Maryland), J. Rappaz (Lausanne), January 2000) Geometry and Physics, Castelvecchio V. Thomee (Goeteborg), L. Wahlbin [For details, see EMS Newsletter 34] Pascoli, Italy (Cornell) 11-17: DMV Seminar on Quantum Aim: to bring together some of the leading Organisers: G. Akrivis (Ioannina, Greece), Cohomology, Mathematisches Institut experts in this very active area of research, M. Crouzeix (Rennes, France) Oberwolfach, Germany to report and exchange information on the Local coordinator: Susanna Papadopoulou Organisers: Bernd Siebert (Bochum) and most influential and exciting new develop- ([email protected]) Gang Tian (MIT) ments, and to stimulate new ideas and Grants: contact local coordinator Information: Prof. Dr Matthias Kreck, approaches for further advances Information: Universität Heidelberg, Mathematisches Topics: harmonic maps and variational e-mail: [email protected] Institut, Im Neuenheimer Feld 288, 69120 problems; non-linear PDEs for special class- 25-28: IMACS-ACA2000 6th International Heidelberg, Germany es of Riemannian metrics; Seiberg-Witten Conference on Applications of Computer 11-17: DMV Seminar on Nonparametric equations and their applications in geome- Algebra, St Petersburg, Russia Function Estimation, Neural Nets and try; Dirac equations and Killing spinors; Scope: actual or possible applications of Risk Asymptotics, Mathematisches Institut spectral theory of Dirac and Laplace opera- non-trivial computer algebra techniques to Oberwolfach, Germany tors other fields and substantial interactions of Organisers: Andrew Barron (Yale), László Organiser: J. Eichhorn (Greifswald) computer algebra with other fields Gyröfi (Budapest) and Michael Nussbaum Speakers: Uwe Abresch (Bochum), Michael Sessions: the meeting will be run in the (Cornell) T. Anderson (Stony Brook), Boguslaw standard IMACS format where individuals Information: Prof. Dr Matthias Kreck, Broda (Lodz), Gilles Carron (Lyon), Harold are invited to organise a special session. Universität Heidelberg, Mathematisches G. Donnelly (Purdue), Tom Ilmanen Individuals can propose a special session by Institut, Im Neuenheimer Feld 288, 69120 (Zürich), Thomas Friedrich (Berlin), Dieter contacting the programme chairs. All paper Heidelberg, Germany Kotschick (Munich), Claude LeBrun (Stony submissions must be directed to an organis- 12-17: 3rd International Conference on Brook), Joachim Lohkamp (Augsburg), er of an appropriate special session Differential Equations and Applications John Lott (Ann Arbor), Matilde Marcolli General chair: Nikolay Vassiliev (DIFFEQ 2000), St Petersburg, Russia (MIT), Werner Müller (Bonn) Richard ([email protected]) 13-16: First AMS-Scandinavian Schoen (Stanford), Stephan Stolz (Notre Programme chairs: Victor Edneral (edner- International Mathematics Meeting, XXIII Dame), Mikhail Shubin (Boston) Michael [email protected]), Richard Liska Scandinavian Congress of Mathematicians, Struwe (Zürich), Gabriella Tarantello ([email protected]), Michael Wester Odense, Denmark (Rome), Gang Tian (MIT), Andrzej ([email protected]) Information: contact Hans J. Munkholm, Trautman (Warsaw), Neil S. Trudinger Scientific committee: Bruno Buchberger Odense University, Campusvej 55, DK 5230 (Canberra), Shing-Tung Yau (Harvard) (RISC-Linz), Jacques Calmet (Univ. of Odense M, Denmark, tel: Site: International Conference Centre Il Karlsruhe), Arieh Cohen (Eindhoven Univ. (+45)65572309/(+45)65932691 Ciocco, Castelvecchio Pascoli (northern Tech), Rob Corless (Univ. of Western e-mail: [email protected] Tuscany), Italy Ontario), Andre Deprit (Ntl. Bureau of Web site: http://www.imada.ou.dk/~hjm/ Grants: available for young scientists from Standarts), Sam Dooley (IBM Yorktown AMS.Scand.2000.html European Community countries and associ- Heights), Keith Geddes (Univ. of Waterloo), 26 EMS March 2000 CONFERENCES Vladimir Gerdt (Inst. of Nuclear Res.), its applications to mechanics and control Aim: to contribute to the dissemination of Gaston Gonnet (Zurich), Richard Jenks processes, geophysics, chemistry, mathemat- new results in operator theory, operator (IBM Yorktown Heights), Erich Kaltofen (N. ical models for processes in the atmosphere, algebras and applications and to facilitate Carolina State Univ.), Deepak Kapur (Univ. oceans and reservoirs, related problems of direct contact between researchers in differ- of New Mexico), Ilias Kotsireas (ORCCA- conversion, biology, ecology, engineering ent countries Univ. of Western Ontario), Wolfgang mathematics, and economics Organisers: Institute of Mathematics of the Kuechlin (Univ. of Tübingen), Bernard Topics: differential equations, geometry Romanian Academy, University of the West, Kutzler (BK Techware), Luis Laita (Univ. and analysis, theory of functions, mechanics Timisoara Politecnica Madrid), Richard Liska (Tech. and control processes, numerical methods, Programme: 40 minute lectures from invit- Univ. Prague), Yuri Matiyasevich (Steklov informatics, cubature formulas and solu- ed speakers, 20 or 30 minute communica- Inst. of Math. St Petersburg), Alexander tions of integral equations, informatics in tions from other participants Michalev (Moscow State Univ.), Michael education and teaching methods, mathe- Call for papers: abstracts should be e- Monagan (Simon Fraser Univ.), Matu- matical geophysics, mathematical simulation mailed to [email protected], mailed to OT18, Tarow Noda (Ehime Univ.), Mohamed O. and modelling, mathematical methods in Institute of Mathematics, PO box 1-764, Rayes (Texas Instruments Dallas), Tomas chemistry, mathematical models for Bucharest 70700, Romania, or submitted Recio (Univ. de Cantabria), Eugenio processes in the atmosphere, oceans and online at http://at.yorku.ca/cgi-bin/amca/ Roanes-Lozano (Univ. Complutense de reservoirs and related problems of conver- submit/caeo-01. They can be viewed at Madrid), Tateaki Sasaki (Univ. of Tsukuba), sion, mathematical biology, methods of dis- http://at.yorku.ca/cgi-bin/amca/caeo-01 Stanley Steinberg (Univ. of New Mexico), covering of the regularities, regional prob- Languages: English (preferably), French David Stoutemeyer (Soft. Warehouse), lems of the development of Siberia and the Programme committee: W. B. Arveson Nikolay Vassiliev (Steklov Inst. of Math. St. Far East, mathematical problems of ecology, (Berkeley), N. K. Nikolskii (Bordeaux/St Petersburg), Anatoli Vershik (Steklov Inst. information processing and device control, Petersburg), N. Salinas (Lawrence), S. of Math. St Petersburg), Emil Volcheck mathematical models in geodesy, cadastre Statila (Bucharest), F.-H. Vasilescu (Lille) (National Security Agency), Volker and optical engineering, mathematical Organising committee: D. Gaspar, T. Weispfenning (Passau), Franz Winkler (J. modeling in high technologies, mathemati- Ceausu, A. Craciunescu, C. Pop, N. Suciu, Kepler Univ. Linz) cal economics, algebra, computing algebra, F. Turcu (all Timisoara), A. Gheondea, R.- Local arrangements: Elena Novikova, mathematical logic N. Gologan, D. Timotin (all Bucharest) Nikolay Mnev, Vyacheslav Nesterov, Sergei Invited speakers: plenary lectures will be Deadlines: for registration, 30 March 2000; Slavyanov given by a number of invited speakers who for abstracts, 1 June 2000 Sponsors: Steklov Institute of Mathematics are internationally recognised for their con- Information: at St Petersburg, Euler International tribution to the field e-mail: [email protected] Mathematical Institute, St Petersburg Programme: twenty plenary lectures, sixty Web site: http://www.imar.ro/~ot/conf.html Mathematical Society, St Petersburg State interdisciplinary lectures in parallel ses- 28-1 July: First World Congress of the University sions, a large number of talks in the sections Bachelier Finance Society, Paris, France Language: English of the Congress, and round table discus- Information: Information: sions e-mail: [email protected] e-mail: [email protected] Languages: English and Russian 29-3 July: International Workshop on Web site: http://www.pdmi.ras.ru/EIMI/ Call for papers: it is recommended that Nonlinear Spectral Theory, Würzburg, 2000/imacs/ abstracts be submitted electronically using Germany 26-28: 5th Workshop on Numerical the Web site http://www.math.nsc.ru/ Information: contact Jurgen Appell, Ranges and Numerical Radii, Nafplio, conference. They may also be e-mailed to Department of Mathematics, University of Greece [email protected], with the subject field Würzburg, Am Hubland, D-97074 Information: containing exclusively the relevant section Würzburg, Germany; tel: +49-931-8885017; Web site: http://www.math.uregina.ca/~tsat title (see topics above). They should be writ- fax: +49-931-8885599 /nr/nr.html ten in English or Russian, and prepared in e-mail: [email protected]. 26-28: 6th International Conference on LaTeX2e, using only standard commands de Advanced Computational Methods in Heat and AMS macros, symbols and fonts. They Web site: www.mathematik.uni-wuerzburg. Transfer, Madrid, Spain should have the following structure: 1. Title de/~appell/nlst.html Information: contact Conference of the talk. 2. Name and affiliation of [For details, see EMS Newsletter 33] Secretariat, Heat Transfer 2000, Wessex author(s). 3. Text of the abstract, not 30-2 July: 2000 Centennial Vranceanu, Institute of Technology, Ashurst Lodge, exceeding 1500 symbols in hard copy, Bucharest, Romania Ashurst Southampton, SO40 7AA, tel: +44- including references Topics: Riemannian and pseudo- (0)23-80-293223, fax: +44-(0)23-80-292853 Programme committee: Prof. M. M. Riemannian geometry, submanifold theory, e-mail: [email protected] Lavrentév (Novosibirsk, Chairman), Prof. Chen invariants, affine differential geome- Web site: http://www.wessex.ac.uk/ Yu. G. Reshetnyak (Novosibirsk, Vice- try, relativity, Lie groups, applications of conferences /2000 Chairman), Prof. P. I. Plotnikov differential geometry [For details, see EMS Newsletter 34] (Novosibirsk, Vice-Chairman), Prof. M. V. Information: 26-30: Formal Power Series and Algebraic Fokin (Novosibirsk, Vice-Chairman), Prof. e-mail: [email protected] Combinatorics (FPSAC ‘00), Moscow, L. A. Bokut’ (Novosibirsk, Vice-Chairman), Russia Prof. V. S. Belonosov (Novosibirsk, July 2000 Information: Scientific Secretary), Dr V. L. Vaskevich Web site: http://www.liafa.jussieu.fr/ (Novosibirsk, Scientific Secretary), Dr A. I. 2-7: 6th International Conference on p- ~fpsac00/ Rylov (Novosibirsk, Scientific Secretary), Adic Analysis, Ioannina, Greece [For details, see EMS Newsletter 34] and others (see the Web site) Information: contact A. K. Katsaras, Dept. 26-30: POISSON 2000, France Deadlines: for submission of abstracts, 1 of Math., Univ. of Ioannina, 45110, Information: April. Decisions on acceptance will be made Ioannina, Greece, tel: +30-651-98289, fax: e-mail: [email protected] by 15 April. Acceptance will be confirmed +30-651-46361 montp2.fr by 1 May. An official invitation will be sent e-mail: [email protected] 26-1 July: 4th Siberian Congress on by airmail on request Web site: http://www.uoi.gr/conf_sem/p-adic Industrial and Applied Mathematics, Information: [For details, see EMS Newsletter 34] Novosibirsk, Russia Web site: http://www.math.nsc.ru/ conference 2-15: NATO Advanced Study Institute [dedicated to the memory of M. A. 27-1 July: 18th International Conference 20th Century Harmonic Analysis-a Lavrentév (1900-80)] on Operator Theory, Timisoara, Romania Celebration, Tuscany, Italy Aim: to provide a forum for discussing Theme: operator theory, operator algebras Information: research in contemporary mathematics and and applications Web site: http://www.cs.umb.edu/~asi/ EMS March 2000 27 CONFERENCES analysis2000 Ganter (Dresden), Pierre Hansen vision and medical imaging, computer sys- 3-7: ALHAMBRA 2000, Granada, Spain (Montréal), Itzhak Gilboa (Tel-Aviv), R. tems, engineering sciences, finance and Information: contact ALHAMBRA 2000 Duncan Luce (Irvine), William T. Trotter risk, the humanities, physics and condensed Conference eurocongres Avda. (Arizona U.), Philippe Vincke (Bruxelles), matter, quantum computing, weather fore- Constitución, 18 - Blq.4 E-18012 - Granada, Uta Wille (Jelmoli Ag) casting and climatology Spain, tel: +34-958-209-361, fax: +34-958- Sessions: a special session in symbolic data Speakers: J. M. Brady (Oxford), R. Catlow 209-400 analysis will be organised by Edwin Diday. (Royal Institution of Great Britain), M. e-mail: [email protected], Other special sessions are under review Dempster (Cambridge), P. Embrechts [email protected] Language: English (Zürich), C. Frenk (Durham), R. Horgan Web site: http://www.ugr.es/local/ alhambra Call for papers: see the conference Web site (Cambridge), J. C. R. Hunt (London), S. 2000 Programme committee: Edwin Diday Popescu (Bristol) [For details, see EMS Newsletter 34] (Paris), Jean-Paul Doignon (Bruxelles), Sessions: the format will be based on a 3-7: ANTS IV Algorithmic Number Melvin F. Janowitz (Amherst), Bernard series of lectures by the invited speakers, Theory Symposium, Leiden, the Monjardet (Paris), Marc Pirlot (Mons), Fred together with refereed poster sessions Netherlands S. Roberts (New Brunswick), Rudolf Wille Language: English Information: (Darmstadt) Scientific and organising committee: H. e-mail: [email