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NEWSLETTER 

OF THE EUROPEAN MATHEMATICAL SOCIETY 

Feature 

Combinatorial Algebraic Topology 

p. 13 

S E 

M M 

E S 

History 

Mittag-Leffl er at Wernigerode 

p. 23 

June 2008 

Issue 68 

ISSN 1027-488X 

European 

Mathematical 

Society 

Interview 

Mihăilescu, Brezis 

p. 27, 37 

ERCOM 

Oberwolfach 

p. 41

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Editorial Team European 

Editors-in-Chief 

Martin Raussen 

Department of Mathematical 

Sciences 

Aalborg University 

Fredrik Bajers Vej 7G 

DK-9220 Aalborg Øst, 

Denmark 

e-mail: raussen@math.aau.dk 

Vicente Muñoz 

IMAFF – CSIC 

C/Serrano, 113bis 

E-28006, Madrid, Spain 

vicente.munoz @imaff.cfmac.csic.es 

Associate Editors 

Vasile Berinde 

Department of Mathematics 

and Computer Science 

Universitatea de Nord 

Baia Mare 

Facultatea de Stiinte 

Str. Victoriei, nr. 76 

430072, Baia Mare, Romania 

e-mail: vberinde@ubm.ro 

Krzysztof Ciesielski 

(Societies) 

Mathematics Institute 

Jagellonian University 

Reymonta 4 

PL-30-059, Kraków, Poland 

e-mail: Krzysztof.Ciesielski@im.uj.edu.pl 

Robin Wilson 


The Open University 

Milton Keynes, MK7 6AA, UK 

e-mail: r.j.wilson@open.ac.uk 

Copy Editor 

Chris Nunn 

4 Rosehip Way 

Lychpit 

Basingstoke RG24 8SW, UK 

e-mail: nunn2quick@qmail.com 

Editors 

Giuseppe Anichini 

Dipartimento di Matematica 

Applicata “G. Sansone” 

Via S. Marta 3 

I-50139 Firenze, Italy 

e-mail: giuseppe.anichini@unifi .it 

Chris Budd 

(Applied Math./Applications 

of Math.) 


Sciences, University of Bath 

Bath BA2 7AY, UK 

e-mail: cjb@maths.bath.ac.uk 

Mariolina Bartolini Bussi 

(Math. Education) 

Dip. Matematica – Universitá 

Via G. Campi 213/b 

I-41100 Modena, Italy 

e-mail: bartolini@unimo.it 

Ana Bela Cruzeiro 

Departamento de Matemática 

Instituto Superior Técnico 

Av. Rovisco Pais 

1049-001 Lisboa, Portugal 

e-mail: abcruz@math.ist.utl.pt 

Ivan Netuka 

(Recent Books) 

Mathematical Institute 

Charles University 

Sokolovská 83 

186 75 Praha 8 

Czech Republic 

e-mail: netuka@karlin.mff.cuni.cz 

Mădălina Păcurar 

(Conferences) 

Department of Statistics, 

Forecast and Mathematics 

Babes, -Bolyai University 

T. Mihaili St. 58-60 

400591 Cluj-Napoca, Romania 

madalina.pacurar@econ.ubbcluj.ro; 

madalina_pacurar@yahoo.com 

Frédéric Paugam 

Institut de Mathématiques 

de Jussieu 

175, rue de Chevaleret 

F-75013 Paris, France 

e-mail: frederic.paugam@math.jussieu.fr 

Ulf Persson 

Matematiska Vetenskaper 

Chalmers tekniska högskola 

S-412 96 Göteborg, Sweden 

e-mail: ulfp@math.chalmers.se 

Walter Purkert 

(History of Mathematics) 

Hausdorff-Edition 

Mathematisches Institut 

Universität Bonn 

Beringstrasse 1 

D-53115 Bonn, Germany 

e-mail: edition@math.uni-bonn.de 

Themistocles M. Rassias 

(Problem Corner) 


National Technical University 

of Athens 

Zografou Campus 

GR-15780 Athens, Greece 

e-mail: trassias@math.ntua.gr. 

Vladimír Souček 

(Recent Books) 

Mathematical Institute 

Charles University 

Sokolovská 83 

186 75 Praha 8 


e-mail: soucek@karlin.mff.cuni.cz 

Contents 


Society 

Newsletter No. 68, June 2008 

EMS Calendar ............................................................................................................................. 2 

Editorial ................................................................................................................................................. 3 

Meetings in Copenhagen ............................................................................................ 5 

Petition ................................................................................................................................................... 9 

Abel Prize 2008 to Thompson and Tits .................................................. 10 

Jahr der Mathematik 2008 – G. Ziegler ............................................................ 11 

Combinatorial Algebraic Topology – D. Kozlov ................................13 

On Platonism – R. Hersh ............................................................................................. 17 

Mathematical Platonism and its Opposites – B. Mazur ....... 19 

The meeting at Wernigerode – A. Stubhaug ................................... 23 

Interview with P. Mihăilescu .................................................................................. 27 

Personal Column .................................................................................................................... 35 

JEMS – an interview with H. Brezis .............................................................. 37 

100 th anniversary of ICMI – M. Bartolini Bussi ............................... 39 

ERCOM: Oberwolfach .................................................................................................... 41 

Book Review: Mittag-Leffl er biography – A. Jensen ............ 44 

Forthcoming Conferences .......................................................................................... 46 

Recent Books .............................................................................................................................. 51 

The views expressed in this Newsletter are those of the 

authors and do not necessarily represent those of the 

EMS or the Editorial Team. 

