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December 2004 Issue 54
ISSN = 1027 - 488XERCOM
ICMS Edinburgh
EMSMathematical WeekendPrague 2004
FeatureSymmetries
p. 11
SocietiesPolskie TowarzystwoMatematyczne
p. 24
p. 30
p. 5
EuropeanMathematical Society
NEWSLETTER
NEWSLETTER No. 54December 2004
EMS Agenda ........................................................................................................... 2
Editorial - Luc Lemaire ......................................................................................... 3
EMS Executive Committee....................................................................................... 4
2nd Joint Mathematical Weekend .......................................................................... 5
EMS Executive Committee Metting ........................................................................ 7
Applied Mathematics and Applications of Mathematics ........................................ 9
Symmetries - Alain Connes ................................................................................. 11
Mathematics is alive... - Luc Lemaire .................................................................. 19
Polish Mathematical Society - Janusz Kowalski .................................................. 24
ERCOM - ICMS .................................................................................................... 30
Problem Corner - Paul Jainta .............................................................................. 33
Personal Column ................................................................................................... 37
Forthcoming Conferences ..................................................................................... 38
Recent Books .......................................................................................................... 41
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Published by European Mathematical SocietyISSN 1027 - 488X
The views expressed in this Newsletter are those of the authors and do not necessari-ly represent those of the EMS or the Editorial team.
NOTICE FOR MATHEMATICAL SOCIETIESLabels for the next issue will be prepared during the second half of February 2005.Please send your updated lists before then to Ms Tuulikki Mäkeläinen, Department ofMathematics and Statistics, P.O. Box 68 (Gustav Hällströmintie 2B), FI-00014 University ofHelsinki, Finland; e-mail: [email protected]
INSTITUTIONAL SUBSCRIPTIONS FOR THE EMS NEWSLETTERInstitutes and libraries can order the EMS Newsletter by mail from the EMS Secretariat,Department of Mathematics and Statistics, P.O. Box 68 (Gustav Hällströmintie 2B), FI-00014University of Helsinki, Finland, or by e-mail: ([email protected] ). Please includethe name and full address (with postal code), telephone and fax number (with country code) ande-mail address. The annual subscription fee (including mailing) is 80 euros; an invoice will besent with a sample copy of the Newsletter.
EDITOR-IN-CHIEFMARTIN RAUSSENDepartment of Mathematical Sciences, Aalborg UniversityFredrik Bajers Vej 7GDK-9220 Aalborg, Denmarke-mail: [email protected] EDITORSVASILE BERINDEDepartment of Mathematics,University of Baia Mare, Romania e-mail: [email protected] KRZYSZTOF CIESIELSKIMathematics InstituteJagiellonian UniversityReymonta 4, 30-059 Kraków, Polande-mail: [email protected] MARKVORSENDepartment of Mathematics, TechnicalUniversity of Denmark, Building 303DK-2800 Kgs. Lyngby, Denmarke-mail: [email protected] WILSONDepartment of Pure MathematicsThe Open UniversityMilton Keynes MK7 6AA, UKe-mail: [email protected] EDITOR:KELLY NEILSchool of MathematicsUniversity of SouthamptonHighfield SO17 1BJ, UKe-mail: [email protected] EDITORSINTERVIEWSSteen Markvorsen [address as above]SOCIETIESKrzysztof Ciesielski [address as above]EDUCATIONTony GardinerUniversity of BirminghamBirmingham B15 2TT, UKe-mail: [email protected] PROBLEMSPaul JaintaWerkvolkstr. 10D-91126 Schwabach, Germanye-mail: [email protected] ANNIVERSARIESJune Barrow-Green and Jeremy GrayOpen University [address as above]e-mail: [email protected] [email protected] Berinde [address as above]RECENT BOOKSIvan Netuka and Vladimír SouèekMathematical InstituteCharles University, Sokolovská 8318600 Prague, Czech Republice-mail: [email protected] [email protected] OFFICERVivette GiraultLaboratoire d’Analyse NumériqueBoite Courrier 187, Université Pierre et Marie Curie, 4 Place Jussieu75252 Paris Cedex 05, Francee-mail: [email protected]
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CONTENTS
EMS December 2004
EDITORIAL TEAM EUROPEAN MATHEMATICAL SOCIETY
EXECUTIVE COMMITTEEPRESIDENTProf. Sir JOHN KINGMAN (2003-06)Isaac Newton Institute20 Clarkson RoadCambridge CB3 0EH, UKe-mail: [email protected]
VICE-PRESIDENTSProf. LUC LEMAIRE (2003-06)Department of Mathematics Université Libre de BruxellesC.P. 218 – Campus PlaineBld du TriompheB-1050 Bruxelles, Belgiume-mail: [email protected]. BODIL BRANNER (2001–04)Department of MathematicsTechnical University of DenmarkBuilding 303DK-2800 Kgs. Lyngby, Denmarke-mail: [email protected]
SECRETARY Prof. HELGE HOLDEN (2003-06)Department of Mathematical SciencesNorwegian University of Science and TechnologyAlfred Getz vei 1NO-7491 Trondheim, Norwaye-mail: [email protected]
TREASURERProf. OLLI MARTIO (2003-06)Department of MathematicsP.O. Box 4, FIN-00014 University of Helsinki, Finlande-mail: [email protected]
ORDINARY MEMBERSProf. VICTOR BUCHSTABER (2001–04)Steklov Mathematical InstituteRussian Academy of SciencesGubkina St. 8, Moscow 117966, Russiae-mail: [email protected] Prof. DOINA CIORANESCU (2003–06)Laboratoire d’Analyse NumériqueUniversité Paris VI4 Place Jussieu75252 Paris Cedex 05, Francee-mail: [email protected]. PAVEL EXNER (2003-06)Department of Theoretical Physics, NPIAcademy of Sciences25068 Rez – Prague, Czech Republice-mail: [email protected]. MARTA SANZ-SOLÉ (2001-04)Facultat de MatematiquesUniversitat de BarcelonaGran Via 585E-08007 Barcelona, Spaine-mail: [email protected]. MINA TEICHER (2001–04)Department of Mathematics and Computer ScienceBar-Ilan UniversityRamat-Gan 52900, Israele-mail: [email protected]
EMS SECRETARIATMs. T. MÄKELÄINENDepartment of Mathematics and StatisticsP.O. Box 68 (Gustav Hällströmintie 2B)FI-00014 University of HelsinkiFinlandtel: (+358)-9-1915-1426fax: (+358)-9-1915-1400telex: 124690e-mail: [email protected] Web site: http://www.emis.de
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EMS NEWS
EMS December 2004
20051 FebruaryDeadline for submission of material for the March issue of the EMS NewsletterContact: Martin Raussen, email: [email protected]
19-27 FebruaryEMS Summer School at Eilat (Israel)Applications to Geometry, Cryptography and Computation.Web site: http://www.cs.biu.ac.il/~eni/ann1-2005.html
16-17 AprilEMS Executive Committee meeting, at the invitation of the Unione Matematica Italiana, inCapri (Italy)Contact: Helge Holden: [email protected]
25 June – 2 JulyEMS Summer School at Pontignano (Italy)Subdivision schemes in geometric modelling, theory and applications
17-23 JulyEMS Summer School and European young statisticians’ training camp at Oslo (Norway)
11 - 18 September EMS Summer School and Séminaire Européen de Statistique at Warwick (UK)Statistics in Genetics and Molecular BiologyWeb site: http://www2.warwick.ac.uk/fac/sci/statistics/news/semstat/
13-23 SeptemberEMS Summer School at Barcelona (Catalunya, Spain)Recent trends of Combinatorics in the mathematical contextWeb site: http://www.crm.es/RecentTrends/
16-18 SeptemberEMS-SCM Joint Mathematical Weekend at Barcelona (Catalunya, Spain)
18-19 SeptemberEMS Executive Committee meeting at Barcelona
12-16 DecemberEMS-SIAM-UMALCA joint meeting in applied mathematicsVenue: the CMM (Centre for Mathematical Modelling), Santiago de Chile
20063-7 July (to be confirmed)Applied Mathematics and Applications of Mathematics 2006 at Torino(Italy)
22-30 AugustInternational Congress of Mathematicians in Madrid (Spain)Web site: http://www.icm2006.org/
200716-20 July6th International Congress on Industrial and Applied Mathematics (iciam 07) at Zürich(Switzerland)Web site: http://www.iciam07.ch/
200814-18 July5th European Mathematical Congress at Amsterdam (The Netherlands) Web site: http://www.5ecm.nl
EMS AgendaEMS Committee
Cost of advertisements and inserts in the EMS Newsletter, 2004(all prices in British pounds)
AdvertisementsCommercial rates: Full page: £230; half-page: £120; quarter-page: £74 Academic rates: Full page: £120; half-page: £74; quarter-page: £44 Intermediate rates: Full page: £176; half-page: £98; quarter-page: £59
Inserts Postage cost: £14 per gram plus Insertion cost: £58 (e.g. a leaflet weighing 8.0 grams willcost 8x£14+£58 = £170) (weight to nearest 0.1 gram)
The project of giving the responsibility of dis-tributing research grants to a European ResearchCouncil (ERC) is currently the subject of intensediscussions in scientific and political circles inEurope. The EMS supports this project, andencourages all its members to promote it at polit-ical and scientific levels in their countries.
Briefly put, the ERC would be both an advi-sory council and a funding agency at theEuropean level, with the specific aim of devel-oping fundamental research in all disciplines.
A two-year historyThe idea of an ERC, catering for fundamentalresearch in Europe, was officially launched at aninternational conference in Copenhagen inOctober 2002, during the Danish EU presidency.An expert group was created by the DanishMinister for Science, Technology andInnovation, Helge Sander, under the leadershipof Federico Mayor, and the group made explicitproposals in December 2003.
Meanwhile, thanks to the role taken by theEuropean Life Science Forum (ELSF) and otherorganisations of life scientists and to theunbounded energy of Luc Van Dyck, the scien-tific community caught on to the idea andformed an informal platform of discussion, theInitiative for Science in Europe (ISE), and hadfurther debates on the aims and the means ofsuch a structure. They quickly had the idea toexpand their vision to all sciences, includingsocial sciences and humanities, and naturallymathematics.
At the time, the idea was wild: the EuropeanUnion’s activities were limited to researchapplicable to industrial development, with acomponent of training through the Marie Curieactions, and it was believed that the EU couldnot do anything else without violating the sub-sidiarity principle, implying that basic researchshould remain national. However, at a meetingin Dublin in October 2003, Achilleas Mitsos,Director General with the EC DirectorateGeneral (DG) Research, announced that theCommission supported the idea of an instrumentfor basic research and, for this purpose, wouldrequest a specific credit line from the EU budget.
The question of budget is of course a majorproblem, even though the EU member statescommitted themselves to reach the Barcelonaobjective (3% of the National Product to beinvested in research - 1% public and 2% privatefunding) by 2010. However, if in 2002 all statesagreed to have a much increased research budgetby 2010, most decreased theirs within a fewmonths!
Still, rowing against many currents, the grouppursued its effort in favour of establishing anERC, with the effect that it now appears as a realand solid perspective.
Clearly, the European Commissioner forResearch, Philippe Busquin, has played animportant role in this process by introducing theidea of a European Research Area and promot-ing it vigorously inside the EU. Under his guid-
ance, the Commission made the explicit requestthat for the next financial period starting in 2007,the budget of research handled by the EC shouldbe doubled. The budget of the FrameworkProgramme (around four billion euros per year)would thus be supplemented by an equivalentbudget, a large part of which would be attributedto the ERC.
At this stage, the research ministers of Europesupport the project, but the finance ministers willhave to include it in their global perspectives.
Basic principlesFor the time being there is reasonable agreement(at all levels) on the principles to be implement-ed in the ERC.
The ERC would be in charge of basic, funda-mental, scientist-driven research. Indeed, it is(finally!) admitted that this type of research is anecessary investment with a long-term perspec-tive that also has to be supported at the Europeanlevel.
Researchers would apply for grants from theERC, and the selection would be based only onscientific quality, by a rigorous peer reviewprocess.
There would be no notion of “juste retour” forthe states funding the ERC through their EUcontributions, and no criteria other than the sci-entific excellence, which in itself provides theadded value for Europe.
A scientific council would be appointed, andbe put in charge of managing the ERC withoutpolitical interference, for instance on the choiceof subjects.
The ERC should be funded thanks to theincrease in the part of the EU budget going toresearch, and absolutely not by diminishing thecorresponding amounts from those of the nation-al research agencies.
The budget should be commensurate with theambitions of the project. Current discussions(but not by those who will have to find themoney!) mention amounts between one and fourbillion euros per year.
A serious problem is the risk of oversubscrip-tion. In the EU Framework Programmes, the rateof success of applications is sometimes down to5%, which leads to the discouragement of thebest scientists. The scientific council of a futureERC will have to handle this problem, but avoidartificial rules resulting in the exclusion of someproposals. It may think of putting a limit to thesize, rotating subjects or using other means - butovercoming this difficulty without perverting theproject will be necessary.
EMS and ISEThe ISE started informally with a dozenEuropean associations. On October 25th 2004,alongside a meeting held in Paris, it was moreofficially created, and the EMS is one of itsmembers. In fact, it is the only body representingmathematicians in this structure.
The ISE was created around the idea of sup-porting the ERC and will continue monitoring
this initiative. It will also act as a reflection plat-form for the development of science in Europe inthe future.
Its web site is http://www.initiative-science-europe.org
What you can do?Right now we are in a position of great uncer-tainty. The idea of an ERC has moved fromnothing to a fully-fledged project, with backingfrom the European Commission and (probably)the Council of European Ministers. However,whether the necessary budget will be madeavailable is doubtful. A strong political argumenthas to be developed by scientists, in particular inthe larger states of the EU, which at this stage donot approve of the idea of increasing the EUbudget, thereby preventing the doubling of theresearch budget. The next few months will becrucial, and a determined action is required inthe member states whenever an opportunity aris-es.
The idea that basic research is key to the futuresuccess of Europe is now better acknowledged,but further efforts are needed to convince thelarger states that part of the scientific activitiesare better handled at a European level, and that afinancial effort to make this possible at the rightlevel must be made now. It is almost sure that theEuropean Parliament will have a say in theprocess, and at this moment very few of themembers have formed an idea on the project. Itmay be a good time to approach them, and dis-cuss this issue with them. This could be a goodstarting point for further contacts since, as math-ematicians, we could make our case in the con-text of a comprehensive project where all sci-ences are involved, but nevertheless of promi-nent significance for us, in view of the almostevanescent support now available for mathe-maticians in the Framework Programme.
ReferencesMore information on the project and its evolu-tion can be found in the ELSF brochurehttp://www.elsf.org/elsfbrochures/elsferc03.pdf,and the site http://www.elsf.org/elsfercc.htmlprovides access to a large collection of docu-ments.
A letter of support for the ERC plan, signed byrepresentatives of 52 associations, was publishedin August in Science, see http://www.initiative-science-europe.org/forms_maps/Science.pdf
A web search on ERC and ISE will show thatthe debate on ERC is developing, and that theadvisory board of the Commission (EURAB) isvery much in favour of the project, but sadly thatsome scientific bodies in large member statesrefuse to give up some of their national power.
EDITORIAL
EMS December 2004 3
EditorialEditorialTowards a European Research Council (ERC)
Luc Lemaire (EMS Vice-President, Bruxelles)
-At its meeting in Uppsala (Sweden) inJune this year, EMS council elected threenew members to the societies executivecommittee for a four-year period begin-ning in January 2005. Victor Buchstaber(Newsletter 38) was reelected. Thankswere given to Bodil Branner, Marta Sanz-Solé and Mina Teicher who leave the com-mittee after several years of service.
Olga Gil Medrano received a Doctoradofrom the Universitat de Valéncia in 1982and a Doctorate from the Université deParis VI in 1985. She has been associateprofessor at the Universitat de Valénciasince 1986.
Her research interests lie in differentialgeometry and global analysis (in particu-lar, variational problems on Riemannianmanifolds). She has visited several univer-sities, while working in cooperation withcolleagues from Austria, Belgium, Braziland France.
Olga has been a member of theExecutive Committee of the RealSociedad Matemática Española (RoyalSpanish Mathematical Society, RSME)during the last four years, and she is nowone of the vice-presidents. Amongst otherthings, she has been the Editor-in-Chief ofthe newsletter of the RSME from itsbeginning.
Carlo Sbordone was born in 1948 inNaples, Italy. He was a visiting researcherin the Scuola Normale Superiore of Pisafor two years, working with Ennio De
Giorgi. He has also been a visiting profes-sor at the University of Marlyland,University of Syracuse, University ofParis VI and the University of Umea. Hebecame a full professor at the Universityof Naples in 1980.
His research interests include: homoge-nization of linear and nonlinear partial dif-ferential operators, relaxation of integralfunctionals of calculus of variations, exi-stence and regularity theory for solutionsof variational problems, reverse Holderinequalities, weak minima of variationalintegrals, regularity theory of Jacobiansand quasiconformal mappings.
He has organized several conferences inItaly and France, and he is a member of theeditorial board of various journals inclu-ding “Annali di Matematica”, “Rendicontidell’ Accademia Nazionale dei Lincei”,“Ricerche di Matematica” and “Bollettinodell’Unione Matematica Italiana”, whichhe currently edits.
Carlo is a Fellow of the AccademiaNazionale dei Lincei, Vice-President ofthe Accademia Pontaniana and Secretaryof the Società Nazionale di ScienzeLettere e Arti in Naples. He is thePresident of the Unione MatematicaItaliana and a member of the TechnicalSecretariat of the Italian Minister ofEducation, University and Research. He isalso a member of the Italian Committeefor the OECD-PISA (Programme for
International Student Assessment).In his opinion the EMS has a very
important role in promoting mathematicalresearch and supporting the Europeanmathematical community. The EMS alsogives considerable attention to increasingthe level of understanding by the public ofthe nature and importance of mathematics.The decline in the number of mathematicsstudents in some European countries isone of his main concerns. He believes weneed to continue and expand our efforts toattract good students into our subjects bydiversifying graduate programs and pro-moting excellence in the teaching ofmathematics. His main hobby is singingNeapolitan songs.
Klaus Schmidt received a PhD from theUniversity of Vienna in 1968. In 1969 hewent to Great Britain, first as a one-yearpost-doc at the University of Manchester,then as a lecturer at Bedford College,University of London, and in 1973/1974he joined Warwick University, where hewas promoted to professor in 1983. In1994, he returned to the University ofVienna.
Klaus Schmidt’s research is mainly inthe area of ergodic theory and its connec-tions with other branches of mathematics(like operator algebras and arithmetic). In1993, he was awarded the Ferran Sunyer iBalaguer Prize for the monograph‘Dynamical Systems of Algebraic Origin’.He became a full member of the AustrianAcademy of Science in 1997.
From 1995 - 2003 he was one of the twoScientific Directors of the ErwinSchrödinger International Institute forMathematical Physics (ESI), Vienna,Austria, and he is now President of theESI.
EMS NEWS
EMS December 20044
New members New members of the of the
EMS executive committeeEMS executive committee
The frameworkIn order to support the growing sense ofEuropean identity that follows the politicalchanges in Europe, the EuropeanMathematical Society organizes many regu-lar events, the European MathematicalCongresses being the largest and most impor-tant. To maintain more frequent contact withnational societies and to help promote theideas of cross-European collaboration, theEMS has launched a new series of annualmeetings called the Joint MathematicalWeekends. The three-day meetings are finan-cially supported by the EMS, under therequirement that the organizing national soci-ety arranges additional funds so that a largenumber of local, as well as other Europeanmathematicians, may participate withoutpaying an expensive conference fee. Themeeting consists of several parallel minisym-posia whose topics are chosen to representthe strongest areas of the organizing society.Each minisymposium is coordinated by alocal expert who chooses one plenary speak-er to represent the area and to give a generallecture introducing the subject to all partici-pants of the conference. Together they selectseveral leading European experts in the areaas further invited speakers. Contributed talksare also accepted as the time schedule allows.The first meeting of this type was organizedby the Portuguese Mathematical Society inLisbon in September 2003.
The genius lociThe 2004 Joint Mathematical Weekend wasorganized by the Czech MathematicalSociety in Prague, with further financial andlogistic help from the Faculty of Mathematicsand Physics of the Charles University inPrague and from the Czech national researchcentre Institute for Theoretical ComputerScience (ITI), which is a platform for thecooperation of research teams from severalinstitutions. The meeting took place in a fac-ulty building at the Lesser Town Square inthe close vicinity of Prague Castle, theCharles Bridge, the Astronomical Clock andother celebrated historical monuments. Theparticipants enjoyed the beautiful atmosphereof the historical building, a former Jesuit
seminar, blending with all the advantages ofmodern technology. A short explanationabout its history was given by Ivan Netuka,the dean of the faculty, in his welcomingspeech. The participants admired the skillsthe architects had performed during therecent renovation works, allowing the Schoolof Computer Science to fit so well in a build-ing with old stone doorframes and ceilingsdecorated with 300-year-old frescos.
Minisymposia and plenary lecturesThe meeting itself was opened by JanKratochvíl, the president of the CzechMathematical Society. The EuropeanMathematical Society was represented by thewelcoming address of Luc Lemaire, a vice-president of the EMS. The Friday afternoonprogramme started with the plenary lecture“Some recent applications of the classifica-tion of finite simple groups” by Jan Saxl fromthe University of Cambridge. His plenarytalk introduced the minisymposium“Discrete Mathematics and Combina-torics”, organized by Jaroslav Nešetøil fromCharles University in Prague and director ofITI. As the title of the talk suggests, it pre-sented interactions of algebra and combina-torics, presenting applications in variousareas of graph theory and discrete mathemat-ics. In the programme of the following days,the invited speakers in this minisymposiumwere Peter Cameron (University of London),Gábor Tardos (Alfréd Rényi Institute ofMathematics, Budapest), Oriol Serra(Universitat Politècnica de Catalunya,Barcelona) and Patrice Ossona de Mendez(CNRS, Paris). Five contributed talks werealso included.
The minisymposium “MathematicalFluid Mechanics” was coordinated byEduard Feireisl from the MathematicalInstitute of the Academy of Science of theCzech Republic in Prague. Since his plenaryspeaker, H. Beirao da Veiga (Università diPisa), could not attend the meeting due to aninjury, E. Feireisl himself presented this min-isymposium’s plenary talk “On the mathe-matical theory of viscous compressible flu-ids” on Saturday morning. He introduced theaudience to methods used for modellingdynamical aspects of liquid and gas media.Invited speakers in this minisymposium wereJosef Málek (Charles University, Prague),Victor Starovoitov (CAESAR, Bonn),Marius Tucsnak (Université Nancy), andAntonín Novotný (Université du SudToulon-Var). Moreover, there were threemore contributed talks.
Mathematical models of the real world
were also the subject of the Saturday after-noon plenary talk “Probabilistic and varia-tional methods in problems of phase coexis-tence”, given by Errico Presutti fromUniversita degli Studi di Roma Tor Vergata.This talk introduced the minisymposium“Mathematical Statistical Physics”, coordi-nated by Roman Kotecký from CharlesUniversity in Prague. The pattern of this min-isymposium differed slightly from the otherthree. It had only three invited speakersMarek Biskup (University of California, LosAngeles), Dmitry Ioffe (TechnionUniversity, Haifa), and Kostya Khanin (IsaacNewton Institute, Cambridge), but the modusoperandi hinged on informal discussions fol-lowing each of the invited talks.
The last plenary talk “Some current trendsin proof complexity” was given on Sundaymorning by Alexander Razborov fromSteklov Mathematical Institute in Moscowand the Institute for Advanced Studies inPrinceton. This talk belonged to the min-isymposium “Complexity of Computationsand Proofs”, organized by Jan Krajíèek fromthe Mathematical Institute of the Academy ofScience of the Czech Republic in Prague.The audience was entertained not only by hisvery colourful presentation of methods andreasoning in contemporary theoretical com-puter science, but also by a malicious coinci-dence that some software-hardware commu-
EMS NEWS
EMS December 2004 5
2nd Joint Mathematical 2nd Joint Mathematical WWeekendeekendPrague, September 3-5, 2004
Jan Kratochvíl, Jiøí Rákosník (Prague)
The mathematical weekend (almost) got itsown stamps!
nication problems delayed the beginning ofthis very computer science oriented talk. Theinvited talks of this minisymposium weregiven by Peter Buergisser (UniversitätPaderborn), Oleg Verbitsky (LvivUniversity), Peter Bro Miltersen (Universityof Aarhus), Albert Atserias (UniversitatPolitecnica de Catalunya, Barcelona), StefanDantchev (University of Leicester), andPavel Pudlák (Mathematical Institute AS CR,Prague) and further six contributed talks wereincluded.
After the last plenary session, Sir JohnKingman, the president of the EMS, official-ly addressed all participants. He appreciatedthe importance and significance of this newseries of meetings, emphasized the role of theEMS, invited the participants to join the soci-ety, and thanked the Czech MathematicalSociety for hosting the Weekend.
Conference banquet and photoOn Saturday evening all participants wereinvited to a conference banquet in the base-ment of the building, to experience roomsthat during the academic year serve as a cafe-teria at lunch time and as the faculty club inthe afternoons. The pleasant atmosphere wasenhanced by toasts of Jaroslav Nešetøil, BodilBranner and David Salinger, and later also bylive music performed by some participantsand organizers of the meeting.
Earlier that day the official conferencephoto was taken during the lunch break fromthe windows of the conference building. Theparticipants were lined up in front of a histor-ical plague column, one of the baroque sculp-ture jewels in Prague.
Additional meetingsThree important work meetings took placebehind the scene at the weekend. The CzechMathematical Society made use of thisopportunity and called its general assemblymeeting for Friday evening. One of the mainagenda items was the change of name for thesociety. For almost one and a half centuriesmathematicians and physicists of this regionhave been organized in the Union of Czech
Mathematicians and Physicists, and theyhave always enjoyed this coexistence andcooperation of closely related sciences.Within the Union, they are grouped in pro-fessional units, one of them having beencalled the Mathematics Research Sectionsince its foundation in 1970’s. This namereflected the era of mainly Eastbound orient-ed international links, governed by the politi-cal situation in Europe in the second half ofthe 20th century, and always required tediousexplanations to foreign colleagues. Thereforethe name “Czech Mathematical Society” hasbeen frequently, informally and somewhatillegally used in foreign correspondence inthe last years. Now, after rigorous voting dur-ing the general assembly meeting, this namehas become official and legal (with its Czechequivalent Èeská matematická spoleènost).This does not, however, change the fact thatwe are still organized within the frames of theUnion together with our sister CzechPhysical Society. The first formal act accom-plished under the new name of the CzechMathematical Society was signing the agree-ment of collaboration and reciprocal mem-bership with the Catalan MathematicalSociety during the EMS weekend banquet,an act which fitted well the spirit of the EMSactivities and of the Joint Weekend in partic-ular. (The next agreement on cooperationwas signed with the Royal SpanishMathematical Society shortly after theWeekend.)
The other event associated with theWeekend, the meeting of the EMS ExecutiveCommittee, was perhaps less ceremonial butmore busy. Most members of the Committee
enjoyed the mathematical programme of theentire weekend, while their working meetingwas scheduled on Sunday afternoon andMonday morning in hotel Vaníèek. In recog-nition of the importance of the EMS, a festivedinner was given for the ExecutiveCommittee members by the president of theAcademy of Sciences of the Czech RepublicHelena Illnerová, the president of the CzechMathematical Society Jan Kratochvíl, the
dean of the Faculty of Mathematics andPhysics of the Charles University IvanNetuka and the director of the MathematicalInstitute AS CR Antonín Sochor.
Finally, there was a meeting of the CzechNational Committee for Mathematics - thebody consisting of leading Czech mathemati-cians, established under the auspices of theAcademy of Sciences of the Czech Republicto deal with basic conceptual and organiza-tional questions of mathematical sciences inthe country and to represent the Czech math-ematical community in the IMU. One of themain topics was to discuss the somewhatdangerous features of a new evaluation sys-tem for the R&D being prepared by the gov-ernment.
We can proudly admit that the organizingteam, Jan Kratochvíl, JiøíRákosník, Jiøí Fiala,Daniel Hlubinka and Pavel Exner (the CzechMathematical Society), Anna Kotìšovcová(Conforg Agency), Hana Polišenská (secre-tary of the ITI) and Jaroslav Nešetøil, EduardFeireisl, Roman Kotecký and Jan Krajíèek(the minisymposia coordinators), did a goodjob and prepared pleasant and friendly work-ing conditions for the participants. As LucLemaire mentioned in his welcoming speech,one meeting does not make a series, twomeetings in a row do. We in the CzechMathematical Society feel honoured by thecommission to organize the second jointweekend and proud of being present at thebirth of a new tradition of what we all believewill prove to become a useful and importantseries of meetings of European mathemati-cians.
Jan Kratochvil [[email protected]. cz],born in 1959, did his Ph. D. in discrete math-ematics at Charles University in Prague.Since 1994, he has been at the Department ofApplied Mathematics of this university, andsince 2003 he is the head of the departmentand a full professor. He has also lecturedseveral times as a visiting professor atComputer and Information ScienceDepartment of the University of Oregon inEugene, OR. His main research interests aregraph theory and computational complexity.He is currently president of the CzechMathematical Society.
Jiøí Rákosník [[email protected]], bornin 1950, graduated in mathematical analysisfrom Charles University in Prague. Hisresearch interest concentrates on theory offunction spaces and theory of integral anddifferential operators. He is associated to theMathematical Institute of the Academy ofSciences of the Czech Republic. Since 2001,he is also a member of the Academy Councilof the AS CR, responsible for economic andfinancial matters of the Academy and forcooperation of the Academy in preparinglegislation concerning research and develop-ment. He heads the Czech Editorial Unit ofZentralblatt MATH.
EMS December 20046
Meetings of the Executive would lookdull to a traditional newspaper. Nopunch-ups, a calm business-like atmos-phere predominates, but underlying it allis a sense that the committee is workingfor the good of mathematics and mathe-maticians. It was a particular pleasure towelcome future members, Olga Gil-Medrano and Carlo Sbordone, to thePrague meeting.
