121/04

NYHEDSBREV FOR DANSK MATEMATISK FORENING

NR. 21 OKTOBER 2004

M A T I L D E

TEMA:ECM 2004 og ICME-10 2004

2 21/04

Leder

Temaet for dette nummer af Matilde er sommerens tostore internationale matematiske kongresser: EuropeanCongress of Mathematics (ECM) i Stockholm, og 10’thInternational Congress on Mathematical Education (IC-ME) i København. ECM får en fyldig omtale af Ulf Pers-son, og ICME dækkes med Carl Winsløv som temare-daktør af Claus Michelsen og Vagn Lundsgaard Hansen.Man kunne også sige, at temaet denne gang er matema-tikkens internationale natur; dette aspekt belyses bl.a. afet interview med vinderne af dette års norske Abelpris:Sir M. Atiyah og I. M. Singer, især citeret for deres arbej-de med indekssætningen. Atiyah holder foredrag 2. no-vember ved Dansk Matematisk Forenings årsmøde, ogSinger holdt ligeledes et foredrag i Århus i forbindelsemed det naturvidenskabelige fakultets 50 års jubilæumfor nylig.

Dansk matematik har i dag særdeles stærke kontaktertil det internationale matematiske miljø, en tradition derhistorisk fortaber sig i Københavns Universitets førsteekperimenter med professorer i matematik; men i hvertfald havde Julius Petersen gode vekselvirkninger medSylow, mens Harald Bohr stod for fine kontakter til sintids førende tyske og engelske matematikere.

Ved 50 års jubilæet i Århus holdt Niels Andreas Baasen tale - som repræsentant for norsk matematik, og somoplæg til en overrækkelse af Abels samlede værker tilMatematisk Instituts bibliotek - hvori han netop under-stregede, hvordan Svend Bundgaard i sin tid havde haftdet internationale som ledetråd i opbygningen af insti-tuttet i Århus. “We will have to do something about that”- “Det må vi gøre noget ved”, var Bundgaards reaktion tilden daværende unge matematiker Baas, der var løbet indi et stillingsmæssigt dødvande. Og det var vel et typisksvar fra Bundgaard, og endnu mere typisk at det snartblev fulgt op af handling, der som sædvanligt optimistiskgik løs på fremtiden.

Der er naturligvis mange udfordringer for dansk ma-tematik af mere national karakter, Især den foreståendereform af gymnasiet - i sig selv en kæmpe opgave for læ-rerne, deres organisationer, og universiteterne, men velogså en mulighed for at vise/bevise matematikkens an-vendelighed og levende væsen. Her vil det blive vigtigtat kunne få andre fag til at inddrage deres matematiskerødder. En anden udfordring er matematikkens rolle somet naturvidenskabeligt grundforskningsfag; både i et hi-

storisk perspektiv og som øjebliksbillede af moderne na-turvidenskab, er matematikkens begreber helt centrale iselve den måde vi tænker på - så forhåbentlig vil detteogså blive og forblive klart, selv i konkurrence med na-no, it og bio, i kommende runder hos den danske Grund-forskningsfond.

Matilde vil fortsætte med at beskrive matematikkensmange roller, og hvad det internationale angår, vil vi for-søge at høste erfaringer fra de andre nordiske lande, bl.a.ved at have temaer om matematik i norden. Desuden vilvi prøve at se på matematikken med øjne udefra, bl.a.hvad vores nærmeste naturvidenskabelige familie måttese og beskrive: fysik og kemi f.eks. - men måske også hø-re hvad mere fjerne slægtninge har at sige om deres synpå vores fag: for endeligt at forstå hvorfor det stadig erhelt i orden og ganske almindeligt at høre dannede folkerklære, at matematik siger dem mindre end ingenting.

Det må vi gøre noget ved.

Det må vi gøre noget ved

Af Bent Ørsted

321/04

Matilde – Nyhedsbrev forDansk Matematisk Foreningmedlem af EuropeanMathematical Society

Nummer 21 – Oktober 2004

Redaktion:

Bent Ørsted, SDU(ansvarshavende)

Carl WinsløvBent Ørsted(TEMAREDAKTØRER)

Carsten Lunde Petersen, RUCJørn Børling Olsson, KUPoul Hjorth, DTUMikael Rørdam, SDUCarl Winsløw, KU

Adresse:

MatildeMatematisk AfdelingKøbenhavns UniversitetUniversitetsparken 52100 København Ø

Fax: 3532 0704e-post: matilde@mathematics.dkURL:www.matilde.mathematics.dk

ISSN: 1399-5901

Matilde udkommer 4 gange omåret

Indlæg til næste nummer skalvære redaktionen i hændesenest 15. februar 2005

TEMA: ECM 2004 og ICME-10 2004

Vagn Lundsgaard HansenNogle personlige betragtninger over ICME-10 .............................. 4

Claus MichelsenICME 10:Matematik og de andre fag ............................................................ 7

Ulf PerssonStockholm 2004:The Fourth European Congress of Mathematics ............................ 9

Nils A. BaasThe Spirit of Bundgaard ........................................................................... 13

Matematiske Institutioner præsenterer sigMorten WillatzenMads Clausen Institute ............................................................................. 14

InterviewMartin Raussen og Christian SkauInterview with Michael Atiyah and Isador Singer ................................... 15

Uddannelsesfronten ................................................................................. 23

Boganmeldelser ........................................................................................ 25

MatematikerNyt ....................................................................................... 27

Begivenheder............................................................................................ 28

Aftermath ................................................................................................. 29

Indhold:

4 21/04 Tema

I anden uge af juli 2004 blev ”10thInternational Congress on Mathema-tical Education”, med forkortelsenICME-10, afholdt på Danmarks Tek-niske Universitet i Kongens Lyngby.Den internationale konference ommatematikundervisning finder stedhvert fjerde år, og denne sommerhavde de nordiske lande fået over-draget opgaven i fællesskab. Somforskningsområde betegnes feltet“Mathematics Education”, eller, somdet er mest almindeligt i Europa,“Didactics of Mathematics”, der iDanmark bliver til ”MatematikkensDidaktik”.

Næsten 2300 forskere i matema-tik, forskere i matematikkens didak-tik og lærere fra alle niveauer i un-dervisningssystemet fra folkeskole(primary school) til universitet, dis-kuterede de nyeste faglige udviklin-ger i matematik og forskningsresul-tater med tilknytning til hvordan mankan forbedre og berige undervisningi matematik i det store perspektiv. Dervar deltagere fra 119 forskellige lan-de. Registreringen fandt sted søndagden 4. juli i den smukke festsal påKøbenhavns Universitet tæt på kon-greshotellerne. Det var en smuk sol-rig dag; senere i kongressen opleve-de deltagerne det lunefulde danskevejrlig! Ved registreringen fik delta-gerne udleveret et kort til det offent-lige transportsystem for kongrespe-rioden, som i de følgende dage hur-tigt og let bragte dem til DTU.

Et særkende for den internationa-le kongres om matematikundervis-ning er, at den bringer aktive forske-re i matematik, som er dybt optagetaf at udvikle og formidle deres fag,sammen med matematikundervisere,der er stærkt optaget af at skabe et

velfungerende og rigt undervisnings-miljø i matematik i de enkelte lande.Alt under vejledning af forskere i ma-tematikkens didaktik, som har bidra-get væsentligt til at strukturere tænk-ning vedrørende problemer forbun-det med undervisning og læring imatematik. Under kongressen gik detop for mig, hvor vigtigt det er at for-skere i matematik og forskere i mate-matikkens didaktik ikke mister kon-takten med hinanden med deraf føl-gende forvirring hos lærerne i klas-seværelserne, som bliver stillet overfor konkrete problemer med under-visning i specifikke emner fra mate-matikken. Mere end nogensinde fin-der jeg, at det er nødvendigt at beggegrupper af forskere bliver hørt i de-batten om indhold, metoder, præsen-tation og evaluering af matematik iskolesystemet.

Mange deltagere vil huske åb-ningsceremonien ikke mindst fordi topolitikere holdt meget gode taler ogvar tilstede under hele ceremonien.Undervisningsminister Ulla Tørnæsomtalte problemer for matematik iskolen uden at gå ind i almindeligpolitisk polemik, og borgmesteren iLyngby-Tårbæk kommune RolfAagaard-Svendsen, som har en ph.d.i matematisk statistik, holdt en talemed stor vid, hvor han endog visteen side fra sin afhandling. Andre ta-lere inkluderede præsidenten for IC-MI (International Commission onMathematical Instruction), professorHyman Bass, formanden for den in-ternationale programkomite (IPC),professor Mogens Niss (se billede 1),formanden for den lokale organisati-onskomite (LOC), lektor MortenBlomhøj, forskningsdekanen vedDTU, professor Christian Stubkjær,

og generalsekretæren for ICMI, pro-fessor Bernard Hodgson. Ceremoni-en blev yderligere oplivet ved musi-kalsk underholdning af Det Konge-lige Danske Blæserensemble.

Åbningsceremonien omfattedeogså tildeling af den første Felix KleinMedalje til professor Guy Brousseau,Frankrig, for hans livslange bidrag tilmatematikkens didaktik, (se billede2) og den første Hans FreudenthalMedalje til professor Celia Hoyles,England, for fremragende bidrag tilforskningen i området teknologi ogmatematikundervisning. Med valge-ne af de første modtagere har man saten høj standard for disse hædersbe-visninger, som vil gøre de to medal-jer særdeles prestigefyldte inden forforskning i matematikkens didaktik.

Og så lød startskuddet for kon-gressen. Programmet var enormt: 8plenary sessioner, omkring 80 størreforedrag (regular lectures), 29 topicstudy groups, 24 discussion groupsog – som en ny ting – en tematiskopbygget eftermiddag med 5 valg-muligheder. Der var 5 nationale præ-sentationer over en fuld eftermiddag:Korea, Mexico, Rumænien, Ruslandog de nordiske lande (Danmark, Fin-land, Island, Norge, og Sverige). Hvernational præsentation havde et detal-jeret og omfattende program. De fle-ste aktiviteter foregik naturligvis iparallelle sessioner med mange valg-muligheder. Alt i alt var der mere enden halv million muligheder for atsammensætte sit eget personlige pro-gram. For at lette den individuelleplanlægning fik hver deltager udle-veret en ekstra kopi af skemaet, hvorman kunne notere hvilke aktiviteterman ville gå til og hvilke lokaler defandt sted i.

NOGLE PERSONLIGEBETRAGTNINGER OVERICME-10

Af Vagn Lundsgaard Hansen

521/04ECM 2004 ogICME-10 2004

For et program af denne størrelsekan ingen deltager fange det hele.Men alle kunne fornemme ånden ogmærke den energi og lidenskab hvor-med de mange bidragydere havdeforberedt deres bidrag. Når man skalvære fair over for helheden, er det sik-kert ikke rimeligt at udpege enkeltebidrag i en generel beskrivelse. Menman har vel lov til at fremhæve, atdanskere var fremme i mange sam-menhænge, og blandt meget andetbidrog med to regular lectures (OleSkovsmose og Vagn LundsgaardHansen) og stod som medarrangøreraf mange delaktiviteter i program-met. Mere end 20 danskere bidrog tilden nordiske præsentation, der blevkoordineret af Carl Winsløw. Jo, Dan-mark kunne godt være det bekendt.Det bør også nævnes, at såvel dennordiske præsentation som den loka-le organisationskomite mobiliseredeen stor bredde på tværs af nogle klas-siske institutionsskel og involveredetæt på 100 danskere i mange forskel-lige roller. Men igen indser man, attilhørerne og de ivrigt engageredebidragydere til diskussionerne erdem, der virkeligt udgør rygraden i

en god kongres. Og jeg fornemmer,at ICME-10 var en succes i den hen-seende.

Jeg deltog selv i den tematiske ef-termiddag om ”mathematics andmathematics education”, som gav an-ledning til en lettere ophedet debat.Efter min opfattelse må kravene tilundervisning i matematik i skolenvære, at den omfatter et udvalg afsamfundsmæssigt vigtige matemati-ske begreber og metoder, og at manfaktisk kan undervise i de emner, derinddrages. Der blev overraskendenok præsenteret enkelte (fra et under-visningsmæssigt synspunkt) mate-matiske luftkasteller af matematikdi-daktikere. Tonen i debatten var påvisse punkter ikke ligefrem høflig,men debatten var meget direkte ogeksplicit, og jeg tror den vil føre tilovervejelser hos deltagerne i dennetematiske eftermiddag om den rolleforskere i matematik og forskere i ma-tematikkens didaktik kan og bør spil-le i arbejdet med at få god undervis-ning i faget matematik ind i klasse-værelserne.

I sammenhæng med kongressenvar der arrangeret et matematisk cir-

kus “Circus Mathematicus” for etgenerelt publikum. Den udviste kre-ativitet var enorm, og det matemati-ske cirkus bød blandt meget andet på:bowling med brøker, en labyrint, pus-lespil i stor størrelse, boomeranger,jonglering, origami, træskæring,smukke juledekorationer, forskelligereb tricks, konstruktion af drager, sebillede 3. Lørdag den 3. juli havdeCircus Mathematicus forpremiere iLyngby Storcenter, hvor en lille del afprogrammet blev præsenteret for etinteresseret publikum. Det matema-tiske cirkus var en stor succes og hjalpmed til at henlede nyhedsmediernesopmærksomhed på hele kongressen.

Kongressen blev således smuktomtalt i landsdækkende aviser og lo-kalblade, samt i TV-avisen på DR1 ogi programmet Sommermorgen på ra-dioens P3. I tilknytning til kongres-sen var der også premiere på en inte-ressant udstilling “Why mathema-tics” udviklet i et samarbejde mellemICMI og UNESCO.

Endvidere fandt den 2. nordiskefinale i KappAbel konkurrencen for8. klasser sted under ICME-10.

KappAbel, der er udviklet i Nor-

Fra venstre Morten Blomhøj (formand for LOC), Elin Emborg (sekretær for LOC og IPC), Mogens Niss (formand for IPC)alle tre IMFUFA, Roskilde Universitetscenter, og Gerd Brandell (formand for den nordiske kontakt komité for ICME-10),Lund Universitet, Sverige.

6 21/04 Tema

ge, er blevet gjort til en fælles nordiskkonkurrence i forbindelse med detnordiske samarbejde om ICME-10.Under ICME-10 oplevede omkring300 kongresdeltagere vinderholdenefra hver af de fem nordiske landefremlægge særdeles spændende pro-jekter om årets tema, matematik ogmusik, på et sprudlende engelsk ogmed en for 8. klassetrin dybt impo-nerende faglig indsigt. Finalen omfat-tede også en problemløsningskon-kurrence med udfordrende opgaversom også tilskuerne kunne forsøgesig med. Det danske hold. 8b fra Ris-skov Skole ved Århus, vandt bådeprojekt- og problemløsningskonkur-rencen. Se www.KappAbel.com foren nærmere omtale af konkurrencen.

Som en af de lokale arrangører afkongressen bør jeg være tilbagehol-dende med lovprisning. Jeg føler dogalligevel trang til at sige, at kongres-sen efter min opfattelse havde succesmed at opfylde sit egentlige mål,nemlig at stimulere den kontinuerli-ge proces hen imod god matematik-undervisning. Som professor JeremyKilpatrick forsikrede os under enmeget interessant plenary sessionmed fire fremtrædende forskere imatematikkens didaktik: “The strivefor good teaching will never stop.

There will always be new challen-ges.”

Afslutningsceremonien fandt stedsøndag den 11. juli med taler og mu-sikalsk underholdning. Herefter varder afskedsparty, og arrangørernekunne endelig ånde lettet op eftermere end fire interessante år medplanlægning i et fortræffeligt samar-bejde med vores effektive og hjælp-somme kongresbureau: CongressConsultants.

