Societat Catalana de Matematiques

President: Carles Casacuberta VergesVicepres.: Josep Grane ManlleuSecretary: Josep Maria Font LlovetTreasurer: Joan C. Artes FerragudMembersat Large: Jaume Amoros Torrent

Martı Casadevall PouAntoni Goma NasarreIgnasi Mundet Riera

Carles Romero ChesaOriol Serra Albo

Enric Ventura CapellJoan Verdera Melenchon

IECDelegate: Joan Girbau i Bado

Contact: Carrer del Carme, 4708001 BarcelonaPhone: 932 701 620

Fax: 932 701 180

E-mail: scm@iec.catSecretary: Nuria Fuster

from 10.00 to 17.00Phone: 933 248 583

SCM/NotıciesAugust 2006. Special issue

Edited by:Societat Catalana de Matematiques(filial de l’Institut d’Estudis Catalans)

Chief Editor:Enric Ventura Capell

enric.ventura@upc.edu

Design: Teresa Sabater

LATEX: Maria Julia

Cover photo:

ICM 2006 Postcards

ISSN: 1696-8247Legal Deposit: B.9480-2003

Index

Welcome 1

Catalonia and the ICM 2006 2

Presentation of the SCM 2

The Parliament of Catalonia supports the ICM 2006 3

The Catalan Mathematical Community 6

Mathematics in Catalonia: 1996–2002 6

The Centre de Recerca Matematica, Barcelona 17

The Ferran Sunyer i Balaguer Prize 21

Mathematics at the UB 23

Mathematics at the UAB 24

Mathematics at the UPC 26

Mathematics at ICREA 28

FEEMCAT 29

Societat Catalana de Matematiques 31

SCM Funds 31

The Czech-Catalan venture: a fruitful collaboration 32

La prova Cangur, the Catalan Kangaroo 33

Brief history of the SCM/Notıcies 36

Reproduction of Featured Articles 38

Jean-Pierre Serre, doctor honoris causa at the UB 38

Xavier Tolsa received a prize at 4ECM 42

Miguel de Guzman, in memoriam 43

Problem Section 44

Mathematics PhD theses in Catalonia 46

Welcome

The International Congress of Mathematiciansis without doubt the most important mathe-matical event in the world. Having the oppor-tunity to organise an ICM is an unparalleledhonor for a community of mathematicians. Thishonor has been bestowed to Spain in 2006.

The Catalan Mathematical Society (So-cietat Catalana de Matematiques, SCM) isproud to participate in the organisation of thisCongress. The SCM president is one of the vi-cepresidents of the Executive Committee of theCongress, together with the president of RSMEand former presidents of SEIO and SEMA.This privilege was agreed in 2002 as a recog-nition of the fact that the four societies RSME,SCM, SEIO and SEMA made up the SpanishIMU Committee when it was entrusted withthe responsibility to organise the ICM 2006 bythe General Assembly of IMU in Shanghai. Atpresent, the Spanish IMU Committee encom-passes seven societies, after the incorporationof FESPM, SEHCYT and SEIEM in 2004.

This special issue of the journalSCM/Notıcies is dedicated to the ICM 2006and has been designed as a homage to theCongress and as a demonstration of enthusi-asm of the Catalan mathematical community.The collective response to the call for contribu-tions to this issue has been rapid and generous.We are thus delighted to present Mathematicsin Catalonia on the occasion of the ICM 2006.

This issue contains a presentation of severalinstitutions that lead mathematical researchand education in Catalonia. In addition, quan-titative and qualitative information is given inthe report entitled Mathematics in Catalonia1996–2002 , which was elaborated by the Insti-tute for Catalan Studies and published as anarticle of the Institute’s journal Contributionsto Science in 2005. A brief summary of interna-tional cooperation is given, together with otheraims and actions of the Catalan Mathemati-cal Society. This offers altogether a substantialoverview of the community’s activity.

The institutions presented in this issue donot exhaust of course the list of centres wheremathematical research, education or divulga-

tion is made in Catalonia. Among the univer-sities, only the three that award a degree inMathematics are displayed here. A number ofother universities exist in Catalonia, and allof them host active mathematicians. Moreover,if the scope of our survey is enlarged to thewhole territory where the Catalan language isspoken and the Catalan culture is alive in itsmany forms, then the list of universities to bementioned would be much longer. The readeris referred to the website of the Institute JoanLluıs Vives (http://www.vives.org), a networkof 20 universities within the territories of Cata-lan culture, for complete information.

Besides the universities and the Centre deRecerca Matematica (CRM), other institutionswork for Mathematics in Catalonia. In fact,a big effort is devoted to the promotion ofMathematics at schools. This task is under-taken by the SCM and also by several as-sociations that are assembled together in theFederacio d’Entitats per a l’Ensenyament deles Matematiques a Catalunya (FEEMCAT).The FEEMCAT shares some activities with theSCM, principally the ESTALMAT programmefor the care of talent among mathematicallygifted students aged 11 to 13, a successful ini-tiative started by Miguel de Guzman in 1998 inMadrid.

We certainly do not want to forget any insti-tution deserving to be mentioned in this issue.There are many, since there are in fact manypeople who love Mathematics and work for itsprogress and divulgation. For this reason, we donot attempt to list all of them, and apologizehere for any deficiency in emphasis that mayhave occurred.

The ICM 2006 is accompanied by a largenumber of satellite events (more than sixty).This is another demonstration of the impor-tance of the Congress and its strong effect onthe mathematical people and institutions, notonly of Spain, but also of some of the surround-ing countries. The benefit of this Congress hasalready been clearly perceived and is expectedto become even more evident in the comingmonths and years.

Carles CasacubertaSCM President

1

The SCM remarks that the opinions expressedand the comments made in the following pagesbelong solely to the authors of the correspond-ing articles and/or to the institutions they rep-resent. The SCM does not necessarily sharethese opinions or comments.

The SCM wishes however to sincerely thankall the contributors to this special issue of

SCM/Notıcies for their prompt and effectiveresponse to the call for contributions. We apol-ogise for the fact that, because of calendar con-straints, the time given to authors for writingtheir contributions has been much less than theusual one. If, after all, the present issue is editedand printed is, doubtless, thanks to the effortof all of them.

Enric VenturaEditor of SCM/Notıcies

Catalonia and the ICM 2006

Presentation of the SCM

The Catalan Mathematical Society aims to pro-mote all aspects of Mathematics in the territo-ries where the Catalan language is official ornormally used. It was established in 1931 as asection of the Society for the Promotion of Sci-ences, which had been created by the Institutefor Catalan Studies (IEC). In 1986, the Societyfor the Promotion of Sciences split into threedifferent societies, namely Chemistry, Mathe-matics, and Physics. Nowadays, the SCM is oneof the 26 member societies of the IEC. Withmore than 1,000 individual members, most ofwhom are researchers or teachers at one orother level of the educational system, it is oneof the most active, prestigious, dynamic and in-fluential among the scientific and scholarly so-cieties affiliated to the IEC.

The SCM is a member of the EuropeanMathematical Society (EMS) since 1992. Aftera successful bid, it was entrusted by the EMSwith organising the Third European Congressof Mathematics in Barcelona in July 2000. Thiswas a major event, which gathered togethermore than 1,500 participants. The proceedingsof this congress were published by BirkhauserVerlag in the Progress in Mathematics series(volumes 201 and 202).

The SCM organises activities of manykinds, publishes books and journals, givesprizes, and represents the Catalan mathemat-ical community before educational authori-ties and other institutions, both at home andabroad. Some of the courses and lectures organ-

ised by the SCM are aimed at researchers, whileothers are geared toward secondary schoolteachers or a general mathematical audience.

The SCM holds reciprocal agreements withseveral societies from outside Spain, includ-ing the American Mathematical Society, theSociete Mathematique de France, and theCeska matematicka spolecnost (Czech Mathe-matical Society). Several joint events have beenorganised with some of these institutions. InMay 2005, the First Czech-Catalan Congressof Mathematics took place in Prague and gath-ered together more than 100 participants. Thesecond event of this series will be held inSeptember 2006 in Barcelona. In 2003, theSCM and the RSME jointly organised a two-day meeting on Cryptography in Barcelona.The Spanish societies RSME, SCM, SEIO andSEMA organised a major conference in Valen-cia in 2005, which served as an advertisementand a collaborative rehearsal for the ICM 2006.

The SCM awards an annual prize for stu-dents (the Evariste Galois prize), which is givento the author of the best research or essay ona mathematical topic, usually at a postgrad-uate level. The Institute for Catalan Studiesalso awards a biannual prize in Mathematicsfor the best PhD thesis written in Catalan (theJosep Teixidor prize) and an annual interna-tional prize (the Ferran Sunyer i Balaguer prize)for a mathematical monograph of an exposi-tory nature in English that presents the latestdevelopments in an active area of research in

2

Mathematics in which the applicant has madeimportant contributions. The prize, amountingto 12,000 euros, is financed by the Ferran Sun-yer i Balaguer Foundation of the IEC. The win-ning monographs are published in the Progressin Mathematics series of Birkhauser Verlag.

In 1996 the SCM took part for the first timein the annual contest of the international associ-ation Le Kangourou sans Frontieres. This com-petition, called Cangur in Catalan, is aimed atstudents aged 14 to 18, and has become ex-tremely popular in Catalonia, where the 10thanniversary of the event was happily celebratedin 2005. In 2006, close to 16,000 selected stu-dents from 487 centres of Catalonia, Valen-cia and Andorra have participated in the Can-gur contest. The SCM organises other activitiesaimed at secondary school students, namely, anonline problem competition awarding a prize tothe centre whose team answers all the questionsin a minimum time, and a literary contest of-fering a prize for the best written compositionin Catalan with Mathematics as its theme.

A selection phase for the InternationalMathematical Olympiad takes place every yearin December in Barcelona. Since 2003 the SCMis also responsible for the local organisation ofthe Olimpiada Iberoamericana Universitaria deMatematicas.

The Butlletı de la Societat Catalana de Ma-tematiques is a journal devoted to mathemat-ical research and high-level expository articleswritten in Catalan. A volume consisting of twoissues is printed every year. Since 1996 the SCMalso publishes the newsletter SCM/Notıciesand, following the launch in 2005, a new se-ries of monographs entitled Publicacions de la

SCM is now freely offered in electronic format.All volumes considered so far in this series arewritten in Catalan, although other languagesare allowed.

Thus, the SCM shares the main goals of theInstitute for Catalan Studies, namely, to pro-mote scientific research, in particular researchrelated to all aspects of Catalan culture, whilefurthering the development of society in gene-ral and, when necessary, acting as an advisorto the Catalan government and other institu-tions. In order to achieve this, and in additionto the main activities described above, the SCMorganises yearly debates and lectures address-ing different aspects of science and culture. Onseveral occasions, SCM officers have led actionsaimed at raising the profile of Mathematicsamong the general public, especially during theWorld Mathematical Year 2000. The interest ofthe SCM in launching wide-ranging actions isillustrated by several events, the most recentof which was a Joint Mathematical Weekendheld with the European Mathematical Societyin September 2005 in Barcelona.

For more information, see the SCM websitehttp://scm.iec.cat.

Carles CasacubertaSCM President

The Parliament of Catalonia supports the ICM 2006

In March 2006, the Commission of Culture ofthe Parliament of Catalonia adopted a resolu-tion of support for the ICM 2006. The initia-tive that led to this resolution was started inJuly 2005 by Carme-Laura Gil, a former Cata-lan minister of Education and currently a mem-ber of the Commission of Culture of the Parlia-

ment. The chief officers of the Catalan Mathe-matical Society were called by the Commission,where they explained the aims and importanceof the Congress and gave updated informationabout it. After this session, the following textwas agreed by all political parties representedat the Parliament in November 2005.

3

In August 2006, Madrid will host the Interna-tional Congress of Mathematicians (ICM), aquadrennial event that is the most important ofall mathematical events held in the world since1897. This is the first time that an ICM is or-ganised in Spain. This honor and responsibilitywas awarded by the International Mathemati-cal Union in 2002 to a committee formed byfour mathematical societies, one of which is theCatalan Mathematical Society.

This is an excellent occasion to increase theinfluence of the Catalan mathematical commu-nity among Spanish scientists, while reinforcingits international prestige and its representationcapability, thanks to the worldwide impact of theICM. At the same time, this is an opportunityto make Mathematics more visible within theCatalan society and especially among the stu-dent population.

It is convenient and even necessary that theCatalan Government assumes these goals asproper in the framework of its scientific, educa-tional and cultural policy, by programming andstarting suitable actions on the occasion of thisCongress.

On these grounds, the Parliament adoptedthe following resolution on March 1, 2006.

The Parliament of Catalonia asks the Gov-ernment:

a) To give institutional and financial supportto the Catalan universities, research centres,the Catalan Mathematical Society and theFEEMCAT in order to make possible theirjoint participation at the ICM 2006 as na-tional representatives of Catalonia.

b) To give institutional and financial supportto the Catalan Mathematical Society, theFEEMCAT and the Catalan universities inorder that they ellaborate jointly a pro-gramme of actions that, starting from the ex-isting initiatives, leads to a further develop-ment of the mathematical capabilities of thestudents of Catalan educational centres andtheir interest for Mathematics and all sci-ences.

c) To give institutional support to the CatalanMathematical Society in order to promote itspresence at international entities.

The recognition awarded to the ICM 2006by such a high political body is emphasised and

acknowledged. It is of course much deserved.On the more practical side, this support hasmade possible that the main mathematical cen-tres and associations of Catalonia are jointlypresent at the ICM 2006 by sharing a boothand displaying information on their aims andactivities.

The building of the Parliament of Catalonia

History of the Catalan Parliament

(Text retrieved and summarised from the Par-liament’s website.)

The Parliament of Catalonia has its roots in thedistant past. These roots were the Assembliesof Peace and Truce and the Court of Counts.The Catalan Court of Counts was founded inthe eleventh century in line with the Frenchroyal Curia, as the counts of Barcelona consol-idated their position as reigning princes. It wasmade up of civil magnates and the clergy, highcouncilors and judges. The Assemblies of Peaceand Truce were promoted by the church andtaken on by the reigning power, the counts ofBarcelona, with the aim of ensuring breaks inthe continual private wars between feudal lords,establishing a system of guarantees and com-pensation for the clergy and civil populationand also making agreements of a fiscal nature.

During the reign of James I the Conqueror(1213–1276), the Court of Counts became theGeneral Courts of Catalonia, by gradually in-creasing the number of members invited and,above all, by consolidating the incorporation ofthe bourgeois state, represented by leading menfrom towns and cities. But the decisive stepwas taken during the reign of his son, Peter IIthe Great (1276–1285), when, at the BarcelonaCourts of 1283, the system of negotiated rule

4

characteristic of Catalan medieval and modernconstitutional law was established.

The Parliament of Catalonia

During the reign of Peter III the Ceremo-nious (1336–1387), the institution of the Dipu-tacio del General was created (the meeting ofthe Courts was called the General de Catalu-nya), with delegated powers to collect and ad-minister the funding the Courts granted to theking. Gradually, the Diputacio del General orGeneralitat became more autonomous and ac-cumulated executive and governmental powers,and by the sixteenth and seventeenth centuriesit was acting as the government for the Princi-pality.

The first monarch of the Bourbon dynastyin Spain, Philip V (1700–1746), respected therights of the Catalan people by holding theCourts in 1701 and 1702. But, in the war of Suc-cession, Catalonia opted to support the arch-duke Charles of Austria and recognized himas king. On losing the war and with the con-sequent fall of Barcelona in 1714, the victorabolished all public Catalan laws and the insti-tutions representing it via the Decree of NovaPlanta or New Proposition (1716), the most im-portant of these institutions being the Courts.

Since that time, the Catalan people have as-pired to recover their lost freedom. The Man-comunitat de Catalunya or Commonwealth ofCatalonia from 1913 to 1925 was the first in-stitutional step in recovering self-government.Once the Second Spanish Republic was pro-claimed, a provisional Catalan government wasset up in 1931, taking the name of the Genera-litat de Catalunya in remembrance of the for-

mer institution. Chosen by the majority of theCatalan people, the Statute of Autonomy thatdefinitively legitimized and structured Catalanself-government was approved by the Courts ofthe Republic in 1932.

The parliamentary tradition of Catalonia,interrupted in 1714, was therefore reinitiated.The first elections for the Parliament of Cat-alonia took place on November 20, 1932, andthe constitutive sitting was held on December6 of the same year. The first President of theParliament was Lluıs Companys.

Franco’s dictatorship destroyed the hopesof the Catalan people for self-government fora long time (1939–1975). During the process ofre-establishing democracy in the Spanish State,Catalonia was able to recover the Generali-tat, provisionally, via the return of its Presi-dent (1977), Josep Tarradellas, who had keptthe institution alive in exile. Once the SpanishConstitution was approved in 1978 and the newStatute of the Autonomy of Catalonia was ap-proved and endorsed in 1979, the Generalitatwas re-established definitively. The first elec-tions for the re-established Parliament of Cat-alonia took place on March 20, 1980, and theconstitutive sitting was held on April 10 of thesame year.

Room sessions of the Parliament

Five political parties are currently repre-sented at the Parliament of Catalonia (Con-vergencia i Unio, Socialistes - Ciutadans pelCanvi, Esquerra Republicana de Catalunya,Partit Popular, and Iniciativa per CatalunyaVerds - Esquerra Alternativa), after the elec-tions held in November 2003.

5

The Catalan Mathematical Community

This section gives a brief overview of Mathe-matics in Catalonia. Without intending to beexhaustive (and with apologies for any deficien-cies in this regard) we provide a list of arti-cles on the main Catalan institutions and or-ganizations connected with the world of Math-ematics. The first article is a reproduction ofthe report on mathematical research in Catalo-nia from 1996 to 2002 that was published lastyear in the journal Contributions to Science ofthe Institute for Catalan Studies. The readerwill find a detailed qualitative and quantitativeanalysis of the state of mathematical researchin Catalonia today.

The second article is about the Centre deRecerca Matematica, a prestigious research cen-tre created in 1984, and the next one is aboutthe Ferran Sunyer i Balaguer prize for mathe-matical research.

The next three articles are about the threeCatalan universities that award degrees inMathematics and have their own PhD pro-gramme in this field, namely the Universityof Barcelona, the Autonomous University ofBarcelona and the Technical University of Cat-alonia. We point out that there are also othermathematicians pursuing active research whobelong to other universities than the three dis-cussed above. The following article is aboutthe recently created ICREA, the body ofhigh-level research scientists (mathematiciansamong them) supported by the Catalan gov-ernment.

Finally, the last article is about FEEM-CAT, one of the main Catalan organizationsdevoted to the teaching and training of stu-dents in Mathematics, and to the divulgationof Mathematical Education.

Mathematics in Catalonia: 1996–2002

Report published in Contributions to Science 3

(2005), no. 1, 103–110, Institut d’Estudis Cata-lans, Barcelona. An extended version of thisreport is available in Catalan on request togguasp@mat.uab.es. [Reports de la recerca aCatalunya: Matematiques. Barcelona. Institutd’Estudis Catalans. 2004.]