Mathematics and the EUCertain issues come round each time. It isnot just a routine complaint to observethat mathematics is under-represented oninfluential EU committees: members ofthe executive committee also do some-thing about it, from lobbying ministers tositting on extra committees, and generally,grabbing opportunities to do things when-ever they arise. At Prague, Luc Lemaire
was able to report on the panel sessionorganised by the EMS and on his plenarylecture to the EuroScience Open Forum inStockholm. Some good contacts weremade, but attendance at sessions waspoor. The Executive Committee resolvedto encourage local mathematicians to getinvolved, when the Forum, now intendedto be a two-yearly event, came their way.
Recognising the importance of contactswith the EU and the burden it placed onthose members who were involved, theExecutive Committee resolved to broadenthe Group on Relations with Europe to thewhole Executive Committee and to wel-come the help of others who were pre-pared to lobby for mathematics inBrussels.
Membership and RepresentationIndividual membership remains stubborn-ly on a plateau of about 2,300, but institu-tional and corporate membership hasgrown. Nearly all the mathematical insti-tutes which are ERCOM members havenow joined. Statistical Societies havebegun to join the Society, too and ourSummer School programme includesevents involving the Bernoulli Society.The EMS, naturally, believes that it’s to
the advantage of mathematics to have onesociety representing it at the Europeanlevel and is working hard to include allstrands of mathematics, including particu-larly applied mathematics and statistics, inits work (and in the committee member-ship). It has decided that, as appliedmathematics is central to the Society’swork, responsibility should no longer be
devolved to a subcommittee, but shouldrest with the Executive Committee itself.A new committee will, however, be set upto look at applications of mathematics.
MeetingsThe next ‘mathematical weekend’ will
be in Barcelona in September 2005, joint-ly organised by the Catalan MathematicalSociety and the EMS. In 2006 there willbe a conference organised by the EMS andthe French and Italian mathematical soci-eties in Turin. We shall submit an appli-cation for further Summer Schools to theEU and have already advertised a call forproposals from members.
4ecm had been a high quality meeting,but had attracted rather too few partici-pants. There was general satisfaction withthe selection of EMS prize-winners, butsome regret that it had proved impossibleto award a Felix Klein Prize.
SubcommitteesMina Teicher would become chair of theEducation Committee. She submitted alist of suggested members and an outlineplan of action. The committee on specialevents would be abandoned and its worktaken over by the General MeetingsCommittee, which would lose the word‘general’ from its name. Andrzej Pelczarwas stepping down from the Committeeon Eastern European mathematicians aftermany years of service to the Society. JanKratochvíl of the Czech MathematicalSociety would take over the chair. Thework of the Committee would takeaccount of changing circumstances inEastern Europe. Pavel Exner wouldbecome chair of the Committee onElectronic Publications: it was felt that anExecutive member should take on thatrole because of the growing importance ofthe committee’s remit and, in particular,because it oversees the Electronic Libraryon EMIS.
The Publications Committee was dis-banded: the Executive itself would take onthe task of dealing with issues concerningthe Publishing House. It agreed to trans-fer the publication of the Newsletter fromthe Society to the Publishing House, aspart of the task of persuading other math-ematicians to publish with EMSph.
Closing MattersOur stay in Prague had been most enjoy-able, thanks in no small part to PavelExner and the Czech MathematicalSociety, who had organised an exemplaryMathematical Weekend beforehand. ThePresident also thanked the retiring mem-bers of the Committee: Bodil Branner,Marta Sanz-Solé and Mina Teicher. Thenext meeting would be in Capri (Italy) inApril at the invitation of the UnioneMatematica Italiana.
EMS NEWS
EMS December 2004 7
EMS executive committee EMS executive committee meeting at Praguemeeting at Prague
David Salinger, EMS Publicity Officer
The August floods in 2002 dramaticallyaffected the mathematical and informationscience's part of the library that is situatedin the Faculty's building in Karlin. Thewater reached a height of 2.90 metersabove pavement level. 13000 books, 468titles of journals, 6800 textbooks and 2000diploma theses were damaged (see theinformation in EMS Newsletter, Issues 45and 49).
In the period after the floods we receivedgreat support from foreign and home orga-nizations, institutions, societies, universi-ties and individual donors. Financial dona-tions reached almost 104,000 euros. Theprincipal financial support was provided byThe Ministry of Education, Youth andSports, and Charles University. In addition,the Faculty received 5907 books and 156journal titles from 165 donors. The donorsthat were identified were being sent lettersof thanks. Two press conferences wereheld on the occasion of the presentation ofa large collection of books and journalsfrom French and German mathematicians.
In September 2004, the construction of
the new library was finished. After the pre-vious unfortunate experience, it was decid-ed to situate it on the first floor. The libraryhas just been opened for the beginning ofthe academic year 2004/2005.
The Faculty of Mathematics and Physicswould like to thank all the people and insti-tutions that contributed to the renewal ofthe library with their understanding andhelp.
EMS NEWS
EMS December 20048
The Committee for Raising PublicAwareness of Mathematics of the EuropeanMathematical Society (acronym RPA)believes that it is of vital importance for therecognition of mathematics in society thatarticles popularising mathematics are writ-ten. The experience gained from the firstarticle competition organized by the RPA-committee of the EMS, completed in 2003,show that it is beneficial to distinguishbetween articles addressing the educatedlayman and articles addressing a generalaudience; see http://www.mat.dtu. dk/peo-ple/V.L.Hansen/rpa/resultartcomp. html.Therefore the EMS now wishes to encour-age the submission of articles on mathe-matics to two competitions, one for articlesfor the educated layman and one for articlesfor a general audience. It is hoped that valu-able contributions will be collected, whichdeserve translation into many languages.The EMS is convinced that such articleswill contribute to raising public awarenessof mathematics.
The RPA-committee of the EMS invitesprofessional mathematicians, or others, tosubmit manuscripts for suitable articles onmathematics for one of two competitions.
Articles for the educated laymanTo be considered an article must be pub-
lished, or be about to be published, in aninternational magazine or a specializednational magazine, bringing articles onmathematics to an educated public. Articlesfor the competition shall be submitted bothin the original language (the published ver-sion) and preferably also in an Englishtranslation. Articles (translations) may alsobe submitted in French, German, Italian orSpanish. The English (or alternative lan-guage) version should be submitted bothelectronically and in paper form.
Deadline for submission to this competi-tion: August 1, 2005.
Articles for the general publicTo be considered an article must be pub-lished, or be about to be published, in adaily newspaper or some other widely readgeneral magazine, thereby providing someevidence that the article does catch theinterest of a general audience. Articles forthe competition shall be submitted both inthe original language (the published ver-sion) and preferably also in an Englishtranslation. Articles (translations) may alsobe submitted in French, German, Italian orSpanish. The English (or alternative lan-guage) version should be submitted bothelectronically and in paper form.
Deadline for submission to this competi-
tion: January 1, 2006.
Prizes and conditions for submissions For each of the two competitions there willbe prizes for the three best articles, to thesum of 200, 150 and 100 Euros respective-ly, and many of the winning articles will bepublished in the Newsletter of the EMS.Other articles from the competition mayalso be published if space permits.
By submitting an article for one of thecompetitions it is assumed that the authorgives permission for the translation of thearticle into other languages and for its pos-sible inclusion on a web site. Translationsinto other languages will be checked bypersons appointed by relevant local mathe-matical societies and will be included onthe web site.
Articles should clearly indicate to whichone of the two competitions the article isbeing submitted and should be sent beforethe given deadlines to the chairman of theRPA-committee of the EMS:
Professor Vagn Lundsgaard Hansen, Department of Mathematics, Technical University of Denmark, Matematiktorvet, Building 303, DK-2800 Kongens Lyngby, Denmark. e-mail: [email protected]
EUROPEAN MAEUROPEAN MATHEMATHEMATICALTICAL SOCIETYSOCIETYARARTICLE COMPETITIONSTICLE COMPETITIONS
The new librarThe new library of the Faculty of Mathematics and Physics,y of the Faculty of Mathematics and Physics,Charles University in Prague
Ivan Netuka and Vladimir Souèek (Prague)
The mathematical library after the floods ... and after reconstruction
In order to increase the place of AppliedMathematics in the EMS, its Councilproposed to organize a conference onthis subject, jointly with the Frenchmathematical societies SMF and SMAI.This conference Applied Mathematicsand Applications of Mathematics tookplace in February 2003 in Nice at thePalais des Congrès. The co-presidentsof AMAM 2003 were Rolf Jeltsch(EMS), Michel Théra (SMAI) andMichel Waldschmidt (SMF). TheScientific Committee was co-chaired byPierre Louis Lions (France) and SergeyNovikov (Russia), both Fields medal-ists. The Organizing Committee wasco-chaired by Doina Cioranescu (SMAI)and Mireille Martin-Deschamps (SMF).
The Scientific Committee establishedthe following list of topics:1) Applications of number theory,
including cryptography and coding.2) Control theory, optimization, opera-
tions research and system theory.3) Applications of mathematics in biol-
ogy, including genomics, medicalimaging, models in immunology,modeling and simulation of biologi-cal systems.
4) Scientific computation, including abinitio computation and moleculardynamics.
5) Meteorology and climate, includingglobal change.
6) Financial engineering.7) Signal and image processing.8) Nonlinear dynamics.9) Probability and statistics, inverse
problems, fluid dynamics, materialsciences and other applications.
The conference consisted of 9 plenaryspeakers, 38 mini-symposia, two roundtables (Mathematics in developingcountries, and Education) and twoposter sessions (with 92 presentations).
More than 500 mathematicians from 25countries took part into the conference.Almost 20% of the total budget wasdedicated to 30 grants which supportedthe participation of young researchers,students or mathematicians from EastEuropean countries, Asia and Africa.
The conference showed the unity ofmathematics and their role in variousfields of science and technology. It alsoemphasized the interest of the EMS inpromoting Applied Mathematics in thecommunity of mathematicians in gener-al. It should be mentioned that moreand more women are now choosing sci-entific careers. Among the participants,27 % were women. The conference alsoexposed to young European students,various topics in mathematics.
In conclusion, AMAM 2003 rein-forced the links between Pure andApplied Mathematics and presented newchallenges and issues for mathemati-cians, arising from varied areas of sci-ence and technology. It also showed theunity of mathematics and their rolewithin the modern world. (taken fromthe report to the EMS Council onhttp://www.math.ntnu.no/ems/council04/uppsalapapers/Report_AMAM_typeset.pdf ).
Inaugural AddressSergey Novikov (University of
Maryland, College Park, USA, andLandau Institute for Theoretical
Physics, Moscow)
Dear Colleagues,
I apologize that for personal reasons Icannot attend this Conference.However, I would like to say a fewwords for the Opening Ceremony.
It was a great pleasure for me towork jointly with my co-chair,Professor P.-L. Lions, ScientificSecretary, Professor A. Damlamianand other members of the Committee,in selecting the main speakers. Iremember it as a parade, demonstratinga series of great achievements in manydifferent areas of applied mathematics,displaying many beautiful mathemati-cal applications. It covers a broadspectrum of subjects including thepractical use of mathematics in busi-ness and finance, practical cryptogra-phy, and the impressive developmentof computational and theoretical meth-ods in the natural sciences. Manytimes I felt sorry that we were restrict-ed in the number of invited talks. Ibelieve we made extremely goodchoices and left many excellent candi-dates for future meetings.
The mathematicians who founded theEuropean Mathematical Society over10 years ago always believed in theunity of mathematics, artificiallydivided into pure and applied parts. Itis the duty of mathematics to supportits applied component. I always treat-ed the opposing point of view as some-thing non-serious, a sort of philosophymade from scientific weakness no mat-ter how broad it became distributed.
Our unity is especially importantnow. Mathematical education hasreached a state of terrible crisis in allcivilized countries. What is going tohappen to the most fundamental exacttheoretical sciences of the past centurylike mathematics and theoreticalphysics? According to my observa-tions, mathematics has a better chanceat survival than theoretical physics, butour unity is necessary for that.
It was already obvious to everybodyin the 1990s that biology had becomethe main candidate for the position of“Miss Science - XXIst Century”.Unfortunately, we have also been wit-nesses to the decay of theoreticalphysics during the last decade. Whatdoes this mean for those of us whohave dedicated their scientific activityto the interaction of mathematics withthe natural sciences? Of course, we arehappy to support all realistic and use-ful mathematically based investiga-tions made for the needs of biology.Some of them are represented in thisconference. No doubt we should helpto increase their number.
However, the connection of mathe-
EMS NEWS
EMS December 2004 9
Applied Mathematics andApplied Mathematics andApplications of MathematicsApplications of Mathematics
(AMAM, Nice, February 10-15, 2003)
matics and physics was so deep that weshould say a few words about today’scrisis. In a sense, theoretical physicshas always been considered as themathematics of the real world, a mainsource of mathematical ideas since theXVII century, the main driving forcefor the development of 90% of mathe-matics, and the main road joiningmathematics with other natural sci-ences. In the XXth Century, theoreti-cal physics reached its highest level. Itbecame a leading exact theoretical sci-ence. It was era of dinosaurs forphysics. Its great leaders were capableof using and sometimes of creatingvery deep abstract mathematics when itwas needed for the study of the realworld. New fundamental laws ofnature were discovered and theyinvented great new technology. Itchanged our world forever.
At the same time, there was a split-ting of the communities of physicistsand mathematicians. As a corollary,several important mathematicalachievements made by physicists (suchas quantum field theory, for example)remain, until now, outside the mathe-matics community. Mathematical edu-cation knew nothing about them evenin the best times. Its language becameincompatible with the mathematicallanguage of physics of the early stages.Pure and applied mathematics haveboth lost contact with high-level mod-ern physics in the past century.Mathematical language and the tech-nique of theoretical physics were espe-cially designed as the best mathemati-cal tools for the solution of real worldproblems. In trying to replace themwith something absolutely formal andrigorous, you make them useless.Therefore, pure mathematics alone willnot be capable of adjusting itself to“physical mathematics”.
We see now that the era of dinosaursis probably over and theoreticalphysics is going down. Maybe it hap-pened as a consequence of overdevel-opment? Is it possible that some unex-pected great achievement will returnits momentum?
If not, let me ask the following ques-tion: Who is going to preserve thisgreat knowledge? Certainly it willremain important for many engineeringapplications and it would be dangerousfor humanity to forget it.
To my opinion, only the joint forcesof pure and applied mathematics mayhelp here. I do not know any other partof science capable of preserving thisgreat mathematical knowledge.
I have no doubts that this conferenceis going to be very successful. I wishyou great working days in Nice.
The following is a slightly edited version of apaper presented by Tsou Sheung Tsun to theIMU Developing Countries Strategy Groupmeeting, 16-17 October 2004, at ICTP,Trieste, Italy.
What are we doing?We have active individual members working
with their own programmes/organizations:CIMPA, ISP, Vietnam, etc. I hesitate toadd China, as her status of being a devel-oping country is unsure. We exchangeexperience and liaise closely.
We have a well-supported book/journal dona-tion scheme, the shipping costs of whichare almost entirely funded by ICTP. Thereis a vast source of books and journals, fromretirees and libraries going electronic in thedeveloped world.
We have a good network of distribution cen-tres, particularly in Africa. These aredepartments or institutes, mostly wherethere is some personal contact with mem-bers of our committee so that we can makesure that the books and journals are usedproperly, e.g. made available to all mathe-maticians. If material is donated that a par-ticular centre cannot use, or already have,they promise to send that to nearby institu-tions.
We have shipped about one and a half tonnesof such material, with a couple of 100kgbeing processed at the moment.
The recipients where we have contacts arewell monitored: sub-Saharan Africa,Zimbabwe, and Vietnam.
Referees’ honorarium donation: this is ascheme by which referees donate their hon-oraria from publishers, usually augmentedif exchanged for books. PrincetonUniversity Press has donated some.Zentralblatt has donated twice, amountingto some 1500 euros. We have talked withJohn Ewing of the American MathematicalSociety and we are currently devising somevariant of the above scheme with him.
A good point about this programme is that therecipient, typically a department with asmall active group of researchers, canchoose the books they need, even though itis a small number of books.
This could involve the whole mathematicalcommunity, and would raise awarenessabout situations in the developing world.
What do we hope (in the short- and medi-um-term)?Expand the book donation scheme: to include
more distribution centres, particularly to
have a wider geographical spread.Lecture notes distribution: we have just start-
ed this scheme with the help of ICTP. Theidea is to make a collection of undergradu-ate lecture notes and make them availableto the developing world, in the form of aCD for example. I have already collectedthose available from Oxford. We hope toextend this scheme to other languages, e.g.French and Spanish. We shall be lookingfor good universities in these languagegroups in the developed world.
Organize short-term placements (say 3months) for visitors from the developedworld to give a lecture course in a develop-ing country. This may interest mathemati-cians who have recently retired, or freshPhDs.
Mobilize support for mathematicians in adeveloping country to attend conferences: Isuppose this is done by the IMU-CDE.
Consolidate and expand existing MSc andPhD programmes in countries regularlyvisited by our members.
Closer liaison with IMU, ICTP, CIMPA,LMS, AMS, IMPA, AIMS, etc.
What do we need?Continued support from ICTP for the book
donation scheme.Help to start and develop the lecture notes
distribution in English, French andSpanish.
Funds for sending lecturers to developingcountries for short periods. Subsistencecan usually be provided locally.
Conference support for participants fromdeveloping countries.
What can we offer?Good field knowledge and contacts in Africa,
South East Asia, and some parts of LatinAmerica, through personal experience andthe contacts of our members.
Good network of book distribution, withmonitoring.
Enthusiasm and some new ideas.
Finally, an appealWe are in need of funds for our various activ-ities and urge our colleagues in the developedworld to help with donations, however small.Donations can be made directly to the EMS-CDC account:
Account number: 157230-381160 Nordea Bank Finland PlcSwift: NDEAFIHHIBAN: FI78157230000381160
EMS NEWS
EMS December 200410
COMMITTE FOR DEVELOPINGCOMMITTE FOR DEVELOPINGCOUNTRIES: COUNTRIES:
AA SUMMARSUMMARYY OFOF WORK WORK ANDANDPLANPLAN
(Chair: Herbert Fleishner; vice-chair: Tsou Sheung Tsun)
FEATURE
EMS December 2004 11
Symmetries
Alain Connes (Paris)
This article appeared originially (in French) in the magazine Pour la Science, no. 292, Feb. 2002. It was submitted to thearticle competition of the EMS Raising Public Awareness Comittee by the journal’s director Philippe Boulanger and wasgiven a runner-up award, cf. issue 50 of the Newsletter. The Newsletter thanks Alain Connes and Philippe Boulanger forthe permission to republish the article.
The concept of symmetry goes well beyond that of simple geometric symmetries. From the fair organisation of thefinal stages of soccer competitions to the solving of equations, via the icosahedral game and Morley’s theorem, we’lldiscover the multiple aspects of this concept.
The purpose of this article is to in-troduce the reader to the mathematicalnotion of symmetry, by way of a fewillustrative examples.
To demonstrate the ubiquity of thisconcept, as a mathematician under-stands it, we’ll start by evoking theconnection between the final stages ofsoccer competitions and the way wesolve quartic equations.
As we move to equations of higherdegree, we describe the icosahedralgame and the ‘icosions’, defined in thenineteenth century by the Irish mathe-matician, William Hamilton.
We finish with a commentary on atheorem in geometry, proved by FrankMorley around 1899, where the sym-metry of an equilateral triangle arisesmiraculously from an arbitrary trian-gle, by taking the intersection of con-
secutive ‘trisectors’ (the two straightlines which divide an angle into threeequal parts). In 1988, I gave an alge-braic formulation of this result and aproof which we shall see below.
The final stages of a soccer cup
Let’s start with the organisation of asoccer competition, for example, the
FIRST DAY
FIRSTDAY
interchanging F and D interchanges
days 2 and 3,
interchanging D and C interchanges
days 1 and 2,
interchanging F with D and C with H
leaves each day the same.
C F
H
SECONDDAY
SECOND DAY
THIRDDAY
THIRD DAY
D
D DHH
HF F F
C C
C
1. THE SYMMETRIES OF THE FINAL STAGES OF A SOCCER COMPETITION
D
F D C H
F D C H
F D C H
1 2 3
1 2 3
1 2 3
1 2 3
F D C H
F D C H 1 2 3
1 2 3
F D C H
The three days, 1,2,3 remain
globally unchanged by the
permutations of F, D, C, H. Thus:
Sy m m e t r i e sSy m m e t r i e sA l a i n C on n e s (Pa ri s)
FEATURE
EMS December 200412
millennial European Cup. During thefinal stage, teams were placed in poolsof four, and arrived at an order of meritwithin each pool. For example, onegroup consisted of Denmark, France,Holland, and the Czech Republic (hereabbreviated to D, F, H, and C).
To arrive at an order of merit fairly,each team had to play each of the threeothers, which meant that games hadto played on three days. For instance,when D and F met, then H and C couldmeet on the same day and so three dayswere enough for all possible configura-tions of matches.
In that European Cup, the matcheswere, FD and CH on the first day, FCand DH on the second, and FH andDC on the third. Intuitively we cansee that this is a fair procedure, be-cause none of the teams has an advan-tage. One checks indeed that, if we ar-bitrarily permute certain teams, for in-stance, if we interchange D and H, thisamounts to a simple permutation of thefirst and third days.
We can visualise the symmetrywhich is at work by putting the lettersD, F, C, H (representing the teams) atfour points of the plane. The line join-ing two points represents a match be-tween the corresponding teams. Eachof the three days corresponds to the
point of intersection of the lines repre-senting the matches on that day. Thusthe matches FD and CH are associatedwith the intersection of the lines FDand CH. Continuing for the two otherpairs, the second day is at the intersec-tion of the lines FC and DH and thethird is at the intersection of the linesFH and DC.
The figure thus constructed, formedby four points and six lines, is called acomplete quadrilateral. It is perfectlysymmetrical (in an abstract sense, evenif none of the usual geometric symme-tries – symmetries with respect to apoint or a line – are present), becauseeach of the four points F, D, C, and Hplays exactly the same role as the oth-ers, and the same is true for the pointsof intersection representing the threedays.
Having visualised this completequadrilateral, we can also give an alge-braic formulation of the symmetry un-der discussion. In the following way:the quantity α, determined from thefour numbers a, b c and d by the for-mula α = ab + cd only takes three val-ues altogether when we permute a, b, cand d. The other values are β = ac + bdand γ = ad + bc.
Resolution of quartic equationsby radicals
This surprising symmetry underliesthe general method for solving quar-tic equations ‘by radicals’. The solu-tion amounts to expressing the zerosa, b, c, and d of the polynomial x4 +nx3 + px3 + qx + r = (x − a)(x − b)(x −c)(x − d) as a function of the coeffi-cients n, p, q, and r using the extractionof roots.
To understand this assertion, wemust go back in time and look at thesolution of equations of degree lowerthan four.
Though the technique of solvingquadratic equations goes back to fur-thest antiquity (Babylonians, Egyp-tians. . . ), it wasn’t extended to cubicequations until much later and waspublished only in 1545 by GirolamoCardano in Chapters 11 to 23 of hisbook Ars magna sive de regulis alge-braicis. In fact, it wasn’t realised untilthe 18th century that the key to the so-lution by radicals of the cubic equationx3 + nx2 + px + q = 0, with zeros a, b,and c (the ‘roots’), lay in the existenceof a polynomial function f (a, b, c) of a,b, and c, which takes only two differentvalues under the action of the six pos-sible permutations of a, b, and c.
Solving polynomial equations by
radicals requires the construction of
auxiliary functions of the roots, which
display symmetries when the
roots are permuted.
The set of these auxiliary functions is
globally unchanged by
permutations of the roots of the
equations. For cubic and quartic
equations, the auxiliary functions enable
us to reduce the solution to that of a
"reduced" equation of lower degree.
1) The cubic equation:
x3 + 3px + 2q = (x – a)(x – b)(x – c) = 0.
Reduced equation: x2+ 2qx – p3 = (x – α)(x – β).
2) The quartic equation:
x4 + px2 + qx + r = (x – a)(x – b)(x – c) (x – d) = 0
Reduced equation:
x3 – px2 – 4rx + (4pr – q2) = (x – α)(x – β)(x – γ) = 0
2. SYMMETRY AND THE SOLUTION OF CUBIC AND QUARTIC EQUATIONS BY RADICALS
AUX. FCT.ROOTS
a b c d α β γ
α β γ a b c d
Three
roots:
a, b, c.
Two auxiliary functions:
α = [(a + bj + cj2) /3]3
β = [(a + bj2 + cj) /3]3
where j = (–1+i √3)/2,
j2 = (–1– i √3)/2,
with i being a square root
of –1, so that
j3 = 1 and j2+ j +1=0.
Symmetry:
The symmetry given
by any permutation of
a, b, and c leaves the
set of auxiliary
functions α,β globally
invariant.
Solutions:
Let u = √α and v = √β
such that uv = – p.
Then
a = u + v
b = j2u + j v
c = ju + j2v.
3
3
Four
roots:
a, b, c, d
Three auxiliary
functions:
α = ab + cd
β = ac + bd
γ = ad + bc
Symmetry:
The symmetry given by
any permutation of a, b,
c, and d leaves the set
of auxiliary functions α, β, γ globally invariant.
Solutions:
1) Knowing α=ab+cd and r=(ab)(cd),
gives the products ab and cd.
2) If ab=cd, the system (a+b)+(c+d)= 0
and cd(a+b) + ab(c+d) = – q,
gives a+b and c+d.
3) Knowing ab and a+b gives
a and b. Similarly for c and d.
FEATURE
EMS December 2004 13
Cardano’s method amounts to writ-ing α = [(1/3)(a + bj + cj2)]3, wherethe number j is the first cube root ofunity, namely (−1 + i
√3)/2, with i de-
noting the square root of −1. The cyclicpermutation taking a to b, b to c, andc to a simultaneously, clearly leaves αinvariant and the only other value ob-tained from the six possible permuta-tions of a, b and c is β = [1/3(a + cj +bj2)]3, where, for example, b and c havebeen transposed. As the set consistingof the two numbers α and β is invari-ant under all the permutations of a, b,and c, it is easy to express the quadraticequation whose roots are α and β, interms of the coefficients of the initialequationx3 + nx2 + px + q: it is x2 + 2qx − p3 =(x + q + s)(x + q − s), where s is one ofthe square roots of p3 + q2 and wherewe have also rewritten the initial equa-tion in the equivalent form x3 + 3px +2q (without the square term) by an ap-propriate translation of the roots. Weintroduced the factors 2 and 3 to sim-plify the formulas.
A simple calculation then shows thateach of the roots a, b, and c of the ini-tial equation can be expressed as thesum of one of the three cube roots ofα and one of the three cube roots of β.These two choices are connected by thefact that their product must equal −p(so there are only three pairs of choicesto account for, which is reassuring, in-stead of the nine possibilities which wemight have thought of a priori).
These formulas logically requiredthe use of complex numbers. Indeed,even in the case where the three rootsare real numbers, p3 + q2 can be nega-tive, and then α and β are necessarilycomplex.
The solution of cubic equations, de-scribed above, took a long time toreach its final form (we know that atleast one of the special cases was be-ing worked out by Scipione del Ferrobetween 1500 and 1515). On the otherhand, the solution of quartic equa-tions followed quickly, because it isalso in the Ars magna (Chapter 39),where Cardano attributes it to his sec-retary Ludovico Ferrari, who appar-ently found it between 1540 and 1545(Rene Descartes would publish an-other solution in 1637). And it is thissolution which brings us back to thefirst symmetry we met, that of the or-ganisation of soccer finals, the com-plete quadrilateral, and the expression
ab + cd. Here again, we can start witha polynomial with no term in x3 (usingthe same technique as before), namelyx4 + px2 + qx + r = (x − a)(x − b)(x −c)(x − d). The set of three numbers α =ab + cd, β = ac + bd, and γ = ad + bc,is invariant under each of the 24 per-mutations acting on a, b, c, and d. Thenumbers α, β and γ are thus the rootsof a cubic equation whose coefficientsare easily expressed as a function ofp, q, and r. A calculation shows thatthe polynomial (x − α)(x − β)(x − γ)equals x3 − px2 − 4rx + (4pr − q2). Itcan thus be decomposed, as we haveseen above, to yield α, β and γ. In-deed, it’s enough to find one of theseroots, α say, to determine a, b, c, andd (because we then know the sum αand the product r of the two numbersab and cd, thus giving these numbersvia a quadratic equation; all we thenhave to do is to exploit the equations(a + b) + (c + d) = 0 and ab(c + d) +cd(a + b) = −q to determine a + b and
3. THE ORDER OF A PERMUTATION
The order of a permutation is the
least integer n such that if the
permutation is applied n times, we
get the identity permutation. Thus
the permutation u has order 2 and
the permutation v has order 3 :
a b c d e
a b c d e
a b c d e
a b c d e
u =
u =
v =
v =
v =
a b c d e
uv = =
a b c d e
a b c d e
a b c d e
Checking that
the permutation
uv has order 5.
(uv)5 =
c + d and hence, finally, a, b, c, and d).The fundamental role of the permu-
tations of the roots a, b, c . . . and ofthe auxiliary quantities α, β . . ., wasbrought to light by Joseph Louis La-grange in 1770 and 1771 (publishedin 1772) and, to a lesser degree, byAlexandre Vandermonde in a memoirpublished in 1774, but written around1770, as well as by Edward Waringin his Meditationes algebraicae of 1770and by Francesco Malfatti. Today werightly call those auxiliary quantities‘Lagrange resolvents’.
The resolvents are not unique (wecould equally well have putα = (a + b − c − d)2 in the case of thequartic equation, which corresponds toDescartes’ method), but they are thekey to all general solutions by radicals.
Abel and Galois
Of course, mathematicians wanted togo further: Descartes certainly tried,and he wasn’t alone. The next stepwould clearly be that of the quinticequation. This was found to poseapparently insuperable obstacles, andsince the time of Abel and Galois (whoobtained their results around 1830) wehave known why the search was invain.
In all the previous cases, we wereable to find a family of n − 1 numbersα, β, γ . . ., determined as polynomialsof the n roots a, b, c, and d . . . (withn less than or equal to 4). This familywas globally invariant under the per-mutations of the roots. More precisely,if we let Sn denote the group of bijec-tions of the set a, b, c, d . . . with itself,what can be done for n strictly less than5 is to define a mapping of Sn onto Sn−1which preserves compositions of per-mutations.