Præsidenten for ICMI Hyman Bassoverrækker den første Felix KleinMedalje til professor Guy Brousseau(th).

Børn og kongresdeltagere ivrigt optaget af matematiske aktiviteter i Circus Mathematicus.

ICME-11 finder sted i Mexico i 2008.Forhåbentlig får kongressen mangedeltagere. En ICME kongres er an-strengelserne værd.

721/04ECM 2004 ogICME-10 2004

Da Danmarks Radio i foråret viste deførste afsnit af Krønikken, var det formig som at rejse tilbage til barndom-mens land. Jeg husker tydeligt, at derpludselig stod et fjernsyn i vores stue,og at familien hver aften klokken 20samledes foran det nye ”møbel”.Højdepunktet var, når en udsendel-se blev indledt med, at der ud af høj-taleren kom en flot fanfare, og skær-men viste noget, som lignede enhalvfærdig tagplade forsynet medteksten Nordvision. Så vidste vi, atdet var en fælles nordisk udsendel-se. Nu er det nordiske samarbejde påmange områder ikke længere, hvaddet har været. Men heldigvis er derstadig områder, hvor de nordiskelande har et tæt og udbytterigt sam-arbejde, som er under stadig udvidel-se. Det gælder bl.a. forskningsfeltetmatematikkens didaktik. Et forelø-bigt højdepunkt i dette samarbejdevar afholdelsen af ICME 10 på DTUsom et fælles nordisk arrangement.De fem med nordiske flag blafredelystigt, når konferencedeltagernehver morgen ankom til DTU, hvorder ventede et spændende og alsidigprogram. Programmet var så omfat-tende, at det ikke er muligt at giveen udtømmende beskrivelse af kon-ferencen, og jeg vil derfor i det føl-gende blot forsøge at videregive nog-le få af mine indtryk fra programmetstre faste daglige punkter, som varplaceret inden frokostpausen.

Første punkt på konferencens pro-gram var hver dag en plenumforelæs-ning i Oticon Hallen. Matematikkensdidaktik er et forskningsfelt i riven-de udvikling, men også et relativtungt forskningsfelt, som stadig søgerefter sin identitet. De daglige plenum-forelæsninger gav et godt billede af,hvor forskningsfeltet befinder sig og

i hvilken retning, det netop nu bevæ-ger sig. Specielt forelæsninger tirsdagog torsdag, hvor resultaterne af toSurvey Teams arbejde blev præsen-teret, gav nogle væsentlige bidrag tilen beskrivelse af ”state of the art” formatematikkens didaktik.

Tirsdag blev rapporten The relati-ons between research and practice inmathematics education præsenteret afAnna Sfard fra University of Haifa.Sfard beskæftigede sig med forsknin-gens manglende betydning for dendaglige undervisningspraksis i mate-matik og pegede bl.a. som en årsagtil dette på, at forskningens resulta-ter bevæger sig gennem en lang ræk-ke af medierende led før beskedennår frem matematiklæreren. Torsdagvar det Jill Adler fra University of theWitwatersrand, der præsenterederapporten Research on mathematics tea-cher education: Mirror images of an emer-ging field. Adler kommenterede domi-nansen af angelsaksiske bidrag i Jour-nal of Mathematics Teacher Educationsåledes: ”The local (USA, UK, ..) be-comes global. The local (DK. S, ..) re-mains local” og pegede på nødven-digheden af at overskride kulturelleog sproglige barrierer med henblikpå, at forskning i matematiklærerud-dannelse kan profitere af multiple tra-ditioner.

Efter plenumforelæsningen bødprogrammet på regulære forelæsnin-ger ved prominente forskere i mate-matikkens didaktik fra hele verden.Der var hver dag op til 15 forskelligeforelæsninger at vælge imellem. Sådet var ikke noget let valg, og detforekom, at man valgte forkert. Ma-tematikkens didaktik er en videnskabder inddrager mange videnskabsdi-scipliner, som fx psykologi, filosofi,sociologi og videnskabsteori. Det er

både en styrke og en svaghed vedmatematikkens didaktik, at den erinterdisciplinær. Svagheden blivertydelig, når et i øvrigt særdeles spæn-dende forskningsprojekt præsenteresmed en indledning om Piaget og Vi-gotskijs teorier for læring, som villefå enhver lærerstuderende til atkrumme tæer, og som er uden nogenforbindelse til projektets tema. Så harvi den situation som Sfard omtalte isin forelæsning: for mange forsk-ningsprojekter er præget af en frag-menteret teoretisk infrastruktur, hvorgrundlæggende begreber anvendes,uden at de er defineret operationelt.

Men heldigvis foretog man i langtde fleste tilfælde et godt valg. Såle-des valgte jeg lørdag at overværeforelæsningen Hands-on-mathematics.Concepts and experiences from ”Mathe-matikum”. Her fik vi beretningen omet initiativ i Giessen – Konrad Rönt-gens fødeby – som har til formål, atgøre matematik tilgængelig for al-mindelige mennesker. Der er indret-tet en lokal udstilling i Giessen ogetableret en rejsende udstilling, somsiden 2002 har besøgt mere end 100tyske byer og har haft mere end500.000 besøgende. Der lægges vægtpå at præsentere et bredt perspektivpå matematik, og der fokuseres isærpå relationer til dagligdagen og an-vendelser. Jeg kommer her til at tæn-ke på, at der overalt dukker sciencecentre op, som skal gøre naturviden-skaben og teknologien tilgængelig oginteressant for specielt børn og ungemennesker. Men det er sjældent, atder som i Giessen sættes fokus påmatematikkens betydning. Hvorforer det matematiske aspekt nærmestblevet usynligt når naturvidenskabog teknologi skal præsenteres forbørn og unge mennesker? Har man

Af: Claus Michelsen, Lektor, Ph.D., Institut forMatematik og Datalogi/Dansk Institut for

Gymnasiepædagogik, Syddansk Universitetemail: cl.mich@dig.sdu.dk

ICME 10: MATEMATIKOG DE ANDRE FAG

8 21/04 Tema

glemt at naturens store bog er skre-vet i matematikkens sprog, og at ma-tematik er teknikkens sprog?

Sidste punkt i dagens programinden frokost var Topic Study Groupsessionerne. Her var der ikke mindreend 29 emner at vælge imellem. Menjeg havde intet valg, idet jeg sammenmed Marta Anaya fra University ofBuenos Aires var leder af Topic Stu-dy Group 21 Relations between mathe-matics and other subjects of art andscience. Der var 12 præsentationermed meget forskellige perspektiverpå relationerne mellem matematik ogandre fag. Tager man et øjeblik denationale briller på, så er det værd atbemærke, at reformen af de ung-domsgymnasiale, som træder i kraftfra august 2005, vil betyde et øgetsamspil mellem fagene. Men som derpeges på i rapporten Kompetencer ogMatematiklæring, så er der problemermed at bringe matematikfaget i sam-spil med andre fag. Det er vanskeligtfor lærere i andre fag at se, hvad ma-tematik gør godt for i netop deres fag.Og dette på trods af, at flere og flerefag rummer matematikholdige ingre-dienser i stadig stigende omfang.Man må søge længe efter ordet ma-tematik i Undervisningsministerietsrapport Fremtidens Naturfaglige Ud-dannelser. Omvendt værner mangematematiklærere om matematikfa-gets logisk stringente begrebsapparatog vægrer sig mod at indgå i et sam-arbejde med andre fag. Denne pro-blemstilling blev tydeliggjort af PaulDrijvers fra Freudenthal Institute iUtrecht, der i et interview i ICME avi-sen, som udkom hver dag under kon-

ferencen, udtalte: ”There is a tenden-cy towards integrating different sub-jects.

It is a big challenge for mathema-tics education to find it’s way – Onone hand you don’t want students’knowledge to be split up in little com-partments, on the other hand, youwant to keep something you see asrelevant for mathematics itself”. Er-faringerne med en bevidst inddragel-se af matematiske kompetencer i an-dre fag er begrænsede. Og der blevpå sessioner i Topic Study Group 21efterlyst en konceptuel ramme forforskning i matematiks relationer tilandre fag. Sammen med Astrid Beck-man fra University of SchwäbischGmünd og Bharath Sriraman fraUniversity of Montana har jeg tagetinitiativ til at danne et netværk, derskal tage de mange udfordringer op,som knytter sig til matematiks relati-oner til andre fag. Netværkets førsteinitiativ bliver et symposium i Schwä-bisch Gmünd fra 19.til 21. maj 2005.

Dannelse af netværket er blot etaf mange synlige beviser på, at et ar-rangement som ICME 10 bidrager tilat nye kontakter inden for forsknings-feltet matematikkens didaktik etab-leres og dermed medvirker til at ud-vikle feltet. Formår vi samtidig atmindske afstanden fra forskningen tilundervisningens praksis, så må forsk-ningen i matematikkens didaktik ef-terhånden få den opmærksomhed,som den er berettiget til. Set i lyset afdebatten om de matematisk-naturvi-denskabelige fags krise er det tanke-vækkende at hverken medier eller depersoner og organisationer, som oftehøjlydt gør opmærksom på dennekrise, ikke ofrede ICME-10 større op-mærksomhed. Men vi er mange, derser frem til ICME 11 i Mexico om fireår.

Fra åbningsceremonien

921/04ECM 2004 ogICME-10 2004

How it came about

In spite of a difficult start, the fourth European Congressin Mathematics (ECM) took place after all. As its Presi-dent Ari Laptev revealed in an earlier issue of this new-sletter, its eventual location in Stockholm was not initial-ly planned but was the result of a crisis perceived in thesummer of 2000. Despite what many outsiders may ha-ve assumed, finding sufficient funding from Swedishsources was not a straightforward matter (as the first bidmade painfully clear). It was only due to the daring and‘savoire faire’ of Ari Laptev and his colleague at Kungli-ga Tekniska Högskolan (KTH), Anders Lindquist, thatsufficiently promising leads were extracted from someof the major Swedish funding institutions, thus enablingthe Swedish Mathematical Society to send a small dele-gation to the executive meeting in London later that fallfor negotiations. However, a final commitment to holdthe congress was not made until just a week before Chri-stmas - after a rather tense encounter with the then Pre-sident of the European Mathematical Society (EMS), RolfJeltsch. The meeting only came to a desired conclusionafter a final short interview with the President of the KTH,Anders Flodström, in which the latter agreed to the ne-cessary financial underwriting. The upshot was that theplanning and organization of the event was shortenedby at least a year from what has been customary for thiskind of event.

Why big congresses? Was it worth it?

This is obviously not the place to delve deeper into thatphilosophical question, yet a few reflections may not beamiss. In the good old days, which meant up into thefifties, conferences were few and small, and in particularthe International Congresses of Mathematicians (ICM)were rather select and as a consequence not bigger thanallowing group portraits to be taken. But when the ICMwas held in Stockholm back in 1962, it was the largestscientific congress that had ever been held in Sweden.Since then the ICM have turned from rather exclusivemeetings to large affairs attended by the ‘mathematicalmasses’. This fact, together with the recent accelerationin the number of specialized conferences, often referred

to as workshops to emphasize their focused and busi-nesslike character, has sown doubts in the minds of manyas to the scientific relevance of such gargantuan meet-ings, which, cynics protest, are more the occasion for tou-rism than serious scientific interchange. In such a per-spective, the establishment of a European Congress mayseem redundant. One may point out though that the si-tuation of mathematics is not unique, but is shared bymost academic disciplines, and that it is very important,among other things, to counter the fragmentation of adiscipline with opportunities for the contemplation of itas a whole. For this to be fully successful there is both aneed for the lectures to be directed to a general mathe-matical audience and for that audience to seek out lectu-res not primarily within their own speciality. Popularsurveys do not rank highly among the priorities of re-search mathematicians. They are notoriously difficult todo and the rewards are marginal. A scientific committeeselects their choices primarily on the basis of scientificexcellence and topicality of the subject matter, not onexpository skills; and notwithstanding the desirability ofthe latter this is basically a sound principle, tamperingwith which would court disaster and seriously undercutthe legitimacy of the whole enterprise. Thus selection asa speaker is seen primarily as recognition of scientificmerit and not as an opportunity, as well as an obligation,to communicate effectively.

What is needed is a change of culture, something thatcannot be decreed but has to evolve slowly. On what con-stitutes a good lecture, one can of course argue, there ob-viously being no specific rules on which everyone canagree. And besides, in all kinds of human interaction ru-les are simply there to be broken. It suffices to point outthat a scientific lecture, and especially a mathematical one,is not necessarily made more accessible by the simpleprocess of ‘watering down’ (i.e. removing technicalitiesand making unwarranted simplifications). What is re-quired of a lecture is the conveying of an idea (usuallyone is enough) and some motivation (which should notbe confused with justification often of the type: ‘this hasapplications to physics’) and placing the subject matterin a wider mathematical context. Apart from that, a goodlecture can contain highly technical material, and there isno reason why the audience should understand most of

Af Ulf Persson(Göteborg, Sweden)Stockholm 2004:

The Fourth EuropeanCongress of Mathematics

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it as long as they have gotten something to take homewith them. It would be very dangerous if mathematici-ans were to abandon their tradition of honesty and aimfor merely the ‘flashy’. To reflect on why your work isinteresting and to try to truly motivate it, should not beseen as a concession to ignorance, but rather as an additi-onal source of inspiration for your own research.

New features

Obviously this is not the place to comment upon howsuccessfully the various speakers performed their taskof communicating to a general mathematical audience.Fortunately, although few lectures can be expected toplease everyone, the lecture is rare indeed that does notbring rewards to at least a token number in the audience,and one may argue that this is indeed all that is neededto make it worthwhile. However, there was a feelingamong the organizers that something extra was neededto justify yet another general conference in mathematics,as well as serving as a rejuvenation of the concept. Twonew features were added. One was to invite scientists innatural science, not only in chemistry and physics, butalso mathematical biologists, to give lectures. The otherwas to latch on to the pre-existing structure of the Euro-pean Networks and thus give to the ECM a natural Euro-pean anchoring. The first may be seen as somewhat con-troversial, wedding mathematics and its ultimate justifi-cation too tightly to applications. Yet, whether we like itor not, practical mathematical applications are whatbrings in material resources, and for public relations theimportance of the willingness of mathematicians to inte-ract externally, and in the process maybe also acquiringgreater public visibility, should not be underestimated.From a less opportunistic standpoint, applications shouldnot merely be seen as necessary justifications, but also assources of inspiration. The second new feature may hop-efully ensure that the new tradition of ECM’s continues.The funding and organization of a big international con-gress is indeed a major undertaking and the difficultiesinvolved may be expected to increase with time ratherthan decrease. The European Union has already investeda formidable apparatus of networks replete with theirown special conferences. What would be more naturalthan a unifying one, which should bring with it a largebody of active participants, and hopefully also channelsfor necessary funding? At Stockholm, admittedly thenetwork lectures played a rather marginal role, but if theidea is accepted and developed in future ECM’s they mayprovide the core activity onto which various extras maybe attached as embellishments.