protected] Organising committee: Jean-Paul Doignon Liddell (London), J. M. Brady (Oxford), P. Web site: http://www.math.leidenuniv.nl (Bruxelles), Samuel Fiorini (Bruxelles), Dasgupta (Cambridge), P. Embrechts /ants4/ Marc Pirlot (Mons) (Zurich), R. Horgan (Cambridge), J. C. R. [For details, see EMS Newsletter 34] Sponsors: F.N.R.S., Belgium, and possibly Hunt (London), K. Moffatt (Cambridge), D. 3-7: Functional Analysis Valencia 2000, other granting institutions B. Mumford (USA). Valencia, Spain Proceedings: the abstracts will appear as a Sponsors: co-sponsored by the Institute of Information: contact: K.D. Bierstedt or J. volume in Electronic Notes in Discrete Physics Bonet, Univ. Paderborn, FB 17, Math., D- Mathematics. Refereed papers will be pub- Site: Cambridge, UK 33095 Paderborn, Germany or Universidad lished in a special issue of Discrete Applied Grants: not available Politècnica de Valencia, Departamento de Mathematics Information: Matemática Aplicada, E-46071 Valencia, Site: the campus of the Université Libre de Web site: http://www.ima.org.uk/ Spain Bruxelles mathematics/confmillennium.htm e-mail: [email protected] Information: 17-20: IUTAM Symposium 2000/10 Web site: http://math-www.uni-paderborn. Web site: http://www.ulb.ac.be/sciences/ Diffraction and Scattering in Fluid de/VLC2000 ulbmath/osda2000 Mechanics and Elasticity, Manchester, UK [For details, see EMS Newsletter 32] 6-8: 6th Barcelona Logic Meeting, Information: contact Professor David 3-9: Euro-Summer School on Barcelona, Spain Abrahams, Department of Mathematics, Mathematical Aspects of Evolving Information: University of Manchester, Oxford Road, Interfaces, Madeira, Portugal e-mail: [email protected] Manchester M13 9PL, UK, tel: +44-(0)161- Information: Web site: http://www.mat.ub.es/~logica/ 275-5901, fax: +44-(0)161-275-5819 e-mail: [email protected] news.html or http://www.crm.es/ e-mail: [email protected] Web site: http://maei.lmc.fc.ul.pt [For details, see EMS Newsletter 34] Web site: http://www.keele.ac.uk/depts/ma/ [For details, see EMS Newsletter 34] 10-14: IUTAM Symposium on Free iutam/ 4-6: Catop 2000, Fribourg, Switzerland Surface Flows, Birmingham, UK [For details, see EMS Newsletter 33] Scope: categorical topological methods Information: 17-21: 9th International Conference on Information: Web site: http://www.mat.bham.ac.uk/ Fibonacci Numbers and their Web site: http://www.unifr.ch/math/catop research/iutam.htm Applications, Luxembourg-City, 2000/ [For details, see EMS Newsletter 33] Luxembourg [For details, see EMS Newsletter 34] 10-14: ICMS Workshop: Dynamical Information: 4-7: 2nd International Conference on Systems, Edinburgh, UK e-mail: [email protected] Mathematical Methods in Reliability, [satellite meeting of the International 17-22: Colloquium on Lie Theory and Bordeaux, France Congress in Mathematical Physics, 17-22 Applications, Vigo, Spain Information: contact Dr. Valentina July, London, UK] Information: contact I Colloquium on Lie Nikoulina, Université Victor Segalen - Information: Theory and Applications, E. T. S. I. Bordeaux 2, Statistique Mathematique, UFR e-mail: [email protected] Telecomunicación, Universidad de Vigo, MI2S, B.P. 69 33076 Bordeaux Cedex, 10-14: 3rd European Congress of 36280 Vigo, Spain; tel: +86 81 21 52 // +86 FRANCE; tel: +33 (0) 5 57 57 10 70 & (0)5 Mathematics (3ecm), Barcelona, Spain 81 24 45; fax: +86 81 21 16 // +86 81 24 57 57 14 25; fax: +33 (0) 5 56 98 57 36 & Information: contact Societat Catalana de 01 +33 (0) 5 57 57 12 63 Matemátiques, Carrer del Carme 47, E- e-mail: [email protected] e-mail: [email protected], 08001 Barcelona, Spain; Web site: http://www.dma.uvigo.es/~clieta/ [email protected] tel: (+34)-270-16-20; fax (+34)-93-270-11- [For details, see EMS Newsletter 33] Web site: http://www.mass.u-bordeaux2.fr 80 17-22: International Congress on /MI2S/MMR2000/ e-mail: [email protected] Mathematical Physics, London, UK [For details, see EMS Newsletter 34] Web site: http://www.iec.es/3ecm/ Information: 5-7: Scandinavian Workshop on Algorithm [For details, including satellite conferences, see Web site: http://icmp2000.ma.ic.ac.uk/ Theory, Bergen, Norway Second Announcement in EMS Newsletter 34] 19-26: 3rd World Congress of Non-linear Information: 13-14: Computational Challenges for the Analysts (WCNA-2000), Catania, Italy e-mail: [email protected] Millennium, Cambridge, UK 22-28: New Mathematical Methods in Web site: http://www.ii.uib.no/swat2000 Theme: a landmark international confer- Continuum Mechanics, Anogia, Crete 5-8: Ordinal and Symbolic Data Analysis ence, organised by the Institute of [part of the series Euroconferences in (OSDA 2000), Brussels, Belgium Mathematics and its Applications in associa- Mathematics on Crete] Aim: the OSDA meetings, of which this is tion with the Isaac Newton Institute, to cele- Main speakers: A. Bressan (Trieste), G. the sixth, are motivated by the fact that brate the new Millennium, which will span a Francfort (Paris-Nord), G. Friesecke ordinal and symbolic data occur quite fre- wide range of major computational science (Oxford), R. James (Minnesota), V. Sverak quently, but theoretical tools for handling areas, which are already, or will become, (Minnesota) such data require further development increasingly more important over the next Organisers: J. Ball (Oxford, UK), S. Invited speakers: Michel Chein decades and beyond Mueller (Leipzig, Germany) (Montpellier), Andreas Dress (Bielefeld), Topics: astrophysics and cosmology, chem- Local coordinator: Susanna Papadopoulou Jean-Claude Falmagne (Irvine), Bernhard istry and molecular modelling, computer ([email protected]) 28 EMS March 2000 CONFERENCES Grants: contact local coordinator Faculty of Mathematics and Physics, there is a special rate for students Information: Institute of Numerical Mathematics, Information: e-mail: [email protected] Sokolovska 83, 186 00 Praha 8, Czech Address: University of Stuttgart, 23-31: ASL European Summer Meeting Republic, tel: +420 2 21911111, +420 2 Mathematisches Institut B/3, (Logic Colloquium 2000), Paris, France 21914223, fax: +420 2 535229, +420 2 Pfaffenwaldring 57, 70550 Stuttgart, Information: 23233994 Germany e-mail: [email protected] Information: Fax: +49-(0)711-685-5322 Web site: http://lc2000.logique.jussieu.fr e-mail: [email protected] e-mail: [email protected] [For details, see EMS Newsletter 33] Web site: http://www.karlin.mff.cuni.cz/ stuttgart.de 24-3 August: EMS Summer School, New katedry/knm/nmicm2000 Web site: http://web.mathematik.uni- Analytic and Geometric Methods in 31-4 August: Workshop on Partial stuttgart.de/~ovid Inverse Problems, Edinburgh, UK Differential Equations: Thermo, Visco and 3-5: Recent Development in the Wave Information: contact Erkki Somersalo, Elasticity, Konstanz, Germany Field and Diffuse Tomographic Inverse Helsinki University of Technology, Finland Topics: partial differential equations relat- Problems, Edinburgh, UK e-mail: [email protected] ed to elasticity, thermoelasticity and vis- Information: 29-4 August: Curves and Abelian Varieties coelasticity e-mail: [email protected] over Finite Fields and their Applications, Invited speakers: H.-D. Alber, D. Andrade, 8-12: XVIII Nevanlinna Colloquium, Anogia, Crete S. Antman, A. Benabdallah, C. Chelminski, Helsinki, Finland [part of the series Euroconferences in C. M. Dafermos, G. Dassios, C. Eck, M. Information: Mathematics on Crete] Fabrizio, J. Ferreira, H. Frid Neto, V. e-mail: [email protected] Main speakers: N. Elkies (Harvard), G. van Georgiev, J. Gwinner, L. Hsiao, S. Jiang, S. Web site: http://www.math.helsinki.fi/ der Geer (Amsterdam), R. Pellikaan Kawashima, J. U. Kim, H. Koch, I. Lasiecka, ~analysis/NevanlinnaColloquium/ (Eindhoven), R. Schoof (Rome), M. Z. Liu, O. Lopes, T. F. Ma, S. A. Messaoudi, [For details, see EMS Newsletter 33] Tsfasman (Marseille) M. Nakao, G. Perla Menzala, M. Reissig, M. 17-3 September: EMS Summer School in Organisers: G. van der Geer (Amsterdam), Renardy, Y. Shibata, M. Slemrod, Y.-G. Probability Theory, Saint-Flour, Cantal, R. Schoof (Rome) Wang, S. Zheng France Local coordinator: Susanna Papadopoulou Organisers: J. E. Muñoz Rivera (Petrópolis, Information: contact: P. Bernard, ([email protected]) Rio de Janeiro), R. Racke (Konstanz) Laboratoire de Mathematiques Appliquees, Grants: contact local coordinator Site: University of Konstanz Univ. Blaise Pascal, F-63177 Aubiere, Information: Information: local organiser: tel/fax: +33 4 73 40 70 64 e-mail: [email protected] [email protected] e-mail: [email protected] 31-3 August: 3rd Conference of Balkan Web site: http://www.mathe.uni-konstanz. [For details, see EMS Newsletter 34] Society of Geometers, Bucharest, Romania de/~racke/announ/ws2000.html 19-25: Discrete and Algorithmic Information: contact V.Balan, University Geometry, Anogia, Crete Politehnica of Bucharest, Department August 2000 [part of the series Euroconferences in Mathematics I, Splaiul Independentei 313, Mathematics on Crete] RO-77206, Bucharest, Romania; fax: (401) 2-9: Summer School on Mathematical Main speakers: G. Kalai (Jerusalem), R. 411.53.65 Physics (emphasis on Quantum Field Seidel (Saarbruecken), J. Snoeyink e-mail: [email protected] Theory), Sandbjerg Manor, Denmark (Vancouver), E. Welzl (Zürich), G. M. [For details, see EMS Newsletter 33] Information: Ziegler (Berlin) 31-4 August: Numerical Modelling In Web site: http://www.maphysto.dk/events/ Organisers: G. M. Ziegler (Berlin), E. Welzl Continuum Mechanics (Theory, 2-18: Rings, Modules and (Zurich) Algorithms, Applications), Prague, Czech Representations-Constanta 2000, Local coordinator: Susanna Papadopoulou Republic Constanta, Romania ([email protected]) Aim: to bring together specialists in fluid Scope: A workshop and conference on alge- Grants: contact local coordinator dynamics, structural mechanics and related bra: 2-12 August, Workshop on Algebra - Information: areas Representation theory (NATO Advanced e-mail: [email protected] Programme: 50-minute invited lectures and Study Institute); 14-18 August, conference 20-23: 3rd International Workshop on 20-minute communications on Rings, Modules and Representations Scientific Computing in Electrical Topics: fluid dynamics, structural mechan- Invited speakers: Henning Andersen Engineering SCEE-2000, Warnemünde, ics, material, structures and optimization, (Aarhus, Denmark), Michel van den Bergh Germany environmental problems (Limburg, Belgium), Jon Carlson (Georgia, Information: Invited plenary speakers: I. Babuska USA), Alexandr Kemer (Moscow State Web site: http://www.SCEE-2000.uni-rostock. (USA), J. W. Barrett (Great Britain), D. University, Simbirsk, Russia), Susan de Braess (Germany), L. Demkowicz (USA), P. Montgomery (Southern California, USA), 21-25: International Association for Fraunie (France) R. Glowinski (USA), T. Claudio Procesi (Rome, Italy), Idun Reiten Mathematics and Computers World Hou (USA), K. Kunisch (Austria), Yu. (Trondheim, Norway), Jeremy Rickard Congress (IMACS 2000), Lausanne, Kuznetsov (USA/Russia), M. A. Leschziner (Bristol, UK), Wolfgang Soergel (Freiburg, Switzerland (Great Britain) Germany) and Efim Zelmanov (Yale Information: contact Prof. Robert Owens, Organising committee chairmen: Miloslav University, USA). The workshop will also IMACS Congress 2000, DGM-IMHEF-LMF, Feistauer, Faculty of Mathematics and include talks by Michel Broué (Paris, Swiss Federal Institute of Technology, CH- Physics, Institute of Numerical France), Steffen König (Bielefeld, 1015 Lausanne, Switzerland; tel: +41-21- Mathematics, Charles University, Prague; Germany), Klaus Roggenkamp (Stuttgart, 693.35.89; fax: +41-21-693.36.46 Karel Kozel, Faculty of Mechanical Germany) and Toby Stafford (Michigan, e-mail: [email protected] Engineering, Department of Technical USA) Web site: http://imacs2000.epfl.ch Mathematics, Czech Technical University, Organising committee chairman: Klaus [For details, see EMS Newsletter 32] Prague; Rolf Rannacher, Institute of Roggenkamp (Stuttgart, Germany) 27-1 September: 9th Summer St Applied Mathematics, Ruprecht-Karls- Local organiser: Mirela Stefuanescu Petersburg Meeting in Mathematical Universitaet Heidelberg (Constanta, Romania) Analysis, St Petersburg, Russia Proceedings: to be published Scientific Committee: László Márki Information: Site: Charles University in Prague (Budapest, Hungary), Fred van Oystaeyen Web site: www.pdmi.ras.ru/EIMI/2000/ Call for papers: abstracts of 15 lines should (Antwerp, Belgium), Klaus Roggenkamp analysis9/index.html be sent to the contact address by 30 March (Stuttgart, Germany) 30-2 September: Innovations in Higher Contact address: Prof. Dr. Miloslav Registration fee: $50 for the workshop and Education 2000, Helsinki, Finland Feistauer, DrSc., Charles University Prague, $50 for the conference ($40 before 31 May); Information: EMS March 2000 29 CONFERENCES e-mail: [email protected] Oxford, Oxford OX1 3LB, UK Departamento de Matematica, Instituto Web site: http://www.helsinki.fi/inno2000 e-mail: [email protected] Superior Tecnico, U.T.L., 1049-001 Lisboa, Web site: http://www.mathematik.uni- Portugal, tel: +351-21-8417095, fax: +351- September 2000 bielefeld.de/~sek/summerseries.html 21-8417598, e-mail: [email protected] [For details, see EMS Newsletter 