Since the beginning of the nineteenthcentury, we have known that this is im-possible for n larger than 4. The same istrue for a (non-constant) composition-preserving mapping of the group An ofpermutations of even order (the prod-ucts of an even number of transposi-tions) onto the group Sm when m is lessthan n and n is greater than 4. Thisshows that Lagrange’s method can’tbe extended to the case n = 5 or tohigher values of n, but is, of course, notenough to show that a solution by rad-icals is impossible for the general equa-tion of degree 5 or higher: other, moregeneral, methods might succeed where
FEATURE
EMS December 200414
4. THE GROUPS A4 AND A5
a
b
c
d
s
d
a
b
c
t
[ [[ [
[ [
[[
[
[[ [
[ [ [ [ [ [ [
[
u
6.1
4
510
2
1
6. 5
4311b a d c
a c d b
[[ b d c a
A) THE GROUP A41) The group A4 is the group of the even permutations of
the four letters (a,b,c,d). This group is generated by the
permutations s = a b c d , which maps a to b, b to a, c
to d and d to c, and t = a b c d , which maps a to a, b
to c, c to d and d to b. They obey the rules:
s2 =1, t3 =1, and (st)3 =1.
2) There's a geometric representation of the group A4 : it's
the group of rotations preserving the regular tetrahedron
a,b,c,d .
s is represented by the symmetry with respect to the line
joining the midpoints of ab and cd.
t is represented by a rotation through an angle 2π/3 about
the axis of the tetrahedron passing through a.
st = a b c d is the rotation through 2π/3 about the axis
through c.
The patient reader may check that the rules of
simplification s2 =1, t3 =1, (st)3 =1 form a presentation of
the group, that is, together with the group law, they are
enough to show that there are only twelve distinct "words"
formed out of the letters s and t.
B) THE GROUP A51) The group A5 is the group of the even permutations of
the five letters (a,b,c,d,e). This group is generated by the
permutations u = a b c d e , which maps a to b, b to
a, c to d, d to c, and e to e, and v = a b c d e , which
maps a to e, b to b, c to a, d to d, and e to c. These obey
the rules: u2 =1, v3 =1, (uv)5 =1 (of course, u and v do not
commute).
2) This group has 60 elements and is isomorphic to the
group of rotations of a regular dodecahedron.
Thus u is one of the 15 symmetries with respect to a line
joining the midpoints of two edges which are themselves
symmetric about the centre.
Similarly, v is one of the 20 rotations through 2π/3 about a
line joining two vertices which are symmetrically placed
about the centre.
C) PRESENTATION OF A5 The patient reader may check that the simplifying rules u2
=1, v3 =1, (uv)5 =1 together with the group law, are enough
to show that there are only 60 distinct words formed out of
the letters u and v. We start by putting s = u and t = k–2uk
(where k = uv), and then show that s and t are generators
of A4, that is, they obey the simplifying rules s2 =1, t3 =1,
(st)3 =1. We then show, using the simplifying rules above,
that every word formed from the letters u and v can be
written in the form km h, where m equals 0,1,2,3,4 and h is
a word formed from the letters s and t.
As there are precisely 12 distinct words h, we see that the
group A5 is given by the above relations.
D) A5: A GROUP OF MATRICES1) Let F5 denote Z/5Z, the field of remainders modulo 5. In
this field, 4 + 2 =1, 3 + 2 =0, 4 x 2 = 3, 3 x 2 =1, etc.
2) We represent u and v as the following mappings of the
projective space P1(F5). This projective space contains the
five points of F5, together with a point "at infinity", denoted
by 1/0. Let us put u(z) = –1/z, for a point z in P1(F5).
Clearly, u2(z)=z, that is u2=1. Now let us put v(z) =
–1/(z+1) : we can check that v3=1. We see that we have a
representation of A5 because k=uv is given by k(z) = z +1
and km(z) = z+m, so that k5=1 since 5 equals 0 in F5.
3) We give a matrix representation of the elements u and v.
If a,b,c,d are elements of F5, with ad – bc =1, we associate
the matrix a b , with the mapping f of P1(F5), given by:
f(z) =
a b z .
1 c d 1
The group of mappings obtained in this way is called
PSL(2,F5), for "Projective Special Linear" group of F5.
Then u, v, k, and t, y are represented by the matrices:
u = 0 –1 v = 0 1 km = 1 m and t = –2 –3
1 0 –1 –1 0 1 1 1
14 6
.
6. 4
311
51
2 v
1
4
510
c d
[[b a d c e
[ [e b a d c
FEATURE
EMS December 2004 15
Figure 5: Lagrange’s text on 3rd degree equations(1772)
Figure 6: Icosahedral game and Hamiltonian circuitaccording to Sainte Lague
Lagarange had failed. Nowadays, be-cause of Abel and Galois, we knowthat even such a generalisation is im-possible.Many of the most celebratedmathematicians were interested in thisfundamental and complex problem,among them Leonhard Euler, who re-turned to it several times and, aboveall, Karl Friederich Gauss (1801) andLouis-Augustin Cauchy (1813).
We’ll stay with the case of the quinticequation. Descartes, for one, was con-vinced that no formula like that of Car-dano could be found. In 1637 Descartessuggested a graphical solution – us-ing the intersection of circles and cubiccurves – which he had invented for thepurpose. From 1799 to 1813 (the dateof publication of his Riflessioni intornoalla solutione delle equazioni algebraichegenerali), Paolo Ruffini published di-verse attempts at proofs, each of themmore refined than the last, attempt-ing to demonstrate the impossibility ofsolving the general quintic equation byradicals. He had the correct idea ofassigning, to each rational function ofthe roots, that group of permutations of
the roots which left the function invari-ant. However, he assumed incorrectlythat the radicals involved in solving theequation necessarily had to be rationalfunctions of the roots.
In the event, it was 1824 before NielsAbel, in his Memoire sur les equationsalgebriques, justified Ruffini’s intuition.Abel, having at first thought, on thecontrary, that he had found a generalmethod of solution, proved the impos-sibility of solving the general quinticby radicals in his 1826 Memoire sur uneclasse particuliere d’equations resolublesalgebriquement, in which he sketched ageneral theory which would only befully worked out by Galois, towards1830. Galois’ work inaugurated a newera in mathematics where calculationsgave way to the consideration of theirpotential, and concepts, such as thoseof abstract group or of algebraic exten-sion, occupied the foreground.
Galois’ great insight was to associateto an arbitrary equation a group of per-mutations which he defined in theseterms:
Let an equation be given which has rootsa, b, c . . . m. It will always have a groupof permutations of the letters a, b, c . . . mwith the following properties:
1. every function of the roots whichis invariant by substitutions of thisgroup, will be rationally determined;
2. conversely, every rationally deter-mined function of the roots will beinvariant under these substitutions.
Galois then studied how this group of‘ambiguities’ could be modified by ad-joining auxiliary quantities thenceforthconsidered ‘rational’. Solving an equa-tion by radicals was then reduced tosolving its Galois group.
The impossibility of reducing thequintic equation to equations of lowerdegree comes from the ‘simplicity’ ofthe group A5 of the sixty even permu-tations of the five roots a, b, c, d, and eof the quintic. We say that an abstractgroup is ‘simple’ if there is no non-constant composition-preserving map-ping of the group into a smaller group.
FEATURE
EMS December 200416
The group A5 is the smallest non-commutative group which is simple,and it arises very frequently in math-ematics. This group can be describedvery economically: it is generated bytwo elements u and v satisfying the re-lations u2 = 1, v3 = 1 and (uv)5 = 1,which gives us an excuse to move toHamilton’s icosions.
Hamilton’s icosions
After discovering quaternions, WilliamHamilton tried, in 1857, to constructa new algebra of generalised numbers,which he called icosions. Two of them,denoted by u and v, which Hamil-ton termed ‘non-commutative roots ofunity’, were to satisfy u2 = 1, v3 = 1and (uv)5 = 1. A childishly simple cal-culation shows that, if uv = vu, thenwe have v = 1v = u5v5v = u5v6 =(u2)2u(v3)2 = u, so u = v = vu2 = v3 =1. So we can’t represent u and v as ele-ments of the groups Sn for n less thanor equal to 4. To represent u and v as el-ements of the group A5 of even permu-tations of the five letters a, b, c, d, ande, it is enough to put u = (b, a, d, c, e),the permutation which fixes e but ex-changes a with b and c with d, and toput v = (e, b, a, d, c), the permutationwhich fixes b and d but changes a toe = v(a), c to a = v(c), and e to c = v(e).The product uv is then the cyclic per-mutation (eabcd) which is indeed of or-der 5. In fact, we could similarly repre-sent u and v in altogether 120 separate(but pairwise isomorphic) ways as ele-ments of A5.The group A5 is isomorphic to thegroup of rotations preserving a regu-lar icosahedron or, which amounts tothe same thing, a regular dodecahe-dron (these are the two most interest-ing of the five platonic solids, whoseother members are the regular tetrahe-don, the cube and the regular octahe-dron, formed by the six centre of thefaces of the cube. These are the onlyregular convex polyhedra which existin our usual three-dimensional space).To construct the isomorphism referredto above, it is enough to associate uwith one of the 15 rotations of order two(a symmetry whose axis is one of the 15lines joining the midpoints of paralleledges) and to associate v with one of the20 rotations of order three (whose axisof symmetry connects one of ten pairsof two diametrically opposite verticesof the dodecahedron, or the centres oftwo parallel faces of the icosahedron) insuch a way that the product uv is one of
7. THE UNIVERSAL COVERING AND NON-EUCLIDEAN GEOMETRY
To get a geometric understanding of the group generated by two elements
u and v and presented by the relations u2=1, v3=1, and (uv)5=1, we start by
considering the two first relations (u2=1, v3=1). The group thus generated
is the group PSL(2,Z) which can be understood by looking at its action on
an infinite tree T in which three edges exit from each vertex. The third
relation ((uv)5=1) can then be understood by identifying the tree T with the
universal covering of the graph in box 8 below. The universal covering, in
the sense of Poincaré , is obtained by considering all the paths which follow
the edges of the regular dodecahedron.
The infinite tree T is represented by two models of non-Euclidean geometry:
Klein's model (A) and Poincaréé 's model (B). In each model, the set of points
of plane geometry lie in the interior of a disk. In Klein's model, the straight
line joining two points is the same straight line as in Euclidean geometry, only
the length of the line is changed. In this model, the length of a straight line is
given by the logarithm of the cross-ratio (ab,cd) of ab with the points c and d
where the line ab intersects the circle C. Thus, [a,b] = log ac x bd . The
edges of the tree T are line segments of equal length.
In the Poincaré model, the "straight lines" are arcs of circles otrhogonal to the
circle C, and the idea of angle remains the same as in Euclidean geometry.
The distances are given by 2log(ab, cd) where the cross-ratio is calculated
on the circle through (a,b,c,d), and where the factor 2 arises in comparing (A)
with (B).
The group PSL(2, Z) is represented by isometries of non-Euclidean
geometry. A presentation of this group is given by the relations u2=1 and
v3=1. The element u is given by symmetry about the origin O, and the
element v, by the non-Euclidean rotation with centre J and angle 2π/3. We
operate on the edge JJ' by the transformations represented by the words
(such as uvvuuuvuv...) whose letters are the elements u and v. This gives
exactly the tree T of the universal covering of the graph of Hamilton's icosion
game. In particular, each Hamiltonian path is a point of the universal
covering.
We get the dodecahedron by identifying the edges of the tree T which are
congruent modulo 5. This congruence means that we pass from one edge to
another by a non-Euclidean isometry given by an element of the group
PSL(2, Z) satisfying a = 1 (modulo 5), b = 0 (modulo 5), c = 0 (modulo 5),
and d = 1 (modulo 5). In this way, the group A5 whose presentation requires
the extra relation (uv)5 = 1, arises from quotienting PSL(2, Z) by the normal
subgroup G generated by (uv)5.The quotient of T by G is precisely the graph
formed by the edges of the dodecahedron. The quotient group PSL(2, Z)/G
is the group PSL(2, F5) of box 4.
ad x bc
[ ]a bc d
c
O
J J'
v
v2
uv2vuv2
uv
vuv
a bd
ca bd
Length of segment [a,b] = Log ac x bd
ad x bc
A KLEIN'S MODEL
B POINCARÉ'S MODEL
J J'
v
v2
uv2vuv2
uv
vuv
O
FEATURE
EMS December 2004 17
the 24 rotations of order five (whoseaxis of symmetry connects one of thesix pairs of centres of parallel faces ofthe dodecahedron, or pairs of diamet-rically opposed vertices of the icosa-hedron). The 60 rotations preservingeither solid can be expressed simplyas products of the generators u and v.Even though the two icosions u and vgenerate the group A5 and satisfy therelations u2 = 1, v3 = 1 and (uv)5 = 1,it’s not immediate that these relationsconstitute a presentation of the group,that is, it’s not immediate that everyrelation between u and v follows fromthese. There are two ways, algebraic orgeometric, of showing this (boxes 4 and7).
The graph of the edges of the dodec-ahedron, which has the same symme-tries as that of the icosahedron, gaverise to Hamilton’s ‘icosahedral game’which he also called the ‘game of non-commutative roots of unity’. Thisgame is the first example of what isnow called the search for a Hamilto-nian circuit, which is a very importantconcept in modern graph theory (box8). The challenge is to pass preciselyonce through each vertex of the dodec-ahedron, using only the edges, and tofinish at a vertex which is joined byan edge to the starting point. A re-markable account of this is given inAndre Sainte-Lague’s 1937 essay Avecdes nombres et des lignes, reissued byVuibert in 1994.
In his book Avec des nombres et
des lignes Sainte-Lagu brought the
game of icosions, invented by the
Irish mathematician Hamilton (1805-
1865), back to life. The game
consists of completing the circuit
going through all the vertices of an
icosahedron once and once only: to
start, the first six vertices are given.
Here's and example: (1, 2, 3, 4, 5, 6,
19, 18, 14, 15, 16, 17, 7, 8, 9, 10,
11, 12, 13, 20).
1
220
3
45
6
7
8
9
11
10
1213
14 15
16
17
1819
8. THE GAME OF ICOSIONS
Morley’s triangle
It is wrong to oppose the ’angel ofgeometry’ with the ’devil of alge-bra’. What takes place is a fruitful co-operation between the visual parts ofthe brain, which can perceive the har-mony of a configuration at a glance,and those parts of the brain which pro-cess language and distil that harmonyinto algebraic formulae. We shall endthis introduction to the idea of sym-metry with a nice example of that co-operation by way of Morley’s theorem.This is a subject where concrete sym-metries arising from geometry becomeabstract and algebraic when we lookat them from a different angle. Theirforceful interplay gives a genuine senseof beauty.
The British mathematician FrankMorley was one of the first universityteachers in America. At the end of thenineteenth century, while pursuing re-search into families of cardioids tan-gent to the three sides of a given trian-gle, he discovered the following prop-erty: the three pairs of trisectors of theangles of a triangle (that is, the straightlines which divide the interior anglesinto three equal parts) intersect in sixpoints, of which three are vertices of anequilateral triangle.
The original proof is quite difficultand depends on ingenious calculationsbased on a profound mastery of ana-lytic geometry. There are now manyproofs of this result, as well as gener-alisations which produce 18, or 27 (oreven more) equilateral triangles whichcan be constructed from the 108 pointsof intersection of the 18 trisectors ob-tained from the original ones by rota-tions of π/3. These proofs include onesby trigonometric calculation as well aspurely geometrical ones, such as thatgiven by Raoul Bricard in 1922.
There is a proof which is entirelydifferent, which illuminates the resultfrom an interesting angle, because itlets us extend the result (a priori en-tirely Euclidean) to the geometry ofaffine lines over an arbitrary field k.The purely algebraic result, which in-cludes and extends the trisector prop-erty, is so general that its proof be-comes a simple verification (a verygeneral result is often easier to provethan a particular case, because the gen-erality can reduce the number of hy-potheses).
The statement is as follows:
If G is the affine group of a commu-tative field k (that is, the group of map-pings g of k into k which can be writ-ten in the form g(x) = ax + b, wherea = a(g) is non-zero), then for eachtriple ( f , g, h) of elements of G suchthat j = a( f gh) is not the identity andsuch that f g, gh, and h f are not transla-tions, the following two assertions areequivalent:
a) f 3g3h3 = 1 (the identity mapping1(x) = x);
b) j3 = 1 and α + jβ + j2γ = 0,where α is the unique fixed pointof f g, β that of gh, and γ that ofh f .
We need to show how this very ab-stract property helps us better under-stand (and, at the same time, prove)Morley’s theorem. We shall take k tobe the (field of) complex numbers. Itsaffine group is that of the direct simi-larities and has the rotations as a sub-group (precisely when a has modulus1). We let f , g and h be the rotationsabout the three vertices of the trianglewith each angle of rotation being two-thirds of the relevant angle of the tri-angle. Thus f is the rotation with cen-tre A and angle 2a (the internal angleat A being denoted by 3a), g is thatwith centre B and angle 2b and h thatwith centre C and angle 2c. The prod-uct of the cubes f 3g3h3 is 1, because, forinstance, f 3 is the product of the twosymmetries with respect to the sides ofthe angle at A, so that these symme-tries simplify pairwise in the productf 3g3h3.
The equivalence above shows us thatα + jβ + j2γ = 0, where α, β, and γare the fixed points of f g, gh, and h f ,and where the number j = a( f gh) isthe first cube root of unity, which wehave already met in the course of thisarticle. The relation α + jβ + j2γ = 0 isa well-known characterisation of equi-lateral triangles (it can be written in theform (α − β)/(γ − β) = −j2, whichshows that we pass from the vector βγto the vector βα by a rotation throughπ/3).
An old recipe, well-known to thosethoroughly trained in the rigours ofclassical geometry, shows that thepoint α, defined by f (g(α)) = α, isnone other than than the intersection ofthe trisectors of the angles A and B ly-
FEATURE
EMS December 200418
9. MORLEY'S THEOREM
BA
C
aaa
bb
b
c c c
γ β
α
a
ab
b
α
α'
A B
g f
Morley's theorem states that the three meeting points α, β, γ of the trisectors of an arbitrary triangle ABC, as
indicated in the diagram, form an equilateral triangle. (n
red).
f, g, h are the three rotations about the vertices of the
given triangle, through angles which are two thirds of the
angles at the vertices.
So f is the rotation through 2a about A, g is the rotation
through 2b about B, and h is the rotation through 2c about
C. We shall look at the properties of these rotations. The
rotation g takes the point α to α', which is the reflection of α through AB. The rotation f takes the point α∋ to α: so α is a
fixed point of the product of rotations fg. Similarly, β is a
fixed point of the product of rotations gh, and γ is a fixed
point of the product of rotations hf.
Now we consider the product f3g3h3. The rotation f3
through an angle 6a about A is the product s(AC) s(AB)
and the reflection s(AB) in the side AB, and the reflection
s(AC) in the side AC. Similarly, g3 is the product s(AB)
s(BC) and h3 is the product s(BC) s(AC). So f3g3h3 =
s(AC) s(AB) s(AB) s(BC) s(BC) s(AC). As the square of a
reflection is equal to 1 this, together with the algebraic
theorem, shows that the triangle is equilateral.
f 3
BA
C
x'
x
x"
Reflectio
n AC
Reflection ABv
ing closest to the side AB. The readercan be convinced of this by checkingthat the rotation g, centred on B, of an-gle 2b, transforms the point of intersec-tion into its symmetric image with re-spect to the line AB and that the rota-tion f , centre A, of angle 2a, returnsthe point to its original place. Thesame applies to the points β and γ. Wehave thereby shown that the triangleABC is equilateral. As a bonus, we cansee that, in this order, the triangle isdescribed in a positive direction (anti-clockwise). This proof applies equallyto the other Morley triangles: the 18 tri-sectors obtained from the interior tri-sectors by rotations through π/3 en-able us to modify f , g and h withoutchanging the product of their cubes,thus giving new solutions of equationa), and new equilateral triangles.
The evident duality of algebra andgeometry in the examples above al-
lows us to expand the boundaries ofour knowledge of geometry, alreadyliberated from its Euclidean chains bythe arrival of non-Euclidean geome-tries (box 7).
The discovery of quantum mechan-ics and of the non-commutativity ofthe coordinates of phase space for anatomic system has given rise, duringthe past twenty years, to what can beseen as an equally radical evolution ofgeometric concepts, freeing the idea ofspace from the commutativity of co-ordinates.
In non-commutative geometry, theidea of symmetry becomes more sub-tle, the groups sketched in this arti-cle being replaced by the algebras in-vented by the mathematician HeinzHopf, exemplifying Hermann Weyl’sfine definition, taken from his bookSymmetries ‘. . . the idea [of symmetry]is by no means restricted to spatial
objects;. . . symmetric means somethinglike well-proportioned, well-balanced,and symmetry denotes that sort of con-cordance of several parts by whichthey integrate into a whole.’
Alain Connes [[email protected]] is awinner of the Fields Medal and theCrafoord Prize. Recently, he receivedthe Gold Medal 2004 of the FrenchCNRS. He is a professor at the Collegede France and the Institut des HautesEtides Scientifiques. He would liketo thank Andre Warusfel for his in-valuable help in preparing this arti-cle, originally a lecture organised byJean-Pierre Bourgignon at the CentreGeorges Pompidou in September 2000.
The Newsletter would like to thankDavid Salinger (Leeds, UK) for thetranslation of the article into English,and also Martin Qvist (Aalborg, Den-mark) for help with the management ofthe figures.
Apology
Unfortunately, a very disturbingmistake occurred during the printingprocess of the recent issue 53 of theNewsletter. In the feature article 25years of Kneser’s conjecture by Marc deLongueville, p.16 – 19, all minus-signs
in the formulas and also the line break-signs disappeared. The Newsletterwishes to apologise sincerely for theinconvenience that this has caused forthe readership; its apologies go also tothe author. A corrected version of this
article can be read and downloadedas a PDF-file from the following URLhttp://www.emis.de/newsletter/current/current9.pdf.
FEATURE
EMS December 2004 19
Mathematics is alive and well and thriving in Europe
Luc Lemaire (Universite Libre de Bruxelles)
This is the text of a lecture given at the first Euroscience Open Forum in Stockholm in August 2004. The text will keepunashamedly the characteristic of the lecture: no previous mathematical knowledge is assumed.
Numbers
This being about mathematics, I mayas well start with numbers, and shallwrite three short formulas.
The first formula will have essen-tially no ingredients, except the posi-tive integers : 1,2,3,4,... Let us write :
1 +122 +
132 +
142 + . . . = A.
Whenever I stop summing after a finitenumber of terms, I just have a sum offractions that can be computed by handor with a pocket calculator.
But the ... means that I want tocontinue summing forever, writing thesum of an infinite number of terms.This should not frighten anybody, weknow of other examples of such infinitesums yielding a finite number, like
310
+3
100+
31000
+ . . . = 0, 3333... =13
.
In the first formula, the sum also givesa number, which we’ll call A.
The second formula will have morecomplicated ingredients, namely thecomplete list of prime numbers. Re-call that a prime number is an inte-ger ≥ 2 which has no divisor except1 and itself. So 6 is not prime be-cause 6 = 2 × 3, and 7 is prime be-cause such a decomposition does notexist. The first prime numbers are2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31... Primenumbers have always fascinated math-ematicians and Euclid, 24 centuriesago, proved that there is an infinitesupply of primes.
Note that his proof is so perfect,short and elegant that it is still the bestproof of this result, in fact it can be usedas a first illustration of what a proof is.
And now we consider the infiniteproduct(
1 − 122
) (1 − 1
32
) (1 − 1
52
) (1 − 1
72
) (1 − 1
112
)...
Like the infinite sum above, this yieldsa (finite) number and we write :
B =1(
1 − 122
) (1 − 1
32
) (1 − 1
52
) (1 − 1
72
)...
The third formula will use an in-gredient from a totally different area,namely classical geometry. As we allknow, the length of a circle of radius Ris given by 2πR, where π has approxi-mate value
π = 3.14159265358979323846...
Let us write
C =π2
6
So we have written three numbers,coming from different areas, with noapparent relation. Yet, one can showthat A = B = CA = B = CA = B = C, they are the samenumber.
It you pause to think about it (do it!)this is unbelievable. It means in partic-ular that some facetious god of mathe-matics has encoded the length of a cir-cle in the list of prime numbers, totallyunrelated a priori.
We draw three first conclusions:1) Mathematics has beauty and
magic.2) Mathematics is not about specific
fields like algebra, geometry, analy-sis,... but is about the relations be-tween different areas, allowing us touse methods of one to solve problemsof the other.
3) Ancient notions and proofs are asfresh today as 24 centuries ago.
Let us push a bit further. In theabove formulas, could we replace thesquares by cubes? In fact we have
1 +123 +
133 +
143 + . . . =
1(1 − 1
23 )(1− 133 )(1 − 1
53 ) . . .
but this is no longer a fraction timesπ3.
Likewise, provided x is a real num-ber greater than 1, we have
1 +12x +
13x + . . . =
1(1 − 1
2x )(1 − 13x )(1 − 1
5x ) . . .
This has now become a function of thenumber x, called the zeta function ofRiemann : ζ(x).
In 1859, in a prodigious paper, Rie-mann analyses this function and showsits deep relation with number theory,and the distribution of prime numbersin particular.
For that, he shows that ζ can be ex-tended as a function defined also forx < 1, and even to the case of ζ(z),where z = x +
√−1y is a complexnumber. He states his belief that thezeros of zeta, i.e. the values of z forwhich ζ(z) = 0, beyond the “easy”ones −2,−4,−6, ... are all situated ona line, namely are all of the form z =12
+√−1y.
Unable to prove it, he assumes it asa hypothesis, leaving to others the taskof proving it.
The “Riemann hypothesis”, still un-proven, is considered by many as themost important single open question inmathematics today.
Why is this obscure looking questionabout the zeros of a specific compli-cated function so important? From
Mathematics is alive and wellMathematics is alive and welland thriving in Europeand thriving in Europe
Luc Lemaire (Universit´e Libre de Bruxelles)
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EMS December 200420
Figure 1: Graphs of the Riemann zeta function
the first formulas, we get a glimpsethat it is related to questions of numbertheory. In fact, hundreds of theoremsabout prime numbers, or about the-oretical computer science, have beenproven under the assumption that Rie-mann’s hypothesis is true, and werenot proven otherwise. So when - or if- the hypothesis is proven, hundreds ofother results will be proven at the sametime.
In 1900, the great German math-ematician David Hilbert drew a vi-sionary list of 23 open problems thatshaped part of the development ofmathematics in the twentieth century.
Riemann’s hypothesis is one ofthem, and the only specific question ofthe list unsolved after one century.
In 2000, the Clay foundation (estab-lished by philanthropist Landon Clay)announced a list of 7 Millennium prob-lems and, times having changed, of-fered a reward of one million dollarsfor each. Riemann’s hypothesis is - ofcourse - one of them.
The main effect of trying to solvesuch a difficult problem is to relate it tonew results and theories - a very posi-tive result even when it does not solve
the problem. Today, the obscure ques-tion of Riemann appears to be at thecentre of a spider web of mathematicalfields and theories.
Physics has also joined the web:Michael Berry related the zeros of zetato quantum chaos, and Alain Connes tothe eigenvalues of an operator, similarto the absorption spectrum of a star.
But is all this confined to the realmof “pure” or fundamental mathemat-ics, highly amusing to mathematiciansbut unrelated to real life?
G. H. Hardy, who contributed to theRiemann hypothesis in 1914, claimedthat number theory’s beauty was re-lated to its uselessness, so that it wasnot related to dull realities or applica-tions.
Well, Hardy was wrong, and all ofyou use prime numbers regularly, butwithout knowing it.
Indeed, whenever you use a bankmachine, or order anything throughthe internet, your bank data is of courseencrypted. And the only totally safeencrypting code today is based onprime numbers.
Basically, it works as follows:
Anybody who wants to receive en-coded messages will choose two verylarge prime numbers - say of one hun-dred digits. He will not reveal to any-one the two chosen numbers, the “se-cret key”.
But he will compute the product ofthe two numbers, obtaining a two hun-dred digit number that he will adver-tise freely (the public key).
Using some clever mathematics(available as a software), anybody whowants to send him a message will en-crypt it using the 200 digit numbersand can send it without much care.Indeed, the only possibility to decodeit is to know the two prime numbers,known only to one person.
This is the principle of RSA cryp-tography, created in 1977 by Rivest,Shamir and Adleman (and slightly ear-lier by Ellis, Cocks and Williamson ofthe British secret service, but they keptit secret).
But, you might say, the two hun-dred digit number is decomposable ina unique way as the product of the twooriginal prime numbers, and it wouldbe sufficient to find that factorisation todecode all messages that use this num-
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ber.The point is that it would take cen-
turies for the larger computer to findthe decomposition, because of the sizeof the numbers.
Could one maybe find by chance oneof the prime numbers, hence the other?The odds would be no better than try-ing to pick up at random a specific par-ticle of the known universe.
So we get two more conclusionsabout mathematics
4) Old notions still give rise to themost pressing open problems of todayand tomorrow.
5)“Pure” mathematics, studied onlyfor the sake of elegance and beauty,suddenly finds crucial applications toscience or economic development. Toquote the physicist Eugene Wiegner,this is the “unreasonable effectivenessof mathematics applied to natural sci-ences”.
Nobel prizes, Fieldsmedals, Abel prizes andeconomic development
There are no Nobel prizes in mathe-matics. The mathematicians managedto keep alive for decades the legendthat Alfred Nobel’s girlfriend elopedwith the Swedish mathematician GostaMittag-Leffler, so that Nobel would notcreate a prize that might go to his ri-val. However, Nobel’s sex life is not sowell documented, and the fun went outof the story when geologists and oth-ers tried to spread similar stories abouttheir science.
The point is much more probablythat Nobel - as an industrialist - wasinterested in “inventions and discover-ies” and didn’t see mathematics fittingthere.
Anyway, the mathematicians createda prize recognised today as the “No-bel of Mathematics”: the Fields medal,first awarded in 1936.
There are three main differences be-tween Nobel prizes and Fields medals.
Compared to Nobel prizes, themedal comes only with a minimal fi-nancial prize.
