European Matl,ematical Society

.. Interview -�-,·�Sir� Michael Atiyah p. 8

SL ETTER ISSN 1027 488X CONTENTS tfi EDITORS Pro( Roy Bradley EUROPEAN MATHEMATICAL SOCIETY Department o( Mathematics Glasgow Caledonian University GLASGOW G4 OBA, SCOTLAND

Editorial Tea.m Glasgow: R. Bradley, J. Gomatam, V. Jha, G. Kennedy, M. A. Speller, J. 'Nilson

Editor - Mathematics Education Prof. Vinicio Villani Dipartimento di t,.,tatematica Via Bounarroti, 2 56127 Pisa, Italy e-mail [email protected] NEWSLETTER No. 27 Editors Brief Reviews I Netuka and V Soucek Mathematical Institute March 1998 Charles University Sokolovska 83 18600 Prague, Czech Republic e-mail; [email protected],cuni.cz Editorial by the President of the Society 4 soucek@k:.arlin.mff.cuni.cz · Interview: Sir Michael Atiyah USEFUL ADDRESSES 8

President: The Jahrbuch Project . 10 Jean-Pierre Bourguignon IHES, Route de Chartres, F-94400 Meeting of the Executive Committee • 10 Bures.. sur-Yvette, France. e-mail: [email protected] Council Elections 12 Secretary Peter W. Michor EMS Agenda 15 lnstitut fiir Mathematik, Universitiit Wien, Strudlhofgasse Diderot Mathematical Forums . 16 4, A-1090 Wien, Aus:tda. e-mai1; michorGe.si.ac.at World Mathematical Year 2000 18 'TI-ea.surer A. Lahtinen Euronews . 19 Department of Mathematics, P.O.Box 4 . FIN-00014 University of Helsinki Mathematical Education 32 Finland e-mail: IahtinenOcscJi Problem Corner . 41

EMS Secretariat Reviews 48 Ms. T. Miikeliiinen University of Helsinki (address above) e-mail [email protected] tel: +358-9-1912 2883 telex: 124690 fax: +358-9-1912 3213 • ·' (\ ,:,1',;.\ i, •. ,·,: �; .... � j Newsletter editor Produced at the Department of Mathematics, Glasgow �Jonian Univ��l!'\J R. Bradley, Glasgow Caledonian . Printed by Armstrong Pressr Southampton�·Y,:ri,'.»"·:·.\'\�··,/· University (address above) .... e-maii [email protected] __ NOTICE FOR MATHEMATICAL SOCIETIES Newsletter advertising officer M. A. Speller, Glasgow Caledonian University (address above) Please note labels are prepared during the second half of the month e-mail msp@gcaLac,uk before the next issue. Would you please send your updated lists before this time. Website http://www.emis.de Many thanks. Ms T Miikeliiinen E.M.S. : Why and How Jean-Pie1Te Bourguignon President of the Buropean Mathematical Society

I am happy to inaugurate this new column of the almost harmful, and we have to argue for r Ncwslr:ttcr, and t.hus to be given the opportunity of the preservation of fee thinking, precisely in order presenting the way in which the EMS is developing. to be in a good position to answer the pressing This occasion gives me the pleasure of thanking questions of tomorrow. Mathematics has a much Profes;;or Martin Speller, and the new team of the longer timeframe than other sciences. Ncwsle!.!.cr from Glasgow Caledonian University, for The need for a structure bringing together all the work done, aml for their willingness to adopt mathematicians on the European level has another a 1ww format in spite of the extra work involved. root. I strongly feel that it is only by doing things Born only 8 years ago, l,he EMS is sWl a young together that we can really learn from each other, societ.y. Thi:, i8 at the 8ame 1.ime a strength and and measure a wcaknc;;s. It means that, within the EMS, new the richness and the complexity that diversity brings iniliativc8 can be easily developed \Vithout being to human actions. Indeed Europe is diverse, obstructed by long established bad habits, and at culturally, linguistically, structurally, some of its I.he Harne tinm that the guiding principles inherited countries have a long tradition of central from a Jong t.radit.ion may be missed when important organizations, others cherish county or village d<\ciHiom, are t.o be takeu. In more concrete terms : structures. For me, this is an advantage, and is to iis being a ucw structure should help convince be preserved. Tbe key word is compatibility, not colleagues that. UH\ EMS can address questions of uniformization. co11m1on intcr<'BI, 1101 tackkd :;o far ; on the other By working together, we will not only acquire some Hide of l.lw coin, t.oo many colleagues have not even new ways of doing things, but we should be able heard of the EMS. to address certa.in questions at tile critical level. It. i:,; therefore of paramount. importance to review The basis of modern society is communication and 11ow why the Enropca11 IVIathernaticalSocic1,y should netvmrking. On this front too, the EMS is the 1�xisl, a11d how its present. structure allow;; it to meet appropriate structure foraction. It is light enough to the drnllm1ge.'i t.hat. bronghi it into being. move quickly when nece.ssary, decentralized enough to catch information where and when it is useful. It Why a European Mathematical Society ? is evident that the EMS needs a large membership Tim ccnl.rn.J rni1;sio11 of t.Jie EIVIS is to help and the active involvement of its member societies. t.Jw cnwrgcncc of ,rn idc11lit.y among European It should be made easy for them to work with 11wl,J1cwnlicimm. This parallels the steady trend the EMS. They should find there visibility for that hm, been r<\,haping t.he life of all European their endeavours of wider interest than their normal .socid.ic:,; .since the end of \,Vorld Vv'ar IL and is constituency, and support for some of their actions bringing EnropP.,UI rmt.iorn,closer. \Ve are witneHsing that naturally have a European dimension. The tlw construdiou of a European Hcene in all facets EMS has to develop its actions at a level which, of .social activity, am! this procf'.SS takes many without ambiguity, must be different from that of diffP.rc11t fonm,. The Enropcun llnion is gaining its member societies. It also relies on them to weight y<•.ar aft.er ye1:1r. and its influence is growing, relay some of its activities, typically those which not onl.v in cmmtries which already belong to it, imply collecting information in various countries or bnt also in t.hoH\1 which aspire to join it. For regions. mathem at icians. a. profos,;iomi.l socic(i · a.ppea.red as If all possible national or regional societies in Europe Uw 11wsl. 11ppropri;1.le sl.nu:lure /,o allow del,a.le 011 have now joined the EMS, a lot remains to be done Uwsc pcrspcd.ivcs. l,o pro1110l,e Mallwmalics and l,o to attract individual members, the most serious rcprnsc11I, Uw corn1m111it,.,· wilh U1e newly eslahlished problem being to overcome economic obstacles for sl.rnc/.11rns. mul also lo Jwlp our discipline meet. the colleagues from less favoured regions. However, we dw.Jlongos Urn/. ii. fac:es armrnd t.he world. recently succeeded in introducing the possibility for Nowaday;;, Society puts much tougher questions colleagues who act as referee.sfor the Zentralblatt fur to scie11tisls in general, ,llld to mat.hema l.icians Mathemalik to pay their feethrough their royalties. iu particular. \Ve must come to grips with Another front on which mathematicianH are involved these demands, without harming the long-term the world around is education. If students are developme11l of our discipline. Short-term views are to circulate easily in the European job market,

4 the various education systems have to be put in bibliographic data basis, Elib-Math, the electronic phase, again while preserving the diversity that is library which now contains 2:3 journals (and 10 more perhaps one of the main resources that Europe soon to come), 7 proceedings of conferencf's, and 1 has to offer. Mathematics is likely to play an monograph. On EMIS will soon be available the important role in these discussions as its teaching CuHent Awareness Progra.mme, which will enable is often controversial. A structure representing the any mathematician to have acces:,, free of charge, mathematical community for discussions on these to a large number of abstracts of article.:,; that questions is certainly . important since we do not have appeared in the last. year. Resume.:, from the want closed lobbies to gain control over such issues. Annales de l'lnstitut Fourier and of publications of A final important point : a the Societe Mathematique de France are aln�ady EMS does provide automatically loaded in CAP. good frameworkfor the development of Mathematics in its unity. Let us remember that this was the Very early, it was decided by tlw l·�xecut.ive motto adopted by the Scientific Committee of the Commit.tee to foi.md JEMS, the Journal of the last European Mathematics Congress in Budapest. EMS. This is just one of the direct.ions in which The speakers selected provided a vivid illustration the EMS is developing it.s publication policy. lt iH of the rightness of this choice. Pure and applied the responsibility of the EMS Publications Offieer, mathematicians have to feel equally at lwme in the formerly Stuart Robertson from Southampt.on, who EMS, and it should keep this in mind in all its was also the animator of 1.he previous Newsletter actions and developments. This is a must if one team, and now Carles Casacuberta, from Barcelona. is to properly address many questions having to do JEMS now has reached a critical stage: the Edit.or­ with the training -of students, relations with society in-Chief, Jiirgen Jost, one of the diredorn of t.hc and with executives from the private sector as well newly founded Max-Planck-Institut fiir Mat,hematik as from public organizations. in den Naturwissenschaften based in Leipzig, has started to work, in close connection with t.hc main How the European Mathematical Society ? editors, Luigi Ambrosio (Pavia), Gerard Beu Arons In order to fulfil its objectives the EMS has set up (EPFL, Lausanne), .John Coates (Cambridge), a number of tools a.nd is pursuing actions on several Helmut Hofer (Courant Institute) and Alexander fronts. Let me quickly describe them, taking this Merkurjev (UCLA). The editorial boarBerlin next August. The EMS Newsletter is of course the natural link between members, and the new series inaugurated The most visible manifestations that the EMS in this issue aims at improving contacts. Each issue has been involved in are the European Congresses will contain regular columns : an editorial by a of Mathematics. Paris anProfessor Giovanni MATH-Zenfralblatt fiir Mathematik, the electronic Monegato. The EMS makes a great effort to have

5 these schools held in Eastern European countries a widely distributed position paper available on to help colleagues from these countries develop EMIS. It prompted very positive reactions from their activities. We have a good record as far mathematicians and scientists of other disciplines as as our applications for support from the European well. We fought of course for a better recognition Commission are concerned. Special funds have to of the role of Mathematics in EC progra1mnes, and be found for schools held in Central and Eastern have been heard on some points. The energetic Europe. The UNESCO Venice office has so far presence in Brussels of the EMS Liaison Officer with shown great interest in a partnership in this. the European Cmmnission, Luc Lemaire, helped the The Diderot Mathematical Forum series was set up EMS become a well recognized partner in these with the explicit aim of opening up contacts with circles. other scentists and other components of In our declaration, we stated as first priority the Society. The scheme, a 2-day conference held recognition by the European Commission of the simultaneously in 3 different cities linked for part data base MATH-Zentralblatt fiir Mathematik as a of the manifestation by audio-visual means, aims large infrastructure for European mathematicians. at maximizing the confrontation between different Thanks to the openmindedness of the mathemati­ points of view and sensibilities while keeping the cal section of the Heidelberger Akademie, chaired by local organization light, since at each node only Professor Dieter Puppe, of the FachinformationZen­ about 100 people are expected. It is true that trum, and of Springer-Verlag, considerable progress putting up the telecommunication part turned out has been achieved. It all started by a French­ to be a non-trivial task. We may just be a bit ahead German collaboration, encouraged on the German of our time. Topics treated so far are "Mathematics side by the Editor-in-Chief of the data base, Profes­ and Finance", "Mathematics and Environment : sor Bernd Wegner, and conducted on the Frenchside problems connected to water" ; the ones which by the newly established structure MathDocCell, are in the making deal with "Mathematics as a joint structure of the University Joseph Fourier a leverage for cultural evolution", "Mathematics (Grenoble) and the Centre National de la Recherche and Music" (these two are by now well defined), Scientifique, under the responsibility of Professor "Mathematics and Telecommunications : problems Pierre Berard and Professor Laurent Guillope, now connected to mobile telephones", "Mathematics and at the University of Nantes. A major contribution of Risks", "Mathematics and Medicine". For this this structure to the database is the new search and again, your input is crucial, both for topics, retrieval software used for the local servers, for the potential partners and locations. For me showing main server in Berlin and for its international New the willingness of the mathematical community to York and Strasburg mirrors. On his side, the Editor­ discuss issues in which Mathematics is concerned in-Chief has considerably developed partnerships in both at the fundamental and at the applied levels Eastern European countries, with the aim of a truly is crucial. The hope is to make the series a major distributed constitution of the data base. l,Fromthe tool forrenovating the image of Mathematics with a moment it was associated· to the discussions, the broad public in mind, and to call the attention of our EMS insisted on having critical opinions received, colleagues on the breadth of possible interactions of analyzed and turned into improvements of the data­ our discipline. base. This mechanism will be achieved through an This series complements other activities such as Innovation Committee, chaired by Professor John the EMS Lectures which are geared towards our Coates, that has been put immediately to work. community, and have twice already functioned Since November 1997, the EMS shares the copyright successfully. The scheme is to support a series of of the database. The aim is now well defined : go lecturesby an outstanding scientist in a not too large much beyond the present partnerships, and make institution with the obligation of making the course MATH-Zentralblatt fiir Mathematik a truly Euro­ readily available to a wider audience than the local pean endeavour, involving many countries through one. The University of Besanc,:on, hosting Professor diverse entities, and many colleagues through their Hendrik Lenstra in 1995, and the University of contributions. Helsinki, hosting Professor Nigel Cutland in 1997, EMPRESSA, the EMS Press Agency, has been set did play their role remarkably well. up in Strasburg. It is still in its infancy. The The science policy of the European Commission lives objective is to help mathematicians whose function on a 4-year tempo, corresponding to the approval in' the community ·is to circulate information of of the Framework Programmes for Research and common interest, both technical and general, by Development. The 5th will soon be approved and putting together different sources. It should also will come into operation in early 1999. The EMS provide journalists with first-hand information on took part in the preparatory discussions through what is going on in the mathematical world and who

6 are the people most likely to help them when they while keeping its diversity displayed in the woven intend to cover mathematical topics. colours of the background. Dossiers giving a full ac­ As I mentioned earlier, mathematicians pay special count of EMS projects is available in French and in attention to questions connected to teaching. This English, an idea of Professor l'viarc Bnmaud. Col­ is why the EMS accepted a request from Ms. Edith leag11es who are interested in using them to make Cresson, the commissioner in charge of science the El'vIS better known can request them. at the European Commission, to participate in a As you can imagine, all this conldnot be achieved project aiming at establishing European reference without the active involvc1rn�nt of a fairly large levels for mathematical teaching at ages 16, 18 and after two years of university. A special working number of colleagues. H will stay that way in group has been set up under the responsibility of the ftitme. Mmit. of the work is carriell out by Professor Villani from Pisa University. It has just conm1it.tee.s (the EMS has 14 by now, besille.s its beg1m its activity, and will asi;ociaLe colleague8 Executive Commit.tee), which make suggest.ions, from many different countries to have a concrete prepare calls, select applications, collect the and documented view of the situation, which is information necessary for applications to sponsoring very diverse, as it should be. Here again, the agencies, act when necessary, etc. Conditions aim is to ease compatibility, and certainly not plan for the success are an efficient management: and uniformity. sound finances. Keeping t.hem in order has been remarkably achieved by Tuulikki Miikclainen, Considerable efforts have gone into helping col­ secretary ·of our Helsinki office, and the EMS leagues in appreciating EMS activities : members Treasurer, Aatos Lahtinen, also based at Helsinki of the Executive Committee presented EMS actions University. This gives me a good opportunity to at scientific meetings, an agenda de.scribing the 1998 remind you of the critical role that this university EMS activities has been sent to many mathematical has played in the life of the EMS, by hm;ling I.he departments and institutes, identifiable posters for firstmeeting ever of the European Council, the EMS EMS events are now systematically produced. Most incubator, and by offering to host. the office of the of the tools that were developed for these purposes society, which is incorporated under Finnish law. are due to the commitment of the EMS Publicity Officer, Mireille Chaleyat-Maurel, and the talent of The EMS can live only because it has many hard Marie-Claude Vergne, secretary at the IHES, but working people behind it. This is also, in my also a dedicated painter. The aim is to build an view, the only way in which we can progTe.ss in image of the EMS around its logo, which represents our endeavour to help the emergence of an identity the European mathematical community through the among European mathematician8. We nee

7 An interview with Sir Michael F. ATIYAH, FRS by Jean-Pierre Bourguig11011

JPB : Sir Michael Atiyah, could you give us your the same time, have an essential simplicity. I think view about the role that the index theorem plays in K-theory satisfied these criteria. many different areas of Mathematics? Did you and Singer foresee the possibility of such developments JPB : You have been a great advocate of a when you first worked at the proof? scheme of development of Mathematics in which mathematicians borrow ideas from Theoretical The index theorem provides a very concrete MFA: Physics to explore new frontiers. What are your topological answer to a basic analytic question. It opinions on the present impact of these ideas? on is not therefore too surprising that it has turned the future of this interaction? out to have many ramifications. It grew out of the Riemann-Roch Theorem so we always knew it MFA : My advocacy of using ideas from Theoretical was important in algebraic geometry and complex Physics in Mathematics was born out of experience analysis. Its more general role in differential and not out of some a priori reasoning. I find geometry, through spinors, was latent in the the success of Quantum Field Theory in Geometry formulas of Hirzebruch which had provided one spectacular, totally surprising and very mysterious. of our starting points. The move into abstract What is clear is that it has been a continuing functional analysis was an attempt to replace sheaf process for the past 20 years, going from strength theory and was stimulated by Singer's background to strength. The impact on many branches of in analysis. We did not foresee the scale . of Mathematics has been profound, and we are still developments in non-commatative c· -algebras, but trying to come to terms with it. In a sense were certainly interested in the interaction with it represents a marriage between Geometry and group reprC'.sentation theory. This ·.vas partly F\mctional Analysis at a much deeper and more stimulated by Serre's introduction of equivariant subtle level than anyone could have predicted. K-theory for finite groups and partly by formulas Internal mathematical development by itself would emerging from the Selberg trace formula. never have discovered the rich structures that are Where we were totally taken by surprise, and in now being unravelled. this we were not alone, was the relevance of the For the future the impact of these ideas in index theorem for theoretical physics. This was a Mathematics will leave a permanent mark. No very remarkable development and a key part of the doubt mathematicians will polish and simplify the recent rapprochement between Theoretical Physics story to the point where the original Physics gets and modern Differential Geometry, as demonstrated well hidden. in particular by Edward Witten. The more general lesson is that Mathematics grows I have to say that, since I worked at various by importing ideas from outside, so that we must developments and generalizations of the index never close our doors. Who knows where the most theorem for at least 10 years, it is hard now fruitful mathematical ideas of the future will come to remember what our ideas were at the start. from? They evolved as we dug deeper into the various applications and extensions. Many people also JPB : Other scientific disciplines are al.so coming influenced the development. up with big challenges for mathematicians, (to name just two, Computer Science and Biology). How do JPB : You have been one of the founders of I<­ you value these challenges? Do you foresee these theory which has expanded in so many different leading to the emergence of really new chapters directions. Do these multi-faceted developments of Mathematics? Would you favour a shift in suggest anything to you on the way Mathematics the training of mathematics students to meet these evolves and grows? challenges? MFA : My experience with K-theory is that many MFA : I agree that new scientific disciplines offer major developments in mathematics come from opportunities for future mathematicians. I feel unexpected and unpredictable directions. However, sure that they will generate new problems and these are likely to emerge out of a deep study of concepts which mathematicians should explore. One classical problems of a quite concrete kind. Abstract cannot guarantee automatic success but I would generalizations can be very powerful, provided they be surprised if subjects like Computer Science and are grounded in important classical results and, at Biology did not provide promising material for

8 mathematicians. In fact one can already point time. As you know, a process of Europeanisation to particular areas, such as Complexity Theory or of this major bibliogmphical database is under· way Evolutionary Biology, where serious work is being with the involvement of the EMS. How do you see the done. future of such tools for the working mathernatician? As far as the training of mathematicians is concerned, I would argue for a sound training MFA : I am sure the new electronic era of the in Pure Mathematics, as in the past, together 21st century will affect the way mathematicians with a more open attitude towards applications. work aud communcate. This will be of special Moreover, at a later stage, doctoral or post­ benefit to those who are in isolated environments. doctoral, students should be exposed to a multi­ This applies to parts of Europe but even more disciplinary environment, as provided for example to other parts of the world. I hope European by the programmes at the Newton Institute. mathematicians wiH be in the forefrontof those who encourage colleagues in less fortunate parts of the JPB In 1978 you started the European world. Imagine Ramannjan with e-mail! Coitncil, which finally gave birth to the European Mathematical Society in 1990. Fritz Hirzebruch likes JPB : You ha11e de·uoted a I.at of tirne and energy to to say that you were President nurnher O of the found the Isaac Newton Institute of Mathemat.ical EMS. How do you see the present development of Sciences in Cambridge. What place do you see the EJvlS? VVhat, in your opinion, should be its main for institiLtes of this nature among m.athem.atical priorities? institutions? MFA : I was naturally very pleased that the MFA : The Newton Institute has e.ssentially no EMS was finally established and got off to such permanent staff. It is a meeting place, designed to a good start. I would like to see i.t, in b1·1ng together mathematical scientists for extended particular, encourage an outward-looking attitude in periods of several months. This is long enough for the European mat.hematical comnmnity. Concern a serious exchange of ideas. It does not replace with education, with applications and with the university departments. On the contrary, it offersan general role of Mathematics in society are among external stimulus t.o mathematicians in universit.ie.-;. the priorities I would like to see. My view that the health and prosperity of Mathematics in the future will be greatly enhanced JPB : Have you a definite vision of the role of by working with other disciplines,and placeslike the mathematicians in the emerging European global Newton Institute offerone way that t.his can happen. society? Do you see differences between this role Collaboration is necessary both for the internal and that of our colleagues in Northern America for health of Mathematics and for the continuance of example ? external support. Society will not pay the bill unless MFA : Each country has its own history and its mathematicians show a willingness to be involved own educational system. Europe has much diversity with others. and much strength, different in may ways from To be effective on the political level mathematicians the American model. I think we should build on must work with other scientists and must emphasise our strengths but also learn from the American the social and economic benefits of Mathematics. experience.

JPB : You have been a memher of the Advisory JPB : Thank ymi for your willingness to inaiigurate Board of the Zentralblatt fur JvlathematiJ.: for a long this new column of the EJvlSNewsletter.

9 The Jahrbuch-project and Digital Math Archive

The Jahrbuch ueber die Fortschritte der Mathe­ EMIS web servers. matik (JFM) was founded in 1868 by the math­ ematicians Carl Ohrtmann and Felix Mueller. It The Jahrbuch-project has started and the keyboard­ contains more than 200,000 reviews of mathemati­ ing of volume 1 of the JFM is now complete. The cal publications and was published from 1868-1943. firstsets of data are available fordistribution to vol­ The goal of this project is to make the contents of unteer mathematical experts forenhancement: Eng­ the JFM freely available as a searchable database on lish keywords and MSC classifications are to be as­ the WWW. This project will also serve as a tool to signed, title.s are to be translated into English, com­ select about 20,000 of the most important of the.se ments on the value of the work are to be added, and publications (1,200,000 pages) for digitization and they are to be ranked in importance for scanning. wide public distribution at the SUB Goettingen. In addition the librarians in Goettingen will assign standard names, identify and give the location of The JFM-project is sponsored by the Deutsche a library where the item may be found. The final Forschungsgemeinschaft (DFG). The editors for product will also include a link to the scanned image the project are Keith Dennis (Cornell University) of the articles selected for the electronic archive. and Bernd Wegner (TU Berlin). They gratefully acknowledge the permission of de Gruyter Verlag For additional information, see Berlin to use the contents of the JFM for the htt ://www.emis.de/ rojects/JFM/ production of a public database. p p Additional volunteer mathematicians, knowledge­ The structure of the database and the search able in German, are needed for the project. If you interface will be the same that for Zentralblatt as are interested, please send an e-mail to one of the fuer Mathematik. In particular both internal editors: and external linking will be possible via unique identifying numbers. The preliminary version and Keith Dennis: [email protected] the enhanced version will be acce.ssible from the Bernd Wegner: [email protected]

MEETING of the EXECUTIVE COMMITTEE Capri (Italy), October 10-11, 1997

SCIENTIFIC ACTIVITIES Two of the three sites will be Lisbon and Paris. A preliminary workshop is planned in Torino m Third European Congress of Mathematics, 1998. 3ECM July 10-14, 2000, Barcelona (Spain) Other topics The Scientific Committee has been appointed, after - "Mathematics and Telecommunications" concen­ consultation of the Chairman. trating on mobile telephone systems; The members of the Prize Committee and the - "Mathematics and Cryptology"; Round table Committee should be chosen at the - "Mathematics and Risks": note that the OECD is next EC meeting. interested in such an event; - Corporate members are asked to appoint contact "Mathematics and Robotics"; persons with the Barcelona Congress. - "lviathematics and 'D-affi.c Regulation". Contact: Mireille Chaleyat-Maurel [email protected] Information: on the Web site www.iec.es/3ecm Summer Schools Diderot Mathematical Forums Summer Schools 1998 DMF3, Mathematics as a Force of Cultural The one in pure Mathematics is organized by Evolution (June 5-6, 1998) N. Teleman (Ancona) in Cluj, Romania, July 27 - The three sites will be and Berlin, Florence Krakow. August 14, "Singularities in Geometry", and may DMF, Mathematics and Music (Autumn 1999) be supported by UNESCO.