Introduction

The aim of this article is to present the mainconclusions of the report on research in Catalo-nia for the area of Mathematics. The report waspublished in 2005 by the Institute for CatalanStudies. This is a report of the scientific activityin Catalonia in the field of Mathematics duringthe period 1996–2002. It deals with two aspectsof mathematical research. First, it provides ananalysis of the country’s research potential, asdefined by its universities and other centers ofinvestigation, their staff, as well as the eco-nomic resources that they obtain for research.Second, it offers an assessment of the scientificproductivity of researchers from these institu-tions, as measured by analyzing, both quanti-

tatively and qualitatively, articles published inspecialized Mathematics journals and doctoraltheses submitted to universities. The data fromour study are often compared with those fromthe report on research in Mathematics corre-sponding to the period 1990–1996. A summaryof that information and the conclusions drawnfrom that earlier report can be found in the ar-ticle by Joan Girbau entitled Mathematics inCatalonia [Contributions to Science 2 (2001),no. 1, 111–121].

Universities and their human resources

In Catalonia, research in Mathematics is mostlycarried out at universities. In this section, weprovide quantitative information about math-ematicians working at these institutions. How-ever, only mathematicians working in Mathe-matics departments have been considered; thus,mathematicians that carry out their researchin other departments, such as Economics andComputer Science, are not included in thisstudy. However, their productivity is discussedin the data reported in the section entitled Sci-

6

entific productivity. Table 1 presents the fig-ures concerning the staff at Catalan universi-ties. The data correspond to the academic year2001–2002, which is the last year of the pe-riod analyzed. During the period 1996–2002,there was an increase of 15 % in the numberof teaching and research staff employed in theMathematics departments of Catalan univer-

sities. This total increase, which was not uni-form for all categories, shows that there hasnot been stagnation during this period; rather,the number of fellowship holders at the Univer-sity of Barcelona, the Autonomous Universityof Barcelona, and the Technical University ofCatalonia has increased significantly.

Permanent Term-contract FellowshipUniversity teaching staff teaching staff holders Total

Autonomous University of Barcelona 55 48 10 113

University of Barcelona 79 30 15 124

University of Girona 14 9 2 25

University of Lleida 17 10 — 27

Technical University of Catalonia 193 79 22 294

University Pompeu Fabra 16 8 — 24

University Rovira i Virgili 14 — 3 17

Total 388 184 52 624

Table 1: Permanent teaching staff, term-contract teaching staff, and fellowship holders

Scientific productivity

Doctoral thesesTable 2 shows the number of doctoral theses onMathematics submitted to Catalan universitiesbetween the academic year 1995–1996 and theacademic year 2001–2002. There was a remark-able increase (ca. 37 %) in the number of thesessubmitted during the more recent period com-pared to the period 1990–1996. The increase inthe number of fellowship holders (158 %) dur-ing this time most likely accounted for the in-crease in the number of theses. Despite this in-crease, the rate of theses supervised by perma-nent teaching staff was 0.45, which is markedlylower than during the period 1990–1996, indi-cating a decrease in productivity in relation tothe potential of Catalan universities.

University Theses

Autonomous University of Barcelona 32

University of Barcelona 46

University of Girona 1

University of Lleida 0

Technical University of Catalonia 78

University Pompeu Fabra 5

University Rovira Virgili 0

Total 162

Table 2: Doctoral theses submitted to each uni-versity during the period 1995–2002

Scientific publicationsScientific productivity in the field of Mathemat-ics in Catalonia during the period 1996–2002is analyzed from two points of view. First, theevolution of productivity of Catalan Mathemat-ics is compared with that worldwide; second,productivity in Catalonia is compared withthat of countries having the highest produc-tivity in Mathematics. Catalan productivity iscompared with worldwide productivity, first ofall, by taking into account, for each year of theperiod analyzed (Table 3), all papers publishedin Mathematics. After this, the data are evalu-ated following the UNESCO codes (Table 4) forthe classification of subfields in Mathematics.All the studies carried out include only workthat was reviewed in the journal Mathemati-cal Reviews during the period 1996–2002 andwhich have been published in journals, congressproceedings, or books. Note that, after a manu-script is sent for publication, it takes an averageof 2-3 years for it to be listed in Mathemati-cal Reviews. The papers that enter in the ac-count of Catalan productivity are those havingat least one author from a Catalan institution.Table 4 shows the distribution of papers pub-lished around the world and in Catalonia, inaccordance with the UNESCO codes. The dif-ferences that can be seen between the totals in

7

Table 3 and those in Table 4, both in Cataloniaand around the world, are due to the fact thatmany articles in Mathematical Reviews corre-spond to what could be called border fields inMathematics, such as physics and economics,and thus have UNESCO codes different fromthose we have considered. The total numberof publications during the period 1996–2002 in

Catalonia is 2, 584, compared to 472, 633 world-wide. Catalan productivity in Mathematics rep-resents 0.5 % of global production, a percentagehigher than during the previous period (0.4 %).In addition, compared with the period 1990–1996, the number of publications in the worldhas increased by a factor of 1.28, while in Cat-alonia it has increased by a factor of 1.71.

1996 1997 1998 1999 2000 2001 2002 Total

Catalonia

293 321 397 395 384 383 411 2,584

World

63,064 66,590 67,886 70,025 70,217 69,170 65,681 472,633

Table 3: Total number of mathematical publications in Cataloniaand around the world

Area and UNESCO code World % Catalonia %

Logic (1101–1104) 11,627 3.16 132 6.45

Algebra (1201) 34,374 9.34 149 7.28

Analysis (1202) 92,179 25.05 501 24.46

Computer Science (1203) 23,344 6.35 225 10.99

Geometry (1204) 32,285 8.78 176 8.59

Number Theory (1205) 14,505 3.94 58 2.83

Numerical Analysis (1206) 22,228 6.04 50 2.44

Operations Research (1207) 6,560 1.78 12 0.59

Probabilities (1208) 16,702 4.54 125 6.10

Statistics (1209) 23,100 6.28 87 4.25

Topology (1210) 15,553 4.23 65 3.17

Total 292,457 100.00 1,580 100.00

Table 4: Partial distribution of mathematical publicationsaround the world and in Catalonia according to UNESCOcodes

Analysis of the distribution according toclassification of the works in the different fields,as defined by UNESCO codes, shows that therelative volume of Catalan production is com-parable to that worldwide in all eleven areasconsidered, except Logic, Numerical Analysis,and Operations Research. In the Logic cate-gory, the percentage of Catalan productivity isalmost twice as high as the global one, while inNumerical Analysis and Operations Researchit is less than half of total global production.Productivity in Mathematics in Catalonia was

compared with that in countries having thehighest levels of productivity, defined as thosecountries with more than 2, 000 publicationsreviewed in Mathematical Reviews during pe-riod 1996–2002. The list of Table 5 comprises33 countries, almost all of them considered inthe 1990–1996 study. Most of the countries witha high degree of development appears in thislist. In Table 5, the 2002 population is givenin millions of persons and the 2002 GDP indollars per inhabitant. Publications is the sumof articles, proceedings and books. The data

8

from the 33 countries are sorted according tothe figures for the column Number of publica-tions by million inhabitants, i.e., from the high-est to the lowest. An analysis of these figures(and those corresponding to the previous pe-riod) shows that mathematical production inCatalonia in 1996–2002 relative to populationincreased compared to the period 1990–1996 bya factor of 1.66. Note that Israel ranks first,far ahead of the other countries, while othermathematically productive countries, such asJapan or Russia, occupy lower positions. Cat-alonia ranks 15, which is the same positionthat it occupied during the previous period, andit has surpassed other countries, including theUnited States, Germany, the United Kingdom,and Italy. The relationship between GDP andpublications by million inhabitants is heteroge-neous. For some countries this relation tends tobe linear. However, in the majority of countries,the number of articles per million inhabitants ishigher than would be expected if the relation-ship was directly proportional.

Analysis of the quality of publicationsIn the previous section, we considered all ar-ticles on Mathematics published in Cataloniaand worldwide, taking into account only thequantity, not the quality of these articles. Wenow turn our attention to the quality of thework produced. One indicator of the quality ofa scientific article is its citation index, which isbased on the number of citations by other au-thors over a given period of time. However, inour opinion, this index, which may be adequatefor many scientific disciplines, should not be sogood for Mathematics. We believe that there isno direct relationship between the quality andsignificance that an article may have for thedevelopment of Mathematics and its low termcitation frequency. Thus, we have chosen to usean indirect measure of the quality of an arti-cle: the level of prestige of the journal in whichit is published. Needless to say, any approachto measuring a journal’s prestige can be objec-tionable, but our aim is to obtain data thatallow comparisons with other countries to beestablished, rather simply reflecting the level ofexcellence of Catalan productivity in any field,and using as many approaches as possible.

We have defined prestigious journals asthose with the highest impact factor amongthose with an index of persistence greater than

10 years. We then added to this group a limitednumber of specialized journals in particular ar-eas to ensure that all areas were represented.The total number of articles published world-wide in the journals listed below (referred toas outstanding articles), over the period 1996–2002, amounts to 6.16 % of the total numberof articles published in all journals reviewed inMathematical Reviews over the same period.

List of prestigious journals:

• Acta Mathematica• Advances in Applied Probability• Advances in Mathematics• American Journal of Mathematics• Annales de l’Institut Fourier• Annals of Mathematics• Annales Scientifiques de l’Ecole Normale

Superieure• Archive for Rational Mechanics and Analysis• Arkiv for Mathematik• Artificial Intelligence• Biometrika• Bulletin of the American Mathematical

Society• Celestial Mechanics and Dynamical

Astronomy• Combinatorica• Commentarii Mathematici Helvetici• Communications on Pure and Applied

Mathematics• Duke Mathematical Journal• Econometrica• Ergodic Theory and Dynamical Systems• Indiana University Mathematical Journal• Inventiones Mathematicae• Journal d’Analyse Mathematique• Journal de Mathematiques Pures et

Appliquees• Journal fur die Reine und Angewandte

Mathematik• Journal of Algebra• Journal of Differential Equations• Journal of Differential Geometry• Journal of Functional Analysis• Journal of Mathematical Analysis and

Applications• Journal of Mathematical Biology• Journal of Number Theory• Journal of the American Mathematical

Society• Journal of the American Statistical

Association• Journal of the Royal Statistical Society• Mathematics of Computation• Mathematische Annalen• Mathematische Zeitschrift• Nonlinearity

9

• Proceedings of the London MathematicalSociety

• SIAM Journal on Applied Mathematics• SIAM Journal on Control and Optimization• SIAM Journal on Discrete Mathematics• SIAM Journal on Numerical Analysis

• Studies in Applied Mathematics• The Annals of Probability• The Annals of Statistics• Topology• Transactions of the American Mathematical

Society

2002 Publications/Country Population 2002 GDP Art. Publications Population

1 Israel 6 16,710 6,828 8,344 1,284.83

2 Switzerland 7 38,930 3,425 4,298 594.67

3 Canada 31 22,300 13,627 16,324 519.64

4 France 59 22,010 23,779 29,878 502.64

5 Singapore 4 20,690 1,863 2,072 497.60

6 Finland 5 23,510 2,004 2,516 483.94

7 Denmark 5 30,290 2003 2,572 478.67

8 Belgium 10 23,250 3,985 4,823 467.34

9 Austria 8 23,390 2,866 3,716 456.46

10 Sweden 9 24,820 3,246 4,024 450.92

11 The Netherlands 16 23,960 5,555 7,164 443.76

12 Australia 20 19,740 7,337 8,590 438.69

13 Czech Republic 10 5,560 3,404 4,224 413.72

14 Hungary 10 5,280 3,573 4,153 408.52

15 Catalonia 6 20,458 2,057 2,525 388.10

16 Germany 82 22,570 24,290 31,969 387.53

17 United Kingdom 59 25,250 18,668 22,715 385.93

18 Italy 58 18,960 18,365 21,920 378.46

19 USA 288 35,060 79,824 97,432 337.87

20 Spain∗ 34 16,475 9,195 11,158 327.79

21 Bulgaria 8 1,790 2,072 2,511 319.14

22 Greece 11 11,660 2,864 3,138 295.18

23 Poland 39 4,570 8,810 10,299 266.63

24 Rumania 22 1,850 5,238 5,943 265.85

25 Japan 127 33,550 22,902 26,251 206.47

26 Russia 144 2,140 22,211 25,764 178.83

27 Ukraine 49 770 6,697 7,632 156.66

28 Korea 48 9,930 6,189 6,555 137.59

29 Brazil 174 2,850 4,923 5,482 35.50

30 China 1,281 940 43,087 44,839 35.00

31 Turkey 70 2,500 2,202 2,348 33.54

32 Mexico 101 5,910 2,283 2,873 28.44

33 India 1,048 480 9,406 10,400 9.92

*Without Catalonia

Table 5: List of countries arranged by total mathematicalproduction

On this basis, we have counted the numberof outstanding articles published in any of thejournals listed above by authors in the 33 coun-tries listed in Table 5. The results are shown inTable 6, in which the countries have been sortedaccording to the ratio of number of outstand-

ing articles by million inhabitants. In Table 7the countries are sorted according to the per-centage of articles published in the above listof prestigious journals compared with the totalnumber of articles published.

10

Outstanding

Outstanding articles/

Country Population articles Population

1 Israel 6 854 131.50

2 France 59 3,179 53.48

3 Switzerland 7 352 48.70

4 Canada 31 1,351 43.01

5 Denmark 5 222 41.32

6 Catalonia 6 254 39.04

7 Sweden 9 343 38.44

8 USA 288 11,020 38.21

9 Australia 20 679 34.68

10 United Kingdom 59 2,004 34.05

11 Finland 5 167 32.12

12 Singapore 4 133 31.94

13 Germany 82 2,584 31.32

14 Belgium 10 297 28.78

15 The Netherlands 16 456 28.25

16 Austria 8 225 27.64

17 Spain∗ 34 915 26.88

18 Italy 58 1,377 23.77

19 Hungary 10 213 20.95

20 Greece 11 215 20.22

21 Czech Republlic 10 172 16.85

22 Poland 39 440 11.39

23 Japan 127 1,396 10.98

24 Bulgaria 8 80 10.17

25 Rumania 22 169 7.68

26 Korea 48 328 6.83

27 Russia 144 489 3.40

28 Brazil 174 448 2.57

29 Ukraine 49 123 2.51

30 Mexico 101 135 1.34

31 China 1,281 1,613 1.26

32 Turkey 70 64 0.91

33 India 1,048 361 0.34

*Without Catalonia

Table 6: List of countries by number of outstanding articles

11

Outstanding articles/ Outstanding articles/

Country total articles Country total articles

1 USA 13.80 18 Italy 7.50

2 France 13.37 19 Belgium 7.45

3 Israel 12.50 20 Singapore 7.14

4 Catalonia 12.34 21 Japan 6.10

5 Denmark 11.08 22 Hungary 5.96

6 United Kingdom 10.73 23 Mexico 5.91

7 Germany 10.63 24 Korea 5.30

8 Sweden 10.57 25 Czech Republic 5.05

9 Switzerland 10.27 26 Poland 5.00

10 Spain∗ 9.95 27 Bulgaria 3.86

11 Canada 9.91 28 India 3.84

12 Australia 9.25 29 China 3.74

13 Brazil 9.10 30 Rumania 3.23

14 Finland 8.33 31 Turkey 2.91

15 The Netherlands 8.20 32 Russia 2.20

16 Austria 7.85 33 Ukraine 1.84

17 Greece 7.78

*Without Catalonia

Table 7: List of countries ordered by outstanding articles/total articles (%)

USA ES FR UK IT DE CA NL MX AR JP CN RU BE Cat

USA 692 2,281 2,507 1,642 2,812 2,739 846 235 148 1,039 1,773 893 470 221

Spain 692 337 260 282 211 116 101 96 93 39 77 150 216 214

France 2,281 337 532 916 884 496 200 89 49 269 167 606 256 136

UK 2,507 260 532 490 810 480 261 61 20 213 339 365 161 87

Italy 1,642 282 916 490 677 290 185 34 43 169 120 512 146 70

Germany 2,812 211 884 810 677 499 401 97 25 350 408 984 183 67

Canada 2,739 116 496 480 290 499 169 51 16 228 601 204 80 47

The Netherlands 846 101 200 261 185 401 169 5 7 84 78 116 143 33

Mexico 235 96 89 61 34 97 51 5 5 72 14 106 9 33

Argentina 148 93 49 20 43 25 16 7 5 3 1 1 9 33

Japan 1,039 39 269 213 169 350 228 84 72 3 371 193 47 30

China 1,773 77 167 339 120 408 601 78 14 1 371 64 54 27

Russia 893 150 606 365 512 984 204 116 106 1 193 64 62 27

Belgium 470 216 256 161 146 183 80 143 9 9 47 54 62 21

Catalonia 221 214 136 87 0 67 47 33 33 33 30 27 27 21

Table 8: Number of articles written in collaboration

12

An analysis of the figures provided showsthat 254 articles were published by Cataloniain prestigious journals during the period 1996–2002, indicating a 1.59-fold increase comparedto the previous period. In the rest of the world,there was a 1.16-fold increase. For Catalonia,the Number of outstanding articles by millioninhabitants ratio (Table 6) increased from 19.18(1990–1996) to 39.04 (1996–2002), thus mov-ing from the 14th place to the 6th place onthe list. Note that this ratio may be influencedby the fact that the list of prestigious journalsused in this study comprises a few more jour-nals than the list used in the previous study(1990–1996). Catalonia has surpassed some Eu-ropean countries, including Germany, Swedenand the United Kingdom, which, like Catalo-nia, have not had large population changes. So,the rate of increase of outstanding articles inCatalonia was higher than in those three othercountries. From the data shown in Table 7, itcan be seen that Catalonia ranks quite high inthe list, namely in the fourth place. The onlysurrounding European country that is ahead ofCatalonia is France. For Catalonia, the Out-standing articles/total articles ratio increasedby a factor of 1.22 compared to the previousperiod.

Analysis of the degree of collaboration in scien-tific publicationsWhen analyzing the state of the art of mathe-matical research in Catalonia, we felt it was im-portant to also consider both the number of ar-ticles written in collaboration with mathemati-cians from outside Catalonia and with whichcountries this kind of collaboration most fre-quently occurred. The results of this type ofanalysis may be interpreted in several ways.Some readers may think that collaborating withforeign colleagues means that the capabilities ofCatalan mathematicians are limited and helpis needed from outside sources. Alternatively, ahigh collaboration index can be viewed as cor-responding to a high level of mathematical ca-pabilities, which allows interactions in variousMathematics-related subjects and with differ-ent research groups from around the world. Re-gardless of the interpretation and subsequentconclusion, analysis of the degree of collabora-tion provides further insight into the state ofmathematical research in Catalonia. Table 8lists the 15 countries that collaborated most

frequently with Catalonia in publishing Math-ematics articles. The data reflect the degree ofcollaboration between the countries in absolutenumbers. Most of the collaborations were withmathematicians in the United States, Canada,and European countries close to Catalonia.Nonetheless, Catalan mathematicians collabo-rated more often with colleagues in Mexico,Argentina, Japan or China than with those incountries such as Belgium or Scandinavia.