Every four years, 2, 3 or 4 medals areawarded and they are never shared be-tween mathematicians for a joint dis-covery.
The major difference is an upper age
limit of 40 for the award of a medal,illustrating the fact that in most casesmathematical genius can be detected ata young age.
For example, Jean-Pierre Serre wasawarded the medal in 1954 at the ageof 27 - because he was already an ac-knowledged master with impressivediscoveries.
On the other hand, Andrew Wiles,who achieved eternal (and even news-paper) fame by proving Fermat’s lasttheorem (a question open since 1637),did not get a Fields medal because hewas 41 when his proof was completed.
Now, many political and economi-cal studies comparing the effectivenessof scientific research in various conti-nents refer to Nobel prizes as an indi-cator and draw a negative conclusionabout Europe.
Indeed, between 1980 and 2003, theNobel prizes in biology and medicine,physics and chemistry give :
68 for Europe154 for the USA
with the gap growing with time.For mathematics, the Fields medals
give a much better picture for Europe.In the same period, Fields medals
were awarded to :9 Europeans (one working in the
USA)5 U.S. citizens4 Russians (one working in Paris,
two in the US)1 Japanese1 New Zealander (working in the
US)All together, including Russia in Eu-
rope where it should be, we get10 working in Europe9 working in the USA
a success story for Europe.Fields medals are awarded for recent
developments, so we see no growinggap at all in mathematics.
More recently, to celebrate the twohundredth anniversary of the birthof Niels Henrik Abel, the Norwegiangovernment created a prize for math-ematics similar to the Nobel prize, fit-tingly called the Abel Prize.
After two awards, the laureates areJean-Pierre Serre (49 years after hisFields medal), Michael Atiyah andIsadore Singer - two Europeans andone US citizen.
To sum up, mathematics in Europe isat top level, quite cheap to run and ex-tremely efficient.
It must be acknowledged that theusefulness of fundamental mathemat-ics will not always appear quickly(even if 24 centuries from Euclid to in-ternet is an extreme example).
But as stated by Timothy Gowersin his millennium address at the Clayfoundation: “If you were to work outwhat mathematical research has costthe world in the last hundred years,then work out what the world hasgained in crude economic terms, youwill discover that the world has re-ceived an extraordinary return on avery small investment”.
Since return is not immediate, it fallsoutside the scope of an industrial com-pany that needs to meet its objectiveswithin a few years. Needless to say, itis outside the aims of a financial groupbuying a company with the sole aim tosell it after five years, after raising itsstock exchange value and nothing else.
Thus, funding for fundamental cu-riosity driven mathematics must comefrom public money, for everybody’slong term interest.
This should be managed by pro-grammes better suited for mathematicsat the level of the European Commis-sion and the national policies.
To include this better in the overallscientific planning of the E.U., a mini-mum step would be to include a math-ematician in EURAB, the 45 memberadvisory committee of the EuropeanCommission on science (so far, 67 peo-ple have been members of EURAB -none of them mathematicians).
It is sometimes said that mathemati-cians work by themselves in their of-fice, with a pen and a piece of paper.First one should not forget the wastepaper basket, much used in mathemat-ical research.
And then there are the serious needsfor mathematical work.
First, of course, we need positions,either in universities or research insti-tutes. Mathematics is done by people -and this is the priority.
In Europe, we face the same para-dox as in other sciences. Indeed, weeducate high level mathematicians upto the doctoral level, then have somepost-doctoral positions, but usually donot provide the bridge between thesepositions and the tenured ones. At that
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stage, the USA step in and offer tenure-track jobs to scientists fully educated inEurope.
We should strongly promote the cre-ation of timely tenure-track positionsin our universities.
In this respect, I believe the Euro-pean Commission is making a mistakein putting the accent on “training” ofdoctoral students (up to four years re-search experience) and not more ex-perienced ones (up to 10 years) in allits Marie Curie activities. Doesn’t thissimply educate more scientists to beswallowed by the American system ?
A special consideration should begiven to Central and Eastern Europe.The extremely high level of the math-ematical tradition there explains whyfor instance we have four RussianFields medallists out of twenty. But thedramatic economic situation explainsthat only one remained in Russia.
A smallish investment to preservethat tradition would be in everyone’sinterest.
Secondly, mathematicians need reg-ular contacts with other researchers,the world over. This happens duringconferences, and by short or long termvisits.
These provide an immense accelera-tion of research. In one or two hours,a specialist can explain the basics andrecent trends in his field, whereas itwould take months to get that informa-tion from the literature. Also mathe-maticians are not working together invery large centres, so they regularlyneed to see specialists in their domain,often rather thinly spread.
Thirdly, easy access to the literatureis necessary. We have seen that olderarticles keep all their value, and accessis needed to all good level literaturepresent and past.
No university library today can keepup with the rising cost of journals, butelectronic access offers new unprece-dented opportunities.
Most new papers are typed in thesoftware TEX, and can be made acces-sible to all.
Another objective is to digitise thewhole literature - an operation es-timated at around 50 million euros.Good co-ordination is needed, so thatall digitised papers are accessible withthe same standard. Also, the owner-ship of the database should not be leftto purely profit-making organisations,
and this again requires public fundingnow.
Note that this database could bemade available at minimal cost in lessdeveloped countries, where high levelmathematics is present but faces majoreconomic difficulties.
Finally, note that mathematical re-search is accomplished both in univer-sities and in a string of high level re-search centres, with regular movementof researchers from one to the other, sothat all are necessary.
Differential equations
Since the invention of calculus by New-ton and Leibnitz, differential and par-tial differential equations have beenthe central tool in mathematics appli-cable to science - first astronomy andphysics, then chemistry, biology andnow economy and finance.
It is a huge subject, both in funda-mental mathematics and its applica-tions.
To take but a specific example, I’llconsider the Navier-Stokes equations.They are a system of partial differentialequations that model the movementsin fluid dynamics. They were writtenby Navier in 1822, then justified moreprecisely by Stokes a few years later.
They are both extremely useful in ap-plications and extremely hard to studywith mathematical rigour.
After almost two centuries, we haveno formula giving the solutions (this isusual for partial differential equations),and we don’t even have decent resultson their existence and properties.
In fact, their mathematical study isthe object of one of the seven Millen-nium problems of the Clay foundation.
But still they are used in applicationslike shaping cars and planes, mod-elling the flow of blood in the cardio-vascular system and many others.
I shall briefly describe an applica-tion developed at the Fraunhofer Insti-tute of Applied Mathematics in Kaiser-lautern, Germany: the conception ofairbags for cars.
For many years, it was an inaccessi-ble idea. Indeed, in case of an accident,the bag has to be inflated fully in 1/20of a second or it is too late.
Everything will count: the way thegas is injected, the shape of the bag, the
way it is folded.
Figure 2. Airbag simulation
The most efficient (and cost efficient)way to find the best shape is by com-puter modelling. The researchers inKaiserslautern developed a computersoftware that allows them to modelany shape and gas speed, and look foran efficient one. They can vary theshape and folding many times and ob-serve the blowing up of the bag.
Comparing the “films” of theirmathematical bags with the film of con-crete realisation of the bags built af-terwards shows that they obtain a re-markable precision.
But this achievement required thedevelopment of new mathematics.
The standard way to model partialdifferential equations on a computer isto choose a mesh on the domain ofthe equation, then approximate deriva-tives by differences of the values at thedifferent vertices.
Here the domain (the airbag) variesquickly, because of the varying gaspressure. Therefore, “mesh free” meth-ods have to be developed.
We see here the same scheme ap-pearing in an endless series of exam-ples. There is first a mathematical the-ory, maybe fairly old. It is still the ob-ject of difficult theoretical questions. Italso gives rise to unexpected applica-tions which in turn give rise to newmathematical questions. Their studywill provide new theories which againwill motivate new problems and giverise to new unexpected applications.
Talking in Stockholm on partial dif-ferential equations, I should mentiona grand master of the subject: LarsHormander.
His early work was so exceptionalthat he was appointed full professor inStockholm at the age of 25.
At the age of 31 he got a Fieldsmedal, but moved to the Institute for
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Advanced Study in Princeton.This brain drain provoked a strong
reaction in Sweden, and the parliamentvoted the “Lex Hormander”, a law al-lowing the creation of personal chairsin exceptional case.
Hormander came back to Lund andthe law also allowed them to attractback Lennard Carlesson, another majorfigure.
Together, they had 36 PhD studentsin Sweden, who in turn had PhD stu-dents, so that the total number of theirmathematical descendants in Swedenis now over 180.
Thus, their presence in Swedenhelped not only the general reputationof the country, but more concretely thelocal development of science and in-dustry.
Let us have a Lex Hormander at theEuropean level!
Pour l’honneur de l’esprithumain
So far, this lecture has concentratedon the far reaching economic benefitsfollowing the development of funda-mental mathematics. I described onlytwo examples, for lack of space, butcould go on endlessly in most if notall fields of human activity. Fast devel-oping examples include medical imag-ing, image compression for storage ortransmission, epidemiology, biostatis-tic, mathematical genomics, finance,control theory for plane safety or en-ergy saving ...
In this presentation, I followed thepresent day trend of having to provethe economic value of all activities, in-cluding art, mathematics, and culture.
But this is a sad evolution of our so-ciety, and mathematics should also bepursued “for the honour of the humanmind”, to coin the phrase of Carl Gus-tav Jacobi.
So I want to point out now thatmathematics, like art, philosophy andscience, are essential parts of civilisa-tion.
The obvious example is the Greekcivilisation, which left us as heritage(with contributions of the Arabic civil-isation) art, mathematics, philosophyand the beginnings of science. It isthese four aspects, and not the value oftheir stock exchange or whatever theyhad, that founded the rebirth of ourcivilisation after the Renaissance.
Civilisation as we live it is not de-fined in economic terms, and is ourmost precious asset.
The princes of Medicis, wealthy asthey were, will be remembered foreverfor triggering that Renaissance.
Likewise, how should we remem-ber Carl Wilhelm Ferdinant, Duke ofBrunswick?
In my dictionary, he is described asa duke soldier who was beaten by theFrench in Valmy, then again in Iena.
But I must say I looked only in aFrench dictionary. Still, an uninspiringnotice.
But one day, he got a report froma school teacher that a young boyseemed remarkably gifted in mathe-matics. The boy was the son of a poorgardener and bricklayer, so his futureshould have been rather bleak.
But the Duke liked mathematics,saw the boy and was convinced by hisobvious talent (if not by his good man-ners). Thus he supported his studiesand career throughout his life.
The boy’s name was Carl FriedrichGauss, and we owe to him (and theDuke) the Gauss law of prime num-bers, the Gauss distribution in prob-ability, the Gauss laws of electromag-netism, most of non-Euclidean geom-etry, and the Gauss approximation inoptics.
Obviously, we need more Dukes ofBrunswick in our governments.
More to the point, we need a Euro-pean Research Council to cater for fun-damental research.
Going back to the idea of civilisa-tion, it seems that abstract mathemat-ics - disjoint from practical applications- appears early in the development ofthe human mind.
The first mathematical ideas werepractical: counting objects (like cattle),computing the area of a field, the vol-ume of a pyramid.
But then why did Euclid care aboutprime numbers, or abstract proofsof geometric theorems? Why didthe Babylonians, 1000 years beforePythagoras, engrave his theorem ontheir clay tablets? Why did the Egyp-tians develop a sophisticated and quiteuseless system of fractions?
On a bone, found by anthropologistJean de Heinzelin in Ishango, Africa,a series of scratches provides the be-ginning of a multiplication table and ashort list of prime numbers. The boneis dated between 7000 and 20000 yearsBC and the scratches could be a coinci-dence - or one of the first appearancesof abstract mathematics.
I believe that the human mind, earlyin its evolution, has the need to thinkin abstract terms, and that mathemat-ics unavoidably appears at this stage.
Finally, why do mathematicians domathematics?
Why did Michelangelo paint andsculpt, why did Beethoven compose?
All for the same reason: because theymust. As David Hilbert put it in 1930:Wir mussen wissen, wir werden wis-sen (we must know, we shall know).
Asked why he insisted in try-ing to climb Mount Everest, the fa-mous mountaineer George Mallory an-swered: “Because it is there”.
Likewise, mathematicians attacktheir own Everest (like Riemann’s hy-pothesis) because it is there.
But when they reach the top, theyhave changed the scenery by theirachievement.
Looking backward, they see the hardpath that they have followed, but alsosome much easier and simplified pathsnow open to others.
Looking forward, they see highermountains which were invisible ormaybe did not exist before. Mountainsthat must now be climbed.
Because they are there.
Luc Lemaire [[email protected]] received a Doctorate from the Universite Libre de Bruxelles in 1975 and a Ph.D. fromthe University of Warwick in 1977. From 1971 to 1982 he held a research position at the Belgian F.N.R.S., and has been aprofessor at the Universite Libre de Bruxelles since then. His research interests lie in differential geometry and the calculusof variations, with a particular interest in the theory of harmonic maps.
A former chairman of the Belgian Mathematical Society, he has been associated with the European Mathematical Societysince its creation in 1990, being a member of the Council from 1990 to 1997, a member of the group on relations withEuropean Institutions since 1990, Liaison Officer with the European Union since 1993, and Vice President since 1999.
PTM in the early yearsThe Polish Mathematical Society wasestablished in 1919. The Reader can findinformation regarding the circumstancesof its rise, as well as a description of itsactivity during the first year of its exis-tence, in an article by Józef Piórek in theEuropean Mathematical SocietyNewsletter ([4]). Stefan Banach,Franciszek Leja, Otto Nikodym, Stanis³awZaremba and Kazimierz ¯orawski wereamong its founder members, andStanis³aw Zaremba was elected as thePresident of the Society. In 1921, theMathematical Society in Cracow(Towarzystwo Matematyczne w Krakowie)was transformed into a national PolishMathematical Society (PolskieTowarzystwo Matematyczne; PTM) withits headquarters in Kraków (Cracow).According to its first statute, the
Society’s aim was “a comprehensive cul-tivation of pure and applied mathematicsby means of scientific sessions combinedwith lectures“. The first change to thestatute - still in 1921 - resulted in thefollowing insertion: “publication of aperiodical and maintenance of contactswith the mathematical scientific move-ment“.Members not residing in Cracow were
admitted to PTM through local sections.Within the period 1921-1939, the follow-ing sections existed: one in Lwów(Lvov), since 1921, and three others since1923 in Warszawa (Warsaw), Poznañ andWilno (Vilnius). Since 1937, mathemati-cians from Cracow have constituted partof the Cracow section (after the Society’s
headquarters had been moved to Warsawand its statute changed). The number of members of the Polish
Mathematical Society equaled 49 personsin 1921 and 155 persons in 1939. TheSociety was of a scientific character, asstated in the statute, where active andpassive voting powers were given exclu-sively to authors of mathematical publi-cations. A new statute, resolved in Lvovin 1936, established a federal organizationof five sections with its headquarters inWarsaw. The General Meeting of PTM,which included delegates from all sec-tions and the President of PTM, electeda president, a secretary and a treasurer -all of them constituting the GeneralManagement - as well as a Board ofControl. The General Management alsoautomatically included the presidents ofthe sections, who held titles of “vice-presidents of PTM“. Sections held localmeetings, where local GeneralManagement members were elected aswell as those of local boards of controland delegates to the General Meeting ofPTM. The aforementioned statute of 1936substantially extended the aims of PTM.
Apart from those adopted before, newones were added, among others to orga-nize competitions, to gather collections ofpublications, to improve work conditionsfor mathematicians, to maintain contactswith scientific institutions both within thecountry and abroad and to invite mathe-maticians from abroad to give lectures.The basic rules under this statute havebeen in force until present times.Until 1936, when a Council for Exact
and Applied Science (Rada NaukŒcis³ych i Stosowanych) and its organcalled the Mathematical Committee were
called into being by the government, thePolish Mathematical Society had been theonly central institution representing Polishmathematics at home and abroad. Apartfrom assemblies of Polish mathematiciansorganized by this right, PTM was in con-tact with public institutions, voiced opin-ions on subjects related to science andeducation and issued its own periodical -“Annales de la Société Polonaise deMathématique“ - distributed nationallyand internationally.
The post war yearsIn the years 1919-39, the time when thefamous Polish mathematical school wasestablished, Polish mathematics was agreat success and met with a high esteemon the international forum. During theSecond World War, when Poland fellunder occupation, all official activity ofthe Polish Mathematical Society came toa standstill; only clandestine scientificsessions were held in Cracow and inWarsaw.The second period of PTM’s activity
involves the years 1945-53, whenKazimierz Kuratowski acted as President.In 1945 the Cracow section was reacti-vated and in 1946 sections in Poznañ andWarsaw started to operate again. Withinthe period 1946-53, six new sectionswere established. The number of mem-bers of PTM grew from 144 persons in1946 to 339 in 1953.During the first years of the post-war
period, the Polish Mathematical Societywas, similarly to the situation before1936, the only institution actively cover-ing the whole range of issues related toPolish mathematics. As such, it cooperat-ed with Polish authorities on the recon-struction of the 3rd level education sys-tem (among others, it prepared a reformregarding mathematical studies and M.Sc.degrees in mathematics). PTM also coop-erated on a regular basis with theMinistry of Education as well as othereducational authorities, being a foundingbody of the first Olympic Games for sec-
SOCIETIES
EMS December 200424
The Polish MathematicalThe Polish MathematicalSociety (PTM)Society (PTM)Janusz Kowalski (Warsaw)
Stanis³aw Zaremba,PTM’s first president
Stefan Banach, elected president in 1939
ondary school students in the country, i.e.the Mathematical Olympic Games. TheSociety was also an initiator of researchwork and systematically convened scien-
tific sessions. Within the years 1946-49,four assemblies of Polish mathematicianstook place, the last one together withCzechoslovak counterparts. At the sametime, annual competitions for the bestworks within the field of mathematicswere organized. Editorial activities werelaunched by PTM as early as in 1945,when the 18th volume of “Annales de laSociété Polonaise de Mathématique“ waspublished.In 1948, the National Mathematical
Institute (Pañstwowy Instytut Matema-tyczny) was established. That meant thatPTM ceased to be the only central math-ematical institution; therefore its activitystarted to be gradually limited to theprofit of the Institute. After the PolishAcademy of Sciences (Polska AkademiaNauk) had been called into being in 1952(the National Mathematical Institutebecoming a part of it after being renamedto: Mathematical Institute of the PolishAcademy of Sciences - InstytutMatematyczny PAN), a third nationalmathematical institution came into being:the National Mathematical Committee ofthe Polish Academy of Sciences (KomitetNauk Matematycznych PAN). Statutorytasks of the Mathematical Institute cov-ered many fields, previously being with-in the scope of activity of the PolishMathematical Society. In bigger scientificcentres, numerous specialist seminarswere established and, as a consequence,scientific sessions of PTM lost much oftheir appeal.However, there still existed important
scientific and social projects, which couldonly be carried out within the frame ofa national scientific society that wouldassemble all scientific and didacticemployees from within one field. A clearindication of this was a spontaneousestablishment of PTM’s new sections incities and towns where new scientific andacademic centres emerged. These were
the sections in Toruñ, Katowice andSzczecin - established in the years 1952-55 - and those in Bia³ystok, Rzeszów,S³upsk, Czêstochowa, Kielce, Olsztyn,Opole, Nowy S¹cz and Zielona Góra -established in the years 1970-75. The sat-uration of big national scientific centreswith specialist seminars made it necessaryto do research work within a much widerscope that might be of interest to allmathematicians. There was also a need tohold a council dedicated to social andorganizational issues. This type of activ-ity was conducted by successive sections,as well as by the General Managementof PTM, during section meetings, coun-cils, conferences and assemblies of Polishmathematicians. The Polish MathematicalSociety also fulfilled important taskswithin the scope of cooperation with theeducational authorities, and introducednew forms of teaching young people witha talent for mathematics. It also cooper-ated with foreign mathematical centersand initiated new publications. An aspectof significant value was PTM’s help tonewly established sections, effectuatedmainly through delegating lecturers.At the end of the 1960s, the Polish
Mathematical Society got involved in acampaign to establish a new professionfor mathematicians working within vari-ous branches of science, economy andstate administration. This problem wasthe focus subject for the 10th Assemblyof Polish Mathematicians - organized inKatowice in 1970, together with theNational Mathematical Committee of thePolish Academy of Sciences. ThisAssembly, being the largest one inPTM’s history as far as the number ofparticipants is concerned (547), formulat-ed new assignments for the PolishMathematical Society and desiderataaddressed to other institutions as well asa definite activity program. In this way,PTM managed to be a focal point forthe majority of mathematicians workingin various branches of economy.Members of the Polish MathematicalSociety are employees of institutes of thePolish Academy of Sciences as well aseducational centers such as various typesof 3rd level education schools (includingtechnical, pedagogic, economic, medicaland agricultural ones), various teachingcolleges and secondary schools.
Financial resources are derived frommembership fees, subsidies granted by theMinistry of Scientific Research andInformation Technology and by theMinistry of National Education and Sport.According to the provisions of the first
statute, PTM was supposed to “compre-hensively cultivate pure and appliedmathematics“, but in practice was con-fined to “scientific sessions combinedwith lectures“. A change in PTM’s pro-
file to a Society involved in large-scalescientific and social activity, becamereflected in the development of variousorganizational structures, created withinthe Society’s frame to fulfill definedassignments. By the end of 1975, sixcommittees operated under the supervi-sion of PTM’s General Management: theCommittee for Popularization ofMathematics and Higher Education, theCommittee for Mathematics atUniversities, the Committee for SchoolHandbooks, the Committee forApplication of Mathematics, theCommittee for Publication and theCommittee for an Information andService Centre. Apart from these, therewas a Main Committee for MathematicalOlympiads, which operated together withits local committees, seven editorial com-mittees, one editorial board and fourcompetition jury boards. Analogous teamsoperated under the supervision of localGeneral Managements.Roman Sikorski, who in the years
1953-75 impacted PTM’s activity anddevelopment the most, was the Society’sPresident between 1965 and 1977.Another important character in thoseyears was Tadeusz Iwiñski, Secretary inthe years 1960-1981.
Conferences and AssembliesSince the very beginning of PTM’s exis-tence, scientific lectures and discussionsduring local meetings, conferences andassemblies have been the basic form ofactivity. In the years 1919-39, as many
as 1143 lectures were given, while in theperiod of 1949-75, the correspondingnumber was 5998 (there are no data cov-ering the period of 1945-48). During theperiod between 1976 and 2003, as manyas 3860 lectures were given. It is worthmentioning that members of other sec-
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EMS December 2004 25
Kazimierz Kuratowski
Roman Sikorski, PTM’s president 1965 – 1977
tions and mathematicians from abroadconstituted a significant part of all lec-turers. Foreign lecturers represented over
40 countries from all over the world.The Polish Mathematical Society has so
far organized 15 assemblies of Polishmathematicians (including 3 in the inter-war period), which usually take placeevery 5 years; in the remaining years,regular PTM meetings have been held. In1929, PTM organized the First Congressof Mathematicians of the SlavicCountries. The assemblies are of scientif-ic character. Their programmes involvelectures and reports of mathematical sub-stance: cross-thematically during plenarymeetings and those of a more specializedcharacter during meetings in sections.
These assemblies create an opportunity toponder on more general issues regardingscience, the system of education andsocial matters. These issues could evenbe a motive to convene a meeting (e.g.in the years 1969, 1970 and 1972).Since 1962 the Polish Mathematical
Society has been organizing 2- to 4-day-long scientific sessions that, together with
debates of the General Meeting of PTM,constitute the key elements of PTM’sassembly. Forty sessions were held before
2004, which were dedicated to a generaloverview of selected issues related tocontemporary mathematics, being ofinterest to all mathematicians. The the-matic content of scientific sessions heldin the years 1999-2002 embraced anoverview of the most significant achieve-ments in mathematics throughout the 20thcentury.
PublicationsThe publication of periodicals has beenthe second basic form of activity of thePolish Mathematical Society. In 1921, anorgan of PTM entitled “Dissertations ofthe Polish Mathematical Society“ wascalled into being, and one year later itwas changed into a periodical named“Annales de la Société Polonaise deMathematique“. 25 volumes of this peri-odical were issued during the period1922-52. In the years 1948-53, PTMstarted publishing other periodicals: a bi-monthly for teachers entitled “Mathe-matics“ (“Matematyka“), launched byPTM in 1948 but taken over by theMinistry of Education in 1953, and aseries called “Mathematical Library“(“Biblioteka Matematyczna“), 1953. Apartfrom the ones mentioned above, the fol-lowing titles were also published:“Fundamenta Mathematicae“, “StudiaMathematicae“, “Colloquium Mathemati-cum“, “Mathematical Monographs“ and“Mathematical Dissertations“ (since1952). In the years 1948-53, PTM super-vised - by the order of the Ministry ofEducation - all Polish mathematical pub-lications.In 1953 the Mathematical Institute of
the Polish Academy of Sciences took
over all these publications, includingPTM’s official organ “Annales de laSociété Polonaise de Mathematique“,which was then renamed to “AnnalesPolonici Mathematici“. The Polish Mathematical Society start-
ed a new phase of publishing activity in
1955, when the first of each of twoseries of “PTM’s Annals“ were issued.Series 1: “Mathematical Papers“ (“PraceMatematyczne“), which became“Commentationes Mathematicae“ in 1967,published 43 volumes up to the year2003. Series 2: “Mathematical News“(“Wiadomoœci Matematyczne“) published
39 volumes up to the year 2003. In1973, Series 3 was launched entitled“Applied Mathematics“, which became“Applied Mathematics. Mathematics forthe Society“ in 2000, with 45 volumespublished up to the year 2003. In 1977,Series 4 was launched, “Fundamenta
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EMS December 200426
Tadeusz Iwiñski, PTM’s secretary 1960 – 1981
PTM’s Assembly in Wroc³aw, 1946
“Functions of a Complex Variable”
Wiadomósci Matematyczne
Informaticae“, publishing 56 volumes upto the year 2003. Finally, in 1982 Series5 was launched, “Didactics ofMathematics“ (“Dydaktyka Matematyki“);25 volumes were published up to the
year 2003.Since 1971, PTM has been distributing
among its members an “InformationBulletin of the Polish MathematicalSociety“, which includes current newsregarding scientific and organizationalissues. At first it was published severaltimes per year, but later this frequencydiminished. After a two-year break (2000-2001) its publication was relaunched andcontinued on a more regular basis. Since1974, further to an initiative of the PolishMathematical and Physical Societies, apopular mathematical/physical monthly,entitled “Delta“, has been published. Thethematic scope of this publication wasextended in 1979 by issues regardingastronomy. For more information about“Delta“, see [4]. Two other periodicalsare published with PTM’s cooperation:“Mathematics“ (a periodical for teachers)and “Gradient“ (a periodical for teachers,pupils and their parents).
Competitions and prizesThe Polish Mathematical Society is alsobusy with organizing competitions andawarding prizes. There are two competi-tions open for entry to all Polish mathe-maticians, one for young mathematiciansunder 28 years of age and three for uni-versity students. Special PTM prizes havebeen set up, named after outstandingPolish mathematicians: Grand Prizes forscientific achievements (named afterStefan Banach, Stefan Mazurkiewicz,Stanis³aw Zaremba, Wac³aw Sierpiñski,Tadeusz Wa¿ewski, and ZygmuntJaniszewski); for achievements in the
field of applied mathematics and practi-cal elaborations (named after HugoSteinhaus and Wac³aw Pogorzelski); final-
ly, for achievements for the benefit ofmathematical culture (named after SamuelDickstein). PTM Prizes for young math-ematicians have also been set up. TheToruñ Section organizes the “JózefMarcinkiewicz Competition“ for the best
student’s paper, the Wroc³aw Sectiondoes similarly for the best student’s paperin the field of probability theory andmathematical applications, and the edito-rial board of “Didactics of Mathematics“organizes the “Anna Zofia KrygowskaCompetition“ for the best student’s paperin the field of didactics of mathematics.The Polish Mathematical Society is alsoinvolved in organizing and supervisingcompetitions for prizes named afterKazimierz Kuratowski, Stanis³aw Mazurand W³adys³aw Orlicz. It also takes partin carrying into effect the idea of lectureswhich are awarded with the “Wac³awSierpiñski medal“. On the whole, 836prizes and 25 medals have been award-ed so far.The Polish Mathematical Society is a
patron of activity aimed at bringing tolight pupils with a talent in mathematics.This activity takes the form of a com-petition for schoolchildren’s mathematical
papers, organized by the editorial boardof “Delta“ monthly. This competition’sfinals take place during the annualScientific Session of the PolishMathematical Society. In the period 1976-2004, nearly one hundred schoolchildrenwere awarded with prizes and distinc-tions. First of all, they were from com-prehensive secondary schools in bigcities; however, among them there werealso some pupils from secondary voca-tional schools in smaller towns.
CommitteesIn order to carry out its statutory tasks,PTM brings into being specialized com-mittees. In 1953 a Committee forPopularization of Mathematics and in1958 a Committee for SecondaryEducation were called into being by theGeneral Management of the PolishMathematical Society. In the period 1959-61, a subcommittee for a reform of pro-grammes and teaching methods workedout projects and drafts regarding someschool handbooks. This programme wasintroduced to schools with minor amend-ments and PTM cooperated with theMinistry of Education in its being carriedout. In 1968 a Committee for SchoolHandbooks was called into being to orga-nize debates in working teams (in theperiod 1968-71), publish articles, deliverrelevant materials to authors, and preparereviews of handbooks (including orga-nized debates over them). TheCommittees for Secondary and PrimaryEducation and for Popularization ofMathematics worked within three spheres:scientific activity (focusing on the mostrecent results of scientific research relat-ed to didactics of mathematics), analysisof documents and cooperation in theirbeing elaborated (legal acts and instruc-tions concerning educational policy), andworking out methods of modernizing theprocess of teaching mathematics as wellas preparing the mathematics teachers fornew tasks.
The Committees for Secondary andPrimary Education and for Popularizationof Mathematics, in cooperation with theAssociation of Teachers of Mathematics,organized, among others, a national sci-entific seminar devoted to didactics ofmathematics. During this seminar, someproposals of methodical and didactic solu-tions were put forward which concernedvarious levels of education in relation tomathematics. Special attention was paidby the Committee to the role of mathe-matics at the “matura“ examination (i.e.the final high school examination). Thiswas exemplified by the “Open letter bythe Polish Mathematical Society“, accept-ed by PTM’s General Meeting in Lublinin 1992 (published in numerous dailiesand weeklies).