10 The one is applied Mathematics is organized number of permanent and long-term staff and that by A. Cohen (Paris) on "Wavelets and their cover Mathematical Sciences broadly are eligible for Applications" in Orsay, France, June 20-July 10. An representation in ERCOM. application to the European Commission is pending. This Committee aims to contribute to the unity of Summer Schools 1999 Mathematics, from fundamentals to applications. A call for application forsummer schools 1999 has Publications to been sent to member societies. C. Casacuberta(Barcelona) is appointed chair of the Contact: Giovanni Monegato [email protected] Publications Committee and Publications Officer of EMS. INFORMATION SERVICES World Mathematical Year 2000 Server EMIS A resolution on WMY 2000 has been adopted by There are 28 mirrors operating and, now, EMIS the General Conference of UNESCO. Au IMU­ mirrors IMU servers, increasing the visibility of UNESCO committee has been set up by the UMI EMS. In the electronic library, ElibEMS, there with M. Chaleyat-Maurel as chairperson (see the are now 22 journals, with 7 more in preparation. article on events prepared in this context in this The server also hosts the service Euro-Math Job, issue). pointing to job opportunities in the different The EMS project of having posters in the subwaysof countries. some European cities has been accepted in principle Address of the main site: www.emis.de by the Paris subway company. Other cif.ies are encouraged to do the same. EMS member Societies are asked to inform the EMS EMS COMMITTEES of their plans for the year 2000. Applications of Mathematics Contact: H. Engl (Austria) is appointed as chair of the Vagn Lundsgaard Hansen [email protected],1I.dk Committee. Eastern Europe RELATIONS WITH MATHEMATICAL IN­ H. Zieschang (Germany) has accepted to serve as STITUTIONS chair. International Mathematical Union (IMU) The International Mathematical Union is proposing Education to its adhering bodies to change its statuf,e.s to allow Z. Semadeni (Poland) is appointed as member to the affiliate members; EMS could become one of t.hen1. committee. The President will represent the EMS as observer at Electronic Publications the next IMU General Assembly in August 1998. Because of the complaint of many mathematicians that there is a lot of discussion but nearly no LIFE OF THE SOCIETY action on the problem of rising costs of journals, Membership P. Michor had proposedthat the EMS should launch If membership payments are collected during the a series of electronic journals, in a concerted action European or International Congresse.,;; for individual with some national societies. This proposal will be members, it should be made for two years. discussed at the Electronic Publications Committee of EMS, with the request to come up with a concrete Council Meeting proposal. The Council meeting will be held at Humboldt University, Berlin, on August 28-29, 1998. EuropeanResea rch Centres Committee (ER­ COM) An EMS committeeconsisting of ScientificDirectors NEXT EXECUTIVE COMMITTEE MEET­ of European Research Centres in the Mathematical INGS sciences has been created. Only centres for which o March 21-22, 1998, Helsinki (Finland) the number of visiting staffsubstantially exceeds the o August, 1998, Berlin (Germany), during ICM98.

11 COUNCIL ELECTIONS

The EMS Secretariat received 6 nominations for Statement: Council delegates representing individual members As a director of a research institute for in the years 1998-2001. Since the number of visitors (CRM) and at present president of the candidates is smaller than the number of seats to Academy (IEC) I am in a position to promote be attributed, all candidates are declared elected, more joint activities among the European without the need for a ballot: mathematical community, especially at graduate CASTELLET, Manuel, Barcelona, Spain and postdoctoral levels. Moreover in the last few JAIANI, George, Tbilisi, Georgia years Barcelona has played an important role in the development of mathematicsand the city will MARCHISIO, Marina R., Torino, Italy host the 3rd European Congress of Mathematics. MICHOR, Peter, Vienna, Austria MILMAN, Vitali, Ramat--Aviv, Israel JAIANI, George, Professor, Dr. SLOVAK, Jan, Brno, Czech Republic Vekua Institute of Applied Mathematics of The biographical information as well as the Tbilisi State University, statements provided by the candidates are printed University St. 2, 380043 Tbilisi 43, Republic below. of Georgia Proposer: R. Bochorishvili, Prof. Dr. Head CASTELLET, Manuel of Department of Vekua Institute of Applied Professor, Universitat Autonoma de Barcelona, Mathematics of Tbilisi State University Centre de Recerca Matematica, Seconder: A. Shapatava, Dr. Assistant-Professor of Apartat 50; E-08193 Bellaterra, Spain Tbilisi State University I certify that I am an individual member of the Proposer: Dr. Sebastia Xambo (Universita Politec­ nica Catalunya) EMS and that I am willing to stand for election as a delegate of individuel members to the Council. Seconder: Sir Mid1ael Atiyah (University of Edin­ burgh) Biography: Born on June 19, 1945 Biography: r in Tbilisi, Georgia, Manuel Castellet born in Barcelona in 1943, graduated from the Tbilisi State University Ph. D. studies at the ETH Zurich, Professor of for Mathematics and Mechanics, January 1968, Mathematics at the Universitat Autonoma de Diploma in Mechanics. Ph.D. (Candidate of Sci.) Barcelona since 1976, President of the Catalan in Mechanics of Deformable Bodies, Razmadze Mathematical Society (1979-1983), Director of Institute of Mathematics, 1974. Thesis: "Some the Centre de Recerca Matematica since 1984, Problems for Prismatic Shells with Cusp Edge", Member of the Institut d'Estudis Catalans (the under supervision of Academician, ProfessorIlya Catalan National Academy of Sciences) since Vekua. Dr. Hab. (FullProfe ssor) in Differential 1982 and President of the Institut d'Estudis Equations and Mathematical Physics, Razmadze Catalans since 1995. Institute of Mathematics, 1986. Since 1987 Deputy Director, Vekua Institute of App�i�� Council delegate representing individual mem­ Mathematics, since 1989 Professor of Tb1hs1 bers of the European Mathematical Society State University. Member of Council of Georgian since 1994, Member of the Societat Catalana Mathematical Union (GMU) and delegation de Matematiques, the Schweizerische Mathema­ of GMU at the inaugural meeting of the tische Gesellschaft, the Deutsche Mathematiker­ EMS. Member of the Georgian Academy of Vereinigung, the European Mathematical Soci­ Natural Sciences. EMS Council delegate of ety, the American Mathematical Society and individual members {1994-1997). Fields of Member of the Committee ERCOM (European interest: Differential Equations, Mechanics of Research Centres on Mathematics). Solids. Visiting positions, research stays and Research on Algebraic Topology, a field strongly participation in Congresses, conferences and developed in the universities of Barcelona under symposia in Germany, Italy, France, Gree?e, his leadership. President of the organizing Switzerland, Hungary, Poland, Czech Republic, conunittee of the Barcelona Conferences on former Soviet Union and USA. Author of 2 Algebraic Topology in 1982, 1986, 1990 and 1994, monographs and about 50 papers. one of the most prestigious conferences in this field.

12

; 1,I\ Statement: secretary of the EMS 1995-1998. My main According to my statement for elections for contribution to the EMS was the conception and 1994-1997, the Tbilisi International Centre of (together with Bernd Wegner) implementation Mathematics and Informatics (TICMI) have of the electronic server EMIS and its electronic been founded by support of EMS ( about library formathematics. 1 am willing to continue activities of TICMI see BULLETIN of TICMI, to work for EMIS for some more years. vol. 1, 1997 or Website: www.viam.hepi.edu.ge). If re-elected, as Chairman of International Sci­ MILMAN, Vitali, Profe,ssor of Mathematics entific Committee of TICMI, I could contribute School of Mathematics, Tel-Aviv (Jniversity, to the efforts of EMS to encourage quality re­ Ramat-Aviv, Israel search and teaching in mathematics in Transcau­ Proposer: Lawrence Zalcman casia and adjacent region. Seconder: Mina Teicher Born: MARCHISIO, Marina R., Dr. 1939 Via Primula, 20; I-12012 Boves (CN); Italy Education: Ph.D. 1965, Institute for Low Tempera­ Proposer: Professor A. Conte ture Physic,s, Kharkov; D.Sc. 1970, Iustitut.e for Low Seconder: Professor G. Anichini Temperature Physic,s, Kharkov Biography: Specialization: F\mctional Analysis and Convexity Born: Cuneo (Italy), 20th December 1969. P1thlications: has published over 100 papers and Title: Laurea in Matematica, 110-/110- e lode, edited 7 books Universita' degli Stndi di Torino, 4h July 1994. Journals: Editor of Geometric and F'unclional Present Position: 4th year Ph. D. student, Analysis Universita' degli Studi di Torino ( advisor: Invited Addresses: ICM 1986 and 1998; EMC Hl96 Professor A. Conte). I am also acting as assistant to- the President of the UniOHe Matematica Statement: Italiana and editorial secretary for the Bollettino I view the EivIS as potentially the most dell'Unione Matematica Italiana. I was a important body representing and lobbying for member of the Italian Delegation at the EMS mathematics in a united Europe. Jn the Council Meeting which was held in Budapest in past, various national mathematical societies July 1996. have had strong connections with the scieutific Research .interests: I am working in. Algebraic ministries of their respective countries, leading Geometry and published already two papers on in many case,s to strong support for research abelian surfaces and on a special congruence of in mathematics at the national level. In the 3 lines of P • I am now working on my Ph. D. thesis current era of a w1ited Europe, scientific funding on "Smooh unirational quartic threefolds". I am will depend increasingly on the decisions of expecting to defend at the end of 1998. a supranational bureaucracy that is not. easily Statement: accessible via national societies. Maintaining If I shall be elected inside the Council of EMS, and enlarging support for mathemat.ics within I shall do my best in order to strengthen the Europe will require vigorous lobbying efforts action of the Society to solve the problems of on the part of a body that can speak clearly Ph. D. students and to enlarge the number of and forcefully for the European mathematical jobs positions available for them. community as · a whole. The only credible candidate for this important role is the EMS. MICHOR, Peter, Prof. Dr. On the international level, I see the EMS as Institut fiir Mathematik, Universitiit Wien, providing a useful and healthy balance to other Strudlhofg�sse 4, A-1090 Wien, Austria groups wilhin the Internalional Malhernalical Union which can serve to further the intere.c;ts Proposer: Max Karoubi of European mathematics and mathematicians Seconder: Jan Slovak within that body. Biography: Finally, as a professional society, the EMS musl For my vita see my home page: continue and expand its activities in support http://radon.mat.univie.ac.at/�michor of joint European mathematical conferences, Statement: scientific exchange, and the easy movement of I was delegate of individual members to the mathematicians between the countries of Europe. EMS council 1994-1997, and serve as elected

13 SLOVAK, Jan, Doc. Dr. versity, Erno, Czech Republic; (1996-97 ARC Department of Algebra and Geometry, Senior Research Fellow, University of Adelaide, Masaryk University, Janackovo n. 2a, 662 95 South Australia); Erno, Czech Republic Research: I am interested mainly in differential Proposer: Peter Michor geometry and its interaction with representation Seconder: Vladimir Soucek theory ( cf. Slovak & Michor & Kolar: Biography: Natural operations in differential geometry, Springer, 1993; Slovak: Invariant operations Education and degrees: RNDr. (MSc) - 1983, on a manifold with connection or metric, J. Mathematical Analysis, Masaryk University in Diff. Geom., 36 (1992), 633-650; Eastwood & Erno, Czech Republic; CSc. (PhD) - 1990, Slovak: Semiholonomic Verma modules, Journal Masaryk University in Erno, Czech Republic; of Algebra, 197 (1997), 424-448) Doc. (Associated Professor) - 1993, Masaryk University in Erno, Czech Republic Statement; Positions: 1983-91 Research Fellow, Mathemat­ As a Council Delegate, I would like to follow ical Institute of the Academy of Science, Erno, the importance of electronic publishing and the Czech Republic; 1991-92 Visiting Professor, Uni­ close relations between all European countries, versity of Vienna, Austria; 1992-93 Assistant across the border of the European Community. Professor, Masaryk University, Erno, Czech Re­ Also I would focus on specific problems of small public; 1993- Associate Professor, Masaryk Uni- language groups.

Delegates for 1996-1999

Anichini, Giuseppe, Modena, Italy Kuusalo, Tapani, Jyvaskyla, Finland Bolondi, Giorgio, Sassari, Italy Lahtinen. Aatos, Helsinki, Finland Branner, Bodil, Lyngby, Denmark Marki, Laszlo, Budapt>.st, Hungary Deshouillers, Jean-Marc, Bordeaux, France Piccinini, Renzo, Milano, Italy Habetha, Klaus, Aachen, Germany Puppe, Dieter, Heidelberg, Germany Karoubi, Max, Paris, France

14 EMS AGENDA

1998 ( Contact: Martin Speller, [email protected]) March, 21st-22nd August 18-27th Executive CommitteeMeeting in Helsinki (Finland) EMS booth at the International Congress (ICM98) May, 15th in Berlin (Germany) Deadline for submission of information or papers to Launching of JEMS (the Journal of EMS) at the June issue of EMS Newsletter ICM98 (Contact: Martin Speller, [email protected]) August 28-29th May, 31st EMS Council in Berlin (Germany)hosted by Deadline for submission of proposals for the the Deutsche Mathematiker Vereinigung 1999 EMS Summer Schools Election of the President, a Vice-President, the ( Contact: Giovanni Monegato, [email protected]) Treasurer, the Secretary, EC members. June, 5-6th November, 15th Third Diderot Mathematical Forum, "'Math­ Deadline for submission of information or papers to ematics as a force of cultural evolution" the December issue of EMS Newsletter in Berlin (Germany), Florence (Italy)·· and ( Contact: Martin Speller, [email protected]) Krakow (Poland) ( Contact: Mireille Chaleyat--Maurel, [email protected]) 1999 June, 20th - July, 10th Autumn EMS Summer School in Orsay (France) in Applied Mathematics DMF "Mathematics and Music" in Lisbon (Portugal), Paris (France) "Wavelets and their applications". Organiser: A. Cohen (University Paris 6, France) ( Contact:Mireille Chaleyat-Maurel, [email protected]) July, 27th - August, 14th EMS Summer School in CLUJ (Romania) in Pure Mathematics "Singularities in Geometry". Organiser: N. 2000 Te leman (Ancona, Italy) July 10-14 August, 15th Third European Congress of Mathematics Deadline for submission of information or papers to (3ECM) in Barcelona (Spain) the September issue of EMS Newsletter ( Contact: S. Xambo--Descamps, [email protected])

15 (

DIDEROT MATHEMATICAL FORUMS

Committee I. Opening Ceremony Jochen Bruning (Berlin, Germany) With, successively, Jean-Pierre Bourguign011 (EMS Mireille Ohaleyat-Maurel (Paris, France) President) in Madrid, Michael Keane (Professor in Alberto Conte (Torino, Italy) (Chair) CWI) in Amsterdam and Gianni Mattioli (Vice­ Minister of Italian Public Work) in Roma. Format II. Scientific Conferences This cycle of conferences, launched in September 1996, with a forum in London, Moscow and With, successively, 0.J. Van Duijn ;CWI, (Salt wa­ Zurich, on the theme of "Mathematics and ter intrusion) in Amsterdam, A. Bermudez de Cas­ Finance", features two conferences a year taking tro, University of Santiago de Compostela, (Math­ place simultaneously in three European cities ematical modelling and optional control methods in exchanging information by telecommunication and a waste water) in Madrid and A. Quarteroni,I. Po­ addressing in their programmes three different litecnico di Milano and CRS4of Cagliari, (Physical­ aspects : fundamental mathematics, applications numerical modelling of environmental processes ) in of mathematics and their relation to society ( e.g., Venice. ethical and epistemological dimensions). Local programme The second Diderot Mathematical Forum Amsterdam "MATHEMATICS AND ENVIRONMENT" B.H. Gilding, University of Twente, ( On the wet­ December 19-20, 1997 Amsterdam, Madrid, ting front, transport of moisture in soiQ, M. de Venice Gee, Landbouw Universiteit Wageningen, ( Semi­ Organizers numerical methods for groundwater contaminant transport) and B. Sommeijer, CWI, (Numerical mod­ Amsterdam: Prof. MichaelKEANE (Centrum voor eling of Three-Dimensional Bio-Chemical Transport Wiskonde en Informatice, CWI) in Shallow Seas) Madrid: Prof. Jesus-Ildefonso DIAZ (Universidad Madrid Complutense, Madrid) Cafl'era, Universidad Politecnica de Catalunya, Venice : Prof. Elio CANESTRELLI (Universita Ca' J. ( Undergroundwater mathematical models), R. Bermejo, Foscari, Venise) Universidad Complutense de Madrid, (A compari­ The second Diderot Mathematical Forum was held son between semi-Lagrangian and Eulerian scheems in Amsterdam, Madrid and Venice on the more for ocean circulation) and J. I. Diaz/ J.L. Lions, focused theme of "problems related to water" UCM/College de France, (Mathematical treatment under the general heading of "Mathematics and of a model arising in lake pollution problems). Environment". Venice This meeting was supported by the European G. Gambolati, Universita de Padova, (Il modello Commission (DG Xll D). The use of sophisticated matematico del sottosuolo di Venezia) and A. Sper­ telecommunication devices was made possible anza, Ministry of Environment in Roma, ( I pre­ thanks to the support of Telecom Italia. supposti delta previsione idro-meteo marina nell'alto Adriatico: storia, attualitd, proiezioni future"). The local organizers had made properarrangements for the success of the conference, in particular selected excellent speakers. The programme was International programme very interesting. Round Table "The role of mathematics in Environ­ ment: the point of view of Industrials and Mathe­ Telecommunication means, provided by Telecom maticians" Italia, functioned well but we are still learning how to make the most out of these new media. All the With M. Knaap(Shell Research), D. Maslen, S. Ver­ teleconference has been recorded. duyn Lune} (Vrije Universiteit van Amsterdam) in Amsterdam; T.. Estrella (Ministerio de Industria), FRIDAY, DECEMBER 19, 1997 R. Llamas (Real Academia de Ciencias de Madrid), G. Parrilla (Instituto Espagna de Oceanografica) International programme in teleconference in Madrid and V. Casulli (Universita de Trento), provided by Telecom Italia G. Gambolati (Universita de Padova), G. Italiano

16 (Universita de Venezia), A. Quarteroni (I. Politec­ A. Cecchini, Universita de Venezia, (Ambiente e Ter­ nico di Milano and CRS4 de Cagliari), A. Speranza ritorio; alcuni modelli di facile uso (e di molte im­ (Ministry of Environnement, Roma) in Venice. plicazioni) realizzati con gli automi cellulari).

SATURDAY, DECEMBER 20, 1997 International programme between Amster­ International programme between Amster­ dam and Madrid dam and Madrid Round Table "The role of Mathematics in With R.D. Gill, Rijksuniversiteit Utrecht, (Lies, Environment : the point of view of Public damned lies, or stat'istics of the environment; use Institutions and Mathematicians" and abuse of statistics in environmental issues) in With M. van Calmthout (Journalist "de Volk­ Amsterdam and C. Pares, Universidad de Malaga, skrant"), M. Keane (CWI) in Amsterdam and (Numerical simulation of the Alhorean sea and the M. Ruiz de Elvira (Diario El Pais), JVfanueJ Tolmria. Gibraltar circulation) in Madrid. (Corespondent of the European Union of Scientific Journalists), A. Ribera (Diario El Pais), J.I. Diaz Local programme (UCM) in Madrid. Amsterdam A. Stein, Landbouw Universiteit Wageningen, IN PREPARATION (Point processes, random sets, and geostatics for ThirdDiderot Mathematical Forum "MATH­ analyzing patterns of methylene blue coloured soiO, EMATICS AS A CULTURAL FORCE OF L. de Haan, Erasmus Universiteit Rotterdam, (Sea EVOLUTION" June.5-6, 1998Berlin, Florence, and Wind : Multivariate Extremes at work) and Krakow R.M. Cooke, Technische Universiteit Delft, (Expert Judgement and the theory of dry water). Madrid Organizers J. Samper, Universidad de La Coruna, ( Ground­ water pollution modelling: a challenge for the next Berlin : Prof. Jochen BRljNING (Humboldt century), G. Winter/J. Betancor and G. IVIontero, University, Berlin) Universidad de Las Palmas de Gran Canarias, ( 3D­ Florence : Prof. Enrico GllJSTI (University of simulations in the lower troposphere: d'ispersion of Florence) air pollutants from industrial emissions of Sulfure Krakow : Prof. Andrzej PELCZAR (Jagellouian and Nitrogen Oa;ides with field mades in connec­ University, Krakow) tion to meteorology) and F.J. Elorza, Universidad Politecnica de Madrid ; ( Transport of pollutants in Diderot Mathematical Forum "MATHE­ ground water and low permeability rocks). MATICS AND MUSIC" Autumn 1999 Lisbon, Venice Paris and a third town yet to be determined V. Casulli, Universita de Trento, ( Un modello matematico per la Laguna di Venezia), A. Berga­ Organizers masco, Universita de Venezia, (Modello numerico Lisbon : Prof. Jose-Francisco RODRIGUES d'interazione tra Laguna di Venezia e Mare Nord­ (University of Lisbon) Adriatico), A. Marani, Universita de Venezia, (Con­ Paris: Dr. Gerard ASSAYAG (IRCAM) and Prof. trollo di qualite dei dati con reti neuronali) and Laurent MAZLIAK (University Paris 6)

17 WORLD MATHEMATICAL YEAR 2000 A world event at the initiative of the International Mathematical Union (IMU) o Sponsorship of UNESCO Regular Program.me and Budget in support of preparatory activities. In its November 11, 1997 plenary meeting, the UNESCO General Conference followed the <> IMU Committee for the Year 2000 / recommendations of Commission III and approved UNESCO draft resolution 29 C/DR126 related to the World Mathematical Year 2000, allocating US $20,000 to This Committee has been set up to coordinate plans this series of events. for WMY 2000 with UNESCO. The following have agreed to serve on the committee: The following 15 countries co-sponsored the draft resolution: Belgimn, Benin, Brazil, Colombia, Cote A. Ashour (Egypt), M. Chaleyat-Maurel, chair (France), M.S. Narasimhan (India-Italy), M. Niss d'Ivoire, Denmark, France, Ireland, L1Lxembourg, A. Philippines, Netherlands, Russian Federation, Spain, (Denmark), R. Rebolledo (Chile), Sierpinska Thailand, Uzbekistan. (Canada) and G. 'Ii:onel (France). The UNESCO resolution <> Projects for the Year 2000 The General Con/erence Many mathematical societies and countries are preparing the WMY 2000 with various projects. Considering the central importance of mathematics Plans of EMS for this operation have been described and its applications in today's world with regard to by V.L. Hansen in the 25th issue of the EMS science, technology, com:rnunications, economics and Newsletter (September 1997). nwnerousother fields, A preliminary Agenda for Year 2000 can be found Aware that mathematics has deep roots in many in the 5th issue of the WMY 2000 Newsletter. cultures and that the most outstanding thinkers over For more details, please contact Mireille Chaleyat-­ several thousand years contributed significantly to Maurel [email protected] . their development, and numerous other fields, <> WMY 2000 Newsletter Aware that the language and the values of mathe­ matics are universal, thus encouraging and making Five issues of the WMY 2000 Newsletter have be it ideally suited for international cooperation, published and worldwide distributed to all IMU affiliated institutions. Stressing the key role of mathematics education, in For a printed version, please send a request to VV1vlY particular at primary and secondary school level, 2000 Newsletter, Institut Henri Poincare, 11, rue both for the understanding of basic mathematical Pierre et Marie Curie, F-75005 Paris (France). concepts and far the development of rational thinking, The five issues are available on the We b at the site www.ma.th.jussieu.Er with links to the servers of Welcomes the initiative of the International Math­ IMU, EMS, AMS, and the French societies, SMF ematical Union (!MU} to declare the year 2000 the and SMAI. World Mathematical Year and carry out, within this framework, activities to promote Mathematics at all <>WMY 2000 booth at ICM98 levels world-wide, The operation "WMY 2000" will be presented at Decides to support the World Mathematical Year the International Congress of Mathematicians in 2000 initiative, Berlin (ICM98) via a booth where doc1m1ents on all projects will be available. The objective is to make Requests the Director General to collaborate with the the operation known through various components international mathematics community in planning of the mathematical community and to convince the World Mathematical Year 2000 and to contribute people to cooperate for the organization of this during 1998-1999 funds of £ 20.000 from the event.