Aula Magna

After those general preambles it may be appropriate todescribe the Fourth European Congress of Mathematici-ans as an actual event tied to a physical location at a gi-ven slot in time. The basic problem of organizing a con-ference for one thousand odd expected participants is tofind a lecture hall big enough. Unlike the case of theOlympics, the erection of new buildings is not an option.Of course big conferences are legion these days, but com-mercially available space often comes with price-tags not

within the capabilities of mathematical meetings. TheRoyal Swedish Institute of Technology (Kungliga Tekni-ska Högskolan - KTH) was not able to provide such ahall on its premises, making a direct collaboration withthe University of Stockholm a necessity on this basis alo-ne. The fact that there are two separate departments ofmathematics in Stockholm has incidentally been a boneof contention for at least fifty years, and may continue tobe so for another fifty years, but this is of course only amatter of local interest. The Aula Magna is the officialgrand lecture hall of the University of Stockholm, andhas of course no direct connection, physical or not, to itsdepartment of mathematics, nor to that of the KTH, beingabout equally (physically) distant from both. It is of fair-ly recent vintage, located on the main campus of the Uni-versity of Stockholm, easily accessible by the StockholmUnderground (‘T-banan’, ‘T’ as in tunnel). Shaped likean amphitheatre, with options of subdivision, the Aulaactually boasts a capacity well in excess of the number ofactual participants (around 800), which resulted in thesomewhat unfortunate impression of not only the lectu-res but also the opening ceremony being sparsely atten-ded. As expected, the Aula along with its adjacent hall-ways became the locus of the meeting. Walking along itsperimeter you had immediate access to the young assi-stants donning yellow T-shirts, as well as to the variousbookstalls providing not only opportunities for browsingbut also seducing discounts. The Springer stall also sup-plied piles of copies of the Stockholm Intelligencer (stillsmelling of fresh print), featuring short articles on Swe-den and mathematics, including a short but morbid listof distinguished mathematicians who died here. Clim-bing a few stairs you could also inspect the various po-ster-sessions. Additional smaller lecture halls were avai-lable at the proverbial stone-throws distance, as well aswhat in recent years has become an absolute necessityfor wayward mathematicians - access to e-mail. A fairlyspacious room, accessible only by pressing a code at theentrance, was filled with a sufficient number of compu-ters to keep waiting lines rather than tempers short.Furthermore, a small staff of knowledgeable yellow-shirtswas always at hand during opening hours. For those ableto resist the allure of the screen at lunchtime, there wasthe temptation of the official cafeteria situated halfwaybetween the two locations (actually accessible from theAula Magna remaining indoors the entire way, a gods-end in the case of inclement weather). It provided thestandard Swedish lunch fare to be expected from thatkind of self-serve establishment, confidently assured ofa captive audience, not only by its isolated existence butalso due to pre-paid lunch coupons.

The opening ceremony

The conference got a head start on Sunday afternoon, onJune 27, by providing registration outside the Aula. Thisinvolved getting a handy black briefcase, doubling as arucksack, with the logo of the 4ECM sufficiently discreetto encourage post-conference use. It would be tedious tolist its contents of ‘goodies’, but I am sure that most peopleappreciated the free public transport passes intended tocover the entire period. One of its more trivial items wasa coupon to be exchanged for a single glass of wine (of

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optional colour) to be served at the hour-long receptionstarting at six o’clock. The next day, Monday June 28,involved the official opening of the meeting at nine thir-ty in the morning, preceded by an opportunity for last-minute registrations. Back at the ICM of 1962, the oldSwedish King Gustavus VI attended, giving out the FieldsMedals. Such a spectacle of royalty at a mathematicalmeeting was, alas, not repeated this time. The presenceof the Swedish Majesty back then has been attributed tothe above mentioned fact that at the time, it was the lar-gest scientific meeting ever to have taken place in Swe-den. No such distinction could be given to the 4ECM, soneither the old King’s grandson, the present King CharlesXVI, nor the great-granddaughter, the crown princessVictoria, were able to squeeze the events into their busyschedules. (One should not overestimate the love thegeneral public, including that of royalty, feels for mathe-matics) Instead two academic notables, Bremer and Fran-ke, provided the required official lustre. The latter, Sig-brit Franke, the chancellor of the university system inSweden, also served the function of awarding the EMSprizes to the young mathematicians, to which we willreturn shortly. Both Bremer and Franke, not surprising-ly, made a special point in referring to Sofia Kovalevska-ya in their short speeches. Kovalevskaya, as can haveescaped the notice of few mathematicians, was the firstever woman professor in mathematics, holding her po-sition at the precursor of Stockholm University at the endof the 19th century, and ever since then being a mathe-matical role model for half the population of the world.Kingman, in his capacity as the President of the EMS,gave the mathematical speech with commendableaplomb, stressing the importance of mathematics, andurging young mathematicians to go for the hard pro-blems, irrespective of whether pure or applied, becauseyou can never tell. He also encouraged the new genera-tion by reminding them that they are as great and imagi-native as the great ones of the past, because what puzz-led our ancestors does not necessarily puzzle us. The chairof the Prize Committee, Nina Uraltseva from St. Peters-burg, who complained that she was a bit too short forthe microphone, explained that the work of the prize com-mittee had been very hard. There were fifty nominees,out of which ten had to be selected. To those who did not

make it, there is only one thing to say - work harder andtry to outdo those who were selected. This is not the pla-ce to give a list of the winners, along with short descrip-tions of their accomplishments. It suffices to add that inaddition to the traditional EMS prizes, a so called BITprize was awarded for work in numerical analysis. Noofficial ceremony is complete without some kind of en-tertainment. The musical interlude in this case was pro-vided by a small Swedish ensemble specializing in Eli-zabethan music but also performing, fittingly, some clas-sical Swedish songs. Dressed in period costumes, theydid their thing, singing with their own instrumental sup-port, concluding their act by strutting around the stagepilfering the belongings of those unfortunate enough tohave the privilege of sitting on the first row.

The week in review

The conference was kicked off by the first plenary lectu-rer, Oded Shramm, associated with Microsoft Research.An appropriate beginning in view of the fact that Presi-dent Laptev, earlier in the ceremony, got stuck on his po-wer-point presentation, cursing modern computer tech-nology. Schramm, however, did not address such word-ly issues of practicality, but expounded on conformallyinvariant random processes instead, albeit with many acomputer visualization. And then there was time forlunch, and in spite of the customary denial of the exi-stence of such entities, free to all participants. The after-noon was devoted to parallel sessions, four in fact, in-volving twelve lectures in total. The day was capped offby an EMS reception at the old location for the depart-ment of mathematics at KTH, a building commonly re-ferred to as Sing-sing, due to its intimidating lay-out. Thereception wisely took place on the ground floor, the li-mited space of which quickly got extremely crowded.One surmises that afterwards there were only emptywine-bottles among the left-overs. The next day startedout with presentations of the prize winners and theirwork (it should be noted that some of them had also,independently one assumes, been invited as regular spea-kers as well.) Those were followed by invited lecturesand then in the afternoon there were three Science lectu-res. The last one was that of R.Ernst, a Nobel Prize win-ner in Chemistry, giving a survey from Fourier to Medi-cal imaging, constituting no doubt a very instructive lec-ture to the mathematicians, giving them, among otherthings, a sense of the somewhat alien culture of bigscience. Tuesday was capped off by a visit to the TownHall of Stockholm replete with a buffet courtesy of thecity of Stockholm. The Town Hall, designed by theSwedish architect Ragnar Östberg and built in the twen-ties, is one of the most commonly pictorially reproducedlandmarks of the city, located at the edge of an island,and commanding a presence on the main waterway. Itsdesign was inspired by the palace of the dodges in Ve-nice and its main feature is the great banquet hall inside,somewhat puzzlingly known as the Blue Hall (guides ofthe building are just delighted to explain to you the hi-storical reason why), which every December is host tothe Nobel banquet following the prize awards at theConcert Hall. This time however, the setting was so-mewhat less sumptuous. Two grand buffets (presumably

From the opening ceremony

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identical) were displayed on two long tables on one si-de. Food was plentiful, but to savour it you needed to dothe customary juggle of balancing your wineglass, yourplate, and your knife and fork with just two hands, whi-le standing firmly on your feet and trying to make co-herent conversation. Afterwards Kingman rose to theoccasion thanking the hosts referring to the great suc-cess of the conference (so far). He concluded by delive-ring a splendid celebration of the importance of mathe-matics, reminding everyone that while back in 1900, so-me of the scientific accomplishments honoured at thisvery place involved no mathematics, nowadays thiswould be almost impossible, and that all scientists shouldacknowledge this fact. In fact, he reminded us all thatthe entire human race will benefit from the developmentof mathematics. In order to gently usher out the mathe-matical crowd from the premises, a guided tour was of-fered at the end providing a natural conclusion. For tho-se of us who afterwards lingered on by the waterfront,we may late forget the glorious view made sublime (asone used to say) by the slowly setting sun. The traditio-nal association of Stockholm with Venice, made particu-larly palpable by the Town Hall, is not plucked out ofthin air, but rests on water. Wednesday was taken overby plenary talks in the morning and three additionalscience lectures in the afternoon. The Austrian Nowakexpounded on mathematical biology with special emp-hasis on evolutionary processes, and Berry presented acascade of computer generated pictures relating to op-tics and concomitant singularities. Thursday was onlyhalf a day, with invited lectures in the morning and sche-duled excursions in the afternoon. In the evening, theFrench ambassador gave a reception for the notables ofthe EMS, the organizers and last but not least the youngprize-winners. France as a country and culture shouldbe commended for the official respect it accords mathe-matics and for always recognising mathematical achie-vement. I fear that the 4ECM may very well have recei-ved more publicity in France than in Sweden. Friday wasthe closing up, with parallel network lectures in the mor-ning and a series of plenary talks in the afternoon, thelast fittingly delivered by a local speaker, Johan Håstadat KTH, talking on the difficulty of proving the generallybelieved NP≠P.

Wrapping up

The concluding ceremony was, as such things tend tobe, rather short. Kingman expressed the pride of the EMSto be associated with the ECM and thanked the organi-zers, and especially its President. There was a lot of ap-plause. Then there was a reference to the upcoming cen-tenary birthday of Henri Cartan, who had an honoraryrole in the first ECM which took place in Paris 1992, andto whom the entire congress relayed its congratulations.Laptev then referred to the statistics of the event, withthe final tally of 930 members, over three hundred po-sters, and sixty three scheduled talks, of which only onehad to be cancelled. As expected he could not refrain fromcongratulating himself on having arranged such niceweather for the entire duration. No mean feet indeed,and a definite contribution to the general pleasure. Fi-nally the congress was treated to an invitation to the fif-th ECM to take place in Amsterdam in 2008. After theusual problems with an unresponsive machine, theaudience was eventually exposed to a power-point showby the representative of the next ECM, promising futuredelights. By the time the audience emerged out of theAula Magna for the last time, bookstalls were being dis-mantled, posters taken down, and the last remaining con-ference briefcases sold off at very reasonable prizes. Clea-rly within hours, no physical traces would be left of thecongress. On the other hand, one would hope that men-tal traces, preferably of the positive and edifying kind,will remain for a very long time.

Ulf Persson [ulfp@math.chalmers.se] has beenprofessor at Chalmers University of Technology inGothenburg (Göteborg) since 1989. He earned his Ph.Dat Harvard 1975 as a student of David Mumford. Worksin Algebraic Geometry, especially compact complexsurfaces. Hobbies include mathematical picturesprogrammed directly in PostScript. Has served as thePresident of the Swedish Mathematical Society andhas been actively engaged in public debate onmathematical education. Publishes regularly reviewson scientific-philosophical matters in the Swedish pressand is the editor of the newsletter of the aforementionedSwedish Mathematical Society.

4ecm staff studying new books at the booth of the EMSpublishing house

Starting the race towards 5ecm atAmsterdam in 2008. Fromleft to righ: H.J.J. te Riele, J.O.O. Wiegerinck, C.M. Ran(http://sta.sience.uva.nl/brandts/5ECM/)

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First of all, my sincere congratulations on the 50th anni-versary of the Division of Science. I am very happy to bepresent here today, and I consider myself as a represen-tative of the many foreign visitors during these 50 years.The number of foreign visitors has always been excepti-onally large, and it may be appropriate today to ask: why?

I can think of three very good reasons:· The emphasis on a very high level of scientific quali-

ty.· Internationality· Friendliness

These three factors together form the foundation of a toplevel research and teaching institution which will alwaysattract scientists from the entire world.

They also characterize what I would like to call the“Spirit of Bundgaard” – after the legendary founder ofthe Division of Science and the Department of Mathema-tics here in Aarhus, Professor Svend Bundgaard.

Allow me to illustrate this spirit with a personalexample.

I came to the University of Aarhus for the first time in1968 as a graduate student at the Department of Mathe-matics. During the year 1968-69, Bundgaard sent some ofmy results to Professor Frank Adams in England. He re-sponded by inviting me to spend the following year work-ing with him in England.

The natural thing for me to do was to apply for a Fel-lowship to the Norwegian Research Council, since Nor-way was my home country. My application was rejectedwith the message that I could continue my studies in Os-lo!

I was very disappointed and told Bundgaard about it.His reaction was:

“That is a shame, we’ll have to do something aboutit!”

A few weeks later, I was called into his office, and he toldme that he had obtained – in a to me miraculous way – afellowship from the Danish Research Council in order tostudy with Adams in England!

I was thrilled by the good news, but suddenly mymoral spine-reaction told me that this wasn’t right: TheDanish Research Council supporting me – a Norwegiancitizen – going to England!

I told Bundgaard so, and then he looked at me seri-ously, smoking his big cigar and said:

“You should always remember that science is not anational affair, but an international one!”

What a wonderful statement!This is an example of “The Spirit of Bundgaard” on

which the Division of Science and the Department ofMathematics have been built, and I am glad to see thatthis spirit is still alive.

Today I am very glad to say that I think the Abelprizehas been created very much in the spirit of Bundgaard: tohonour outstanding mathematical achievements indepen-dent of nationality, race, sex or religion, but also to stimu-late the general interest in mathematics.

The Abelprize in mathematics was established by theNorwegian Storting in 2002 in honour of the Norwegianmathematical genius Niels Henrik Abel (1802-1829). It isalready being considered as an equivalent prize in ma-thematics to the Nobelprizes in other fields.

The Abelprize has received much attention this yearhere in Aarhus. In particular, it is a great pleasure for usthat one of this year’s Abel-laureates, Professor IsodoreSinger, will give the main speech here today in a fewmoments.

However, having spent the spring here this year, I ha-ve noticed that in the excellent mathematical library, N.H.Abel’s collected works are missing.

I imagine that Bundgaard’s reaction to this would ha-ve been:

“That is a shame, we’ll have to do something aboutit!”

Therefore it is a great pleasure for me to express the gra-titude of all Norwegian visitors during these 50 years,and present as a gift from The Norwegian MathematicalSociety:

Abel’s collected works.

May it serve as an inspiration to future generations ofDanish mathematicians and visitors!

Finally, may I ask the Chairman of the Department ofMathematics – Johan P. Hansen – to come forward andreceive the gift.

The Spirit ofBundgaard

By Nils A. Baas 1

Rector of the UniversityDean of Science

ColleaguesLadies and Gentlemen

1 Professor of Mathematics, NTNU, Trondheim, NORWAY.Speech given at the 50th anniversary of the Division ofScience at the University of Aarhus on June 4, 2004.

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Matematiske Institutioner præsenterer sig:

Matematisk Modelleringved Mads ClausenInstituttet, SyddanskUniversitetMads Clausen Instituttet for ProduktInnovation (MCI) ved Syddansk Uni-versitet blev etableret i 1999 baseretpå en donation fra Bitten og MadsClausen Fonden (15 mio. kr samt eks-tra 30 mio. kr til det nye universitets-byggeri). MCI er det nyeste institutved Syddansk Universitet Sønder-borg og fungerer som et regulærtuniversitetsinstitut under det Natur-videnskabelige-Tekniske Fakultet.

MCI har bl.a. som formål at ud-danne civilingeniører herunder kan-didater i IT Product Design gennem2-årige Masters overbygningsforløbeller 5-årige kandidatuddannelser.Desuden udbyder MCI Ph.D. uddan-nelsen.

MCI er et erhvervsrettet universi-tet, der gerne vil være karakteriseretved utraditionelle arbejdsmetoder ogomgivelser. Foruden de universitets-ansatte er fem fuldtidsansatte Dan-foss-medarbejdere placeret ved MCIindenfor arbejdsområdet brugerori-enteret design (se beskrivelsen ne-denfor).