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EMS December 2004 27
Delta
Hugo Steinhaus
Wac³aw Sierpiñski
In 1962 the Polish MathematicalSociety called into being a Committee forUniversity Education which worked intwo teams: one dedicated to non-univer-sitary schools and one dedicated to uni-versities and pedagogic schools. Theseteams embraced among other activities: areform of the course and the programmesof studies (1964-67), educating mathe-maticians at technical schools (1962-65),the inter-school exchange of students andcandidates for a doctor’s degree, methodsof educating future mathematicians,assessment of experimental programs,modernization of teaching methods, andeducation by the media. In the followingyears, smaller specialised committeesemerged from this Committee.In 1993, the Committee of Mathematics
at Universities, Pedagogical Academiesand Teaching Colleges, together with itscounterparts in the Polish Physical,Chemical and Biological Societies, hand-ed to the Ministry of Education a“Memorial regarding education within thescope of non-major basic subjects at ter-tiary-level schools“. The Committee ana-lyzed the phenomenon of intensifieddiversification of programmes related toteaching mathematical subjects in mathe-matical faculties at individual universities.On the initiative of the Committee of
Mathematics at Technical Universities,tests were conducted in 1995 to verifythe level of mathematical knowledge of1st year students at several technical ter-tiary schools. The results, which turnedout to be somewhat alarming, werepassed on to educational authorities andmade public. Following this, theCommittee insisted that an entry exami-nation in mathematics be compulsory atall technical tertiary schools until a com-pulsory “matura“ examination in mathe-matics was introduced.The Committee for Mathematics in
Economic Studies was busy with prob-lems related to “the New Matura“ exam-ination in mathematics and a “programmebase“ for mathematics in economic stud-ies. Since 1987, the Committee for the
History of Mathematics, under the aus-pices of PTM’s General Management andin cooperation with mathematical insti-tutes of schools of higher education, hasbeen organizing annual Schools ofHistory of Mathematics. 18 such schoolswere organized until the year 2004.During the PTM’s assembly in 1970, a
Committee for Application ofMathematics was established in responseto the fact that many “non-academic“(working in various branches of econo-my) mathematicians had joined the PolishMathematical Society. This obliged theCommittee to organize (in cooperationwith other institutions) annual
Conferences of Application ofMathematics, where mathematical modelsapplicable to specified practical issueswere acquired and presented. 32 suchconferences took place until the year2003.In 1972, an Information and Service
Centre was set up. Up until the end ofthe 1980s, it was responsible for solvingproblems submitted by various scientificand economic institutions, acting as advi-sor and conducting training for groups ofemployees in their workplace. DuringPTM’s scientific sessions, assemblies andconferences, problems from applied math-ematics have been the subject of a muchmore detailed scrutiny now than in thepast.
PopularizationThe Polish Mathematical Society under-takes various types of actions to com-memorate Polish mathematicians. Manystreets in Cracow, Warsaw and Wroc³awhave been named, on PTM’s initiative,after outstanding mathematicians:Stanis³aw Go³¹b, Bronis³aw Knaster,Miros³aw Krzy¿añski, KazimierzKuratowski, Franciszek Leja, EdwardMarczewski, Zdzis³aw Opial, WitoldPogorzelski, Marian Rejewski, Wac³awSierpiñski, Stefan Straszewicz, JacekSzarski, Tadeusz Wa¿ewski and Stanis³awZaremba. Further to PTM’s application, on 23rd
November 1982 the Polish Post-Officeissued four stamps in a “Polish mathe-maticians“ series with portraits of StefanBanach, Zygmunt Janiszewski, Wac³awSierpiñski and Stanis³aw Zaremba. On theinitiative of PTM’s Cracow Section, amonument dedicated to Stefan Banachwas erected, to be unveiled with due cer-emony on 30th August 1999 during theXV Assembly of Polish Mathematicians.The Polish Mathematical Society carries
on its activities in the school environ-ment. It organizes lectures for teachersand schoolchildren, as well as the gener-al public. 3637 such lectures were givenin the years 1952-2003. Since 1954, PTMhas been managing inter-school mathe-matical circles and competitions for tal-ented pupils. 1100 such forms of workhave been recorded so far.Occasionally, other forms of popular-
ization of mathematical knowledge areused: scientific camps for young people,distance learning studies (delivery ofsource materials and assignments to stu-dents and sending back corrected assign-ments), and guidance units for schools’mathematical circles.The Mathematical Olympic Games
have been operating under the auspicesof the Polish Mathematical Society. 55Olympiads were held in the years 1949-2004: 81972 pupils took part in 1st
degree competitions, 20718 in 2nd degreecompetitions and 3846 in 3rd degreecompetitions. The total number of laure-ates amounted to 817, and of thoseawarded with a distinction title, to 444.Since 1959, a 6-8 person delegation hasbeen chosen to participate in internation-al mathematical Olympiads. Three ofsuch Olympiads were organized in Polandin 1963, 1972 and 1986. Since 1977,
there has also been a 6-person delegationchosen to participate in the Polish-Austrian mathematical competitions, orga-nized each year alternately in Poland andin Austria. Since 1992, a 5-person dele-gation has been chosen to participate inmathematical competitions of the BalticStates, organized by Poland in 1998. Several mathematicians have been
working on the organization of theOlympic Games. The first Chairman ofthe Main Committee was StefanStraszewicz, one of the creators of theMathematical Olympic Games, who heldthis position for 20 years (1949-1969).The Toruñ, Wroc³aw and Nowy S¹cz
sections, popularized among pupils andstudents, as well as among adults, theidea of participation in three internation-al mathematical competitions:“Kangaroo“, “International FrenchChampionship in Mathematical andLogical Games“ and “Mathématiques sansFrontières“.
International CooperationThe Polish Mathematical Society hasbeen actively cooperating with the math-ematical community abroad. Since thefirst years of PTM’s existence, lectureshave been organized to be given by for-eign mathematicians at various occasionsor at a special invitation to give a lec-ture in Poland. These lectures are givenduring scientific sessions in PTM’s localsections, during PTM’s conferences orassemblies. Nearly 2000 such lectures
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EMS December 200428
Stefan Straszewicz, organiser ofMathematical Olympic Games
were held in the years 1949-2003. The Polish MathematicalSociety made agreements for an exchange with the followingsocieties: Czechoslovak (1962), Bulgarian (1968), Hungarian(1973) and Greek (1980). Within this framework, in the years1976-1990, 390 persons profited from the exchange program onboth sides, spending both in Poland and in the other abovementioned countries a total of 2641 so-called “exchange days“.In the case of Greece, such cooperation took place in the years1980-1982.Stanis³aw Zaremba, the President of the Mathematical Society
with its seat in Cracow, represented Polish mathematics duringthe International Mathematical Congress, held in 1920 inStrasbourg. The International Mathematical Union (IMU) wasestablished there by representatives from 11 countries: Belgium,Czechoslovakia, France, Greece, Japan, Poland, Portugal, Serbia,United States of America, Great Britain and Italy. Among themembers of the Executive Committee of the InternationalMathematical Union were Kazimierz Kuratowski (1959-1962), apatron of one of PTM’s prizes, and Czes³aw Olech (1979-1982,1983-1986). In the years 1963-66, K. Kuratowski acted as Vice-President of the Union.The Polish Mathematical Society sent its own delegations to
4 international congresses organized by the InternationalMathematical Union (Stockholm in 1962, Moscow in 1966,Nice in 1970 and Helsinki in 1978), as well as to 16 scien-tific conferences in the years 1950-75. In 1983, Poland was theorganizer of the International Mathematical Congress inWarsaw. Due to the Martial Law imposed in 1981, theCongress, initially planned to take place in 1982, could onlytake place as late as 1983. The Organizing Committee waspresided by Czes³aw Olech, who was also elected President ofthe Congress. The Polish Mathematical Society has also been cooperating
with the American Mathematical Society (AMS) and theCanadian Mathematical Society (CMS) on the basis of a reci-procity agreement. The Polish Mathematical Society is also one of the founder
members of the European Mathematical Society (EMS), calledinto being in December 1990 during a meeting held in thePolish Academy of Sciences conference centre in M¹dralin nearWarsaw. EMS was established thanks to the initiative of about30 mathematical societies representing nearly all European coun-tries. During the founding meeting, the Polish Mathematical Society
was represented by Bogdan Bojarski (then director of theMathematical Institute of the Polish Academy of Sciences) andAndrzej Pelczar, PTM’s President in the years 1987-1991. Oneof EMS’s first Vice-Presidents, elected for the term ending in1992, was Czes³aw Olech. During the term 1992-96, AndrzejPelczar acted as a member of the Executive Committee of EMSand later, in the years 1997-2000, as EMS’s Vice-President.Another organ of EMS’s authority is the Society’s Council,where the Polish Mathematical Society holds a two-person rep-resentation. During the term 1999-2002, its representatives wereJulian Musielak and Andrzej Pelczar. Kazimierz Goebel andZbigniew Palka have been elected for the term 2002-2006.
This elaboration has been written on the basis of an article byTadeusz Iwiñski [2]
References[1] K.Ciesielski, Z.Pogoda, Interview with Marek Kordos,
Newsletter of the European Mathematical Society 41 (2001).[2] T.Iwiñski, Polskie Towarzystwo Matematyczne [The Polish
Mathematical Society] in: S³ownik polskich towarzystwnaukowych [Dictionary of Polish scientific societies],Warszawa 1975.
[3] T.Iwiñski, Ponad pó³ wieku dzia³alnoœci matematyków pol-
skich [More than 50 years of activities of PolishMathematicians], Pañstwowe Wydawnictwo Naukowe[Polish Scientific Publishers], Warszawa 1975.
[4] J. Piórek, From the minutes of the Mathematical Societyin Cracow, Newsletter of the European MathematicalSociety 32 (1999).
Janusz Kowalski [[email protected]] graduated in math-ematics at Warsaw University in 1969. He currently works atthe £azarski Law and Trade University in Warsaw. For manyyears he taught at secondary schools in Warsaw. Since 1988he has been the Vice-Secretary of the Polish MathematicalSociety. He is also a member of the Polish MathematicalSociety Commission of Teaching and Popularization.
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Honorary Members In recognition for contributions to the development of math-ematics, its being taught, applied and popularized, as well asin acknowledgement of devoted participation in PTM’s activ-ities, the Polish Mathematical Society confers the dignity ofHonorary Member of PTM. The following mathematicianshave been given this dignity so far: Pawe³ S. Aleksandrow,Donald W. Bushaw, Karol Borsuk, Zygmunt Butlewski,Zbigniew Ciesielski, Mieczys³aw Czy¿ykowski, SamuelEilenberg, Pàl Erdõs, Eugeniusz Fidelis, Stanis³aw Go³¹b,Tadeusz W. Iwiñski, Wiktor Jankowski, Leon Jeœmanowicz,Bronis³aw Knaster, Andriej N. Ko³mogorow, Jan Kozicki,Anna Krygowska, W³odzimierz Krysicki, KazimierzKuratowski, Andrzej Lasota, Jean Leray, Franciszek Leja,Stanis³aw £ojasiewicz, Edward Marczewski, Stanis³aw Mazur,Jan Mikusiñski, Julian Musielak, Jerzy Sp³awa-Neyman,Witold Nowacki, W³adys³aw Orlicz, Franciszek Otto,Aleksander Pe³czyñski, Helena Rasiowa, Marian Rejewski,Edward S¹siada, Wac³aw Sierpiñski, Hugo Steinhaus, RomanSikorski, Stefan Straszewicz, W³adys³aw Œlebodziñski, AndrzejTurowicz, Eustachy Tarnawski, Kazimierz Urbanik, AntoniWakulicz, Tadeusz Wa¿ewski, Lech W³odarski, ZygmuntZahorski, Antoni Zygmund.
Presidents The function of President of the Polish Mathematical Societywas performed by: Stanis³aw Zaremba (1919-21, 1936-37),Wiktor Staniewicz (1921-23), Samuel Dickstein (1923-26),Zdzis³aw Krygowski (1926-28), Wac³aw Sierpiñski (1928-30),Kazimierz Bartel (1930-32), Stefan Mazurkiewicz (1932-36,1937-39), Stefan Banach (1939-45), Karol Borsuk (1946),Kazimierz Kuratowski (1946-53), Stefan Straszewicz (1953-57), Edward Marczewski (1957-59), Tadeusz Wa¿ewski (1959-61), W³adys³aw Œlebodziñski (1961-63), Franciszek Leja(1963-65), Roman Sikorski (1965-77), W³adys³aw Orlicz(1977-79), Jacek Szarski (1979-81), Zbigniew Ciesielski (1981-83), Wies³aw ¯elazko (1983-85), Stanis³aw Balcerzyk (1985-87), Andrzej Pelczar (1987-91), Julian Musielak (1991-93),Kazimierz Goebel (1993-99), Boles³aw Szafirski (1999-2003),Zbigniew Palka (2003- ).
PTM’s presidents (1946 – 1977)
ERCOM: ICMS EdinburghThe inaugural event of Edinburgh’sInternational Centre for MathematicalSciences (ICMS) was a meeting onGeometry and Physics held at NewbattleAbbey near Edinburgh in March 1991.Alain Connes, Roger Penrose and SimonDonaldson were among the speakers andJacques-Louis Lions (Collège de France)was chair of the ICMS ProgrammeCommittee.
ICMS had been founded by a partner-ship that involved Edinburgh and Heriot-Watt universities, Edinburgh CityCouncil, and the International Centre forTheoretical Physics in Trieste, with sup-port from some of Edinburgh’s financialinstitutions. An important figure duringthose early days of ICMS was John Ball,currently president of the InternationalMathematical Union. The Nobel Prizewinner Abdus Salam also took a keeninterest in its development.
In 1994 ICMS moved to its currentpremises, 14 India Street, which is thehouse in Edinburgh’s New Town wherethe celebrated physicist James ClerkMaxwell was born on 13 June 1831.
Purpose of ICMSFrom the outset the purpose of ICMSwas to • Create an environment in which the
mathematical sciences will develop innew directions
• Encourage and exploit those areas of
mathematics that are relevant to othersciences , industry and commerce
• Promote international cooperation withparticular reference to mathematiciansworking in developing countries
As a joint venture of its two foundinguniversities, ICMS has, under the leader-ship of Angus MacIntyre, its firstScientific Director, established a strongreputation for running high quality inter-national workshops across the full rangeof mathematical sciences and for bring-ing the best mathematicians in the worldto Edinburgh and the UK. This record isall the more remarkable because it has,until now, been achieved in the absenceof systematic core funding from govern-ment agencies, using only funds raisedthrough the initiative of organisers whobrought their projects to Edinburgh.
Editorial services, workshops andmeetingsTo compensate for the lack of core fund-ing for its activities, ICMS has beendeveloping a modest commercial arm byoffering editorial services to journals andmanagement services to research agen-cies. Currently, editorial work for theProceedings of the Royal Society ofEdinburgh and Proceedings of theEdinburgh Mathematical Society aredone at ICMS headquarters, and ICMSmanages the Environmental Mathematicsand Statistics Workshops on behalf of the
Natural and Environmental, and theEngineering and Physical Sciences,Research Councils.
Determined that ICMS should succeedwith its mission despite limited means,over the past decade a small but highlycommitted and resourceful staff has pro-duced a range of meetings of outstandingquality across the entire range of theMathematical Sciences, from NewMathematical Developments in FluidMechanics (1995) organised byConstantin, P-L Lions and Majda to therecent celebration of the Centenary of SirWilliam Hodge who was born inEdinburgh in 1903. This meeting, whichwas organised by Atiyah, Bloch,Donaldson, Griffiths and Witten, withmembers of Hodge’s immediate familyattending the historical lectures and thebanquet, was described as ‘stellar’ byone distinguished participant. Details ofmeetings held since 1995 are available athttp://www.icms.org.uk/previous/index.html.
In addition to research workshop activ-ity ICMS has played a full part in themathematical life of Scotland and of theUK. On the one hand it had a pivotal rolein the organisation of ICIAM whichbrought over 2000 delegates to
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EMS December 200430
Delegates at Geometry and Physics, Newbattle Abbey, 1991
Edinburgh in 1999 and, on a totally dif-ferent scale, it runs workshops and sup-port meetings for research students inScottish universities. It also encouragesworkshops on industrial problems fromthe local region. Most recently, in 2004ICMS joined with the Faculty ofActuaries to set up a major meeting forsenior professional actuaries on the prob-lems currently facing the pensions indus-try in the UK and worldwide.
The FutureWith its strong record of past achieve-ment on a limited budget, in November2004 ICMS and its two founding univer-sities entered a partnership with theScottish Higher Education FundingCouncil (SHEFC) and the UK’sEngineering and Physical SciencesResearch Council (EPSRC) with the aimof having guaranteed funds for work-shops from now until 2008/9. Duringthat period the ICMS ProgrammeCommittee, under the chairmanship ofJerry Bona (University of Illinois atChicago), will have at its disposal fundsfor about eight international workshopsper year, each with about 40 participants,and at the same time will continue tosolicit proposals for workshops of vari-ous sizes from organisers who can makeother funding arrangements. The newarrangement will enable ICMS toimprove its facilities and very soon pastvisitors to India Street will see changes,with new decoration, modernisation of
the lecture-room facilities and the intro-duction of new break-out rooms andcomputer equipment. Workshop organis-ers and participants will still benefit fromthe experience of the Centre Manager,Tracey Dart, who will now be able todesignate one of the ICMS staff as aConference Coordinator to each work-shop. ICMS will continue to offer edito-rial and managerial services to the acad-emic community.
With the new arrangements in placethe ICMS Scientific Director, John
Toland (University of Bath), and theProgramme Committee are now solicit-ing high quality adventurous proposals inall branches of the mathematical sciencesand scientific areas with significantmathematical content and, to be fair tointer-disciplinary proposals, will findways to counter the inherent conser-vatism of the peer review process. Theywill also continue to look for ways tosupport participants from developingcountries, an important part of the ICMSmission since it was founded, but stilldifficult because of a lack of availablefunding.
Within the United Kingdom the role ofICMS in promoting and supporting shortworkshop activity in Edinburgh comple-ments that of the Isaac Newton Institutein Cambridge where support forresearchers in residence in Cambridge isthe key to the success of its long-termresearch programmes. In fact the twoinstitutes are developing a good under-standing and a close working relation-ship.
Institutional Structure ICMS is a joint venture of Heriot-Wattand Edinburgh universities, is an institu-tional member of the EuropeanMathematical Society and is representedon European Research Centres onMathematics (ERCOM). The scientificprogramme is controlled by theProgramme Committee with membershipdrawn from across the UK and abroad.Resources from EPSRC, SHEFC, Heriot-Watt and Edinburgh Universities, andelsewhere are allocated to individualworkshops by a Management Committeeof the two universities, following advice
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EMS December 2004 31
New Mathematical Developments in Fluid Mechanics, Edinburgh, 1995 - part of theUNESCO 50th anniversary celebrations.
Distinguished participants at the Hodge Centenary Conference, Edinburgh, 2003, includingpast and current scientific directors Angus MacIntyre (bottom row, second left) and John
Toland (middle row, third left).
from the Programme Committee. ICMSis advised by the ICMS Board, whosemembership is drawn from organisationswith an interest in ICMS, such as theEdinburgh and London MathematicalSocieties, the Royal Society ofEdinburgh, ICTP, and the City ofEdinburgh.
International Centre for MathematicalSciences14 India Street, Edinburgh EH3 6EZ
Scientific Director & Chair of theManagement Committee: John Toland (University of Bath)
Programme Committee Chair: Jerry Bona (University of Illinois atChicago)
Management Committee Deputy Chairs:Chris Eilbeck (Heriot-Watt University)Tony Carbery (Edinburgh University)
Centre Manager:Tracey Dart (ICMS)
The classical Georgian house in centralEdinburgh that is the headquarters ofICMS has proved an attractive venue forworkshops and with the new fundingarrangements will, it is hoped, continueto be so far in the future.
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EMS December 200432
Call for ProposalsProposals are invited for workshops to be held at ICMS in Edinburgh in 2005/6.The International Centre for Mathematical Sciences (ICMS) is based in central Edinburgh, in the birthplace of JamesClerk Maxwell. Following new funding arrangements with EPSRC for the period 2005 to 2008 ICMS is able to offersupport to run workshops and symposia on all aspects of the mathematical sciences in new or traditional subjects andinterdisciplinary areas with significant mathematical content.The core of ICMS activity will be the rapid-reaction research workshop programme (R3WP). ICMS therefore particu-larly welcomes proposals for workshops in rapidly-developing and newly-emerging areas where there is a need to eval-uate new developments quickly. ICMS will respond quickly to such proposals. Organisers can expect preliminary com-ments from reviewers in 8 weeks. Decisions will be made by the Programme Committee four times a year (December,March, May and September). Small meetings can be organised in 6-8 months from acceptance.Potential organisers should contact ICMS as early as possible to discuss ideas, before submitting a firm proposal. Theproposal document should not normally exceed five pages and should be submitted electronically (PDF, PS, Word orDVI). Proposals may be submitted at any time. Full instructions on how to submit a proposal, together with details of the refereeing process and criteria for selectioncan be found on the webpages:
http://www.icms.org.uk/call/index.htmlAnyone unable to read these pages or download documents can order print versions from ICMS.If your application is successful, you will be offered a funding package to contribute to the travel and subsistence of aproportion of the participants. ICMS staff will undertake all non-scientific administration connected with the workshop(such as issuing invitations, processing registrations, organising accommodation, preparing material, financial adminis-tration). One of the Scientific Organisers (often an author of the initial proposal) will be appointed Principal Organiserand be the main point of contact.For all enquiries about ICMS or the procedures for submitting a proposal, please contact Tracey Dart, Centre Manager,ICMS, E-mail [email protected] (14 India Street, Edinburgh EH3 6EZ, Tel +44 (0)131 220 1777; Fax +44(0)131 220 1053).
Ramanujan Prize for Young Mathematicians from Developing Countries
The Abdus Salam International Centre for Theoretical Physics (ICTP) ispleased to announce the creation of the Ramanujan Prize for young mathe-maticians from developing countries. The Prize is funded by the Niels HenrikAbel Memorial Fund.
The Prize will be awarded annually to a researcher from a developing countryless than 45 years of age at the time of the award, who has conducted out-standing research in a developing country. Researchers working in any branchof the mathematical sciences are eligible. The Prize carries a $10,000 cashaward and travel and subsistence allowance to visit ICTP for a meeting wherethe Prize winner will be required to deliver a lecture. The Prize will usually beawarded to one person, but may be shared equally among recipients who havecontributed to the same body of work.
The Prize will be awarded by ICTP through a selection committee of five emi-nent mathematicians appointed in conjunction with the InternationalMathematical Union (IMU). The first winner will be announced in 2005. Thedeadline for receipt of nominations is July 31, 2005.
Please send nominations to [email protected] describing the work of the nomi-nee in adequate detail. Two supporting letters should also be arranged.
Unquestionably, the peak of most sportsevents culminates in a grand finale. Andit’s often been this way with intellectualchallenges, say, maths contests. Finalistsin these meetings have to clear the hurdlesof divers selection criteria and/or comethrough several qualifyings, i.e. local andregional qualifying rounds bearing partic-ularly exotic names, and observe specificrules on the whole. For example, inBulgaria they are trading under unusualnames such as the ‘Virgil Krumov’ contestor the ‘Chernorizets Hrabar Tournament’and so on .
The last three issues of Problem Cornerhave dealt with the wealth of differentcompetitions. Here, all roads lead to thenational highlight for mathematically ableyoungsters, the National MathematicsOlympiad (NMO). But it would go beyondthe scope of this Corner to enumerate thewhole plethora of contests that Bulgariaoffers to its adolescents in this field. For,in spite of this diversity, the separatemaths trials are distinguishing from oneanother only by nuances. The NationalBulgarian Mathematics Olympiad repre-sents a country-wide showdown for nativepupils to decorate themselves with theunofficial title of champion of maths, asreported by Prof. Sava Grozdev, Instituteof Mechanics, Bulgarian Academy ofSciences, Sofia. Here comes his final partof a long story that describes the effortsmade in his country towards getting youngpeople really interested in mathematics.
The National Mathematics OlympiadThe Bulgarian National MathematicsOlympiad (BNMO) originates in 1949 andwas actually the first competition orga-nized in Bulgaria. Because of a two yearinterruption during the period 1957-59, the53rd National Olympiad took place inMay, 2004. Initially, only students fromgrades 8 to 11 were permitted to partici-pate in the Olympiad. At present the con-test is open to grades 4 to 12. The BNMOis run in 3 rounds. Naturally, the firstround, better known as ‘school’ round, isconquered by participants. About ten yearsago, the number of starters evened out at150 000 individuals, almost 10 per cent ofall participating students. Currently, theinitial number is reduced to about 20 000.One of the reasons for this decrease is areduction in the total number of studentsdue to a strongly abating birth-rate.
During the first round students have to
solve three questions within four hours.All problems are compiled and worked outby the Regional Inspectorates ofEducation while the present teachers areasked to correct the examination papers oftheir own charges. About 30 per cent of allparticipants in this round will pass to thesecond stage, the regional round. Studentsof different grades are gathered in region-al centres and have to solve three problemswithin four hours again. This time, theresponsibility for the set of questions liesin the hands of the National OlympiadCommission and the examination papersare marked under the supervision ofregional inspectors. The third round, or theNational round, is reserved for studentsfrom grades 8 to 12. In former times therewas an additional round for 7-graders, andthese results were partially used as a per-mit to enter a Mathematics-, ForeignLanguage-, or Technical School. Alas, thistradition does not exist any more becauseof a complete restructuring of the veteraneducational system, which is still goingon. The third round is a two-day event likethe International Olympiad. Students areasked to solve three problems each daywithin 4 or 5 hours. The NationalCommission in Mathematics creates theproblems and is also responsible for mark-ing the examination papers. The coordina-tion of the results is carried out in the pres-ence of both teachers and students. Thisprocedure is a fully objective, fair and thusdemocratic element of the construct,called BNMO. Usually, the number of par-ticipants in the third round is about 100,with a growing tendency to drop further.For instance, in May 2002 the total num-ber of students in the third round of theNational Olympiad reached a minimum of38 ‘survivors’. The champions of the finalround are awarded the possibility to studyMathematics at a Bulgarian University oftheir choice, without passing entranceexaminations. The universities in Bulgaria(about 40) follow an autonomous policy,which includes such individual tests, butthey respect the results achieved at theNational Olympiad, and thus are acknowl-edging high level talents. Besides this, the12 students with the highest scores in thethird round are invited for further selectioneach year. The selection includes two two-day tests consisting of six problems in all.When the six constituents of the Nationalteam have finally been identified, theyhave to undergo a fortnight preparation for
the International Olympiad afterwards. Bulgaria is one of the founder countries
of the International MathematicsOlympiad (IMO) and has participated inall its 42 editions. Only two other coun-tries can look back to a similar constance:Romania and the Czech Republic.Bulgarian students have won 32 gold, 73silver and 81 bronze medals altogether.During the last 10 years, Bulgaria hasranked among the top ten countries withbest performance, and regarding the last 4years, Bulgaria has even moved into thebest five. France, Germany, England, Italyand other countries that are world centresof Mathematics, have much lower rank-ings. In 1998 in Taiwan, Bulgaria was sec-ond, in 1999 and in 2000 - fifth, in 2001with the participation of exactly 83 coun-tries - third. The International Jury awardsexceptional prizes to students with extra-ordinary achievements. Since 1987, onlytwo prizes have been awarded. Both ofthese were addressed to Bulgarians: inAustralia in 1988 and in Canada in 1985.The recent winner of a special prize wasNikolay Valeriev, who represents a realphenomenon of the IMO: from four partic-ipations he has won three gold and one sil-ver medal. During the long history of theIMO, the mother of all maths contestsworldwide, there have been only five otherparticipants with comparable achieve-ments. Four of them were before 1974,when the number of participating coun-tries was limited. The fifth was RideBurton from the USA team, who obtainedfour gold medals out of four and maintainsa world record in this branch until today.Nowadays, the Bulgarian studentAlexander Lishkov has the possibility toimprove this mark, since he has one silvermedal and four more participations tocome through.
What is expected from those who haveworked their way from a large field ofstarters up to the finale can best be learntby an examination of the problems posedin my recent Corner. The new set is abundle of questions which were usedwhile preparing the Bulgarian team forparticipating in the InternationalMathematical Olympiad, held in Glasgow,Scotland, July 17 - 30, 2002. All problemsare original and are worked out by mem-bers of the Team for Extra CurriculaResearch in cooperation with the Union ofBulgarian mathematicians.
PROBLEM CORNER
EMS December 2004 33
PPrroblem Corneroblem CornerContests frContests from Bulgaria Pom Bulgaria Part IVart IV
Paul Jainta
PROBLEM CORNER
EMS December 200434
164 Strange animals live in a building with 3n floors. The roof is considered the (n+1)st floor. Exactly one
animal is living on the first floor, while one animal at most is living on each of the other floors. Within a
month curious things will happen: Exactly once a month each animal gives birth to a new animal. Immedi-
ately after his birth the new animal moves to the nearest upper floor and remains there if the floor is unoccu-
pied by another animal. If not, the newborn animal eats the previous inhabitant and walks on to the next
upper floor repeating the same procedure. But a giant is dwelling on the roof and if an animal enters the
roof, it will be eaten by the giant. The same things will happen the next month and so on. It is known that
all the animals give birth to new animals simultaneously. What is the smallest number of months, after
which the building will be settled the same way as it was at the beginning?
165 Consider a regular polygon with 2002 vertices and all its sides and diagonals. How many different ways
can you choose some of them, such that they form the longest continuous (= connected) route?
166 Given the sequence: 2
1 1, 1, ( 1)
2n n n
ax x x x n , where a is a positive integer. Prove that
nx
is irrational for every 3n .
167 All points on the sides of an acute triangle ABC are coloured white, green or red. Prove that there exists 3
points of the same colour, which form the vertices of a right triangle, or rather there exists 3 points of differ-
ent colours, which are vertices of a triangle similar to the given one.