18 r EURONEWS

EUROPEAN NEWS: Country by Country

CZECH REPUBLIC Local Organisers: J. Chvalina {Masaryk University, Rmo), F. Neuman and .T. 8imsa (Math. lnst. Acad. Conference: Sci. Rrno) "Partial Differential Equations - The­ Information: Participation is by invitation only. ory and Numerical Solutions" Those who wish to attend should write to The conference "Partial Differential Equations - 36th ISFE Theory and Numerical Solutions" will be held in Math. Inst. Acad. Sci Prague {Czech Republic) in August, 10-16, 1998 as Zizkova 22 a satellite conference to the International Congress 616 62 Brno of Mathematics {ICM'98) under the auspices of the Charles University of Prague and Ruprecht­ Czech Republic Karls University in Heidelberg. It is organized Fax ++420-5-41218657 in the framework of the 650-th anniversary of the E-mail: [email protected] foundation of the Charles University. Organising committee W. Jager, J. Necas {chairmen), M. Feistauer, 0.John, R.Rannacher, J.Stara Programme Committee FRANCE H. Berestycki, M. Feistauer, A. Friedman, H. Gajewski, M. Golubitsky, J. Haslinger, G. C. Hsiao, XXVIIIth Probability Summer School T. J. R. Hughes, J. Kacur, D. Kroner, J. M. Morel, Saint-Flour ( Cantal) R. Rannacher, M. Renardy, I. Vrkoc 17th August 2nd September, 1998 Important Information INVITED SPEAKERS The number of participants is limited by the Michel EMERY, Directeur de Recherches C.N.R.S. capacity of lecture halls to 250. Registration form "Martingales continues dans les variett'Bdiffernnt.iables" as well as further details can be found on http://kmal4.karlin.mff.cuni.cz/pdeconf ArkadiNEMIROVSKI, Professeur, Technion-farnel Deadline for registration forms and abstracts is Institute of Technology 15 March 1998. The registration fee is 450 US$ "Topics in Non-parametric Statistics" including accommodation in a double room and - Dan VOICULESCU, Professeur, University of breakfasts and lunches. California, Berkeley Contact Address "Lectures on Free Probability Theory" Jana Stara, KMA MFF UK, Sokolovska 83, 18675 For further information, please contact P. BERNARD Praha 8, Czech Republic Universite Blaise pascal e-mail: [email protected] Mathrhatiqucs Appliquees web address: F63177 AUBIERE CEDEX http://kma14.karlin.mff.cuni.cz/pdeconf Tel. 04.73.40.70.52 ou 04.73.40.70.50 Telefax 04.73.40.70.64 36th International Symposium on E-mail bernard@ucfma. univ-bpclermont.fr Functional Equations Hotel Santon, Brno-Bystrc, Czech Republic May 24-30, 1998 GERMANY Scientific Committee: J. A, �,ii (Waterloo, Canada), W. Benz (Hamburg), R. Ger {Katowice), J. Raetz An International Conference devoted to (Bern), L. Reich (Graz), A. Sklar (Chicago) Representation Theory of Algebras

19 EURONE WS

will be held from August 31 - September 5, 1998 at A Euroconference on the University of Bielefeld. Infinite Length Modules The scientific organisation lies in the hands of will be held from September 7 - 11, 1998 at the Dieter Happel �Chenmitz) Helmut Lenzing (Pader­ born) Claus Michael Ringel (Bielefeld) Klaus W. University of Bielefeld (starting on September 7 at 9:00 am and ending on September 11 at 17:00 pm). Roggenkamp (Stuttgart). The scientific organisation lies in the hands of Programme Ken Brown (Edinburgh), Paul M. Cohn (London), There will be 18 one-hour lectures on import.ant Idun Reiten (Trondheim) and Claus Michael Ringel recent developments in the subject. and on (Bielefeld). decisive relations to other parts of mathematics. The following mathematicians have accepted our Progran1me invitation for such a lecture: The aim of the conference is to provide a survey of Hans Joachim Baues (Bonn, Germany) methods and results concerning modules of infinite Raymundo Bautista (UNAM, Mexico) length (in comparison to the well-known properties of finite length modules). A detailed program will Jon Carlson (Athens, USA) be available by the end of February 1998. William Crawley-Boevey (Leeds, England) Gordon James (London, England) Financial Support Mikhail Kapranov (Boston, USA) The Euroconference is financially supported by the Olivier Mathieu (Strasbourg, France) TMR-program of the European Union. Special funds are available for allowing young researchers Sergei Ovsienko (Kiev, Ukraine) from member states of the European Union to Idun Reiten (Trondheim, Norway) participate. Andrzej Skowronski (Torm'!, Poland) Further information concerning the Euroconference Katsuhiro Uno (Osaka University, Japan) (including a registration form) is available in the Michel Van den Bergh (Hasselt, Belgium) internet under: Jie Xiao (Beijing, China) http://YWW.mathematik.uni-bielefeld.de/ sek/E-C-BIE.html The remaining one-hour lectures and additional twenty-minutes lectures (in parallel sessions) will or you may contact be arranged on the basis of abstracts provided by Ms. H. Scharsche, participants. Fakultiit fiir Mathematik, Universitiit, On Tuesday evening (September 1) there will be POBox 100 131, D-33501 Bielefeld, Germany. a presentation of computer algebra packages which handle problems in representation theory, organised Fax: (+49) 521 106-4743, by Peter Draxler (Bielefeld). The final programme E-mail: scharsch@mathematik. uni-bielefeld.de will be available July 31, 1998 in the internet. Abstracts The deadline for the submission of an abstract is June 30, 1998. Such an abstract should be no longer 6th International Conference on Evo­ than one page and should clearly state the result to lution Equations and their Applica­ be presented. The abstracts obtained in time will tions in Physical and Life Sciences be distributed to all participants. Further information concerning the meeting (includ­ September 14 - 19, 1998 ing a registration form) is available in the internet under: Topics range from functional-analytic methods for partial differential equations to the modelling of http://ww,;.mathematik.uni-bielefeld.de/ sek/98rep.html physical and biological processes by evolution equa­ or you may contact tions, with particular emphasis on non-linear evo­ Ms. H. Scharsche, lution equations; (e.g. reaction-diffusion equa­ tions, Hamilton-Jacobi equations), semigroups and Fakultiit fiir Mathematik, Universitiit, PDEs, stochastic evolution equations, mathematical POBox 100 131, D-33501 Bielefeld, Germany. physics ( e.g. fluid mechanics, quantum physics, sta­ Fax: (+49) 521 106-4743, tistical mechanics), and mathematical biology (e.g. E-mail: scharsch@mathematik. uni-bielefeld.de population dynamics, spreading of epidemics).

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EURONEWS

Besides survey talks and individual lectures, there "Mathematics in the Sciences" willalso be workshop discussions dedicated to these subjects. from June 19 until June 23, 1998 in Leipzig. The goal of this symposium is to promote t.he flow of Scientific Committee: S. Albeverio (Honn), H. ideas between mathematics and the sciences in both Amann (Zurich), W. Arendt (Ulm), G. Da Prato directions and to open new perspectives for the (Pisa), G. Lurner (Mons), J. PriiB(Halle), L. Weis future. (Karlsruhe) The followingscientists have agreed to give one-hour lectures: Lecturers include: S. Albeverio, H. Amann, I. Antoniou, W. Arendt, P. Auscher, C.J. Batty, Ph. Abhay Ashtekar (Penn State University) Benilan, J. v. Below, I. Cioranescu, Ph. Clement, Sir Michael Atiyah (University of Cambridge) M.G. Crandall, T. Coulhon, G. Da Prato, E.B. John Ball (University of Oxford) Davies, M. Demuth, 0. Diekmann, K-J. Engel, J. Randolph Bank (Supercomputer Center San Diego) Escher, A.Favini, G. Goldstein, J. Goldstein, M. Alain Connes (IRES Bures-sur-Yvette) Gyllenberg, K:P. Hadeler, M. Hieber, M. Iannelli, W. Jager, N. Jacob, A. Lasota, Y. Latushkin, R. de Sir Sam Edwards (University of Cambridge) Laubenfels, P.-L. Lions, S.-0. Londen, G. Lumer, Manfred Eigen (MPI for Biophysical Chemistry A.. Lunardi, V.P. Maslov, S. Monniaux, R. Nagel, Gottingen) J. van Neerven, S. Nicaise, S. Oharu, I. frigogine, Mikhail Gromov (IHF,S Bure.'l-Snr-Yvet.te) J. Prii.6, F. Rabiger, S. Romanelli, M. Rockner, W. Helmut Hofer {Gourant Institute) Rue.B, B.-W. Schulze, G. Simonett, J. Sprekels, Th. Richard James (University·of Minnesota) Sturm, K. Taira, A. Venni, Q-Ph. Vu, W. v. wahl, Anatole Katok {Penn State University) V. Wrobel, J. Zabczyk Elliot Lieb (Princeton University) Sponsors: Deutsche Forschungsgemeinschaft, (DFG), Bernard Frederik 8chutz (MPI for Gnwitational Landesregiering Bad&Wiirttemberg, Universitat Physics Potsdam) Karlsruhe. Furthermore, the organisers are hopeful Yakov Sinai (Princeton University) that the European Union will grant additionalfunds Luc Tartar (Carnegie Mellon Uuiversity/MIS destined specifically to support the participation of Leipzig) a number of highly promising young researchers. Srinivasa Varadhan (Courant Institute) Information: For more details contact: Julius Wess(MPI for Physics Miinchen) L. Weis Shing-TI.mg Yau (Harvard University) Conf. on Evolution Eq. The symposium will be opened by Mathematisches Institut I Friedrich Hirzebruch {MPI for Mathematics Bonn). Universitat Karlsruhe The organizing committee consists of Jiirgen Jost, Stefan Miiller, Eberhard Zeidler {MIS Leipzig), 76128 Karlsruhe and representingthe Andrejewski Foundation, Bodo Germany Geyer (Leipzig University) and Dietmar Liist Fax: +49-(0)721-608-6177 (Humboldt University Berlin). E-mail: [email protected] . To encourage students and young scientists to attend, no participation fee will be charged. More Updates of this informationwill be given on WWW: information (hotel reservations etc.) can be found http:malserv.mathematik.uni-Karlsruhe.de/c,html on the home page www.mis.mpg.de/conferences/symp98

Any interested person should send an empty email In memoriam, to: symp98©mis. mpg. de A registration form will be sent back. Walter Andrejewski The deadline for applications is April 30, 1998.

The Andrejewski Foundation and the Max­ Planck-Institute for Mathematics in the Sciences (MIS) announce an outstanding interdisciplinary symposium

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ITALY Prof. STEFANO MARMI {Universita di l<'irenze) and International Mathematical Summer Prof. JEAN-CHRISTOPHE YOCCOZ (Universite Centre de Paris Sud, Orsay) Fondazione C.I.M.E. Courses Centro Internazionale Matematico Es­ a) KAM-Theory for Linear Quasi-Periodic Systems tivo (8 lectures in English), Prof. L. Hakan ELIASSON (KTH, Stockholm) Courses for 1998 b) Invariant Tori (8 lectures in English), Prof. C.I.M.E. activity 1998 consist in five courses: Michael HERMAN (Universite de Paris 7, Ecole the first course is jointly organised by C.I.M.E. Polytechnique) and C.I.M. (Centro Internacional de Matematica, c) Geometrical Methods in Small Divisors Problems Coimbra) in Portugal; other four courses will take (8 lectures in English), Prof. Jean-Christopher place at Cetraro (Cosenza), in Italy. YOCCOZ (Universite de Paris-Sud, Orsay)

FIRST C.I.M.E. C.I.M. COURSE Applications "Optimal Shape Design" Those who wisht to attend the Session should fill in Troia (Portugal) from 1 to 6 June, 1998. in an application to the Director of C.LM.E at the address below, not later than April 15, 1998. Scientific Direction: Prof. Arrigo CELLINA (Universita di Milano) THIRD C.I.M.E. COURSE Prof. Antonio ORNELAS (University �f Evora) "l\![athematicalProblems in Semiconductor Pl1ysics" Courses at Grand Hotel San Michele, Cetraro (Cosenza), (4 lecturE>B in English for each course) from 15 to 22 July, 1998. a) Some nonconvex optimal shape problems, Prof. Scientific Direction: Bernd Kawohl (Univ. Koeln, Germania) Prof. MARCELLO ANILE (Universita di Catania), b) Shape Control and Optimal Shape Design, Prof. Prof. PIERRE DEGOND (Universite Paul Sabatier, Olivier Pironneau, Analyse Numerique, Paris 6. c) Toulouse) and Homogenisation methods in Optimal Design, Prof. Luc Tartar, Carnegie Mellon University Prof. PETER A. MARKOWICH (TU, Berlin) d) Explicit solutions in Elastic Optimisation, Prof. Courses Piero Villaggio, Universita di Pisa. a) Drift Diffusion Equations and Applications (6 e) Optimal Shape Design: Theory, Modelling and lectures in English), Prof. Walter ALLEGRETTO numerical algorithms, Prof. Jean Paul Zolesio, (Univ. of Alberta, Canada) CNRS, Sophia Antipolis. b) An Introduction to Kinetic Theory (6 lectures in Applications English), Prof. David LEVERMORE (University of Arizona, Tu cson) A registration fee of 9 OOO Portoguese Escudos (approximately 50 US $). Those who want to c) Transport Modelling in Semiconductors (6 attend the Session should fill in an application lectures in English), Prof. Frederick POUPAUD that _can be obtained by e-mail at the address (Universite de Nice) shape98©hermit e.cii.fc.ul.pt not later than d) Foundations of Mathematical Models for April 15, 1998. Further information can Semiconductor (5 lectures in English), Prof. be obtained through internet at the address Christian RINGHOFER (Arizona State University, ht tp://www.cim.pt Tempe) Applications SECOND C.I.M.E. COURSE Those who want to attend the Session should fill "Dynamical Systems and Small Divisors" in an application to the Director of C.I.M.E at the at Grand Hotel San Michele, Cetraro (Cosenza), address below, not later than April 30, 1998. from 13 to 20 June, 1998.

Scientific Direction:

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FOURTH C.I.M.E. COURSE Applications Those who want to attend the Se.ssion :,;hould fill "Filtration in Porous Media and Induat1·ial Applica­ in an application to the Director of C.I.!:vI.E at the tions" address below, not later than May 15, 1998. at Grand Hotel San Michele, Cetraro, from August 24 to September 1, 1998. Further information: Fonda.zione C.I.M.E. c/o Dipartimento rli l\latemat­ Scientific Direction: ica "U. Dini" Prof. ANTONIO FASANO {Universita di Firenze) Viale Morgagni. 67 / A - 501:N FIRENZE (ITALY) and Tel. +:39-5,5-434975 / +:l9-55-42:J7123 Prof. HANS VAN DUIJN {University of Amsterdam) FAX +39-.55-434915 / +:19-55-4222695 E-mail [email protected]. UNIFJ.IT Courses Informationon CIME can be obtained on the system a) Mathematical Models for Oil Reservoirs Engi­ World-Wide-Web on the file neering {6 lectures in English), Prof. Magne S. ES­ PEDAL (University of Bergen) http: I /v,ww. math. unifi. _it/CIME/welcome . to. CIME b) Filtration Processes in Various Industrial Problems (4 lectures in English), Prof. Antonio The CJME Foundation is spousored by Con­ FASANO (Universita di Firenze) siglio Nazionalc dcllc Riccrdw (C.N.R.), Miuistcro dell'Universita e della Ricerca Sciimtifica e Tecno­ c) Reactive TransportPro cesses in Porous Media {,'3 logica (M.U.R.S.T.) and European Community. lectures in English), Prof. P. KNABER (Universitat Erlangen- N iirnberg) d) Homogenisation Theory and Applications to Filtration Processes (6 lectures in English), Prof. A. MIKELIC (Universite Lyon I) SCUOLA MATEMATICA INTERUNI­ e) Some Nonlinear Models Arising in Subsurface VERSITARIA Transfport (3 lectures in English), Prof. Hans VAN DUIJN (Delft University of Technology) SUMMER COURSE IN MATHEMATICS - PERUGIA 1998 Applications Those who want to attend the Session should fill In Summer 1998 graduate courses in nmthematics in an application to the Director of C.I.M.E at the will be organized under the sponsorship of the address below, not later than May 15, 1998. Italian National Research Council and MURST, at the University of Perugia, Perugia (Italy); the FIFTH C.I.M.E. COURSE courses will take place between July 26 and August 29, 1998. and in "Stochastic PDEs Kolmogorov Equations The courses are directed towards young graduates Infinite Dimensions" wishing to study mathematics at a graduate level. at Grand Hotel San Michele, Cetraro, from August 24 to September 1, 1998. The list of courses offered is attached. Each participant is require.cl to choose two courses for a ScientificDirection: total of 10 hours a week of lecturns in addition t.o Prof. GIUSEPPE DA PRATO (S. N. S., Pisa) problem sessions. Daily attendance is compulsory for participants. Although written tests will be given Courses by the Jectnrers, no certification of proficiency will a) Kolmogorov equations (8 lectures in English), be is..�ued. Prof. N. V. KRYLOV (University of Minnesota, Ten fellowships, of 570.000 Italian lire each, will Minneapolis) be available to foreign participants to help cover b) LP-analysis of finite and infinite dimensional their living expenses during the eourse. All the diffusion operators (8 lectures in English), Prof. M. participants will be lodged, at no cost, in the Casa ROCKNER (Universitiit Bielefeld) dello Studente. Te xt · books, lecture notes and c) Kolmogorov equations with infinite numbers of photocopies will be provided by the School. variables (8 lectures in English), Prof. J. ZABCZYK NO TRAVEL EXPENSES WILL BE REIM­ (Polskiej Akademii Nauk, Warszawa) BURSED

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Applications should be sent by mail to the following -DIFFERENTIAL GEOMETRY Prof. E. Musso, address : Univ. de L'Aquila Professor Graziano GENTILI (lecturesin Italian or English) Scuola Matematica Interuniversitaria -INTRODUCTION TO PROGRAMMING Prof. Summer Course in Mathematics J.R. Hindley, Univ. Wales-UK Via S. Marta 13/ A AND COMPUTER SCIENCE (lectures in English) 50139 - Firenze, Italy -PROBABILITY Prof. H. Teicher, Rutgers Univ. and should reach this address before April 30, 1998. (lectures in English) A selection committee will meet shortly afterwards and all admitted foreign applicants will be notified --MATHEMATICAL STATISTICS Prof. E. of the result in due time. Applications should Regazzini, Univ. ''L. Bocconi" contain the titles of three courses the applicant (lectures in Italian) would like to attend to (in order of preference) a brief curriculum vitae and a. DETAILED CURRICULUM -ALGEBRAIC TOPOLOGY Prof. K. Johnson, STUDIORUM (including a certificate with the Dalhousie Univ. list of university courses taken and corresponding (lectures in English) grades). Applicants are requested to state if their participation is conditioned by the allotment of a fellowship. They should indicate very dearly SCUOLA MATEMATICA INTERUNI­ the exact address to which all correspondence VERSITARIA concerning the Summer Course should be mailed. SUMMER COURSE IN MATHEMATICS - Information - Programmes and further informa­ CORTONA 1998 tion can be found on the WEB page of SMI at the address: During Summer 1998 several graduate courses in http://www.iaga.fi.cnr.it/SMI/index.html mathematics will be held, under the sponsorship of the National Research Council and MURST, in or can l?eobtained by e-mail at: Cortona at the Scuola Norma.le Superiore. The [email protected] courses will take place during the periods June 28- .July 11, 1998; July 12-August 1, 1998 and August Programmes and further information can also be 2-August 15,1998. required by fax at +39-55-47591.5 or by mail at the address of the Scuola Matematica Interuniversitaria The courses are directed towards young graduates (see above). who would like to engage in re.search in one of the The Chairman of S.M.I. following fields : (Prof. Graziano Gentili) June 28 - July 11 LIST OF COURSES - Perugia 1998 -COMBINATORIAL GEOMETRY Prof. G.-C. -ALGEBRA Prof.M. Gutierrez, Tuft.s Univ. Rota (MIT-Cambridge) Prof. A. Beutelspacher (lectures in English) (Univ. Giessen) -COMPLEX ANALYSIS Prof. T. Gamelin, UCLA - DIFFERENTIAL GEOMETRY Prof. S. Salamon (lectures in English) (Oxford Univ.) Prof. P. de Bartolomeis (Univ. Firenze) -FUNCTIONAL ANALYSIS Prof. L. De Michele, Univ. Milano July 12 - August 1 (lectures in Italian) -ALGEBRA Prof. L. Di Martino (Univ. Milano) -NUMERICAL ANALYSIS Prof. C. Lubich, Univ. Prof. A.E. Zalesskii (Univ. East Anglia) Tu bingen ELLIPTIC DIFFERENTIAL EQUATIONS (lecturesin English) (Lecturers to be confirmed) -DIFFERENTIAL EQUATIONS OF Prof. K. Oddson, Univ.Calif. Riverside August 2 August 15 MATHEMATICAL PHYSICS (lectures in English) - OPERATIONS RESEARCH Prof.D. Bertsekas -ALGEBRAIC GEOMETRY Lecturer to be (MIT-Cambridge) - Prof. P. Hansen (Univ. confirmed Montreal)

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Each participant is required to choose one topic for PORTUGAL a total of 12 hours a week of lecture.s. International Conference on Combina­ In addition, the participants themselves will be torial Methods in Mathematics asked to participate in the problem sessions and in the seminars that will be organised in the II Meeting of the project Algebra, afternoons. Geometria e Combinatoria Six fellowships, of the duration of two or three (Praxis 2/2.l/MAT/63/94) weeks ( each consisting of 90.000 Italian lire per week) will be available to foreign participants to 9 - 11 JULY, 1998 help cover their living expenses. The participants Faculdade de Ciencias da Universidade do Porto will be lodged at no cost in the Palazzone or nearby (Portugal) hotels; furthermore breakfast and lunch (the latter only fromMonday through Saturday) will be offered The Conference aims to bring together mathemati­ by the organization. The School will also provide cians working in Ccnnbinatorics and maLhemat.icim1s texts books, lecture notes and photocopies. that use combinatorial methods as a tool in their re­ NO TRAVEL EXPENSES WILL BE REIM­ search. BURSED There will be one hour invited lectures given by I.he Applications should be sent by mail to the following following speakers: address: - ANDERS B.JOERNRR ( l Tuiversity of Stockholm, Professor Graziano GENTILI Sweden) (to be confirmed) Scuola Matematica Interuniversitaria - RAUL CORDOVIL (Technical University of Summer Course in Mathematics Lisbon, Portugal) Via S. Marta 13/ A - KOMEI FUKUDA (Institute for Operations 50139 - Firenze, Italy Re.search, ETH Zentrum, Switerzland) Applications for the courses of the first and second - JOHN MEAKIN (University of Nebraska-Lincoln, period should reach this address before April 15, USA} 1998; for the third period before May 15, 1998. - CHRISTOPHE REUTENAUl�R (Univen,ity of Quebec at Montreal, Canada) A selection committee will meet shortly afterwards - GRANT WALKER (University of Manchester, and all accepted foreign applicants will be notified United Kingdom) of the result in due time. The programme of the Conference will also include Applications should contain a BRIEF CURRICU­ sessions for SHORTER COMMUNICATIONS (30 LUM VITAE and a DETAILED CURRICULUM minutes). Participants wishing to present a commu­ STUDIORUM and should indicate which course the nication are asked to SUBMIT A SUMMARY of no candidate would like to follow. Applicants are re-­ more than two A4 pages (12 point fonts) by 15th quested to state if their participations is conditioned MAY. by the allotment of a fellowship.The exact address to which all correspondence concerning the Summer Please complete and RETURN THE FOLLOWING courses has to be mailed should be clearly stated. FORM, preferably by email, before 30th APRIL.