34] Local organisation: contact N. Manojlovic, 1-4: Constantin Caratheodory Congress, 11-14: International Colloquium in U.C.E.H., 8000 Faro, Portugal, tel: +351- Evros, Greece Honour of Professor Michel Mendès, 28-9800914 ext 7637, fax; +351-289 Scope: measure theory, function theory, Bordeaux, France 818560, e-mail: [email protected] partial differential equations and their [on the occasion of his 65th birthday] Web site: http://www.ualg.pt/cma/iwota applications Topics : number theory, combinatorics and 13-15: International Conference of the Information: physics Royal Statistical Society, Reading, UK e-mail: [email protected] Invited Speakers : Alan Baker, Vitaly Information: 2-9: International Conference on Bergelson, Michel Dekking, Maurice e-mail: [email protected] Topology and its Applications, Ohrid, Dodson, Teturo Kamae, Michael Keane, 15-18: Physical Interpretations of Macedonia Hugh Montgomery, Wladyslaw Narkiewicz, Relativity Theory, London, UK Information: Andrew Pollington, Imre Ruzsa, Andrzej Information: Web site: http://www.pmf.ukim.edu.mk/ Schinzel, Jeffrey Shallit, Chris Smyth, Vera e-mail: [email protected] mathematics/icta2000.html Sós, Alf Van der Poorten, Zhi-Ying Wen, 18-22: International Data Analysis 3-6: 31st European Mathematical Jia-Yan Yao, Don Zagier, Marie-José Bertin, Conference, Innsbruck, Austria Psychology Group Meeting (EMPG 2000), Anne Bertrand, Paula Cohen, Hédi Information: Graz, Austria Daboussi, Bernard Derrida, Etienne Fouvry, e-mail: [email protected] Scope: all areas of mathematical psychology Jean-Pierre Kahane, Yves Meyer, Martine Web site: Topics: measurement theory, psy- Queffélec, Gérard Rauzy, Georges Rhin, http://www.statistik.tuwien.ac.at/ida2000/ chophysics, psychometrics, statistical theory, Bahman Saffari, Gérald Tenenbaum 18-23: International Congress on response times, perception, cognition, deci- Languages : English and French Differential Geometry, Bilbao, Spain sion theory, memory, learning, knowledge Advisory committee: Pierre Cartier (chair- [in memory of Alfred Gray (1939-98)] space theory, neural networks, and others man), Anne Bertrand, Jean-Marc Theme: differential geometry Organising committee: Prof. Dietrich Deshouillers, Pierre Liardet, Jean-François Topics: special Riemannian manifolds, Albert (Graz), Chairman Méla, Michel Olivier, Jacques Peyrière homogeneous spaces, complex structures, Site: University of Graz, Austria Organising committee : Jean-Paul Allouche symplectic manifolds, geometry of geodesic Language: English (chairman), Christophe Doche, Jean- spheres and tubes and related problems, Deadlines: for submission of abstracts and Jacques Ruch geometry of surfaces, computer graphics in early registration, 31 May Sponsors : University Bordeaux I, Réseau differential geometry and Mathematica( Information: Diophante, EMS Main speakers: T. Banchoff (USA), R. e-mail: [email protected] Site : Université Bordeaux I, 351, cours de Bryant (USA), H. Ferguson (USA), T. Web site: http://psyserver.kfunigraz.ac.at/ la libération 33405 Talence, France Friedrich (Germany), K. Grove (USA), S. empg2000/ Deadline: for registration, end of June Gindikin (USA), A. Huckleberry (Germany), 4-6: Mathematics of Surfaces, Cambridge, Information: D. Joyce (England), M. Mezzino (USA), V. UK Web site: http://www.math.u-bordeaux.fr/ Miquel (Spain), E. Musso (Italy), R. Palais Information: ~stan/Colloque/MMF.html (USA), M. Pinsky (USA), A. Ros (Spain), D. Web site: http://www.ima.org.uk 11-15: Boundary Integral Methods: Sullivan (USA), I. Taimanov (Russia), J. 4-8: FGI2000 French-German-Italian Theory and Applications, Bath, UK Wolf (USA) Conference on Optimization, Montpellier, Information: Language: English France Web site: http://www.ima.org.uk/ Call for papers: we are using the facilities Information: contact: Bernard Lemaire, mathematics/conferences.htm of Atlas Mathematical Conference Abstracts. Mathématiques, Université de Montpellier [For details, see EMS Newsletter 34] If you wish to present an oral communica- II, Place Eugène Bataillon, 34095 12-15: Imaging and Digital Image tion or a poster, please submit before 31 Montpellier cedex 05 Processing: Mathematical Methods, May 2000 an extended (up to two pages) e-mail: [email protected] Algorithms and Applications, Leicester, abstract (either plain ASCII or TeX) via Web site: http://www.math.univ-montp2.fr/ UK http://at.yorku.ca/cgi-bin/amca/submit/cadq- [For details, see EMS Newsletter 34] Information: 01. Abstracts accepted by the organising 4-15: Spatial Structures in Biology and Web site: http://www.ima.org.uk/ committee will become available at Ecology: Models and Methods, A mathematics/conferences.htm http://at.yorku.ca/cgi-bin/amca/cadq-01 Biomathematics Summer School, Taranto, [For details, see EMS Newsletter 34] Programme committee: T. Banchoff (USA), Italy 12-15: IWOTA -Portugal 2000 J. P. Bourguignon (France), S. Donaldson Information: International Workshop on Operator (England), J. Eells (England), S. Gindikin Web site: http://www.mat.unimi.it/~miriam Theory and Applications, Faro, Portugal (USA), M. Gromov (France), O. Kowalski /ESMTB/MARTINA-SS/summer_school. Main topics: factorization theory, factoriza- (Czech Republic), M. Mezzino (USA), S. html tion and integrable systems, operator theo- Novikov (USA), M. Pinsky (USA), A. Ros 5-7: Quantitative Modelling in the retical methods in diffraction theory, alge- (Spain), S. Salamon (England), L. Vanhecke Management of Health Care, Salford, UK braic techniques in operator theory, related (Belgium), J. Wolf (USA) Information: topics and applications to mathematical Organising committee: M. Fernández Web site: http://www.ima.org.uk/mathematics physics (chairman, Spain), L.C. de Andrés (Spain), /conferences.htm Steering committee: T. Ando, H. Bart, L.A. Cordero (Spain), A. Ferrández (Spain), [For details, see EMS Newsletter 34] H.Bercovici, R. Dijksma, H. Dym, C. Foias, R. Ibáñez (Spain), M. de León (Spain), M. 5-16: Advanced Course on Algebraic I. Gohberg, J. W. Helton, M. A. Kaashoek, Macho-Stadler (Spain), A. Martinez Naveira Quantum Groups, Bellaterra, Spain H. Langer, R. Mennicken, L. Rodman, J. G. (Spain), L. Ugarte (Spain) Information: Stampfli Sponsors (provisional): Universidad del e-mail: [email protected] Organisers: A. F. dos Santos, F.-O. Speck, País Vasco-Euskal Herriko Unibertsitatea, Web site: http://crm.es/quantum A. B. Lebre, R. Picken (Lisboa), V. G. Bilbao Iniciativas Turisticas, Casco Viejo, 10-17: Summer School on Geometry of Kravchenko, N. Manojlovic (Faro), M. A. Codorniu, Colegio Mayor Miguel Quiver-Representations and Preprojective Kaashoek (Amsterdam) Unamuno, El Correo, Deia, Europen Algebras, Isle of Thorn, UK Registration: use IWOTA web-page or e- Mathematical Society, Iberia, Iparlat-Kaiku, Information: contact Karin Erdmann, mail to receive second announcement Laboratorios Kodak, Metro Bilbao, El Mathematical Institute, University of Information: contact F.-O. Speck, Mundo, Panda Software, Real Sociedad 30 EMS March 2000 CONFERENCES Bascongada de Amigos del Pais, Staedtler, particular those from less favoured regions 2000.htm Wolfram Research Inc of European Community countries, and for Proceedings: to be published participants from Central and Eastern October 2000 Site: the buildings of Facultad de Ciencias Europe Economicas y Empresariales (Avenida Deadline: for applications, 2 May 7-10: International Conference on Lehendakari Agirre 83, Bilbao) Information: Mathematical Modelling and Grants: probably support for participants http://www.esf.org/euresco/00/pc00127a. Computational Experiments (ICMMCE), from countries in a difficult economic situa- htm Dushanbe, Tajikistan tion and young mathematicians 22-27: EURESCO Conference on Number Information: Deadlines: for registration, 30 June; for Theory and Arithmetical Geometry: Web site: http://www.tajnet.com/ abstracts, 31 May Motives and Arithmetic, Obernai (near 15-21: DMV Seminar on The Riemann Information: Strasbourg), France Zeta Function and Random Matrix e-mail: [email protected] Organiser: U. Jannsen (Regensburg) Theory, Mathematisches Institut Web site: www.ehu.es/Gray Information: available soon at Oberwolfach, Germany 18-27: 8th Workshop on Stochastic and Web site: http://www.esf.org/euresco Organisers: Jon Keating (Bristol), Zeév Related Fields, Famagusta, North Cyprus 22-28: 4th International Conference on Rudnick (Tel Aviv) and Kannan Information: Functional Analysis and Approximation Soundararajan (Princeton) Web site: http://mozart.emu.edu.tr/workshop Theory (4th FAAT), Acquafredda di Information: Prof. Dr Matthias Kreck, 19-22: Fractal Geometry: Mathematical Maratea , Potenza, Italy Universität Heidelberg, Mathematisches Techniques, Algorithms and Applications, Aim: to bring together mathematicians and Institut, Im Neuenheimer Feld 288, 69120 Leicester, UK specialists working in functional analysis Heidelberg, Germany Information: and approximation theory, in order to pro- 15-21: DMV Seminar on Motion by Web site: http://www.ima.org.uk/mathematics mote interdisciplinary collaborations and to Curvature, Mathematisches Institut /confractalgeometry.htm enhance exchanges of results, techniques Oberwolfach, Germany [For details, see EMS Newsletter 34] and applications Organisers: Gerhard Huisken (Tübingen) 19-22 SCAN 2000: 9th GAMM-IMACS Main topics: Banach spaces, Banach lat- and Tom Ilmanen (ETH Zürich) International Symposium on Scientific tices, function spaces, (positive) linear oper- Information: Prof. Dr Matthias Kreck, Computing, Computer Arithmetic and ators, semigroups of (positive) linear opera- Universität Heidelberg, Mathematisches Validated Numerics, Karlsruhe, Germany tors, evolution equations, approximate Institut, Im Neuenheimer Feld 288, 69120 Information: quadratures and integral equations, approx- Heidelberg, Germany Web site: http://www.scan2000.de/ imation methods in abstract spaces and in 30-3 November: Clifford Analysis and its 22-27: EURESCO Conference on function spaces, approximation by (positive) Applications, NATO Advanced Research Geometry, Analysis and Mathematical operators, interpolation, polynomial Workshop, Praha, Czech Republic Physics: Analysis and Spectral Theory, approximation, constructive approximation, Information: San Feliu de Guixols, Spain orthogonal polynomials Web site: http://www.karlin.mff.cuni.cz/ Aim: the interaction between physics and Confirmed invited speakers: P. Aiena ~clifford mathematics has become ever more impor- (Palermo), H. Berens (Erlangen), P. tant in recent years. This meeting aims to Clément (Delft), T. Erdelyi (College November 2000 bring together mathematicians and some Station), C. Franchetti (Firenze), G. physicists working in areas related to quan- Godefroy (Paris), D. S. Lubinsky 12-18: DMV Seminar on Computational tum physics and mathematical analysis (Johannesbourg), G. Milovanovic (Nis), G. Mathematics in Chemical Engineering and Scope: recent progress in non-linear scat- Monegato (Torino), M. Neumann Biotechnology, Mathematisches Institut tering theory and evolution equations, soli- (Mississippi), P. L. Papini (Bologna), I. Rasa Oberwolfach, Germany tons, quantum many body problems, reso- (Cluj-Napoca), B. Silbermann (Chemnitz), Organisers: Peter Deuflhard (Berlin), nances (scattering poles), field theory, quan- V. Totik (Szeged), P. Vertesi (Budapest), L. Rupert Klein (Potsdam/Berlin) and Christof tum chaos, and also some related topics Weiss (Karlsruhe) Schütte (Berlin) such as control theory, spectral theory on Scientific programme: survey talks by invit- Information: Prof. Dr Matthias Kreck, manifolds, semi-classical problems. These ed speakers (45 minutes) and short commu- Universität Heidelberg, Mathematisches areas are closely related to each other and nications (20 minutes) Institut, Im Neuenheimer Feld 288, 69120 many modern mathematical tools, such as Sponsors: Center for Studies on Functional Heidelberg, Germany for instance microlocal (or phase space) Analysis and Approximation Theory of the 12-18: DMV Seminar on Characteristic analysis have been successfully applied and University of Basilicata (Potenza, Italy), Classes of Connections, Riemann-Roch developed. The purpose of the conference Department of Mathematics of the Theorems, Analogies with e-Factors, is to promote the interaction between these University of Basilicata (Potenza, Italy), Mathematisches Institut Oberwolfach, themes in a unifying effort Department of Mathematics of the Germany Speakers: Boris Altshuler (Princeton), V. University and of the Polytechnic of Bari Organisers: Spencer Bloch (Chicago) and Bach (Mainz), Nicolas Burq (Orsay), (Italy), University of Basilicata (Italy), Helene Esnault (Essen) Vladimir Buslaev (St. Petersburg), Jan University of Bari (Italy), Polytechnic of Information: Prof. Dr Matthias Kreck, Derezinski (Warshaw), Bruno Eckhart Bari (Italy), National Group of Functional Universität Heidelberg, Mathematisches (Marburg), Maria Esteban (Paris), Jean Analysis and Applications (G.N.A.F.A.), Institut, Im Neuenheimer Feld 288, 69120 Ginibre (Orsay), Bernard Helffer (Orsay), Progetti di Ricerca di Interesse Nazionale: Heidelberg, Germany Victor Ivrii (Toronto), Vojkan Jaksic Analisi Funzionale (M.U.R.S.T.) (Ottawa), Andreas Knauf (Erlangen), Organising committee: F. Altomare December 2000 Alexander Komech (Moscow), Gilles Lebeau ([email protected]), Attalienti (Palaiseau), Galina Perelman (Palaiseau) ([email protected]), M. Campiti 18-20: 5th International Conference on Claude-Alain Pillet (Toulon), Israel Sigal ([email protected]), Della Vecchia Mathematics in Signal Processing, (Toronto), Uzy Smilanski (Rehovot), Jan- ([email protected]), G. Coventry, UK Philip Solovej (Copenhagen), T. Tao (Los Mastroianni ([email protected]), M. R. Information: contact Pamela Bye, The Angeles), Andras Vasy (Berkeley-Boston), Occorsio ([email protected]) Institute of Mathematics and its Giorgio Velo (Bologna), Giorgi Vodev Site: Hotel Villa del Mare, Acquafredda di Applications, Catherine Richards House, 16 (Nantes), Steven Zelditch (Baltimore), Maratea, Potenza, Italy Nelson Street, Southend-on-Sea, Essex SS1 Maciej Zworski (Berkeley) Information: contact any of the organising 1EF, England, fax: +44 (0)1702 354111 Site: Hotel Eden Roc, San Feliu de Guixols committee or visit the website e-mail: [email protected] (Costa Brava), Spain Web site: [For details, see EMS Newsletter 34] Grants: available for young scientists, in http://www.dm.uniba.it/maratea/FAAT EMS March 2000 31 RECENT BOOKS Explicit models for these representations (including Whittaker and Kirillov models) are constructed here and a connection with holomorphic Jacobi forms and their gener- RecentRecent booksbooks alisations is discussed. The book comple- edited by Ivan Netuka and Vladimír Sou³ek ments well the book Automorphic forms and representations by D. Bump (see EMS Newsletter 29, p.39) and The Theory of Jacobi Books submitted for review should be sent to the expository articles based on lecture courses Forms by M. Eichler and D. Zagier. following address: Ivan Netuka, MÚUK, given at the MSRI semester on this subject This book is a very useful addition to the Sokolovská 83, 186 75 Praha 8, Czech in the fall of 1995. A range of topics is literature on the topic and can be recom- Republic. addressed, focusing primarily on opera- mended to readers interested in represen- tor–theoretic aspects. tation theory, the theory of algebraic M. S. Agranovich, B. Z. Katsenelenbaum, The opening paper by D. Sarason gives groups and number theory. (vs) A. N. Sivov and N. N. Voitovich, a beautiful overview of the area. The next Generalized Method of Eigenoscillations in two papers are concerned with Bergman A. Beutelspacher, N. Henze, U. Kulisch Diffraction Theory, Wiley-VCH, Berlin, spaces: factorisation, invariant subspaces and H. Wussing (eds), Überblicke 1999, 377 pp., DM298, ISBN 3-527-40092- and Beurling’s theorem (H. Hedenmalm) Mathematik 1998, Friedrich Vieweg & Sohn, 3 and the harmonic Bergman spaces on the Braunschweig, 1997, 148 pp., ISBN 3-528- This book presents a new method for solv- disc (K. Stroethoff). Four papers are devot- 06944-9 ing various diffraction and scattering prob- ed to Hankel operators and their generali- Following a positive reaction to the first vol- lems in acoustics, electrodynamics, and sations, by V. Peller (a survey of the classi- ume, the Wiesbaden publishing house quantum mechanics. This method is based cal Hankel operators), P. Gorkin, S. Vieweg has already published the second on the representation of the solution of a Saccone (both dealing with Hankel-type volume of Überblicke der Mathematik which stationary diffraction problem by a series operators on uniform algebras), and R. summarises the development and the con- with respect to some orthogonal system of Rochberg (Hankel forms of higher order). temporary state of some mathematical dis- functions. These functions are the eigen- The papers by Z. Wu and by J. B. Conway ciplines and, above all, their application in functions of the auxiliary homogeneous and L. Yang concern operator theory in the the fine arts, natural sciences and other sci- problem in which the spectral parameter is Dirichlet space and subnormal operators, entific disciplines. All essays are well and some electrodynamic parameter (not nec- respectively. Six papers that follow address attractively arranged and written and give a essarily frequency). The choice of the spe- the field of operator models, systems theo- reliable survey that is fairly accurate from cific electrodynamic variable used as the ry and interpolation; namely, a coordinate- the professional point of view and readable spectral parameter depends on the form of free function model approach to spectral for mathematicians active in other disci- the diffraction problem. The book presents theory (N. Nikolski and V. Vasyunin), scat- plines. several versions of the method. The main tering systems (C. Sadosky), feedback stabi- The survey contains contributions by P. part of the book contains the exposition of lization (N. Young), an abstract interpola- Schreiber (Mathematics and fine arts), U. the formal technique of various versions of tion problem (A. Kheifets), a unified Kulisch (Computer, arithmetic and numer- the generalised method of eigenoscillations approach to the classical interpolation ic), R. Weiss (Mathematics and mathemati- (Chapters 1-2), the construction of station- problems and their generalizations (H. cal tools: advice from history), P. Brass ary functionals (Chapter 3), and examples Dym), and the reproducing kernel (Distances of finite point sets), G. Strang of applications to specific problems Pontryagin spaces (D. Alpay, A. Dijksma, J. (Mathematics in satellite navigation), E. (Chapter 4). Chapter 5 deals with a mathe- Rovnyak and H. S. V. de Snoo). The con- Gauss (Fractals and fun – a programme of matically rigorous treatment and justifica- cluding article by Vinnikov exhibits an variants of classic fractals), L. Rüschendorf tion of the technique constructed in interesting relationship between the theory (Stochastics – an interdisciplinary science), Chapters 1-2 and connected with the use of of commuting families of non-self-adjoint B. Polster and A. E. Schroth (Models of non-self-adjoint operators. (dmed) operators and function theory on Riemann semibiplanes), and R. Schreckenberg surfaces. (Mathematics of transport flows). (efu) M. Aigner and G. M. Ziegler, Proofs from Though the papers vary both in length THE BOOK, Springer, Berlin, 1998, 199 pp., (10 to 92 pages) and in the degree of spe- B. Blackadar, K-Theory for Operator DM49.90, ISBN 3-540-63698-6 cialisation (for instance, Peller’s paper may Algebras, Mathematical Sciences Research This is an international bestseller which well serve as an introductory tutorial on Institute Publications 5, Cambridge University presents 30 problems in less than 200 Hankel operators for a newcomer), for the Press, Cambridge, 1998, 300 pp., £19.95, pages which can be solved by means of most part they are accessible even to a non- ISBN 0-521-63532-2 pleasing tricks. No previous knowledge is expert. In my opinion, this is a very nice This is the second edition of a famous book, needed, and yet the problems are taken collection of articles and it is extremely the first edition of which appeared in 1986. from number theory, geometry, analysis, unlikely to disappoint any reader interest- For the second edition the author has made combinatorics and graph theory (of course, ed in the subject. (me) only minor changes. He has corrected and perhaps naturally, most of the solu- some errors, and added new comments and tions have a combinatorial flavour). Paul R. Berndt and R. Schmidt, Elements of the references. The references now contain 250 Erdös, who invented the mythology of ‘The Representation Theory of the Jacobi Group, items and go up to 1998. The only substan- Book’, was reluctant to assign ‘Book’ status Progress in Mathematics 163, Birkhäuser, tial addition is a new section (at the very to any particular proof, but he would prob- Basel, 1998, 213 pp., DM108, ISBN 3-7643- end) on E-theory. ably not protest against the balanced (and 5922-6 K-theory first appeared first in topology as in a way expected) choice of the authors. Parabolic groups and their representations a device for the study of vector bundles. This is a useful book for motivating stu- nowadays play a very important role in Later on, it was realised that instead of a dents and to stimulate mature minds. (jnes) many parts of mathematics. One of the sim- vector bundle one can consider continuous plest and most important examples is the sections of this bundle as a module over the S. Axler, J. E. McCarthy and D. Sarason Jacobi group. It is a semidirect product of algebra of continuous functions on the base (eds), Holomorphic Spaces, Mathematical the symplectic group Sp(2n) with the corre- space. This idea was at the beginning of the Sciences Research Institute Publications 33, sponding Heisenberg group. K-theoretical investigations of operator Cambridge University Press, Cambridge, 1998, In the book, the authors restrict them- algebras (as well as at the beginning of alge- 476 pp., £35, ISBN 0-521-63193-9 selves to the case n = 1. They describe a braic K-theory). This K-theoretic approach The term ‘holomorphic spaces’ is used here classification of the irreducible representa- has brought a revolution into the study of as short for ‘spaces of holomorphic functions of the Jacobi group in the case of real, operator algebras and has led to many tions’, and this book consists of several complex, p-adic and adelic coefficients. interesting applications. 32 EMS March 2000 RECENT BOOKS To understand these parts of mathemat- tional applications of elliptic curves, such as example, the Hochster-Huneke theorem, ics is not very easy. At the time of its first factoring, primality proving or the equiva- that equicharacteristic direct summands of edition Blackadar’s book was the only com- lence between the discrete logarithm prob- regular rings are Cohen-Macaulay, is prehensive introduction to the K-theory of lem and a problem connected to a Diffie- proved. To cover the applications, other operator algebras. Since then several very Helman key exchange. The final chapter extensions were needed. For example, a good books on this subject have appeared, discusses a generalisation to hyperelliptic section developing techniques for reduc- but for a comprehensive one we would most systems. The book closes with an appendix tion to characteristic p, has been rewritten. probably again choose this one. presenting examples of elliptic curves Chapter 4, dealing with Hilbert func- According to the author, the reader of whose groups of rational points contain tions and multiplicities, has been extended the book should be familiar with the rudi- large prime subgroups. by a new section containing Gotzman’s reg- ments of the theory of Banach algebras and The book is written in a very readable ularity and persistence theorem. Chapter 5, C*-algebras, such as can be found in the form and thus can be consulted and used dealing with Stanley-Reisner rings of sim- first part of J. Dixmier’s Les C*-algebras et both by mathematicians and by anybody plicial complexes, now includes a new sec- leurs représentations, or G. K. Pedersen’s C*- wishing to learn more about the mathemat- tion where Hochster’s formula for Betti algebras and their automorphism group, or M. ics behind the implementations of elliptic numbers of Stanley-Reisner rings is proved. Takesaki’s Theory of operator algebras, but curve cryptosystems. Though the book is The authors present the classical mater- sometimes must know more. written for a wide audience, familiarity with ial together with quite recent develop- The book is well written, with a great the main principles of the involved public ments, avoiding more complex structures number of examples, exercises and prob- key cryptography and number theory such as derived categories or spectral lems that help one to understand the theo- would be useful. The book can also be of sequences. This makes the book valuable ry, and the reader can find a good survey of great help also for those who want a quick not only for experts, but also for beginners the theory as well as many references to survey of the main and new results of ellip- equipped only with a basic knowledge of more detailed or more advanced reading. tic curve cryptography. commutative and homological algebra. Section 24 outlines the nice applications of (jtrl) the theory in geometry and topology. The D. A. Brannan, M. F. Esplen and J. J. book will be interesting for specialists in Gray, Geometry, Cambridge University Press, F. Buekenhout, M. Dehon and D. operator algebras, in topology, in group Cambridge, 1999, 497 pp., £18.95, ISBN 0- Leemans, An Atlas of Residually Weakly representations, and for anybody who wish- 521-59193-7 and 0-521-59787-0 Primitive Geometries for Small Groups, es to appreciate the very nice interplay The topic discussed in this book is geome- Mémoire de la Classe des Sciences, Tome XIV, between operator algebras and topology. try in its most classical sense. The authors’ Académie Royale de Belgique, Brussel, 1999, For postgraduate students the book may aim was to create a text addressing students 175 pp., ISBN 2-8031-0161-0 not be easy reading, but will be very help- in a way that shows