Instittuttets indretning er et åbentmiljø, hvor de ansatte fra de tre ho-vedforskningsområder:(I) Bruger-Orienteret Design(II) Apparat Software(III) Matematisk Modelleringi fællesskab deltager i undervisningog forskning i produktinnovation ogmekatronik. Sidstnævnte er en sam-let betegnelse for produkter, der in-deholder både elektronik, hardware

og software og mekaniske/elektro-mekaniske elementer.

I 2003 blev det nye Center for Pro-duktUdvikling startet op som en delaf MCI. Center for ProduktUdviklinger blevet til gennem støtte fra Sønder-jyllands Amt, Syddansk Universitetog EU’s Interreg IIIa program (20mio. kr de første tre år, hvorefter cen-teret skal være selvfinansierende) oghar som formål at fungere som bin-deled mellem virksomheder og forsk-nings-og videninstitutter i Danmarkog Tyskland specielt indenfor områ-det intelligente mekatroniske pro-dukter.

Gruppen for matematisk model-lering ved Mads Clausen Instituttetbestår (pr. 1 september 2004) af syvpersoner:Morten Willatzen, Professor, Ph.D.,Jens Gravesen, Gæsteprofessor,Ph.D.,Roderick Melnik, Adjungeret Profes-sor, Ph.D.,Hemant Kamath, Associate Profes-sor, Ph.D.,Linxiang Wang, Assistant Professor,Ph.D.,Benny Lassen, Ph.D. studerende,Nenad Radulovich, Ph.D. studeren-de, foruden 18 Master’s studerende,der alle arbejder indenfor ét af neden-nævnte hovedområder og ofte i tætsamarbejde med industrien.

Generelle forskningsområder:De ansatte i matematisk modelle-ringsgruppen arbejder med følgen-de områder:(1) Matematisk modellering af inge-

niørmæssige problemer, der om-fatter elektromekaniske, magne-tostriktive og termoelektriske sy-stemer og komponenter herundersensorer, aktuatorer og transduce-re.

(2) Dynamisk beskrivelse af fase-overgange og anvendelser i indu-strien, materialevidenskab og fy-sik herunder matematisk og nu-merisk analyse af randværdipro-blemer.

(3) Matematiske modeller og nume-

riske beregningsmetoder for an-vendelser i materialevidenskabmed vægt på matematisk model-lering af ‘smart materials’ ogkompositmaterialer.

(4) Varme- og masse transportpro-blemer herunder modellering ogregulering af termofluide syste-mer.

(5) Numerisk analyse af faststofme-kaniske og fluiddynamiske pro-blemer herunder problemer, hvorvekselvirkning mellem fluider ogfaste stoffer er af betydning.

(6) Matematiske aspekter i anvendtreologi herunder matematiskemodeller for elektro- og magne-toreologiske systemer/kompo-nenter.

(7) PDE-baserede modeller til beskri-velse af transport fænomenermed anvendelser på ingeniørvi-denskabelige problemstillinger.

(8) Regulering af dynamiske syste-mer i generel fysik og ingeniørvi-denskab specielt flow akustik oganvendelser indenfor forbræn-dingssystemer.

(9) Matematisk modellering i mi-kroelektronik og nanoteknologimed vægt lagt på halvlederfysikog optoelektroniske systemer/komponenter.

(10) Undersøgelse af betydningen afgeometrien af (og strain i) nano-strukturer for deres elektroniskeog optiske egenskaber. Dette om-fatter optimering af nanostruktu-rers geometri med anvendelserindenfor bl.a. halvlederforstærke-re og halvlederlasere baseret på enkvantemekanisk beskrivelse.

Af: Morten Willatzen MCIemail: willatzen@mci.sdu.dk

Mads Clausen Instituttet, SyddanskUniversitet

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Boganmeldelse Interwiewved Martin Raussen og Christian Skau

The Index Theorem

First, we congratulate both of youfor having been awarded theAbel Prize 2004. This prize hasbeen given to you for “thediscovery and the proof of theIndex Theorem connectinggeometry and analysis in asurprising way and youroutstanding role in building newbridges between mathematicsand theoretical physics”. Both ofyou have an impressive list offine achievements in mathe-matics. Is the Index Theoremyour most important result andthe result you are most pleasedwith in your entire careers?

ATIYAH First, I would like to saythat I prefer to call it a theory, not atheorem. Actually, we have workedon it for 25 years and if I include allthe related topics, I have probablyspent 30 years of my life working onthe area. So it is rather obvious thatit is the best thing I have done.SINGER I too, feel that the index the-orem was but the beginning of a highpoint that has lasted to this very day.It’s as if we climbed a mountain andfound a plateau we’ve been on eversince.

We would like you to give ussome comments on the history ofthe discovery of the IndexTheorem.1 Were there precursors,conjectures in this directionalready before you started? Werethere only mathematical moti-vations or also physical ones?

the more the better. For two reasons:usually, different proofs have diffe-rent strengths and weaknesses, andthey generalize in different directions- they are not just repetitions of eachother. And that is certainly the casewith the proofs that we came upwith. There are different reasons forthe proofs, they have different histo-ries and backgrounds. Some of themare good for this application, someare good for that application. Theyall shed light on the area. If you can-not look at a problem from differentdirections, it is probably not very in-teresting; the more perspectives, thebetter!SINGER There isn’t just one theo-rem; there are generalizations of thetheorem. One is the families indextheorem using K-theory; another isthe heat equation proof which makesthe formulas that are topological,more geometric and explicit. Eachtheorem and proof has merit and hasdifferent applications.

Collaboration

Both of you contributed to theindex theorem with differentexpertise and visions – and otherpeople had a share as well, Isuppose. Could you describe thiscollaboration and the estab-lishment of the result a littlecloser?

SINGER Well, I came with a back-ground in analysis and differentialgeometry, and Sir Michael’s experti-se was in algebraic geometry and to-pology. For the purposes of the In-dex Theorem, our areas of expertisefit together hand in glove. Moreover,in a way, our personalities fit toget-her, in that “anything goes”: Make asuggestion - and whatever it was, wewould just put it on the blackboardand work with it; we would both en-thusiastically explore it; if it didn’twork, it didn’t work. But oftenenough, some idea that seemed far-fetched did work. We both had thefreedom to continue without worry-ing about where it came from or whe-re it would lead. It was exciting to

ATIYAH Mathematics is always acontinuum, linked to its history, thepast - nothing comes out of zero. Andcertainly the Index Theorem is simp-ly a continuation of work that, Iwould like to say, began with Abel.So of course there are precursors. Atheorem is never arrived at in theway that logical thought would leadyou to believe or that posteritythinks. It is usually much more acci-dental, some chance discovery inanswer to some kind of question.Eventually you can rationalize it andsay that this is how it fits. Discoveri-es never happen as neatly as that.You can rewrite history and make itlook much more logical, but actuallyit happens quite differently.SINGER At the time we proved theIndex Theorem we saw how impor-tant it was in mathematics, but wehad no inkling that it would havesuch an effect on physics some yearsdown the road. That came as a com-plete surprise to us. Perhaps itshould not have been a surprisebecause it used a lot of geometry andalso quantum mechanics in a way, àla Dirac.

You worked out at least threedifferent proofs with differentstrategies for the Index Theorem.Why did you keep on after thefirst proof? What differentinsights did the proofs give?

ATIYAH I think it is said that Gausshad ten different proofs for the lawof quadratic reciprocity. Any goodtheorem should have several proofs,

Interview withMichael Atiyah andIsadore SingerInterviewers: Martin Raussen and Christian SkauThis interview has appeared also in EMS-Newsletter no.53

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

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work with Sir Michael all these ye-ars. And it is as true today as it waswhen we first met in ’55 - that senseof excitement and “anything goes”and “let’s see what happens”.ATIYAH No doubt: Singer had astrong expertise and background inanalysis and differential geometry.And he knew certainly more physicsthan I did; it turned out to be veryuseful later on. My background wasin algebraic geometry and topology,so it all came together. But of coursethere are a lot of people who contri-buted in the background to the build-up of the Index Theorem – goingback to Abel, Riemann, much morerecently Serre, who got the Abel pri-ze last year, Hirzebruch, Grothen-dieck and Bott. There was lots ofwork from the algebraic geometryside and from topology that prepa-red the ground. And of course thereare also a lot of people who did fun-damental work in analysis and thestudy of differential equations: Hör-mander, Nirenberg... In my lecture Iwill give a long list of names2; eventhat one will be partial. It is anexample of international collaborati-on; you do not work in isolation,neither in terms of time nor in termsof space – especially in these days.Mathematicians are linked so much,people travel around much more. Wetwo met at the Institute at Princeton.It was nice to go to the Arbeitstagungin Bonn every year, which Hirze-bruch organised and where many ofthese other people came. I did notrealize that at the time, but lookingback, I am very surprised how quick-ly these ideas moved...

Collaboration seems to play abigger role in mathematics thanearlier. There are a lot of confe-rences, we see more papers thatare written by two, three or evenmore authors – is that a necessaryand commendable developmentor has it drawbacks as well?

ATIYAH It is not like in physics orchemistry where you have 15 authorsbecause they need an enormous bigmachine. It is not absolutely neces-sary or fundamental. But particu-larly if you are dealing with areaswhich have rather mixed and inter-disciplinary backgrounds, withpeople who have different expertise,it is much easier and faster. It is also

much more interesting for the parti-cipants. To be a mathematician onyour own in your office can be a littlebit dull, so interaction is stimulating,both psychologically and mathema-tically. It has to be admitted that the-re are times when you go solitary inyour office, but not all the time! Itcan also be a social activity with lotsof interaction. You need a good mixof both, you can’t be talking all thetime. But talking some of the time isvery stimulating. Summing up, Ithink that it is a good development –I do not see any drawbacks.SINGER Certainly computers havemade collaboration much easier.Many mathematicians collaborate bycomputer instantly; it’s as if they we-re talking to each other. I am unableto do that. A sobering counte-rexample to this whole trend is Perel-man’s results on the Poincaré conjec-ture: He worked alone for ten to twel-ve years, I think, before putting hispreprints on the net.ATIYAH Fortunately, there are manydifferent kinds of mathematicians,they work on different subjects, theyhave different approaches and diffe-rent personalities – and that is a goodthing. We do not want all mathema-ticians to be isomorphic, we wantvariety: different mountains needdifferent kinds of techniques toclimb.SINGER I support that. Flexibilityis absolutely essential in our societyof mathematicians.

Perelman’s work on the Poincaréconjecture seems to be anotherinstance where analysis andgeometry apparently get linkedvery much together. It seems thatgeometry is profiting a lot fromanalytic perspectives. Is thislinkage between different disci-plines a general trend – is it true,that important results rely on thisinterrelation between differentdisciplines? And a much morespecific question: What do youknow about the status of the proofof the Poincaré conjecture?

SINGER To date, everything is work-ing out as Perelman says. So I learnfrom Lott’s seminar at the Universi-ty of Michigan and Tian’s seminar atPrinceton. Although no one vouchesfor the final details, it appears thatPerelman’s proof will be validated.

As to your first question: Whenany two subjects use each other’stechniques in a new way, frequently,something special happens. In geo-metry, analysis is very important; forexistence theorems, the more the bet-ter. It is not surprising that some new[at least to me] analysis implies so-mething interesting about the Poin-caré conjecture.ATIYAH I prefer to go even further– I really do not believe in the divisi-on of mathematics into specialities;already if you go back into the past,to Newton and Gauss... Althoughthere have been times, particularlypost-Hilbert, with the axiomatic ap-proach to mathematics in the firsthalf of the twentieth century, whenpeople began to specialize, to divideup. The Bourbaki trend had its usefor a particular time. But this is notpart of the general attitude to mathe-matics: Abel would not have distin-guished between algebra and analy-sis. And I think the same goes forgeometry and analysis for people li-ke Newton.

It is artificial to divide mathema-tics into separate chunks, and then tosay that you bring them together asthough this is a surprise. On the con-trary, they are all part of the puzzleof mathematics. Sometimes youwould develop some things for theirown sake for a while e.g. if you de-velop group theory by itself. But thatis just a sort of temporary convenientdivision of labour. Fundamentally,mathematics should be used as a uni-ty. I think the more examples we ha-ve of people showing that you canusefully apply analysis to geometry,the better. And not just analysis, Ithink that some physics came into itas well: Many of the ideas in geome-try use physical insight as well – takethe example of Riemann! This is allpart of the broad mathematical tra-dition, which sometimes is in dangerof being overlooked by modern,younger people who say “we haveseparate divisions”. We do not wantto have any of that kind, really.SINGER The Index Theorem was infact instrumental in breaking barriersbetween fields. When it first appea-red, many old-timers in special fieldswere upset that new techniques we-re entering their fields and achievingthings they could not do in the fieldby old methods. A younger genera-tion immediately felt freed from the

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barriers that we both view as artifi-cial.ATIYAH Let me tell you a little sto-ry about Henry Whitehead, the topo-logist. I remember that he told methat he enjoyed very much being atopologist: He had so many friendswithin topology, and it was such agreat community. “It would be a tra-gedy if one day I would have a bril-liant idea within functional analysisand would have to leave all my to-pology friends and to go out andwork with a different group ofpeople.” He regarded it to be his du-ty to do so, but he would be very re-luctant.

Somehow, we have been very for-tunate. Things have moved in such away that we got involved with func-tional analysts without losing our oldfriends; we could bring them all withus. Alain Connes was in functionalanalysis, and now we interact close-ly. So we have been fortunate tomaintain our old links and move in-to new ones – it has been great fun.

Mathematics and physics

We would like to have yourcomments on the interplaybetween physics and mathe-matics. There is Galilei’s famousdictum from the beginning of thescientific revolution, which saysthat the Laws of Nature arewritten in the language ofmathematics. Why is it that theobjects of mathematical creation,satisfying the criteria of beautyand simplicity, are precisely theones that time and time again arefound to be essential for a correctdescription of the external world?Examples abound, let me justmention group theory and, yes,your Index Theorem!

SINGER There are several ap-proaches in answer to your questi-ons; I will discuss two. First, someparts of mathematics were created inorder to describe the world aroundus. Calculus began by explaining themotion of planets and other movingobjects. Calculus, differential equa-tions, and integral equations are anatural part of physics because theywere developed for physics. Otherparts of mathematics are also natu-ral for physics. I remember lecturingin Feynman’s seminar, trying to ex-

plain anomalies. His postdocs keptwanting to pick coordinates in orderto compute; he stopped them saying:“The Laws of Physics are indepen-dent of a coordinate system. Listento what Singer has to say, because heis describing the situation withoutcoordinates.” Coordinate-free meansgeometry. It is natural that geome-try appears in physics, whose lawsare independent of a coordinate sy-stem.

Symmetries are useful in physicsfor much the same reason they’re use-ful in mathematics. Beauty aside,symmetries simplify equations, inphysics and in mathematics. So phy-sics and math have in common geo-metry and group theory, creating aclose connection between parts ofboth subjects.

Secondly, there is a deeper reason ifyour question is interpreted as in the tit-le of Eugene Wigner’s essay “The Unre-asonable Effectiveness of Mathema-tics in the Natural Sciences3”. Mathe-matics studies coherent systems which Iwill not try to define. But it studies co-herent systems, the connections betweensuch systems and the structure of suchsystems. We should not be too surprisedthat mathematics has coherent systemsapplicable to physics. It remains to beseen whether there is an already develo-ped coherent system in mathematics thatwill describe the structure of string the-ory. [At present, we do not even knowwhat the symmetry group of string fieldtheory is.] Witten has said that 21st cen-tury mathematics has to develop newmathematics, perhaps in conjunctionwith physics intuition, to describe thestructure of string theory.ATIYAH I agree with Singer’s de-scription of mathematics havingevolved out of the physical world; ittherefore is not a big surprise that ithas a feedback into it.