Information - Programmes and further informa­ The second announcement should be available in tion can be found on the WEB page of SMI at the May. address: THERE IS ALSO INFORMATION AT THE http://www.iaga.fi.cnr.it/SMI/index.html FOLLOWING ADDRESS: or can be obtained by e-mail at: [email protected]. it http://www.fc.up.pt/agc98.htm The meeting will be held at the Faculty of Scieuces Programmes and further information can also be of University of Porto, near I.he historical part. of the required by fax at n. +39-55-475!}15 or by town. The city of Porto, standing in the northern mail at the address of the Scuola Matematica coast of Portugal, is over one thousand years old Interuniversitaria (see above). an

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of port wine trade that the wine got its name Organising Committee: from Porto. Recently, the city of Porto has been Maria LeonorMoreira, Maria do Rosario Pinto, Jose appointed by UNESCO a World Heritage. Carlos Santos. For MORE INFORMATION about the meet­ The organisation would appreciate YOUR. FOR­ ing, CONTACT the Organizing Committee at WARDING THIS MESSAGE TO ANYONE WHO agc98©fc.up.pt MAY BE INTERESTED IN PARTICIPATING.

International Conference on Combinatorial Methods in Mathematics II Meeting of the project Algebra, Geometria e Combinatoria 9 -11 July, 1998 Faculdade de Ciencias da U niversidade do Porto (Portugal)

NAME:

Address:

Institution:

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E. MAIL:

FAX:

DO YOU PLAN TO PRESENT A SHORT COMMUNICATION"?

Date:

Signature:

RUSSIA is one of the several international institutes like IRES (France), Max Planck Institute (Germany), Euler International Mathematical In­ Banach Center (Poland), Newton Institute (UK). stitute at St.Petersburg The decision of the Academy of Sciences to create EIMI was first announced in 1988 in recognition of A.Vershik, Yu.Matiyasevich (St.Petersburg, the high contribution of the Russian mathematical Russia) School to international mathematir.s. In this article we describe for the readers of The creation of the Euler Institute was made possi­ EMS Newsletters history, activities, and plans of ble by the support of the Academy of Sciences as well EIMI: Euler International Mathematical Institute as assistence from some international organizations at St.Petersburg. More up-to-date information can including UNESCO, JEC FUND, Japan Association always be found at forMathematical Science, Society for the Support of URL: http://www.pdmi.ras.ru/EIMI the Euler Institute (Berlin). I. History The ·main purpose of the EIMI is to serve as a Euler International Mathematical Institute (EIMI) meeting place for leading scientists of the former

26 15UHUJ\JJ!;WS

Soviet Union and their foreign colleagues. The 1996 activity of the EIMI consists in the organization L Petersburg Workshop on Perturbat.ive QCD and running of scientific progra1mnes, conferences 2. Bmmdary Control and Inverse Problems and workshops on concrete mathematical problems 3. FreeBoundaries with Viscous Flows of fundamental importance for mathematics andits applications. 4. British-Russian Workshop in Functional Analysis The scientific activity of the EIMI began in October, 5. Galois Groups of Local and Global Fid

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October 18-22, 1996 VI St.Petersburg Summer Analysis Meeting ''Galois Groups of Local and Global Fields" Co-chairman: N .Nikolski, S.Kisliakov, V. Vasyunin Co-chairmen: H.Koch (Max-Plank Institute, Ger­ (Steklov Institute of Mathematics in St.Petersburg) many), A.Parshin (Steklov Institute RAS), S.Vostokov 35 participants, including 9 foreigners (St.Petersburg State University) June 24 30, 1997 39 participants, including 17 foreigners (participation in organization) October 24-26, 1996 International Algebraic Conference dedicated to "Efficient Symbolic Computing" D.K.FADDEEV Co-chairmen: Jean-Marie Jacquet (Namur, Bel­ Chairman: A. Yakovlev (St.Petersburg State Univer­ gium) and E.Dantsin (PDMI RAS) sity) 31 participants, including 14 foreigners 233 participants, including 67 foreigners December 13-14, 1996 September 22-27, 1997 "Experience in realization of joint projects IN­ TAS/NIS" Russian-German Geometry Meeting dedicated to of A.D.Alexandrov Co-chairmen: A.Zizcenko (RAS, Moscow) and the 85th birthday Yu.Teterin (PDMI RAS) Chairmen: W.Ballmann (Mathematisches Institut, Bonn), Yu.Burago (Steklov Institute of Mathemat­ 25 participants, including 4 foreigners. ics in St.Petersburg), Yu.Re5hetnyak (Sobolev Insti­ May 19-24, 1997 tute of Mathematics, Novosibirsk) "2nd POMI-Florence workshop on integrable 71 participants, including 38 foreigners systems and quantum groups" Chairman: Anatoly lzergin (Steklov Institute of IV. Programme of the personal visits of Euler Mathematics in St.Petersburg) Institute 27 participants, including 7 foreigners EIMI runs an ongoing program of scientific collaboration: May 22-24, 1997 TETE-A-TETE in RUSSIA ''14th Days on Weak Arithmetics" The goal of this program is to provide facilities for Chairman: Yuri Matiyasevich (Steklov Institute of we.stern and Russian mathematicians (not necessary Mathematics in St.Petersburg) from St.Petersburg) to meet for joint work. Such a 32 participants, including 17 foreigners meeting of two or more colleagues could last from a June 3-5, 1997 couple of days to several months. Annual International Seminar "DIFFRACTION In view of the presentdifficult economic situation, it DAYS - 97" is assumed that visitors will find the required funds from other sources. For its part, EIMI will provide Co-chairmen: V.Babich (Steklov Institute of office.s,(mo dest) computer facilities (including TeX Mathematics in St.Petersburg) and V.Buldyrev and INTERNET connection), and access to the (St.Petersburg State University) libraries of EIMI and PDMI. Visitors are welcome 60 participants, including 8 foreigners to participate in the regular mathematical seminars June 6-7, 1997 of PDMI and the UNIVERSITY as well as in the International meeting "New Asymptotic and Nu­ meetings of the St. Peter&- burg Mathematical merical Methods in Diffraction" Society. ElMJ will issue invitations as required for obtaining Russian visas and will help in finding Chairman: Vasily Babich (Steklov Institute of (reasonably inexpensive) accommodation. Mathematics in St.Petersburg) If you and your colleague(s) in Russia are interested 15 participants, including 5 foreigners in participating in this programme, please fill in the June 16 - 20, 1997 attached formand e-mail it (with any que5tions you (participation in organization) may have) to The First International Conference on Problems admin@euler. pdmi.ras. ru of Dynamic Objects Logic-Linguistic Control Acceptance· of the application is subject to demand DOLLC'97 and to events in other EIMI programmes. Co-chairmen: N.Lyashenko (USA), L.Reznik (Aus­ Application for participation in the EIMI pro­ tralia), A.Gorodetsky (St.Petersburg, Russia) gramme TETE-A-TETE IN RUSSIA: The form can 50 participants, including 5 foreigners also be downloaded from June 23 24, 1997 http://www.pdmi.ras.ru/EIMI/imitet.html

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I. Western participant(s) 1) last name 2) given name(s) 3) citizenship 4) degree, title(s) 5) affiliation 6) position 7) mailing address 8) e-mail address 9) phone number 10) FAX number If visa is required, the following fields should be filled in too: 11) Date of bith 12) Passport number

II. Russian participant( s) 1) family name 2) given name 3) patronymic 4) degree, title(s) 5) affiliation 6) position 7) mailing address 8) e-mail address 9) phone number 10) FAX number

III. Subject and form of proposed joint work (a few phrases)

IV. Desired duration ...... days/weeks/months

in the period from ...... to ......

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The following researchers have visited EIMI in 1996- Introduction to Duration Calculus", he gave a 1997 in the framework of the progranune "Tete--a­ detailed description of this institute and outlined tete in Russia": possible forms of cooperation with it. Prof. Wilfrid Hodges, GreatBritain; In January 1997 the following Agreement was signed Prof. Denis Richard, France; with UNESCO. Prof. VVilliam S.Hatcher, Canada; AGREEMENTBET�'EEN Prof. Valerio Tognetti, Italy; The United Nations Educational Scientific and Prof. Francois Englert, Belgium; Cultural Organisation and Prof. MikhaelBalabane, France; The EULER International Mathematical Institute Prof. Roland Gillard, France; Whereas the United Nations Educational, Scientific Prof. Nicholas Young, GreatBritain; and Cultural Organisation (hereinaft.er referred to Prof. Francois Cuvelier, France; as "UNESCO') is a specialised agency in the area of Prof. Alberto Te.sei, Italy; co-operation in all branches of intellectual activity Prof. Paolo Emilio Ricci, Italy; and especially in international scientific relations, Dr. Paolo Aschieri, Italy; Whereas the Euler International Mathematical Institute (hereinafter referred to as the "EIMI") Prof. Johann Makowsky, Israel; at the Steklov Mathematical Institute of the Prof. Herbert Koch, Germany; Russian Academy of Sciences is directed to promote Prof. Ruediger Goebel, Germany; research and the advanced training of researchers Prof. Anca Muscholl, Germany; and university teaching staff in various areas Olivier Teytaud, France; of fundamental and applied mathematics and mathematical physics, and Prof. Laura Ande.rson, USA; Whereas the EIMI and UNESCO wish to establish Prof. Richard Bishop, USA; an entity to be known as the Euier International Prof. FrancoiseBlanchard, France; Mathematical Institute associated with UNESCO Prof. Zhou Chaochen, China; and acting under the auspices of the Stekiov Prof. Bruno Courcelle, France; Mathematical Institute of the Russian Academy of Prof. Victor Donnay, USA; Sciences, which will specifically: Dr. Thomas Dunker, Germany; * promote international cooperation in mathemat­ Prof. Emmanuel Fontain, France; ical research in modern branches of mathematics such as mathematical logic, including foundations Prof. Leonid Fukshansky, Germany; of mathematics and computing science, nu1nberthe­ Prof. Franco Garrani, Italy; ory, algebra, geometry, topology, differential equa­ Prof. Jose Gonzalez Llorente, Spain; tions, functional analysis, theory of probability and Prof. James Lawry, UK; statistics, and mathematical aspects of theoretical Prof. Vladimir Lifschitz, USA; physics; Prof. Jean Michel Maillet, France; * provide advanced training in mathematical sci­ ences for specialists from developing and industri­ Prof. Grigori Mints, USA; ali7,ed countrieAs; through the organization, in the Prof. Manuel Ojanguren, Switzerland; framework of thematic trimesters, of international Prof. Regimantas Pliuskevicius, Lithuania; workshops, ad-hoe training groups and advanced Prof. Jose Francisco Rodrigues, Portugal; schools; Prof. Zigmund Rogoff, UK; * promote the sharing of knowledge in mathematical Prof. Vesa Ruuska, Finland; sciences through electronic publishing, via INTER­ Prof. Stefan Wa bnitz, Italy; NET and \VvVVv, of research results and training materials available at the EIMI; Prof. Maciej Wo jtkowski, USA. * promote women's careers in mathematical research You can findopinions of visitors to Euler IMI on and university education. http://www.pdmi.ras.ru/EIMI/imioplst. html Whereas within the framework of their terms of VI. Contact with UNESCO. reference UNESCO and the EIMT may co-operate In April 1997 the Institute was visited by with other spes.;ialised scientific organisations and Academician Chaochen ZHOU from Int,ernational institutions with whose goals their own are m Institute for Software Technology of United accord. Nations University. Besides scientific lecture "An UNESCO and the EIMI agree on the following:

30 EURO NEWS

ARTICLE I Small-x Physics and Light Front Dynamics in QCD 1 . The parties will exchange on a regular basis *August 24-29, 1998 information on their activities in the framework of Dynamical Inverse Problems mutual co-operation. 1999: 2. At the request of either party, UNESCO and the EIMI will mutually facilitate participation in *May (First week), 1999 their meetings and provide each other with advice 3rd International \Vorkshop PDMI-Floreuce on on the area in which they are competent, especially Quantum Groups and Integrable Systems regarding the joint organization of projects in the We are ready to consider your proposals about area of mathematical research and training and aid conferences, workshops, summer schools etc. at, to assist activities undertaken by UNESCO or by the the EIMI. Your suggestions could be addressed to EIMI that could be of interest to the other party. [email protected] 3. The EIMI takes full responsibility for organizing activities (scientific research, workshops, se1ninars, selection and hosting of leading· scientists for participation in programme activities, etc.) organized in the framework of mutual agreements. UNITED KINGDOM 4. Both organizations will promote specific programmes aimed at sharing advanced sdentific CONFERENCE ON COMMUTATIVE knowledge, and the preservation of traditions, ALGEBRA and further integration, in world science, of Russian scientific schools, particularly those of the in Honour of David Rees's 80th Year Stekiov Mathematical Institute and its affiliated institutions. Exeter, England, 13 - 16August 1998 5. UNESCO and the EIMI will also promote Organisers : P. Vamos (Exeter, local organiser), the participation of women in the advancement of R.Y. Sharp (Sheffield) mathematical research and education. Location : Exeter is a cit.y in Sout.h West l�ngland, ARTICLE II about 250 kilometres from London. UNESCO and the EIMI will be empowered to Provisional list of invited speakers : Among those conclude any further arrangements necessary forthe who have already accepted invitations to speak are execution of this agreement. C. Huneke (Purdue), D. Katz (Kansas), D.Kirby ARTICLE III (Southampton), L. O'Carroll (Edinburgh), N.V. This agreement may be terminated by one of the Trung (Hanoi), G.Valla (Genoa) and J.K. Verma participating parties by one year's written notice. It (Bombay). may be reviewed as mutually determined between Scope of the conference : The conference, which is UNESCO and the Institute. open to all interested mathematicians, is expected VII. Euler International Mathematical Institute to concentrate on recent and current research iu plans: aspects of commutative algebra related to the * April 20-24, 1998 work of David Rees, including reductions and Computer Algebra in Scientific Computing integral closures of ideals, Rees rings and algebras, uniform Artin-Rees Theorems, mixed multiplicities *May 16-18, 1998 II) and Hilbert functions. Participants from developed Final meeting of INTAS project EmNet (Phase countries will be expected to pay a Registration Fee * June 2-4, 1998 of £35. Annual International Seminar DIFFRACTION Financial support : The conference is supported fi­ DAYS'98 nancially by a grant from the London Mathematical * June 17-20, 1998 Society: this will be used to cover part of the costs of Seventh Summer St.Petersburg meeting in Mathe­ attendance by the invited speakers and some grad­ matical Analysis uate students. * June 24-28, 1998 Further information : Those interested in further Probability Theory and Mathematical Statistics information should send an e-mail message to Asymptotic Methods car-meet©maths.ex. ac. uk or visit the web site * July 6-15, 1998 http://www.maths.ex.ac.uk/conf_rees.html

31 MATHEMATICAL EDUCATION

ICMI Study On the Teaching and Learning of Mathematics at University Level

The purpose of this Discussion Document is to other hand, an increasingly smaller percentage of raise important issues related to the study of the students appears to be opting for studies which teaching and learning of mathematics at university require substantial amounts of mathematics. Thus level and to stimulate discussion and research on university departments are faced with a double these topics as background for a conference to be challenge. On the one hand, they have to cope held in Singapore in December 1998. After this with the influx of students whose preparation, conference, a publication covering the fundamental background knowledge and even attitudes are quite areas of the topic will be published in the ICMI different from those of past students. On the other Study Series. The main aspects of the Study will hand, they have to attract students to pursue studies also be presented at ICME-9 in Makuhari, Japa11 in in mathematics, where employment opportunities the year 2000. It is anticipated that the Study will and well-paying jobs appear not to be as certain as be of interest to those concerned with the teaching in some other disciplines. of mathematics at university level, to mathematics educators undertaking research in related areas, and Some new developments inthe teaching and learning to many other people with an interest in university of mathematics attempt to come to grips with level mathematics. The conference and publication these issues. For example, alternative approaches related to this Study are likely to have a positive to calculus and linear algebra in the United States influence on the understanding and practice of the reflect, in part, attempts to make these subjects teaching and learning of mathematics at university more engaging and meaningful for the majority of level in the early years of the 21st century. students. There have also been content changes, with increased emphases in some universities on 1. Why a Study on the Teaching and applications and modelling, history and philosophy Learning of Mathematics at University of mathematics, and so on. But a general perception Level? remains in some quarters that the teaching of mathematics at the undergraduate level has not A number of changes have taken place in recent to date made sufficient effort to deal with the years which have profoundly affected the teaching backgrounds and needs of present-day students. of mathematics at university level. Five changes, which are still having considerable influence, are There is also often perceived to be a discontinuity (i) the increase in the number of students now between matheniatics education in secondary attending tertiary institutions; schools and mathematics education in universities. (ii) major pedagogical and curriculum changes that Certainly the levels of ambition and demand placed have taken place at pre-university level; on students are increased at the tertiary level. There is not the same attention paid to learning (iii) the increasing differences between secondary theories in the delivery of university mathematics and tertiary mathematics education regarding the as there is in the teaching of the subject at lower purposes, goals, teaching approaches and methods; levels. University teaching methods tend to be more (iv) the rapid development of technology; and conservative. Often university teachers have joint (v) demands on universities to be publicly responsibility for research and teaching. This is accountable. clearly beneficialbut it can cause more emphasis to Of course, all of these changes are general and have be placed on mathematical research in places where had their influence on other disciplines. However, that is the main criterion for promotion. because of its pivotal position in education genefally, and its compulsory nature for many students, it Teachers of university mathematics courses, on the could be argued that these changes have had a whole, have not been trained to, and do not often, greater influence on mathematics than perhaps �n consider educational, didactic or pedagogical issues any other discipline. beyond the determination of the syllabus; few have been provided with incentives or encouragement There is no doubt that, in many countries, to seek out the results of mathematics education. significantly more students are now entering In days gone by, responsibility was placed largely university and taking mathematics courses than on students' shoulders: it was assumed that was the case ten years or so ago. On the faculty's responsibilities were primarily to present

32 MATHEMATICAL EDUCATION material clearly, and that 'good' students would * to identify, review, encourage and disseminate, pass and 'poor' ones fail. The climate today is research in educational matters at the tertiary level; that academic staff are considered to have greater * to identify and describe major approaches overall responsibility for students' learning. The to tertiary mathematics teaching within different role of instruction (specifically, of lectures) and staff cultures and traditions; accountability are both being reconsidered. * to identify obstacles which might prevent the Wo rldwide, increasing use is also being made learning of mathematics; of computers and calculators in mathematics * to discuss equity and other issues relating to instruction. Much mathematical software and many mathematics education at university level; teaching packages are available for a range of * to discuss the goals of teaching mathematics curriculum topics. This, of course, raises such to a range of students with different backgrounds issues as what such software and packages offer and needs, and who should be re.sponsible for that to the teaching and learning 'of the subject, and teaching; what potential problems for understanding and * to find ways to meet changing needs without reasoning they might generate. It would be good to compromising. the integrity of the subject; collect examples of the use of information technology * to identify, publicise, and expose to scrutiny, new and software which enrich students' experience of teaching methods and the positive use of technology; mathematics and result in better understanding and learning. * to discuss the transition and the relations between secondary school and university; Many academic mathematicians are aware of * to consider ways to improve the preparaLion of changes occurring around them, and of experimen­ teachers of mathematics at university level. tation with different teaching approaches, but they have limited opportunities to embrace change owing Leading up to and during the Conference relating to faculty structures and organisation. Further, the to this Study, it is expected that there will be relationships between mathematicians in mathemat­ debate as to why mathematics is taught and what icsdepartments and their colleaguesin mathematics mathematics education is at university level. In education are often strained, with less productive di­ addition, consideration will be given as to what alogue between them than there might be. The same is the current teaching and learning situation in can be said of relationships between mathematicians universities, what it is believed that the situaLion and engineers, economists, etc., even though math­ should be, and how desired changes can be effected. ematics service teaching to students in other disci­ plines is an enormous enterprise. These general fac­ 2. Themes and Issues Pertaining tors tend to work against, or delay, improvements in to Research on the Teaching. and the teaching and learning of mathematics, particu­ Learning of Mathematics at University larly for those students whose main interests are in Level. other disciplines. Most academic mathematicians know lit.tie about As a result of the changing world scene, ICMI feels the research that has been undertaken in mathemat­ that there is a need to examine both the current ics education iu general, or at the tertiary level in and future states of the teaching and learning of particular. Generally speaking, they are unaware of mathematics at university level. The primary aim the methods 1ised by researchers in education. One of this ICMI Study is therefore to pave the way of the most valuable aspects of the current study is for improvements in the teaching and learning of that it could collect together the major findings of mathematics at university level for all students. mathematics education research, review them, and To achieve. this aim it is important for the make them readily accessible to a wide audience. professionals involved to The potential usefulness and limitations of this re­ search should then be considered in the light of the * exchange views and experiences from a wide practice of teaching. At the same time, it would be variety of places and backgrounds; valuable to determine researchareas which have not * report about developments and projects that have yet been explored and to encourage work in them. taken place; * consider the contributions from theory and The following questions are of particular interest. for research, and identify areas still to be investigated. the Study. More specifically the Study will cover the following What is mathematical understanding and learning, and related points: and how are these achieved? What are the

33 MATHEMATICAL EDUCATION

underlying theories behind these and how do they using mathematics as a service course, as relate to teaching at university level? prospective mathematics teachers, or as recipients of some form of general 'mathematics appreciation' What research methods are employed in mathe­ course. Hence we are addressing the needs of not matics education? What are the major research only future research mathematicians but. also other findings of mathematics education? What are the categories of future mathematics professionals as obstacles to having teaching practice become in­ well as graduates in other disciplines who require formed/influenced by research findings? varying amounts of mathematical knowledge, skill or Might insights into the nature of the learningprocess insight. play out differently at different grade levels? Are the For several reasons, in many countries there has been theories that are relevant at school level, relevant at a move to mass education at university level. As a university level as well? Is there a need for theories result many mathematics departments are providing that are specific to university level? courses for a much wider range of ability and needs than was formerly the case. Simultaneously with ,vhat research has there been into traditional and this increase in student numbers, there has been alternative methods of teaching and what do the a change in the kind of student preparation in results of such research tell us? secondary schools as well as in students' interests In what ways can teaching change to take into and motivation. Consequently many students have account the different background, abilities and not met material which was in most secondary interests of the learner? What methods are effective school curricula of the 1970s. In addition they may forteaching large classes? have been taught by an approach which places more emphasis on the intuitive and pragmatic. Some What do we know about the learning and teaching university mathematics departments have been slow of specifictopics such as calculus and linear algebra? in recognising these changes in their student, intake. Are there characteristics which are relevant to Others have developed new courses to cope for specific topics? Are there characteristics which are the range of content needs but have made few pertinent to a number of topics? pedagogical concessions. What alternative forms of assessment exist'? How There are a number of special groups of students can assessment be used to promote better learning including potential teachers of school mathematics, and understanding? scientists, engineers. What should the interaction between mathematical and professional knowledge What are the mathematical competences that are be? To what extent do these groups need specially required in the different professions? designed courses? What are students' attitudes and beliefs concerning Curriculum mathematics? What causes them to change? How By curriculum we mean matters pertaining to do these affect their enrolments and success in the purposes, goals and content. of mathematics courses with substantial mathematical components? education. Current curricula may need to be What are the effects of the use of technology in the reconsidered for at least two reasons. There are the teaching and learningof mathematics? In ,vhat ways different student needs that were mentioned above can technology be used to enhance understanding? and there are the developments in mathematics itself. What important issues are under-represented in the research literatme and how can researchers be As far as the changing clientele is concerned, it is encouraged to work in these areas? not clear that its constitution or its needs have been adequately considered. What are the professional 3. Themes and Issues Pertaining to aspirations of our student population? Will they Practice go on to be teachers, to work in industry, to be academics, etc.? How should the curriculum be We divide this section into four parts: Clientele, shaped t.o meet. the needs of these groups? Curriculum, Student Activity and Pedagogy. What. changes arp, or should be, taking place in the Clientele curriculum? Some mathematical subject areas are The students who are of interest for this on the decline while others are in the ascendancy. Study include all those students who are taught \�1rnt is the rationale for the changes? Are some mathematics at university level, whether as content areas now less important and should other mathematics majors, as students of other subjects areas take their place?