the beauty of the classi- This atlas analyses geometries induced by ful. (jiva) cal geometrical ideas concerning conics, 32 small groups. The groups can be divid- affine and projective geometries, inversive ed into three families: Sn and An for n ≥ 7; 2 I. Blake, G. Seroussi and N. Smart, Elliptic and spherical geometries, and non- PSL2(q), PGL (q) and PΓL2(q) for q ≥ 8; and Curves in Cryptography, London Euclidean geometries. The Kleinian philos- affine-type groups which include (among Mathematical Society Lecture Note Series 265, ophy of geometry is used through the book, others) the groups AGL1(q) for q ≥ 13. Cambridge University Press, Cambridge, 1999, but is discussed explicitly only in the last For each of these groups G, the authors list 204 pp., £24.95, ISBN 0-521-65374-6 chapter, when most of the previous treat- geometries Γ = Γ (G, (Gi)i∈I), where Gi are This book summarises the latest knowledge ments are revisited. subgroups of G, and the incidence relation on the theoretical aspects and practical The book is designed for a student work- is given by non-empty intersection of left implementation related to public key cryp- ing without further help, and the text is cosets of Gi, i∈I. Only those geometries Γ tography based on elliptic curves. The produced very carefully from the point of are included that satisfy at least one of the authors start with the discussion of issues view of contents, splitting of the material properties F, RC, FT, (IP)2 and RWPRI; the about basic arithmetic operations on curves into individual blocks (each one evening’s property F (firm) means that every flag of and finite fields. work) and the excellent graphical design. rank |I|–1 is contained in at least two The first chapter surveys some standard Many historical comments and links are chambers, RC stands for residually con- protocols of public key cryptography based added. All expositions are accompanied by nected, FT for flag transitive, (IP)2 refers to on groups and discusses some practical several solved and unsolved problems and the 2 intersection property and RWPRI implications of using groups. The second exercises, whose solutions may be found at stands for residually weakly primitive (each chapter addresses problems connected with the end of the text. The book will be help- residue of a flag is weakly primitive: the implementation of the underlying field ful for all students of mathematics, as well group acts primitively on the set of i-ele- arithmetic; the material is treated separate- as for their teachers. (jslo) ments for at least one i∈I.) ly for fields of odd characteristic and fields For each group the authors give the list of characteristic 2. Chapter 3 introduces W. Bruns and J. Herzog, Cohen-Macaulay of subgroup conjugacy classes, together the basic concepts from the theory of ellip- Rings, Cambridge Studies in Advanced with the inclusion relations. For each tic curves needed for the rest of the book. Mathematics 39, Cambridge University Press, geometry (with some restrictions pertain- Besides the basic notions, the reader will Cambridge, 1998, 453 pp., £24.95, ISBN 0- ing to geometries of higher rank), the find notions like the division and modular 521-56674-6 and 0-521-41068-1 authors draw the diagram, the incidence polynomials and the Weil pairing. The This is a revised edition of a monograph graph and the collinearity graph, and com- fourth chapter is devoted to efficient algo- first published under the same title in 1993. pute groups of automorphisms and corella- rithms for computation of point addition, Its topic, Cohen-Macaulay rings and mod- tions. The graphs are equipped with vari- their doubling, and of integer multiples of ules, is central to modern commutative ous numerical and symbolic data, in order points on an elliptic curve. Chapter 5 deals algebra. As important particular cases, it to convey as much information as possible. with various attacks (e.g. MOV, baby includes the regular local rings (those of The atlas also contains a list of geometry step/giant step, time and wild kangaroos, finite global dimension), complete intersec- diagrams that have been induced by the etc.) on the converse question of the elliptic tions, and Gorenstein local rings (those of investigated groups. One can thus get from curve discrete logarithm problem. Chapter finite injective dimension). So the focus is a group to geometries and from a geome- 6 introduces the problem of determining naturally on homological methods, but the try to groups. (ad) the order of the groups of rational points. more recent combinatorial aspects due to More advanced methods are then present- Hochster and Stanley are also presented. S. C. Coutinho, The Mathematics of ed in two chapters entitled ‘Schoof’s algo- The main revisions concern Part III, Ciphers: Number Theory and RSA rithm and extensions’ and ‘Generating now called ‘Characteristic p methods’. Cryptography, A. K. Peters, Ltd., Natick, curves using complex multiplication’. The There is a new Chapter 10 dealing with 1999, 196 pp., £19, ISBN 1-56881-082-2 penultimate chapter discusses some addi- tight closures and their applications; for From a background of the RSA cryptosys- EMS March 2000 33 RECENT BOOKS tem, the author develops various elemen- parabolic equations (a paper by Yuan-Wei This is a nice English edition of Dirichlet’s tary algorithmic aspects of number theory Qi) and Boltzmann-Poisson systems; singu- famous Vorlesungen über Zahlentheorie, and algebra. The book is centred more lar perturbation methods; and numerical including the nine Supplements by around mathematics than cryptography, analysis. (jmal) Dedekind, translated by John Stillwell. The which makes the book suitable to all novices book opens with an introduction by the interested in number theory. Thus the H. G. Dales and G. Oliveri (eds), Truth in translator. As one of the most important reader can find here a description of the Mathematics, Clarendon Press, Oxford, 1998, number-theoretical and mathematical principle of finite induction, introduced by 376 pp., ISBN 0-19-851476-X books of the 19th century this book needs the well-known puzzle called the Tower of This book contains the lectures given at a no further description, and can be recom- Hanoi. As this example shows, the author conference on this theme, held in Sicily in mended to those who have problems with avoids the ‘dry’ theorem-proof style, and September 1995; the book is dedicated to the German language, or to those who can- devotes a lot of space to historical com- the memory of Dr. R. O. Gandy. not find the German original in the library. ments. Many branches of mathematics (founda- This book should certainly have a perma- The topics covered by the book are the tions) are discussed in these papers. nent place on every mathematical book- sieve of Erathosthenes, modular arithmetic, Besides the history of development of shelf. (šp) pseudoprimes, the Chinese remainder the- mathematical truth, we mention construc- orem, the basic theory of groups (including tivism, computability and algorithms, and C. Dorschfeldt, Algebras of Pseudo- Lagrange’s theorem), Mersenne and set theory and natural numbers. differential Operators near Edge and Fermat primes with the Lucas-Lehmer test, One of the papers is an excellent contri- Corner Singularities, Mathematical Research primitive roots, Carmichael numbers, and bution on foundations of the theory of 102, Wiley-VCH, Berlin, 1998, 202 pp., the cumulative applications to the RSA algorithms by Y. N. Moschovakis which can DM128, ISBN 3-527-40118-0 cryptosystem with a discussion of its various be understood as to be written quite for- The general theory of elliptic operators on elementary aspects. The book is easy to malistically without considering such pecu- manifolds has a long history, and was read, and is suitable also forthose studying liar things as mathematical truth. On the recently extended to manifolds with singu- independently. The book contains no other hand, in another excellent paper on larities of certain types. The simplest and answers to the exercises closing each chap- mathematical evidence by D. A. Martin, the best-understood is the case of conical sin- ter. notion of mathematical truth is used to con- gularities, modelled by a cone whose base is vince the reader that it is reasonable to a closed smooth manifold. This research R. Curtis and R. Wilson (eds), The Atlas of accept projective determination as a new monograph treats more general types of Finite Groups: Ten Years On, London valid statement of set theory. singularities. Two recent monographs by Mathematical Society Lecture Note Series 249, This book can be especially recommend- B.-W. Schulze described a pseudodifferen- Cambridge University Press, Cambridge, 1998, ed to those who do not want to use only the tial calculus of operators on manifolds with 293 pp., £27.95, ISBN 0-521-57587-7 safe formalism created by others, but want cone and edge singularities. This book is the proceedings of a confer- also to consider the area of mathematical The first part of the book (Chapters 2 ence organised in Birmingham in July truth, which cannot be fully formalised, and 3) contains a parameter-dependent 1995, to mark the tenth anniversary of the such as by the Tarski theorem. (k³) version of elliptic edge operators; the main Atlas of Finite Groups. It contains twenty arti- tool used here is the Mellin transform. cles by leading experts in the field. Besides B. Davies and Y. Safarov (eds), Spectral Another type of singularity is the so-called research papers, we note a historical article Theory and Geometry, London Mathematical ‘corner singularity’; a model example is a on the development of the Atlas project Society Lecture Note Series 273, Cambridge cone over a close compact manifold with since 1970 by three of its authors, J. H. University Press, Cambridge, 1999, 328 pp., conical singularities. The last chapter of the Conway, R. T. Curtis and R. A. Wilson. Of £27.95, ISBN 0-521-77749-6 book treats the Mellin-type operators on particular interest are survey papers on This Lecture Note appeared as the final manifolds with corner singularities. applications of character theory to surfaces product of the ICMS Instructional This book presents material that has by G. A. Jones, on recent advances in the Conference, held in Edinburgh in 1998. previously been available only in papers or representation theory by G. Hiss, and on The volume contains most of the (extend- PhD dissertations. (vs) Zassenhaus conjectures on integral group ed) lectures of the invited speakers. rings by W. Kimmerle. (jtu) It is organised according to the level of L. van den Dries, Tame Topology and O- the courses: Introductory courses (F. E. Minimal Structures, London Mathematical H.-H. Dai and P. L. Sachdev (eds), Recent Burstall, I. Chavel and E. B. Davies); Society Lecture Note Series 248, Cambridge Advances in Differential Equations, Pitman Medium-level courses (M. Ashbaugh, A. University Press, Cambridge, 1998, 180 pp., Research Notes in Mathematics Series 386, Grigor’yan and M. Shubin); and Advanced £24.95, Addison Wesley Longman Ltd, Harlow, 1998, courses (only the lecture by S. Zelditch is The study of properties of sets A ⊆ Rn given 243 pp., £35, ISBN 0-582-32219-7 included). The titles of the lectures are: by a finite set of inequalities A =∩i{x | fi(x) This volume of the Pitman Research Notes Basic Riemannian geometry, The ≥ 0} (where fi belongs to a given class of in Mathematics Series contains eighteen Laplacian on Riemannian manifolds, functions) has a long tradition. Typical lectures from the first Pan-China Computational spectral theory, examples are the theory of semi-algebraic Conference on Differential Equations, held Isoperimetric and universal inequalities for or subanalytic sets. Grothendieck has indi- in Kunming in May-June 1997. The aim of eigenvalues, Estimates of heat kernels on cated in his ‘Esquisse d’un programme’ a the editors was to present recent research Riemannian manifolds, and Spectral theo- definition of ‘tame topology’ (topologie in China in the area of ordinary and partial ry of wave invariants. modérée) which would be a generalisation differential equations. The first two chapters, by F. E. Burstall of these important examples. The papers can be split into the follow- and I. Chavel, start practically from zero, The main aims of the book are to show ing topics: large-time behaviour, including and introduce the reader to elements of that a simple set of axioms (defining the so- exponential stability, the existence of glob- Riemannian geometry and the basic prop- called o-minimal structures on R) gives al attractors and further qualitative proper- erties of the Riemannian Laplace operator. such a generalisation and to prove basic ties; asymptotic theory of linear ordinary They are self-contained and can be recom- properties of sets in an o-minimal struc- differential equations, including a construc- mended as an excellent short text for those ture. Every semi-algebraic or subanalytic tion of hyperasymptotic expansions and who need only basic information about the set can be stratified into a union of sub- error bounds (with a contribution by