Funding and infrastructure

In analyzing the sources that fund mathemati-cal research, the cost of maintaining infrastruc-ture, including buildings, libraries, computerequipment, and lecture rooms, should be takeninto account. There are also personnel costs,that is, the salaries of everyone working inmathematical research at any institution inCatalonia. Besides this, direct grants for re-search, which fund many research groups, fi-nance research projects, the organisation ofcongresses, fellowships for visiting foreign re-searchers, scholarships for training researchers,etc. In the following, we refer mostly to directgrants for research and less frequently to bib-liographic support and computer equipment.There are essentially three sources of subsidiesfor Catalan mathematical research: the Cata-lan Government, the Spanish Government, andthe European Union. Table 9 summarizes thefunding by theses three sources during the pe-riod 1996–2002. The Catalan Government hasestablished several diverse programmes to fundresearch groups, projects, and individual re-searchers. Among these programmes, the oneset up to fund consolidated research groups inCatalonia stands out since it aims to strengthenresearch groups by requiring that projects becofunded by the group. At the end of the period1996–2002, there were 18 consolidated groups inMathematics in Catalonia, comprising 263 re-searchers from Catalan universities, a numberthat represents 60 % of the potential mathe-matical researchers in Catalonia. These groupsreceived a total amount of 1,089.567 Euros dur-ing that time. The programme of the Span-ish Government that provides much fundingfor mathematical research is the Sectorial pro-gramme to promote general knowledge. Whilea few Spanish programmes fund research in

13

areas that we considered to be related, theyhave little impact on funding in the field ofMathematics itself. Only a few of these pro-grammes, such as the Telematic Services andApplications and Space Research, finance workin Mathematics, and most of the funding is in-direct. Even though Mathematics is not a pri-ority for the European Union, there are sev-eral programmes that support research. Thesehave been funded by the European Union withthe goal of creating thematic networks that fa-cilitate researcher mobility and aid in the or-ganisation of courses and congresses. In Cat-alonia, during the period 1996–2002, researchprojects centered around Mathematics receivedmore than 10 million Euros. Regarding thesources of funding, only 13.8 % came from theCatalan Government. The main source of fund-ing is the Spanish Government, with 49.5 %,followed by the European Union with 36.7 %.Comparing these figures with those from theperiod 1990–1996, there was an increase of 4 %in the funds from the Catalan Government,and a decrease of 5 % in the funding receivedfrom Spanish Government, while the EuropeanUnion’s funding remained at the same level.Funding provided for Mathematics libraries isanother area of consideration. The investmentthree libraries of this type in Catalonia: at thefaculties of Mathematics at the University ofBarcelona (UB) and the Technical Universityof Catalonia (UPC), and at the Department ofMathematics at the Autonomous University ofBarcelona (UAB). The investment in books and

journals at the Mathematics library of the Uni-versity of Barcelona during the period 1996–2002 amounted to 1,213,700 Euros. At the De-partment of Mathematics at the AutonomousUniversity of Barcelona, 498,942 Euros were in-vested, while at the Statistics and MathematicsFaculty of the Technical University of Catalo-nia this investment was 286,244 Euros. Com-paring these figures with those from the pre-vious period, investments in books and jour-nals at Catalan universities from 1996 to 2002markedly increased. Nonetheless, both at theUniversity of Barcelona and the AutonomousUniversity of Barcelona, the number of journalsubscriptions decreased. The Technical Univer-sity of Catalonia doubled its outlay for booksand started investing in journals. The total in-vestment in Catalonia in bibliographic materialrelated to Mathematics during the period 1996–2002 was almost 2 million Euros, an increase ofmore than 40 % compared with the previousperiod. This amount was less than 20 % of thetotal direct financial help received by researchgroups. Together with books and journals, com-puter resources are among the most impor-tant forms of support required by researchers inMathematics. Without entering into a lengthydescription of the types of computer equipmentfound in Catalan universities and other researchcenters, the equipment of the Dynamics Sys-tems Group of the UB and UPC, which in-cludes a HIDRA parallel computer (Beowulfcomputer) that provides service to a large num-ber of users, should nonetheless be mentioned.

1996 1997 1998 1999 2000 2001 2002 Total

Catalan Government 223,641 330,166 476,607 394,258 596,485 890,247 561,403 3,472,808

Spanish Ministry 493,974 607,851 377,927 614,377 801,832 952,552 1,366,702 5,215,215

European Union 851,495 419,661 385,726 342,062 589,333 529,365 823,216 3,940,858

Table 9: Funding

Other centers and institutions involved in mathematical research

As noted above, mathematical research in Cat-alonia is carried out essentially at universities,but there are other research centers that eitherdo research in Mathematics or are involved inMathematics-related collaborations in Catalo-nia. In this section, we will briefly describe thesecenters and institutions.

Centre de Recerca Matematica (CRM). TheCRM is a consortium between the Catalan Gov-ernment and the Institute for Catalan Studiesthat invites distinguished mathematicians fromaround the world to spend time as visiting re-searchers. It sponsors research programmes andorganises thematic semesters, congresses, and

14

seminars. The activities of the CRM are car-ried out in coordination with research groupsfrom Catalan universities.

Mathematics Institute of the University ofBarcelona (IMUB). The IMUB is a research cen-ter linked to the University of Barcelona thatis dedicated to research and basic training inall the fields of pure and applied Mathemat-ics. It develops and supports research in Math-ematics, including the organisation of scientificactivities such as advanced courses, workshops,conferences, and seminars.

Institute of Research on Artificial Intelligence(IIIA). The IIIA is a center of the Spanish Scien-tific Research Council, located at the campus ofthe Autonomous University of Barcelona. Someof its scientific activity is directly devoted toMathematics but it is also involved with Infor-mation Technology, which also has an impor-tant mathematical component.

Catalan Mathematical Society (SCM). The SCMis a subsidiary of the Institute for Catalan Stud-ies that promotes both theoretical and appliedresearch in Mathematics. It acts as a meetingpoint between mathematicians from the vari-ous Catalan universities and those working asMathematics teachers in high schools.

Institute of Spatial Studies of Catalonia (IEEC).The IEEC is a private foundation that pro-motes activities, studies, and projects relatedto scientific research and space technology.

Statistics Institute of Catalonia (IDESCAT). TheIDESCAT belongs to the Ministry of Eco-nomics and Finance of the Autonomous Gov-ernment of Catalonia. Its objective is to pro-mote research and development in the field ofstatistics.

Ferran Sunyer i Balaguer Foundation. This isa private foundation, sponsored by the Insti-tute for Catalan Studies, that grants an annualaward bearing the name of the Catalan mathe-matician Ferran Sunyer i Balaguer. The awardis given to a monograph on recent developmentsin any field of Mathematics.

Edition of scientific publications

One of Catalonia’s contributions to the devel-opment of Mathematics is the publication ofseveral periodical journals of Mathematics. For

the period 1996–2002, these publications in-cluded:

Collectanea Mathematica. This is the oldestCatalan mathematical journal. Its publication,by the University of Barcelona, began in 1948.The journal is exchanged by the Mathemati-cal Library of the University of Barcelona withseveral international journals.

Publicacions Matematiques. Published by theDepartment of Mathematics of the Au-tonomous University of Barcelona. Its publica-tion began in 1976 and it has been in the ISIdatabases since 2002.

SORT. It is published by IDESCAT, sponsoredby the University of Barcelona and the Techni-cal University of Catalonia. Since 2003 its nameis Statistics and Operations Research Transac-tions.

Mathware & Soft Computing. This journal wascreated in 1994 and is published by the Math-ematics and Computing Section of the Schoolof Architecture of Barcelona, which publishesworks in the fields of cognitive science and ar-tificial intelligence.

Butlletı de la Societat Catalana de Matematiques.This journal publishes, in Catalan, articles inany field of Mathematics that may be of inter-est to a large number of readers.

Qualitative Theory of Dynamical Systems. TheUniversity of Lleida has published this journalsince 2000. It consists of articles concerning thetheory, methods, and applications of dynamicalsystems.

Advanced Courses in Mathematics CRMBarcelona. The CRM, by means of an agree-ment with the Birkhauser publishing house,produces a collection of books based on thecontents of the advanced courses organised bythe CRM.

Publications of the CIMNE. The InternationalCenter of Numerical Methods in Engineering,which is a consortium between the CatalanGovernment and the Technical University ofCatalonia, publishes Archives of ComputationalMethods in Engineering and Revista Interna-cional de Metodos Numericos para Calculo yDiseno en Ingenierıa.

15

Conclusions

1. Among the universities with Mathematicsprogrammes, permanent teaching staff in-creased only at the Autonomous Universityof Barcelona, wich previously had the lowestrate of permanent teaching staff.

2. The number of permanent teaching staff hasstabilized. While this can have a positive ef-fect on research dynamics, there is also anegative effect on the incorporation of youngresearchers. New professional paths for math-ematicians through connections between uni-versity and the industrial and business worldsshould be developed.

3. The number of fellowship holders at the threeuniversities with Mathematics programmessignificantly increased compared with theprevious period. This probably resultedin an increase in the number of doctoral the-ses in Catalonia. The number of doctoraltheses submitted by visiting students also in-creased. However, the total number of thesessubmitted was low when compared with thescientific potential of Catalan universities.

4. An analysis of mathematical productivity ac-cording to country showed that, in math-ematical research, Catalonia maintained orimproved its position among the mostadvanced countries. It should be noted, how-ever, that the most rapid increase in produc-tivity occurred up until the mid 1990s. Thenumbers seem to point out that a steadystate was subsequently reached and thatmathematical research in Catalonia has nowentered a period of maturity.

5. The most productive fields in Mathematicsaround the world are also highly productivein Catalonia, except for Numerical Analy-sis and Partial Differential Equations, whichneed to be strongly promoted in the future.

6. The number of outstanding publications in

relation to the size of the population inCatalonia and the country’s position com-pared with surrounding European countriesincreased over the previous period.

7. Catalan mathematicians frequently collabo-rated with mathematicians from other coun-tries. More than 40 % of the mathe-matical publications in Catalonia includedauthors from other countries, especially fromthe United States and surrounding Europeancountries.

8. Catalonia should have more weight in theadministration of resources dedicated to re-search. In the line to assisting research groupsaccording to the quality of their productiv-ity, the resources available for mathematicalresearch should be increased, thereby pro-moting both scientific and technologic de-velopment. Catalonia has the appropriateconditions for making a qualitatively impor-tant advance in mathematical research, pro-vided that it receives the necessary financialsupport.

9. Although investments in Mathematics li-braries at Catalan universities increased inabsolute numbers, the amount devoted tojournals subscriptions remains insufficientdue to large increases in subscription costs.

10. Despite the high level of mathematical re-search in Catalonia, its international recogni-tion remains low. This may be probably dueto the lack of a mathematical tradition. It isnecessary to increase the presence of Catalanmathematicians on the committees of out-standing journals and in the decision-makinggroups of international institutions that for-mulate scientific policies. A major step toraise the prestige of Catalan Mathematicswould most certainly be to increase the qual-ity of Catalan mathematical journals to thelevel of outstanding journals.

Julia Cufı (UAB)Gerard Gomez (UB)

Gregori Guasp (UAB)Agustı Reventos (UAB)and Oriol Serra (UPC)

16

The Centre de Recerca Matematica, Barcelona

Institutional framework

The Centre de Recerca Matematica (CRM), theonly Mathematics research institute in Spain,was created in 1984 as a personal initiativeof Manuel Castellet by the Institut d’EstudisCatalans (Institute for Catalan Studies), anacademic, scientific and cultural body whoseaim is to promote research in science, technol-ogy and humanities, while supporting all as-pects of Catalan culture. In this year 1984, pre-cisely when the CRM was born, Peter Hiltonwrote in The Mathematical Intelligencer: “Aresearch institute is at least two things at thesame time: it is a building and it is an organisa-tion of people working together and dedicatedto the pursuit and support of research. But atits best it is more. It starts life as an idea in themind of one person —or, rarely, of persons—of insight and imagination, and then lives andgrows by spreading the spirit, imbued by theirfounders, through the hearts and minds of allthose benefiting from its presence and at thesame time contributing to its future”.

Since 2002, the CRM is a consortium, withits own legal status, integrated by the Insti-tut d’Estudis Catalans and the Catalan Gov-ernment. It is a research institute associatedwith the Autonomous University of Barcelona(UAB) and located on the Bellaterra campus,about 15 km from Barcelona.

The Governing Board of the CRM is chairedby the Minister of Universities, Research andthe Information Society of the Catalan Gov-ernment, and has eight members. In 2002, theGoverning Board re-elected Manuel Castellet asDirector, who is assisted by Carles Casacubertaand Jordi Quer as Associate Directors. A Sci-entific Advisory Board meets three times peryear to advise the Directorate Team.

Aims and scope

The CRM gives support to local researchgroups in all areas of Mathematics and fos-ters emerging research directions, by invit-ing outstanding mathematicians and organisingspecialised research programmes, conferences,workshops, and advanced courses.

Since 1989 it hosts post-doctoral fellows (11

in 2005), including several Marie Curie grantholders, and it also hosts doctoral students asa Marie Curie Training Site of the EuropeanUnion from 2000 to 2004. The CRM has beena node of three different European CommunityResearch Training Networks entitled “EllipticCohomology” (1994-1996), “Homotopy Theoryand its Applications” (1995-1997) and “ModernHomotopy Theory” (2000-2003), jointly withteams in Aarhus, Aberdeen, Bonn, Cambridge,Glasgow, Lille, Louvain-la-Neuve, Milano, Parisand Sheffield.

Research programmes

The Governing Board of the CRM approved in2002 a quadrennial plan that includes two re-search programmes every year, together withcomplementary activities. Each research pro-gramme offers the following positions dur-ing one academic year: one full-time local re-searcher, one full-time visiting researcher, twopost-doctoral fellows, and 24 months of visitingresearchers for periods of one to three months.Activities include a weekly seminar, a confer-ence or a workshop and an advanced course ata doctoral or post-doctoral level. Partial fund-ing for visitors and activities is provided by theDepartment of Universities, Research and theInformation Society (DURSI) of the CatalanGovernment, under a contract programme thatis revised every year, and by means of com-petitive calls. Other sources of funding throughcompetitive applications are the Spanish Min-istry of Education and Science and the Euro-pean Commission.

An open call for research programmes ismade every year. Each programme has to be ap-

17

proved by the CRM Governing Board. Propos-als are presented by the Director on the groundsof evaluation reports prepared by the ScientificAdvisory Board.

CRM Research Programmes

2003–2004 – Set Theory

2004–2005 – Geometry of the Word Problem

2005–2006 – Arakelov Geometry and ShimuraVarieties

– On Hilbert’s 16th Problem

2006–2007 – Enumerative Combinatorics andRandom Structures

– Discrete and Continuous Methods inRing Theory

2007–2008 – Homotopy Theory and HigherCategories

– Geometric Flows. EquivariantProblems in Symplectic Geometry

Recent CRM Research Thematic Quarters

2004-2005 – Control, Geometry and Engineering

– Contemporary Cryptology

2005-2006 – Fourier Analysis and GeometricMeasure Theory

2006-2007 – Non-Smooth Complex Systems

– Group-Based Cryptography

Scientific meetings

In 1986 the CRM started the organisationof conferences (29 ever since) and workshops(25) and in 1995 a prestigious series of ad-vanced courses (27). Amongst the conferenceswe should point out the series “BarcelonaConference on Algebraic Topology” (BCAT),organised in 1986, 1990, 1994, 1998 and 2002,and “Barcelona Logic Meeting” (BLM), organ-ised in 1994, 1995, 1997, 1998, 1999 and 2000,two reference points in these fields.

Recent CRM Conferences, Workshops andAdvanced Courses

2004 – Advanced Course on Ramsey Methods inAnalysis

– Workshop on the Foundations of SetTheory

– Advanced Course on ContemporaryCryptology

– Conference on Mathematical Foundationsof Learning Theory

– Workshop on Non-Linear DifferentialGalois Theory

– Advanced Course on Automata Groups

– HYKE Conference on Complex Flows

2005 – 4th Congress of the European Society forResearch in Mathematics Education

– Barcelona Conference on Geometric GroupTheory

– Workshop on Mathematical Problems andTechniques in Cryptology

– Advanced Course on the Geometry ofthe Word Problem for Finitely GeneratedGroups

– Advanced Course on Recent Trends onCombinatorics in the MathematicalContext

– Advanced Course on Shimura Varieties

– Workshop on Graphs, Morphisms andApplications

– 2nd Workshop on Tutte Polynomials andApplications

2006 – Advanced Course on Arakelov Geometryand Shimura Varieties

– Barcelona Conference on Planar VectorFields

– Workshop on Fourier Analysis, GeometricMeasure Theory and Applications

– Advanced Course on Limit Cycles ofDifferential Equations

– Advanced Course on Combinatorial andComputational Geometry Trends andTopics for the Future

– Conference on Mathematical Neuroscience

2007 – Advanced Course on Analytic and Proba-bilistic Techniques in Combinatorics

– Advanced Course on Quasideterminantsand Noncommutative Symmetric Polyno-mials

– Conference on Cryptography and DigitalContent Security

– Advanced Course on Group-BasedCryptography

– Barcelona Conference on C∗-Algebras andtheir Invariants

– Conference on Enumeration andProbabilistic Methods in Combinatorics

– 2007 Barcelona Conference on AsymptoticStatistics

Around the International Congress of Math-ematicians (ICM 2006) in Madrid, the CRMorganises a three months research programmeon “Fourier Analysis” and an advanced courseon “Combinatorial and Computational Geom-etry”, both co-ordinated jointly by researchersfrom Barcelona and Madrid, and a conferenceon “Mathematical Neuroscience”, the last twoactivities as ICM satellite conferences.

18

Links with European entities

Since 2003, the CRM is an institutional mem-ber of the European Mathematical Society(EMS). It is also a foundational member of ER-COM (European Research Centres on Mathe-matics), a committee of EMS consisting of sci-entific directors of European research centresin Mathematics. In fact the current CRM Di-rector, Manuel Castellet, has been the Chair-man of ERCOM from January 2002 to Decem-ber 2005.

As done by other ERCOM centres, theCRM undertakes actions to reinforce the roleof Mathematics in the thematic priority ar-eas of the 6th Framework Programme of theEuropean Commission. Funding is offered toyoung mathematicians in order to foster thedevelopment of the following topics, whichwere selected on the basis of reports preparedby local teams: Life sciences, genomics, andbiotechnology for health; nanotechnologies andnanosciences; information society technologies;sustainable development, global change andecosystems. Several activities, including doc-toral training and workshops, are planned onneuroscience and cryptology for the years 2004,2005 and 2006.

In December 2005, a project entitled “Shap-ing New Directions in Mathematics for Sci-ence and Society” (MATHFSS) started as oneof the Support Actions that were approvedin 2005 within the NEST (New and Emerg-ing Science and Technology) programme ofthe European Commission. It will last twoyears and has been possible by the collabo-ration of the following ERCOM centres: Cen-tre de Recerca Matematica (CRM), project co-ordinator; Emmy Noether Research Institute

for Mathematics (ENI), Israel; European In-stitute for Statistics, Probability and Opera-tions Research (EURANDOM), The Nether-lands; and Institut des Hautes Etudes Scien-tifiques (IHES), France. The goal of the projectis to foster international contacts and drawtraining contents in Mathematics around thefollowing emerging research topics: Systems Bi-ology, Risk Assessment, Mathematical Neuro-science, Digital Content Security.

The CRM is a member of the EPDI (Eu-ropean Post-Doctoral Institute for the Mathe-matical Sciences), a network of nine Europeanresearch institutes which jointly offer post-doctoral fellowships in Mathematics and math-ematical physics every year.

Publications

Besides its twenty-year old preprint collection,the CRM publishes since 2001 a monograph se-ries entitled Advanced Courses in Mathemat-ics CRM Barcelona, which is produced anddistributed by Birkhauser Verlag (Basel). Theseries is especially addressed to doctoral andpost-doctoral students. Volumes contain care-fully edited notes written by lecturers at CRMadvanced courses. The following nine volumeshas been published:

– Homotopy Theoretic Methods in Group Co-homology, by W. Dwyer and H.-W. Henn(2001).