More fundamentally: to underst-and the outside world as a humanbeing is an attempt to reduce comple-xity to simplicity. What is a theory?A lot of things are happening in theoutside world, and the aim of scienti-fic inquiry is to reduce this to as simp-le a number of principles as possible.That is the way the human mindworks, the way the human mindwants to see the answer.

If we were computers, whichcould tabulate vast amounts of allsorts of information, we would neverdevelop theory – we would say, just

press the button to get the answer. Wewant to reduce this complexity to aform that the human mind can un-derstand, to a few simple principles.That’s the nature of scientific inquiry,and mathematics is a part of that.Mathematics is an evolution from thehuman brain, which is responding tooutside influences, creating the ma-chinery with which it then attacks theoutside world. It is our way of try-ing to reduce complexity into simpli-city, beauty and elegance. It is reallyvery fundamental, simplicity is in thenature of scientific inquiry – we donot look for complicated things.

I tend to think that science andmathematics are ways the humanmind looks and experiences – youcannot divorce the human mind fromit. Mathematics is part of the humanmind. The question whether there isa reality independent of the humanmind, has no meaning – at least, wecannot answer it.

Is it too strong to say that themathematical problems solvedand the techniques that arosefrom physics have been thelifeblood of mathematics in thepast; or at least for the last 25years?

ATIYAH I think you could turn thatinto an even stronger statement. Al-most all mathematics originally aro-se from external reality, even num-bers and counting. At some point,mathematics then turned to ask in-ternal questions, e.g. the theory ofprime numbers, which is not direct-ly related to experience but evolvedout of it.

There are parts of mathematicswhere the human mind asks internalquestions just out of curiosity. Origi-nally it may be physical, but eventu-ally it becomes something indepen-dent. There are other parts that rela-te much closer to the outside worldwith much more interaction back-wards and forward. In that part of it,physics has for a long time been thelifeblood of mathematics and inspi-ration for mathematical work. Thereare times when this goes out of fashi-on or when parts of mathematicsevolve purely internally. Lots of ab-stract mathematics does not directlyrelate to the outside world.

It is one of the strengths of mathe-matics that it has these two and not a

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single lifeblood: one external and oneinternal, one arising as response toexternal events, the other to internalreflection on what we are doing.SINGER Your statement is toostrong. I agree with Michael that ma-thematics is blessed with both an ex-ternal and internal source of inspira-tion. In the past several decades,high energy theoretical physics hashad a marked influence on mathema-tics. Many mathematicians have be-en shocked at this unexpected de-velopment: new ideas from outsidemathematics so effective in mathe-matics. We are delighted with thesenew inputs, but the “shock” exagge-rates their overall effect on mathema-tics.

Newer developments

Can we move to newer develop-ments with impact from theAtiyah-Singer Index Theorem?I.e., String Theory and EdwardWitten on the one hand and onthe other hand Non-commu-tative Geometry represented byAlain Connes. Could youdescribe the approaches tomathematical physics epitomizedby these two protagonists?

ATIYAH I tried once in a talk to de-scribe the different approaches toprogress in physics like different re-ligions. You have prophets, you ha-ve followers – each prophet and hisfollowers think that they have thesole possession of the truth. If youtake the strict point of view that the-re are several different religions, andthat the intersection of all these the-ories is empty, then they are alltalking nonsense. Or you can takethe view of the mystic, who thinksthat they are all talking of differentaspects of reality, and so all of themare correct. I tend to take the secondpoint of view. The main “orthodox”view among physicists is certainlyrepresented by a very large group ofpeople working with string theorylike Edward Witten. There are asmall number of people who havedifferent philosophies, one of themis Alain Connes, and the other is Ro-ger Penrose. Each of them has a ve-ry specific point of view; each of themhas very interesting ideas. Within thelast few years, there has been non-trivial interaction between all of the-

se.They may all represent different

aspects of reality and eventually,when we understand it all, we maysay “Ah, yes, they are all part of thetruth”. I think that that will happen.It is difficult to say which will be do-minant, when we finally understandthe picture – we don’t know. But Itend to be open-minded. The pro-blem with a lot of physicists is thatthey have a tendency to “follow theleader”: as soon as a new idea comesup, ten people write ten or more pa-pers on it and the effect is that every-thing can move very fast in a techni-cal direction. But big progress maycome from a different direction; youdo need people who are exploringdifferent avenues. And it is very goodthat we have people like Connes andPenrose with their own independentline from different origins. I am infavour of diversity. I prefer not to clo-se the door or to say “they are justtalking nonsense”.SINGER String Theory is in a veryspecial situation at the present time.Physicists have found new solutionson their landscape - so many that youcannot expect to make predictionsfrom String Theory. Its original pro-mise has not been fulfilled. Never-theless, I am an enthusiastic suppor-ter of Super String Theory, not justbecause of what it has done in ma-thematics, but also because as a co-herent whole, it is a marvellous sub-ject. Every few years new develop-ments in the theory give additionalinsight. When that happens, you re-alize how little one understood aboutString Theory previously. The theoryof D-branes is a recent example. Of-ten there is mathematics closely as-sociated with these new insights.Through D-branes, K-theory enteredString Theory naturally and resha-ped it. We just have to wait and seewhat will happen. I am quite confi-dent that physics will come up withsome new ideas in String Theory thatwill give us greater insight into thestructure of the subject, and alongwith that will come new uses of ma-thematics.

Alain Connes’ program is verynatural – if you want to combine ge-ometry with quantum mechanics,then you really want to quantize ge-ometry, and that is what non-commu-tative geometry means. Non-commu-tative Geometry has been used effec-

tively in various parts of String The-ory explaining what happens at cer-tain singularities, for example. I thinkit may be an interesting way of try-ing to describe black holes and to ex-plain the Big Bang. I would encoura-ge young physicists to understandnon-commutative geometry moredeeply than they presently do. Phy-sicists use only parts of non-commu-tative geometry; the theory has muchmore to offer. I do not know whetherit is going to lead anywhere or not.But one of my projects is to try andredo some known results using non-commutative geometry more fully.

If you should venture a guess,which mathematical areas doyou think are going to witness themost important developments inthe coming years?

ATIYAH One quick answer is thatthe most exciting developments arethe ones which you cannot predict.If you can predict them, they are notso exciting. So, by definition, yourquestion has no answer.

Ideas from physics, e.g. QuantumTheory, have had an enormous im-pact so far, in geometry, some partsof algebra, and in topology. The im-pact on number theory has still beenquite small, but there are someexamples. I would like to make a rashprediction that it will have a big im-pact on number theory as the ideasflow across mathematics – on oneextreme number theory, on the otherphysics, and in the middle geometry:the wind is blowing, and it will even-tually reach to the farthest extremi-ties of number theory and give us anew point of view. Many problemsthat are worked upon today with old-fashioned ideas will be done withnew ideas. I would like to see thishappen: it could be the Riemann hy-pothesis, it could be the Langlandsprogram or a lot of other relatedthings. I had an argument with An-drew Wiles where I claimed that phy-sics will have an impact on his kindof number theory; he thinks this isnonsense but we had a good argu-ment.

I would also like to make anotherprediction, namely that fundamentalprogress on the physics/mathematicsfront, String Theory questions etc.,will emerge from a much morethorough understanding of classical

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four-dimensional geometry, of Ein-stein’s Equations etc. The hard partof physics in some sense is the non-linearity of Einstein’s Equations. Eve-rything that has been done at themoment is circumventing this pro-blem in lots of ways. They haven’treally got to grips with the hardestpart. Big progress will come whenpeople by some new techniques ornew ideas really settle that. Whetheryou call that geometry, differentialequations or physics depends onwhat is going to happen, but it couldbe one of the big breakthroughs.

These are of course just my spe-culations.SINGER I will be speculative in aslightly different way, though I doagree with the number theorycomments that Sir Michael mentio-ned, particularly theta functions en-tering from physics in new ways. Ithink other fields of physics will af-fect mathematics - like statistical me-chanics and condensed matter phy-sics. For example, I predict a newsubject of statistical topology. Rat-her than count the number of holes,Betti-numbers, etc., one will be moreinterested in the distribution of suchobjects on noncompact manifolds asone goes out to infinity. We alreadyhave precursors in the number of zer-os and poles for holomorphic func-tions. The theory that we have forholomorphic functions will be gene-ralized, and insights will come fromcondensed matter physics as to what,statistically, the topology might looklike as one approaches infinity.

Continuity of mathematics

Mathematics has become sospecialized, it seems, that onemay fear that the subject willbreak up into separate areas. Isthere a core holding thingstogether?

ATIYAH I like to think there is a co-re holding things together, and thatthe core is rather what I look at my-self; but we tend to be rather egocen-tric. The traditional parts of mathe-matics, which evolved - geometry,calculus and algebra - all centre oncertain notions. As mathematics de-velops, there are new ideas, whichappear to be far from the centre go-ing off in different directions, whichI perhaps do not know much about.

Sometimes they become rather im-portant for the whole nature of themathematical enterprise. It is a bitdangerous to restrict the definition tojust whatever you happen to un-derstand yourself or think about. Forexample, there are parts of mathema-tics that are very combinatorial.Sometimes they are very closely re-lated to the continuous setting, andthat is very good: we have interestinglinks between combinatorics and al-gebraic geometry and so on. Theymay also be related to e.g. statistics.I think that mathematics is very dif-ficult to constrain; there are also allsorts of new applications in differentdirections.

It is nice to think of mathematicshaving a unity; however, you do notwant it to be a straitjacket. The cen-tre of gravity may change with time.It is not necessarily a fixed rigid ob-ject in that sense, I think it shoulddevelop and grow. I like to think ofmathematics having a core, but I donot want it to be rigidly defined sothat it excludes things which mightbe interesting. You do not want toexclude somebody who has made adiscovery saying: “You are outside,you are not doing mathematics, youare playing around”. You neverknow! That particular discoverymight be the mathematics of the nextcentury; you have got to be careful.Very often, when new ideas come in,they are regarded as being a bit odd,not really central, because they looktoo abstract.SINGER Countries differ in their at-titudes about the degree of speciali-zation in mathematics and how totreat the problem of too much spe-cialization. In the United States I ob-serve a trend towards early speciali-zation driven by economic conside-rations. You must show early pro-mise to get good letters of recommen-dations to get good first jobs. Youcan’t afford to branch out until youhave established yourself and havea secure position. The realities of li-fe force a narrowness in perspectivethat is not inherent to mathematics.We can counter too much specializa-tion with new resources that wouldgive young people more freedomthan they presently have, freedom toexplore mathematics more broadly,or to explore connections with othersubjects, like biology these days whe-re there is lots to be discovered.

When I was young the job marketwas good. It was important to be at amajor university but you could stillprosper at a smaller one. I am distres-sed by the coercive effect of today’sjob market. Young mathematiciansshould have the freedom of choice wehad when we were young.

The next question concerns thecontinuity of mathematics.Rephrasing slightly a questionthat you, Prof. Atiyah are theorigin of, let us make thefollowing gedanken experiment:If, say, Newton or Gauss or Abelwere to reappear in our midst, doyou think they would understandthe problems being tackled by thepresent generation of mathe-maticians – after they had beengiven a short refresher course?Or is present day mathematicstoo far removed from traditionalmathematics?

ATIYAH The point that I was tryingto make there was that really impor-tant progress in mathematics is so-mewhat independent of technical jar-gon. Important ideas can be expla-ined to a really good mathematician,like Newton or Gauss or Abel, in con-ceptual terms. They are in fact coor-dinate-free, more than that, techno-logy-free and in a sense jargon-free.You don’t have to talk of ideals, mo-dules or whatever – you can talk inthe common language of scientistsand mathematicians. The really im-portant progress mathematics hasmade within 200 years could easilybe understood by people like Gaussand Newton and Abel. Only a smallrefresher course where they weretold a few terms – and then theywould immediately understand.

Actually, my pet aversion is thatmany mathematicians use too manytechnical terms when they write andtalk. They were trained in a way thatif you do not say it 100 percent cor-rectly, like lawyers, you will be takento court. Every statement has to befully precise and correct. Whentalking to other people or scientists, Ilike to use words that are common tothe scientific community, not neces-sarily just to mathematicians. Andthat is very often possible. If you ex-plain ideas without a vast amount oftechnical jargon and formalism, I amsure it would not take Newton, Gauss

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and Abel long – they were brightguys, actually!SINGER One of my teachers at Chi-cago was André Weil, and I remem-ber his saying: “If Riemann werehere, I would put him in the libraryfor a week, and when he came outhe would tell us what to do next.”

Communication ofmathematics

Next topic: Communication ofmathematics: Hilbert, in hisfamous speech at the Inter-national Congress in 1900, inorder to make a point aboutmathematical communication,cited a French mathematicianwho said: “A mathematicaltheory is not to be consideredcomplete until you have made itso clear that you can explain itto the first man whom you meeton the street”. In order to passon to new generations of mathe-maticians the collective know-ledge of the previous generation,how important is it that theresults have simple and elegantproofs?

ATIYAH The passing of mathema-tics on to subsequent generations isessential for the future, and this isonly possible if every generation ofmathematicians understands whatthey are doing and distils it out insuch a form that it is easily under-stood by the next generation. Manycomplicated things get simple whenyou have the right point of view. Thefirst proof of something may be verycomplicated, but when you underst-and it well, you readdress it, andeventually you can present it in away that makes it look much moreunderstandable – and that’s the wayyou pass it on to the next generati-on! Without that, we could nevermake progress - we would have allthis messy stuff. Mathematics doesdepend on a sufficiently good grasp,on understanding of the fundamen-tals so that we can pass it on in assimple a way as possible to our suc-cessors. That has been done remar-kably successfully for centuries.Otherwise, how could we possibly bewhere we are? In the 19th century,people said: “There is so much ma-thematics, how could anyone makeany progress?” Well, we have - we

do it by various devices, we genera-lize, we put all things together, weunify by new ideas, we simplify lotsof the constructions – we are verysuccessful in mathematics and havebeen so for several hundred years.There is no evidence that this hasstopped: in every new generation,there are mathematicians who makeenormous progress. How do theylearn it all? It must be because wehave been successful communicatingit.SINGER I find it disconcertingspeaking to some of my young colle-agues, because they have absorbed,reorganized, and simplified a greatdeal of known material into a newlanguage, much of which I don’t un-derstand. Often I’ll finally say, “Oh;is that all you meant?” Their newconceptual framework allows themto encompass succinctly considera-bly more than I can express with mi-ne. Though impressed with the pro-gress, I must confess impatiencebecause it takes me so long to un-derstand what is really being said.

Has the time passed when deepand important theorems inmathematics can be given shortproofs? In the past, there aremany such examples, e.g., Abel’sone-page proof of the additiontheorem of algebraic differentialsor Goursat’s proof of Cauchy’sintegral theorem.

ATIYAH I do not think that at all!Of course, that depends on what fou-ndations you are allowed to startfrom. If we have to start from theaxioms of mathematics, then everyproof will be very long. The commonframework at any given time is con-stantly advancing; we are already ata high platform. If we are allowedto start within that framework, thenat every stage there are short proofs.

One example from my own life isthis famous problem about vectorfields on spheres solved by FrankAdams where the proof took manyhundreds of pages. One day I disco-vered how to write a proof on a post-card. I sent it over to Frank Adamsand we wrote a little paper whichthen would fit on a bigger postcard.But of course that used some K-the-ory; not that complicated in itself.You are always building on a higherplatform; you have always got more

tools at your disposal that are part ofthe lingua franca which you can use.In the old days you had a smaller ba-se: If you make a simple proof nowa-days, then you are allowed to assu-me that people know what group the-ory is, you are allowed to talk aboutHilbert space. Hilbert space took along time to develop, so we have gota much bigger vocabulary, and withthat we can write more poetry.SINGER Often enough one can dis-til the ideas in a complicated proofand make that part of a new lan-guage. The new proof becomessimpler and more illuminating. Forclarity and logic, parts of the origi-nal proof have been set aside and di-scussed separately.ATIYAH Take your example of Abe-l’s Paris memoir: His contemporariesdid not find it at all easy. It laid thefoundation of the theory. Only lateron, in the light of that theory, we canall say: “Ah, what a beautifully simp-le proof!” At the time, all the ideashad to be developed, and they werehidden, and most people could notread that paper. It was very, very farfrom appearing easy for his contem-poraries.