168 Given is a sequence of polynomials: P1(x) = x, P2(x) = 4x3+3x, …, Pn+2(x)= (4x2+2)·Pn+1(x) - Pn(x), n 1.
Prove that there are no positive integers k, l and m, such that Pk(m) = Pl(m+4).
169 Find all continuous functions f: R R, such that f satisfies the functional equation: ( ( )) ( )f x f x f x for
all real x.
It remains for me to present solutions to questions 152 to 157, published in Issue 49 of the Corner. All problems
come from Hungarian sources, which stand for high quality of course.
152 The first four terms of an arithmetic progression of integers are a1, a2, a3, a4.
Show that 1 a12 + 2 a2
2 + 3 a32 + 4 a4
2 can be expressed as the sum of two perfect squares.
Solution by J.N. Lillington, Wareham, UK .
Let a1 = a, a2 = a+d, a3 = a+2d, a4 = a+3d, where a,d are integers. Then 1 a12 + 2 (a+d)2 + 3 (a+2d)2 + 4 (a+3d)2
= a2 +2a2 +4ad+2d2+3a2+12ad+12d2+4a2+24ad+36d2 = 10a2+40ad+50d2 = a2+(3a)2+40ad+d2+(7d)2 = (3a+7d)2 +
(a-d)2 .
Also solved by Niels Bejlegaard, Copenhagen, Denmark; Pierre Bornsztein, Maisons-Laffitte, France; Erich N.
Gulliver, Schwäbisch-Hall, Germany; Gerald A. Heuer, Concordia College, Moorhead, MN, USA, and Dr Z
Reut, London, UK.
153 Is it possible to get equal results if 2
10 10n n
and 2
10 10 1n n
are rounded to the nearest interger?
(n is a positive interger).
Solution by Gerald A. Heuer, Concordia College, Moorhead, MN, USA .
No. From the fact that 2 2 21
10 10 10 10 10 10 14
n n n n n n it follows that
2 21
10 10 10 10 10 12
n n n n n. Therefore the nearest integer to
210 10
n n is at most 10n-1,
while the nearest integer to 2
10 10 1n n
is 10n.
Also solved by Niels Bejlegaard, Pierre Bornsztein, E.N. Gulliver, J.N. Lillington, and Dr Z Reut, London.
PROBLEM CORNER
EMS December 2004 35
A B
C D
E F
G H
I J
K
16
81
600
154 An Aztec pyramid is a square-based right truncated pyramid. The length of the base edges is 81 m, the top
edges are 16 m and the lateral edges 65 m long. A tourist access is designed to start at a base vertex and to
rise at a uniform rate along all four lateral faces, ending at a corner of the top square. At what points should
the path cross the lateral edges?
Solution by J.N.Lillington.
(Ed. We refer to the opposite figure. It can easily be seen that the lateral faces of the truncated
pyramid form trapezoids with 600 base angles. Line segments CD, EF, GH, IJ are drawn parallel to base AB.)
First, we show that triangle ABK is equilateral. Triangles ABK and IJK are similar according
to the preface, thus 65
16 81
KJ KJ (because the lateral edges measure 65 m) or KJ = 16,
which gives the result.
Let the tourist walk be the path ADCFEHGJI crossing the right lateral edge at D,F,H.
Applying the sine rule on triangles ABD, and ACD gives: 0 0
81
sin 120 sin 60
AD and
0 0sin120 sin(60 )
AD CD or (because of sin1200 = sin600 and sin(1200- ) = sin[1800-(1200- ) = sin(600+ ))
this leads to 0 0
81
sin 60 sin 60
CD or
0
0
sin 6081
sin 60CD .
Then applying successively to triangles CDF, CEF, EFH, EGH, GHJ and GIJ we get
40
0
sin 6081 16
sin 60
or 3 3 tan
2 3 tan which finally simplifies to 5 tan 3 .
Applying the sine rule on triangle ABD gives 0
81
sin sin 60
DB or
81
3 1cot
2 2
DB . Hence DB = 27
and applying successively we yield DF = 18, FH = 12, and HJ = 8.
Also solved by Niels Bejlegaard.
155 The billposters of the Mathematician’s Party observed that people read the posters standing 3 meters away
from the centres of the cylindrical advertising pillars that have a 1.5 m diameter. The Party wants to
achieve that, after sticking the posters around a pillar, a whole poster will be visible from any direction.
How wide should the posters be?
Solution by Niels Bejlegaard, Copenhagen, slightly revised by the editor. (Ed. A person standing at M, a distance of 3 meters away from the centre, is able to see the
part of the cylindrical advertising pillar that is bounded by two tangent planes drawn to it.
Clearly, it will be sufficient to view a circle as a cross-section of the pillar). According to
the figure a simple calculation shows:
2 23 .75
tan 15.75
or 75.52o
. The length of a bill along the cylinder is s =
arctan 15 1.98 m (or s0
2 75.52.75 2
360
o
).
Obviously two equal bills are not sufficient to be viewed completely because the visual angle would be 0o then.
If we denote the width of the posters by w, then we must have w 2
s. Since we are looking for the maximal
width, we may assume the posters to tile the pillar, i.e. the circumference of the pillar is an integer multiple of
the poster width. If we have, say, n posters, then .75 2
5 2
s must hold, from which we get n > 4.767 … . So
the smallest possible value of n is 5, and the poster width must be w = .75 2
5 .942 m.
Also solved by J.N. Lillington.
PROBLEM CORNER
EMS December 200436
156 Find a simpler expression for the sum S = 3 + 2·32 + 3·33 + … + 100·3100.
Solution by Dr Z Reut, London.
The sum can be written as
100 100 100 100
1 1 1 1
3 ( 1 1) 3 ( 1) 3 3n n n n
n n n n
S n n n . The first term can now
be written as follows:
100 100 99 1001 100
1 1 1 1
( 1) 3 3 ( 1) 3 3 3 3 ( 3 100 3 )n n n n
n n n n
n n n n
= 3·(S - 100·3100). The second term is a geometric series, which is reduced as follows: 100100
100
1
3 1 33 3 3 1
3 1 2
n
n
. The first equation becomes 100 1003
3 100 3 3 12
S S . Solv-
ing for S gives the result 100 1003 3 3
150 3 199 3 14 4 4
S .
Also solved by Pierre Bornsztein, Niels Bejlegaard, Erich N. Gulliver, Gerald A. Heuer, and J.N. Lillington.
157 Given are two non-negative numbers x,y, that satisfy the inequality x3 + y4 x2 + y3. Prove that x3 + y3 2.
Solution by Pierre Bornsztein.
First, the result is trivially true for x = y = 0. Thus, we assume (x,y) (0,0).
Suppose, for a contradiction, that x3 + y3 > 2.
The inequality between means of order 2 and order 3 gives
2 2 3 3
3
2 2
x y x y.
Thus, x2+y2
2 12 2 2 11
3 3 3 3 3 3 3 3 3 33 33 3 3 32 2x y x y x y x y x y .
It follows that x2 – x3 < y3 – y2 .
On the other hand we have y3-y2 y4-y3 0 y2·(y-1)2, and this inequality is generally true.
It follows that x2-x3 < y3-y2 y4-y3, which contradicts the statement of the problem.
Thus, x3 + y3 2, as desired.
Also solved by Niels Bejlegaard and J.N. Lillington.
(Ed. Marcel G. de Bruin, Faculty of Electrical Engineering, Mathematics and Computer Science, Department of
Applied Mathematics, Delft, The Netherlands, has proposed an improvement of the outcome of problem 147,
given in the NewsletterNo. 51. The question reads as follows:
Given 100 positive integers x1, x2, …, x100, such that
1 2 100
1 1 1... 20
x x x , prove that at least two of
the integers must be equal. A simple sharpening of the integral-test allows one to establish a better estimation, for the following inequality
can be easily shown:
1
21
2
1 n
n
dx
n x, \ 0 .n Assuming x1, … ,xm to be distinct positive integers, we find
1
21
1 1 2
1 1 12 2.
2
m mm
n kn
dxm
x n x The ‘best’ integer m follows easily from
12 2
2m
20, i.e. m = 114.)
That completes this issue of the Corner. Send me your nice solutions and generalizations.
Alain Connes (IHES) was awardedthe Gold Medal 2004 of the FrenchCNRS for the creation of non-com-mutative geometry, revolutionizing thetheory of operator algebras, and forhis contributions to the mathematicalproblems arising from quantumphysics and relativity theory.
Pierre Deligne (Princeton) has beenawarded the 2004 Balzan Prize inMathematics for major contributionsto several important domains of math-ematics.
Mikhael L. Gromov (IHES) and JayGould (Courant NYU) received the2003-2004 Frederic Essers NemmersPrize in Mathematics.
Gérard Laumont and Bao-Châu Ngô(Paris-Sud) were presented a 2004Clay Research Award in recognitionof their work on the FundamentalLemma for unitary groups.
Guy David (Paris-Sud) received theFerran Sunyer i Balaguer Prize ofthe Institut d’ Estudis Catalans, inrecognition of his monographSingular Sets of Minimizers for theMumford-Shah Functional. He wasalso awarded the Prix Servant of theFrench Academy of Sciences.
Henri Berestycki (Paris) was award-ed the Grand Prix Sophie Germain ofthe French Academy of Sciences forhis fundamental contributions to theanalysis of nonlinear partial differen-tial equations.
Colette Moeglin received the PrixJaffé of the French Academy ofSciences.
Laurent Stolovitch was awarded thePrix Paul Doistau-Emile Bluter of theFrench Academy of Sciences.
Paul Biran (Tel Aviv) has receivedthe 2003 Oberwolfach Prize for out-standing research in geometry andtopology, particularly symplectic andalgebraic geometry.
AwardsGabriele Veneziano (CERN) has beenawarded the Dannie Heineman Prizefor Mathematical Physics for his pio-neering work in dual resonance mod-els.
Frank Allgöwer (Stuttgart) has beenawarded a 2004 Leibniz Prize by theDeutsche Forschungsgemeinschaft forhis work in nonlinear systems andcontrol theory.
Jerzy Jezierski (Warsaw) andWac³aw Marzantowicz (Poznañ) wereawarded the Banach Great Prize ofthe Polish Mathematical Society fortheir research papers on topology andnonlinear analysis.
Witold Wiês³aw (Wroc³aw) wasawarded the Dickstein Great Prize ofthe Polish Mathematical Society forhis achievements in the history ofmathematics and popularization.
Dariusz Buraczewski (Wroc³aw) wasawarded the Kuratowski Prize.
Jakub Onufry Wojtaszczyk (Warsaw)was awarded the first MarcinkiewiczPrize of the Polish MathematicalSociety for students' research papers.
Sir Roger Penrose OM FRS (Oxford)was awarded the De Morgan Medalof the London Mathematical Societyfor his wide and original contribu-tions to mathematical physics.
Boris Zilber (Oxford) was awardedthe Senior Berrick Prize for his paper“Exponential sums equations and theSchanuel conjecture”, J. LondonMath. Soc. (2), 65, (2002).
Richard Jozsa (Bristol) was awardedthe Naylor Prize of the LMS for hisfundamental contributions to quantuminformation science.
Ian Grojnowski (Cambridge) is thefirst recipient of the LMS FröhlichPrize for his work on problems inrepresentation theory and algebraicgeometry.
The Whitehead Prizes of the LMSwere awarded to Mark Ainsworth(Strathclyde), Vladimir Markovic
(Warwick), Richard Thomas(London) and Ulrike Tillmann(Oxford).
Amongst those elected to Fellowshipof the Royal Society in May 2004were: Samson Abramsky (Oxford),Julian Besag (Seattle), David Epstein(Warwick) and David Preiss(London).
Björn Sandstede (Surrey) has beenawarded a five-year Royal SocietyWolfson Research Merit Award forresearch in coherent structures in sci-ence and biology: dynamics, stabilityand interaction.
David Acheson (Oxford) has beenawarded a National TeachingFellowship.
Ben Green (Cambridge) was present-ed a 2004 Clay Research Award inrecognition of his work on arithmeticprogressions of primes.
Jens Marklof (Bristol) has receivedone of the five 2004 Marie CurieExcellence Awards of the EuropeanCommission for his work in quantumchaos and number theory (Semi-Classical Correlations in QuantumSpectra).
We regret to announce the deaths of:
John Chalk (28.6.2004)
Janusz Jerzy Charatonik (11.7.2004)
Klaus Doerk (2.7.2004)
Anatoolii Ya. Dorogovtsev(22.4.2004)
Günther Hellwig (17.6.2004)
Arno Jaeger (24.2.2004)
Olga Alexandrovna Ladyzhenskaya(12.1.2004)
Thomas Maxsein (10.5.2004)
Jerzy Norwa (14.6.2004)
Flemming Damhus Pedersen23.2004)
Gert Kjærgaard Pedersen (13.4.2004)
Helmut Pfeiffer (2.7.2004)
Deaths
EMS December 2004 37
Personal columnPersonal columnPlease send information on mathematical awards and deaths to the editor.
Please e-mail announcements of European confer-ences, workshops and mathematical meetings ofinterest to EMS members, to one of the followingaddresses [email protected] or [email protected]. Announcements should be written in astyle similar to those here, and sent as MicrosoftWord files or as text files (but not as TeX input files).Space permitting, each announcement will appearin detail in the next issue of the Newsletter to go topress, and thereafter will be briefly noted in eachnew issue until the meeting takes place, with a ref-erence to the issue in which the detailed announce-ment appeared.
6–9: 24th Nordic and 1st Franco-NordicCongress of Mathematicians, Reykjavik,IcelandInformation: e-mail:[email protected] Web site: http://www.raunvis.hi.is/1FrancoNordicCongress/[For details, see EMS Newsletter 53]
28–30: 4th Mediterranean Conference onMathematics Education, Palermo-ItalyInformation: e-mail: [email protected] Web sites: http://math.unipa.it/~grim/ mediter-ranean_05.htm; www.cms.org.cy[For details, see EMS Newsletter 53]
5–13: Applications of braid groups and braidmonodromy, EMS Summer School, Eilat, IsraelInformation: Web site: www.emis.de/etc/ems-summer-scho-ols.html
14-19: Topology, analysis and applications tomathematical physics, Moscow, RussiaDedicated to the memory of Prof. Yu.P. Solovyov(8.10.1944-11.9.2003)Topics: Geometry and topology, analysis, applica-tions to mathematical physics including the follo-wing topics of particular interest of Yu.P. Solovyov:algebraic and differential topology, noncommutati-ve geometry, elliptic theory, operator algebras, pathintegrationFormat: Plenary lectures (1.5 hour) and sessionlectures (45 min)Organizing and Programme Committee: V.A.Sadovnichij (chairman), A. Bak, P. Baum, V.V.Belokurov (vice-chairman), V.M. Buchstaber, D.Burghelea, S.M. Gussein-Zade, A.A. Egorov, J.Eichhorn, A.T. Fomenko (vice-chairman), M.Karoubi, G.G. Kasparov, R. Krasauskas, A. Lipko-vski, A.S. Mishchenko (vice-chairman), Th.Yu.Popelensky, V.V. Sharko, V.M. Tikhomirov, E.V.
February 2005
January 2005
Troitsky, V. ZollerInformation: e-mail: [email protected], [email protected] site: dgfm.math.msu/conf/
17-21: CERME 4 (Fourth Congress of the Euro-pean Society for Research in Mathematics Edu-cation), Sant Feliu de Guíxols, SpainAim: CERME is a congress designed to foster acommunicative spirit. It deliberately and distincti-vely moves away from research presentations byindividuals towards collaborative group work. Itsmain feature is a number of thematic groups whosemembers will work together in a common researcharea. Researchers wishing to present a paper at thecongress should submit the paper to one of thesegroups. In addition to the thematic working groups,there will be plenary sessions, including plenaryspeakers and debates, poster sessions, and ERMEpolicy sessions.Programme Committee: A. Arcavi (IL); C. Berg-sten (S); R. Borromeo Ferri (G); M. Bosch (E); J.-L. Dorier (F); N. Gorgorió (E); B. Jaworski (UK-N); J. Pedro Ponte (P); H. Steinbring (G); N. Steh-likova (CH); E. Swoboda (PL); R. Zan (G)Speakers: Y. Chevallard (F); M. Brown (UK); J.D.Godino (E); M. Artigue (F); P. Ernest (UK); F.Furinghetti (I); J. Matos (P); C. Tzanakis (GR)Working Groups: The congress is organised aro-und some thematic groups working in parallel.Over 4 days, groups will have at least 12 hours tomeet and progress their work. Participants areencouraged to attend and contribute to the work ofjust one group.Information:Web site: http://cerme4.crm.es
9–April 1: 14th INTERNATIONAL WORK-SHOP ON MATRICES AND STATISTICS(IWMS-2005), Massey University, AlbanyCampus, Auckland, New ZealandInformation:Web site: http://iwms2005.massey.ac.nz/[For details, see EMS Newsletter 53]
4-8: International conference on SelectedProblems of Modern Mathematics, dedicated tothe 200th anniversary of K.G. Jacobi, and the750th anniversary of the Koenigsberg founda-tionAim: Kaliningrad State University hosts an interna-tional conference, organised under the aegis of theRussian Academy of Sciences (RAS). Leadingmathematicians from Russia and other countries areexpected to participate.Topics: Algebra and Geometry; MathematicalPhysics; Applied Mathematics, MathematicalModelling and IT; Theoretical Mechanics; History
April 2005
March 2005
of Mathematics International Programme Committee: L.D.Faddeev, chair; A.B. Zhizhchenko, deputy chair;S.K. Godunov; V.V. Kozlov; A.P. Klemeshev; S.P.Novikov; V.A. Sadovnichii; V.M. Buchstaber(Russia); O. Pironneau (France); F. Hirzebruch; W.Jäger; C. Zenger (Germany); M. Atiyah (UK); L.Karleson (Sweden)Organising Committee: A.P. Klemeshev, chair;S.A. Ishanov, deputy chair; B.N. Chetverushkin,deputy chair; G.M. Fedorov; L.V. Zinin; A.I.Maslov; S.I. Aleshnikov; K.S. Latyshev Format: Plenary lectures and working groups.Apart from plenary sessions and working groups,the scholars are welcome to make presentations andgive lectures for students and post-graduates atKSU departments. Kaliningrad State University covers the lecturers’royalties, accommodation costs and per diem. Abstracts: are to be published before the confer-ence. Conference fee: 200 €; for students and post-grad-uates: 100 €Deadline: January 10, 2005Information:e-mail: [email protected]
9-20: The Third Annual Spring Institute onNoncommutative Geometry and Operator alge-bras & 20th Shanks Lecture, VanderbiltUniversity, Nashville, Tennessee, USAMain speakers: A. Connes (College de France,IHES & Vanderbilt University), V. Jones (UCBerkeley), U. Haagerup (Univ. of SouthernDenmark), S. Popa (UCLA) and D. Voiculescu(UC Berkeley).Organisers: A. Connes (director), D. Bisch, B.Hughes, G. Kasparov and G. Yu.Information:e-mail: [email protected] site: http://www.math.vanderbilt.edu/~ncgoa05
16-18: Conference “Algorithmic InformationTheory”, University of Vaasa, FinlandTheme: Algorithmic Information Theory is aninternational meeting bringing together mathemati-cians, computer theorists, logicians, mathematicalphysicists and scientists in related fields to assessrecent development in the Foundations ofMathematics and the limits to Computability andComputation and to stimulate further research inthese and related fields.Topics: Algorithmic Information Theory; FormalSystems; Symbolic Computation; Cellular andQuantum Automata; Operator and Spectral TheoryMain speakers: B. Buchberger (RISC Institute inLinz); G. Chaitin (IBM Thomas Watson ResearchCenter, New York); G. Dasgupta (Columbia Univ.,New York); J. Kari (Univ. of Turku); H. Langer(Technische Univ. Wien); H. de Snoo (Univ. ofGroningen); K. Sutner (Carnegie-MellonUniversity)Organising committee: M. Laaksonen (Univ. ofVaasa); M. Linna (Univ. of Vaasa); S. Mäkinen(Vaasa Polytechnic); Jari Töyli (Univ. of Vaasa);M. Wanne (Univ. of Vaasa), chairmanProgram Committee: S. Hassi (Univ. of Vaasa);V. Keränen (Rovaniemi Polytechnic); C.-G.Källman (Vaasa Polytechnic); chairman, M. Linna(Univ. of Vaasa); H. Niemi (Vaasa Polytechnic); T.Vanhanen (Stadia)Deadline: for extended abstracts January 31, 2005
May 2005
CONFERENCES
EMS December 200438
Forthcoming conferencesForthcoming conferencescompiled by Vasile Berinde (Baia Mare, Romania)
Information: e-mail: [email protected] Web site: http://www.uwasa.fi/ait05
31-June 3: 4th Kortrijk Conference on DiscreteGroups and Geometric Structures, withApplidations, Oostende, BelgiumTopics: Recent developments concerning all inter-actions between group theory and geometry includ-ing geometric group theory, group actions on man-ifolds, crystallographic groups and all possiblegeneralizations (affine, polynomial, projective crys-tallographic groups, almost-crystallographicgroups), discrete subgroups of Lie groups, graphs ofgroups Scientific Committee: Y. Félix (Univ. Catholiquede Louvain), W. Goldman (Maryland, CollegePark), F. Grunewald (Univ. Düsseldorf), P. Igodt(K.U.Leuven/Kortrijk), K.B. Lee (Oklahoma)Main Speakers: O. Baues (Univ. Karlsruhe), Y.Benoist (ENS, Paris), M. Bridson (ImperialCollege, London), B. Farb (Univ. of Chicago), O.Garcia-Prada (Univ. Autonoma de Madrid), E.Ghys (Ecole normale Supérieure, Lyon), D. Toledo(University of Utah)Proceedings: will be published as a part of a spe-cial issue in Geometriae DedicataLocation: Hotel Royal Astrid, OostendeConference fee: 145€Deadline: with abstract: April 12, 2005; without:May 1, 2005Information: e-mail: [email protected] site: http://www.kulak.ac.be/ workshop
12–24: Foliations 2005, £odz, PolandInformation:e-mail: [email protected] Web site: http://fol2005.math.uni.lodz.pl [For details, see EMS Newsletter 53]
13–17: Computational Methods and FunctionTheory CMFT 2005, Joensuu, FinlandAim: The general theme of the meeting concernsvarious aspects of interaction of complex variablesand scientific computation, including related topicsfrom function theory, approximation theory andnumerical analysis. Another important aspect is topromote the creation and maintenance of contactswith scientists from diverse cultures.Format: Invited one-hour plenary lectures, invitedand contributed session talks, and poster sessions.Plenary speakers: L.A. Beardon (Oxford), P.Clarkson (Canterbury), H. Farkas (Jerusalem), A.Fokas (Cambridge), H. Hedenmalm (Stockholm),D. Khavinson (Fayetteville), A. Martinez-Finkelshtein (Almeria), M. Seppälä (Helsinki), N.Trefethen (Oxford) and R. Varga (Kent)Scientific committee: St. Ruscheweyh(Würzburg), E. B. Saff (Nashville), O. Martio(Helsinki) and I. Laine (Joensuu)Grants: Limited amount of support available forparticipants coming from developing countries. Information: e-mail: [email protected] site: http://www.joensuu.fi/cmft
20-22: Workshop on Mathematical Problemsand Techniques in Cryptology, Centre deRecerca Matemàtica, Campus of the UniversitatAutònoma de Barcelona
Speakers: A. Lenstra (Eindhoven Univ. ofTechnology); R. Cramer (Leiden Univ.); T.Okamoto (NTT Laboratories); H. Krawkzyc
June 2005
(Technion Univ.); S. Galbraith (Royal HollowayUniv. of London)Program Co-chairs: R. Cramer; T. OkamotoCoordinators: J.L. Villar and C. Padró (Univ.Politècnica de Catalunya)Information:e-mail: [email protected] site: http://www.crm.es/jornadas_ criptologia
25–July 2 : Subdivision schemes in geometricmodelling, theory and applications, EMSSummer School, Pontignano, ItalyInformation: Web site: www.emis.de/etc/ems-summer-scho-ols.html[For details, see EMS Newsletter 53]
28-July 2: Barcelona Conference on GeometricGroup Theory, Centre de Recerca Matemàtica,Campus of the Universitat Autònoma deBarcelonaSpeakers: M. Bestvina (Univ. of Utah); T. Delzant(Univ. de Strasbourg); G. Levitt (Univ. of Caen); K.Vogtmann (Cornell Univ.)Programme Committee: N. Brady; J. Burillo; E.Ventura; J. Burillo, coordinatorInformation:e-mail: [email protected] Web site: http://www.crm.es/GeometricGroupTheory
5-15: Advanced Course on the Geometry of theWord Problem for finitely generated groups,Centre de Recerca Matemàtica, Campus of theUniversitat Autònoma de BarcelonaSpeakers: H. Short (Univ. d’Aix-Marseille I); N.Brady (Oklahoma Univ.); T. Riley (Yale Univ.)Coordinator: Josep BurilloRegistration Fee: 160 EURDeadline: May 15, 2005Grants: The CRM offers a limited number ofgrants for registration and accommodationaddressed to young researchers. The deadline forapplication is April 5, 2005.Information: e-mail: [email protected] site: http://www.crm.es/Word Problem
17-August 14: Summer School of AtlanticAssociation for Research in the MathematicalSciences, Campus of Dalhousie University inHalifax, Nova Scotia, CanadaAim: The Summer School is intended for graduatestudents and promising undergraduate students inMathematics from all parts of the world. Each par-ticipant is expected to register for two courses. Thecourses will be: 1. Convexity and Fixed PointAlgorithms in Hilbert Space (H. Bauschke, Univ. ofGuelph); 2. Integral Geometry of Convex Bodiesand Polyhedra (D. Klain, Univ. of Massachusetts atLowell); 3. The Mathematics of Finance (W.Runggaldier, Univ. di Padova); 4. MathematicalStatistics (B. Smith, Dalhousie Univ.)Format: Each course consists of four sixty-minutelectures and two ninety-minute problem sessionseach week, for four weeks;Organizer: The Atlantic Association for Researchin the Mathematical Sciences” (Canada)Location: Campus of Dalhousie University inHalifax, Nova Scotia, Canada. Accommodation isavailable on CampusGrants: Some financial support in Halifax is possi-ble.
July 2005
School Director: T. Thompson, Dalhousie Univ. Information:e-mail: [email protected]; [email protected]; [email protected]
17-23: European young statisticians trainingcamp, EMS summer school, Oslo Norwayassociated with the European Meeting of Statistics(see below).Information: Web site: www.emis.de/etc/ems-summer-scho-ols.html#2005
24-28: 25th European Meeting of Statistics,Oslo, NorwayOrganizer: University of Oslo and the NorwegianComputing CenterFormat: 8 invited lectures, 24 invited sessions andcontributed sessionsInvited lecturers: D. Donoho (Stanford), Y. Peres(Berkeley), O. Aalen (Oslo), J. Bertoin (Paris), W.Kendall (Warwick), N. Shephard (Oxford), L.Davies (Essen), K. Worsley (Mc Gill)Programme Committee: A. van der Vaart (chair,Amsterdam), Ø. Borgan (Oslo), U. Gather (Dort-mund), S. Richardson (London), G. Roberts (Lan-caster), T. Rolski (Wroc³aw), J. Steif (Göteborg)Deadlines: for grant application: March 1, 2005;for submission of contributed papers and posters:March 31, 2005; for early registration: May 1,2005; for registration: June 30, 2005.Conference fee: 310€, 280€ (Bernoulli Society ofISI members), 200€ (students)Information: e-mail: [email protected] Web site: www.ems2005.no;
30-August 6: Groups St. Andrews 2005,University of St. Andrews, St. Andrews,ScotlandAim: This conference, the seventh in the series ofGroups St Andrews Conferences, will be organisedalong similar lines to previous events in this series.Speakers: P.J. Cameron (Queen Mary, London);R.I. Grigorchuk (Texas A&M); J.C. Meakin(Nebraska-Lincoln); A. Seress (Ohio State)Programme: The speakers above have kindlyagreed to give short courses of lectures. In additionthere will be a programme of one hour invited lec-tures and short research presentations.Topics: The conference aims to cover all aspects ofgroup theory. The short lecture courses are intend-ed to be accessible to postgraduate students, post-doctoral fellows, and researchers in all areas ofgroup theory.Location: Mathematical Institute, St. Andrews.Scientific Organising Committee: C. Campbell(St. Andrews); N. Gilbert (Heriot-Watt); S. Linton(St. Andrews); J. O’Connor (St. Andrews); E.Robertson (St. Andrews); N. Ruskuc (St.Andrews); G. Smith (Bath)Information:e-mail: [email protected] site: http://groupsstandrews.org
21-26: SEMT ‘05 (International Symposium onElementary Mathematics Teaching), CzechRepublic, Charles University in Prague, Facultyof EducationTheme: Understanding the environment of themathematics classroom.Format: Plenary lectures, presentation of papers,workshops and discussion groups.