– Lectures on Algebraic Quantum Groups, byK. Brown and K. Goodearl (2002).

– Symplectic Geometry and Integrable Hamil-tonian Systems, by M. Audin, A. Cannas daSilva and E. Lerman (2003).

– Global Riemannian Geometry: Curvatureand Topology, by S. Markvorsen and M. Min-Oo (2003).

– Proper Group Actions and the Baum-ConnesConjecture, by G. Mislin and A. Valette(2003).

– Polynomial Identity Rings, by V. Drenskyand E. Formanek (2004).

– Kolmogorov Equations for Stochastic PDE,by G. Da Prato (2004).

– Ramsey Methods in Analysis, by S. A. Argy-ros and S. Todorcevic (2005).

– Contemporary Cryptology, by D. Catalano,R. Cramer, I. Damgard, G. Di Crescenzo,D. Pointcheval and T. Takagi (2005).

19

– String Topology and Cyclic Homology, byL. Cohen, K. Hess and A. A. Voronov (2006).

Two more volumes are in preparation: Au-tomata Groups, by R. I. Grigorchuk, and TheGeometry of the Word Problem, by N. Brady,T. R. Riley and H. Short.

Facilities and infrastructure

The CRM occupies 1,300 square metres in theScience Building of the UAB. Office space al-lows to allocate up to 28 guests. Two lecturerooms are used for seminars and meetings. Alloffices and rooms are fully equipped and airconditioned. The CRM has a LAN Ethernetwith 30 working stations and four printers. Ac-cess to major bibliographic data bases is pro-vided.

In addition, visitors have free access to thescientific infrastructure of the Catalan universi-ties, including the use of the UAB Science andEngineering Library. The CRM has several fur-nished apartments at its disposal in Barcelona,in the nearby town of Sant Cugat and in theVila Universitaria of the Bellaterra campus.

20th anniversary

In the Autumn of 2004 the CRM celebratedits 20th anniversary. On this occasion, Jean-Pierre Serre, first Abel Prize, delivered a lec-ture during an academic event on November9, which was presided by the foremost acad-emic and political authorities of Catalonia andSpain. The history of the CRM during its firsttwenty years of existence has been publishedin a commemorative volume and a CD. In thisperiod, the CRM hosted 969 visitors from 58different countries, including 44 post-doctoralfellows. Twenty-four congresses were organised,together with 21 workshops and 23 advancedcourses. These events were attended by a to-tal number of 3,480 participants, coming from72 countries. Many of them —hopefully all—enjoyed Barcelona, the Catalan countryside, orthe hospitality of their Catalan colleagues, andkeep pleasant memories from their stay.

Jean-Pierre Serre, College de FranceGroupes finis: Choix de theoremesNovember 9, 2004Institut d’Estudis Catalans, Barcelona

A few final comments

In mathematical research the exchange of ideasplays a central role; the contact and the trans-mission of knowledge is the true laboratoryfor mathematicians. They work in one of themost international sciences, since, comparedwith other disciplines, it is based less on theuse of instruments and more on a strong hu-man contact. This is where the research insti-tutes play a crucial role. They allow not onlythe exchange of ideas between specialists in thesame field, but also the establishment of pro-found and sometimes surprising links betweendifferent lines of research.

For the first time in Catalonia the wordsof Konrad Knopp, pronounced at an inaugurallecture at the University of Tubingen, are be-ing understood: “Mathematics is the basis ofall knowledge and contains all other culture”.The complexity of any system and our worldis one, increases with the degree of intercon-nection. A more interconnected world —moreglobal—, is therefore a more complex systemand at the same time a more fragile and unsta-ble one. Mathematics has an increasingly deci-sive role to play in the management of complexsystems, (be they technological, financial or so-cial), and therefore it will increasingly be, forthose countries which develop high quality re-search, an instrument of power. Our country,Catalonia, can be one of them.

There have alwaysbeen internationallyrecognised Catalan re-searchers in some scien-tific areas, and in recentyears this recognitionhas extended to areaswhere previously we hadno presence, as is thecase for Mathematics. Itis also true that our research groups are be-coming more and more visible in the literature.But it is one thing to be known and recognisedindividually and quite another for the coun-try to be recognised as a scientific community.Without the former we could not achieve thesecond, but the later must always be an objec-tive of the Catalan mathematical communityand of the society in general. In this sense, theCentre de Recerca Matematica has also been

20

an instrument for the international promotionof our country, an instrument that many othercountries wish for but do not have.

Today we believe that our mathematicalcommunity is richer, that our country is scien-tifically more developed, and we congratulateourselves for this and thank all of those whohave made it possible.

Let us end this article with a sentence of theletter addressed by Sir John Kingman, Presi-

dent of the European Mathematical Society inoccasion of the 20th anniversary of the CRM:“You have put Catalan Mathematics firmly andpermanently on the map”.

Website

Additional information about the CRM, includ-ing full lists of visitors and activities, can befound at http://www.crm.cat.

Manuel CastelletCRM Director

The Catalan mathematician Ferran Sunyer i Balaguer (1912-1967) andthe Prize in his honor

Ferran Sunyer i Balaguer was a self-taughtCatalan mathematician very active in clas-sical Mathematical Analysis in the period1940–1967. Each year, in honor of his mem-ory, the Ferran Sunyer i Balaguer Foundationawards a mathematical research prize, whichwas awarded for the first time in April 1993.

Ferran Sunyer i Balaguer

A biographical sketch

Ferran Sunyer i Balaguer was born in Figueras(Girona) in 1912, with a practically completephysical disability that confined him for all hislife to a wheelchair, and died in Barcelona in1967. After the early death of his father, hemoved in with Ferran’s maternal grandmotherand his cousins Maria, Angels and Ferran, whoprovided the pleasant and suitable environmentin which the mathematician grew up. The sickly

boy was left with the option to learn by him-self or through his mother’s lessons. After aperiod of intense reading, arousing a first in-terest in astronomy and physics, his passionfor Mathematics awoke and dominated his fur-ther life. Ferran’s physical handicap did not al-low him to write down any of his papers byhimself. He dictated them to his mother un-til her death in 1955, and when, after a pe-riod of grief and desperation, he resumed re-search with new vigor, his cousins took care ofthe writing. His working power, paired with ex-ceptional talents, produced a number of resultswhich were eventually recognized for their highscientific value and for which he was awardedvarious prizes. In 1938, he communicated hisfirst results to Prof. J. Hadamard of the Acad-emy of Sciences in Paris, who published one ofhis papers in the Academy’s “Comptes Ren-dus” and encouraged him to proceed in his se-lected course of investigation. From this mo-ment, Ferran Sunyer i Balaguer maintained aconstant interchange with the French analyticalschool, in particular with Szolem Mandelbrojtand his students. In the following years, his re-sults were published regularly. His research wasrecognized with a significant number of prizes:Agell Prize, of the Academia de Ciencies i Artsde Barcelona (1948); Premi Prat de la Riba, ofthe Institut d’Estudis Catalans (1949); Premide l’Academia de Ciencias de Zaragoza (1950);two prizes “Torres Quevedo” and “Francisco

21

Franco” (1956) of the Consejo Superior de In-vestigaciones Cientificas; Premi de l’Academiade Ciencias de Madrid (1959), and the PrizeMartı d’Ardenya of the Institut d’Estudis Cata-lans (1966). Despite all honour it was muchmore difficult to reach the social and profesionalposition corresponding to his scientific achieve-ments. At times, his economical situation wasnot the most comfortable either. It wasn’t untilthe 9th of December 1967, 18 days prior to hisdeath, that his confirmation as a scientific mem-ber was made public by the Division de CienciasMatematicas, Medicas y de Naturaleza of theConsejo Superior de Investigaciones Cientıficas.Furthermore, his election was qualified onlyas “de entrada”, in contrast to the first classelection. He always participated and supportedguest lectures in Barcelona, many of them hav-ing been prepared or promoted by him. On theoccasion of a conference in 1966, H. Mascart ofToulouse publicly pronounced his feeling hon-oured by the presence of M. Sunyer i Balaguer,“the first, by far, of Spanish mathematicians”.On December 27, 1967, still fully active, FerranSunyer i Balaguer unexpectedly passed away.

The Ferran Sunyer i Balaguer Foundation

In June, 30 of 1983 Mrs. Maria Assumpcio Car-bona i Balaguer and Maria dels Angels Carbonai Balaguer, cousins of Ferran Sunyer i Bala-guer, created the Fundacio Sunyer i Balaguer.This Foundation became inoperative for sev-eral years, until it was remodeled by Ferran’scousins in December 1991, entering the Insti-tut d’Estudis Catalans. Since then, the Foun-dation is presided by the President of the Ins-titut d’Estudis Catalans. The main objectiveof the Ferran Sunyer i Balaguer Foundationis to award a mathematical monograph of anexpository nature presenting the latest devel-opments in an active area of research in Mathe-matics, in which the applicant has made impor-tant contributions. The Foundation may awardalso other works related to the mathematicalresearch. The Ferran Sunyer i Balaguer Prizeis awarded yearly in the Prize Cerimony of theInstitut d’Estudis Catalans after the proposalof an International Jury. The conditions of thePrize are:

• The monograph must be original, writtenin English, and of at least 150 pages. The

monograph must not be subject to any previ-ous copyright agreement. In exceptional cases,manuscripts in other languages may be consid-ered.

• The winning monograph will be published inBirkhauser Verlag’s series Progress in Mathe-matics, subject to the usual regulations con-cerning copyright and author’s rights.

• The prize, amounting to 12,000 euros, is pro-vided by the Ferran Sunyer i Balaguer Founda-tion.

For the 2007 call of the Prize, the scientificcommittee is formed by: A. Cordoba (Universi-dad Autonoma de Madrid), P. Malliavin (Uni-versite de Paris VI), J. Oesterle (Institut deMathematiques de Jussieu), O. Serra (Univer-sitat Politecnica de Catalunya), A. Weinstein(University of California at Berkeley).

Here is the list of winner monographs fromthe set up of the Prize in 1992:

1992 Alexander Lubotzky, Hebrew University ofJerusalem Discrete Groups, Expanding Graphsand Invariant Measures, (Progress in Mathemat-ics, vol. 125, Birkhauser)

1993 Klaus Schmidt, Warwick University. Dynam-ical Systems of Algebraic Origin, (Progress inMathematics, vol. 128, Birkhauser)

1994 The scientific committee decided not toaward the prize.

1995 As from this year prizes will bear the yearthey are awarded, rather than the year they wereannounced.

1996 V. Kumar Murty and M. Ram Murty, Uni-versity of Toronto. Non-Vanishing of L-Functionsand Applications (Progress in Mathematics,vol. 157, Birkhauser)

1997 A. Bottcher, T. U. Chemnitz and Y. I.Karlovich, Marine Hydrophysical Institute.Carleson Curves, Muckenhoupt Weights, andToeplitz Operators, (Progress in Mathematics,vol. 154, Birkahuser)

1998 Juan J. Morales-Ruiz, Technical University ofCatalonia. Differential Galois Theory and Non-integrability of Hamiltonian Systems, (Progress inMathematics, vol. 179, Birkhauser)

1999 Patrick Dehornoy, Universite de Caen. Braidsand Self-Distributivity, (Progress in Mathemat-ics, vol. 192, Birkhauser)

22

2000 Juan-Pablo Ortega and Tudor Ratiu, EcolePolytechnique Federale de Lausanne. Hamil-tonian Singular Reduction, (Progress in Mathe-matics, vol. 222, Birkhauser)

2001 Martin Golubitsky and Ian Stewart, Univer-sity of Houston, Warwick University. The Sym-metry Perspective, (Progress in Mathematics,vol. 200, Birkhauser)

2002 Alexander Lubotzky and Dan Segal, HebrewUniversity of Jerusalem and All Souls College,Oxford. Subgroup Growth, (Progress in Mathe-matics, vol. 212, Birkhauser)

Andre Unterberger, Universite de Reims. Auto-morphic Pseudodifferential Analysis and Higher-level Weyl Calculi (Progress in Mathematics,vol. 209, Birkhauser)

2003 Fuensanta Andreu-Vaillo and Jose M. Mazon,Universitat de Valencia, and Vicent Caselles,Universitat Pompeu Fabra. Parabolic Quasilin-ear Equations Minimizing Linear Growth Func-

tionals, (Progress in Mathematics, vol. 223,Birkhauser)

2004 Guy David, Universite de Paris-Sud. Singu-lar Sets of Minimizers for the Mumford-ShahFunctional, (Progress in Mathematics, vol. 233,Birkhauser)

2005 Antonio Ambrosetti and Andrea Malchiodi,SISSA, Italy. Perturbation Methods and Semilin-ear Elliptic Problems on Rn, (Progress in Math-ematics, vol. 240, Birkhauser)

Jose Seade, UNAM, Mexico On the Topol-ogy of Isolated Singularities in Analytic Spaces,(Progress in Mathematics, vol. 241, Birkhauser)

2006 Xiaonan Ma, Ecole Polytechnique Palaiseauand George Marinescu, Johann-Wolfgang-GoetheUniversitat. Holomorphic Morse Inequalities andBergman Kernels.

For more information on the Ferran Sunyeri Balaguer Prize and on the Foundation, see theweb page http://ffsb.iec.cat.

Pere Pascual GainzaDirector of the

Ferran Sunyer i Balaguer Foundation

Mathematics at the University of Barcelona

Building of the UB The Cloister

The University of Barcelona (UB) has recentlycelebrated its 555 anniversary. It is the univer-sity of Catalonia which has the biggest numberof students and offers the widest and most com-plete range of courses including undergraduatedegrees, masters and doctorate programs. It isalso the leading center for university researchin Spain in terms of number of research pro-

grams and excellence of results. The facultiesof the University of Barcelona are distributedalong four main campuses in different places ofthe city of Barcelona.

The University of Barcelona has been offer-ing studies of Mathematics since the 16th cen-tury. In modern times the Faculty of Mathe-matics of the University of Barcelona was cre-

23

ated in 1974 as a result of the splitting of theformer Faculty of Sciences into several new fac-ulties.

It is located at the historical building atPlaca Universitat in the center of the city. Thebuilding was begun in 1863 and was declared anational monument of historic and artistic in-terest in 1970.

The Faculty of Mathematics offers studies ofMathematics, Computer Science and Doctorateof Mathematics.

In the academic year 2006–2007 the newmaster of Advanced and Professional Mathe-matics will start in the framework of the Euro-pean Space of High Education.

The faculty has astaff of 117 people ded-icated to both teachingand research. They areorganised in severalwell-established andinternationally recog-nized research groupsin the following re-search areas: Alge-braic Geometry, Commutative Algebra, Com-

plex Analysis, Differential Geometry, Dy-namical Systems, History of Mathematics,Mathematical Logic, Number Theory, Realand Functional Analysis, Statistics, Stochas-tic Processes, Topology, Volume Visualizationand Artificial Intelligence.

In association with the Faculty there isa research institute called IMUB (Institut deMatematiques de la Universitat de Barcelona).The Institute is devoted to advanced researchand training in any field of pure and appliedMathematics. More precisely, its main objec-tives are the support to research in Mathemat-ics through the organisation of scientific activi-ties and the promotion of interdisciplinary workamong researchers from different fields of Math-ematics. It also edites the journal CollectaneaMathematica, which was founded by ProfessorJose M. Orts in 1948 and is the oldest mathe-matical journal in Spain. The journal publisheshigh quality original research papers in pureand applied Mathematics.

The mathematical library of the Universityof Barcelona is one of the best in Spain andthe biggest of Catalonia. It has about 18,000monographs and 669 journals.

Mathematics FacultyUniversity of Barcelona

Mathematics at the Autonomous University of Barcelona

Catalonia holds a significant place in interna-tional mathematical output, with a 0.5 % shareof all publications worldwide. According to theReport on Research in Catalonia (reproducedin this issue of SCM/Notıcies), Catalonia isin sixth place —after Israel, France, Switzer-land, Canada and Denmark— in terms of thenumber per inhabitant of mathematical arti-cles published in quality journals. In Spain,the university rankings consistently place theAutonomous University of Barcelona (UAB)(www.uab.cat) in the top three or four placesand, according to the analysts, the UAB’s De-partment of Mathematics holds the top posi-tions in the different established classifications,due to the quality of its research and its teach-ing staff.

The UAB’s Department of Mathematicshas 56 full-time professors, a total teach-ing and research staff of 129 mathematiciansand is home to 13 highly competitive re-search groups. The department offers gradu-ate studies in Mathematics and statistics anddouble degrees in Mathematics and computersciences and Mathematics and statistics, aswell as several master’s and doctoral courses.The fact that all the areas of Mathemat-ics are grouped in a single department andthat the UAB is a campus university stronglyfavours the multidisciplinary nature of the re-search and the courses. As a result of this,we boast a department that is unique in theCatalan scientific ambit. It is located at theheart of one of the country’s most dynamic

24

R&D areas, surrounded by 33 important re-search centres, such as the Centre for ComputerVision, the Artificial Intelligence Centre, theCatalan Nanotechnology Institute and the fu-ture Synchrotron Light Laboratory, which willbe one of the main scientific facilities of south-ern Europe. Furthermore, our department isclose to the UAB’s sports facilities, the univer-sity village apartments and to the metro stationthat links us to the centre of Barcelona.

The fact that the facilities of the Centrede Recerca Matematica (CRM) are adjacent toour department and that the CRM is a centreattached to the UAB has given an enormousboost to the quality of research at the depart-ment and has helped to get our research groupsbetter known around the world. The close co-operation between the UAB’s Department ofMathematics and the CRM is partly responsi-ble for the leadership of our research groups inCatalonia.

Jose A. Carrillo

We are proud to includeamong our researchers theonly Spanish Salem Prizewinner (Xavier Tolsa), theonly Spanish Richard vonMises Prize winner (Jose A.Carrillo) and two of the fourmost cited Spanish math-ematicians (Pere Ara and

Jaume Llibre, according to the ISI). Apartfrom these noteworthy individuals, our 13 re-search groups also hold important places on thecutting edge of international mathematical re-search. We have research groups in ring theory,algebraic and arithmetic geometry, geometricgroup theory, function theory, harmonic andcomplex analysis, stochastic analysis, math-ematical statistics, algebraic topology, differ-ential geometry, dynamical systems, partialdifferential equations, mathematics educationand low-dimensional discrete dynamics. Thesegroups participate actively in projects at theCatalan, Spanish and European levels and havean annual research budget of 1.2Me.

The UAB’s Depart-ment of Mathematicshas published the inter-national research jour-nal, Publicacions Mate-matiques, since 1976. Inthe 2005 JCR impactfactor index this journalis ranked 49 out of 181journals in pure math-ematics and, over theyears, has gained a reputation for the qualityof its articles and its relatively short waitingtime. The journal is published simultaneouslyon paper and online.

We feel that all of this makes the UAB’sDepartment of Mathematics a sound choice forthe best students in Europe when deciding ontheir choice of master’s and doctoral studies.It is also the right choice for young researcherswho, through the different research scholarships(Ramon y Cajal, ICREA, Marie-Curie, etc.),seek the opportunity to work and improve ina dynamic and friendly institution. We inviteyou all to visit our website (mat.uab.cat) and tocontact us for more information on study andresearch in Mathematics at the UAB.

Jaume AguadeDirector of the Mathematics Department

UAB

25

Mathematics at the Technical University of Catalonia

The mathematical community at UPC

The Technical University of Catalonia (Univer-sitat Politecnica de Catalunya, UPC) employsover 300 mathematicians, for the most part as-sociated with one of the four departments inApplied Mathematics, with the Department ofStatistics and Operations Research or with theDepartment of Architechture Structures.