F. W. topic. (ok) manifolds. An analogue for o-minimal J. Olwer); bifurcation theory, including the structures is a ‘Cell decomposition theo- conditions for existence and non-existence P. G. L. Dirichlet, Lectures on Number rem’, which leads to a definition of dimen- of limit and heteroclinic cycles; global exis- Theory, History of Mathematics 16, American sion and Euler characteristic of a set in the tence theory, providing a unified approach Mathematical Society, Providence, 1999, 275 structure. If new axioms (related to addi- to the study of a general class of non-linear pp., $49, ISBN 0-8218-2017-6 tion and multiplication) are added to the 34 EMS March 2000 RECENT BOOKS definition of an o-minimal structure, it is dice games and playing cards led to many This book is based on extended versions of possible to prove a suitable triangulation interesting problems, the solution of which papers of the author. The fibring is a theorem. The book offers a systematic and originated probability theory. Today, we method of how to combine two or more self-contained treatment of o-minimal have more opportunities for gambling. The logic systems or, more precisely, how to structures which needs almost no prerequi- book describes lotteries, football pools, establish axioms and semantics of an sities. It is a nicely written summary of roulette, and horse racing. Elements of appropriate combination of such systems. interesting results. (vs) probability theory are demonstrated by The basic idea of fibring can be presented interesting examples concerning balls and by the following example: L. C. Evans, Partial Differential Equations, urns, birthdays, coincidences, conditional Let S1, S2 be systems, where S1 is modal Graduate Studies in Mathematics 19, American probabilities, and Bayes’ theorem. Further, logic K1 with the modality M1 and S2 is Mathematical Society, Providence, 1998, 662 the author shows that people cannot always modal logic S4 with the modality M2, and pp., ISBN 0-821-80772-2 make rational choices about the differing assume that the logics are presented via This excellent textbook is meant as an risks. The last part of the book is devoted to classes C1, the class of all Kripke models of introduction to mathematical analysis of statistics in medicine. The role of randomi- the form m = (S, R, a, h) with R transitive partial differential equations. The book is sation is explained and a description of and aRa, and C2, the class of all Kripke split into three parts. It starts with the clas- experiments is given. It is demonstrated models of the form m’ = (S’, R’, a’, h’) with sical theory, with the emphasis on model that this scientific approach is usually miss- R’ transitive and reflexive. Assume that A = (prototype) equations. While in many text- ing in alternative therapies. M1M2q is a mixed formula with an atom q books this means a study of the Laplace, This easily understood book can be rec- of S2; then A = M1p with an atom p of S1, heat and wave equations and their general- ommended to everybody. The mathematics since S1, where p = M2q. We see now this isations, but here model cases include also is kept to a minimum and the author stress- relation as sat2(a’, M2q) with F(t) = (S’, R’, the transport Hamilton-Jacobi equations es the philosophical features of uncertainty. a’, h’). A generalisation of this idea and and equations of hyperbolic conservation If you read this book, you will better under- applications of results obtained are pre- laws. The representation formulas are stand the impact of chance on your life. (ja) sented in twenty-one (slightly independent) given, which makes it possible to obtain chapters. (jmlc) qualitative properties of solutions, as well as P. Eymard and J.-P. Lafon, Autour du to introduce the notion of a weak solution. nombre π, Actualités scientifiques et industrielles N. Guicciardini, Reading the Principia. The second part is then devoted to the 1443, Hermann, Paris, 1999, 318 pp., FF148, The Debate on Newton’s Mathematical theory of linear second-order elliptic, para- ISBN 2-7056-1443-5 Methods for Natural Philosophy from 1687 bolic and hyperbolic equations, based on This is an interesting book on mathematics to 1736, Cambridge University Press, energy methods and linear functional and history surrounding the number π. Cambridge, 1999, 285 pp., £50.00, ISBN 0- analysis. Questions of the existence of weak Five chapters dealing with π are ordered 521-64066-0 solutions, their uniqueness, regularity, or according to decreasing difficulty. Whereas This book is divided into three parts blow-up are systematically discussed. The Chapter 1 is accessible to students starting (Newton’s methods, Three readers, Two introductory chapter to the second part their university studies, Chapter 5 requires schools). contains a very well written description of a substantial knowledge of advanced cours- The first part (82 pp.) is an introduction properties of functions from the Sobolev es in mathematical analysis. to Newton’s Principia. In the second chap- spaces, not restricted only to the case p = 2. Chapter 1 is devoted to the ter we find a concise presentation of The third, main part of the book is devoted Archimedean method of approximating π, Newton’s methods of series and fluxions. to the modern theory of non-linear second- volumes and areas involving the number π, The author’s aim is to give an idea of math- order elliptic and evolutionary equations, the asymptotic Gauss formula for number ematical method, the ‘new analysis’ that Hamilton-Jacobi equations and equations of lattice points contained in large circles, Newton had already developed before writ- of hyperbolic conservation laws. Buffon’s needle problem, and curves of ing the Principia. The third chapter of the Throughout the book the reader is constant width. Chapter 2 presents various first part is devoted to the mathematical acquainted with various approaches and expressions of π in terms of infinite series methods employed by Newton in the techniques to initial and boundary-value and products. The results presented go Principia; it is not an easy chapter. This part problems. In particular, in the final part back to Viète, Stirling, Gregory, Leibniz, is not to be taken as an introduction to the the methods of variational calculus, includ- Euler, Machin, etc.; however, new formu- Principia or a critical analysis of the ing the mountain pass theorem, the theory las, such as those of Bailey, Borwein and Principia, but as a work devoted to the of monotone operators, the techniques of Plouffe from 1997, are also discussed. reception of Newton’s magnum opus. sub- and super-solution, fixed-point meth- Chapter 3 reveals the position of ( in analy- The second part (70 pp.) explains the ods, the method of non-linear semigroup, sis (Fourier series, Euler formulas, the reactions to the Principia of three giant and viscosity solutions, are outlined and Gamma function, Bernoulli numbers, and readers: Newton himself, Huygens and applied to specific problems. The necessary the behaviour of certain arithmetic func- Leibniz. This part is divided into three knowledge of linear functional analysis and tions). Chapter 4 deals with squaring the chapters. The first one describes Newton’s measure theory is surveyed in an circle, classical geometric constructions, evaluation of his own published master- Appendix. I recommend this book as the irrational numbers, algebraic numbers, the piece, the worries of the calculus priority first textbook for anyone who wants to irrationality and transcendence of π, and dispute with Leibniz, Newton’s attempt to learn the theory of partial differential equa- Liouville numbers. Chapter 5 is on ( and defend the mathematical methods of the tions. (jmal) elliptic integrals: arithmetic-geometric Principia against the criticism of the mean, algorithms for calculating elliptic Leibnizians. The second chapter discusses B. S. Everitt, Chance Rules. An Informal integrals, and algorithms of Brent and Huygens’ reaction and the third describes Guide to Probability, Risk and Statistics, Salamin, as well as of J. M. and P. B. Leibniz’s reaction to the Principia; for Springer, New York, 1999, 202 pp., DM49, Borwein for the calculation of π, theta func- example, Huygens was dissatisfied with ISBN 0-387-98776-2 tions, Abel’s function, modular functions, Newton’s use of proportion theory, and Statistics and probability theory are scien- Ramanujan’s formulas, and so on. criticised the use there of this classic ingre- tific disciplines describing effects caused by Solutions of the exercises in individual dient of ancient geometry. The author also chance. It is known that people have used chapters are included in Chapter 6. explains the important differences between chance for amusement for a long time. Students and teachers of mathematics Leibniz and Newton. Board games involving chance were per- on various levels will find the book interest- The third part (94 pp.) pays attention to haps known in Egypt some 5000 years ago. ing and useful. (in) the two schools that divided Europe, the so- The first part of this book describes the called British Newtonian school and the history of chance. The author points out D. M. Gabbay, Fibring Logics, Oxford Logic Continental Leibnizian school. The author that even in the Bible lots were used to Guides 38, Clarendon Press, Oxford, 1998, 471 shows their different mathematical prac- ensure a fair division of property. Later on, pp., ISBN 0-19-850381-4 tices, their mathematical methods, the pri- EMS March 2000 35 RECENT BOOKS ority dispute on the invention of the calcu- applications, such as the so-called group of (p, n)-bounded index which is lus, etc. The third part concludes with a homogenisation. Chapter 8 is independent nilpotent of class at most 2, and abelian if p short characterisation of Euler’s Mechanica of the preceding ones, and deals with bifur- = 2. and new edition of the Principia enriched cation phenomena and minimal surfaces of 2. If a finite p-group P admits an automor- by an extensive commentary, because after revolution. The last section discusses the phism of order p with exactly pm points, Euler the Principia’s mathematical methods unstable critical points of variational prob- then P has a subgroup of (p, m)-bounded became obsolete. (mbec) lems, and is also independent of Chapters index which is nilpotent of an m-bounded 4-8. class. V. P. Havin and N. K. Nikolski (eds), The reader should enjoy the straightfor- The style has a textbook character to the Commutative Harmonic Analysis II, ward exposition, with clean and exact argu- end, and each of the fourteen chapters is Encyclopaedia of Mathematical Sciences 23, ments. The whole book is nearly complete- supplemented by exercises. There are also Springer, Berlin, 1998, 325 pp., DM158, ly self-contained, the prerequisites involv- many remarks pointing towards generalisa- ISBN 3-540-51998-X ing only the basic calculus of one and sev- tions, related results, open problems and This part of the Encyclopedia is written by V. eral variables. Many relevant links to other authorships of various theorems and P. Gurarii and is a translation of the 1988 textbooks or research monographs are proofs. (ad) Russian edition. Its subtitle, ‘Group given throughout the text. The book is Methods in Commutative Harmonic warmly recommended to a wide class of Y. Kitaoka, Arithmetic of Quadratic Forms, Analysis’, gives a true picture of its con- mathematicians. (jslo) Cambridge Tracts in Mathematics 106, tents. Cambridge University Press, Cambridge, 1999, The first part of the book consists of E. I. Khukhro, P-Automorphisms of Finite 270 pp., £18.95, ISBN 0-521-40475-4 and 0- twelve paragraphs, devoted to ‘classical’ P-Groups, London Mathematical Society 521-64996-X harmonic analysis (integral transforms in Lecture Note Series 246, Cambridge University The aim of this book is to provide an intro- Rn. The emphasis in Sections 1-7 is mainly Press, Cambridge, 1998, 204 pp., £24.95, duction to the arithmetic theory of qua- on L2-theory. Positive definite functions ISBN 0-521-59717-X dratic forms. The book starts from the and kernels and their connections with The aim of this book is to expose the inter- basics and proceeds to some very recent functional analysis and probability theory play of nilpotent groups and Lie rings in results. It covers many aspects of the sub- are discussed in Sections 8-10. The rest of the modular case, and to deduce results ject, including lattice theory, Siegel’s for- the first part contains deeper results on about p-groups that admit p-automor- mula, and tensor products of positive defi- Tauberian theorems and spectra of bound- phisms with few fixed points. nite quadratic forms. Quadratic forms are ed functions. Applications to number theo- The first half of the book presents stan- mainly considered over the rationals or the ry and information theory are also given. dard material, ranging from definitions ring of rational integers and their comple- The second part of the book deals with and basic properties of groups, nilpotent tions. general locally compact abelian groups and and soluble groups, Lie rings, nilpotent The reader is required to have only an special examples of them. Integration the- and soluble Lie rings, free Lie rings and elementary knowledge of algebraic number ory, invariant means, characters and associated Lie rings, to their connections by fields. This makes the book ideal for grad- abstract Fourier transforms, duality, and the Three-Subgroup Theorem and to some uate students and researchers from other structural theorems are described here. more advanced features such as the charac- fields interested in quadratic forms. (jtu) The paragraph on duality also contains terisation of soluble varieties as varieties Bohr compactification and the abstract where M’ ≠ M for all non-trivial members V. F. Kolchin, Random Graphs, Poisson formula, together with applications M, or certain estimates of the nilpotency Encyclopedia of Mathematics and its to number theory via numerical characters. class and of the factor orders, or some start- Applications 53, Cambridge University Press, There is also a paragraph on commutative ing observations concerning automor- Cambridge, 1999, 252 pp., £50, ISBN 0-521- Banach algebras, where basic information phisms and their fixed points. 