34 MATHEMATICAL EDUCATION

Mathematics as a rapidly developing research field applying them in controlled i;ituutiolls. Thii; hi

011 content knowledge. 1 ne emphasis seems to affect individual departments. How are these be on learning certain algorithms or theorems and policies formed? What influence can and should

35 MATHEMATICAL EDUCATION mathematicians and mathematics educators have on mathematics departments.) Should departments of them? mathematics be responsible for all of the students taking mathematics at its university or should The previously mentioned increasing number of it concentrate on its traditional clientele, the students at the university level has, in many nations, mathematics majors? 'Will departments which occurred either explicitly or implicitly as the result do not teach a range of students remain viable of government policy. Is there cause for satisfaction in an environment where a balanced budget, with the result of this policy or is there a need to rather than education, is the main concern of change or modify it in some way? administrators? What cooperation can there be with other disciplines for whom mathematics is The mathematical community is convinced of a service comse? In some cases there is an the importance of mathematics both for its own overlap in the material being taught in courses by a sake and for the contribution that it ultimately mathematics department and a service department. makes to society. It is not clear that society Are there good reasons for continuing this practice? in general also holds this position. Perhaps it does not realise what it takes to generate the Clearly no university department. can teach all contribution mathematics can make. What does branches of mathematics. Are there fundamental the mathematical community need to do to make branches of the subject which should be in all society aware of the mathematical requirements of programmes? How should the balance be struck society and how these can be achieved? What between suitable major components? does the mathematical community need to do to make mathematics more visible in a competitive How strongly are incoming students influenced by environment? In what ways should. society provide career prospects in mathematics? How should this its citizens with the basic ideas and philosophy of affect the courses offered and the advice given to mathematics and its impact on our lives, both from prospective students? a philosophical and practical point of view? 5. Call for Reactions University The work of this Study will tal{e place in two In some countries the difference between universities parts. The first consists of a conference which is and other tertiary institutions is the fact that to be held in Singapore from December 8 to 12, research takes place in universities. In such 1998. Engl'ish will be the language of the conference. countries, universities have a research culture in The conference will be a working one where every which it is assumed that most lecturers will engage participant will be expected to be active. Current in research. To what extent should the teaching planning is for a limited attendance of about 75 of mathematics be delivered by lecturers who are persons. engaged in some form of resear�h? Given the style of the conference, we anticipate In some countries, university degrees are of a general a variety of types of contributions that will be nature and cover a range of topics. In other presented in plenary sessions, working groups, panels and short presentations. Presentations may countries, there are more directed programmes for students to follow. What is more, some of the include position papers, discussion papers, surveys more applicable areas of mathematics may be taught of relevant areas, reports of projects, or research papers of an educational nature. outside a mathematics department by engineers, statisticians, physicists, etc. To what extent should We invite you to make a submission for consider­ courses be general and to what extent should they be ation by the International Programme Committee specific to each user group? To what extent should no later than 1 May 1998. Submissions should be courses be taught by mathematicians and to what up to three pages in length and may be emailed, extent should they be taught by experts from other faxed or sent as hard copy. They should be related appropriate fields? to the problems and issues identified in this docu­ ment but need not be limited to these alone. You What then is the role of a department of might also draw to the attention of the Committee, mathematics at the end of the twentieth century, the names of other people whom you feel ought to given that there is a tendency for non-mathematics be invited, stating the type of the contribution they departmentsto teach their own mathematics? (This might make. We would appreciate knowing the na­ is not only for bureaucratic reasons but also ture and results of related studies in this area. because these departments are often dissatisfied with the gap between the content and approach Participation in the conference is by invitation only. they require and the content and approach of Invitations to those whose submissions have been

36 MATHEMATICAL EDUCATION

accepted will be made in July 1998. At the same email: artig11e,Q\mathp7.jussieu.fr time invitees will be asked to produce a longer fax: ( +33) 1 -1427 5608 version of their submission for publication in the pre­ conference proceedings. The St.ucly organisers are Frank BARRJNGTO:'.\, Department of l\·htthemut.ics seeking funds to provide partial support to enable and Statistics, UniYersiiy of Ivlelbourne, ParkYille, participants from non-affluent countries to attend Victoria 3052, Australia the conference but it is unlikely that full support email: [email protected] will be available forany one individual. fax: ( + 61) :3 g:3,14 ,1599 All contributions and suggestions concerning the Mohamed E A EL TOl\I, Departmen1 of Mathemat­ content of the study and the conference programme ics, UniYersity of Qatar, Doha. Qatar should be sent to email: ssct.Q\africamail.com Derek HOLTON, Chair, IPC, IClVU Study, Depart­ fax: (+20-2) 578 7],12 (+20-2) 242 8789 ment of Mathematics and Statistics, University of Otago, POBox 56, Dunedin, New Zealand Joel HILLEL, Department of l\Iatliemalic�-; ,m

3i MATHEMATICAL EDUCATION

Mathematics from K to 12 : about consistency and continuity Nicolas Rouche

This paper presents a study, carried out in Belgium, is likely to be of some interest to people in other cow1tries facing the difficulty of teaching mathematics coherently throughout elementary and secondary schools. This study has some antecedents also worth reporting. In 1988, Yvan Ylieff, the Minister of Education for the French-speaking part of Belgium, set up a commission to study the difficultiesof mathematical teaching. This commission has been named after its president as: "la Commission Danblon". We shall concentrate here on two contributions of the report1 submitted to the Minister in 1990. The first contribution we choose to emphasize is the remark that, in the French-speaking Community of Belgium (Communaute Franc,-:aise de Belgique), mathematical education is traditionally conceived on a very strong basis of horizontal divisions. These divisions are present in three sectors of the school system. Firstly, the curricula for Kindergarten, elementary and secondary schools (three levels) are elaborated by distinct commissions, with fewcoor dination efforts. Secondly, the future teachers forKindergarten, elementary schools, junior and senior high schools (four levels) are trained in different institutions, without cmmnunication. At last, as far as research on mathematical learning is concerned, it is taken in charge mostly at the K and elementary levels by the psychology and pedagogy departments of the universities, without substantial contacts ·with mathematicians, and at the high school level by the mathematics departments, often without substantial knowledge or even concern about what happens at the elen1entary level. These multiple divisions crossing the system horizontally result, among other deficiencies, in a lack of coordination of the subjects taught at the various levels, with severe difficulties of adaptation for the students. The second contribution we want to emphasize is in dose connection with the first. It is a recommendation to create in Belgium a centre for research in mathematics education. During the new math period, we had the well known C.B.P.M. ("Centre Belge pour la Pedagogie de la Mathematique") chaired by Georges Papy. But the Government had ceased to fund it a number of years ago, and the Commission considered that the Belgian school system needed the support of a strong research group similar to the IREMs in France or the Freudenthal Institute in the Netherlands. Such a group should include researchers familiar with all the levels of mathematics education, as well as professional mathematicians with a strong interest in teaching questions. About this recommendation, the Minister did nothing. That is why a group of colleagues originating from all French-speaking universities of Belgium (some of them former members of the "Commission Danblon") created in 1993 a nonprofit--making organisation called the OREM ("Centre de Recherche sur l'Enseignement desMathematiques"). By the autumn of 1993, the OREM had a board of trustees composed of representatives of all school levels, but had no money; it had only energy and good will. It applied to the Minister of Education, Elio Di Rupo, for an initial research grant. The Minister accepted, but imposed the research subject: we had to develop a general plan formathematical education fromK to 12 in 10 months. We thought it was a joke. But it wasn't and our choice was to accept it or disappear. We accepted and it took just over two years. The rest of this paper is a presentation of the book which resulted from this

1 Perspectives sur l'enseignement des mathematiques dans la GammW1aute fra.nr;aise de Belgique premier rapport de la Commission scientifique sur l'enseignement des mathematiques et des sciences, presente a Monsieur le Ministre Yvan Ylieff, le 7 juin 1990, Ministere de l'Education et de la Formation, Bruxelles, Belgique. This report is still available upon request to the CREM, see address at footnote 2.

38 MATHEMATICAL EDUCATION

2 3 research • It is an attempt to partially bridge the multiple gaps mentioned abovc . Its authors are Bernard Honclaire, Marisa Krysinska, Nicolas Rouche, Fran<;oise Van Dieren and Marie-F'rarn;oise Van Troeye, with the indispensable collaboration (on a voluntary basis) of a dozen teachers and professors of all levels. Needless to say that the leitmotiv is : mathematical education has absolutely to be conceiwd in its natural continuity from the prime to the adult age. The study begins with three chapters of a general character.

The first one analyses the idea of ability (competence in French), a Yery popular notion nowa(lays. It intends to show that (at least in the mathematical field) eYery ability is a capacity to put sunw knowledge to work. Considering abilities in the abstract, apart. from the knowledges, would be nonsensical.

The second chapter entitles Teaching mathematics today: it is a critical survey of which topic� to drop from the curriculum and which ones to introduce. It further explains the main trends and i

The third chapter outlines what might b!;l a minimal curriculum for today's citizens, not those hound to advanced mathematics, but the future consumers, voters, readers of c-;nquiries,reports and statistics, holders of insurance policies, Lotto players, etc. The rest of the book consists in six chapters, all structured alike, successively devoted to magnitudes, numbers, geometry, algebra, data processing with statistics and probability, and at last analysis. To explain the common structure of these chapters, let us take geometry as an example (in what follow:.-;, geometry can be replaced by magnitudes, or algebra, etc.) The chapter consists of two parts, and the first part of six sections. The first section explains what is r 4 geometry at the basic level (an expression b01Towed f om Freudenthal ) : in other words, it highlights the everyday phenomena which appear to be in need of a geometrical explanation, which mobiliS<" geometrical thinking. The title of the second section is The construction of geometry : it 011tli11es (nothing more) a plausible content for the learning of geometry from K to 12. Then comes, as a third section, a glimpse at what geometry is at college and university levels, as well as it.s place and rok in today's research. The fourth section is of particular importance : it tries to e.xplain what geometry is as a form of tlwught, with distinct traits as compared to arithmetic thought, algebraic thought., stochastic thought, etc. The most relevant applications of geometry are outlined in the fifthsection. The sixth and last section explains to some extent what geometry is or has been in history, philosophy and culture. In each of these sections, an effort is made to convey the essentials, of course within a small number of pages. The second part of the chapter consists of three sections : geometry from K to 6, then from 6 to about 9, andlastly from 9 to 12. No one of these sections proposes detailed topics to teach. Each one develops and discusses a few principles which are thought to be appropriate for teaching and learning geometry during the corresponding years. The principles are illustrated. by some typical problematic situations. A kind of continuity is made apparent in the evolution of the principles from one age to the next.

r 2Bernard Honclaire et al. Les matbematiques de la maternelle jusqu'a 18 a.us {Matliema.tics fom K lo .12/, Centre de Recherche sur l'Enseignement des Mathematiques (CREM), Nivelles, 1995, 327 pages, available at CREM, 5 rue Emile Vandervelde, B-1400 Nivelles, Belgium ; price : FEB 300 (about $8) ; not available in bookshops. 3 Another document, written in an analogous intention, is the well-known Curriculum and EvaluationStandards for School Niathematics, National Council of Teachers of Mathematics, Reston, 1987. It seems however that the vertical continuity of the Standards could still be improved since, according to Gail Burri!, president of the NCTM, "there is a need �n build stronger articulation between the grade-level bands." (See Interview with Gail Burril (conducted by Allyn Jackson), Notices of the Am. Math. Soc. 45 (1998), 87-90. 4Hans Fteudenthal, Mathematics as an Educational Task, D. Reidel, Dordrecht, 1973.

39 MATHEMATICAL EDUCATION

Let us repeat that similarly structured chapters are devoted to magnitudes, arithmetic, algebra, statistics and analysis. No doubt this list of subjects will appear natural, or even necessary, except probably for magnitudes. Why magnitudes? In a fewwords, although magnitudes disappearedfrom 20th Century mathematics, they are present by necessity, before being measured, already in the lives of small children, and along with their measures, in the lives of everybody, not to mention all the natural sciences. Allowing room for magnitudes in the mathematical curriculum is merely recognizing their presence at the roots of mathematical thinking. Further, it is a contribution to reducing a harmful gap between mathematics and physics teachers. Another question: why did the authors deal with each subject from K to 12, and not, as in the NCTM Standards, with all subjects from K to 4, followed by all subjects from 5 to 8, and then all subjects from 9 to 12? The only but important reason is that they wanted to emphasize the continuity of each subject through the entire school period. Considering each subject throughout the entire childhood and adolescence resulted for the authors in a strong conscience that teaching is not about putting science as it is into the mind of the pupil, but about working with the pupil as he or she is, while aiming in the long nm at science as it is. A conspicuous example is orientation : for a baby, it concerns distinguishing above and below, before and behind, left and right, the clockwise sense, etc. But there is a continuous thread from these questions to the orientation of the bases in a vector space. What is the expected readershipof this study? Written as far as possible in straightforwardlanguage, a large part of it should be accessible to' the reader without any particular mathematical training. The authors hope that it will allow (1) each teacher to recognize her situation and role between those before her and those after; (2) each future teacher at any level - to appreciate the scope of the matters they have to learn; (3) the teachers of other disciplines to clarify their relation to mathematics and the role of mathematics in general education; (4) the parents to realize where their chidren come from and in which direction they are moving; (5) the students of a certain age and maturity to understand the meaning of their efforts; (6) the curriculum commissions and textbook authors to coordinate their contributions with the preceding and following ones; (7) the managers of education and the politicians to better appreciate the scope and some global implications of their decisions.

On the first page of the book, one reads : "This work was conceived as a source of ideas anda basis for discussion : let us hope that nobody will receive it as a dogma." But the question may be asked : to which extent will the school system be improved as a consequence of this publication ? It is hard to say. One of the authors of the report has already been consulted as an expert for the curricula coordination. The schools for teachers preparation will certainly not be coordinated in the near future, but some of the professors read the report. At last, as far as research is concerned, there is now a group in Belgium (the CREM) where teachers of all levels and mathematicians collaborate closely all the year round. There is some hope. All this however may appear overambitious ! But remember the scope of the study wasimposed bythe Minister. He received the report rather well : 5000 copies were disseminated in the French-speaking school system of Belgium. All criticisms will, of course, be welcomed.

Nicolas Rouche, Emeritus Professor, University of Louvain, Belgium

40 PROBLEM CORNER

Problem Corner Paul Jainta, Werkvolkstr. 10, D-91126 Schwabach, Germany Proposing Problen1s is a Practical Art ... The Vlaainse Wiskunde Olympiade (Part III) A familiar 'face' is honouring the Corner for the third time: The Vlaamse \Viskuude Olympiade {VWO). My thanks go to Prof. Dr Paul lgodt, Katholieke UniYersiteit Leuw,n, Campus Kortrijk, Belgium, the Executive Secretary of the Flanders competition who has provided me with the annual reports on the VWO as regular as clockwork. Consequeutly, this Belgian mathematics contest is almost part of the furniture of the Problem Corner. The latest dispatch received presents the problems of the llth and 12th editions of the VWO together with an overview of some statistics rela!,espectrum of areas of undergraduate mathematics and levels of difficulty. Sometimes there are problems or parts of problems (generally marked with an asterisk) for which the problem-selection committee itself doE>.s not know the solution (see for instance Question 91 which is taken from the set of problems proposfld for the competition in the recent year). Each student intending to engage in the competition is issued with problem-papers through local contacts appointed specifically for this purpose (and at the same time they canvass potential customers for the contest in their local Alma Mater). The table shows all those persons responsible for implementing the competition at Belgian Universities in accordam:c with regulations:

K.U.Leuven Prof. Roger Holvoet Departement Wiskunde Celestijnenlaan 200 B 3001 Heverlee E-mail: [email protected] K.U.Leuven Prof. Paul Igodt Campus Kortrijk Faculteit Wetenschappen Universitaire Campus, E. Sabbelaan 53 8500 Kortrijk E-mail: [email protected]

41 PROBLEM CORNER

Limburgs Universitair Prof. Noel Veraverbeke Centrum Departement Wiskw1de Universitaire Campus 3590 Diepenbeek E-mail: [email protected]. be Universiteit Antwerpen Prof. Jan van Casteren Departement Wiskunde Universiteitsplein 1 2610 Wilrijk E-mail: [email protected]. be Universiteit Gent Prof. Frank de Clerck Vakgroup Fundamentele en Cornputergerichte Wiskunde Galglaan 2 9000 Gent E-rnail: [email protected] Vrije Universiteit Brussel Prof. Francine Grandsard Departement Wiskunde Pleinlaan 2 1050 Brussel E-mail: [email protected]

Only these cited lecturers have the authority to permit participation in the contest. Students who are involved in the competition are normally allowed 6 to 7 weeks for solving the questions, and while brooding over them they are allowed to consult reference or textbooks. The appropriate faculty member additionally watches carefully to ensure that students meet the deadline and he or she collects all 'preparations' and forwards them to the c01mnittee for assessment. Stress is explicitly laid on the fact that participants are not expected to crack all questions set: complete solutions of one problem (or more) are always worth entry.

A praiseworthy search for problems Towards the end of the 12th year of VWO ( 1996-97) the organising committee decided to renovate the competition in order to enhance its attraction. One cannot equate a mathematics contest solely with problem solving but a competition often requires to come up with new questions. Anyone concerned with conducting a national or even regional mathematics competition can tell you a thing or two about that: it's a hard way to earn your living stocking up each year on unfamiliar but appealing and challenging problems. Fortunately, the need for new problem proposals was not so serious as at first. thought. Nevertheless, the organisers from Flanders have made a virtue out of necessity. As a further stimulant for the lure of the VWO, they have established another new contest around it: the 'Zoekerswedstrijd Fernand Goethals' and its purpose is to help 'invent' new sets of problems. The jury will-vet and rank all submitted proposals against well-established criteria (such as authenticity, clarity). Finally, the best submissions will be awarded prizes in a formal ceremony. Of course, members of the jury are prohibited from submitting questions. The executive committee of the VWO wants to make use of the Corner to repeat its call for problems from all interested colleague.s, teachers, professional or amateur creators of Olympiad-like problems, but students at school and their parents are warmly invited to proffer home-made teasers, too. The problem selection committee accepts everything from multiple choice problems to unsettled questions. Yet there is one stringent restriction imposed on all proposals: they have to be set down in plain words and should be suitable for students (of average ability) in years 11 and 12. The selection committee would welcome written questions which become progressively difficult until they are a serious challenge even for those students in the Belgian team at the IMO. Each submission should be accompanied by a solution (multiple choice questions must offer five possible answers each).

42 1 PROBLEM CORNER

I The board of organisation has deliberately set no deadline for submitting problems; the Fernaud Goethals Contest runs continuously. Proposals which are not adopted, for whatever reason, are still eligible for a prize.

The contest year 1996-97 brought the problem selection committee a reasonable muuber of nice questions. However, there was no use for some of them because they drifted in too belatedly to be considered forthe competition about to appear. Such being the case, it is sometimes couveuient. to be able to fall back on previous proposals and the organisers have been re-assessing them; they also cherish hopes that more than a sprinkling of colleagues will engage in 'inventing' questions helpful to the competition. Possibly enthusiasts of puzzles, in addition, might submit their own problems to the committee so that a stock of problems can be accumulated.

There are very many mathematical competitions, at many levels, being given around t.he world each year. As a consequence of these numerous competitions, there have been quite a number of duplications of the problems, either inadvertently or in some cases by direct copying. Also, many contest problems have appeared previously in well known books and journals. In view of all this, competition examination committees now have to be much more vigilant than ever before in settiug their competitions. They must now continually keep abreast of problems set in other competit.ions, and they have to be very careful about duplicating problems from books and journal problem sections. Even if the book or journal used is not so well known, the problems used (',0uld have already been cloned in other compendia and journals that are widely known. To play safe;problf'n1s should either be new or else based on some nice result from some non-recent mathematical paper. This brings us back to the theme of the heading, "Problem Proposing is a Practical Art ...... ".

hl creating problems it is certainly quite helpful to have a good memory and to be observant. The guru of inventing new problems, George Polya, made the analogy of finding a precious uncut stone on the shore and tossing it away as it is not recognised as being valuable. One has to do a certain amount of cutting and polishing before the value of the stone is acknowledged, although an expert usually can get away with just a careful examination. So, to become proficient in problem creating, as in-other non-trivial activity, one must have lots of practice. For example, at the U.S.A Mathemat.ical Olympiad, students in training sessions were required not only to solve challenging problems but. also to submit original, reasonable, proposals for problems. As most of these students had had little or no practice in problem-creation, their first efforts were usually poor. The next quotation aclmowledges this deficiency; it is a fragment froma famous saying coined by George Polya: "Pr·oposing problems is a practical art like, sW'imming or skiing, or playing the piano: you can learn it only by imitation and practice . .. If you wish to learn swimming you have to go into the water, and if you W'ish to become a problem proposer you have to propose proble1ns".

In closing I should like to give the address where your artful ideas will be received with open arms: V1aamse Wiskunde Olympiade Zoekerswedstrijd Fernand Goethals Algemeen Secretariat Universitaire Campus Etienne Sabbelaan 53 8500 Kortrijk The distress signal from Flanders is no isolated occurrence. Even on a world-wide scale (the IMO for example) organisers are sorely in need of fresh ideas. And I am not affected, too. Ideally, one needs to have a huge 'Problem Bank' with enough variety of materials, from relatively easy to moderately difficult. Fortunately, I am not yet at the point of having to declare bankruptcy, for Im fortunately able to take advantage of numerous collections of problems that I have laid in. Nevertheless, the constant struggle to feature a healthy mix of interesting problems in the Problem Corner is becoming more and more arduous. I really need your help!

Before the next cycle of seasons with problems commences, I would like to express my thanks t.o all readers for their every good wish for the Problem Corner. I hope that you all had a wonderful year, and that 1998 will bring you even more joy and happiness - and that the problem section in the Newsletter will continue to be a great source of amusement for you.

43 PROBLEM CORNER

Q. 86 Consider the triangle with Yertices (-6, 0), (0, 12) and (16, 0). How many points with integer coordinates lie on the sides of the triangle and form together with the points (0, 0)

and (5, 0) an obtuse triangle'? (VWO, 1996-971 Flr•t Round) Q. 87 Let a, band c be the lengths of the medians of a right-angled triangle, such that a � b � c. 2 2 0 b Find the numerical value of ; , (VWO, 1996-97, Second Round ) Q. 88 ABG and DAG are two isosceles triangles with BAG 20" A and ADC= 100° (see figure). Prove that AB BO+ (Final Round, 1996-96) *�·"D \ 100'\ I I \ '/ ; \ I Q. 89 The number 1997 is expressed as the sum of some natural I numbers, not necessarily differentfrom one another. "What I \ I \ is the greatest possible value of the sum? (Final Round, I \ I 1996-97) 8 c Q. 90 \Vith respect to an orthononnal basis, we consider the curves determined by the equations x2 + y2 = r2 (r � 0, r ER) and (x.y)2 = 1.

Let Fr denote the convex polygon having the intersection points of these curves as vertices (assuming that they exist). (a) Find an expression f(r) which describes the area of the polygon Fr. (b) For which values of r is Fr a regular polygon?

(Final Round, 1996-97)

or any positive numbers 4 and any positive real numbers a , ••. , a,,. Q. 91 Prove that f n � 1, a2 the following inequality holds:

Please note: there was a printing mistake in Q.85 (Newsletter 26). The quantity in small brackets in the second term of the final expression should be the scalar product of vectors v and F, namely (v.F), rather than their difference. \Ve apologise for this mistake.

Now we are going to proceed to solve the problems given in issues 25 and 26. But before then I am able to greet some new and inquisith·e guests who dared to sniffat this corner for the first time. It is to be wished that they have acquired a taste for it.

Q. 74 At the beginning of a month a shop has 10 different products for sale, each with equal prices. Every day the price of each product is either doubled or trebled. By the beginning of the following month all the prices have become different. Prove that the ratio maximal price/minimal price is greater than 27. Solution (Eoin Coleman, King's College, London) The price of a product at the beginning of the followingmonth is determined by an ordered n-tuple x in which each component is 2 or 3. Since multiplication is commutative, it does not alter the price if we permute the components (pushing the 3s to the left of the vector), so that, without loss of generality, we may assume Xi+i ::::;; x1 for 1 ::::;; i k we must have Z = Xk < Yk = 3 and Xr ::::;; Yr· Suppose now that X = X1 < ... < X10 Y are the different terminal prices of the ten items. Then

44 PROBLEM CORNER

- (. 7 3 ' ::;i, as k(X, X2) < k(X2, X3) < ... < k(Xg, Y). Since -- T2 ) :J > . 3 - , -- 27 tl1e iesu· It , follows.