Their teaching covers the programs of theFaculty of Mathematics and Statistics, and theMathematics/statistics of the Faculty of Infor-matics, of the Schools of Industrial Engineering(schools in Barcelona and in Terrassa, the latterincluding Aeronautical Engineering), Telecom-munications Engineering (schools in Barcelonaand in Castelldefels), Civil Engineering, andArchitecture (schools in Barcelona and in Ter-rassa), and of the bachelor degrees offered byseveral “polytechnic schools”.

With 37 established research groups, fundedfor Research and Technology Transfer Projectswith over 3M Euro (2004), its “scientific pro-duction and impact factor” was ranked 3rd outof 40 in the “Report 2000” on mathematicalproduction in Spain.1

Two Doctoral Programs are offered: Ap-plied Mathematics and Statistics and Opera-tions Research, both distinguished by the Qual-ity Award of the Spanish Ministry of Educa-tion. They output an average of 6-7 and 3-4PhD per year, respectively.

The FME

Founded in 1992, the UPC Faculty of Mathe-matics and Statistics (FME) offers degrees inMathematics and in Statistics and OperationsResearch.

Mathematics degrees

The FME offers a variety of degrees to suit thewidest range of interests and needs, from themost application-oriented to the most theory-oriented.

Higher degree in Mathematics. Students enter-ing the university are offered a 5-year Degree inMathematics (Llicenciatura en Matematiques),comparable to a BSc+MSc. It is limited to 50students. The lowest mark for admission is 7/10and the first year is selective (required to bepassed in at most two academic years).

Double degrees. Students of the Higher Degreein Mathematics can opt to enrol in a special6-year program to earn a double degree, theHigher Degree in Mathematics and a degreein Engineering (Industrial, Computer Science,Telecomunications or Civil Engineering). En-rolment in these special programs is about onein six of the students in the Higher Degree inMathematics (to qualify, they have to take andpass a special examination).

Graduate courses. With an enrolment of 30students, the FME offers a graduate coursein Mathematical Techniques for the FinancialMarkets.

1La investigacion matematica en Espana en el periodo 1990–1999. Report prepared by the Spanish Committeefor the World Mathematical Year (2000), under the coordination of C. Andradas and E. Zuazua. Edited by the RealSociedad de Matematicas (RSME), the Catalan Mathematical Society (SCM), the Spanish Society for Statistics andOperations Research (SEIO) and the Spanish Society for Applied Mathematics (S~eMA).

26

New master’s programs. Starting in Septem-ber 2006, and in the framework of the Euro-pean Space for Higher Education (ESHE), theFME offers, in cooperation with the depart-ments of Applied Mathematics, the following2-year Master’s degrees:

• Applied Mathematics.

• Mathematical Engineering.

Statistics and Operations Research

degrees

The degrees offered by the FME in these areasgrant students optimal flexibility in designingtheir careers.

Degree in Statistics. A 3-year degree in Statis-tics (Diplomatura en Estadıstica). It is a BScdegree totally adapted to the ESHE.

Higher Degree in Statistics. A 2-year degree inhigher Statistics (Llicenciatura en Ciencies iTecniques Estadıstiques). It is comparable toa MSc on “statistical sciences and techniques”.

New master’s program. Starting in September2006, and in the framework of the ESHE, theFME offers, in cooperation with the Depart-ment of Statistics and Operations Research,a Master’s degree in Statistics and OperationsResearch.

Graduate course. With an enrolment of 30 stu-dents, the FME offers, in cooperation withthe Statistics and Operations Research Depart-ment, a graduate course in Biostatistics.

New organization

Starting in September 2006, the Doctoral Pro-grams, and the double-degree special pro-grams, are adapted to the new ESHE organi-zation. This ensures that the degrees are validacross the European Union, that the studentswill have unprecedented mobility opportunities,and that they will be able to follow curriculathat truly meet their individual scientific inter-ests and needs.

Student mobility

About 20 students per year take the oppor-tunity to study for some period in other uni-versities. In exchange the FME receives aboutthe same number of students. In general, theseexchanges are regulated by a number of bilat-eral agreements with other universities in theframework of some global exchange program.At present, FME exchanges students with:

• Universities in Madrid, Santiago de Com-postela, Sevilla, Cadiz, Granada, Tenerife,Murcia and Zaragoza, in the framework ofthe Spanish Sicue/Seneca program.

• European Universities in Finland, Norway,United Kingdom, France, Italy, Greece, theNetherlands, Belgium, Switzerland, Portu-gal and Germany, in the framework of theSocrates/Erasmus European program.

There are also exchanges with Latin-American universities and visitors from the In-stituto Tecnologico Autonomo de Mexico.

Cultural life

At the FME there is a rich range of activitiesbeyond Mathematics: music, theatre, sports(including Go and Chess), creative writing, ...

With regard to the lectures’ program, ineach of the last three academic years it hasbeen catalyzed by its association with a his-torical figure: Poincare in 2003–2004, Einsteinin 2004–2005 and Gauss in 2005–2006. The lec-tures can be found in the FME Web page (clickat Butlletı Digital FME ). Moreover, the inten-tion is to publish in book form a selection ofthe lectures of each academic year, as it hasbeen already the case with Poincare and Ein-stein (Conferencies FME, Volums I and II).

Key sources for further information– UPC web page: www.upc.edu– FME web page: www-fme.upc.edu– FME masters’s programs: mastersfme.upc.edu– FME Office for Mathematics Research (OSRM):

www-fme.upc.edu/osrm/

Sebastia XamboDean of the FME2

2For the preparation of this note, the author wants to thank the efficient assistance of the associated deans (JaumeFranch, Pere Grima, Margarida Mitjana, Eduard Recasens and Oriol Serra) and the useful comments of professorsMiguel Munoz Lecanda and Manuel Martı Recober.

27

Mathematics at ICREA

The Catalan Institution of Research and Ad-vanced Studies (ICREA) is a foundation cre-ated jointly by the Generalitat de Catalunya(the autonomous government of Catalonia),and the Catalan Foundation for Research andInnovation (FCRI). ICREA was born in 2001with the aim of promoting scientific researchin Catalonia. Its main instrument towards thisgoal is the annual call for permanent senior re-search positions. In addition to this, ICREAis also giving the opportunity to young re-searchers to join the R+D Catalan Systemthrough the ICREA Junior program, dividedinto Junior Academia and Junior Empresa, thelatter aimed at bringing in scientists to the pri-vate sector.

ICREA researchers work in universities andresearch centers in Catalonia, and in the case ofjunior researchers also in companies. Currently,there are 119 senior and 13 junior researchers.Two thirds of them were working abroad at thetime of being hired by ICREA. Since half ofthem have Spanish nationality, this means thatat least one third of the hirings made by ICREArepresent a clear recovery of scientific talent forthe country.

ICREA has been able to attract distin-guished researchers in many fields and frommany different countries. The selection processis based exclusively on scientific and acad-emic excellence, and is carried out by exter-nal expert evaluation committees. The annualICREA calls for positions are announced world-wide, both on-line and in journals like Natureor Science, and are open to researchers of anynationality. Since ICREA receives over 400 ap-plications from highly qualified researchers, thecompetition is very strong.

Approximately, 33 % of ICREA researchersbelong to the area of life sciences and medi-cine, another 30 % to experimental sciences andMathematics, 17 % to technology, 15 % to hu-manities, and 5 % to social sciences. Mathemat-ics represents a 5 % of the total. This distrib-ution does not obey a pre-determined systemof quotas by areas, but it is simply the resultof the evaluations of candidates by the expertcommittees, based only on the need to hire sci-entific talent.

Periodically, ICREA researchers undergo a

thorough evaluation of their scientific work, aswell as of their degree of adaptation to theirhost institution and of the impact of their ac-tivity on the local community. The evaluation isbased on confidential reports by peers, all fromoutside Catalonia. Positive evaluations usuallylead to a salary increase, while repeated nega-tive evaluations may lead to a termination ofthe contract.

Besides research, other important activitiesof ICREA researchers are: teaching of gradu-ate courses and seminars, supervision of doc-toral theses, direction of research groups, man-agement of research projects, popularization ofscience, etc.

At ICREA’s web page www.icrea.eu one canfind much more information, both on the Insti-tution and its researchers.

The mathematicians of ICREA

Currently, ICREA has 5 senior researchers inMathematics: Xavier Cabre, Jose Antonio Car-rillo, Sy D. Friedman, Xavier Tolsa, and theauthor of this article. There is also one juniorresearcher: David Aspero.

I was hired in the first call, back in 2001.In 1992 I returned to Catalonia after a 7 yearsstay at the University of California at Berke-ley, where I earned my PhD degree and workedas a post-doctoral researcher. After 9 years inseveral Catalan universities, always with tem-porary or visiting positions, ICREA gave mea unique opportunity to stabilize my positionand remain in Catalonia doing research undersuitable conditions. I work at the Departmentof Logic and History and Philosophy of Scienceof the University of Barcelona and my area ofresearch is Set Theory and its applications toAnalysis (Measure Theory and Banach SpaceTheory) and General Topology.

In the second call, in 2002, ICREA hiredJose Antonio Carrillo and Xavier Tolsa. Bothwork at the Department of Mathematics of theAutonomous University of Barcelona (UAB).The research of J. A. Carrillo focuses on Par-tial Differential Equations and MathematicalPhysics. At the moment of signing his con-tract with ICREA, J.A. Carrillo was a profes-sor at the University of Granada. In 2003 hewas awarded the price of the Spanish Societyof Applied Mathematics. And very recently he

28

has also been awarded the Richard-Von-MissesPrize 2006 of the Gesellschaft fur AngewandteMathematik und Mechanik (GAMM).

Xavier Tolsa was a Ramon y Cajal re-searcher at the UAB, but after some spectacu-lar work in which he solved a famous problemof Painleve’s and proved the additivity of theanalytic capacity, ICREA offered him a perma-nent position so he could stay and carry outhis work in Catalonia. In 2002 he received theprestigious Salem price, and in 2004 the price ofthe European Mathematical Society. He is oneof the invited speakers at the ICM 2006. Hisareas of research are Harmonic and ComplexAnalysis.

In the 2003 call for positions ICREA hiredXavier Cabre and Sy D. Friedman. X. Cabreworks at the Department of Applied Mathe-matics I of the Technical University of Catalo-nia, in Barcelona, and his area of research arethe Partial Differential Equations. He obtainedhis PhD at the Courant Institute of New Yorkand was Associate Professor at the Universityof Texas at Austin until 2003, when he returnedto Catalonia.

Sy D. Friedman was for 26 years Profes-sor of Mathematics at the Massachussetts Insti-tute of Technology, and is currently OrdentlicheUniversitats-Professor and Director of the KurtGodel Institute for Mathematical Logic at theUniversity of Vienna. He is joining ICREAgradually and his total incorporation at theCentre de Recerca Matematica, in Bellaterra(Barcelona), is expected to take place duringthe academic course 2006–2007. He works onMathematical Logic.

Finally, in 2005, ICREA hired a junior re-searcher in Mathematics, David Aspero. He gothis doctorate at the University of Barcelona andhas been a postdoctoral researcher in Vienna,Helsinki, and Bristol. He is currently at the Uni-versity of Barcelona and his research interestsare in Set Theory, especially forcing axioms andcombinatorics.

It is expected that in the next years ICREAwill continue hiring mathematicians, both se-nior and junior, who will contribute with theirwork to the increase in quality of the mathe-matical research done in Catalonia.

Joan BagariaICREA Director’s Scientific Advisor and

ICREA Research Professor

Federacio d’Entitats per a l’Ensenyament de les Matematiquesa Catalunya (FEEMCAT)

Introduction

The Federation of Or-ganisations for theTeaching of Mathe-matics in Catalonia(FEEMCAT) is a non-profit association andits objectives are toimprove the teaching of Mathematics at alllevels of education, by improving teacher train-ing, the mathematical training of students andsociety’s view of Mathematics.

Apart from the groups that currently formpart of FEEMCAT, at the beginning the Feder-ation also included the participation of groups

of Mathematics teachers and lecturers work-ing in Girona (Grup Perımetre, Grup +3),Osona (Grup + o −), Lleida and Tarrega(Grup Nombres de Ponent), and Barcelona(Grups Almosta de l’Associacio de Mestres deRosa Sensat) The Societat d’Ensenyants deMatematiques del Garraf (Society of Math-ematics Teachers of El Garraf, SEMG) alsoformed part of the Federation between 1999 and2005. Most members of these groups have grad-ually joined the associations that make up theFederation today:

• Associacio d’Ensenyants de Matematiques deles Comarques Gironines (ADEMGI)

• Associacio de Professors de Matematiques deles Comarques Meridionals (APMCM)

29

• Associacio de Professors de Matematiquesdel Maresme (APaMMs)

• Associacio de Barcelona per a l’Estudi i l’A-prenentatge de les Matematiques (ABEAM)

Twelve years of activities

Journal Biaix

BIAIX was set up in April1992 and is currently theonly journal on Mathemat-ical Education written en-tirely in Catalan. To findout all about BIAIX today,you can visit the website:www.xtec.es/entitats/BIAIX,where you will find the on-

line version of the most recent issues.

Meetings and Congresses on MathematicalEducationIn recent years, there have been more than40 meetings with a total participation of morethan 5,000 Mathematics teaching professionals.

Courses, seminars and work groupsOpen to all teachers in Catalonia.

Maria Antonia Canals PrizeIn 2002, FEEMCAT created the Maria AntoniaCanals Prize with the aim of promoting inno-vation in education in the area of Mathematicsat all levels of teaching, from pre-school to uni-versity.

Fem MatematiquesFem Matematiques (Let’s Do Mathematics)is currently an annual activity organised byFEEMCAT with the aim of contributing to-ward developing mathematical competence inall students in the sixth year of primary educa-tion and the first year of secondary education.It also provides for promoting experiences andsharing proposals by the teaching staff from dif-ferent parts of Catalonia, and aims to raise theawareness of Catalan society in general about

the need to improve a mathematical educationthat encourages the personal development andsocial integration of free and responsible citi-zens.

ESTALMATFEEMCAT and the SCM agreed in 2002 tojointly organise the Estalmat activity in Cat-alonia under the auspices of the Royal Academyof Sciences and with the support of the Voda-fone Foundation. For more information, visitthe project’s website: www.estalmat.org

Working for the future

The Federation recently took part in thecurricular debate at the Department of Ed-ucation. For more information, you canvisit the site www.gencat.net/educacio/debat.The document on technical and scien-tific education is available at the websitehttp://xtec.net/e13−debatcurricular/docs/3.cientific.pdf. We have also contributed tothe National Agreement on Education inCatalonia. You can view the document athttp://www.xtec.net/entitats/feemcat/historia/activitats05/pne.pdf.

We are working for Catalonia to host aJAEM congress soon (Meeting on Teaching andLearning Mathematics). The Federation alsocontinues with its aim to maintain the highstandards achieved by the work groups in ouractivities and to take the first steps to form ateam of teachers to design and create mathe-matical exhibitions.

Coordination work and relations with otherorganisations and between research groups, in-novation teams and teachers associations re-quire considerable development in order to beable to cooperate in research on adapting math-ematical education to today’s social challenges.

Pilar RoyoFEEMCAT President

30

Societat Catalana de Matematiques

SCM Funds

Activites Promotion Fund

The SCM General Assembly, in its meeting ofJune 3, 2003, approved the creation of a fundcalled Fons de Promocio d’Activitats (Activi-ties Promotion Fund). This fund had an ini-tial amount of 102,372.90 euros, and is devotedto supporting new activities undertaken by theSCM itself, as well as supporting other activi-ties involving the teaching or research in Math-ematics, organised by Catalan institutions orworking groups, provided that these activitiesbenefit the Catalan mathematical communityand conform to the statutory goals of the SCM.

In all cases alternative or partial financialsupport should be seeked for the activities thatapply to this fund. One of the goals is to helporganisers in the early stages of preparationof their activities, hence a bigger sum can beinitially delivered, and when the final budgetis established part of it will be eventually re-turned. Proposals are submitted to the Execu-tive Board of the SCM, which decides on theapplications. If necessary the Scientific Com-mittee or the Teaching Committee of the Soci-ety can be consulted by the Board. The GeneralAssembly of the SCM is informed, once a year,of the activities organised with the fund’s help,and of the financial managing of the fund itself.

Among the external activities supported bythe fund up to now, we can mention:

– XI Topology Meeting (Barcelona, March2004)

– Special Session on Algebraic Geometry inthe conference MAT.ES (Valencia, January-February 2005)

– 4th Congress of the European Society of Re-search in Mathematical Education (Sant Fe-liu de Guıxols, February 2005)

– Algebraic and Topological Methods in Non-Classical Logics II (Barcelona, June 2005)

– Primeras Jornadas de Teorıa de Numeros(Vilanova i La Geltru, June-July 2005)

– Winter School on Commutative Algebra andApplications (Barcelona, January-February2006)

– Godel Centenary Day (Barcelona, May 2006)

– Geometric and Asymptotic Group The-ory with Applications (Manresa, September2006)

– Barcelona Analysis Conference (Barcelona,September 2006)

– Topology of Complex Algebraic Varieties(L’Alguer, Italy, September 2006)

– Eurocrypt (Barcelona, 2007)

Cooperation Fund

The SCM General Assembly, in its meeting ofJune 29, 2005, approved the creation of a fundcalled Fons de Cooperacio (Cooperation Fund).The aim of this fund is to promote the par-ticipation of mathematicians from developingcountries in the activities organised in the scopeof the SCM, either by the SCM itself or by oth-ers. The amount devoted yearly to this fundshould be not less than the 0.7 % of the Soci-ety’s income from membership fees. The Exec-utive Board will determine the list of eligiblecountries, in accordance with the criteria usedin similar organisations.

This fund will be available both to individ-uals who want to attend the eligible activities,and to the organisers of these who want to pro-mote the participation of mathematicians fromthe eligible countries in the activity they areorganising.

This fund will be also used in direct ac-tions that share the same goal, including co-operation with the Committee of InternationalCooperation and Development of the SpanishCommittee of Mathematics, of which the SCMis one of the four founding societies; followingits initiative, at the end of 2005 a shipment ofmathematical textbooks was sent to Morocco,Nicaragua and Paraguay.

Josep M. FontSecretary of the SCM

31

The Czech-Catalan venture: a fruitful collaboration

The Czech Republic and Catalonia share somecharacteristic features: relatively small Euro-pean countries with a healthy sense of nationalidentity and a strong belief in culture and sci-ence.

Stamps published in occasion of the Congres

The connection between the mathematicalcommunities in the two countries can be tracedback to the early 70’s, usually based on per-sonal collaboration. These contacts were strongenough to consider the formalization of an offi-cial agreement between the Czech and Catalanmathematical societies, which was signed in theoccasion of the the second Joint EMS Mathe-matical Weekend held in Prague in September2004.

From the very beginning there was a def-inite will to go beyond the usual reciprocityagreements and to create a real framework ofcollaboration which would enrich the mathe-matical and human connections between thetwo communities.

One of the highlights of this agreementhas been the organisation of CSM-SCM JointMathematical Meetings. The first edition washeld in Prague in May 2005 with great suc-cess. The meeting gathered some 100 mathe-maticians from both communities in areas withexisting collaboration: Computational Statis-tics and Data Analysis, Discrete Mathemat-ics and Computer Science, Homotopy The-ory, Logic, Functional and Real Analysis andRing and Module Theory. A special series ofmail stamps was produced to commemorate theevent. Moreover, the Catalan partners were in-vited as observers to the Czech-Slovak competi-tion for young researchers in order to study theorganisation of a similar competition involvingyoung researchers of the two countries.