Individual work style

I heard you, Prof. Atiyah,mention that one reason for yourchoice of mathematics for yourcareer was that it is not necessaryto remember a lot of facts byheart. Nevertheless, a lot ofthreads have to be woven to-gether when new ideas aredeveloped. Could you tell us howyou work best, how do new ideasarrive?

ATIYAH My fundamental approachto doing research is always to askquestions. You ask “Why is thistrue?” when there is something my-sterious or if a proof seems very com-plicated. I used to say – as a kind ofjoke – that the best ideas come to youduring a bad lecture. If somebodygives a terrible lecture, it may be abeautiful result but with terribleproofs, you spend your time tryingto find better ones, you do not listento the lecture. It is all about askingquestions – you simply have to havean inquisitive mind! Out of ten que-stions, nine will lead nowhere, andone leads to something productive.

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You constantly have to be inquisiti-ve and be prepared to go in any di-rection. If you go in new directions,then you have to learn new material.

Usually, if you ask a question ordecide to solve a problem, it has abackground. If you understand whe-re a problem comes from then itmakes it easy for you to understandthe tools that have to be used on it.You immediately interpret them interms of your own context. When Iwas a student, I learned things bygoing to lectures and reading books– after that I read very few books. Iwould talk with people; I would lea-rn the essence of analysis by talkingto Hörmander or other people. Iwould be asking questions because Iwas interested in a particular pro-blem. So you learn new things becau-se you connect them and relate themto old ones, and in that way you canstart to spread around.

If you come with a problem, andyou need to move to a new area forits solution, then you have an intro-duction – you have already a pointof view. Interacting with other peopleis of course essential: if you move in-to a new field, you have to learn thelanguage, you talk with experts; theywill distil the essentials out of theirexperience. I did not learn all thethings from the bottom upwards; Iwent to the top and got the insightinto how you think about analysis orwhatever.SINGER I seem to have some built-in sense of how things should be inmathematics. At a lecture, or readinga paper, or during a discussion, I fre-quently think, “that’s not the way itis supposed to be.” But when I tryout my ideas, I’m wrong 99% of thetime. I learn from that and fromstudying the ideas, techniques, andprocedures of successful methods.My stubbornness wastes lots of timeand energy. But on the rare occasionwhen my internal sense of mathema-tics is right, I’ve done something dif-ferent.

Both of you have passed ordinaryretirement age several years ago.But you are still very activemathematicians, and you haveeven chosen retirement or visi-ting positions remote from youroriginal work places. What arethe driving forces for keeping upyour work? Is it wrong that

mathematics is a “young man’sgame” as Hardy put it?

ATIYAH It is no doubt true that ma-thematics is a young man’s game inthe sense that you peak in your twen-ties or thirties in terms of intellectualconcentration and in originality. Butlater you compensate that by experi-ence and other factors. It is also truethat if you haven’t done anythingsignificant by the time you are forty,you will not do so suddenly. But it iswrong that you have to decline, youcan carry on, and if you manage todiversify in different fields this givesyou a broad coverage. The kind ofmathematician who has difficultymaintaining the momentum all hislife is a person who decides to workin a very narrow field with greatdepths, who e.g. spends all his lifetrying to solve the Poincaré conjec-ture – whether you succeed or not,after 10-15 years in this field youexhaust your mind; and then, it maybe too late to diversify. If you are thesort of person that chooses to makerestrictions to yourself, to specializein a field, you will find it harder andharder – because the only things thatare left are harder and harder techni-cal problems in your own area, andthen the younger people are betterthan you.

You need a broad base, fromwhich you can evolve. When thisarea dries out, then you go to that area– or when the field as a whole, inter-nationally, changes gear, you canchange too. The length of the timeyou can go on being active withinmathematics very much depends onthe width of your coverage. Youmight have contributions to make interms of perspective, breadth, inter-actions. A broad coverage is the se-cret of a happy and successful longlife in mathematical terms. I cannotthink of any counter example.SINGER I became a graduate stu-dent at the University of Chicago af-ter three years in the US army duringWorld War II. I was older and far be-hind in mathematics. So I wasshocked when my fellow graduatestudents said, “If you haven’t provedthe Riemann Hypothesis by age thir-ty, you might as well commit suici-de.” How infantile! Age means littleto me. What keeps me going is theexcitement of what I’m doing and itspossibilities. I constantly check [and

collaborate!] with younger collea-gues to be sure that I’m not deludingmyself – that what we are doing isinteresting. So I’m happily active inmathematics. Another reason is, in away, a joke. String Theory needs us!String Theory needs new ideas. Whe-re will they come from, if not fromSir Michael and me?ATIYAH Well, we have some stu-dents...SINGER Anyway, I am very excitedabout the interface of geometry andphysics, and delighted to be able towork at that frontier.

History of the EMS

You, Prof. Atiyah, have been verymuch involved in the estab-lishment of the EuropeanMathematical Society around1990. Are you satisfied with itsdevelopment since then?

ATIYAH Let me just comment a littleon my involvement. It started anawful long time ago, probably about30 years ago. When I started tryingto get people interested in forming aEuropean Mathematical Society inthe same spirit as the European Phy-sical Society, I thought it would beeasy. I got mathematicians from dif-ferent countries together and it waslike a mini-UN: the French and theGermans wouldn’t agree; we spentyears arguing about differences, and– unlike in the real UN – where even-tually at the end of the day you aredealing with real problems of theworld and you have to come to anagreement sometime; in mathema-tics, it was not absolutely essential.We went on for probably 15 years,before we founded the EMS.

On the one hand, mathematicianshave much more in common thanpoliticians, we are international in ourmathematical life, it is easy to talk tocolleagues from other countries; onthe other hand, mathematicians aremuch more argumentative. When itcomes to the fine details of a consti-tution, then they are terrible; they areworse than lawyers. But eventually– in principle – the good will was the-re for collaboration.

Fortunately, the timing was right.In the meantime, Europe had solvedsome of its other problems: the Ber-lin Wall had come down – so sudden-ly there was a new Europe to be in-

22 21/04

volved in the EMS. This very factmade it possible to get a lot morepeople interested in it. It gave anopportunity for a broader base of theEMS with more opportunities andalso relations to the European Com-mission and so on.

Having been involved with theset-up, I withdrew and left it to othersto carry on. I have not followed indetail what has been happening ex-cept that it seems to be active. I getmy Newsletter, and I see what is go-ing on.

Roughly at the same time as thecollapse of the Berlin Wall, mathema-ticians in general – both in Europeand in the USA – began to be moreaware of their need to be socially in-volved and that mathematics had animportant role to play in society.Instead of being shut up in their uni-versities doing just their mathematics,they felt that there was some pressu-re to get out and get involved in edu-cation, etc. The EMS took on this ro-le at a European level, and the EMScongresses – I was involved in the onein Barcelona – definitely made an at-tempt to interact with the public. Ithink that these are additional oppor-tunities over and above the old-fashi-oned role of learned societies. Thereare a lot of opportunities both interms of the geography of Europe andin terms of the broader reach.

Europe is getting ever larger:when we started we had discussionsabout where were the borders ofEurope. We met people from Geor-gia, who told us very clearly, that theboundary of Europe is this river onthe other side of Georgia; they werevery keen to make sure that Georgiais part of Europe. Now, the politici-

ans have to decide where the bordersof Europe are.

It is good that the EMS exists; butyou should think rather broadlyabout how it is evolving as Europeevolves, as the world evolves, as ma-thematics evolves. What should itsfunction be? How should it relate tonational societies? How should it re-late to the AMS? How should it rela-te to the governmental bodies? It isan opportunity! It has a role to play!

Apart from mathematics...

Could you tell us in a few wordsabout your main interests besidesmathematics?

SINGER I love to play tennis, and Itry to do so 2-3 times a week. Thatrefreshes me and I think that it hashelped me work hard in mathema-tics all these years.ATIYAH Well, I do not have his ener-gy! I like to walk in the hills, the Scot-tish hills – I have retired partly toScotland. In Cambridge, where I wasbefore, the highest hill was about this(gesture) big. Of course you have goteven bigger ones in Norway. I spenta lot of my time outdoors and I liketo plant trees, I like nature. I believethat if you do mathematics, you ne-ed a good relaxation which is not in-tellectual – being outside in the openair, climbing a mountain, working inyour garden. But you actually domathematics meanwhile. While yougo for a long walk in the hills or youwork in your garden – the ideas canstill carry on. My wife complains,because when I walk she knows I amthinking of mathematics.

SINGER I can assure you, tennisdoes not allow that!

Thank you very much on behalfof the Norwegian, the Danish,and the European MathematicalSocieties!

The interviewers were Martin Raussen,Aalborg University, Denmark, andChristian Skau, Norwegian Universityof Science and Technology, Trondheim,Norway.

References

1 More details were given in the laure-ates’ lectures.

2 Among those: Newton, Gauss, Cau-chy, Laplace, Abel, Jacobi, Rie-mann, Weierstrass, Lie, Picard,Poincaré, Castelnuovo, Enriques,Severi, Hilbert, Lefschetz, Hodge,Todd, Leray, Cartan, Serre, Ko-daira, Spencer, Dirac, Pontrjagin,Chern, Weil, Borel, Hirzebruch,Bott, Eilenberg, Grothendieck,Hörmander, Nirenberg

3 Comm. Pure App. Math.13(1), 1960

From left to right: I. Singer, M.Atiyah, M. Rasmussen, C. Skau

Bemærkninger tilinterviewet1) Yderligere informationer om

abelprisen 2004 kan findes påInternetadressen:hhtp://www.abelprisen.no/

2) Den i interviewet omtalte artikel:„The Unreasonable Effectivnessof Mathematics in the NaturalSciences“ af Eugene Wigner kanlæses på Internettet:http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

2321/04

Uddannelsesfronten

ved Carl Winsløw

Det er reformtid. Gymnasiet begynder næste år en nyæra med ”studieretningsgymnasiet”, hvor fagene

skal arbejde langt mere sammen end tidligere, ligesomder indføres nye undervisningsformer og forløb [1]. Påflere af landets naturvidenskabelige fakulteter (herunderAarhus og København) er semesterstrukturen ved at bli-ve erstattet med et system med fire undervisningsperio-der om året (se fx [2]). I princippet er det sidste i højeregrad blot en rammeforandring, men når rammerne for-andres så radikalt, sker der også noget med indholdet.Matematik skal finde sin plads i nye omgivelser: bliverdet bedre eller værre?

Det afhænger selvfølgelig ikke kun af rammerne.Rammeforandringer kan tværtimod være en (gan-

ske vist påtvungen) lejlighed til at se på nogle fundamen-tale forhold. F.eks. til en genovervejelse af, hvad der ervæsentligt for at lære matematik. Alle er vist enige om, atman lærer matematik ved selv at udøve matematik (læsematematisk tekst, tænke over matematiske problemer/opgaver, skrive foreløbige og mere polerede løsningerned, forklare sin tankegang til andre, etc.) – og at manlærer ret lidt af blot se eller høre andre udøve matematik.Alligevel ved enhver matematikunderviser, hvor let manender i det sidste, dvs. selv at udøve matematik (somregel med stor fornøjelse), mens de studerende ikke gørdet – desværre ofte også med en vis fornøjelse. (NB: I detflg. kaldes de, som skal lære matematik, ”studerende”,men jeg tænker både på gymnasie- og universitetssam-menhængen.)

I udkastet til læreplan for gymnasiets A-niveau [1]hedder det først i §3.1: I centrum for undervisningen skal

stå elevernes selvstændige håndtering af matematiske problem-stillinger og opgaver.

Hvordan kan reformerne så bruges til at sætte destuderendes matematikudøvelse i centrum? Et bud,

som ofte fremhæves, er projektarbejdsformen. Den præsen-teres undertiden som om der var tale om noget veldefi-neret og entydigt. Mange får associationer til en traditi-on, hvor matematikprojekter er meget åbne mht. det ma-tematiske indhold, og hvor det er problemstillinger ”uden-for matematikken” som sætter dagsordenen. De store in-ternt matematiske sammenhænge kan være svære at ram-me med den type af projekter.

Men projektarbejdsformen kan være så meget andet, og for mange formål bedre, i forbindelse med

matematik. Det skyldes at en rent matematisk problem-

Matematik i nye (re)former:Brousseau i praksis

stilling kan formuleres med en meget høj grad af fleksi-bilitet, og med en (i forhold til andre fag) relativt høj gradaf forudsigelighed mht. de forudsætninger som kan mo-biliseres i arbejdet med den. I ovennævnte læreplan (§3.2)hedder det da også: En betydelig del af undervisningen til-rettelægges som projekt- eller emneforløb over forskellige deleaf kernestoffet og det supplerende stof eller problemstillinger,der er genstand for fagsamarbejde. Og projekter i matema-tik kan sagtens være tilrettelagt sådan at arbejdet vedrø-rer et ganske veldefineret matematisk indhold – som afden ene eller anden grund er på dagsordenen – samtidigmed at studenterarbejdet udfordrer til selvstændighedog fantasi. Der er tre afgørende momenter i lærerens op-gaver i denne forbindelse:· overgive arbejdet til den studerende,· følge og i passende omfang vejlede det undervejs· forestå evalueringen af arbejdet når det er tilendebragt.

I Brousseau’s terminologi, som blev omtalt i sidste nummer af Matilde, tales om devolution, didaktisk situation

og institutionalisering.

Lad mig som illustration antyde en problemkreds,som også svarer til velkendte emner i flere sammen-

hænge. Hvordan definerer man tangenten til en kurve,og hvordan finder man den? Hvilke antagelser skal mangøre? Man kan evt. præcisere og supplere med flerespørgsmål, fx ved at starte med velkendte kurver (så-som linier, cirkler og grafer for konkrete funktioner).Pointen er at begrebsdannelsen starter med forståeligespørgsmål, også selvom man ikke forlanger at den stude-rende skal besvare dem alle sammen selv (se dog [2] foret universitært forsøg i den retning).

Billedtekst

24 21/04

En hovedopgave for matematiklærere (på alle niveauer!), som desværre ofte gives for lidt opmærk-

somhed, er netop tilrettelæggelse af problemstillinger somstuderende kan lære af at arbejde med. I en del klassiske un-dervisningssammenhænge løses denne læreropgave vedat give nogle lærebogsopgaver for. Sådanne opgaver hardet med at være ret rutineprægede, idet de typisk er be-regnet på at blive løst af den studerende uden yderligerehjælp. I den model for ”projektarbejde”, som er skitseretovenfor, fjerner vejledningselementet denne begrænsning.Alligevel er det vigtigt at tilrettelægge den studerendesudgangsposition sådan, at arbejdet i det mindste har enchance for at komme i gang. Man vil naturligvis altidoverveje de studerendes forudsætninger for at løse enopgave, inden man stiller den – men vejledningselemen-tet åbner mulighed for ”modigere” valg, og for at opga-ven løbende tilpasses dem, som arbejder med den.

Vejlederopgaven er nok det, som i mange sammenhænge vil være mest kritisk – og mest nyt. Denne

opgave drejer sig om den delikate balancegang mellempå den ene side at hjælpe de studerende videre, så vidtmuligt med at forfølge de relevante ideer og ressourcerde selv har – og på den anden side at undgå at lærerenovertager arbejdet. Vejlederrollen forbinder mange nokmed relativt avancerede studiesammenhænge – fx spe-cialeskrivning – men skal den studerendes arbejde foralvor sættes i centrum, bliver håndteringen af denne rol-le betydningsfuld i enhver matematikundervisning. I for-hold til klassisk tavleundervisning er vejledning i selv-stændig matematikudøvelse særdeles krævende – bådefagligt og personligt. Til gengæld er vejledningsfasen og-så en fagligt og personligt meget givende fase i samspilletmellem studenter og lærere.