Q. 75 A sequence a-.i is determined by the rules ao = 9 and for any non-negative k,

Prove that a10 contains more than 1000 nines in decimal notation. Solution (Dr J N Lillington, Winfrith Technology Centre, Dorch.est.er) Computing the first term we find a 3(10 -1)4 + 4(10 - 1)3 and expanding it. by the 1 = binomial theorem gives that the coefficient of 10 vanishes (for -:3 ( i) + 4 ( :: ) '"" 0). Hence we can write a in the form a1 c .102 + 102 1, where c is an integer. Number 1 = 1 1 a1 has the last two terms in its decimal representation as 99. Similarly, 2 + 2 4 + 2 2 3 a2 3(c1.10 (10 -1)) 4(c1 .10 + (10 - 1)) 2 4 2 3 · 3((c1 + 1).10 - 1) + 4((c1 + 1).10 1) hasthe property that the coefficentof (c1 + 1).102 vanishes when expanded by the binomial 4 4 theorem. Hence we can write a2 c2.10 + 10 -1, where c2 is an integer. 2n 2,. Proceeding inductively, an Cn .10 + 10 - 1 for all integers n. 10 1024 1024 Specifically, for n = 10, 2 1024 and a10 = c10.10 + 10 1 with the last 1024 t.ernm in its decimal representation equal to 9.

Q. 77 Fifteen elephants stand in a row. Their weights are expressed by integer numbers of kilograms. The sum of the weight of each elephant ( except the one on the extreme right) and the doubled weight of its right neighbour is exactly 15 tonnes. Determine the weight of each elephant. Solution (Dr J N Lillington) Let the weights be n1, .. . , n15. Then n; + 2n;+1 = 15, 1 ::; i < 15. (*) it follows from (*) that n; is odd for each i, 1 :s;i < 15. Then

n1 = 1 ==> n2 7 ==> na 4, which contradicts (*) n1 =3 ==> n2 = 6, which contradicts (*) n1 5 ==> n; = 5 for 2 ::::; i < 15 n1 = 7 ==> n2 = 4, which contradicts (*) n1 = 9 ==> n2 = 3 ==> na 6, which contradicts (*) n1 11 ==> n2 = 2, which contradicts (*)

n4 = 4, which contradicts (*) again

The weight of the first 14 elephants is 5 tonnes. It is not possible to deduce the weight of the elephant on the far right.

Q. 79 A square is constructed on the side AB of triangle ABC ( outside the triangle). 0 is the centre of the square. M and N are the midpoints of the sides BC and AC. The lengths of these sides are a and b respectively. Find the maximalpossible value of the stun OM +ON (when the angle ftt C changes).

45 PROBLEM CORNER

c \ Solution {Dr J � Lillington, slightly modified by the editor) b /

N /, \M Let L be the foot of the perpendicular through O on AB. .. . 1\ a ,/'\\ .· \ Lis the midpoint of side AB. Put AB = c. Then LM % A/ / I L.· .1 ,9 OLM A+ 1· Applying the cosine law to triangle c/ and \\ ll I OLA! ==> \, I ,if·v 0 2 2 2 ., � b bcsinA c . LM +oL--2LM.OL.cosOLM= + + forcos(A+f) sinA. 4 4 By sine rule on triangle ABC, c sin A. a sin C. By cosine rule on triangle ABC, c? a2 + b2 2abcos C. Hence,

b absinC a2 b 2 abcosC a2 b2 ab OM2 : + + + + + (sin C-cosC). 2 4 2 2 Now, sin c-;;o·c = sin Ccos f �cos C sin :f = sin ( c - :f) attains its maximum value of 1 at · 3 C - :f = ; or C = ;. Hence the maximum value of sin C - cos C is ../2. / 2 It follows that the maximum rnlue of O A·P = + � + � ( 1 + ./2) and therefore OM t + �· Similarly, ON= ! + ==;. OM+ON !( ../2l).(a + + b).

Q. 80 How many positive integers n with 1 s n:::; 1996 exist that solve the equation

([x] is the greatest integer not exceeding x.) Solution (Niels Bejlegaard, Stavanger, Xorway) Since [:1:] s x with equality only when x is an integer, the given equation has only solutions when n 12q, q an integer as well as n (or n = 0 mod 12). With 1996 = 12.166 + 4 there are 166 positive integers n, 1 :5. n :5 1996 satisf:ying the equation. Also solt•ed by Dr J.N. Lillington

Q. 81 Let a and b be non-negative real numbers. Determine all sets of solutions {x1 ,x2, ... ,x96} of the following systems of equation (all x, non-negative integers):

X1 - QJ.'3 = b

X1 + X2 + .. , + Xgs 96

Solution (Niels Bejlegaard) (Ed. I have to make an apology for a misprint in the last row of the system. Instead of b there must come the integer 96. Owing to this mishap, the solution has become trivial.)

46 PROBLEM CORNER

Addition of all rows but the last one gives

96 90 Lx; - a Lx; 96b ==> b(l - a) 96b (**). i=l i=l

If b =I O we arrive at a = -95 in contradiction with a 2:: 0. Therefore b = 0. Since I::�! x; b, we have I:;�! x; 0, but :r, 2:: 0 means that the only solution is 1 X95 1 a. x1 = x2 = 0 irrespective of there is no soltition when b i= 0 under the given restrictions. (Ed. Thus far Bejlegaard. I want to complete his reasoning. From (**) we can deduce that b a+ b. a and must satisfy the relation 1 Then obviously (:r 1, :r2, ... , ;rgo) ( l, 1, ... , 1) is a solution of the system of simultaneous equations (for Xi E N). Otherwise the seL of solutions is empty. Rider. The solution of system (*) turns out the be more difficult if we alter the domain of the x;'s. So one can refrmulate question 81 as follows: Determine all real roots of system (*) if x;, 1 :5 i :5 96, are non-negative real n·umher.s. I'm keen on your new solutions!)

That completes the Corner for this isslle. Send me your nice solutions and contest materials. Finally, you are invited to propose problems for which readers will send in solutions. Proposals should, whenever possible, be accompanied by a solution, references, and other insights which are likely to be of help for the editor. They can be anything from elementary to advanced, from easy to difficult. Original problems are particularly sought. So, please submit any interesting problems you come across, especially those from (problem) books and contests that are not easily accessible. But other interesting problems may also be acceptable provided they are not too well known and references are given as to their provenance. J hereby invite my readers to share them with their colleagues and students. I welcome your input, most especially problem sets and solutions for future use!

Oberwolfach Prize for Young European Scientists

Verein zur Forderung de.s Mathematischen Forschungsinstituts Oberwolfach (Friends of Oberwolfach) and Gesellschaft fiir mathematische Forschung are awarding a prize for excellent achievements in Stochastics The prize amounts to DM 10,000. Candidates must be nominated, come from Europe and be under the age of 35. All European senior scientists in Stochastics are requested to submit proposals by :n May 1998 to: Matthias Kreck Mathematisches Forschungsinstitut Oberwolfach Schwarzwaldstr. 9 - 11, 77709 Oberwolfach

47 'I!

REVIEWS

BRIEF REVIEWS·

Edited by Ivan Netuka and VladimirSoucek. Books submitted for review should be sent to the following address: Ivan Netuka, MUUK, Sokolovska 83, 186 75 Pmha 8, Czech Republic.

M.Giaquinta, S.Hildebrandt: Calculus of Varia­ another extension is explained. The book relies on tions II. The HamiltonianFormalism, Grundlehren the notion of a vector lattice and the corresponding der mathematischen Wissenchaften 311, Springer­ theory is developed in order to describe property Verlag, Berlin, 1996, xxix+652 pp., 82 fig., DM spacesof integrable functions and spaces of measures 198.00, ISBN 3-540-57961-3 as well. Extension of Daniell space is the main The second volume of this excellent survey of the method for constructing an integral. However, a calculus of variations consists of two parts, Part III measure theoretic approach is compared with the "Canonical Formalism and Parametric Variational Daniell-type method. The context of the theory, as Problems" and Part IV "Hamilton Jacobi Theory treated in the book, makes it possible to develop a nice measure theory on Hausdorff topological spaces. and Canonical Transformations", and they are P covered by Chapters 7 - 10 of the book. In Chapter Two chapters deal with L spaces. Spaces of signed 7 the authors describe an important involutory measures are also studied, the main tool being a transformation, which is used to derive a dual vector lattice approach. Elements of the theory picture of the Euler-Lagrange formalism and the so­ of functions of bounded variation and absolutely called canonical formalismof field theory. It forms a continuous functions are included in the last chapter. part of Hamilton-Jacobi theory presented in Chapter The book is well written and contains a large number 9 together with a brief introduction to symplectic of exercises. It can be used as a textbook for geometry. The connection between canonical ( or a university course in integration. The book is symplectic) transformations and Lie's theory of recommended to mathematics teachers as well as contact transformations is discussed in Chapter students and can also serve for a second reading to 10, where relations between Fermat's and Huygens those who have already studied integration based on principles are also investigated. Cauchy's method another approach. (in) of integrating partial differential equations of first A. W.Knapp: Lie Groups Beyond and Introduc­ order by the method of characteristics is described, tion, Progress in Mathematics, vol.140, Birkhauser, and connections of this technique with Lie's theory Basel, 1996, 656 pp., sFr 78.00, ISBN 3-764-33926-8 are also illustrated here. The field theory for parametric integrals corresponding to Weierstrass The theory of Lie groups, Lie algebras and their rep­ field theory for non-parametric variational problems resentations have had an important influence on all considered in Chapter 6 is presented in Chapter 8. fields of mathematics and theoretical physics dur­ As in the first volume, the reader can find here ing the last century. The classification of complex detailed introductions to each chapter and section semisimple Lie algebras and of their finite dimen­ of the book, many examples, historical references sional representations has already become a stan­ and comments. Beside the vast bibliography dard part of a good mathematical education. There the authors completed the book with a useful is a certain number of well written textbooks avail­ overview of textbooks on the classical calculus of able on this subject. A study of infinite dimensional variations. Both broad and deeply rooted contents representationsof reductive Lie groups is much more and remarkably clear style of presentation make the difficult, although a huge amount of important re­ book highly recommended for all analysts. (jsta) sults has beenaccumulated after a half century of in­ tensive research. Prerequisites necessary for an un­ W.Filter, K.Weber: Integration Theory, Chap­ derstanding of these results are Cartan's structure man & Hall Mathematics Series, Chapman & Hall, theory of real semisimple Lie algebras and groups London, 1997, xii+294 pp., GBP 35.00, ISBN 0-412- and various forms of decompositions of reductive 57680-5 Lie groups. However these topics cannot be found A Daniell space is a triple consisting of a set (with one exception) in standard textbooks on Lie X, a vector lattice L of real-valued functions on algebras and Lie groups. The important feature X and a null-continuous positive linear functional of the present book is that it starts from the be­ on L. Unlike the traditional approach, the book ginning (with only a very modest knowledge as­ presents an extension of a Daniell space in two steps. sumed) and covers all mentioned important topics- The second extension gives nothing new in the a­ the connection between Lie groups and Lie algebras; finite case. However, there are examples showing a complex semisimple Lie algebras, their universal en­ weakness of the classical construction which stops veloping algebras and their finite dimensional repre­ after introducing the closure and a justification for sentations; compact Lie groups and their represen-

48 REVIEWS

tations; Cartan andIwasawa decompositions, Vogan & Applications, Birkhauser, Boston, 1996, xviit339 diagrams and classification of simple real Lie alge­ pp., DM 108.00, ISBN 3-764-33877-6, ISBN 0-817- bras; reductive Lie groups, I

49 REVIEWS

book starts with a very nicely written introduction This a paperback classic dating back to 1927 explaining basic facts about integrable 5Ystems in and reprinted many times since! Nevertheless, general as well as specific features and aims of it is still very valuable book containing essential the book. In the Appendices, the reader can find facts of analysis (basics of real analysis and of a summary of facts used in the book (Poisson complex function theory, asymptotic expansions, structure.s, the AKS theorem, complex curves and Fourier series, ordinary differential equations in their Jacobians, Prym varieties). This small book the complex domain and integral equations) in a has very rich contents which can be recommended short, condensed and precise form approved by to any mathematician or physicist interested in the years of practical use. The second part of the subject. (vs) book describes in detail the main special functions A.Krause, M.Olson: The Basic of S and S-Plus, used in applications and mathematical physics Statististics and Computing, Springer-Verlag, New (Gamma function, zeta function, hypergeometric York, 1997, xiv+242 pp., 34 fig., DM 48.00, ISBN functions, Legendre, Bessel, Mathieu and Lame 0-387-94985-2 functions, elliptic and theta functions). A wealth of S is a powerful tool for interactive data examina­ mathematical ideas with a touch of old time.s make tion, creation of graphs and implementation of cus­ this book a pleasure to read. ( vs) tomized routines based on the S language. This B.Hughes, A,Ranicki: Ends of Complexes, Cam­ books gives a solid but quick introduction to the bridge Tracts in Mathematics, vol.123, Cambridge S environment. If you have never used S before, this University Press, Cambridge, 1996, xxv+353 pp., book will get you up to speed quickly, as it is full of GBP 45.00, ISBN 0-521-57625-3 examples and insights into how S works, Although This is a highly specialized monograph which is very the book is primarily intended for the S novice, the clearly written and made as accessible for the reader material covered extends to more advanced topics as possible. Its short title not specifying the notion and contains many hints beneficialalso to those who of complex comprises two aspects of the theory: already have some knowledge of S or are not aware of namely, the complex means here either a CW­ the many enhancements that have been made to the complex in topology or a chain complex in algebra. system over the past few years. The structure of the Accordingly, we meet ends of CW-complexes and book, with its detailed exerdses and solutions at the ends of chain complexes which represent the end of each chapter, reflects its origin from lecture algebraic counterpart of the topological situation. notes and homework exercises. The best way to use The authors have succeededin presenting practically the book is to read it while sitting at the computer a complete theory of ends of all kinds and their and repeat all of the examples. As the authors state, interrelations which are relevant to the topology of forgetting an overview two days of readingwould be high dimensional manifolds. The book contains a sufficient, while working through the examples will great amount of information, many new results and take surely much more time. Using the book to lay also new approaches. It is absolutely indispensable the foundations, the reader will be able to use S with for any specialist in the field. The authors declare all its basic capabilities. (jan) that they are assuming some familiarity with high P.llibenhoim: The New Book of Prime Nwnber dimensional manifold theory and with the standard Records. 3rd ed., Springer-Verlag, New York, 1996, applications of algebraic K- and L- theory to xxiv+541 pp., DM 88.00, ISBN 0-387-94457-5 manifolds. But beyond these prerequisities they The book is an extension and an improved version introduce and explain almost all necessary notions. of the 2nd edition of "The book of prime number In the text are to be found many historical remarks records" published in 1989. In six chapters and that clarify the development of the theory of ends on more than 500 pages, almost all problems which and help in their understanding. At the end of the were studied in connection with prime numbers are book there are also short appendices describing the discussed in detail. The author carefully describes history of end spaces and the history of wrapping the historical evolution of the problems, including up. The references extend to 170 items. (.iva) last results obtained using computer techniques. J .Casey: Exploring Curvature, Friedrich Vieweg The book contains almost 70 pages of a carefully & Sohn, Braunschweig, 1996, xv+291 pp., 141 fig., compiled bibliography, index, as well as 53 tables in $ 49.00, ISBN 3-528-06475-7 text. It is a nicely readable book which will be useful It is easy to use a formula to compute the curvature both for students and specialists. (bn) of an analytically given curve. Nevertheless, what is E.T.Whittaker, G.N.Watson: A Course of the geometrical meaning of the curvature and how is Modern Analysis. Fourth Edition, Cambridge it obtained by measurements? Likewise, in the case Mathematial Library, Cambridge University Press, of a surface - how can we estimate the curvature Cambridge, 1996, 608 pp., GBP 29.95, ISBN 0-521- of a surface if we only have its model? We get the 58807-3 length of a curve from the length of broken lines by

50 REVIEWS

a limiting process; the case of the surface area is axiom of .determinacy, applications of forcing). The more difficult. What is the deformation of a surface monograph contains both useful and interesting and how does one demonstrate it? Such problems historical notes and a guide to the bibliography. are dealt with in this book. The author shows The book covers major areas of modern set theory: how, with help of experiments and measurements, cardinal arithmetic, constrnctible sets, forcing we can obtain an understanding of the geometrical and Boolean-valued models, large cardinals and meaning of several notions of elementary differential descriptive set theory. It contains results on various geometry. Physics helps us in the case of minimal topics, i1icludi11g properties of the reals, infinitary surfaces. The author presents the well-known "soap combinaLorics, ultrapowers, Suslin's Problem and solution" of Plateau's problem: there is a surface Martin's Axiom, measurabk cardinals, axiom of with a minimal possible area among all surfaces determinacy and many others. The book is aimed with boundary equal to a fixed closed curve. In primarily at graduate sfaidents and researchers in set the reviewer's opinion, it is important to show a theory. It can be used as a textbook for a graduate relationship with physical reality both to motivate course in set theory and also can serve as a reference mathematical assumptions and to present questions book. (pp} from which theorems arise. The book is very H.McKean, V.Moll: Elliptic Curves: Function useful, especially for teachers, and serves to make Theory, Geometry, Arithmetic, Cambridge Uni­ their students acquainted with the interrelations versity Press, Cambridge, 1997, xiii-1-280 pp., GBP between mathematical ideas and real things. The 40.00, ISBN 0-521-58228-8 only drawback of the book is a certain immber of This book is a full and sufficiently detailed incorrect figures (e.g. on pp.158, 161) - they are introduction t.o the theory of elliptic curves the false from the point of descriptive geometry. On beauty of which has bewitched mathematicians the other hand, the reader will appreciate many at least since the last ceutmy. The book is historical comments. Special chapters are devoted divided into seven chapters: 1. First ideas: complex to C.F.Gauss, B.Riemann and T.Levi-Civita. (lbo) manifolds, Riemann surfaces, and projective curve.s; T.Jech: Set Theory. Second Corrected Edition, 2. Elliptic integrals and functions; :t Theta Perspectives in Mathematical Logic, Springer­ functions;4. Modular groups and modular functions; Verlag, Berlin, 1997, xiv+634 pp., DM 168.00, ISBN 5. Ikosaeder and the quintic; 6. Jmaginai:y quadraLic 3-540-63048-1 number fields; 7. Arithmetic of elliptic curves. This The monograph is devoted to a detailed study of indicates that all three pillars, complex function axiomatic set theory. It contains comprehensive theory, geometry and arithmetic, on which t,he material about all main topics. Chapter 1 is about theory of ellipLic curves rests are represenLt'line. The descriptive set through the Black-Scholes model, and up Lo recent theory is studied (Borel and analytic sets, Baire developments in the numerical analysis of such space, projective sets, constructible universe, the American put options. Unfortunately, this useful

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survey is spoiled by a completely wrong numbering A large number of exercises accompanies each of references. (jd) cliapter. Besides the main text, additional material A. van der Poorten: Notes on Fermat's Last is often included in order to provide the reader Theorem, Canadian Mathematical Society Series of with further results or information. A nice Monographs and Advanced Texts, J.Wiley & Sons, feature of the book is the inclusion of historical Inc., New York, 1996, xv+222 pp., GBP 35.00, ISBN notes and comments giving a competent historical 0-471-06261-8 perspective of the subject. There are also many Based on the author's lecture series at Macquarie original quotations fromimportant work on measure University in Sydney, the book covers, in its 17 and integration, and short biographies of the chapters, the history of the solution of Fermat's most important personalities are also included last theorem from first results, through Kummer's (e.g. Borel, Radon, Caratheodory, Hausdorff, work, up to the proof of the theorem worked Vitali, Levi, Young, Fatou, Lebesgue, Fubini and out by A.Wiles. This is a very attractively Tonelli, F.Riesz, Fischer, Jegorov, Hahn, Nikodym, written publication containing extended remarks, Lusin, Haar). The textbook can be recommended comments and stories that complement every to mathematics students and also to teachers of chapter, illustrate the evolution of the problem and mathematical analysis who will appreciatean access are often more interesting (for beginners as well as to facts not easily available in such a form in specialists) than the contents themselves. The book traditional textbooks on the subject. (in) is very warmly recommended to everybody. (bn) J.R.Schott: Matrix Analysis for Statistics, Wiley J.Stoyanov: Counterexamples in Probability. Series in Probability and Statistics, J. Wiley & Sons, Second EdJtion, Wiley Series in Probability and Inc., New York, 1997, xii+426pp., GBP 50.00, ISBN Statistics, J. Wiley & Sons, Inc., Chichester, 1997, 0-471-15409-1 xxviii+342 pp., GBP 55.00, ISBN 0-471-96538-3 Matrix methods are among to the most important The author started to collect counterexamples in tools used in statistics. Their role in statistical 1970. The first edition of this book which appeared theory has been even more enhanced by the use in 1987, has become very popular and today it is a of computers in data analysis. The book unq.er classic reference work. In the second edition, some review brings an accessibleand modern background examples are replaced by more attractive ones and in matrix analysis required by statisticians. new examples are included. The book contains 25 Introductory chapters review elementary matrix sections grouped into the following parts: 1. Classes algebra and vector spaces. Further topics are of random events and probabilities. 2. Random eigenvalues, eigenvectors, matrix factorizations, and variables and basic characteristics. 3. Limit matrix norm. The chapter about generalized theorems. 4. Stochastic processes. Undoubtedly, inverses is mostly devoted to the Moore-Penrose every teacher needs examples to clarify that some generalized inverse. The chapter devoted to systems sufficient conditions are not necessary, that some of linear equations is oriented to the least squares necessaryconditions are not sufficient, and that the method. Then the author deals with Kronecker converse to a statement is not true. This book product, vec operator; Toeplitz matrices, matrix is a very important source of such examples in derivatives and some other subjects. The last probability theory. (ja) chapter describes some properties of quadratic forms J ,Elstrodti Maf3..und lntegrationstheorie,Grund­ which are useful in statistics. Some theorems are wissen Mathematik, Springer-Verlag, Berlin, 1996, stated with a reference substituting the proof (e.g., xvi+398 pp., DM 58.00, ISBN 3--540-15307-1 Theorem 3.12, which is then used as a tool for This is a carefully written textbook which covers proving continuity of eigenvalues). Each chapter in detail the most important topics in measure contains a section with problems. The book can theory and integration. The contents are more be used as a course of matrix theory for students of or less standard: set systems (including semirings, statistics. (ja) mqnotone classes, Dynkin systems), set functions R.B.Bapat, T.E.S. Raghavam Nonnegative Ma­ (Fourier transform, £P -spaces, convergence An integrated treatment of the theory of nonneg­ of sequences of measurable functions, absolute ative matrices with the emphasis on its applica­ continuity (measures, functions), measures on tions to game theory, combinatorics, optimization topological spaces (regularity, Kisiiiski extension and mathematical economics. About half of the theorem, Riesz representation theorems, Haar material in the book presents standard topics in a measure on locally compact topological groups). novel fashion. The remaining portion brings many