44081-5 on the Gelfand representation, analytic While the style is relaxed, it gives suffi- The random graphs discussed in this book functions of elements, spectral synthesis of cient amount of detail, and, generally are very sparse: for example, random ideals, and involutions are presented. speaking, the ease with which the author forests, random graphs with unicyclic com- The book covers a rather large part of has introduced so many different concepts ponents, and random graphs with average commutative harmonic analysis and can coherently in a relatively small space is degree close to 1; further topics include therefore serve as a good source of infor- quite admirable. As in most books, slips of random systems of linear equations over mation for non-experts. Unfortunately, the the pen were not completely avoided, and a GF(2) and random permutations. Graph- book contains few references to recent few of them could mislead a beginner; for theoretical considerations seldom play a results (obtained since the Russian 1988 example, Lemma 1.13 does not hold in the role in their investigation; rather, the cen- edition). (jmil) stated form. tral technique is the generalised allocation The core of the book lies in its second scheme. This method, whose theory is elab- J. Jost and X. Li-Jost, Calculus of half, which starts with Kreknin’s and orated in the first chapter, deals with situa- Variations, Cambridge Studies in Advanced Higman’s theorems about Lie rings that tions where the joint distribution of N ran- Mathematics 64, Cambridge University Press, possess an automorphism of a finite order. dom variables can be described as the dis- Cambridge, 1998, 323 pp., £37.50, ISBN 0- From Higman’s theorem the author proves tribution of N independent random vari- 521-64203-5 that any finite p-group that admits an auto- ables conditioned on the sum of these vari- This textbook guides the reader through morphism of order p fixing pm points con- ables being a given number n. The book various areas of the calculus of variations. tains a characteristic subgroup of (p, m)- covers almost exclusively the work of The first part, split into five chapters, is of bounded index and of nilpotency class not Russian mathematicians (with some refer- an introductory nature and covers the clas- exceeding h(p), where h denotes Higman’s ences to alternative approaches by other sical Euler-Lagrange theory, symmetries, function. After presenting several topics, authors), mostly from the last twenty years. the saddle-point constructions, the theory such as the Baker-Hausdorff formula for The material is well presented and offers a of Hamilton and Jacobi, and some links to free nilpotent groups, nilpotent Q-powered new perspective to readers interested in the control theory; all of this is developed very groups, the Mal’cev and Lazard correspon- above topics. (jmat) carefully in less than 120 pages. dence, and the basic properties of powerful The rest of the book is more advanced. p-groups, the above-mentioned theorem S. Konyagin and I. Sharplinski, Character After three chapters on the background on automorphisms of order p serves as one Sums with Exponential Functions and their (Lebesgue integration and functional of the ingredients for the two theorems in Applications, Cambridge Tracts in analysis), the show goes on with standard which the book culminates: Mathematics 136, Cambridge University Press, and abstract methods for the existence of 1. If a finite p-group P admits an automor- Cambridge, 1999, 163 pp., £30, ISBN 0-521- the minimisers, and the next two chapters phism of order pn and if the number of 64263-9 are devoted to the Γ-convergence and some fixed points equals p, then P has a sub- The book covers various aspects related to 36 EMS March 2000 RECENT BOOKS the distribution problem for integer powers gx of some positive integers g modulo a W. Krawcewicz and J. Wu, Theory of J. Matoušek and J. Nešetøil, Invitation to power of a prime p with p and g coprime. Degrees with Applications to Bifurcations Discrete Mathematics, Clarendon Press, The authors also consider applications of and Differential Equations, John Wiley & Oxford, 1998, 410 pp., £19.50, ISBN 0-198- this problem to problems from algebraic Sons, 1997 50207-9 and 0-198-50208-7 number theory, the theory of function This book is devoted to the theory and This is more than a textbook for under- fields over finite fields, and linear congru- applications of the degree of non-linear graduates. The book stems from the ential pseudorandom number generators, mappings in finite- (Brouwer degree) and authors’ long experience with teaching a including cryptography and coding theory. infinite-dimensional spaces (Leray- beginners’ course of discrete mathematics After two short preparatory chapters Schauder degree and its generalisations to at Charles University. They cover selected forming the first part of the book, the first condensing maps, including the coinci- topics in considerable depth and often ‘mathematical’ chapter is devoted to esti- dence degree). Attention is paid to maps from several points of view. The authors mates of characters sums with sufficiently with symmetries for which the S1-degree aim at cultivating mathematical and logical large number of terms; the main result of and the Dold-Ulrich equivariant degree are reasoning by the reader rather than merely the chapter gives bounds for Gauss sums defined. teaching facts. It really is an invitation proved recently by Heath-Brown and The explanation in the book is self-con- worth accepting. The book is suitable for Konyagin. The next chapter gives an upper tained, and chapters on algebraic and dif- virtually everyone – beginners will learn the estimate for maximal absolute value of ferential topology and transformation basics of discrete mathematics, graduate short exponential sums proved by the first groups are included. The degree theory is students will find interesting applications author. These chapters, together with applied to local and global bifurcation, and connections with other branches of chapters containing bounds of character including the Hopf bifurcation. The second mathematics, and everybody will enjoy the sums for almost all moduli and bounds for type of application is towards ordinary, authors’ lively style and subtle humour. maximal absolute values of Gaussian sums delay and neutral differential equations (jkrat) with general non-prime denominator, form (boundary value and eigenvalue problems, the second part of the book. The third part periodic solutions). This part of the book is M. Overbeck-Larisch and W. Dolejsky, is devoted to multiplicative translations of closely related to the authors’ investiga- Stochastik mit Mathematica, Vieweg, subgroups of Fp* and of arbitrary subsets of tions. The book is clearly written and can Braunschweig, 1998, 370 pp., ISBN 3-528- Fp by an element a∈ Fp* modulo p. The rest be recommended to graduate students 06921-X (paperback) of the book contains the above-mentioned interested in differential equations. There This book is an introductory course in applications. are more than 170 exercises of medium dif- probability theory and mathematical statis- The book is written in a concise, but ficulty. (jmil) tics. The style of presentation is somewhat good readable manner. The exposition different from the classical textbooks on contains references to related problems J. Madore, An Introduction to the subject, however. The reader equipped and can therefore also be useful to those Noncommutative Differential Geometry and with the Computer Algebra System who want an insight in the further develop- its Physical Applications, London Mathematica can download the correspond- ments of the presented and related ideas. Mathematical Society Lecture Note Series 257, ing Mathematica notebooks (free) from the (šp) Cambridge University Press, Cambridge, 1999, Vieweg web site and see almost all of the 321 pp., £24.95, ISBN 0-521-65991-4 and 0- concepts and methods throughout the S. G. Krantz, How to Teach Mathematics, 521-59838-9 book in the working environment of American Mathematical Society, Providence, The first edition of this book appeared in Mathematica. 1993, 76 pp., £11.95, ISBN 0-8218-0197-X 1995. This is the second edition, in which The book consists of six chapters. Chapter Unlike secondary school teachers, college some errors have been corrected and a few 0 is a brief introduction into sets and func- and university teachers usually have no pre- recent results included. The second edition tions under Mathematica. Chapters 1 and 2 liminary theoretical background in the appearing so soon after the first suggests deal with the elements of probability. teaching of mathematics. Not only those that the book is of particular interest. Chapter 3 introduces random samples. The who feel such a gap in their education The book is built on examples that com- basic methods of estimation (principle of should read this booklet, which presents a prise more than half of the text; this is maximum likelihood and least squares personal perspective. However, the reader probably its most interesting feature. The method) are presented in Chapter 4; the will find much more than a particular per- reader has a good feeling that the non- construction of confidence intervals for the sonal opinion. commutative generalisation of differential binomial and normal distribution is also The booklet is a practical guide to the geometry makes sense and finds reasonable given here. Chapter 5 is devoted to hypoth- teaching of mathematics, with an emphasis applications. The book is designed for esis testing. Besides the classical parametric on specifics. What Chapter I (Guiding beginners but takes the reader rather far. tests, tests of goodness of fit and non-para- Principles) includes can be seen from the Moreover, at the end of each chapter we metric tests are mentioned here. In the following selected keywords: Respect, find Notes with many interesting comments Appendix, the reader will find the solutions Prepare, Speak up, Inductive vs. deductive and hints for further reading. to selected exercises. method, Time, Why do we need mathemat- It requires from the reader some knowl- The book is clearly written and easily ics teachers?, Math anxiety, How the stu- edge from several branches of mathematics readable with plenty of carefully selected dents learn, Computers, Applications. such as differential geometry, the theory of examples. It seems that this way of presen- Chapter II contains practical matters Hilbert and Banach spaces, the theory of tation is a prospective method of teaching (voice, eye contact, blackboard technique, rings and algebras, and also some knowl- the subject. The book is recommended to handouts, exams, transparencies, etc.). edge of physics, especially of classical field lecturers and students and to all with an Chapter III (Sticky Wickets) touches sensi- theory. But in all these areas it requires interest in computer-aided theory of prob- tive subjects, from problems caused by non- only very basic knowledge. It seems that ability and statistics. (jh) native English speakers, through difficult according to its style and language the questions and mistakes in the lecture, to book was written more for physicists, and A. Quarteroni and A. Valli, Domain cheating, bribery and sexual harassment. that mathematicians accustomed to a high Decomposition Methods for Partial The booklet is written in a lively and level of abstraction will have slight prob- Differential Equations, Numerical humorous style, even though the points dis- lems in understanding some points. Mathematics and Scientific Computation, cussed are entirely serious and sensible. Nevertheless, I strongly recommend the Cambridge University Press, Cambridge, 1999, The author succeeds in elucidating the fine book to mathematicians. It represents an 360 pp., £55, ISBN 0-19-850178-1 points of excellent teaching and offers a lot ideal opportunity to familiarise oneself with This book is concerned with the underlying of important practical advice. The book is physical language, to learn a new branch of mathematical concepts of domain decom- strongly recommended to everybody who mathematics, and to see some nice physical position methods. For any given partial dif- teaches mathematics. (in) applications. (jiva) ferential equation, the authors derive its EMS March 2000 37 RECENT BOOKS multidomain formulation, describe suitable 1960, she moved to Cambridge, and passed researchers, students and practitioners. transmission conditions and investigate the away shortly before her 90th birthday. In corresponding Steklov-Poincaré operators. 