In this first meeting it was agreed to have asecond edition in Barcelona, which will be heldin September 2006. The meeting is again organ-ised around special sessions, this time involv-ing Computational Statistics and Data Analy-sis, Discrete Mathematics and Computer Sci-ence, Logic, Mathematical Physics and Ringand Module Theory. The meeting will also wel-come a special session devoted to the young re-searchers competition, gathering the winners ofthe Czech-Slovak and the Catalan ones.

The Czech-Catalan collaboration is sittingon the firm base of a real mathematical ex-change. It shows that such cooperation amongcommunities of manageable size can be ex-tremely fruitful and Europe can draw greatbenefits from its characteristic diversity.

Oriol SerraUPC

32

La prova Cangur, the Catalan Kangaroo

By the end of 1995, the Catalan Mathemat-ical Society had been organising the Catalanstage of the Mathematical Olympiad within theframework of the IMO (International Mathe-matical Olympiad) for many years. The thengoverning board of the SCM got wind of an ac-tivity called the “Kangaroo” that had a verydifferent aim from that of the Olympiad: itsgoal was to stimulate and motivate a broadmajority of students and encourage everyone totake part in a mass scientific demonstration inthe form of a real Mathematics festival. The ac-tivity is a game aimed at attracting the largestpossible number of students without drawingcomparisons between countries.

It was with this goal in mind thatAndre Deledicq created Le Kangourou desmathematiques (Mathematics Kangaroo) inFrance in 1991. He gave it this name because itwas based on the model of the Australian na-tional competition invented in Canberra in thelate 1980s by Peter O’Halloran. At that time,the competition was the only one of its kind andits goal was the participation of vast numbers

of students, regardless of their academic leveland output. This popularizing activity spreadrapidly to several European countries and wasconsolidated by the founding of the interna-tional association Kangourou Sans Frontieres(Kangaroo without Frontiers), created in 1995with the sponsorship of the Council of Europe.Other countries in Asia and America have sincejoined.

Francisco Bellot created the AsociacionCanguro Matematico (Kangaroo MathematicalAssociation) in Castilla-Leon and the first edi-tion was held in 1994. It was he who passed thenews on to the members of the SCM and withhis formal support, the first Kangaroo competi-tion in Catalonia was held in late 1995, almostas an experiment. The level of results was in-teresting and there was very positive feedbackabout the activity and so the board of the SCMdecided to give it strong backing.

Participation grew and the Kangaroo be-came consolidated as an annual event in Catalo-nia. For this reason, the SCM applied to have itsown national representative in the Kangourousans Frontieres association and has been amember since 1999. Our Kangaroo has growntremendously. The SCM Kangaroo committeehas moved from being merely Catalan to repre-senting Catalonia, Valencia and the Balearic Is-lands, thus covering the three Catalan-speakingterritories. Between the first and eleventh edi-tions, the number of participating centres mul-tiplied by 4 and the number of participatingstudents by more than 11. The following tableis quite explicit:

SCM Kangaroo (I: 1996; XI: 2006)

I II III IV V VI VII VIII IX X XI

Level 1 648 666 833 1,663 2,255 2,425 3,031 3,659 4,113 4,839 5,288

Level 2 453 696 867 1,323 2,072 2,544 2,098 3,082 4,049 4,621 5,260

Level 3 212 473 576 742 961 1,311 1,590 1,802 2,309 2,942 3,176

Level 4 - - - 275 329 414 619 599 765 967 1,219 1,410 1 ,774

Students 1,313 2,110 2,605 4,142 5,907 6,879 7,484 9,510 11,690 13,812 15,498

Centres 106 126 167 244 299 297 314 372 406 450 487

33

We say that out Kangaroo has grown verybig, though anyone comparing this data withthat of other countries would draw the oppositeconclusion. But there is an aspect of the organi-zation of the SCM’s Kangaroo that is quite dif-ferent from that of other countries, where theproblems are sent out to each centre and theKangaroo takes place almost as a regular classactivity.

Here, the idea materialised of doing mathstogether, at the same time as colleagues fromother centres, other towns and other countriesand the competition was organised in head-quarter centres that took in participants fromother centres. Mathematical fun activities com-plementary to Kangaroo were often preparedand lunch provided. And so, on the whole, theevent became the Mathematics celebration.

This philosophy, aimed at sharing Kangarooheadquarters among several centres, often on arotational basis from one year to the next, wasa resounding success. In following editions, sev-eral regional civic centres and the universities ofCatalonia and Valencia offered their classroomsso that groups of students could come togetherto take part in the Kangaroo. The idea of a fes-tival does not prevent, but rather encouragesthe desire to enjoy oneself while getting to gripswith the mathematical challenges posed. DoingMathematics is unquestionably about solvingproblems. And if you can enjoy yourself at thesame time, as most Kangaroo participants do,the aim has been achieved.

At the Kangaroo committee, we are happyto see how, year after year, the enormous ge-ographical diversity of the participants is re-flected in the prize winners. Among the prizesawarded this year in a ceremony presided overby Catalan educational authorities (held in re-cent years in the Aula magna of the Catalan

universities with Mathematics degree courses),the mentions that acknowledge up to 1 % of thebest participants in each level and the prizesfor other activities, which we will discuss later,have provided 165 distinctions and we can saythat there are no “specialist centres” as the stu-dents receiving the awards belonged to 116 dif-ferent centres. We believe that this diversity islargely responsible for the success of the Kan-garoo.

Another aspect that may make the organ-isation of our Kangaroo different from that ofother nations is the constant feedback. The or-ganising committee is open to suggestions fromthe entire school community that drives partici-pation in the different centres. This exchange ofopinions makes it easy to collect many interest-ing ideas and we will always be more likely toimplement them if we really listen to the teach-ers of the participating centres than if we justlimit ourselves to the ideas of the committeemembers.

For example, this was how the idea of hold-ing the Kangaroo in university centres cameabout and these teacher meetings have alsogiven birth to some interesting complementaryactivities, such as sprint problems, poster com-petitions and short-story competitions.

In 2000, the World Mathematical Year be-gan with a competition to solve problems aimedat teams in the centres and designed to be runonline over the Internet. This activity is nowcalled Sprint problems because the first teamto send all the correct answers is, naturally, thewinner. To date there have been 10 editionswhich were highly rated by the participatingcentres. Each centre organises the team as theysee fit, with the sole condition that the teammust be of mixed levels (ages between 15 and 18

34

years). There are problems with different lev-els of difficulty and the groups of students whotackle the “easier” problems must pass on infor-mation to the groups doing the “harder” prob-lems, thus establishing complicity between stu-dents from different classes-something everyonesees as very positive.

Here is the opinion of one of the centres thattook part in the latest edition:

We felt it was very important that no boyor girl should feel excluded from an activ-ity like problem solving. We can also seethat the different levels of difficulty of theproblems helped us to handle the diversityof students. We insisted that everyone wascapable of solving some of the problems.In fact, we were not thinking of a prize.The students were distributed in groupsthroughout the school (which is linked bycable) and each group started with differ-ent problems and then compared results.

At the organising committee, we believethat the statement that the students were all“capable of solving” mathematical problemsmore than covers the goal of the activity andencourages us to continue.

Another area where the Kangaroo aims topromote activities based on suggestions fromthe different schools is the interdisciplinaryarea, interrelating Mathematics with other ar-eas of education and culture.

With this objective in mind, since 2002there have been three calls for posters to goalongside the official logo of Le Kangourou,work of Raoul Rabba, in order to person-alise our Kangaroo. The results have revealedan outstanding quality of design and this hassometimes made us reluctant to organise a newcompetition as we do not wish to abandonsomething we have made our own.

Finally, the first Kangaroo short story com-petition was held in 2005 for short stories witha Mathematics-related theme. As with all ac-tivities in the framework of our Kangaroo,this competition was aimed at the 15-18 agegroup. In Catalonia, there are other associa-tions grouped in FEEMCAT (Federation of Or-ganisations for the Teaching of Mathematicsin Catalonia) that carry out similar work foryounger children aged 11 to 14 who are in thelast years of primary school and the first yearsof secondary school. The success of both thesecompetitions in terms of quantity and, espe-cially, quality has encouraged us to continue.

Who knows? Perhaps, in the future, we willsign up to the linguistic Kangaroo being organ-ised by our Romanian colleagues or maybe wewill organise a Kangaroo fun run, though in-deed the real race to run is ensuring that eachyear the Kangaroo shows many boys and girlsthat doing Mathematics can also be a real fes-tival. Antoni Goma

Chair of the Cangur commission

Winners of the 10th Cangur

35

Brief history of the SCM/Notıcies

Number 0 of the journal SCM/Notıcies ap-peared in February 1995. It was designed tobe a periodical that would expand and improveon Noticiari de la SCM, a two-page publica-tion containing the most relevant Catalan andinternational Mathematics news.

Carles Casacuberta was the editor of theoriginal Noticiari. The first four issues, pub-lished in 1993 and 1994, announced the Leib-nitz scholarships, the Ferran Sunyer i BalaguerPrize, ICM 94, the CRM’s postdoctoral schol-arships and semesters on Dynamical Systemsand Differential Geometry, Europroj 94, the 6thPedagogical Meeting, Temu 95, the awardingof the Catalan autonomous government’s Crossof Saint George to Lluıs Santalo, the winnersof the Fields Medal and the publication of thecomplete works of Pere Menal.

In February 1995, Sebastia Xambo waselected chairman of the Catalan MathematicalSociety (SCM). One of the milestones of thechairman and the board was to expand Noti-ciari and so SCM/Notıcies was born. The firsteditorial board was made up of Carles Casacu-berta, Joaquın Ortega and Antoni Goma, andwas coordinated by Sebastia Xambo himself.The aim was not just to produce a journal ofCatalan and international Mathematics newsand inform SCM members about the activi-ties being promoted by the new board, butalso to provide a means of popularising Math-ematics in Catalonia and a natural mediumof communication for the Catalan Mathemat-ics community —“our newspaper”.

Looking back over more than 10 years of his-tory, these goals have all largely been achieved.Today, SCM/Notıcies is a well-known period-ical in Catalan mathematical circles, both inuniversities and secondary schools, and is ful-filling its role as a nexus and information pointfor Catalan mathematicians.

But let us go back to its beginnings. J. L.Ruiz of the Technical University of Cataloniabegan designing SCM/Notıcies and Butlletı inLATEX2e in 1995. The result was an scm.cls filethat made editing the journal very easy andgave it its own individual style. Other peoplehave also made decisive contributions to theseediting and formatting tasks: J. Trias wrote aPostScript program to help with editing fig-

ures, Toni Escola took part as the IEC’s head ofpublications and later, in 2001, Teresa Sabaterdesigned the journal’s current format, whichmade its debut in Number 15, and Joan Tor-regrosa put it into LATEX. Finally, Maria Juliahas, with much care, dedication and efficiency,taken charge of composition and final publica-tion since Number 9 and Nuria Fuster at theSCM Secretariat is always on call for all theadministrative work.

SCM/Notıcies introduced permanent sec-tions that have been consolidated over timeand are still featured. The Problems section ap-peared in Number 0, as a space for readers tosolve and propose problems, together with theBook Review section. The Theses section wasintroduced in Number 1, containing systematicreviews of all Mathematics theses read in Cat-alonia.

The Problems section was initially coor-dinated by Antoni Goma, with the help ofJosep Grane, coordinator of the MathematicalOlympiad in Catalonia, who provided the ma-terial for these tests. Josep Grane has also beenon the SCM/Notıcies editorial board since itsinception.

Antoni Goma, the coordinator of the Kan-garoo contest in Catalonia, also writes regularlyfor SCM/Notıcies providing all the informationregarding this contest-dates, winners, partici-pants, etc. The Kangaroo and Olympiad sec-tions are permanent and, at this stage, classicsections of SCM/Notıcies.

Pelegrı Viader and Anna Pol replacedJoaquın Ortega on the editorial board ofSCM/Notıcies in Number 7.

At the request of Sebastia Xambo, I tookcharge of SCM/Notıcies from Number 9 on-wards, with a board of editors made up of Se-bastia Xambo, Antoni Goma, Josep Grane andCarles Casacuberta. The Theses and Book Re-view sections were kept on and became my re-sponsibility. The Problems section was handedover to Pelegrı Viader. A special section wasstarted on 3ecm, Barcelona 2000 and was runby Carles Casacuberta.

For a few issues around Number 10, a sec-tion was run on language considerations thatdealt with the correct way of translating intoCatalan those technical words or expressions

36

that we as mathematicians commonly use in theoriginal English or for which we devise rough-and-ready translations.

From Number 16, the Problems section wastaken over by Carles Romero and in Number18, the Column of the European Mathemati-cal Society, now the International section, wasbegun under Marta Sanz.

I was replaced as editor by Enric Venturain Number 20. Antoni Goma, Carles Romero,Marta Sanz and Josep Pla, who has lately takenover the Book Review section with a large doseof initiative, continue to work with him. Thefour issues published under his direction (thelatest of which is the special ICM 2006 issueyou are now reading) continue the general linesof the journal, maintaining the traditional sec-tions and introducing new and interesting ones,such as the Debate Forum for debate betweenmembers, the Universities Inform and Activi-ties with the Collaboration of SCM sections andthe new page on mathematical websites.

Below is a list of the most noteworthy newspublished in the 22 issues of SCM/Notıcies overthese past 11 years:

• Tribute to William K. Clifford(SCM/Notıcies 1).

• The Taylor and Wiles Proof of Fermat’s The-orem (SCM/Notıcies 3).

• Realism in Mathematics (SCM/Notıcies 7).Translation by Antoni Goma of an article byMorris W. Hirsch in the AMS Bulletin issue32, 1995.

• The 1998 Fields Medals (SCM/Notıcies 9).

• 3ECM (SCM/Notıcies 10).

• Interview with Professor Josep Vaquer(SCM/Notıcies 11).

• Interview with Shiing Shen Chern(SCM/Notıcies 12). Translation by ImmaGalvez of the interview published in AMSNotices, 45, Number 7.

• Gerbert d’Orlhac and Scientific Culture inthe Year 1000 (SCM/Notıcies 12).

• World Mathematics Year. Day at theCongress of Deputies (SCM/Notıcies 13).

• Third European Mathematical CongressClosing report (SCM/Notıcies 14).

• Meeting of Deans and Directors of Mathe-matics in Barcelona (SCM/Notıcies 15).

• Tribute to William Tutte(SCM/Notıcies 17).

• Tribute to Lluıs Antoni Santalo(SCM/Notıcies 18).

• 20th Anniversary of the CRM(SCM/Notıcies 20).

• Debate Forum on the Training of Mathemat-ics Teachers (SCM/Notıcies 21).

• Honorary Doctorate conferred on Jean-Pierre Serre by the University of Barcelona(SCM/Notıcies 21).

• EMS-SCM Joint Mathematical Weekend(SCM/Notıcies 22).

• Let’s Talk Books (SCM/Notıcies 22).

The evolution of the quality of this popu-larising journal throughout its 11 years of ex-istence (in terms of both form and content) isnothing more than the reflection of the vitalityof the Catalan Mathematical Society and theCatalan mathematical community in general,far more than the few nuggets contributed bythose of us who have worked on it in the pastand those who work on it today. I cannot there-fore finish without saying thanks to the manySCM members for their selfless collaboration inthe publication of SCM/Notıcies over the years.Without your contribution (as authors, editors,translators, reviewers, etc.) we could not havemade this journal happen. Thank you all.

Agustı Reventos TarridaFormer Editor of SCM/Notıcies

37

Reproduction of Featured Articles

For the sake of completeness, in this section we provide English translations of three articles thathave appeared recently in SCM/Notıcies. The first article appeared in SCM/Notıcies 21 on theoccasion of the doctorate honoris causa for J.-P. Serre at the University of Barcelona. The second,also from number 21, is an article on a prize awarded to a young Catalan mathematician at 4thEuropean Congress of Mathematics. Finally, the third article, from SCM/Notıcies 20 is a tributeto Miguel de Guzman, a distinguished and highly acclaimed Spanish mathematician who passedaway in April 2004.

Jean-Pierre Serre, doctor honoris causa at the UB

At about midday on 10 November 2005 in arather intimate ceremony presided by the rec-tor and held in the meeting room of the rec-tor of the University of Barcelona (UB), Jean-Pierre Serre was made an honorary doctor. Thehonorary doctorate was proposed by the UBnumber theory research group and was put for-ward for approval to the Board of Governorsby the Faculty of Mathematics, as stipulated inthe regulations.

Thus far, the news could have come froman agency. However, I think it would be a goodidea to give it further coverage, though with amore personal take.

Jean-Pierre Serre needs no introduction—at least not for most mathematicians interestedin topology, geometry, algebra or number the-ory. He is well known as a leading light in andmaster of these disciplines. There are many ar-ticles that discuss his personality and his sci-entific work. He is also notable for the numberof prizes and distinctions he has received (andcontinues to receive) throughout his career as amathematician.

But I am not going to talk about that: youwill find a perspective on this in the laudatiospeech made by the sponsor of the event, Dr.Pilar Bayer, in the conferring ceremony andwhich accompanies this article with her permis-sion.

We may agree or disagree with or be indif-ferent to the prizes or distinctions, but there isno question that if any mathematician deservesthe recognition, due to both his scientific workand his personality, it is Jean-Pierre Serre. Ina world in which non-media events are eithercondemned to obscurity or forgotten and wheresimilar events are often programmed solely forthe media coverage they generate, the confer-

ring of an honorary doctorate on Jean-PierreSerre by UB is a breath of fresh air that helpsto keep alive our faith in academic institutionsand in what is expected of them. Those of us inthe group sponsoring the event did not want itto be full of pomp, which would be the case ifit were to be held in the Paranimf theatre, withthe media and cameras everywhere distractingthe attention of the large audience that turnsup to a social affair. This, however, was not theintention and so a small room was chosen thatwould barely take 50 people, and even then noteverybody could get in and some had to listento the speeches from outside the open door.

Jean-Pierre Serre at the UB

The naturalness of Jean-Pierre Serre waspatent, even at this event. He came toBarcelona, where he gave a conference at the In-stitute for Catalan Studies (IEC) to commem-orate the 20th anniversary of the Mathemat-ical Research Centre and two further confer-ences at the Barcelona Number Theory Sem-inar (STNB). The first conference was at theIEC on 9 November. His conferring was on 10November, having given the first of his 2 con-ferences at the STNB first thing that morning.The following day, he gave the last of his series

38

of conferences. His speech before the rector wasnot exactly a speech in response to the lauda-tio. I shall quote from memory the translationof something Jean-Pierre Serre said during his

acceptance speech: “It would be rude to rejectthe laudatory comments in the speech —andworse to accept them”.

Artur TravesaUB

Laudatio

Monsieur le Recteur, Autorites academiques,Mesdames et Messieurs, Professeur Serre,

C’est pour moi un honneur, ainsi qu’une re-sponsabilite, de prononcer la laudatio du pro-fesseur Serre a l’occasion de son investiture entant que docteur honoris causa de l’Universitede Barcelone, sur la proposition de la Facultede Mathematiques.