Med evalueringsopgaven menes her ikke blot enbedømmelse af det udførte arbejdes kvalitet, men

især en organiseret opfølgning og præcisering af dets pointer.Ofte vil der – særlig i forbindelse med problemstillinger,der giver mulighed for at forfølge forskellige strategier –være flere væsentlige pointer end dem, som var tilsigteteller forudset da opgaven blev stillet. Men i det mindstede tilsigtede erkendelser skal præciseres, idet studenter-arbejde i matematik som regel har til formål at udvideden studerendes viden og kunnen indenfor et veldefine-ret og struktureret indholdsområde (”kernestoffet”). Deter med andre ord nødvendigt, men ikke tilstrækkeligt atfå sved på panden – de studerende skal erkende, at de erkommet videre fra udgangspositionen, og hvordan. Fak-tisk bliver også denne fase ofte lidt overset i klassiskearbejdsformer, og derved forspildes muligheden for atudnytte en af matematikkens charmer: at man ret klartkan indse, at arbejdet bar frugt – at man er nået videreend før – og at man herefter kan bygge videre på det.

Både gymnasiets og universitetets reformer indbyder til at tænke i de netop skitserede baner, idet de

åbner mulighed for længerevarende studentersamarbej-der under vejledning. En relateret ide, som netop nu erunder udvikling, er matematiske forskningssituationer i klas-serum [3] – det handler her om at lade de studerende ar-bejde med åbne problemstillinger i velkendte faglige sam-menhænge, hvor formulering af hypoteser og beviser står

centralt, og hvor den fulde problemstilling indebærer ulø-ste matematiske spørgsmål. Mindre kan også gøre det –som nævnt kan selv et ret velafgrænset indholdsområdemed fordel udforskes med udgangspunkt i opgaver, somikke er entydigt knyttet til bestemte teknikker.

Referencer

[1] http://www.uvm.dk/nyheder/gymnasiereform/oversigt.htm [sept. 2004]

[2] S. Horst og C. Winsløw (2004) Undervisning i blokstruktur– potentialer og risici. Didaktips nr. 5, Center for Natur-fagenes Didaktik, KU. On-line på: http://www.naturdidak.ku.dk/arkiv/publikationer/didaktips%205.pdf

[3] L. Alcock og A. Simpson (2001) The Warwick Analysis Pro-ject: practice and theory. I Holton, D. (red) Teaching and Lea-rning Mathematics at University Level: An ICMI Study, Dor-drecht: Kluwer Academic.

[4] D. Grenier og K. Godot (2004) Research situations for tea-ching: a modelling proposal and example. Artikel præsente-ret ved ICME-10. On-line på: http://www.icme-organisers.dk/tsg14/TSG14-02.pdf

Forslag til illustrationer af Uddannelsesfronten (ad libitum,de er ikke direkte nødvendige for at læse teksten; hvisde bringes bør dog nok begge bringes, idet størrelsennaturligvis kan tilpasses):

Kilde: http://www.haverford.edu/math/

Kilde: [1] (se referencer)Uddannelsesfronten, Matilde 21

Billedtekst

2521/04

Boganmeldelserved Carsten Lunde Pedersen

For ikke så længe siden realiseredeEuropean Mathematical Society engammel ide, idet man oprettede siteget forlag kaldet European Mathe-matical Society Publishing House.

På EMS Publishing House’s hjem-meside skriver Thomas Hintermann:

“...there is certainly no denyingthat at present in every companythe order of the day is to meet theshort-term (short-sighted?)targets, most often at the expenseof everything else. Financialconsiderations are clearly givenfirst priority, editorial mattersand publishing aspects ranksecond. One of the aims of theEMS publishing house is toreverse that order. The needs ofthe community are foremost inour minds, admittedly with theobligation to run an econo-mically sound operation.’’

EMS Publishing House udgiver totidsskrifter “Interfaces and Free Bo-undaries” og “Journal of the Europe-an Mathematical Society”. Herfor-uden har man nu udgivet ialt fem bø-ger indenfor fem forskellige serier.Nærværende boganmeldelse be-skæftiger sig med en af disse bøger,nemlig

“A Course In Error-CorrectingCodes” af Jørn Justesen og TomHøholdt, begge DTU.

I denne behandles en bred vifte afemner indenfor kodningsteorien.†Kodningsteorien er den del af infor-mationsteorien, der handler om,hvorledes man i praksis kan kommu-nikere tilnærmelsesvis fejlfrit over enstøjfyldt kanal. Teoridannelsen sigesat være født med Shannons artikel fra1948, hvori han viser, at man løst sagt

kan kommunikere såvel sikkert somtidsmæssigt effektivt over en støjfyldtkanal, hvis ellers man arbejder medmeget store pakker af informationer.Shannons bevis er ikke konstruktivt,og det er netop dette faktum, der gør,at kodningsteorien har været og sta-dig er et aktivt forskningsområde.

Bogen bærer præg af, at forfatter-ne er aktive forskere indenfor områ-det. Således behandles relevante klas-siske emner i et moderne sprogbrug,ligesom en række højaktuelle emnerindgår. Forfatternes brede fagligebaggrund har sat dem i stand til atskrive en bog, som appellerer såveltil matematikstuderende som til in-geniørstuderende med interesse forinformationsteori. Bogen udspringeraf et kursus afholdt af de to forfatte-re. I introduktionen til bogen skriverforfatterne, at

Målgruppen er “graduate andadvanced under graduate students’’.En ganske præcis anmeldelse af bo-gens stil vil være at sige, at den fuldtud opfylder den formulerede politikfor “EMS Textbooks in Mathematics”:

”EMS Textbooks in Mathema-tics” is a book series aimed atstudents or professional mathe-maticians seeking an intro-duction into a particular field.The individual volumes areintended to provide not onlyrelevant techniques, results andtheir applications, but affordinsight into the motivations andideas behind the theory. Suitabledesigned exercises help to masterthe subject and prepare thereader for the study of moreadvanced and specializedliterature.’’

Af: Christian Thommesencthom@math.aau.dk

Lektor ved Institut for MatematiskeFag, Aalborg Universitet

Anmeldelse af:

Jørn Justesen & Tom HøholdtA Course In Error-Correcting Codes.

EMS Textbooks in MathematicsEuropean Mathematical Society 2004ISBN 3-03719-001-9

Forfatterne skriver i forordet, at læ-seren forventes at have kendskab tilgrundlæggende lineær algebra ogalgoritmer, samt at en vis grad afmodenhed forudsættes. Således harJustesen og Høholdt gjort meget udaf kun at anvende den strengt nød-vendige matematik ved præsentati-on af emnerne, uden at opdyrke etteoretisk vildnis, der ville skygge forde fleste læseres udsyn.

Som eksempel kan nævnes, atendelige legemer på vellykket vis in-troduceres næsten uden brug af ab-strakt algebra , og at Hermite koder-ne, der hidrører fra den algebraiske

og Olav Geilolav@math.aau.dk

Lektor ved Institut for MatematiskeFag, Aalborg Universitet

26 21/04

geometri, overvejende behandles vedhjælp af lineær algebra. Med hensyntil Hermite koderne er der dog denpris at betale, at læseren må accepte-re at få serveret Bezouts sætning udenbevis.

Der gives en grundlæggende ind-føring i de basale kodningsteoretiskebegreber, og informationsbegrebernefejlsandsynlighed, gensidig informa-tion og kanalkapacitet indføres.

Reed-Solomon koder og de be-slægtede BCH koder behandles medvægt på kodernes algebraiske dekod-ning . Såvel klassiske dekodnings al-goritmer som de nye og højaktuellelistedekodningsalgoritmer, oprinde-ligt introduceret af Sudan, behandles.Bortset fra et kapitel om dekodningaf Reed-Solomon koder og BCH ko-der ved hjælp af Euklids udvidedealgoritme, er den bærende idé, at for-klare algoritmerne ud fra det ”Sudan-ske setup”, dvs kodepolynomiernefindes som rødder i et interpolationspolynomium. Denne fremgangsmå-de, som iøvrigt også anvendes pådekodning af ovennævnte Hermitekoder, giver en sammenhængendeforståelse af de algebraiske dekod-ningsmetoder for familien af ReedSolomon koder og deres slægtninge.

Introduktionen af cyliske koder oginformationsrammer leder frem til ensærdeles vellykket og utraditionelbehandling af foldningskoder og de-res dekodning.

Sammensatte koder som pro-duktkoder og konkatenerede koderbehandles med vægt på fejlsandsyn-ligheder .

Endelig er der et kapitel om de fortiden varmeste emner i branchen,nemlig LDPC (Low-Density ParityCheck) koder , turbokoder og itera-tiv dekodning.

Til hvert afsnit er der knyttet enrække gode og meget konkrete opga-ver. Opgaveløsninger findes bagest ibogen. Fra bogens hjemmeside kanhentes en række Maple-worksheets ihvilke, der arbejdes med endelige le-gemer, Reed-Solomon og BCH koderog deres dekodning. Enkelte af bo-gens opgaver kan alene løses, hvisman gør brug af Maple eller et andetmatematikprogram. Endelig vedlige-holdes på hjemmesiden en korrekt-ions og trykfejlsliste.

Olav Geil &Christian Thommesen

I august 2002 mødtes 42 matemati-kere, matematikhistorikere, militær-historikere og -analytikere samt filo-soffer til International Meeting on Ma-thematics and War i Karlskrona, Sve-rige i anledning af Bernhelms 60årsdag.

Den foreliggende bog indeholderindlæg fra dette møde.

Bogen er opdelt i 4 dele med over-skrifterne “Perspectives from Mathe-matics’’ (8 artikler), “Perspectivesfrom the Military “(5 artikler), “Ethi-cal Issues” (5 artikler) og “Enlighten-ment Perspectives’’(2 artikler) samten introduktion, der samtidig er enslags opsummering, af redaktørene.

De fire dele behandler spørgsmå-lene

1. I hvilket omfang har militærethistorisk, specielt efter anden ver-denskrig, spillet en aktiv rolle i ud-viklingen af moderne matematik ogfor matematikeres karrieremulighe-der?

2. Er matematisk tankegang, ma-tematiske metoder og matematikun-derstøttet teknologi med til at ændrekarakteren og udførelsen af moder-ne krig, og hvis det er tilfældet hvor-dan påvirker det offentligheden ogmilitæret?

3. Hvad var, under krigen, de eti-ske valg af fremragende individersom Niels Bohr og Allan Turing? Ihvilket omfang kan generelle etiskediskussioner udnyttes af den arbej-dende matematiker?

4. Hvad var den matematiske tan-kegangs rolle i udformningen af mo-derne international lov om krig ogfred? Kan matematiske argumenterbenyttes til at understøtte konfliktløs-ning?

I stedet for at anmelde de 20 ar-tikler individuelt, vil jeg nedenforforsøge at beskrive og vurdere, i hvil-ket omfang de bidrager til en besva-relse af de ovenfor nævnte interessan-te og vigtige spørgsmål.

1. Klassisk anvendt matematik(differentialligninger) blev bestemtstyrket og operationsanalyse og spil-teori blev vel nærmest skabt i krigs-årene. Dette beskrives indgåendemed eksempler fra hele verden. Hi-storien bag brydningen af kryptosy-

Af: Tom Høholdt, Docent, Institut forMatematik, DTU

email: T.Hoeholdt@mat.dtu.dk

Anmeldelse af Mathematics and War(Bernhelm Booss-Bavnbeck,Jens Høyrup eds.)Birkhäuser 2003

temet “Enigma” bliver gennemgået,specielt er det godt at se at den pol-ske indsats, som ikke er alment kendt,endelig bliver fremhævet. A.Turingsindsats er omtalt, mens et ligeså af-gørende bidrag fra W.Tutte ikke næv-nes.

Med hensyn til karrieremulighe-der er det påfaldende at National Se-curity Agency (NSA) ikke nævnessom den største aftager af matematikph.d-er i USA. (Afsnittet indeholderogså en egentlig matematik artikelaf Gamkrelidze)

2. To artikler af Svend Bergsteinmed titlerne : “War Cannot Be Calcu-lated’’ og “Warfare Can Be Calcu-lated” stimulerer jo umiddelbart in-teressen. Han argumenterer fint forsynspunkterne, men man fristes til at

2721/04

sige at hans krigsførelsesmodel , ogdet gælder også de andre matemati-ske modeller i dette afsnit, er tem-meligt primitive. Det er frygteligt atlæse om “Information Warfare”, derhandler om ødelæggelse af fjendenscomputere og informationssystemerfordi det synes meget nemt, og det erendnu værre at læse om “ModernWarfare” fordi systemerne, der benyt-tes åbenbart ikke er gode nok, menmatematikindholdet er da vist be-grænset til, at de matematisk-datalo-giske hjælpemidler der benyttes er altfor ringe. ( Det betyder så at tonsvisaf civile bliver dræbt).

3. Her gennemgås Niels Bohrskamp for en ‘’åben” verden, AllanTurings arbejde i Bletchley Park ogden japanske matematiker K.Ogurasrolle i den japanske krigsførelse. Hanstartede som antifascist og demokrat,men argumenterede under krigenmeget kraftigt for statslig kontrol ogstyring af den videnskabelige indsats.Endelig diskuteres indgående viden-skabsfolks moralske ansvar (specielthvis de direkte arbejder for militæret) og vanskelighederne ved at tillæg-ge dem et sådant ansvar bliver grun-digt belyst.

4. En analyse af H.Grotius’ (1583-1645) “De iure belli ac pacis”, der reg-nes for det egentlige grundlag for in-ternational lov (specielt om krig ogfred) viser at den axiomatiske meto-de i nogen udstrækning kan siges atvære anvendt. Matematiske modellerfor våbenkapløb og mere generellekonfliktmodeller bliver præsenteret,ikke overraskende fører nogle af dis-se til kaos.

Konklusion:

Den foreliggende bog er (måske iføl-ge sagens natur) noget uegal. Ikkedesto mindre er der noget godt og in-teressant at finde for enhver matema-tiker eller for den sags skyld enhversamfundsborger.

BemærkningEn anden anmeldelse af dennebog („Biting the bullit“ af PhilipJ. Davis) kan findes på adressen:http://www.siam.org/siam-news/03-04/war.pdf

Boganmeldelse MatematikerNytved Mikael Rørdam

As of August 1, 2004, Friedrich Huba-lek is employed as a lektor for Ma-thematical Finance at the Mathema-tics Department at Aarhus Uni-versity.Friedrich got his professionaleducation at the Vienna University ofTechnology,diploma (1992) under thedirection of Peter Kirschenhofer onthe Mellinintegral transform and itsapplications, doctoral thesis (1994)under the direction of Peter Kirschen-hofer and Helmut Prodingeron theprobabilistic analysis of bucket digi-tal search trees.Then he joined Wal-ter Schachermayer’s group first atVienna University,then at the Vien-na University of Technology. He isdoing research and teaching in Ma-thematical Finance, Insurance Ma-thematics, and related fields, in par-ticular on analytical and numericalaspects of concrete models with jumpprocesses (Levy processes). Frie-drich’s professional hobby is findingnew applications of “old” specialfunctions, his private hobbies are hi-king and cykling.