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new results in matrix theory. The book is divided proach data analysis and offered the possibility to into seven chapters. The material begins with the perform penetrating analyse.s routinely. The book is basics of the subject such as Perron-Frobenius the­ intended both for would-be users of S as an introduc­ ory and shows the connections with game theory, tory guide and forclass use. The level of a course for M-matrices and the theory of finite Markov chains. which it is suitable ranges from the upper years of Doubly stochastic matrices are treated in the sec­ an undergraduate course to the Master's level. Full ond chapter. Some matrix inequalities as for in­ exercises are provided only for Chapters 2-5. Gen­ stance information inequalities, further Hadamard, erally, this book shows its reader how to extend S, Fiedler-Oppenheim, Levinger-Kingman inequalities and uses this facility to discuss the procedures not and some related problems are studied in the third implemented in S, thereby providing fairly extensive chapter of the book. Chapter 4 is devoted to some examples of how this might be done. Throughout., results concerning conditionally positive matrices the emphasis is on presenting practical problems and and chapter 5 contains some topics in combinatorial full analyses of real data sets. Many of the methods matrix theory (e.g. matroids, a new proof of Alexan­ discussed are state-of-art approaches to topics such droff inequality, Coxeter graphs, matrices over the as linear and non-linear regression models, robust max-algebra, Boolean matrices). Some scaling prob­ and smooth regression methods, survival analysis, lems and their applications are surveyed in Chapter multivariate analysis, tree-based methods, t.ime se­ 6. The last chapter investigates the application of ries, spatial statistics and classification. (jan) special matrices in economic models. The Perron­ P.Hilton, D.Holton, J.Pedersen: Mathematical Frobeniustheorem is used to explain both the Leon­ Reflections. In a Room with Many Mirrors, tief and Sraffasystem; further, the linear expanding Undergraduate Texts in Mathematics, Springer­ economy and the behaviour of market prices are in­ Verlag, New York, 1997, xvi+351 pp., t:l4 fig., DM vestigated. The treatment of the material is rigor­ 49.00, ISBN 0-387-94770-1 ous and almost all the resultsare completely proved. Fibonacci, Fermat and Mersenne numbers are well­ The book is aimed at first year graduate students known; not so much Euler and Lucas ones. The book and researcherswith a minimal background in linear under review deals with all these numbers but this algebra and advanced calculus, although familiarity is not its main purpose. As explained in the preface, with the basics of linear programming and statistics the authors want to show what mathematks is, what will be helpful in understanding some of its sections. it is about, and how it is done. Mathematics must (kz) be taught so that students understand how and why W.N. Venables, B,D,Ripley1 Modem Applied mathematics is done by those who do it with succe..ss. Statistics with S-Plus. Second Edition, The authors present a picture of mathemal,i.e.'l as Statististics and Computing, Springer-Verlag, New an exciting, stimulating activity not as a set of York, 1997, xvii+548 pp., 144 fig., DM 88.00, ISBN procedures set in stone. They use several topk.s 0-387-98214-0 to illustrate their ideas (number theory, fractal S is a powerful tool for interactive data examina­ geometry Koch snowflake, decorative geometry - tion, creation of graphs and implementation of cus­ frieze and wallpaper patterns, cardinality - pigeon­ tomized routines based on the S language. Since hole principle, Schroder-Bernstein theorem). From the first edition of this book in 1994, S has contin­ time to time you find a HRRA K to test. your ued to grow in popularity, due also to the fact that understanding of the matter jm,t present.e.d. At t.he its Windows version now offers equivalent capabili­ end of some chapters there is a FINAL BREAK tiesto the versions running on UNIX workstations. which is followedby a list of references and answers. The second edition of the book is designed for users The last chapter is quite different fromthe preceding of S 3.3 and later. It was written whilst S for MS ones. The authors formulate here certain priueiples Windows was under development, and covers what of mathematical pedagogy. "Never be pedantic the authors see as the most exciting new features. but sometimes be precise" or "Geometry plays a This is not a text in statistical theory, but does special role in mathematics". (lbo) cover modern statistical methodology. After the in­ R.R,Hocking: Methods and Applications of troduction to the S language and the system itself, Linear Models. Regression and the Analysis of there are twelve chapters, each of them covering one Variance, Wiley Series in Probability and Statistics, part of modern statistical methodology. Each chap­ J. Wiley &Sons, Inc., New York, 1996, xxii+731 pp., ter summarizes the method discussed, in order to GBP 70.00, ISBN 0-471-59282-X set out the notation and the precise method imple­ This book provides a thorough presentation of linear mented in S. However, the main aim is rather to model analysis. In the first part the author defines show how to analysedatasets using S. In doing that, the essential distributions encountered in linear the authors show how the availability of a powerful model analysis and derives their basic properties. and graphical system has altered the way people ap- The methods of simultaneous inference are then

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explained. The second part is concerned with source for Ph.D. theses. The present book shows regression models and gives a detailed description that there are also some other reasons based of building regression models, estimating the on real life problems supporting the importance parameters, testing for violations of a model, of this fundamental branch of mathematics. transforming data, dealing vvith outliers, influential Motivating the reader by the construction of observations and collinearity. Calibration, robust the so-called Ramanujan graphs appearing in regression and nonlinear regression are also briefly network communications, the author moves on to mentioned. The theory is accompanied by several important branches of number theory in deeply analysed real-data examples illustrating the a concise form. These parts taken by itself are problems commonly met in practice. The third usually considered "only" of theoretical importance. part is devoted to the analysis of variance models. Chapter 1 recapitulates basics about finite fields, The fixed effects models are studied, including and Chapter 2 is devoted to the Weil conjectures one-way classification, two-way classification with on zeta functions of projective varieties over finite and without interactions, and with balanced or fields. In Chapter 3, the reader is introduced to unbalanced data, and nested models. The local and global fields and adelic language. In remainder of the book develops mixed effects Chapter 4 the Riemann-Roch theorem is proved, and models and describes the AVE method for variance Bombieri's approach to counting points on curves component estimation. I consider this book over finite fields is presented. Analytic behaviour invaluable for graduate students in statistics of the zeta and £-functions attached to idele class and applied statisticians for its comprehensive characters is proved in Chapter 5 based on the explanation of the basic methodologies and the philosophy of Tate's thesis using Fourier analysis on presentation of examples which help to avoid the adelic groups. Chapter 6 is devoted to estimates common misinterpretations of the data. (sk) of various character sums needed for construction of B.Flury: A First Course in Multivariate Ramanujan graphs based on estimates of roots of Statistics, Springer Texts in Statistics, Springer­ £-functions attached to character of the idele class Verlag, New York, 1997, xiii+713 pp., 141 fig., DM group of a function field. Chapter 7 summarizes 148.00, ISBN 0-387-98206-X classical modular forms, including Hecke operators, This book gives a thorough introduction into the £-functions, and the theory of newforms. The area of multivariate statistics at an intermediate reader finds here how Hasse conjecture implies level. After reviewing basic statistical concepts, the Taniyama-Shimura conjecture. The content of the author introduces the multidimensional normal dis­ last two chapters is characterized by the author as tribution and spherical and elliptical distributions. follows: "Automorphic forms and representations The methods of parameterestimation are explained, are discussed in Chapter 8. There we give an concentrating mainly on maximum likelihood esti­ adelic interpretation of the classical modular forms, mation, and the EM algorithm for incomplete data which naturally leads to the adelic definition of is also presented. The central topics of the book automorphic forms and representations for GL(2). are discriminant analysis and classification, includ­ Then we survey the J acquet-Laglands theory of local ing logistic regression. The problem of statistical in­ and globalrepresentation for GL(2) and quaternions ference for means is approached via maximum like­ groups. ..."In Chapter 9 we apply what we learned lihood, the union-intersection principle, and boot­ to give explicit constructions of Ramanujan graphs; strapping. The linear principal component analysis on the other hand, we obtain some information on is presented using self-consistency principle. The fi­ the distribution of eigenvalues of a Hecke operator nal chapter gives an introduction to finite mixture using the limit of the measure attached to certain analysis. Factor analysis and cluster analysis are family of graphs arising from quaternion groups. not presented in the book. The book contains many (sp) exercises, real-data examples, and the data files and A.C,Zaanen: Introduction to Operator Theory in the software instructions fileswritten in S-PLUS and RieszSpaces, Springer-Verlag, Berlin, 1997, xi+312 GAUSS are available via ftp. The book is suitable pp., DM 98.00, ISBN 3-540-61989-5 for advanced undergraduate students in statistics This is an excellent detailed textbook dealing and for a broad class of researchers in other fields. with ordered spaces and operators in these spaces. (sk) Assuming only a knowledge of basic notions in W.C.Winnie Li: Number Theory with Appli­ functional analysis and measure theory, the author cations, Series on University Mathematics, vol. 7, starts with algebraic theory of Riesz spaces ( chapters World Scientific, Singapore, 1996, x+229 pp., GBP on Lattices and Boolean algebras, Riesz spaces, 39.00, ISBN 9-810-22226-2 Ideals and bands, Archimedean spaces, Projections Edmund Landau said that the number theory and Dedekind completeness), and continues with is immortal because it provides an inexhaustible normed Riesz spaces. A major part of the book

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is devoted to linear operators on Riesz spaces. of crests, for most of them ensued from apparently Here are some highlights: order bounded and order spontaneous relations of the zeta-function with a continuous operators, order duals and biduals, the variety of Eisenstein series. Nevertheless such Radon-Nikodym theorem, Freudenthal's spectral glimpses are highly suggestive of a gnmd view theorem, functional calculus, the Hahn-Banach over and far beyond the Eisenstein ridge, and theorem in normed Riesz spaces,the Krein-Rutman bring forth the notion of a kamuy-m.intar where the theorem, a spectral theory of positive operators. entire collection 0f automorphic forms contribute The textbook includes exercises (many with a hint to the formation of the zeta-function. My aim in for the proof) and a short list of other books on the the present monograph is to try to substantiate subject. This hook is warmly recommended to both this belief by demonstrating that the :1..eta�function graduate students and experts in the field. (jl) has indeed a structure tightly supported by all P.-L.Lions: Mathematical Topics in Fluid automorphic forms. The story begins with an Mechanics. Volume I. Incompressible Models, unabridged treatment of the spectral resolution of Oxford Lecture Series in Mathematics and its the non-Euclidean Laplacian, and continues to a Applications, Oxford University Press, Oxford, theory of trace formulas. The fundamental means 1996, x+237 pp., GBP 29.95, ISBN 0-198-51487-5 thus readied are suhseqnently mustered np for the Various continuum mechanics models of complex guest to find an explicit formula for the fourth motions of incompressible and compressible fluids power moment of the zeta-values. Then l,he zet.a­ prove to be a neverending source of deep function emerges as a magnificent peak embracing mathematical problems. Treatment by P.L. Lions infinitely many gems calledautomorphic [.,-functions in two volumnes deals with essentially three representing the spectrum." Some experience with fundamental questions: (i) global existence of integrals involving transcendental functions will (weak, renormalized) solution, (ii) its· uniqueness help the reader, and as the author says "ample and (iii) regularity. The attention is focussed knowledge" of the properties of the zeta-functions is on evolutionary equations, for which three kinds also welcomed. The first three chapters of the book of 'boundary problems' are investigated: (i) the (I. Non-Euclidean harmonic; 2. Trace formulas, and Cauchy problem, (ii) the space periodic problem, 3. Automorphic £-functions) are intended as an and (iii) the Dirichlet problem. The first volume introduction to the subject. The fourth chapt,er summarises and proves significant results relating "An explicit formula" is devoted to the theory of the to various incompressible models. The book power moments Zk(g) = 3D J:°°"1((1/2 + il.)j2kg(l)ideal fluids, i.e., Lie Groups and Special Functions, Mathematics Euler equations, density-dependentEuler equations, and Its Applications, vol.316, Kluwer Academic and hydrostatic approximations. The volume is Publishers, Dordrecht, 19-95, xvi+497 pp., GBP completed by five appendices interesting on their 151.50, ISBN 0-792-33210-5 own. Everybody who is interested in studying Both authors published a comprehensive three mathematical questions arising in fluid mechanics volume treatise with the same title and in the sa:'me should read this book. (joma) series over the period 1991-93. A continuation of Y.Motohashi: Spectral Theory of the Riemann this series describing recent results is presented here. Zeta-Function, Cambridge Tracts in Mathematics, While reading this book, it is often necessary to vol.127, Cambridge University Press, Cambridge, have the previous volumes nearby. The subject, 1997, ix+228 pp.,' GBP 29.95, ISBN 0-521-44520-5 is the same a description of many fundamental The aims of the book are perhaps bestdescribed by facts in the theory of various special functions from a quotation from the author's preface: "Ever since the point of view of representation theory. This is Riemann's use of the theta transformation formula a very natural and efficient wa,y of systemat,ically in one of his proofs of the functional equation explaining various properties of different classes for the zeta-function, number-theoretists have been of special functions. A very interesting part fascinated by various interactions between the zeta­ of the book discusses in detail the properties function and automorphic forms. These experiences, of decomposition of the tensor product of two howeve�, have remained episodic like rare glimpses irreducible representation of a compact group into

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irreducible parts, and related Clebsch-Gordan and sion {)a(u)/fJt - liu 3 f, a doubly nonlinear para­ Racah coefficients. A substantial part of the bolic problems of the type 80.(u)/at div')'(v'u) 3 book contains a discussion of various generalizations f, a(8u/8t) - div')'(v'u) 3 f, or even more general of hypergeometric functions. The connection of ones, are investigated. The book continues with the hypergeornetric functions on Grassrnanians to the classical Stefan problem and its generalizations con­ Radon transform on k-dimensional subspaces in sidering undercooling or superheating, with Muskat Rn is described at the end of the book. Various and Hele-Shaw problems, a phase-field model, nu­ recent generalizations of symmetric polynomials and cleation and growth, mean curvature flow, Stefan­ the so called h-harmonic polynomials (where h is Gibbs-Thomson problem, micromagnetics, etc. The a suitable function depending on a choice of a main ingredients are both mathematical (Sobolev Coxeter group G) are other topics described in spaces, weak solutions, monotonicity method and its the book. There is an extensive bibliography ( 450 generalizations) and physical ( continuum mechan­ items) and bibliographic comments. The book ics, equilibrium and non-equilibrium thermodynam­ includes re.cent results and will be useful for both ics). Basic mathematical tools are summarized in mathematicians and physicists interested in the appendices, which makes the book self-contained in theory of applications of special functions. (jhu) this rnspect. This clearly written monograph will be J,J,Kinney: Probability, An Introduction with found useful by applied mathematicians, both ex­ StatisticalAp plications, J.Wiley & Sons, Inc., New perts and advanced students, and mathematically York, 1997, xiv+513 pp., GBP 21.55, ISBN 0-471- oriented physicists and engineers working in contin­ 12210-6 uum mechanics and thermodynamics. (trou) This live textbook has some remarkable features B.Fine, G.Rosenberger: The Fundamental that could differentiate it from other introductory Theorem of Algebra, Undergraduate Texts in courses on probability theory. Though the readers Mathematics, Springer-Ver lag, Berlin, 1997, xi + 208 are assumed to have some background in algebra pp., DM 59.00, ISBN 0-387-94657-8 and calculus, computational aspects are relieved As explained in preface, the book is based on two by using modem computer algebra system like courses delivered to the upper level undergraduates Mathematica and the readability and the meaning in US and to a class of potential (gymnasium � of results are emphasized. Various applications high school) teachers. It contains a description to manufacturing processes, especially to the of six proofs of the theorem ( calculus of functions quality control and quality production appear in of two variables, complex function theory, algebra the text. Headlines are as follows: Sample and topology form the background for it) in nine spaces and random variables, Discrete random chapters and other five (two based on complex variables and probability distributions, Continuous function theory and three topological proofs) are random variables and probability distributions, described in Appendices; also the first of four Gauss' Functions of random variables, generating functions proofs is included. The book brings a good portion and statistical applications, Bivariate probability of a non-trivial mathematics ranging from Cauchy­ distributions, Recursions and Markov chains. Topics Goursat theorem to Galois theory and Brower fixed from statistics, like acceptance sampling, reliability, point theorem, of course more in a form of account confidence intervals and hypotheses testing as well and sometimes even in very simplified version. as simple linear regression are integrated throughout Inaccuracies in standard terminology: (re,gion is any the book. Each chapter is followed by a summary subset of C, domain is a connected open region and exercises and problems at various level of where connected means arcwise-connected versus difficulty; the readt>.:rs with an access to Mathematica compact domain and continuity in a region (pp. 36- computer system will findvery useful appendix with 7), definition (5.1.1) (p. 53), Morera's theorem commands and hints how to use this system to solve (p. 71) to name a few)and seldom used "proofs" can examples and graph the pictures appeared in the assure prospective teachers that mathematics could text. (zp) be also a vague talk about it. This is illustrated by A, Visintin: Models of Phase Transitions, a "proof" of the Cauchy integral formula (p. 66). A Progress in Nonlinear Differential Equations and version of Cauchy theorem for Star 3D starlike) Their Applications, vol.= 28, Birkhiiuser, Boston, domains is presented in one of Appendices but it 1996, vii+322 pp., DM 138.00, ISBN 3-764-33768-0,.. barely corrects the overall view. The choice of the ISBN 0-817-63768-0 central theme is rewarding and the book in hands of The volume presents mathematical analysis of var­ an experienced teacher will do a good job but for a ious macroscopic, mesoscopic, and two-scale mod­ beginner could do a disservice. (jve) els of phase transitions based on nonlinear par­ F.Smithies:. Cauchy and the Creation of Complex tial diferential equations and inequalities. Start­ Function Theory, Cambridge University Press, ing from a (possibly degenerate) parabolic inclu- Cambridge, 1997, 216 pp., GBP 35.00, ISBN 0-521-

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59278-X the verge of a cancer epidemic?... ) and also some A nicely wTit.t.en book which brings more detailed valuable indications concerning the corresponding information and sometimes corrections to a part software ( e.g. Power Functions on the TI-82 or TI- of the history of holomorphic (analytic) function 83: Graphing Calculator Techniques for Chapter 9). theory like the one presented by Markushevich. The There are some 1ninor points in the book which on book closes the activity in research of Cauchy's e could criticise: The letters in the figures on pages works performed by the author during past 20 years. 554 and 555 are too small to be clearly legible and 2 More than 140 references (45 to Cauchy's works) "taking the square root of 4a " (p. 461) might show the care which was given to the book and deserve some e.xplanation. The reader will find make it a w1ique source not only for the historians many refreshing and non-traditional moments in the of mathematics but to all who are fascinated by the book (like the title Algebra Aerobics (used many beauty of the complex function theory. The author times) or exercises like "Read the chapter entitled does not omit the description of the blind paths "The Cosmic Calendar" from Carl Sagan's book The and hence helps to the reader really understand Dmgons of Eden (p. 817)). The "Explorations in its development. Analysis of the Cauchy's activity College Algebra." are written very well, contain au in this field in the period 1814 - 1831 and his unbelievable quantity of problems and ideas from way to some versions of statements nowadays called the real world and I would like to emphasize that. Cauchy-Goursat theorem and theorem on residua. they are written with a visible love for the subject. (jve) And also with a good taste: "Mathematic,:; is a L.A.Kime, J.Clark: Explorations in College filter, a way of looking at the world. .Just as Algebra. Preliminary Edition, J.Wiley & Sons, musical training increases your awareness of sounds, Inc., New York, 1997, xv+694 pp., GBP 16.99, ISBN or knowledge of history gives a deeper perspective on 0-471-10699-2 the present, mathematics heightens your perception This book represents a new approach to the teaching of underlying order and systematic pat.terns in and learning of the college algebra (in fact, it is numerical information." (p. xi). l think that. you meant to be a course on this subject). A simple would also sign these words and I am nearly sure enumeration of the topics covered by the text that you would surely accept such a course of college could be misleading (linear and quadratic functions, algebra for your own children. (lbe) exponential functions, logarithms, polynomials, ... ) . A,J,Chorin: Vorticity and 'lurbulence, Applied Some quotations from the first pages of the book Mathematical Sciences, vol.103, Springer-Verlag, will quickly correct such judgements: "· · · This book New York, 1994, 174 pp., 45 fig., DM 76.00, ISBN has been developed by a consortimn of schools · · · 0-387-94197-5, ISBN 3-540-94197-5 The course materials have now been tested in fifteen This is an interesting textbook on the mathematical different schools · · · The text has been developed theory of turbulence, reflecting both dassical and refined over the last three years by expertsfrom Kolmogorov theory and its revision based on many different disciplines." Teachers and their pupils the author's original contributions. The book can find interesting problems "· · · that require you starts with the equations of incompressible fluid to reasoncritically · · · ." Some examples: How to find mechanics, and representation formulas for the models to describe data ( e.g. the relationship between energy and velocity in terms of vorticity, while the life expectancy and per capita gross domestic Chapter 2 provide.s an introduction to probability product (p. 83) where one uses the data from 114 and random flows. After introducing the different countries (see the table on pages 510-512) ), appropriate tools, a discussion of the Kolmogorov estimating linear models for data, deep time and deep qualitative theory is presented. This theory space ( e.g. Hubble's Law holds true across the known can capture basic properties of fully developed universe, p. 312), modeling epidemics: an iterative turbulence, but there are many detailed properties approach to exponential functions ( e.g. ". . . Is this a of turbulent flows due to intermittency t.hat are realistic model? Can a disease spread this fast? The inconsistent with Kolmogorov's theory. The bubonic plague · · · called the black death · · · killed remaining chapters are devoted to the vortex about 14% of the population of London ···We still dynamics of turbulence with the aim to building have concerns about epidemics today, like the AIDS a theory consistent bqth with the Kolmogorov and Ebola virus- · · "). At the end of the book one can prediction and with intermittency. Chapter 4 find Anthology of Readings (a sample: Two articles "A concentrates on two dimensional case (serving here Fragmented War on Cancer" and "Promote Cancer more as an illustrative example of an application Treatment, Not Cancer Phobia" from the New York of statistical theory in fluid mechanics), while the Times (May 29, 1996). These articles are related three-dimensional vortex motions are studied in to Exercise 27 (p. 48): What questions are raised the following chapter; the main difference comes [in the first article] about the claim that we are on from vortex stretching, and non-conservation of

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vorticity and entropy. In Chapter 6, further tools of by a chapter on probabilistic and statistical aspects statistical mechanics of polymers, percolation, real­ of cluster analysis (H.-H.Bock). There is a wealth space renormalization, and self-avoiding random of bibliographical data, more than 1500 papers or walks are introduced to be deployed in the analysis books quoted. The book is justwhat it was intended of vortex equilibria in three dimensions. This for: a thorough survey of up to date state of art of analysis is presented in the last chapter where the classification and clustering. (jmac) connection with Kolmogorov theory is explained. B,Kutzler: lnlproving Mathematics Teaching (joma) with DERIVE. A Guide for Teachers, Chartwell­ S.Gong: Integrals of Cauchy Type on the Ball. Bratt, Bromley, 1996, 185 pp., sFr 14.50, ISBN 0- Volume I, International Press, Cambridge, 1993, 862-38422-2 iii+304 pp., ISBN 1-571-46005-5 This book, a subtitle of which is "A Guide for Varioustypes of kernels (Cauchy-Fantappie, Cauchy­ Te achers", is an introduction to using the computer Szego), the corresponding singular integrals and sin­ algebra system DERIVE for teaching mathematics. gular integral equations have been extensively stud­ The text consists of several parts. In one of ied in the theory of several complex variables during them the author explains how DERIVE can be last decades. Due to historical circumstancies, the used for teaching mathematics alongside traditional study of these problems in China was developed in curricula. In other one he thinks about future and sixties and seventies independently, without a close develops a new concept of teaching ma.thematics, in contact with the rest of world. This book collects re­ whicha computer algebra system plays an important sults obtained there during that period; they have a role. Besides these more theoretical parts the book minimal overlap with other results. Two chapters of contains a chapter, in which the author shows the book are devoted tothe study of certain systems practically how it is possible to use DERIVE for of singular integral equations, an attempt to gener­ teaching of several topics of mathematics. There alise Hadamard's 'parties finies'. The· subsequent is an interesting and useful chapter Tips and two chapters contain study of the Cauchy-Szego in­ Tricks too. No a priori knowledge of DERIVE is tegral on certain Cartan domains of a higher rank; required because the first chapter is devoted to an kernels considered here have higher dimensional sin­ introduction to this computer algebra system. At gularities. The book contains interesting new re­ the end of the book we can also finda chapter which sults. (vs) provides a look "inside" computer algebra systems. P.Arable, L.J.Hubert, D.De Soete (Eds.): The book is very nicely written and reconunended Clustering and Classification, World Scientific, to every mathematics teacher. (ml) Singapore, 1996, ix+490 pp., GBP 44.00, ISBN 9- T,O.Moore, E.H.Hadlock: Complex Analysis, 810-21354-9, ISBN 9-810-21287-9 Series in Pure Mathematics, vol.9, World Scientific, This book is a collection of ten papers on cluster Singapore, 1991, x+391 pp., GBP 23.00, ISBN 9- analysis, problems of classification and multidimen­ 810-20246-6, ISBN 9-810-20247-4 sional scaling. The papers included in the volume As explained in the preface the text was written to are written by outstanding contributors to the field; serve both undergraduate and graduate students in they are organized in such a way that they provide engineering, physics, and mathematics. The style of a compendious scholarly review of it. The volume Definition Theorem Proof is used and proofs begins with an introduction (J.A.Hartigan) explain­ are carefully presented in details. The content of ing its scope and organization. The following four the book is more or less standard and includes also chapters are devoted to combinatorial data analysis approx. 100 pages on conformal mapping and the (P.Arabie, L.J.Hubert), hierarchical classifications Riemann Mapping Theorem. Hannonic functions (A.D.Gordon), theory and methods of a hierarchical are treated in about 40 pages. The style is clear, the classes model with some applications (S.Rosenberg, authors try to choose the most economical path to I. van Rosenberg, P. de Boecke) and tree models arrive at results, providing readers with all details for proximity data (G. de Soete, J.D.Carrol). The which must be omitted when the allocated time next chapter brings an analysis of computational for the course is not sufficient. A lot of examples aspects of the application of different algorithms. are solved in the text; a good number of exercises Then there is a chapter on neural networks for clus­ presented in the text will help students to apply the tering (F.Murtagh), an alternative approach to the material explained. Weak points of the book are problem. Two chapters that follow treat the devel­ those parts where the dependence on the intuition opment and advances of clustering and classification is too high (e.g. a counter clock oriented contour methods in two countries: Japan (Akinori Okada) is the one for which its interior remains on the and Russia (B.G.Mirkein, J.Muchnik). The penulti­ left while the parameter varies from a to b); the mate chapter is devoted to clustering validatation winding number (index) should be defined more by simulation (G.W .Milligan). The volume ends precisely. A mixed approach to treatment of curves