Jean-Pierre Serre est l’un des chercheurs lesplus prises et les plus apprecies de la com-munaute mathematique internationale. Ne aBages, ville du sud de la France, il a fait sesetudes de mathematiques a l’Ecole NormaleSuperieure de Paris. Il a collabore au SeminaireCartan pendant quinze ans, en tant qu’etudiantde doctorat puis comme diplome. A partir de1948, il a ete un des redacteurs anonymes del’ouvrage de caractere encyclopedique Elementsde mathematique qui etait publie en Francedepuis 1939 sous le pseudonyme de N. Bour-baki.

Les debuts de sa carriere de chercheur onteu lieu pour l’essentiel au C.N.R.S. (CentreNational de la Recherche Scientifique), ainsiqu’a l’Universite de Nancy et a Princeton. Atrente ans, il a ete nomme titulaire de la chaired’Algebre et Geometrie du College de France,et il s’est consacre exclusivement a la recherche.Aujourd’hui, il est professeur emerite de cetteinstitution.

Jean-Pierre Serre a cultive la topologie,la geometrie, l’algebre et la theorie des nom-bres, traitant souvent ces matieres comme untout et rendant evidente sa competence par ladecouverte de connexions fascinantes. Ses pre-miers travaux sur la topologie algebrique com-prennent sa these sur l’homologie singuliered’espaces fibres, les espaces de lacets et les ap-plications au calcul de groupes d’homotopie despheres. Ses premiers travaux sur la geometriealgebrique comprennent le memoire sur lesfaisceaux algebriques coherents (FAC), qui a

ete source d’inspiration de la future theoriedes schemas. Ces contributions premieres,premonitoires d’une carriere exceptionnelle, luiont valu de recevoir la medaille Fields, la plusgrande distinction mathematique de l’epoque.Serre avait alors vingt-sept ans et, encore au-jourd’hui, il demeure le plus jeune laureat decette recompense qui s’adresse aux chercheurset chercheuses ages de moins de quarante ans.

Il est titulaire de nombreuses autres dis-tinctions parmi lesquelles on remarquera lamedaille d’or du CNRS, qui lui a ete decerneeen 1987. L’annee derniere (2003) il a recu le prixAbel de l’Academie des Sciences de Norvege,lors de sa premiere edition. Instaure avec uncaractere annuel en memoire du mathematiciennorvegien Niels Henrik Abel (1802-1829), ceprix a pour intention de compenser l’inexistencede prix Nobel pour les mathematiques.

Jean-Pierre Serre est l’auteur d’une œuvremathematique profonde et vaste, d’un grandimpact. Etant donne qu’il est absolument im-possible de la synthetiser en quelques mots, jeme limiterai a en presenter une ebauche, met-tant en relief certains aspects particuliers de soninfluence.

Il est l’auteur d’une serie d’ouvrages quisont devenus des classiques dans leurs do-maines de specialisation respectifs. La plupartont ete l’objet de nombreuses traductions eteditions. Les textes, Groupes algebriques etcorps de classes [1959], Corps locaux [1962],Cohomologie galoisienne [1964], Lie Algebrasand Lie Groups [1965], Algebre Locale. Mul-tiplicites [1965], Algebres de Lie semi-simplescomplexes [1966], ont represente pour nom-bre d’entre nous la possibilite de nous famil-iariser avec un materiel specialise d’acces dif-ficile. Leurs sujets, les problemes et les tech-niques qu’ils contiennent ont entraıne la forma-tion de chercheurs en algebre, en geometrie eten theorie des nombres. Dans le Departement

39

d’Algebre et Geometrie, et a partir d’un certainage, chacun de nous a quelque texte de Serre lied’une maniere ou d’une autre a une epoque deson passe.

Ses ouvrages Representations lineaires desgroupes finis [1968] et Cours d’arithmetique[1970] font partie de la bibliographie recom-mandee dans des cours que nous donnons pourla maıtrise. Son cours d’arithmetique a etetraduit en chinois, en anglais, en japonais eten russe. En 1995, vingt-cinq ans apres sapremiere edition, ce petit bijou a recu le prixSteele d’exposition mathematique decerne parl’American Mathematical Society.

D’autres ouvrages de Serre, comme Abelianl-adic representations and elliptic curves(McGill) [1968], Arbres, amalgames, SL2

[1977], Lectures on the Mordell-Weil Theorem[1989], Topics in Galois Theory [1992] ainsi queson apport a Cohomological Invariants in Ga-lois Cohomology [2003], integrent du materielde ses cours specialises. Dans un autre ordred’idees, la Correspondance Grothendieck-Serre,editee en 2001 et annotee par Serre lui-meme,aide a saisir la genese de la remarquable trans-formation qu’a subie la geometrie algebrique apartir de 1955.

Le noyau dur de l’œuvre de Serre est con-stitue par ses articles de recherche, qui ont etereunis par la maison d’edition Springer dans untotal de quatre volumes. Les Œuvres - CollectedPapers [1986 ; 2000] reunissent cent soixante-treize articles ordonnes chronologiquement, etelles constituent l’un des apports les plus en-richissants de la mathematique de notre temps.Les notes techniques qui accompagnent chaquearticle sont aussi d’une grande valeur.

Les articles de theorie des nombres sontdeja presents dans les volumes I et II, et ilsoccupent la presque totalite des volumes IIIet IV. Il ont eu une influence sur de nom-breuses recherches et ont joue un role fonda-mental dans les resultats les plus remarquablesobtenus ces dernieres annees dans cette disci-pline. Je voudrais juste signaler que la situa-tion exacte des representations de Galois modu-le p de degre 2, conjecturee par Serre [Oe 143,1987] et prouvee dans un cas tres particulier parKenneth Ribet [1990], a ete la cle de voute quia permis la demonstration d’Andrew Wiles duTheoreme de Fermat, huit ans plus tard [1995].On resolvait de cette maniere un probleme sur

les nombres qui, enonce au XVIIe siecle par lemathematicien et juriste francais Pierre de Fer-mat (1601–1665), etait etudie depuis trois centcinquante ans.

Serre est un mathematicien de premier or-dre qui ne fuit pas le contact avec les autres. Sesenseignements s’adressent de facon generiqueaux chercheurs et ils s’exercent au moyen deconferences dans des congres, des seminaires,etc., en plus des cours qu’il donnait chaqueannee au College de France. Ses dissertations,toujours des modeles du genre, constituent unabrege equilibre de concepts et de techniques ;ses questions deviennent souvent une sourced’inspiration pour ceux qui l’ecoutent. D’autrepart, son aversion pour l’erreur est proverbiale,ce qui implique un degre eleve d’exigence enverslui-meme et envers les autres. Arrive a ce pointde mon expose, je crois que c’est un devoir degratitude vers lui d’expliciter son influence surnous ainsi que la part de sa production qui nousa le plus frappes.

La premiere fois que j’ai ecoute le professeurSerre de vive voix c’etait en 1974, au cours desJournees arithmetiques de Bordeaux. J’etaisalors une etudiante de doctorat engage a laUB et la UAB et j’assistais a mon premiercongres international. Serre en a ete le premierconferencier : Valeurs propres des operateurs deHecke module l [Oe 104, 1975]. J’en ai tire... queje devais etudier beaucoup plus ! Plus tard, dutemps de mon contrat post-doctorat de plus detrois ans a l’universite allemande de Regens-burg, j’ai eu l’occasion de parler avec lui en di-verses occasions : a Bonn, a Oberwolfach, etc.Mais la rencontre la plus importante s’est pro-duite en 1982, lors d’une visite qu’il faisait aBarcelone, au cours de laquelle il a donne deuxconferences a l’Universite de Barcelone.

Avec Nuria Vila, ma premiere etudiante dedoctorat, nous lui avons expose certaines dif-ficultes qui avaient surgi lors de la resolutionde problemes de plongement galoisien. Sonaide directe en cette occasion s’est traduitepar une formule innovatrice, qui a permis dedeterminer l’obstruction a la resolubilite duprobleme a partir d’une equation definissantl’extension initiale. Cette formule, une desses questions sur la resolution effective desproblemes de plongement, et son appreciationposterieure d’une condition diophantienne is-sue des calculs, servirent a l’elaboration de

40

trois theses, donnant lieu a la premiere lignede recherche en theorie des nombres com-mencee dans l’Universite de Barcelone. La for-mule obtenue par Serre, publiee dans son arti-cle L’invariant de Witt de la forme Tr(x2) [Oe131, 1984], a constitue un point de referencede nombreux autres travaux, aussi bien d’iciqu’ailleurs.

Les articles et les ouvrages de Serre, et lemateriel qu’il nous a transmis pendant plus devingt ans, ont ete indispensables pour entamerchez nous de nouvelles lignes de recherche, et ilsont contribue a mener a bon port des projets derecherche, des publications, ainsi qu’une bonnepartie des quinze theses en theorie des nom-bres qui ont ete soutenues jusqu’a aujourd’huidans notre universite. De ce point de vue, jevoudrais souligner ses articles sur la topolo-gie des varietes algebriques en caracteristiquep [Oe 38, Mexico 1958] ; sur la bonne reductiondes varietes abeliennes [avec J. Tate; Oe 88,1968]; sur les proprietes galoisiennes des pointsd’ordre fini des courbes elliptiques [Oe 94,1972] ; sur les formes modulaires de poids 1[avec P. Deligne; Oe 101, 1974]; sur les ap-plications du theoreme de densite de Cheb-otarev [Oe 125, 1981] ; et sur les representationsgaloisiennes modulaires de degre 2 [Oe 143,1987]. Actuellement, certaines de ces lignes derecherche sont poursuivies avec succes par deschercheurs formes par l’Universite de Barceloneet exportes dans d’autres universites.

L’experience mathematique comporte denombreuses Ithaques, est pleine d’aventures,pleine de projets. Le Seminaire de Theoriedes Nombres issu dans les annees quatre-vingtde la collaboration de trois universites cata-lanes —l’Universite de Barcelone, l’UniversiteAutonome de Barcelone et l’Universite Poly-technique de Catalogne— reunit aujourd’huiune trentaine de personnes. Les travaux deSerre ont ete tout particulierement importantsdans les editions suivantes : de 1986–1987,consacree aux points rationnels des courbesalgebriques ; de 1987–1988 consacree aux con-jectures de Serre sur les representations de Ga-lois de degre 2 ; de 1989–1990, consacree auxstructures de Hodge-Tate ; de 1992–1993, con-sacree aux representations de Galois de degre 2 ;de 1993–1994, introductrice au theoreme deFermat-Wiles ; de 1994–1995, consacree auxformes modulaires et aux groupes de Galois,

ainsi qu’aux points de torsion des courbes el-liptiques ; de 1995–1996, consacree aux courbeselliptiques modulaires ; et de 2003–2004, con-sacree a l’etude des conjectures d’Artin, deSerre et de Fontaine-Mazur.

L’article [Oe 94, 1972], qui ouvre le volu-me III, est cle pour la comprehension des ac-tions de Galois dans les objets arithmetiquesgeometriques ; une partie de sa base theoriqueest fournie par le cours, difficile, sur les re-presentations l-adiques que Serre a donne aMcGill. Serre prouve dans ce travail que lesgroupes de Galois associes aux points d’ordrefini des courbes elliptiques sont ≪ aussi gros quepossible ≫. Le controle de l’action de l’inertiemoderee devient essentielle ; celle-ci se fait pardes produits de caracteres fondamentaux auxexposants bornes.

L’article [Oe 143, 1987], dans le volumeIV, est le fruit de sa connaissance profonde etvaste. Il a un precedent clair dans la conferenceprononcee aux Journees arithmetiques de Bor-deaux, douze ans auparavant. Ribet et Wilesont obtenu leurs resultats grace a un epsilonde la vision que le travail offre. Les conjecturesprecises de cet article sont aujourd’hui un ob-jectif convoite de la recherche d’avant-garde.Comme le dit Mme la docteur Vila, ≪ Serrea pu les formuler parce qu’il connaıt bien Serretout entier ≫.

La recherche mathematique catalane estune recherche jeune, sans tradition, qui a prisson envol au cours des dernieres decennies enpartant de la carence la plus absolue de textesmathematiques catalans ; sa langue de trans-mission habituelle est l’anglais. Diverses lignesde recherche contribuent a pallier ces manques.Pour notre part, la quasi-totalite des theses entheorie des nombres ainsi que certains ouvragesont ete rediges en catalan.

La recherche en theorie des nombres qui esteffectuee a l’Universite de Barcelone est trans-mise au professeur Serre de maniere reguliere etindependamment de la langue qui y est utilisee.L’attention qu’il a bien voulu preter a cetterecherche, evidente dans certaines lettres et ci-tations, a contribue de maniere inestimable asa diffusion et elle constitue pour nous un en-couragement permanent. Personnellement, jene connais aucune autre personne etrangere quiait lu davantage de mathematiques en catalanque lui.

41

L’Universite de Barcelone —fondee en1450—, en concedant a Jean-Pierre Serre letitre de docteur honoris causa, se joint au-jourd’hui aux universites de Cambridge (1978),Stockholm (1980), Glasgow (1983), Athenes(1996), Harvard (1998), Durham (2000), Lon-

dres (2001), Oslo (2002), Oxford (2003) et Bu-carest (2004) qui, depuis 1978, se sont honoreesen reconnaissant sa qualite scientifique et hu-maine.

Professeur Serre, merci.

Pilar BayerUB

Xavier Tolsa received a prize at 4ECM

Several prizes are awardedat the European math-ematical congresses, heldevery four years, to youngmathematicians in recog-nition of particularly rele-vant contributions. At the2004 European Congress,held in Stockholm, a prize was awarded toXavier Tolsa, an ICREA analyst attached toUAB. The only Catalan mathematician to havewon such a prize before Xavier Tolsa is RicardoPerez-Marco, who was awarded his prize at the1996 European Congress in Budapest for hav-ing solved several conjectures (by Arnold, Sad,Siegl, Moser and others) in dynamic systems.The work for which Tolsa was awarded theprize is an article [To1] published in the Swedishjournal, Acta Mathematica, which is known asone of the best in the world. In this article hesolves the problem of the semi-additivity of an-alytic capacity, posed in 1966 in an influentialarticle by Vitushkin. The work is the brilliantculmination of a series of prior contributionsby several mathematicians. David, Journee andSemmes (from the school of Yves Meyer, one ofthe creators of wavelet theory), Christ (Berke-ley), Nazarov, Treil and Volberg (Saint Pe-tersburg), Melnikov and Vitushkin (Moscow),Jones (Yale), Mattila (Helsinki) and Mateu andVerdera (Barcelona). Since one of Tolsa’s mainresults can be expressed very easily in terms in-telligible to any graduate, we provide it below.

The question involves analytical functionson the plane. Remember that Riemann showedthat if a function f , analytic in a disc, exceptperhaps at its centre, has the property that thevalues of f(z) are bounded as z approaches thecentre, then f extends to a function analytic inthe entire disc. Specifically, f has a continuous

extension to the centre. This is a surprising factthat strongly depends on the analyticity andthe dimension, and it is obvious that the realvariable analogue does not hold: the functionthat takes the value 1 on the interval (0, 1) and0 on the interval (−1, 0) does not extend contin-uously to 0. Painleve, a French mathematician,studied the removable sets for bounded analyticfunctions in his doctoral thesis of 1888. Theseare the compact sets K of the plane, with theproperty that if we take an analytic functionbounded at Ω \ K, for any open Ω, then thefunction extends analytically to all Ω. Painleveshowed that every set of null Hausdorff lengthis removable. He thus gained a dimension overRiemann. There was considerable activity re-garding the notion of removability in the firsthalf of the last century until Ahlfors, a Finnishanalyst of a strong geometric bent, asked in1947 whether it was possible to find geometriccharacterisations of the removable sets and, infact, called the question the Painleve problem.

In his article (see also [MTV]), Xavier Tolsashowed that a compact K is not removable ifand only if it is possible to construct a posi-tive measure µ in K, not null, which has thefollowing two properties:

1. For any disc D, the measure of the discdoes not exceed the radius:

µ(D) ≤ radi(D).

2. If R(z,w, ζ) denotes the radius of the cir-cumference that passes through points z, w andζ, then

∫ ∫ ∫

1

R(z,w, ζ)2dµ(z)dµ(w)dµ(ζ) < ∞.

Any reader will be able to see clues to theimportance of the above result in the factthat removability is described in terms that no

42

longer refer to analyticity and that only involvereal variable notions (measures) with geometriccontent (the radius R(z,w, ζ)). We note, how-ever, that it is, in principle, arguable whetherthe condition is geometric because it brings inthe existence of a measure satisfying specificconditions. The real question is the following:is the above condition geometric in the precisesense that it is a bilipschitz invariant? Remem-ber that a homeomorphism Φ of the plane isbilipschitz if it preserves distances modulo con-stants, i.e. if there exists a constant C ≥ 1 suchthat:

C−1|z−w| ≤ |Φ(z)−Φ(w)| ≤ C|z−w|, z, w ∈ C.

Strong evidence was presented in [GV] thatthis had to be true and in [To2] the invarianceof the removable sets in bilipschitz geometrywas confirmed, in another excellent article. ThePainleve problem can therefore be considered

solved and Mathematics has lost an open prob-lem but gained a first-rate mathematician.

C−1|z − w| ≤ |Φ(z) − Φ(w)| ≤ C|z − w|,z, w ∈ C.

References

[GV] J. Garnett and J. Verdera, Analytic capac-ity, bilipschitz maps and Cantor sets. Math.Res. Lett. 10 (2003), vol. 4, 515–522.

[MTV] J. Mateu, X. Tolsa and J. Verdera, The pla-nar Cantor sets of zero analytic capacity andthe local T (b) theorem, J. Amer. Math. Soc.16 (2003), vol. 1, 19–28.

[To1] X. Tolsa, Painleve’s problem and the semi-additivity of analytic capacity, Acta Math.190 (2003), 105–149.

[To2] X. Tolsa, Bilipschitz maps, analytic capacityand the Cauchy integral, Ann. of Math. 162

(2005), 1243–1304.Joan Verdera

UAB

Miguel de Guzman, in memoriam

Miguel de Guzman died suddenly in Madridon April 14, 2004. Having had the privilege tohave been his friend, I would like to share withthe members of the Catalan Mathematical So-ciety the memory of one who was undeniably akey figure in Spanish Mathematics and a goodfriend to Catalan mathematicians.

Miguel de GuzmanOzamiz was born inCartagena in 1936.He studied philos-ophy in Germanyand Mathematics inMadrid and earnedhis doctorate inChicago under thedirection of Profes-sor Calderon. He wasProfessor of Mathe-matical Analysis atthe Complutense University of Madrid, a mem-ber of the Royal Academy of Sciences, Chair-man of the International Commission for Math-ematical Education (1991-1998) and a visitinglecturer in many countries. But apart from the

details of his curriculum vitae, I would like totalk about him as a person. He was a friendlyman with deep ethical convictions and a spe-cial love of his family and friends and was ableto carry out intensive research into analysisand geometry while maintaining a tremendousvocation for education and the popularisationand promotion of everything surrounding theworld of Mathematics. He was an extraordinar-ily well educated and highly trained man whodevoted himself body and soul to transmit-ting his passion for Mathematics to the world.His goal was the future and he wanted to re-serve in it a place of honour for his beloveddiscipline: new research subjects, the new gen-erations that had to be trained, the socialperspective that had to be improved, theprogress of people, etc. He has left us trainedpeople, articles, books and, above all, manymemories with which to continue to promotehis ideas.

His intense life as a lecturer, populariser andpromoter of new initiatives has laid down mile-stones that today, in his eternal absence, serveas beacons.