August 2005
CONFERENCES
EMS December 2004 39
Information:e-mail: [email protected] site: http://www.pedf.cuni.cz/kmdm/index.htm
13-23: Advanced Course: Recent Trends onCombinatorics in the Mathematical Context;Centre de Recerca Matemática, Campus of theUniversitat Autònoma de BarcelonaSpeakers: B. Bollobas (Trinity College Cambridgeand Univ. of Memphis); J. Nesetril (Charles Univ.and ITI, Prague)Coordinator: O. SerraRegistration Fee: 160 EURDeadline: July 10, 2005Grants: The CRM can offer a limited number ofgrants to young researchers covering the registra-tion fee and/or accommodation. The deadline forapplication is June 1, 2005.Information:e-mail : [email protected] site: http://www.crm.es/Recent Trends
11-18: EMS Summer School and SéminaireEuropéen de Statistique, Statistics in Geneticsand Molecular Biology, Warwick, UKGrants: 40 Marie Curie Actions grants availableLocal Organization: B. Finkenstadt (Warwick)Information: e-mail: [email protected] site: http://www2.warwick.ac.uk/fac/ sci/sta-tistics/news/semstat/
17-21: Nonlinear Parabolic Problems, HelsinkiAim: An international program on nonlinear para-bolic partial differential equations will be organizedin Helsinki, Finland, during Sept-Nov 2005.The program will run at the University of Helsinkiand at the Helsinki University of Technology(HUT) and is sponsored by the governmentalagency The Academy of Finland. Within theframework of this programme, a conference onnonlinear parabolic problems will be also held inHelsinki.Organizers: H. Amann (amann@math. unizh.ch);J. Taskinen (jari.taskinen@ helsinki.fi); S.-O.Londen ([email protected])Main topics: Qualitative theory of parabolic equa-tions; Reaction-diffusion systems; Fully nonlinearproblems; Free boundary problems; Navier-Stokesequations; Maximal regularity; Degenerate para-bolic problemsSpeakers: M. Chipot (Zurich); Ph. Clement(Delft); J. Escher (Hannover); M. Fila (Bratislava);M. Hieber (Darmstadt); G. Karch (Wroclaw); H.Kozono (Sendai); Ph. Laurencot (Toulouse); J.Lopez-Gomez (Madrid); S. Nazarov (St.Peters-burg); W.-M. Ni (Minnesota); M. Pierre (Rennes);J. Pruess (Halle); P. Quittner (Bratislava); J.Rehberg (Berlin); G. Simonett (Nashville); H.Sohr(Paderborn); V. Solonnikov (St. Petersburg); Ph.Souplet (Versailles); J.L.Vazquez (Madrid); D.Wrzosek (Warszawa); L. Weis (Karlsruhe); E.Yanagida (Sendai)Location: The conference will take place at HUT,located northwest of downtown Helsinki.Information: Web site: http://www.math.helsinki.fi/research/FMSvisitor0506
October 2005
September 2005
EMS December 200440
M. Anderson, V. Katz, R. Wilson: Sherlock Holmes inBabylon and Other Tales of Mathematical History,Spectrum, The Mathematical Association of America,Washington, 2003, 420 pp., $49,95, ISBN 0-88385-546-1This excellent book contains 44 articles on the history ofmathematics, which were published in journals of theMathematical Association of America (AmericanMathematical Monthly, College Mathematics Journal,Mathematics Magazine, National Mathematics Magazine)over the past 100 years. The articles were written by dis-tinguished past historians of mathematics (e.g., F. Cajori, J.L. Coolidge, M. Dehn, D. E. Smith, C. Boyer) as well assome contemporary ones (including E. Robson, R.Creighton Buck, V. Katz). The articles are divided intofour sections (Ancient mathematics, Medieval and renais-sance mathematics, The seventeenth century, The eigh-teenth century) and they cover almost 4000 years (from theancient Babylonians to the development of mathematics inthe eighteenth century). The most interesting topics fromBabylonian, Greek, Roman, Chinese, Indian as well asEuropean mathematics are included. Each section is pre-ceded by a Foreword, containing comments on historicalcontext, and followed by an Afterword. In some cases, twoarticles on the same topic are included showing theprogress in the history of mathematics. The comparisonshows that although modern research brings new informa-tion and new interpretations, the older papers are neitherdated nor obsolete. The book is not a classical textbook ofthe history of mathematics; the topics included do notcover the whole development of mathematics. The bookcan be recommended to everybody interested in the histo-ry of mathematics and to anybody who loves mathematics.(mnem)
R. Arratia, A.D. Barbour, S. Tavaré: LogarithmicCombinatorial Structures: A Probabilistic Approach,EMS Monographs in Mathematics, vol. 1, EuropeanMathematical Society, Zürich, 2003, 375 pp., €69, ISBN 3-03719-000-0Many combinatorial (and other) objects can be decom-posed into connected components and one can be interest-ed in numbers and sizes of components in a random object.In this monograph, strong results are derived for the com-ponent frequency spectrum in a rather general probabilisticsituation, which is determined by two conditions (axioms):the conditioning relation and the logarithmic condition. Inchapter 1, the decomposition of permutations to cycles andthe decomposition of integers to primes are compared andin chapter 2, many more examples of combinatorial struc-tures and their decompositions are presented. In theremaining eleven chapters, the authors build an abstractprobabilistic approach to the problem of estimation of thecomponent frequency spectrum. Techniques and notionsused include the Wasserstein distance, the Stein method,the Ewens sampling formula, size biasing, the scale invari-ant Poisson process, the GEM distribution and the Poisson-Dirichlet distribution. In the final chapters, the treatisebecomes technical but the reader is rewarded by strengthand generality of obtained theorems. (mkl)
A. Arvanitoyeorgos: An Introduction to Lie Groups andthe Geometry of Homogeneous Spaces, StudentMathematical Library, vol. 22, American MathematicalSociety, Providence, 2003, 141 pp., $29, ISBN 0-8218-2778-2The main topics treated in this small book are semisimpleLie groups, homogeneous spaces and their geometry,invariant metrics, symmetric spaces and generalized flagmanifolds. The book starts with the definition of a Liegroup and its associated Lie algebra, together with a simpleversion of Lie theorems. A study of the Killing form leadsto the notion of a semisimple Lie algebra. To equip Liegroups with a structure of a Riemannian manifold, bi-invariant metrics are introduced, together with the associ-ated connections and expressions for their curvature.Homogeneous spaces are introduced and their Riemannianstructure is defined by means of G-invariant metrics. Twobasic classes of examples - symmetric spaces and general-ized flag manifolds - are classified. The last chapter con-
tains some applications to physics (homogeneous Einsteinand Kähler-Einstein metrics, Hamiltonian systems on gen-eralized flag manifolds and homogeneous geodesics).Notions introduced in the book are nicely illustrated by alot of examples. The size of the book is very small but itcontains a wealth of interesting material. (vs)
M. Audin, A.C. da Silva, E. Lerman: SymplecticGeometry of Integrable Hamiltonian Systems, AdvancedCourses in Mathematics CRM Barcelona, Birkhäuser,Basel, 2003, 225 pp., €28, ISBN 3-7643-2167-9The three contributions contained in the book are based onlectures given by the authors at the Euro Summer School“Symplectic geometry of integrable Hamitonian systems”,held in Barcelona in July 2001. The first paper (by M.Audin) contains a discussion of special Lagrangian sub-manifolds. This notion was studied very intensively duringrecent years in connection with integrable systems and, inparticular, with string theory and the mirror symmetry.Special Lagrangian submanifolds are rare beings, hard toconstruct, and their moduli space is finite dimensional. Inthe paper, explicit examples of special Lagrangian sub-manifolds are constructed and the moduli space of specialLagrangian submanifolds of a Calabi-Yau manifold is dis-cussed. The second paper (by A. C. da Silva) contains anintroduction to toric manifolds using the moment map andthe moment polytope as the main tools. The text has twoparts. The first part uses a classification of equivalenceclasses of symplectic toric manifolds, using their momentpolytopes, and a computation of homology of symplectictoric manifolds using the Morse theory. The second partconsiders toric manifolds in algebraic geometry and it hasa structure parallel to the first part. The third contribution(by E. Lerman) is devoted to the following problem: Let Mbe the cotangent bundle of an n-dimensional torus withoutthe zero section. Suppose that there is an effective actionof the group G=Rn/Zn on M, preserving the standard sym-plectic structure on M and commuting with dilations. Isthe action of G necessarily free? To answer the question,contact toric manifolds are introduced, and contactmoment maps are used together with the Morse theory onorbifolds. The three sets of lectures complement eachother nicely and the book offers a very useful and system-atic introduction to a modern and interdisciplinary field.(vs)
C. Bardaro, J. Musielak, G. Vinti: Nonlinear IntegralOperators and Applications, de Gruyter Series inNonlinear Analysis and Applications 9, Walter de Gruyter,Berlin, 2003, 201 pp., €88, ISBN 3-11-017551-7The book is devoted to quite general approximation meth-ods based on convolution operators (both linear and non-linear) and to the study of such operators. In order to devel-op the right setting for this investigation, the first two chap-ters contain preliminary results on modular spaces, whichare a suitable generalization of the Orlicz spaces as well asof spaces with bounded ϕ-variation. The following twochapters present some error estimates of approximations interms of moduli of continuity. Classical linear convolutionoperators are generalized here to nonlinear convolutionswith kernels, which either satisfy Lipschitz-type conditionsor are homogeneous in the generalised sense. Applicationsto the summability problem of a family of functions aregiven in Chapter 5. The convergence of approximations inϕ-variation is possible only in certain subspaces of func-tions of bounded ϕ-variation. These subspaces are studiedin Chapter 6. Two basic fixed-point theorems are used inChapter 7 to obtain solutions of nonlinear convolutionequations. There is an important application of interpola-tions to the so-called sampling theorem in signal analysis.The classical linear interpolation methods can be replacedby nonlinear ones. Results in this direction for severalclasses of signals are shown in the last two chapters. Thebook, based mainly on results from the authors, can be con-sidered as a nonlinear continuation of the classical book ofP. L. Butzer and R. J. Nickel (Fourier Analysis andApproximation, Academic Press, 1971). The presentationis clear and self-contained so that the book can be recom-mended not only to researchers in approximation theory
but to graduate students as well. (jmil)
H.-J. Baues: The Homotopy Category of SimplyConnected 4-Manifolds, London Mathematical SocietyLecture Note Series 297, Cambridge University Press,Cambridge, 2003, 184 pp., £24,95, ISBN 0-521-53103-9The main aim of this book is to understand the category ofsimply connected closed topological 4-manifolds and thehomotopy classes of mappings between them. It is neces-sary to mention that the problem consisted especially in thedescription of the homotopy classes of maps. As an impor-tant tool the author uses the category CW(2,4). Its objectsare CW-complexes X with one 0-cell, and then with cellsonly in dimension 2 and 4, and its morphisms are homo-topy classes of mappings. Within this category we find theabove-mentioned manifolds in the sense that every simplyconnected 4-manifold is homotopy equivalent with such aCW-complex X with only one 4-cell. The main achieve-ment of the book is a construction of a purely algebraic cat-egory, which is equivalent to the category CW(2,4). This,of course, enables the transform from various topologicalconsiderations into an algebraic setting. The advantage ofsuch a transformation is obvious. The subject of the bookis relatively special, and this naturally brings the necessityof special procedures and special computations. Thereforethe reader may have an impression that the text is rathertechnical. This is not at all true. On the contrary, the read-er will obtain very deep information on the structure of rel-evant categories. The text is very clearly written but theauthor substantially uses many previous results of his ownas well as many other results. This means that the readingis not very easy. Nevertheless, the results are so excellentthat they deserve some patience and effort. (jiva)
M. A. Bennet, B. C. Berndt, N. Boston, H. G. Diamond,A. J. Hildebrand, W. Philips, (Eds.): Number Theoryfor the Millennium, I, A. K. Peters, Natick, 2002, 461 pp.,$50, ISBN 1-56881-126-8M. A. Bennet, B. C. Berndt, N. Boston, H. G. Diamond,A. J. Hildebrand, W. Philips, (Eds.): Number Theoryfor the Millennium, II, A. K. Peters, Natick, 2002, 447 pp.,$50, ISBN 1-56881-146-2M. A. Bennet, B. C. Berndt, N. Boston, H. G. Diamond,A. J. Hildebrand, W. Philips, (Eds.): Number Theoryfor the Millennium, III, A. K. Peters, Natick, 2002, 450pp., $50, ISBN 1-56881-152-7This three volume collection of papers contains more than1300 pages with 72 talks given at the MillennialConference on Number Theory, which was held at thecampus of the University of Illinois at Urbana-Champaignin May 2000. The papers cover the wide range of contem-porary number theory. The conference was one of themost important international meetings devoted to numbertheory, framed by 175 talks. Consequently the interestedreader finds here not only surveys on the most importantcontributions to number theory and its applications, butalso surveys on methods and techniques used in this impor-tant branch of mathematics. The collection offers arespectable view on contemporary number theory given byprominent number theorists, and therefore could be recom-mended not only to number theorists but generally to allmathematicians interested in various aspects of numbertheory. (špo)
S. Bezuglyi, S. Kolyada, Eds.: Topics in Dynamics andErgodic Theory, London Mathematical Society LectureNote Series 310, Cambridge University Press, Cambridge,2003, 270 pp., £30, ISBN 0-521-53365-1The volume collects some of the mini-courses presented atthe International Conference and US-Ukrainian Workshop“Dynamical Systems and Ergodic Theory”, held inKatsiveli (Crimea, Ukraine) in August 2000. The intro-ductory contribution by A. M. Stepin is devoted to thememory of V. M. Alexeyev, one of the best lecturers of theKatsiveli’s school, who died in 1980 at the age of 48. Thepaper “Minimal idempotents and ergodic Ramsey theo-rems”, by V. Bergelson, reviews the construction of theStone-Èech compactification β N of N as the Ellis envelop-ing semigroup of the map σ = x+1 on N. Using minimalidempotents, the celebrated van der Waerden theorem onarithmetic progressions is proved. In “Symbolic dynamicsand topological models in dimensions 1 and 2”, A. deCarvalho and T. Hall present the classical kneading theoryof unimodal systems and extend it to two-dimensional sys-tems like the horseshoe. In “Markov odometers”, A. H.Dooley proves that every ergodic non-singular transforma-tion is orbit equivalent to a Markov odometer on a Bratteli-
BOOKS
EMS December 2004 41
Recent booksRecent booksedited by Ivan Netuka and Vladimír Souèek (Prague)
Vershik system. In “Geometric proofs of Mather’s con-necting and accelerating theorems”, V. Kaloshin treats thewandering trajectories of exact area preserving twist maps.In “Structural stability in 1D dynamics”, O. Kozlovskitreats the structural stability in spaces of smooth and ana-lytic maps. In “Periodic points of nonexpansive maps: asurvey”, B. Lemmas shows that trajectories of nonexpan-sive maps converge to periodic orbits. In “Arithmeticdynamics”, N. Sidorov deals with explicit arithmeticexpansions of reals and vectors that have a dynamicalsense. In “Actions of amenable groups”, B. Weiss gener-alizes much of the classical ergodic theory to generalamenable groups like Zn; in particular he proves theShannon-McMillan theorem. (pku)
N. Bourbaki: Elements of Mathematics. Integration I.Chapters 1-6, Springer, Berlin, 2004, 472 pp., €99,95,ISBN 3-540-41129-1Bourbaki, a collective author in the sixties, wrote the seriesof books ‘Elements of Mathematics’. This is an Englishtranslation of the well-known original French edition. Itcontains the first six chapters (of nine) from the part devot-ed to integration. As in all books in the series, presentationof material is abstract, proceeding from general to particu-lar. Therefore a good knowledge of an undergraduatecourse seems to be almost necessary. The approach to theintegration theory here is functional, based on the notion ofa measure as a continuous linear functional on the space ofreal, or complex, continuous functions with compact sup-port in a locally compact topological space. The six chap-ters cover a detailed exposition of results on extension ofmeasures. The chapter on integration of measures includesthe Lebesgue-Fubini theorem, the Lebesgue-Nikodým the-orem, and results on disintegration of measures. An essen-tial part of the theory of Lp spaces is also covered. The pre-sentation of vector-valued integration is based on the weakintegral. The text contains full proofs of the stated results,many exercises, and worthwhile historical notes. It is writ-ten very carefully to prevent misunderstandings and tomake orientation in the text easy. (ph)
D. Cerveau, E. Ghys, N. Sibony, J.-Ch. Yoccoz:Complex Dynamics and Geometry, SMF/AMS Texts andMonographs, vol. 10, American Mathematical Society,Société Mathématique de France, Providence, 2003, 197pp., $59, ISBN 0-8218-3228-XThe book contains four survey papers on different butclosely related topics in the theory of holomorphic dynam-ical systems. The first paper (by D. Cerveau) is devoted toa study of codimension 1-holomorphic foliations. Themain tool used here is a reduction of singularities.Particular attention is devoted to foliations in dimensionstwo and three. The paper ends with applications to the sin-gular Frobenius theorem and a discussion of invarianthypersurfaces. The second paper (by E. Ghys) contains adiscussion of Riemann surface laminations, arising in thetheory of holomorphic dynamical systems. The Riemannsurface laminations are more general objects than ordinaryfoliations, the ambient space need not have a structure of amanifold. Their leaves are (not necessarily compact)Riemann surfaces. Classical questions for Riemann sur-faces (uniformization, existence of meromorphic func-tions) are studied in this more general setting. The paperby N. Sibony describes an analogue of the Fatou-Julia the-ory for a rational map f from Pk(C) to itself. The dynamicsof f are described using properties of a suitable closed pos-itive current of bidegree (1,1). The second part is devotedto a study of regular polynomial biholomorphisms of Ck,and the last part to holomorpic endomorphisms of Pk(C).For the convenience of the reader, properties of currentsand plurisubharmonic functions are summarized at the endof the paper. The fourth contribution (by J.-C. Yoccoz,written by M. Flexor) describes properties of the simplestnontrivial case of holomorphic dynamics, given by a qua-dratic polynomial in one complex variable. The discussionis centred around hyperbolic aspects, the Jacobsen theoremand quasiperiodic properties, related to problems of smalldivisors. The whole book starts with an introductory paperon holomorphic dynamics (written by E. Ghys). It containsmany interesting comments on the historical evolution ofthe discussed topics and their mutual relations. (vs)
J. W. Cogdell, H. H. Kim, M. R. Murty: Lectures onAutomorphic L-functions, Fields Institute Monographs,American Mathematical Society, Providence, 2004, 283pp., $63, ISBN 0-8218-3516-5This book consists of lecture notes from three graduate
courses given at the special program on authomorphicforms at the Fields Institute in the spring of 2003. Thecourse by Cogdell is an introduction to standard L-func-tions of automorphic forms on GL(n) and the Rankin-Selberg L-functions on GL(m) x GL(n). It covers the fol-lowing topics: Whittaker models, local and global func-tional equations, converse theorems and functorial lifts forclassical groups. The course by Kim is a survey of theLanglands–Shahidi approach to L-functions via constantterms of Eisenstein series. It culminates in the recentproofs of functoriality of the symmetric cube and fourth forGL(2). The course by Ram Murty centers on analyticproperties of automorphic L-functions and their applica-tions to estimates for Hecke (and Laplace) eigenvalues.This book is a wonderful introduction to the Langlandsprogram. It is heartily recommended to students (andresearchers) specializing in number theory and relatedareas. (jnek)
J. K. Davidson, K. H. Hunt: Robots and Screw Theory.Application of Kinematics and Statistics to Robotics,Oxford University Press, Oxford, 2004, 476 pp., £85, ISBN0-19-856245-4This monograph can be regarded as a comprehensive text-book on applications of screw theory in a variety of situa-tions in mechanics and robotics. Special attention is paidto modern applications of screws in the robotics of bothserial and parallel manipulators. A screw is defined eitheras a pair of a force and a couple (a wrench) or as an infini-tesimal space motion (twist). Later on, the connection tothe vector field of velocities of a spatial motion and to theinstantaneous motion is described. The basic properties ofscrews are described in chapter 3, including screw(Plücker) coordinates. The next three chapters deal withcoordinate transformations, screw systems and reciprocity(duality) of twists and wrenches. Properties of serial robotmanipulators are studied in chapters 6 and 7, includingmany examples of commercially used robots. Chapters 7and 8 are devoted to parallel robot-manipulators and theirbasic geometric properties, including the Jacobian and itscomputation. Special attention is given to the 3-3 parallelmanipulator - the octahedral structure. The rest of the bookconcentrates on more advanced topics, for instance onmanipulators combining serial and parallel structures,redundant robotic systems and legged vehicles.Appendices give some useful formulas from line geometryand basic ideas of the projective representation of screws inconnection with the projective line geometry. The Studyrepresentation is mentioned at the very end. The book con-tains useful historical remarks on the origins of screw the-ory and related topics, references contain not only histori-cal sources on the subject but also recent publications rele-vant to considered problems. The main point of the booklies in applications of screws and not much space is devot-ed to the theory. This means that no proofs are given, thealgebraic structure of the screw space is not emphasizedand the style of the book is traditional. It can be recom-mended to all engineers and postgraduate students doingresearch in complicated mechanical systems, in particularin the mechanics of serial and parallel robot manipulators.(ak)
M. Emerton, M. Kisin: The Riemann-HilbertCorrespondence for Unit F-Crystals, Astérisque 293,Société Mathématique de France, Paris, 2004, 257 pp.,€57, ISBN 2-85629-154-6If k is a perfect field of characteristic p > 0 and X a smoothWn(k)-scheme, a well known generalization of theArtin–Schreier theory (due to N. Katz) establishes anequivalent of categories between locally free étale sheavesof Z/pnZ –modules on X and vector bundles E on X satis-fying F*R ≈ E (where F is a lift of the Frobenius to X).The main goal of the book is to generalize this result to aRiemann-Hilbert type correspondence between the derivedcategory Dbctf (Xet, Z/pnZ) and a certain triangulated cate-gory of arithmetic D–modules equipped with an action ofFrobenius. In fact, the case n=1 is treated separately, as itdoes not require any differential operators (nor the perfect-ness of k). (jnek)
L. Fargues, E. Mantovan: Variétés de Shimura espacesde Rapoport-Zink et correspondances de Langlandslocales, Astérisque 291, Société Mathématique de France,Paris, 2004, 331 pp., €66, ISBN 2-85629-150-3This volume contains two articles that generalize someaspects of the recent work of M. Harris and R. Taylor oncohomology of certain unitary Shimura varieties and asso-
ciated moduli space of p-divisible groups. As the authorsconsider unitary groups of more general signature, theyhave to get around the fact that one can no longer useDrinfeld bases to define nice integral models of the rele-vant moduli spaces. Instead, the authors work with the cor-responding rigid analytic spaces defined by M. Rapoportand T. Zink. In the first article, L Fargues shows that onecan realize the local Langlands correspondence (in thesupercuspidal case) in the étale cohomology of certainRapoport–Zink spaces. In the second article, E. Mantovanseries uses the Newton–polygon stratification of the specialfibre of the Shimura variety to relate its cohomology to thecohomology of the associated Rapoport–Zink spaces andgeneralized Igusa varieties. (jnek)
G. N. Frederickson: Dissections: Plane and Fancy,Cambridge University Press, Cambridge, 2003, 310 pp.,£16,95, ISBN 0-521-57197-9, ISBN 0-521-52582-9The book by Frederickson on recreational mathematics isdevoted to the problem of how to cut a square (or triangleor hexagon) into the smallest number of pieces and how torearrange them into two squares (or triangles or hexagons).The book also deals with others figures, e.g. stars, MalteseCrosses, and with solids (polyhedra). Martin Gardner, awell-known expert in recreational mathematics, can beconsidered the unofficial godfather of the book. Even greatmathematicians were interested in dissections. In 1900,David Hilbert presented the famous wide-ranging list oftwenty-three problems and a third of them dealt with dis-sections of polyhedra (the negative result proposed by himwas proven by M. Dehn within a few months). The read-er will find solutions to all problems contained at the endof the book. The bibliography is very comprehensive. Abasic knowledge of high school geometry is sufficient forreading the book. Every puzzle fan will like this interest-ing and amusing book. (lboc)
F. Gesztesy, H. Holden: Soliton Equations and TheirAlgebro-Geometric Solutions, vol. I: (1+1)-DimensionalContinuous Models, Cambridge Studies in AdvancedMathematics 79, Cambridge University Press, Cambridge,2003, 505 pp., £65, ISBN 0-521-75307-4The field of completely integrable systems has developedenormously in the last decades. The book under reviewcovers a part of this broad landscape. Its aim is to discussin detail algebro-geometric solutions of five hierarchies ofintegrable nonlinear equations. The presented class ofsolutions form a natural extension of the classes of solitonand rational solutions, and can be used to approximatemore general solutions (e.g. almost periodic ones). Basictools in the description are spectral analysis and basic the-ory of compact Riemann surfaces and their theta functions.The basic KdV hierarchy is the most famous case, it con-tains the equation for solitary waves on channels, whichwere discovered by Scott Russell in 1834. The solutions ofthe KdV hierarchy are discussed in the first chapter. Thediscussion is presented in more detail for this first case thanin the other four cases. The second hierarchy treated inChapter 2 is a combined sine-Gordon and modifiedKorteweg-de Vries hierarchy. The third chapter contains adiscussion of solutions of the AKNS (Ablowitz, Kaup,Newell, Segur) system and related classical Boussinesqhierarchies. The classical massive Thirring system is treat-ed in Chapter 4. The last chapter describes solutions of theCamassa-Holm hierarchy. Individual chapters are orga-nized in such a way that they can be read independently.To reach this goal, similar arguments in constructions arerepeated in individual cases. Each chapter ends withdetailed notes (e.g. notes for the first chapter have 17pages) with references to literature, comments and addi-tional results. In the Appendix (140 pages), it is possible tofind a summary of many fields (e.g. algebraic curves, thetafunctions, the Lagrange interpolation, symmetric func-tions, trace formulae, elliptic functions, spectral measures),which are used in the main chapters. At the end, the read-er can find an extensive bibliography (30 pages of refer-ences). The book is very well organized and carefully writ-ten. It could be particularly useful for analysts wanting tolearn new methods coming from algebraic geometry. (vs)
V. I. Gromak, I. Laine, S. Shimomura: PainlevéDifferential Equations in the Complex Plane, de GruyterStudies in Mathematics 28, Walter de Gruyter, Berlin,2002, 299 pp., €88, ISBN 3-11-017379-4At the beginning of the 20th century, Painlevé studiedproperties of second order differential equations in thecomplex plane and isolated a certain number of equations
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with distinguished behaviour. Their solutions have theproperty that there are no movable singularities other thanpoles. Later on, attention has concentrated to six equationsP1, …, P6 of Painlevé type with the most interesting prop-erties. The book is devoted to them. In the first chapter,the authors show that the equations P1, P2, P4 and modifica-tions of equations P3 and P5 have meromorphic solutionsonly (with proofs in the first three cases). The growthproperties of solutions of these 5 types of equations togeth-er with the value distribution theory for them are studied inthe next two chapters. The next six chapters are devoted toa study of the properties of solutions of six Painlevé equa-tions. Behaviour of solutions in a neighbourhood of a sin-gularity is studied. Integrable (systems of) equations usu-ally come in whole hierarchies. Higher order analogues ofthe Painlevé equations of the first two types are discussedin the book. For specific values of parameters of the equa-tions, solutions can be constructed using the Bäcklundtransformations. For these values of parameters, solutionscan be expressed in terms of rational functions or classicaltranscendental functions. Relations of the Painlevé equa-tions to hierarchies of integrable systems (KdV,Boussinesq, sine-Gordon, nonlinear Schrödinger, Einsteinand Toda type equations) are discussed in the last chapter.The two appendices offer a summary of basic facts neededabout solutions of ordinary differential equations in thecomplex plane and on the Nevanlinna theory. Interest inproperties of solutions of Painlevé equations is steadilygrowing and they appear in many questions in mathemat-ics and mathematical physics. Hence, the careful treatmentof them in the presented monograph is a valuable additionto the existing literature. (vs)
A. Grothendieck, Ed.: Revêtements étales et groupe fon-damental (SGA 1), Documents Mathématiques 3, SociétéMathématique de France, Paris, 2003, 325 pp., ISBN 2-85629-141-4This is a new edition of the first volume of A.Grothendieck’s SGA (Séminaire de géométrie algébrique),which is devoted to the theory of the algebraic fundamen-tal group. The book incorporates a few minor correctionsand several updating remarks by M. Raynaud. A TEX fileof this text (as well as of the uncorrected version), typesetby a team of volunteers headed by S. Edixhoven, is avail-able from the arXiv.org e-print server. This book is indis-pensable to any serious student of algebraic geometry.(jnek)
B. Hasselblatt, A. Katok: A First Course in Dynamics:with a Panorama of Recent Developments, CambridgeUniversity Press, Cambridge, 2003, 424 pp., £25,95, ISBN0-521-58750-6, ISBN 0-521-58304-7The book has two parts. The first one (A Course inDynamics: From Simple to Complicated Dynamics) canserve as a textbook for a beginner course in dynamical sys-tems for senior undergraduate students of mathematics,physics and engineering. The explanation is slightlyunusual, since it is based on a number of simple examplesthat present basic behaviour of dynamical systems. Theoutset of theory, including topological and probabilisticmethods, begins from the detailed study of these examples.The exposition is accompanied by many valuable com-ments and exercises of different complexity. The first partends with complicated orbit structures like recurrence andmixing for systems on a torus. Chaotic behaviour is pre-sented and applications to coding are also given.
The second part (Panorama of Dynamical Systems) ismore advanced and is intended as an introduction to mod-ern achievements of the theory. Hyperbolic dynamics isstudied, and results like the closing and shadowing lemmaare proved. A lot of attention is paid to the classical exam-ple of the quadratic map. A mechanism that produceshorseshoes and thus gives rise to chaotic dynamics isshown. The famous Lorentz attractor, as a prototype of astrange attractor, is examined in details. Variationalapproach does not belong to usual tools in dynamical sys-tems. The authors present how this approach can beapplied to the study of twist maps and afterwards toRiemann manifolds. The existence of infinitely manyclosed geodesics on the two-dimensional sphere equippedwith a Riemannian metric is proved. The second part endswith the chapter devoted to relations between number the-ory and dynamical systems. The book is written in a pre-cise and very readable style, there are many useful remarksand figures throughout. It can be highly recommended toanybody who is interested in dynamical systems and whohas a basic knowledge of calculus. The book is also an
excellent introduction to the more advanced monographwritten by the same authors (Introduction to the ModernTheory of Dynamical Systems, Encyclopaedia of mathe-matics and its applications, vol. 54, Cambridge UniversityPress, 1995). (jmil)
R. M. Heiberger, B. Holland: Statistical Analysis andData Display, Springer Texts in Statistics, Springer, NewYork, 2004, 729 pp., 200 fig., €79,95, ISBN 0-387-40270-5The book is written as a text for a yearlong course in sta-tistics. The importance of such a textbook follows from thefact that today there are more than 15,000 statisticians inthe United States only, over 100 U.S. universities offergraduate degrees in statistics and a shortage of qualifiedstatisticians is expected to persist for some time. The stu-dents should learn not only purely mathematical tools butalso the use of computers for obtaining numerical resultsand their graphical presentation. The book describes statis-tical analysis of data and shows how to communicate theresults. The topics included in the book are standard ones:an introduction to statistical inference, one-way and two-way analysis of variance, multiple comparisons, simpleand multiple linear regression, design of experiments, con-tingency tables, nonparametrics, logistic regression, andtime series analysis in time domain. An “invisible” butextremely important part of the book is a web page (or CD)with the data for all examples and exercises, which alsocontains codes in S (i.e., S-PLUS and R) and in SAS. Inonline files, the reader also finds the code and thePostScript file for every figure in the text.