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(a subset I( of C is a simple closed (Jordan) curve school up to the 70's. The book under review iff there is a parametrization f : [a, b] -+ K which is presents contributions of the Chine..se school, based continuous, one-to-one on [a, b) and f(a) = 3D f(b)) on articles published mostly in Chinese language sometimes introduces unnecessary complication in up to 1990. The book can easily be used the exposition. Nevertheless, it is a good multi­ (and was used) in one-year comse for seniors or purpose book on complex analysis to have in the graduate students, it covers the main parts of library. For a course with higher aspirations, this the theory: Cauchy-type integrals, Holder space.s, book must be supplemented by additional material Plemelj formulas, singular integral equations for which can be found elsewhere. (jve) closed contours, Riemann, Hilbert and mixed L,C,G.Rogers, D.Tala:y (Eds.): Numerical boundary problems, periodic, doubly periodic and Methods in Finance, Publications of the Newton quasi-periodic problems, boundary value problems Institute, Cambridge University Press, Cambridge, for systems of functions and systems of singular 1997, x+326 pp., GBP 35.00, ISBN 0-521-57354-8 integral equations, and boundary problem..'l with This book contains 16 papers written by specialists shift. The mathematics underlying the presentation in financial mathematics, probability theory, and is classical. Small part of the hook is devoted to numerical analysis. The papers are based on numerical approximations of singular kernels and t.o lectures given at the session 'Numerical Methods the weighted quadrature fonnula.s for Cauchy-type in Finance' at the Isaac Newton Institute in 1995. integTals and it is only the first step to the numerical The reader should have a basic knowledge of the solution of boundary problems. (zv) t�eory of stochastic processes and finance. The E,B,Saff, V.Totik; Logarithmic Potentials with reader approaching the book from the numerical External Fields, Gnmdlehren der mathemat.iscben point of view will find numerical solutions to Wissenschaften, vol.316, Springer-Verlag, Berlin, both partial differential equations (PDE's) and 1997, xv+505 pp., 18 fig., DM 158.00, ISBN 3-540- stochastic differential equations (SDE's) usually 57078-0 used in finance; that finite difference methods, The book studies the extension of classical viscosity solutions of PDE's, Monte Carlo methods, logarithmic potential theory to the case when there lattice methods, etc. Most of the papers pre.sent is a weight (external field) present. The book a large variety of numerical methods used for the is a self-contained introduction t.o the "weighted" computation of option prices (European, lookback, potential theory as well as to its numerous Asian (also called option on the average) and, applications. As a by-product the classical theory especially, American) with various modifications of logarithmic potential is fully developed. The (with or without continuous dividends, with external field problem has its origin in the work constant or stochastic volatility etc.) and ntunerical of C.F. Gauss, and is sometimes referred to as the solutions to portfolio management strategies. Three Gauss variational problem. The book presents a papers are devoted to backward SDE's, their comprehensive treatment of the Gauss variational applications in finance, and numerical solutions problem in the plane. The review of properties of to them. The reader will also find statistical harmonic functions is contained in the Chapter 0. procedures, identification of model= s or simulation Chapter I on weighted potentials presents a detailed of the market in other papers. The articles cover treatment of Frostman type for the existen<'.e and both mathematical matters and practical issues uniquene.ss of the extremal measure (including the in numerical problems. Thus the book can be weight function on an unbounded set, the energy strongly rec01mnended to economists, probabilists, problem, the minimum principle, t,he Dirichlet and applied mathematicians working in finance. problem, the extremal measure, the equilibrium (mk) potential, the fine topology and the continuity Jian-Ke Lu: Boundary Value Problems for of equilibrium potentials, the weighted capacity). Analytic Functions, Series in Pure Mathematics, Chapter II on recovery of measures, Green fund.ions vol. 16, World Scientific, Singapore, 1993, xiii+466 and balayage includes the problem of recovering a pp., GBP 54.00, ISBN 9-810-21020-5 measure from its potential, the unicity theorem, This book is the English version of the original the Riesz decomposition theorem and the principle book which was published in Chinese, Shanghai, of domination, Green's functions and the balayagc 1987. Theory of boundary proble1ns for analytic measures, and Green's potential. Motivations ftmctions and the related theory of singular integral and detailed proofs for many of the basic results equations belong to important branches of complex from potential theory are provided. Chapter analysis and have many applications in mechanics, III on weighted polynomials presents weighted physics and engineering (fra-:ture problems). The Fekete points and weighted Chebychev polynomials. famous monographs of Muskhelishvili and Gakhov Chapter IV on determination of the extremal have collected results of the Georgian and Russian measure uses the random matrix techniques (some

59 _, I

REVlEWS

problems from physics included). Chapter V algebra, and a sound geometrical imagination. The uses the extremal point method for solving some exposition of the book turns around the notion of problems (Dirichlet problem, determination of the so called connection number in a near-linear Green's function and conformal maps). Chapter VI space ( or partial plane in M.Hall 's terminology). deals with weights on the real line (with different Thus in the first chapters the combinatorial theory weights). Chapter VII contains some applications of finite geometries is built up, based on finite concerning orthogonal polynomials. The last sets of points and lines. Then the basic theory Chapter VIII is about signed measures. Appendix of affine and projective spaces is developed. The A contains studies of the Dirichlet problem and second half of the book is devoted to structures harmonic measures (regularity, Poisson formula). which have been introduced in the recent past. Appendix B deals with weighted approximation in The heading of the chapters here are Polar spaces, the multidimensional case ( extending the external Generalized quadrangles, Partial geometries, and field to the pluripotential setting). All chapters are the finalchapter Blocking sets, asalready indicated. provided with clear notes and historical references Every chapter is accompanied by up to fifty to help reader to trace the development of the examples which either can serve to consolidation of field. The book is very well written and can be the reader knowledge, or contains additional related recommended to anyone interested in logarithmic material. The book is written in a very lively and potentials with external fields. (pp) readable style, and must be recommended as the M.Machover: Set Theory, Logic and their Lim­ firstreading for everybody interested in this area of itations, Cambridge University Press, Cambridge, combinatorics and geometry. (sp) 1996, ix+288 pp., GBP 14.95, ISBN 0-521-47493-0, A.S.Fokas, I.M.Gelfand (Eds.): Algebraic As­ ISBN 0-521-47998-3 pects of Integrable Systems. In Memory of Irene This book arose from two of the author's courses Dorfman, Progress in Nonlinear Differential Equa­ on the subject for undergraduate students of tions and Their Applications, vol.26, Birkhiiuser, philosophy and students of mathematics interested Boston, 1996, viii+350 pp., DM 168.00, ISBN 3-764- in philosophy. Hence the book can be understood 33835-0, ISBN 0-817-6:3835-0 as a general introduction. But being written This is a collection of papers on various prob­ by an excellent mathematician, there are no lems in the theory of integrable systems. The ma­ misunderstandings or omissions of justifications of jority of contributions deal with algebraic prob­ assertions which are common in many popular lems connected with such systems. Among top­ books. After a short discussion concerning ics discussed are multi-Hamiltonian structures, the mathematical induction, a course of axiomatic Sato's metho= d of r-function formatrix hierarchies, theory is developed. Then the course of vVDVV system of the associativity equations, the mathematical logic with the standard contents correspondence between classical modular forms and (propositional calculus, first order logic, recursion w DO's, billiard solutions on Riemannian manifolds, theory and !imitative /Godel's/results) is presented. master symmetries, discrete integrable systems, as­ I very much like the way the author explains things, ymptotic integrability, invariance under Laplace­ especially the description of recursion using the Darboux-type. transformations, nonultralocal Pois­ intuitive concept of computability followed by exact son brackets, trace formulas and canonical I-forms. description using Matyiasvic theorem (n-ary domain Most of these contributions are related to work of /or natural numbers/ is recursively enumerable iff Irene Dorfman who died recently. The whole vol­ it is the projection /m-ary domain/ of an n + ume is dedicated to her. (jbu) m.-ary relation described by the equality of two H.-M.Krolzig: Markov-Switching Vector Autore­ polynomials) seem to me to be very efficient for the gressions. Modelling, Statistical Inference, and mentioned type of students. (kc) Application to Business Cycle Analysis, Lecture L.M.Batten: Combinatorics of Finite Geome­ Notes in Economics and Mathematical Systems, tries. Second Edition, Cambridge University Press, vol.454, Springer-Verlag, Berlin, 1997, xiv+357 pp., Cambridge, 1997, xiv+193 pp., GBP 16.95, ISBN 0- 49 , DM 106.00, ISBN 3-540-63073-2 521-,59014-0, ISBN 0-521-59993-8 Univariate autoregression and vector autoregression This is a second edition of an introductory text on (VAR) models are standard tools in time series the combinatoria,l theory of finitegeometries. This analysis. A new class of models is based second edition contains completely new chapter on on the assumption that the parameters of the the so called blocking sets in linear spaces with model depend on an unobservable regime variable. indications of their application in game theory, In Markov-switching (MS) models the regime is testing of statistical sampling of surveys and generated by a discrete-state homogeneous Markov cryptography. The reading of the book requires chain. The book is a revised version of author's only a basic experience with set theory and linear dissertation. It contains a description of MS-VAR

60 REVIEWS

model, its state-space representation, methods for followed by a satisfactory list of references. (tk) forecasting, filtering, and smoothing, maximum­ D.Acheson: From Calculus to Chaos. An likelihood estimation, procedures for model selection Introduction to Dynamics, Oxford University and model checking, and other topics. The author Press, Oxford, 1997, ix+269 pp., GBP 25.00, ISBN presents results of application of MS models to the 0-198-50077-7, ISBN 0-198-50257-5 German, global, and international business cycles. The book is intended as an introduction to The last part of the book is devotedto cointegration elementary calculus built on applications of the analysis of VAR models with Markovian shifts in classical calculus. Using carefully selected examples, regime. Univariate MS models were analyzed iu from the time of Newton to the present day, several papers and monographs. The reviewed book t.he author demonstrates in a highly readable provides a systematic approach to the modelling of form what we usually call the mathematization dynamic systems based on vector MS models, from of physical problems. Tiiese are mainly of which solutions for special problems are derived. the dynamical nature as the title and subtitle (ja) indicate. This selection simultaneously opens the M,Hazewinkel (Ed.): Handbook of Algebra. door for computer orientated readers to start Volume 1, North-Holland, Amsterdam, 1996, their own experimentation with modelling of real xix+915 pp.,$ 187.50, ISBN 0-444-82212-7 processes. The first two chapters recapilnlate The first volume of the intended multi-volume some basic results from the elementary calculus Handbook consists of 8 more or less randomly chosen and its application to particle motion. The l,hird Sections (Linear Algebra; Linear (In )dependence; chapter is devoted to the first and second order Fields, Galois Theory and Algebraic Number linear and nonlinear differential equations. The Theory; Generalizations of Fields and Related simple methods for their solution are demonsl.rnt.cd, Objects; Category Theory; Homological Algebra, among others, on an epidemic model or harmonic Cohomology, Cohomological Methods in Algebra, oscillators. The fourth chapter centres on the Homotopical Algebra; Commutative Rings and Newton method for solution of differential equations. Algebras; Associative Rings and Algebras) and However, the reader finds here an improvement of 24 Articles (Van der Waerden conjecture towards Runge-Kutta method. In the fifth chapter and applications; Random matrices; Matrix the reader encounters various forms of pendulum, equations, Factorization of matrix polynomials; oscillations and the notion of the linearization. The Matrix functions; Matroids; Higher derivation sixth chapter is devoted to the planetary motion Galois theory of inseparable field extensions; and the two and three-body problem. The reader Complete discrete valuation fields, Abelian local is introduced to the partial differenlial equaLion::; class field theories; Infinite Galois theory; Finite through waves and diffusions in the seveut.hchapter. fields and their applications; Global class field The following chapter "The best of all possible theory; Finite fields and error correcting codes; worlds" revolves around the calculus of variations. Semi-rings and semi-fields; Near-rings and near­ In the chapter "Fluid flow" fundamentals of the fields; Topos Theory; Categorical structures; theory of viscous flow are developed. The chapter The cohomology of groups; Relative homological "Instability and catastrophe" is based on the algebra, Cohomology of categories, posets and previous one with the linear stability theory and coalgebras; Homotopy and homotopical algebra; bifurcations. The eleventh chapter "Nonlinear Derived categories and their uses; Ideals and oscillations and chaos" demonstrates the sensitivity modules; Polynomial and power series rings, Free to initial conditions. The reader finds here among algebras, firs and semifirs; Simple, prime and others, the van der Pol and the forced Duffing semi-prime rings; Algebraic microlocalization and equation, or description of the Poincare-Bendixon modules with regular singularities over filteredrings; theorem. The last chapter is devoted to some Frobenius rings). This means that the articles wider aspects of the pendulum motion. 11. are published as they arr1ve from the authors. In includes the author's own research on the inverted some cases, the level is almost undergraduate, but pendulum. The book ends with recommendations basically a graduate-level background is needed. for further reading, with Appendix A on elpmentary To understand well the topics covered, a reader prograrning in QBasic. Appendix B contains listings is assumed to be acquainted with the material of of ten programs related to themes dealt with in this an opulent textbook of Algebra. The Handbook book The industrious reader will find answers to will certainly provide professional mathematicians the exercises following each chapter at the end of with sufficientinformation on the particular parts of the book. (sp) Algebra contained in the firstVolume (and hopefully W.Neutsch: Coordinates, Walter de Gruyter, in the other ones to come). Unfortunately, only a Berlin, 1996, xiii+1365 pp., DM 398.00, fewtopics fromthe Subsections mentioned above are ISBN 3-110-14852-8

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The main purpose of this huge book (more than part (more than 500 pages) contains a systematic 1300 pages) is to offer a comprehensive description review of differential, lliemannian and projective of the most useful and practical coordinate systems geometry and complex analysis; various descriptions in low dimensions. The theoretical description of the group of rotations by means of quaternions, and the corresponding tables form the last two octonions, or more generally Clifford algebras; parts of the book. For each set of coordinates, and a description of the theory of Coxeter and the reader can find in the tables key information Weyl groups. The chapter on lattices contains ( singularities, coordinate surfaces, volume element, a discussion of elliptic functions and modular metric tensor, Christoffel symbols) and formulae for forms, theta functions, lattices and linear codes, the most important operators (gradient, divergence and the chapter on spheres describes harmonic and the Laplace operator). The first six parts and spherical functions and presents two special of the book contain a summary of mathematics numerical integration methods. The whole book needed in the later theoretical description and in is clearly meant for practical use and should be of some practical applications to problems of efficient value mainly to engineers, physicists and applied numerical integration on spheres. The largest mathematicians. (jbu)

UNIVER SITY OF CA MBRIDGE

DEPARTMENT OF PUR E MATHEMATICS AND MATHEMATICAL STATISTICS University Lecturer or Assistant Lecturer in Pure Mathematics

Applications are invited for this post in any fieldof Pure Mathematics to take up appointment from 1 October 1998.

Further particulars may be obtained from the Head of Department, DPMMS, 16 Mill Lane, Cambridge CB2 lSB (telephone (01223) 337996, fax (01223) 337920, e-mail [email protected], or from the Department's website at http://www.dpmms.cam.ac.uk).

Applications should be sent to the Head of Department and should include a curriculum vitae and e-mail address, list of publications, and the names, postal and e-mail addresses of 3 referees. Candidates must ask their referees to send their reports direct to the Head of Department, to reach him by the closing date.

The closing date for applications is 24 April 1998. The University follows an equal opportunities policy and has a policy on arrangements for part-time work.

62 From Kant to Hilbert:. An Invitation to A Source Book in the FoundatioQ$f' ·, Arithmetic Geometry of Mathematics, Two Volu Dino Lorenzini William Ewald "... it is ce11ai11ly well prepareda the way to a profound stlldy of the c subject." Zenlralblatt fur Mathematlk Vol. I: 694 pp; Val.II: 706 pp, 1996 0-19-853271-7�-�---.,;i Hardback£195.00 Geometric Stabil Anand Pillay " ... a much needed and hi recommended textbook deep, and actively d�ve ZentralblattIUr Mathematlk Oxford Logic Guides No. 372 pp, 1996 0-19-853437-X £60.00 � ll:J,a,1,:-11&1,Mo Area, Lattice P Exponential Sum M.N.Huxley "This book [is] very detailed .. is verywell written. It is an excell ction to important work for all mathematicia who deal with exponential sums and er Bases lattice point theory. It is accessible to illiam W. Adams and Philippe graduate students beginning research." oustaunau Zentralblatt fiir Mathematik This book provides a leisurely and fairly London Mathematical Society Monographs No. 13 comprehensive introduction to Grabner 506 pp, 1996 bases and their applications .. 0-19-853466·3 £85.00 Hardback Graduate Studies in Mathematics No. 3 � SALE PRICE £42.00 289 pp American Mathematical Society Vertex Algebras for 1994 Revolutions in Beginners 0-8218-3804-0 £25.00 Hardback Victor Kac � SALE PRICE £18.00 Mathematics This is an introduction to algebraic aspects of Edited by Donald Gillies conformal field theory, which in the past Fixed Points Twelve leading experts in the history of decade has revealed a variety of unusual Yu A. Shashkin mathematical notions. Vertex algebra theory mathematics describe major advances in This book presents a readable exposition of provides an effective tool to study them in a mathematics, and examine, from a variety of fixed poinf theory. theoretical perspectives, whether these should unified way. Mathematical World No. 2 be considered to be revolutionary. University Lecture Serles No. 10 86 pp American Mathematical Society 140 pp American Mathematical Society 1997 364 pp, 1995 1991 0-19-851486-7 £26.00 Paperback 0-8218-0643-2 £18.50 Paperback SALE PRICE £12.50 0-8218-9000-X £11.95 Paperback � SALE PRICE £17.00 � � SALE PRICE £9.95

Wavelets: Over 70 recently published mathematics Mathematical An Analysis Tool titles at huge discounts available from Impressions M. Holschneider good booksellers throughout Europe as well as directly from OUP. Anatolii T. Fomenko "This book is a self-containedand original For your complete salecatalogue A Soviet mathematician with a talent for text on the theo1y of wavelets ... Graduate Tel: +44 (O) 1865 267426 expressing abstract mathematical concepts students of mathematics and physics will Fax: +44 (O) 1865 267782 or through artwork,Fomenko's work can be findit useful, as will interested readers from Email: [email protected] interpreted and appreciated in various ways otherfields." Physics World mathematical, aesthetical, or emotional. Oxford Mathematical Monographs or visit our website: http://www.oup.eo.uk/sale/math98 184 pp American Mathematical Society 436 pp, 1995 1990 0-19-853481-7 £45.00 Hardback Please note the LMS or AMS members 0-8218-0162'7 £14.99 Hardback � SALE PRICE £27 .00 discount does not apply to sale books' � SALE PRICE £11.99

Prices and extentsare subject to change. OXFORD UNIVERSITY PRESS EMS398 Innovation • Excellence • Tradition Reedition du Seminaire N. Bourbaki Annees 1948 a 1968 10 Exposes 1 a 346 9 10 volumes, 4400 pages 8 5 3

EX 1 s

EX

EX AN EX M SEMINAIRE BOURBAKI soc 1948/51 soc ANNE ES EXPOSES 1-49 soc

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SOCIETEMATHEMATIQUEDE FRANCE Reedit.es et rendus plus lisibles, les exposes 1 a 346 du seminaire couvrent de fac;on assez complete l'ensemble des progres realises en mathematiques pures pendant les tres fecondes annees 1948 a 1968 naissance de la geometrie algebrique moderne, developpement de la theorie des groupes de Lie et de leurs espaces homogenes, des esp aces fibres, des methodes cohomologiques, de l'analyse harmonique, naissance de la theorie moderne des formes automorphes, etc. Ils restent a tout jamais un ouvrage dereference pour cette periode de l'histoire mathematique.

Pour· tout renseignement, contacter Societe Mathematique de France Maison de la SMF, avenue de Luminy, BP 67 1327 4 Marseille cedex 09, France. Fax : 04 91 4117 51 mail : [email protected] ...IN MATHEMATICS _,,. oe>CIM

go T.A. Springer nd Jordan Algebras and Theorems in Analysis I Algebraic Groups Series. Integral Calculus. 1997. VIII, 170 pp. C*-Algebras Softcover DM 68,- Theory of Functions. and W*-Algebras 1997. XII, 386 pp. ISBN 3-540-63632-3 SoftcoverDM 68,- ISBN 3-540-63640-4 G. P6lya, G. Szego Problems and Theorems in Analysis II Jordan Algebras Theory of Functions. Zeros. and Polynomials.Determinants. S. Sakai Algebraic Groups Number Theory. Geometry (*-Algebras and 1997. XII, 320 pp. Softcover DM 68,- W*-Algebras ISBN 3-540-63686-2 1997. X, 256 pp. Softcover DM 68,­ From the reviews: ISBN 3-540-63633-1 "Thework is one of the real classics of this century; it has had much influ­ From the reviews: From the reviews: ence on teaching, on research in sev­ "This book is an excellent and "... presents an important and eralbranches of hard analysis, partic­ comprehensive survey of the the­ novel approach to Jordan alge­ ularly complex functiontheory, and it ory of von Neumann algebras. It bras. Springer's work will be of has been an essential indispensable includes all the fundamental service to research workers famil­ source book forthose seriously inter­ results of the subject, and is a iar with linear algebraic groups ested in mathematicalproblems. valuable referencefor both the who findthey need to know These volumes contain many extra­ beginner and the expert:' something about Jordan algebras ordinary problems and sequences of Math. Reviews and will provide Jordan alge­ problems, mostly fromsome time "The specialist in this and allied braists with new techniques and a past, well worth attentiontoday and areas will findthe wealth of new approach to finite-dimen­ tomorrow. Written in the early twen­ recent results and new approach­ sional algebras over fields:' ties by two young mathematicians of es throughout the textespecially American Scientist outstanding talent, taste, breadth, per­ rewarding?' American Scientist "By placing the classification of ception, perseverence, and pedagogi­ "The title of this book at once Jordan algebras in the perspective cal skill,this work broke new ground suggests comparison with the two teaching of mathematicsand of classificationof certain root in the volumes of Dixmier and thefact systems, the book demonstrates how to do mathematical research. that one can seriously make this thatthe structure theories asso­ Bulletin of the American comparison indicates that it is a ciative, Lie, and Jordan algebras Mathematical Society far more substantial work that are not separate creations but others on this subject which have ratherinstances of the one all­ recently appeared" encompassing miracle of root Bull. Lond. Math. Soc. systems ...." Math. Reviews

Problems and Theorems in Analysis I ••••••••••

Strltt, Problems lnltgml Caltulu1. Thet11yofFurt

Springer-Ver1ag, P. 0. Box 3113 401 D-10643 Berlin,German y. Gha.4709/MNT/Vh Journal of Group Theory The Journal of Group Theory is devoted to the publication of original research articles in all aspects of group theory. Articles concerning applications of group theory and articles from research areas which have a significant impact on group theory will also be consid­ ered. Volume 1 • Issue 2 • 1998 M. ASCHBACHER Near subgroups of finite groups G. R. ROBINSON Further consequences of conjectures like Alperin's K. COOLSAET A construction of the simple group of Rudvalis from the group U3(5):2 S. LI The number of conjugacy classes of non­ normal cyclic subgroups in nilpotent groups of odd order P. J. CAMERON On an algebra related to orbit-counting S. A MORRIS, V. PESTOV A topological generalization of the · Higman-Neumann-Neumann theorem Y. SYSAK Radical modules over hyperabelian groups of finite torsion-free rank V. S. GUBA Polynomial upperbounds for the Dehn function of R. Thompson's group F

Managing Editor: J. S. Wilson, Birmingham EditorialBoard: A.]. Berrick • A. V. Borovik• M. Broue• K. A. Brown• R Buekenhout• F. de Giovanni• R. Gobel• R. L. Griess, Jr.• N. D. Gupta• T. 0. Hawkes• A. A. Ivanov• E. I. Khukhro • L. G. Kovacs• V. D. Mazurov • F. Menegazzo• S. A. Morris• A. Yu. Olshansk.ii• C. W. Parker• I. B. S. Passi• R. E. Phillips• D. J. S. Robinson• R. Schmidt• Y. Segev• A. Shalev • W. J. Shi• S. Sidki• D. M. Testerman• B. A. F.Wehrfritz

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