43

He visited Catalonia many times to speakat congresses and conferences, as the authorof books, as a member of doctoral thesis ju-ries, etc.

Here he would find Albert Dou whom heso admired, some of his disciples, many lec-turers with whom he collaborated and manyfriends and followers. His last visits were for thetribute in Girona to our own Lluıs A. Santaloand for the inauguration at IEC headquarters

of the programme for promoting mathematicaltalent —a programme he had led successfullyin Madrid and which is now taking off in Cat-alonia.

Heaven has a new light. It is a star-shapedpolyhedron and anyone looking at it will dis-cover things there. It is him.

Thank you, Miguel Guzman, for your example.We will never forget you.

Claudi Alsina i CatalaUPC

Problem Section

From the first issues of SCM/Notıcies, one of the most popular sections of our Newsletter has beenthe Problem Section, where our readers can find and post mathematical questions and problemsbeing either curious or interesting (or both). And, of course, they can also propose solutions to theproblems announced in previous issues. For each issue, the editorial board chooses and publishesthe nicest problems and the best solutions. As a sample, we reproduce three of the problems thatappeared in this section.

Selection of posted problems with the corresponding solutions:

A52 SCM/Notıcies 16, December 2001.

(59th Annual William Lowell Putnam Mathe-matical Competition)

Let s be an arbitrary arc of the unit cir-cle, situated at the first quadrant. Let A be thearea of the trapezoidal region defined betweenthe arc and the “x” axis, and let B be the areaof the trapezoidal region defined between thearc and the “y” axis. Show that A + B onlydepends on the length of the arc s and not onits position.

Solution: (Redaction)

O

s

E

F

G H

K

C D

1S

2S

Let S1 be the area of the rectangle EGKF ,and S2 be the area of the rectangle CDHG.

It is clear that S1 = 2 · Area (OGK) be-cause the triangle and the rectangle have thesame basis and height. By the same reason,S2 = 2 · Area (OGH). Hence,

A + B = S1 + S2 + 2 · Area (GHK) =

= 2 · (Area (OGK)

+ Area (OGH) + Area (GHK)) =

= 2 · (Area Sector (OHK)) =

= Length of arc (s).

A59 SCM/Notıcies 18, January 2003. (Agerman suggestion for an International Mathe-matical Olympiad)

Let a, b and m be integral numbers suchthat

a2 + b2

ab + 1= m ≥ 0.

Show that, m is a square.

Solution: (Solution by Carles Romero, IES“Manuel Blancafort”, La Garriga)

i) If one of the numbers is zero, the propo-sition is trivial.

44

ii) Assume a = b. Then we have:

0 ≤ m =2a2

a2 + 1=⇒ 2a2 = ma2 + m

=⇒ (2 − m)a2 = m ≥ 0.

The only possibilities are m = 0 and m = 1,both being squares. The corresponding valuesfor a are a = 0 and a = ±1, respectively.

iii) It is not possible that a 6= 0 and b 6= 0have different signs, because a2 + b2 > 0 and sois ab + 1 > 0, hence ab ≥ 0.

iv) Let us assume now 0 < a < b. In thiscase,

b2 − amb + a2 − m = 0

is a degree two equation with respect to theunknown “b”, having the given value of b as asolution. Let b′ be the other solution (possiblyequal to b). They satisfy

b + b′ = am ; bb′ = a2 − m.

This is saying that if a and b produce the integerm > 0, then b′ = am − b and a also do. Hence,by (iii), b′ ≥ 0. Additionally, using a < b, wededuce that

ab′ ≤ bb′ = a2 − m < a2 =⇒ b′ < a.

Summarizing, if 0 < a < b produce the integerm > 0, then 0 ≤ b′ < a also do.

v) Repeating the previous process severaltimes, we obtain a strictly descending list of in-tegral numbers, no two consecutive ones beingof different sign. This implies that the list con-tains the zero. And immediately next to it thereis the positive integer

√m = m.c.d.(a, b).

A63 SCM/Notıcies 20, November 2004.

(Proposed by Pelegrı Viader, Universitat Pom-peu Fabra, Barcelona)

Let f : [0, 1] → R be a continuous functionsuch that f(0) = f(1). Show that:

a) For every positive integer n there exists ahorizontal cord of the graph of f having length1/n (a horizontal cord is a segment between twopoints in the graph of f having the same secondcoordinate).

b) f does not necessarily have horizontalcords of length not being the inverse of a posi-tive integer.

(Theorem of the universal cord)

Solution: (Solution by Albert Ferreiro Casti-lla, student)

The case n = 1 is trivial, by the hypothesisof the problem.

Suppose now n > 1. The problem consistson finding a pair of points, 0 ≤ x1 ≤ 1 and

0 ≤ x2 ≤ 1, in such a way that |x1 − x2| =1

nand f(x1) = f(x2). To show this, let us considerthe following set of points in the real plain:

Cn =

(

x, x +1

n

)

| 0 ≤ x ≤ 1 − 1

n

⊂ R2,

which is nothing else but the set of all pairs ofreal numbers between 0 and 1, and being ex-

actly1

napart from each other (and ordered,

first the small then the big one). Let us observethat this set Cn ⊂ R2 is a segment of a straightline in the plane. Let us consider the followingmap, which is clearly continuous:

F : Cn → R

F (x, y) = f(x) − f(y).

If we find (x, y) ∈ Cn with F (x, y) = 0, here wehave the pair of points generating the cord of

length1

n, and we are done. If we prove that, on

the set Cn, the map F takes simultaneously pos-itive and negative values, then Bolzano’s theo-rem will ensure us the existence of a zero, andwe are also done. Let us assume that F has nozero (and so, does not take simultaneously pos-itive and negative values), and we will find acontradiction.

Pick the pair

(

0,1

n

)

from the set Cn,

and let us apply F . Since F

(

0,1

n

)

6= 0,

we can assume without lost of generality, that

F

(

0,1

n

)

> 0.

Let us consider now the pair

(

1

n,2

n

)

: again

by our assumption,

F

(

1

n,2

n

)

> 0.

Repeating the same argument, we can concludethat

F

(

i − 1

n,

i

n

)

> 0 , i = 1, . . . , n.

45

Putting all these inequalities together, we have:

F

(

0,1

n

)

> 0 ⇒ f(0) > f

(

1

n

)

F

(

1

n,2

n

)

> 0 ⇒ f

(

1

n

)

> f

(

2

n

)

. . .

F

(

n − 1

n,n

n

)

> 0 ⇒

f

(

n − 1

n

)

> f(n

n

)

= f(1)

But then, f(0) > f(1), which contradictsthe hypothesis of the problem. Consequently,the map F is negative or zero on some the pairsin Cn. Thus, we have the result. We remark thefact that we have been looking for points whosedistance to each other is the inverse of a positiveinteger; this is essential sence, otherwise, thelist of points obtained in the argument wouldnot end at 1, which is what we need to contra-dict the hypothesis of the problem. That is, forlengths different from 1/n the given argumentis no longer working.

Let us consider now the second part of theproblem. The claim is obvious for the inverse of

every number between 0 and 1: it is clear that

no cord can exist with length1

0.5= 2 and this

is not the interesting case. We shall constructan example concerning the inverse of a numberbigger than 1:

Consider the following continuous functionon the interval [0, 1]:

g(x) =

x , si 0 ≤ x ≤ 1

4

−x + 1

2, si 1

4≤ x ≤ 3

4

x − 1 , si 3

4≤ x ≤ 1

It clearly satisfies g(0) = g(1) = 0. So the hy-pothesis hold, but there are no cords of lengthbigger than 1

2(see the graph of g).

x

0.2 0.4 0.6 0.8 1

-0.2

-0.1

0

0.1

0.2

g(x)

Mathematics PhD theses in Catalonia

Mathematics PhD theses in Catalonia

Three of the Catalan universities offer Mathe-matics courses, both at undergraduate and atdoctorate level. The Faculty of Mathematics ofUniversity of Barcelona, the Faculty of Sciencesof the Autonomous University of Barcelonaand the Faculty of Mathematics and Statis-tics of the Technical University of Catalonia allhave their corresponding doctoral programmesin Mathematics, in which doctors receive ongo-ing training in the different areas and speciali-ties represented in Catalonia.

The quality of these doctoral programmeshas increased considerably in recent years andis currently comparable to that of many of thebest European and American universities. Theoutgoing students are fully initiated and ready

to carry out research in their correspondingspecialities, as shown by the fact that manyof them participate actively in research groupsboth here and abroad and successfully devotethemselves to mathematical research. Further-more, following the trend of the most pres-tigious universities, there is a growing inter-est on the part of companies in the technol-ogy and financial sectors in hiring increasingnumbers of doctors in Mathematics in order todevelop their more specialised work. This is aclear indication of the vitality of our three doc-toral programmes and of the usefulness of high-level Mathematics, beyond the realm of pureresearch.

To give an idea of the subjects dealt with

46

in the different doctoral theses of recent years,we provide a list of the titles, doctoral studentsand thesis directors that have been mentionedin SCM/Notıcies since Number 18. For moreinformation, see the corresponding issue of thejournal or contact the coordinators of the cor-responding doctoral programme:

• Evolution of sex-ratio in structured popu-lation dynamics, defended on October 5th,2005 by Jordi Ripoll i Misse and directed byAngel Calsina i Ballesta at the University ofBarcelona.

• Hermeneutics of the diferential calculus inthe Europe of the XVIII century: from theAnalyse des infiniment petits de L’Hopital(1696) to Traite elementaire de calculdiffırentiel et de calcul integral de Lacroix(1802), defended on October 28th, 2004by Monica Blanco Abellan and directed byJosep Pla i Carrera at the University ofBarcelona and Autonoma de Barcelona.

• Localization and preservation of structuresin stable homotopy, defended on September10th, 2004 by Javier J. Gutierrez Marın anddirected by Carles Casacuberta at the Uni-versity of Barcelona.

• Consecutive and statistic motives in re-stricted permutations, defended on July 16th,2004 by Sergi Elizalde Torrent and directedby Marc Noy Serano at the Technical Uni-versity of Catalonia.

• Reducibility of quasi-periodic skew-productsand the spectrum of Schrodinger operators,defended on June 22th, 2004 by JoaquimPuig i Sadurnı and directed by Carles Simo iTorres at the University of Barcelona.

• On linear secret sharing schemes and dis-tributed cryptographic protocols, defended onJune 7th, 2004 by Vanesa Daza Fernandezand directed by Carles Padro Laimon at theTechnical University of Catalonia.

• Predictive control systems in irrigationditches: formulation and numerical simula-tion, defended on April 5th, 2004 by JuanAntonio Mantecon Baena and directed byJose Rodellar Benede and Manuel Gomez Va-lentın at the Technical University of Catalo-nia.

• Design and analysis of semantically securepublic key encryption schemes, defended on

April 5th, 2004 by David Galindo Chaconand directed by Sebastia Martı Mollevı at theTechnical University of Catalonia.

• Localization, proper actions and classify-ing spaces for discrete groups, defended onMarch 19th, 2004 by Ramon Jesus FloresDıaz and directed by Carles Broto Blanco atthe Autonomous University of Barcelona.

• Algorithmic problems on proximity and loca-tion under metric constraints, defended onNovember 21th, 2003 by Belen Palop del Rıoand directed by Ferran Hurtado at the Tech-nical University of Catalonia.

• Distribution models on the simplex, defendedon October 10th, 2003 by Gloria MateuFigueras and directed by Vera PawlowskyGlahn and Carles Barcelo i Vidal at the Tech-nical University of Catalonia.

• On symplectic linearization of singular La-grangian foliations, defended on September22th, 2003 by Eva Miranda Galceran and di-rected by Carlos Curras Bosch at the Univer-sity of Barcelona.

• Adaptive dynamics in an infinite dimensionalsetting, defended on July 16th, 2003 by SılviaCuadrado Gavilan and directed by AngelCalsina Ballesta at the Autonomous Univer-sity of Barcelona.

• Curvature integrals and integral geometry inthe hyperbolic space, defended on June 27th,2003 by Gil Solanes Farres and directed byEduard Gallego Gomez at the AutonomousUniversity of Barcelona.

• Algoritms and methods for robust geodetickinematic positioning, defended on June27th, 2003 by Julia Talaya Lopez and di-rected by Ismael Colomina Folch at the Tech-nical University of Catalonia.

• Analytic capacity and Riesz kernels, defendedon June 26th, 2003 by Laura Prat and di-rected by Joan Mateu and Joan Verdera atthe Autonomous University of Barcelona.

• Mathematical modelization and simulationof metal recuperation processes from resid-ual waters using hollow fiber supported liq-uid membranes, defended on June 25th, 2003by Graciela Benzal and directed by AmadeuDelshams and Anna Sastre at the TechnicalUniversity of Catalonia.

• Localizations and completions of asphericalspaces, defended on June 13th, 2003 by

47

Gemma Bastardas i Ferrer and directed byCarles Casacuberta at the Autonomous Uni-versity of Barcelona.

• Set of periods, topological entropy and com-binatorial dynamics for tree and graph maps,defended on June 13th, 2003 by David Juheri Barrot and directed by Lluıs Alseda andPere Mumbru at the Universitat Autonomade Barcelona.

• Improving evaluation codes, defended onJune 5th, 2003 by Maria Bras Amoros anddirected by Sebastia Xambo Descamps andMichael O’Sullivan at the Technical Univer-sity of Catalonia.

• Intervals of marks, defended on May, 2003by Lambert Jorba Jorba and directed byErnest Gardenes Martın at the University ofBarcelona.

• Analysis and construction of topologicalstructures useful for the modelization of in-terconnection networks, defended on April24th, 2003 by Eduardo A. Canale Betan-court and directed by Jose Gomez Martı andXavier Munoz Lopez at the Technical Uni-versity of Catalonia.

• On the dynamics of the Trojan asteroids,defended on April 29th, 2003 by FredericGabern Guilera and directed by Angel JorbaMonte at the University of Barcelona.

• Galois groups over Q with prefixed ramifica-tion conditions, defended on April 30th, 2003by Bernat Plans i Berenguer and directed byNuria Vila at the University of Barcelona.

• Decompositions of graphs into trees, defendedon February 20th, 2003 by Susana ClaraLopez Masip and directed by Anna Llado atthe Universitat Politecnica de Catalunya.

• Abelian varieties with quaternionic multipli-cation and their moduli, defended on January

30th, 2003 by Vıctor Rotger Cerda and di-rected by Pilar Bayer Isant at the Universityof Barcelona.

• Numerical study of Hopf bifurcations in thetwo-dimensional plane Poiseuille flow, de-fended on November 28th, 2002 by JosePablo Sanchez Casas and directed by AngelJorba at the Technical University of Catalo-nia.

• On the quasi-periodic Hamiltonian Andronov-Hopf bifurcation, defended on October 21th,2002 by Juan Ramon Pacha Andujar and di-rected by Merce Olle and Jordi Villanueva atthe Technical University of Catalonia.

• Geometric problems on inscription and par-tition of figures, defended on July 10th, 2002by Roser Guardia Riera and directed by Fer-ran Hurtado Dıaz at the Technical Universityof Catalonia.

• The origins of the nuclear engineering inBarcelona, the Catedrra Ferran Tallada(1955-1962) defended on July 2nd, 2002by Francesc Xavier Barca and directed byGuillermo Lusa at the Technical Universityof Catalonia.

• Cohomology and deformation theory forsemigroupal 2-categories, defended on June21th, 2002 by Josep Elgueta Monto and di-rected by Sebastia Xambo i Descamps at theTechnical University of Catalonia.

• On geometric separability, defended on June3st, 2002 by Carlos Seara Ojea and directedby Ferran Hurtado Dıaz at the Technical Uni-versity of Catalonia.

• Local cohomology modules supported onmonomial ideals, defended on May 24th, 2002by Josep Alvarez Montaner and directedby Santiago Zarzuela at the University ofBarcelona.

48

49

50

www.birkhauser.chTel.: +49 / 6221 / 345 0 Fax: +49 / 6221 / 345 42 29e-mail: orders@birkhauser.ch

Just released:

Darvas, G.

Symmetry2006. 456 p. SoftcoverISBN 3-7643-7554-X

The first comprehensive book on the topic in half a century explores recent symmetry – and symmetry breaking – related discoveries, and discusses the questions and answers they raise in diverse disciplines.

Haase, M.

The Functional Calculus for Sectorial Operators2006. 408 p. HardcoverISBN 3-7643-7697-XOT – Operator Theory: Advances and Applications, Vol. 169

This book contains a systematic (and partly axiomatic) treatment of the holomorphic functional calculus for unbounded sectorial operators. The account is generic so that it can be used to construct and interrelate holomorphic functional calculi for other types of unbounded operators.

Birkhäuser Customer Servicec/o SDCHaberstrasse 7D-69126 Heidelberg

Dragomir, S. / Tomassini, G.

Differential Geometry and Analysis on CR Manifolds2006. XIV, 487 p. HardcoverISBN 0-8176-4388-5PM – Progress in Mathematics, Vol. 246

This monograph provides a unified presentation of several differential geometric aspects in the theory of CR manifolds and tangential Cauchy-Riemann equations. It is motivated by clear exposition, many examples, explicitly worked-out geometric results, and stimulating unproved statements and comments referring to the most recent aspects of the theory.

at ICM 2006

Visit our booth (10-11) and get 20% discount on all available titles!

Clark, J. / Lomp, C. / Vanaja, N. / Wisbauer, R.

Lifting ModulesSupplements and Projectivity in Module Theory

2006. Approx. 408 p. SoftcoverISBN 3-7643-7572-8FM – Frontiers in Mathematics

While the theory of extending modules is well documented in monographs and text books, the purpose of this monograph is to provide a thorough study of supplements and projectivity conditions needed to investigate classes of modules related to lifting modules. The main text is augmented by a plentiful supply of exercises.

Shiryaev, A. / Peskir, G.

Optimal Stopping and Free-Boundary Problems2006. 520 p. HardcoverISBN 3-7643-2419-8LM – Lectures in Mathematics. ETH Zürich

The book aims at disclosing a fascinating connection between optimal stopping problems in probability and free-boundary problems in analysis using minimal tools and focusing on key examples. The general theory of optimal stopping is exposed at the level of basic principles in both discrete and continuous time covering martingale and Markovian methods.

Hu, P.-C. / Yang, C.-C.

Value Distribution Theory Related to Number Theory2006. 556 p. HardcoverISBN 3-7643-7568-X

The subject of the book is Diophantine approximation and Nevanlinna theory. Not only does the text provide new results and directions, it also challenges open problems and collects latest research activities on these subjects made by the authors over the past eight years.

Henrot, A.

Extremum Problems for Eigenvalues of Elliptic Operators

2006. 216 p. SoftcoverISBN 3-7643-7705-4FM – Frontiers in Mathematics

Problems linking the shape of a domain or the coefficients of an elliptic operator to the sequence of its eigenvalues are among the most fascinating of mathematical analysis. This book focusses on extremal problems. It provides also a self-contained presentation of classical isoperimetric inequalities for eigenvalues and 30 open problems.

Bernstein, J. / Hinich, V., / Melnikov, A. (Eds)

Studies in Lie Theory

Dedicated to A. Joseph on his Sixtieth Birthday

2006. XXII, 494 p. 2 illus. HardcoverISBN 0-8176-4342-7PM – Progress in Mathematics, Vol. 243

Dedicated to Anthony Joseph, this volume contains surveys and invited articles by leading specialists in representation theory. The focus here is on semisimple Lie algebras and quantum groups, where the impact of Joseph's work has been seminal and has changed the face of the subject.