The statistical analysis in the book is taught on interest-ing real examples. Sometimes (e.g. in logistic regression)the analysis is quite deep. An extraordinary care is devot-ed to graphical presentation of results. Many of the graph-ical formats are novel. Finally, appendices to the bookcontain useful remarks on statistical software, typography,and a review of fundamental mathematical concepts. Areader who studies the book and repeats the authors’ cal-culations gains basic statistical skills and knowledge ofsoftware for solving similar problems in applications. Thebook contains no theorems and no proofs. The methodsare only explained and then demonstrated. However, if astatistical course contains a theoretical part, then the bookcan serve as a source of interesting examples and problems.By the way, the authors’ web page also contains errata,which is quite comfortable for the readers. It is surely nota final list of misprints. For example, the median of thebinomial distribution with n=2 and p=0.5 is h=1 whichdoes not satisfy condition (3.10) on p. 28. (ja)
T. A. Ivey, J. M. Landsberg: Cartan for Beginners:Differential Geometry via Moving Frames and ExteriorDifferential Systems, Graduate Studies in Mathematics,vol. 61, American Mathematical Society, Providence,2003,378 pp., $59, ISBN 0-8218-3375-8Moving frames and exterior differential systems belong toclassical methods for studying geometry and partial differ-ential equations. These ideas emerged at the beginning ofthe 20th century, being introduced and developed by sever-al mathematicians, in particular, by Élie Cartan. Over theyears these techniques have been refined and extended.The book is a nice introduction into classical and recentgeometric applications of the techniques. It covers classi-cal geometry of surfaces and basic Riemannian geometryin the language of the method of moving frames; it alsoincludes results from projective differential geometry.There is an elementary introduction to exterior differentialsystems, basic facts from G-structures and general theoryof connections. Every section begins with geometricexamples and problems. There are four appendices devot-ed to linear algebra and representation theory, differentialforms, complex structures and complex manifolds and ini-tial value problems. Interesting exercises are included,together with hints and answers to some of them. Theauthors presented it as a textbook for a graduate levelcourse. It can be strongly recommended to anybody inter-ested in classical and modern differential geometry. (jbu)
W. Keller: Wavelets in Geodesy and Geodynamics,Walter de Gruyter, Berlin, 2004, 279 pp., 130 fig., €84,ISBN 3-11-017546-0This excellent textbook gives an introduction to wavelettheory both in the continuous and the discrete case. Afterdeveloping the theoretical fundament, typical examples ofwavelet analysis in geosciences are presented. The bookconsists of three main chapters. Fourier and filter theoryare shortly sketched in the first chapter. The second chap-
ter is devoted to the basics of wavelet theory. It includesthe continuous as well as the discrete wavelet transformboth in one- and in two-dimensional cases. A specialemphasis is paid to orthogonal wavelets with finite support,because they play an important role in many applications.In the last chapter, some applications of wavelets in geo-sciences are reviewed. The book has developed from agraduate course held at the University of Calgary and isdirected to graduate students interested in digital signalprocessing. The reader is assumed to have a mathematicalbackground on the graduate level. (knaj)
B. M. Landman, A. Robertson: Ramsey Theory on theIntegers, Student Mathematical Library, vol. 24, AmericanMathematical Society, Providence, 2003, 317 pp., $49,ISBN 0-8218-3199-2The topic of this book is Ramsey theory on sets of integers.Chapter 1 introduces notation and basic results of the field,van der Waerden’s theorem, Schur’s theorem and Rado’stheorem. The largest portion of the book, chapters 2-7, isdevoted to the first result. In Chapter 2, a proof of van derWaerden’s theorem is given and bounds on van derWaerden’s numbers are discussed, including the break-through by W. Gowers. Chapters 3-7 deal with variousmodifications and variations on the theme of arithmeticalprogressions (subsets and supersets of arithmetical pro-gressions, homothetic copies of sequences, and modulararithmetical progressions). Chapter 8 is devoted to Schur’stheorem, Chapter 9 to Rado’s theorem and Chapter 10 toother topics (Folkman’s theorem, Brown’s lemma and oth-ers). Each chapter is followed by exercises, a list ofresearch problems, and commentary. The book concludeswith a list of notation and an extensive bibliography with275 items. (mkl)
J.-L. Lions: Oeuvres choisies de Jacques-Louis Lions,vol. I, EDP Sciences, Paris, 2003, 722 pp., €70, ISBN 2-86883-661-5J.-L. Lions: Oeuvres choisies de Jacques-Louis Lions,vol. II, EDP Sciences, Paris, 2003, 864 pp., €70, ISBN 2-86883-662-3J.-L. Lions: Oeuvres choisies de Jacques-Louis Lions,vol. III, EDP Sciences, Paris, 2003, 813 pp., €70, ISBN 2-86883-663-1Jacques-Louis Lions, outstanding scientist and leading per-sonality in partial differential equations and many relatedfields, published more than 20 monographs and more than600 research papers. The presented three volumes repre-sent a high level and representative sample of papers andparts of monographs carefully chosen by the scientificcommittee (A. Bensoussan, P. G. Sciarlet, R. Glowinskiand R. Temam, with the collaboration of coordinators F.Murat and J.-P.Puel).
The first volume starts with an introduction (written byR. Temam) and a commentary by E. Magenes about hisjoint work with J.-L. Lions. The volume is devoted to par-tial differential equations and interpolation theory. It cov-ers, roughly speaking, results published in the periodbetween 1950 and 1960 and it contains almost 30 papers.Let us quote at least a few of the most interesting points. J.-L. Lions and E. Magenes developed a theoretical frame-work for solving non-homogeneous boundary value prob-lems in a series of joint papers “Problèmes des limites nonhomogénes I - VII”, whose first five parts are reproducedhere. The Hilbert part of the theory in Hs(Ω) spaces is con-tained in the three volume monograph of J.-L. Lions and E.Magenes. The Wk,p(Ω) theory is very well described in the(fully reproduced) course by J.-L. Lions at the Universityof Montreal. The interpolation between Sobolev spacesplays a very important role here and thus these results areclosely connected with J.-L. Lions’ contributions to inter-polation theory both within Hilbert spaces and Banachspaces. Significant achievements were obtained also innonlinear partial differential equations. J.-L. Lions togeth-er with J. Leray used monotonocity methods and general-ized the results of F. Browder and G. Minty to functionalsof the calculus of variations that are convex in the highestderivatives only. They also presented a new and elegantproof to the results of M. Vishik. An important part of non-linear PDEs is the classical Navier-Stokes system and itsgeneralizations. J.-L. Lions also contributed to the moderntheory of the Navier-Stokes system. He presented a newand simpler variant of Hopf’s proof of the existence ofweak solutions in three space dimensions - the most inter-esting from the point of view of physics - and in a jointpaper with G. Prodi, they proved uniqueness of weak solu-tions in two space dimensions. Together with Q.
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Stampacchia, he introduced the concept of a variationalinequality, which proved to be very useful in many appli-cations.
The second volume of the series with subtitles“Controle” and “Homogenization”, is introduced by A.Bensoussan and contains results published from the end ofthe sixties up to the end of the nineties. J.-L. Lions’ inter-est in optimal control theory started soon after the pio-neering books of L. S. Pontryagin and R. Bellman, whichappeared in 1957. Already 11 years later, Lions’ book“Controle optimal des systèmes ...“ was published, and itbecame soon a standard reference book. It was well knownthat the problem of optimal control for distributed systemscan lead to problems with free boundary of Stefan type. J.-L. Lions used variational inequalities to make this relationstraightforward. In the seventies, he generalized the notionof variational inequalities so that it was well suited todescribe the problems of optimal stopping time and togeth-er with A. Bensoussan, they studied the stochastic controland variational inequalities and impulse control and quasi-variational inequalities. Later he returned to optimal con-trol theory from a new point of view - namely the possibil-ity of using control theory for stabilization of unstable or illposed problems. New aspects of control theory appear alsoin J.-L. Lions later papers, especially his famous HUM -Hilbert uniqueness method. Together with R. Glowinsky,he introduced numerical methods suitable for solving thesequestions. In the second half of the seventies, J.-L. Lionswas interested in the homogenisation theory for a largescale of equations with periodically oscillating coefficients.The method he used was far-reaching and applicable alsoto problems with an oscillating boundary or problemsposed on perforated domains. Thus it made it possible influid mechanics to deduce Darcy’s equation from Stokes’system, or to study the homogenisation of Bingham fluids.At the same time it allowed the study of mechanical prop-erties of composite materials.
The third volume, with an introduction by P. G. Ciarlet,is devoted to numerical analysis and applications of PDEsin a wide scale of problems - the mechanics of fluids andsolid bodies, Bingham fluids, viscoelastic or plastic mate-rials, problems with friction, and plates described by linearas well as nonlinear elasticity theory. Since 1990, J.-L.Lions was attracted by very complex and complicated sys-tems of PDEs, i.e., by climatology models. Even in thisexceptional case, he, together with R. Temam and S.Wand, proved existence and uniqueness of solutions, theirasymptotic behaviour and ways to their numerical solution.These two papers and more than 20 others dealing withhighly interesting problems are contained in Volume III.The papers in the collection are, as well as all works by J.-L. Lions, written in a very clear and concise form and theyare indispensable for researchers in PDEs and in numericalanalysis. (jsta)
Y. Lu: Hyperbolic Conservation Laws and theCompensated Compactness Method, Monographs andSurveys in Pure and Applied Mathematics 128, Chapman& Hall/CRC, Boca Raton, 2003, 241 pp., $84,95, ISBN 1-58488-238-7The book is devoted to the theory of the compensated com-pactness method, which is a principal tool for studyingproperties of systems of hyperbolic conservation laws.Quasilinear systems of hyperbolic conservation laws in onespace dimension are considered. This setting (systems inone space dimension) is more or less the only case forwhich the existence and uniqueness results can be obtainedalso in a classical way, e.g. by the method of wave-fronttracking. In this book a different approach is used, namelythat of compensated compactness. The author introducesbasic elements of the theory of compensated compactness,based on results of Tartar and Murat from the 80’s. Thenotion of a Young measure is introduced and discussed.After these preliminaries, the author studies the Cauchyproblem for a scalar equation with L∞ and Lp data. It is tobe noted that the simplified proof of the existence of thesolution presented here does not need to use a concept ofthe Young measure. In the system case, the author workswith all the usual concepts, such as the strict hyperbolicity,genuine non-linearity, linear degeneracy, Riemann invari-ants, entropy-entropy flux pair and theory of invariantregions to obtain uniform L-infinity estimates, symmetricand symmetrizable systems. The author then studies dif-ferent important systems of hyperbolic equations, namelythe system of Le Roux type, the system of the polytropicgas dynamic, Euler equations of one-dimensional com-pressible fluid flow, systems of elasticity and some appli-
cations of the compensated compactness to relaxationproblems. The book is carefully written and will be appre-ciated both by PhD students and experts in the field as oneof a few books gathering the knowledge until recently dis-persed among research papers. (mro)
G. Mislin, A. Valette: Proper Group Actions and theBaum-Connes Conjecture, Advanced Courses inMathematics CRM Barcelona, Birkhäuser, Basel, 2003,131 pp., €28, ISBN 3-7643-0408-1The book has two parts. The first contribution, by G.Mislin, contains a discussion of the equivariant K-homolo-gy KG*(EG) of the classifying space EG for proper actionsof a group G. The Baum-Connes conjecture states that theK theory of the reduced C* algebra of a group G can becomputed by the equivariant K-homology KG*(EG). Thetools used in the exposition contain the Bredon homologyfor infinite groups. Relations of the Baum-Connes conjec-ture to many other famous conjectures in topology aredescribed in the Appendix. The second part (written by A.Valette with Appendix by D. Kucerovsky) contains a dis-cussion of the Baum-Connes conjecture for a countablediscrete group Γ. Suitable index maps provide the linkbetween both sides of the conjecture. The second part ofthe book contains a careful discussion of these maps. Bothlecture notes clearly cover the area around a beautiful,interdisciplinary field and could be very useful to anybodyinterested in the subject. (vs)
V. Mûller: Spectral Theory of Linear Operators andSpectral Systems in Banach Algebras, Operator TheoryAdvances and Applications, vol. 139, Birkhäuser, Basel,2003, 381 pp., €158, ISBN 3-7643-6912-4The monograph is written as an attempt to organize a hugeamount of material on spectral theory, most of which wasuntil now available only in research papers. The aim is topresent a survey of results concerning various types ofspectra in a unified, axiomatic way. The book is organizedin to five chapters. At the beginning, the author presentsspectral theory in Banach algebras, which forms a naturalframe for spectral theory of operators. The second chapteris devoted to applications to operators. Of particular inter-est are regular functions: operator-valued functions whoseranges behave continuously. A suitable choice of a regularfunction gives rise to the important class of Kato operatorsand the corresponding Kato spectrum. The third chaptergives a survey of results concerning various types of essen-tial spectra, Fredholm and Browder operators, etc. Thefourth chapter contains an elementary presentation of theTaylor spectrum, which is by many experts considered tobe the proper generalization of the ordinary spectrum of asingle operator. The most important property of the Taylorspectrum is existence of a functional calculus for functionsanalytic on a neighbourhood of the Taylor spectrum. Thelast chapter is concentrated on the study of orbits and weakorbits of operators, which are notions closely related to theinvariant subspace problem. All results are presented in anelementary way. Only a basic knowledge of functionalanalysis, topology and complex analysis is assumed.Moreover, basic notions and results from Banach spaces,analytic and smooth vector-valued functions and semi-con-tinuous set-valued functions are given in the Appendix.The monograph should appeal both to students and toexperts in the field. It can also serve as a reference book.(jsp)
M. C. Pedicchio, W. Tholen, Eds.: CategoricalFoundations: Special Topics in Order, Topology,Algebra, and Sheaf Theory, Encyclopaedia ofMathematics and Its Applications 97, CambridgeUniversity Press, Cambridge, 2004, 440 pp., $90, ISBN 0-521-83414-7The book is a result of a collaborative research project of
mathematicians from four European and three Canadianuniversities. During the years 1998 – 2001, small teamswere formed to work on a variety of themes of currentinterest and to develop the categorical approach to them.This book presents the results of their work. The book con-tains 8 chapters devoted in turn to ordered set and adjunc-tion (by R. J. Wood), locales (by J. Picado, A. Pultr and A.Tozzi), general topology (by M. M. Clementino, E. Giuliand W. Tholen), regular, protomodular and Abelian cate-gories (by D. Bourn and M. Gran), monads (by M. C.Pedicchio and F. Rovatti), sheaf theory (by C. Centazzoand E. M. Vitale) and to effective descent morphisms (byG. Janelidze, M. Sobral and W. Tholen). Every chapter isself-contained with its own list of references. The book is
very comprehensive and presents a lot of material on eachof the themes. Moreover, it offers various ways of how tostudy “spaces” or ”algebras”, selecting and accentuatingsome of their features. The knowledge required for thereading of the book varies between the chapters, but only amodest knowledge of category theory is supposed at thebeginning. The authors of each chapter develop the neces-sary categorical techniques themselves. The book will bevery useful for graduate students and teachers, and inspir-ing for the researchers interested in the discussed topics aswell as in category theory itself. (vtr)
G. Pisier: Introduction to Operator Space Theory,London Mathematical Society Lecture Note Series 294,Cambridge University Press, Cambridge, 2003, 475 pp.,£39,95, ISBN 0-521-81165-1The monograph is devoted to the study of operator spacetheory. The book has three parts. The first part contains abasic exposition of the theory and various illustrativeexamples. An operator space is just a (usually complex)Banach space X, equipped with a given embedding into thespace B(H) of all bounded linear operators on a Hilbertspace H. Natural morphisms in this category are com-pletely bounded maps (a linear map is completely bound-ed if the naturally induced mappings between respectivespaces of matrices have uniformly bounded norms). Inview of this, we can have many operator space structureson a given Banach space. One of the basic tools for the the-ory is the Ruan theorem, which shows the correspondencebetween operator space structures on X and some kind ofnorms on the tensor product of X with the space K(L2) ofcompact operators on L2. Basic operations on Banachspaces (dual space, quotient space, direct sum, complexinterpolation, some tensor products, etc.) can be defined inthe category of operator spaces. Some of these definitionsare elementary, some make use of the Ruan theorem. It isalso worth mentioning the existence (and uniqueness) ofthe “operator Hilbert space” - a canonical operator spacestructure on the Hilbert space. The second part is devotedto C*-algebras, which form a subclass of operator spaces(notice that the structure of a C*-algebra on a Banach spaceinduces a unique operator space structure). The mainthemes in this part are C*-tensor products and variousclasses of C*-algebras and von Neumann algebras. Someproperties of C*-algebras are extended to general operatorspaces, and local theory of operator spaces is investigated.The third part deals with non-self-adjoint operator alge-bras, their tensor products and free products. The theory ofoperator spaces is also used to reformulate some classicalsimilarity problems. (okal)
A. Polishchuk: Abelian Varieties, Theta Functions andthe Fourier Transform, Cambridge Tracts inMathematics 153, Cambridge University Press,Cambridge, 2003, 292 pp., £47,50, ISBN 0-521-80804-9The book gives a modern introduction to theory of Abelianvarieties and their theta functions. The text is based on lec-tures by the author delivered at Harvard University (1998)and Boston University (2001) and it provides an up-to-dateintroduction to the subject, oriented to the general mathe-matical community. One of the main goals is to give thefirst introduction to algebraic theory of Abelian varietiesand theta functions, employing Mukai’s approach to theFourier transform in the context of Abelian varieties. Thisapproach is also supported by recently discovered links tothe mirror symmetry problem in algebraic geometry andquantum field theory. The exposition of material present-ed in the book was influenced by the category approach toproblems under consideration. The book is divided intothree main parts, then each of them into seven or eightchapters. Part I (Analytic Theory) discusses classical andrecent aspects of transcendental theory of Abelian vari-eties. Part II (Algebraic Theory) is devoted to generalAbelian varieties over an algebraically closed field of arbi-trary characteristic. Part III (Jacobians) contains theory ofJacobian varieties of smooth irreducible projective curvesover arbitrary fields. The chapters’ text contains manyexercises complementing the material covered. The bibli-ography is up-to-date and comprehensive, consisting of138 titles. The book is primarily intended for anybodyinterested in modern algebraic geometry and mathematicalphysics, with a good background not only in complex anddifferential geometry, classical Fourier analysis, or repre-sentation theory, but also in modern algebraic geometryand categorical algebra. The book is written by a leadingexpert in the field and it will certainly be a valuableenhancement to the existing literature. (špor)
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N. Saveliev: Invariants for Homology 3-Spheres,Encyclopaedia of Mathematical Sciences, vol. 140,Springer, Berlin, 2002, 223 pp., €84,95, ISBN 3-540-43796-7The book can be considered as a fundamental monographon invariants of homology 3-spheres. Let us mention thatwhile the topic may seem rather specialised, these investi-gations have proved to be extremely useful in the manifoldtopology. The text covers almost all invariants from theclassical Rokhlin invariant, through the Casson invariantand its various refinements and generalizations, to recentinvariants of the gauge type. Quite naturally, it also con-tains many results on topology of 4-manifolds. Obviously,the book is designed for specialists in the field who willfind there a practically complete survey of results andmethods. On the other hand, it is written in such a clearstyle that I would like to recommend it strongly to post-graduate students starting to make themselves familiar withthe field. The book will help them to learn basic notionsand will gradually introduce them into contemporaryresearch. The large number of references (311 items goingup to 2001) will enable them the further orientation. (jiva)
L. Schneps, Ed.: Galois Groups and FundamentalGroups, Mathematical Sciences Research Institute 41,Cambridge University Press, Cambridge, 2003, 467 pp.,£50, ISBN 0-521-80831-6This volume is the outcome of the MSRI special semesteron Galois groups and fundamental groups, held in the fallof 1999. The book contains scientific and survey articlesfrom the most important extensions and ramifications ofGalois theory - geometric Galois theory, Lie Galois theoryand differential Galois theory, all in various characteristics.The main focus of the study of geometric Galois theory isthe theory of curves and objects associated with them -curves with marked points, their fields of moduli and theirfundamental group, covers of curves with their ramifica-tion information, finite quotients of the fundamental groupthat are Galois groups of the covers. The articles present-ed in the book include fundamental groups in positive char-acteristic, anabelian theory and Galois group action on fun-damental groups. The subject of Lie Galois theory origi-nates in the geometric situation, whose linearised versionleads to graded Lie algebras associated with profinite fun-damental groups. Special attention is paid to special loci inthe moduli space with a particular group of automor-phisms. Instead of considering finite groups as Galoisgroups of Galois extensions of arbitrary fields, differentialGalois theory treats linear algebraic groups as Galoisgroups of so called Picard-Vessiot extensions of D-fields,which are fields equipped with derivation. (pso)
T. Sheil-Small: Complex Polynomials, CambridgeStudies in Advanced Mathematics 75, CambridgeUniversity Press, Cambridge, 2002, 428 pp., £65, ISBN 0-521-40068-6The book studies geometric properties of polynomials andrational functions in the complex plane. The book startswith a description of foundations of complex variable the-ory from the point of view of algebra as well as analysis.For example, the degree principle is studied in connectionwith the fundamental theorem of algebra, falling out intothe mini-course of plane topology. To mention anotherexample, the Rouché theorem is studied together with itstopological analogues, including the Brouwer fixed-pointtheorem. After the preliminary chapter on the algebra ofpolynomials, the book is divided in to 11 chapters contain-ing a study of the Jacobian problem, analytic and harmon-ic functions in the unit disc, trigonometric polynomials,critical points of rational functions, self-inversive polyno-mials and many other topics in connection with the centralnotion of a complex polynomial. Real polynomials are dis-cussed in a separate chapter, including the Descartes rule ofsigns, distribution of critical points of real rational func-tions, and real entire and meromorphic functions.Blaschke products are also mentioned at the end of thebook, with connection to harmonic mappings, convexcurves and polygons. In general, the book offers a wholevariety of concepts, oscillating among analysis, algebraand geometry. This is clearly one of the advantages of thisnice publication. The book, with (or despite) its more than400 pages, gives the reader a feeling that it is both a con-cise and a comprehensive monograph on the topic. It willsurely be appreciated by graduate and PhD students, aswell as by researchers working in the field. (mro)
J. F. Simonoff: Analyzing Categorical Data, SpringerTexts in Statistics, Springer, New York, 2003, 496 pp., 64fig., €84,95, ISBN 0-387-00749-0The book can be divided into several parts. It starts with anintroduction to regression models, including regressiondiagnostics and model selection. Then the author dealswith discrete distributions and corresponding goodness-of-fit tests. The main topics here are binomial, multinomial,and Poisson distributions complemented by the zero-inflat-ed Poisson model, the negative-binomial model, and thebeta-binomial model. Then regression models for countdata are presented. They are based mainly on generalizedlinear models. The part on contingency tables describeslog-linear models, conditional analyses, structural zeros,outlier identification, models for tables with ordered cate-gories, and models for square tables. The last part of thebook introduces regression models for binary data and formultiple category response data. The chapters end with asection that provides references to books or articles relatedto the material in the chapters.
The author based this book on his notes for a class witha very diverse pool of students. The material is presentedin such a way that a very heterogeneous group of studentscould grasp it. All methods are illustrated with analyses ofreal data examples. The author provides a detailed discus-sion of the context and background of the problem. Forexample, it is known that incorrect statistical analysis ofdata that were available at the time of the flight ofChallenger on January 28, 1986, led to its explosion. It isless known that one of the recommendations of the com-mission was that a statistician must be part of the groundcontrol team from that time on. All statistical modelingand figures in the text are based on S-PLUS. The authorhas set up a web site related to the book, where data setsand computer code are available as well as answers toselected exercises (there are more than 200 exercises in thebook). On the other hand, there are no theorems and proofsin the book. Before using it as a textbook, the instructorshould consult original papers and books to prepare theo-retical background. At the same time, the style is not a”cookbook”. The book is very interesting and it can bewarmly recommended to people working with categoricaldata. (ja)
J. Stopple: A Primer of Analytic Number Theory: FromPythagoras to Riemann, Cambridge University Press,Cambridge, 2003, 383 pp., £ 22,95, ISBN 0-521-81309-3,ISBN 0-521-01253-8The book constitutes an excellent undergraduate introduc-tion to classical analytical number theory. The authordevelops the subject from the very beginning in anextremely good and readable style. Although a wide vari-ety of topics are presented in the book, the author has suc-cessfully placed a rich historical background to each of thediscussed themes, which makes the text very lively. Theauthor covers topics with roots in ancient mathematics likepolygonal numbers, perfect numbers, amicable pairs, basicproperties of prime numbers and all central themes of thebasic analytic number theory. Assuming almost no knowl-edge from complex analysis, he develops tools needed toshow the significance of the Riemann hypothesis for thedistribution of primes. Problems of additive number theo-ry are not covered, as well as the prime number theorem.In the last three chapters of the book, the reader finds a cou-ple of specific examples of L-functions attached toDiophantine equations. The material covered in thesechapters includes solutions of the Pell equation, ellipticcurves and analytic aspects of algebraic number theory.The text contains a rich supplement of exercises, briefsketches of more advanced ideas and extensive graphicalsupport. The book can be recommended as a very goodfirst introductory reading for all those who are seriouslyinterested in analytical number theory. (špor)
W. Tutschke, H. L. Vasudeva: An Introduction toComplex Analysis, Modern Analysis Series, Chapman &Hall/CRC, Boca Raton, 2004, 460 pp., $89,95, ISBN 1-584-88478-9The authors present two parallel approaches to complexfunction theory. One follows the idea that what can beused from the real analysis must be applied to lay the foun-dations of complex function theory, while the other one is“purely complex”. The book starts with a review (86 p.) ofbasic methods and notions from real analysis such as met-ric spaces, liminf and limsup of a sequence of real numbersor the Gauss-Green formula, and basics on the field C ofcomplex numbers and on elementary functions (in C).
Then the theory is developed and this is quite often done ontwo levels. For example, the Cauchy theorem is first donein the version with sufficiently smooth positively oriented(firstly based on the intuition) curves bounding a boundeddomain G and with function f = u + iv holomorphic in Gand u,v in C1-class on the closure of G. Later on it isproved for a rectifiable boundary and f holomorphic in Gand continuous on its closure. Many things, which arebriefly described in other books, in remarks or exercises,are given in full details (at least 5 different proofs of thefundamental theorem of algebra are given). The book con-tains an exposition of analytic continuation, homotopy,conformal mappings, special functions and boundary valueproblems, to name the less frequently treated material. Theauthors will please readers interested mainly in applica-tions as well as those who want to know how things reallywork and prefer deeper and more detailed treatment of thematerial. The book also contains more than 200 examplesand 150 exercises. A certain drawback is that the fontsused in the typesetting of the book are rather small.Regardless of this fact the book is nice and I recommend itfor courses in complex function theory (even on anadvanced level) and also as a reference book. (jive)
C. Villani: Topics in Optimal Transportation, GraduateStudies in Mathematics, vol. 58, American MathematicalSociety, Providence, 2003, 370 pp., $59, ISBN 0-8218-3312-XThe monograph is an exhaustive survey of the optimalmass transportation problem. It gives an overview of therecent knowledge of the subject and it introduces all toolsconvenient for its investigation. One of the principal toolsused in the book is the Kantorovich duality on boundedcontinuous functions. Subsequent chapters introduce rele-vant geometric arguments needed to prove the duality the-orem for the quadratic cost. Furthermore, Brenier’s Polarfactorisation theorem, the Monge-Ampère equation, dis-placement interpolation and the probabilistic metric theory,are all discussed. Relations to physical theories are alsomentioned. The optimal mass transportation problem isreformulated in terms of fluid mechanics. Reformulationand explanation, by means of energy and entropy produc-tion optimisation under variational inequalities, are stated.The monograph grew from a graduate course taught by theauthor. The book is well organized and written in a clearand precise style. The text includes a list of illustrativeproblems helping to understand the theory. A certain levelof mathematical skill is required from the reader. Themonograph can be recommended to researchers and scien-tists working or interested in the field as well as an appro-priate textbook for graduate and postgraduate courses onthe subject. (pl)
C. Voisin: Théorie de Hodge et géométrie algébriquecomplexe, Cours Spécialisés 10, Société Mathématique deFrance, Paris, 2002, 595 pp., €69, ISBN 2-85629-129-5Cambridge University Press published the English transla-tion of the book in a two-volume version in 2002, 2003resp. The review of both parts of the English translationappeared in the EMS Newsletter No. 53, p. 48. (pso)
List of reviewers for 2004
The Editor would like to thank the following for theirreviews this year.
J. Andìl, R. Bashir, M. Beèváøová-Nìmcová, L. Beran,L. Bican, L. Boèek, J. Bureš, J. Dolejší, P. Dostál,J. Drahoš, M. Ernestová, D. Hlubinka, Š. Holub, P.Holický, M. Hušková, O. John, O. Kalenda, A. Kar-ger, T. Kepka, M. Klazar, J. Kopáèek, V. Koubek, O.Kowalski, P. Kùrka, P. Lachout, J. Lukeš, J. Malý, P.Mandl, M. Markl, J. Milota, J. Mlèek, E. Murtinová,K. Najzar, J. Nekováø, J. Nešetøil, I. Netuka, O.Odvárko, D. Praák, Š. Porubský, J. Rataj, B. Rie-èan, M. Rokyta, T. Roubíèek, P. Simon, P. Somberg,J. Souèek, V. Souèek, J. Spurný, J. Stará, Z. Šír, J.Štìpán, J. Trlifaj, V. Trnková, J. Tùma, J. Vanura, J.Veselý, M. Zahradník, M. Zelený.
All of the above are on the staff of the CharlesUniversity, Faculty of Mathematics and Physics,Prague, except: M. Markl and J. Vanura(Mathematical Institute, Czech Academy of Sciences),Š. Porubský (Technical University, Prague), B. Rieèan(University of B. Bystrica, Slovakia), J. Nekováø(University Paris VI, France).
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