Henrik Holm er i perioden 1.august2004 til 31.juli 2006 ansat som adjunktved Institut for Matematiske Fag,Aarhus Universitet.Han er uddannetcand. scient. i matematik og fysik fraKøbenhavns Universitet (september2000), hvorefter han blev forskning-assistent lønnet af Lundbeck Fonden(indtil maj 2001). Han tog også sinph.d. i matematik fra KøbenhavnsUniversitet (juni 2004) under vejled-ning af Hans-Bjørn Foxby.Henriksforskning i kommutativ algebra om-fatter dimensionsteoretiske aspekteraf (hyper)homologisk algebra, som

blandt andet giver vigtig informati-on om den slags ringe og moduler,man støder på i algebraisk geo-metri.Fritiden bruges med familie/venner og med at spille skak og GOpå amatørniveau.

Iver Ottosen er ansat som adjunkt vedInstitut for Matematiske Fag, AarhusUniversitet.Iver fik sin Ph.D.-grad i1997 ved Aarhus Universitet undervejledningaf Marcel Bökstedt. I defølgende år har han været postdocved Oslo Universitet, Paris Univer-sitet 13, Bielefeld Universitet,samtKøbenhavns Universitet. I april 2003blev han ansat som adjunktved Aar-hus Universitet. Ivers forskningsom-råde er Algebraisk Topologi. Han harisær arbejdet med bestemmelse afkohomologi grupper og stabile ho-motopityper affrie løkkerum.Iverstyrketræner og løber i sin fritid.

Kasper K. S. Andersen er ansat somadjunkt ved Institut for MatematiskFag,Aarhus Universitet pr. 1. august2004. Kasper er uddannet cand.scient. (1997) og ph.d. (2001) fra Kø-benhavns Universitet. I 2001-2002havde han en EU postdoc stilling vedUniversity of Malaga og Centre deRecerca Matematica, Barcelona. Det-te blev efterfulgt af postdoc stillingerved Institut for Matematiske Fag, Kø-benhavns Universitet (2002-2003) ogInstitut for Matematiske Fag, AarhusUniversitet (2003-2004).Hans forsk-ningsområde er algebraisk topologi,mere specifikt studiet af homotopi-teorien for klassificerende rum. Hanhar hovedsagligt arbejdet med klas-sifikation og anvendelser af p-kom-pakte grupper.Fritiden bruger Kas-

28 21/04

per på sin kæreste og deres søn. Hanspiller desuden skak og holder af atlæse bøger og se film.

Malene Højbjerre er pr. 1. september2004 ansat som lektor i biostatistikved Institut for Matematiske Fag,Aalborg Universitet, hvor hun er til-knyttet “Center for Sundhedsstati-stik”, og bl.a. sidder som administra-tor af centerets statistiske konsu-lenttjeneste.Malene er uddannetcand. scient. og ph.d. i statistik fraAalborgUniversitet. Hendes forsk-ning omhandler primært stokastiskesimulerings metoder og grafiske mo-deller, specielt med fokus på biologi-ske/medicinske anvendelser. Maleneer gift og har to børn, en dreng og enpige på hhv. 5 og 7 år. Fritiden gårmed familie, venner, hus, have oggerne tre lange gåture om ugen.

Markus Kiderlen er d. 17. maj 2004 an-sat som adjunkt i et 3-årigt Carlsbergstipendium ved Institut for Matema-tiske Fag, Aarhus Universitet. Markushar taget en Ph.D. grad i matematikfra universitetet i Karlsruhe, Tyskland,under vejledning af Wolfgang Weil.Efterfølgende har Markus arbejdetsom Research Associate (postdoc) iKarlsruhe, og han har besøgt Aarhusadskillige gange med henblik påforskningssamarbejde. Markus’ forsk-ningsområde er konveks og stokastiskgeometri. Han undersøger hvilke ge-ometriske informationer om et højeredimensionalt objekt kan bestemmesud fra lavere dimensionale snit ellerfra digitale billeder af objektet. Hanshovedinteresse er de teoretiske pro-blemstillinger, men han deltager ogsåi udviklingen af anvendelser ved ana-lysen af biologisk væv eller af metal-legeringer i materialevidenskab. Mar-kus holder af yoga og udendørsakti-viteter såsom cykelture og svømning.

Begivenhederved Poul Hjorth

Danish Mathematical SocietyInvitation to the Annual Meeting of the Danish Mathematical Society, 2 November 2004,Department of Mathematical Sciences, Auditorium F, Ny Munkegade, Bld. 530, 8000 Århus C

Programme

12:00 – 13:00 LunchAt the mathematical canteen, Ny Munkegade, Bld. 530. Registration is needed.

13:00 – 13:15 Welcome by the chairman of the Danish Mathematical Society, Johan P. Hansen.13:15 – 14:30 Matematik i det nye gymnasium (this part will be conducted in Danish) - Fag-

konsulent i matematik, Bjørn Grøn: Abstrakt: Da valggymnasiet afløste grengym-nasiet i slutningen af 80’erne, skete der en dramatisk forøgelse i andelen af studen-ter med matematik på højeste niveau: Fra under 1/3 af de matematiske studenterfør til over 75% efter reformen. Alligevel beskrives den udvikling ofte som en sænk-ning af det faglige niveau. Hvordan måler man det faglige niveau? Et af den nuvæ-rende regerings udtalte mål med den kommende gymnasiereform er at styrke detfaglige niveau i almindelighed og styrke de matematisk-naturvidenskabelige fag isærdeleshed. Som konsekvens heraf nytænkes hele den almendannende side vedgymnasieundervisningen. Og samtidig afløses sproglig og matematisk linje af stu-dieretninger, indenfor hvilke fagene skal samarbejde.Dette har vidtgående konsekvenser for arkitekturen i læreplanerne. Et A-niveau imatematik har en for hele landet fælles målbeskrivelse af, hvad eleverne skal kun-ne. De faglige mål udmøntes i et kernestof, der er fælles gods for alle, og et supple-rende stof, der vil variere betydeligt i emnevalg fra studieretning til studieretning.Kan der formuleres succeskriterier for reformen? Er det supplerende stof irrelevant for aftagerne? I læreplansgrupperne i matematik arbejdede vi efter følgendeledetråd: Vi ønsker et gymnasium, hvor eleverne møder de matematisk-naturvi-denskabelige fag på en sådan måde, at det giver dem lyst til at læse disse fag, ogsom samtidig giver studenterne et solidt fagligt grundlag for at studere videre.Når vi formulerer opgaven på den måde kalder det næsten på en sådan kombinati-on af kernestof og supplerende stof. Og når de faglige mål skal opgives i dereshelhed til eksamen, og en betydelig del af disse faglige mål udmøntes gennem detsupplerende stof – så må det supplerende stof også være relevant for aftagerne. –Fm. for gymnasierådet, rektor Ove Poulsen: Abstrakt: Den nye gymnasiereformtager stilling til Science-for-all vs. - Science-for-the few. Reformen forholder sig tilbalancen mellem disse to begreber idet den naturvidenskabelige dannelse vil blivestyrket og bedre integreret i et fælles alment dannelsesbegreb. Derigennem tagerreformen indirekte stilling til naturvidenskabelige spidskompetencer i Ungdoms-uddannelserne, idet disse nu mere er overladt til markedskræfter. Spørgsmålet er,om der vil blive uddannet flere eller færre unge med en naturvidenskabelig spids-kompetence i det nye gymnasium?Der tegner sig et broget geografisk mønster med stærke naturvidenskabelige stu-die-retningspakker i dele af landet, medens der i andre dele af landet er en bekym-rende svækkelse af de naturvidenskabelige fagpakker. – Dekan, Erik M. Schmidt-Medlem af læseplansgruppen for HHX, Tage Bai Andersen.

14:30 – 15:15 Professor and Abel Prize Winner 2004, M. Atiyah.Title: The Interaction of Geometry and Physics: an Overview.

15:15 – 15:30 Coffee break.15:30 – 16:15 Professor G. Grubb, Copenhagen. Title: On the Atiyah-Patodi-Singer problem.

Abstract: In 1975, Atiyah, Patodi and Singer initiated the study of the index ofspectral boundary problems (defined by pseudodifferential projections) for first-order elliptic differential operators of Dirac type on compact manifolds withboundary, introducing in particular a nonlocal invariant known as the eta inva-riant. This has led to a wealth of further studies and extensions. We shall try togive an overview of some of the recent progress, mainly from an analysis pointof view.

16:15 – 17:00 Professor B. Ørsted, University of Aarhus. Title: Geometry of the Maslov index.Abstract: The Maslov index plays a key role in connection with the geometricasymptotics of solutions to partial differential equations. It also appears natural-ly in the representation theory of the symplectic group, and in gluing formulasfor certain elliptic boundary value problems. In this lecture we shall give somenew properties of the Maslov index, and at the same time generalize it, in bothfinite and infinite dimension; in particular we relate it to the theory of boundedcohomology of groups.

18:00 Dinner. Registration is needed

Please register for the meeting and the lunch/dinner at: http://www.dmf.mathematics.dk/Danish Mathematical Society, Chairman Johan P. Hansen, matjph@imf.au.dk

2921/04

Boganmeldelse Aftermathved Mogens Esrom Larsen

AFTERMATH

L�SNINGER

Opgaverne er hentet fra Peter Winkler� Mat�

hematical Puzzles� A K Peters� ����� Alle op�

gaver er ogs�a l�st af Ebbe Thue Poulsen�

Eksponentielt

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y� s�a m�a x l�se ligningen

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konvergent Af ligningen f�as� x �maxfy �

y jy � �g e�

e � For � � x � e�

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er f�lgen voksende� s�a den konvergerer mod

y � e� hvis den konvergerer mod e for x e�

e �

Men s�ttes den sidste eksponent i den n�te

potens op fra e�

e til e� s�a kollapser udtrykket

til e� der alts�a er en �vre gr�nse� Svaret p�a

opgaven er alts�a� at for x p� f�ar vi �� mens

vi ikke for noget x kan f�a ��

Soldaterne p�a marken

Opgaven er fra den � sovjetiske matematik�

konkurrence i Voronezh� �����

Et ulige antal soldater st�ar p�a en mark s�adan�

at alle parvise afstande er forskellige� Hver

soldat bliver bedt om at holde �je med den

n�rmeste anden soldat�

Vis� at der er mindst �en soldat� der ikke bliver

holdt et �je med�

Betragt de to soldater� hvis indbyrdes af�

stand er den mindste forekommende� De to

m�a holde �je med hinanden� Hvis der er en

anden soldat� der holder �je med en af disse

to� s�a er der en soldat� som to holder �je med�

S�a kan der ikke holdes �je med alle� Ellers kan

vi fjerne disse to og f�a problemet med to f�rre

soldater� Forts�tter vi reduktionen� m�a vi f�a

en til overs� n�ar der var et ulige antal ialt�

Intervaller og afstande

Opgaven er fra den ��� sovjetiske matematik�

konkurrence i Kishenew� �����

Lad S v�re foreningen af k disjunkte� afslut�

tede intervaller i enhedsintervallet ��� ��� An�

tag� at S har den egenskab� at for ethvert

reelt tal� d � ��� ��� �ndes to punkter i S med

afstanden d�

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Aftermath løsninger

30 21/04

Vis� at summen af l�ngderne af intervallerne

i S er mindst �

k�

Lad l�ngderne af de k intervaller v�re

s�� � � � � sk med summen s� Lad Iij v�re in�

tervallet af afstande mellem et punkt fra det

i�te og et punkt fra det j�te interval� L�ng�

den af dette interval er si � sj� Summerer vi

alle disse� forekommer si k � � gange� s�a ialt

f�as h�jst k � �s� Afstande mellem to punk�

ter i samme interval g�ar fra � til si� ialt f�as

h�jst s� S�a ialt f�as h�jst ks� Men vi ved jo� at

ks � �� s�a s � �

k�

Skabsl�agerne

I et omkl�dningsrum p�a et gymnasium er der

��� skabe p�a r�kke� nummererede fra � til

���� Da den f�rste elev ankommer��abner hun

alle skabene� Den anden elev lukker alle skabe

med lige nummer� Den tredie elev �abner de

lukkede og lukker de �abne� hvis nummeret

er deleligt med �� S�aledes bliver eleverne ved

med at �ndre tilstanden p�a nogle af skabene�

Dette forts�tter� indtil alle ��� elever er g�aet

gennem omkl�dningsrummet� Hvilke skabe

er nu �abne

Tilstanden af l�as nr� n �ndres af den k�te

student� netop hvis kjn� Nu kommer nogle

divisorer i par� s�a de h�ver hinanden� Men

kvadrattallene har jo divisorernepn� s�a de

�abne skabe har numrene �� �� �� ��� ��� ���

��� ��� �� og ����

Nuller� ettaller og totaller

Lad n v�re et naturligt tal� Vis� at a N

har et multiplum �� � hvis repr�sentation i

titalstystemet skrives udelukkende med brug

af � og �� og at b �n har et multiplum ��� hvis repr�sentation i titalstystemet skrives

udelukkende med brug af � og ��

Tallene �� ��� ���� osv� m�a indeholde � tal i

samme restklasse modulo N � Deres di�erens

skrives med kun � og �� ��� � � � ���� og er de�

lelig med N �

Hvis N er primisk med � og �� kan man

droppe ��erne og f�a et multiplum med lutter

��taller� David Gale�

Ved induktion viser vi� at for hvert k �ndes

et multiplum af �k skrevet med lutter ��er og

��er� �� � �� �� � �� Skriver vi � eller �

foran l�sningen� adderer vi enten �k�k eller

�k���k� Begge ny tal er derfor stadig delelige

med �k� Deres di�erens er �k�k� s�a et af de ny

m�a v�re delelig med �k���

Summer og di�erenser

Opgaven er fra den �� sovjetiske matematik�

�

3121/04

konkurrence i Riga� �����

Givet �� forskellige positive tal� Vis� at man

kan v�lge to af dem sadan� at ingen af de

andre er lig med hverken sum eller dierens

af de to valgte�

Lad tallene v�re x� � x� � � � � � x��� Hvis

x�� ikke kan indga i et sadant par� sa ma der

til hvert xi v�re et xj� sa xi � xj � x��� Dvs�

at xi � x���i � x��� Betragter vi x�� sammen

med x�� � � � � x��� sa er summen af hvert par

st rre end x��� sa hvis ingen af disse par kan

bruges� ma de ogsa opfylde x��i�x���i � x���

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NYE OPGAVER

Summer af br�ker

Givet et naturligt tal� n � �� Dan summen

af alle br ker� �

pq� hvor p og q er indbyrdes

primiske� � � p � q � n� og p� q � n�

Vis� at resultatet altid er lig med �

��

De � cirkler

� cirkler har forskellige radier� og ingen lig�

ger inden i en af de andre� F�llestangenterne

til hvert par af cirkler sk�rer hinanden i et

punkt� vi kan kalde �focus��

Vis� at de � foci ligger pa en ret linie�

Y�er i planen

Et Y er en �gur dannet af � liniestykker med

et f�lles endepunkt� Vis� at der h jst er t�l�

lelig mange disjunkte Y�er i en plan�

Cirkel�

Vis� at blandt m�ngder med diameter � er

cirklen den� der har st rst areal�

Vinkler i rum

Vis� at blandt en m�ngde af mere end �n

punkter i Rn �ndes altid �� der danner en

spids vinkel�

Kasser

Lad n � N� En kasse er et krydsprodukt af n

endelige m�ngder� A�� � � � � An� En kasse B �

B� � � � � �Bn kaldes en egentlig underkasse�

hvis hver delm�ngde er �gte� Bi � Ai�

Kan det forekomme� at en kasse kan deles i

f�rre end n egentlige disjunkte underkasser�

Rationelt

En m�ngde S indeholder tallene � og �� samt

middelv�rdien af enhver endelig delm�ngde

i S� Vis� at S indeholder alle rationale tal i

intervallet ��� ���

�

Fra Abelprisuddelingen 2004

En eller anden billedtekst? Kilde: http://www.math.uio.no/~reichelt/Bilder2004/