1992, the element 109 of the periodic table G. Royer, Une initiation aux inégalités de A large variety of boundary value problems was named ‘Meitnerium’. Sobolev logarithmiques, Course Spécialisés 5, are addressed: symmetric elliptic equa- This book is pleasure to read. The Société Mathématique de France, Paris, 1999, tions, convection-diffusion equations, elas- chronological presentation of Lise 114 pp., FF120, ISBN 2-85629-075-2 ticity problems, Stokes problems of incom- Meitner’s life is given in fourteen main This course is a quick (114 pages) introduc- pressible and compressible fluids, the time- chapters. There are also numerous notes to tion to logarithmic Sobolev inequalities harmonic Maxwell equations, parabolic the chapters, the personal chronology of (especially the Gross inequality) and their and hyperbolic problems, and suitable cou- Lise Meitner, the list of her awards and application to ergodicity for a stochastic plings of heterogeneous equations. In the honours, the list of her publications, and a differential equation (the equation of discretisation of the problem, the finite ele- rich bibliography. Still, to my eyes, one Langevin). As an illustrative example the ment method is used, but the analysis pre- important topic is not covered thoroughly unbounded spin system with weak interac- sented can be adapted to any Galerkin-type enough in the book: the question of why tions is studied. The main notions of func- approximation, as spectral method or hp- only Otto Hahn was honoured with the tional analysis (self-adjoint operators and version of the FEM. Both overlapping and Nobel prize. After the usual 50-year delay, semigroups) needed in the text are intro- non-overlapping subdomain decomposi- the records of the Nobel prize deliberations duced. The reader is supposed to be famil- tions are treated, and special attention is of the Royal Swedish Academy of Sciences iar with basic objects of probability theory; paid to the analysis of the convergence of are released to scholars. It is a pity that however, the notions of Kolmogorov several iterative procedures among subdo- these records are not discussed in the book process, Gibbs measure and Markov kernel mains. The algebraic part of algorithms is in more detail. (zpl) are revisited. The book is prepared as a also explained. one-semester course for graduate students, The book contains a number of various S. M. Ross, An Introduction to but would be also of interest to scientists approaches, techniques and results pre- Mathematical Finance: Options and Other interested in probability theory. (efas) sented in a very nice style. It will be of inter- Topics, Cambridge University Press, est to researchers and students of applied Cambridge, 1999, 184 pp., £21.95, ISBN 0- J.-P. Schneiders, Quasi-Abelian Categories and numerical mathematics, specialists in 521-77043-2 and Sheaves, Mémoires de la Société partial differential equations, scientists and This book provides an accessible and rela- Mathématique de France 76, Société engineers engaged in applied mathematics tively deep insight into basic and advanced Mathématique de France, Paris, 1999, 134 pp., and scientific computing. (mf) topics of mathematical finance. No prior FF 150, ISBN 2-85629-074-4 knowledge of probability is assumed. In this memoir, quasi-abelian categories P. Rife, Lise Meitner and the Dawn of the Instead, the first two introductory chapters are introduced in order to extend classical Nuclear Age, Birkhäuser, Boston, 1999, 432 present all the necessary preliminary mate- homological and sheaf-theoretic methods pp., DM78, ISBN 0-8176-3732-X and 3- rial in an understandable and rigorous way. from abelian categories to other exact cate- 7643-3732-X This in turn allows the author to treat the gories occurring in algebraic analysis. Lise Meitner (1878-1968) is one of the most Brownian and geometric Brownian motion Chapter 1 develops a general theory of fascinating figures in the early history of as a limit of a binomial process. The basics quasi-abelian categories. For E quasi- nuclear physics. In this book, Patricia Rife of mathematical finance that are included abelian, an abelian envelope LH(E) of E is presents a very readable and very thorough contain an explanation of interest rates and constructed. Among other things, it is picture of her life and personality. During present value and two well-elaborated proved that the derived categories of E and her life, Meitner witnessed enormous chapters on arbitrage pricing and the arbi- of LH(E) are equivalent. Chapter 2 deals changes both in society and in science. In trage theorem. Also included are valuation with sheaves with values in quasi-abelian 1906, she was awarded a PhD in Physics, by expected utility, the capital assets pric- categories, in particular in the so-called ele- the second woman in the history of Vienna ing model (CAPM) and the portfolio selec- mentary ones (the cocomplete ones with a University to do so. In 1907, she went to tion problem; these are all explained in a small strictly generating set of tiny objects. the Chemistry Institute of the Berlin concise and precise way. The chapter culminates with a proof of the University (and later to the Kaiser Wilhelm The presented method of option pricing Poincaré-Verdier duality and the projec- Institute for Chemistry) to work on radioac- is based on the assumption of geometric tion formula in this general setting. tivity with the radiochemist Otto Hahn. Brownian motion of security prices and on Chapter 3 is dedicated to applications of By the early 1930s, she was an estab- no-arbitrage reasoning. The results cover the general theory to sheaves of Z-filtered lished and respected expert in nuclear the Black-Scholes formula for European abelian groups, and of locally convex topo- physics. Jointly with Hahn, she was repeat- options and its extension to American ones logical vector spaces. (jtrl) edly nominated for the Nobel prize in and no-arbitrage pricing of exotic (barrier, chemistry. Unfortunately, in 1933, the lookback, Asian) options, and simulation B.-W. Schulze, B. Sternin and V. Shatalov, attacks on non-Aryans like Lise Meitner methods for their valuation are discussed. Differential Equations on Singular started. In August 1938, she fled to The last two chapters explain limitations of Manifolds, Mathematical Topics 15, Wiley- Stockholm, and in December she received a the geometric Brownian motion model and VCH, Berlin, 1998, 376 pp., DM198.00, letter from Hahn reporting the mysterious suggest two alternatives: the log ratios ISBN 3-527-40086-9 behaviour of uranium after it had been viewed as a Markov chain and log prices as The theory of differential equations on bombarded by neutrons. In discussions an autoregressive scheme. The situation is manifolds with singularities has a long his- with her physicist nephew Otto Frisch, she illustrated on an extensive set of crude oil tory. The best-studied case is that of mani- concluded that the neutrons had penetrat- data. folds with conical singularities. In this book, ed into the uranium nucleus and split it The text is accompanied by elaborate singularities of more general types are cov- into two parts. Meitner and Frisch called examples and each chapter is augmented ered. The main problems addressed are a this process ‘fissioning’, calculated the asso- by numerous original exercises. description of the asymptotic behaviour of ciated energy release, and published their Throughout the book, the author points solutions and finiteness theorems (that the results in a letter to Nature. This experi- out the limitations and shortcomings of the corresponding differential operators are ment and its interpretation marked the presented methods and presents sugges- Fredholm operators). The study of singu- birth of the nuclear age. In recognition of tions for improvements and extensions. He larities more general than conical ones the discovery, Otto Hahn was awarded the discusses the validity of assumptions, and is brings new effects and needs new tools. The 1944 Nobel prize in chemistry. concerned with the properties of the time so-called ‘resurgent analysis’ used in the During and after the Second World War, series of input data, with statistical methods book deals with a problem of re-summation Lise Meitner worked in Sweden. She was for estimation of parameters. The lucid of divergent series appearing in the theory. remembered and repeatedly honoured by style of the exposition will be appreciated The Maslov non-commutative analysis (a the international scientific community. In by readers interested in the topic, and by construction of algebra of functions of non- 38 EMS March 2000 RECENT BOOKS commuting operators) is needed for finite- comments and supplements, and a list of Cambridge, 1999, 282 pp., £27.95, ISBN 0- ness theorems. The main part of the book 174 references. To read this book knowl- 521-63747-3 (Chapters 4-8) is devoted to a study of ellip- edge of the main properties of orthogonal This book is an outcome of the conference tic operators. Chapter 9 covers a special polynomials in one variable is necessary. ‘UF Galois Theory Week’, held at the case of hyperbolic equations and Chapter Also used are the main concepts and meth- University of Florida in October 1996. The 10 contains a discussion of the evolution ods from the theory of functions of a real or main highlight of the meeting was the equations for thin elastic shells. Two complex variable and the theory of differ- Inverse Galois Problem, one of the most appendices offer a brief summary of facts ential equations. exciting questions of contemporary mathe- needed from other monographs of the The volume represents an extensive sur- matics. Among the different aspects of this authors. vey of the area, and contains new results. It matter one can find here new methods for The book is very well organised and will doubtless be very valuable, not only for determining explicit classes of polynomials clearly written with a lot of commentary specialists but for a general audience inter- with positive characteristic among whose and motivating examples showing main ested in mathematics. (kn) Galois groups appear entire families of features of the theory. (vs) groups of Lie type. Further, a recent result Y. C. de Verdière, Spectres de Graphes, on the realisation of series of Lie type P. K. Suetin, Orthogonal Polynomials in Cours Spécialisés 4, Société Mathématique de groups as Galois groups in characteristic 0 Two Variables, Analytical Methods and France, Paris, 1998, 114 pp., ISBN 2-85629- is presented; in particular the realisation of Special Functions, Vol. 3, Gordon and Breach 068-X the projective symplectic groups PSp(n, q) Publishers, Amsterdam, 1999, 348 pp., $130, This book develops for finite graphs some under a restriction on n and q is obtained. ISBN 90-5699-167-1 analogues of the spectral theory of The study of the entire world of Galois This monograph presents a comprehensive Schrödinger operators on compact mani- extensions of a given field may lead to the theory of orthogonal polynomials in two folds. It covers some spectacular recent structure of certain profinite fundamental real variables and properties of Fourier developments, including the second-small- groups. Here the finite quotients of the series in these polynomials. Cases of est eigenvalue method (Cheeger and fundamental group of an affine curve in orthogonality over a region and a contour Fiedler), expanders of Ramanujan graphs, positive characteristic are determined. are presented, and much attention is paid minor theory (geometric view) and a Some other topics of this volume, such as to the relationship between orthogonal famous invariant (Colin de Verdière). The the comparisons between the absolute polynomials in two variables and differen- book is self-contained and has a lucid Galois group of the rationals and the tial equations. The volume includes the mature style. This is a great book which Grothendieck-Teichmueller group, and classification of differential equations that should be translated into English. (jnes) between the fundamental group of the admit orthogonal polynomials as eigen- punctured line in characteristic 0 and in functions, and several two-dimensional H. Völklein, P. Müller, D. Harbater and J. positive characteristic, should be men- analogies of classical orthogonal polynomi- G. Thompson (eds), Aspects of Galois tioned. (rb) als. Theory, London Mathematical Society Lecture The monograph consists of 11 chapters, Note Series 256, Cambridge University Press,

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