ISSN 1027-488X 

© 2008 European Mathematical Society 

Published by the 

EMS Publishing House 

ETH-Zentrum FLI C4 

CH-8092 Zürich, Switzerland. 

homepage: www.ems-ph.org 

For advertisements contact: newsletter@ems-ph.org 

EMS Newsletter June 2008 1

EMS News 

EMS Executive Committee EMS Calendar 

President 

Prof. Ari Laptev 

(2007–10) 


South Kensington Campus, 

Imperial College London 

SW7 2AZ London, UK 

e-mail:a.laptev@imperial.ac.uk 

and 


Royal Institute of Technology 

SE-100 44 Stockholm, Sweden 

e-mail: laptev@math.kth.se 

Vice-Presidents 

Prof. Pavel Exner 

(2005–08) 

Department of Theoretical 

Physics, NPI 

Academy of Sciences 

25068 Rez – Prague 


e-mail: exner@ujf.cas.cz 

Prof. Helge Holden 

(2007–10) 


Sciences 

Norwegian University of 

Science and Technology 

Alfred Getz vei 1 

NO-7491 Trondheim, Norway 

e-mail: holden@math.ntnu.no 

Secretary 

Dr. Stephen Huggett 

(2007–10) 

School of Mathematics 

and Statistics 

University of Plymouth 

Plymouth PL4 8AA, UK 

e-mail: s.huggett@plymouth.ac.uk 

Treasurer 

Prof. Jouko Väänänen 

(2007–10) 



Gustaf Hällströmin katu 2b 

FIN-00014 University of Helsinki 

Finland 

e-mail: jouko.vaananen@helsinki.fi 

and 

Institute for Logic, Language 

and Computation 

University of Amsterdam 

Plantage Muidergracht 24 

1018 TV Amsterdam 

The Netherlands 

e-mail: vaananen@science.uva.nl 

Ordinary Members 

Prof. Victor Buchstaber 

(2005–08) 

Steklov Mathematical Institute 

Russian Academy of Sciences 

Gubkina St. 8 

Moscow 119991, Russia 

e-mail: buchstab@mi.ras.ru and 

Victor.Buchstaber@manchester.ac.uk 

Prof. Olga Gil-Medrano 

(2005–08) 

Department de Geometria i 

Topologia 

Fac. Matematiques 

Universitat de Valencia 

Avda. Vte. Andres Estelles, 1 

E-46100 Burjassot, Valencia 

Spain 

e-mail: Olga.Gil@uv.es 

Prof. Mireille Martin- 

Deschamps 

(2007–10) 

Département de 

mathématiques 

Bâtiment Fermat 

45, avenue des Etats-Unis 

F-78030 Versailles Cedex, 

France 

e-mail: mmd@math.uvsq.fr 

Prof. Carlo Sbordone 

(2005–08) 

Dipartmento de Matematica 

“R. Caccioppoli” 

Università di Napoli 

“Federico II” 

Via Cintia 

80126 Napoli, Italy 

e-mail: carlo.sbordone@fastwebnet.it 

Prof. Klaus Schmidt 

(2005–08) 

Mathematics Institute 

University of Vienna 

Nordbergstrasse 15 

A-1090 Vienna, Austria 

e-mail: klaus.schmidt@univie.ac.at 

EMS Secretariat 

Ms. Riitta Ulmanen 



P.O. Box 68 

(Gustaf Hällströmin katu 2b) 

FI-00014 University of Helsinki 

Finland 

Tel: (+358)-9-191 51507 

Fax: (+358)-9-191 51400 

Telex: 124690 

e-mail: ems-offi ce@helsinki.fi 

Web site: http://www.emis.de 

2 EMS Newsletter June 2008 

2008 

30 June–4 July 

The European Consortium For Mathematics In Industry (ECMI), 

University College London (UK). www.ecmi2008.org 

11–12 July 

EMS Executive Committee Meeting, Utrecht (The Netherlands) 

Stephen Huggett: s.huggett@plymouth.ac.uk 

12–13 July 

EMS Council Meeting, Utrecht (The Netherlands) 

Stephen Huggett: s.huggett@plymouth.ac.uk; Riitta Ulmanen: 

ems-offi ce@cc.helsinki.fi 

http://www.math.ntnu.no/ems/council08/ 

13 July 

Joint EWM/EMS Workshop, Amsterdam (The Netherlands) 

http://womenandmath.wordpress.com/joint-ewmemsworskhop-amsterdam-july-13 

th -2008/ 

14–18 July 

5th European Mathematical Congress, Amsterdam 

(The Netherlands). http://www.5ecm.nl 

18–22 July 

European Open Forum ESOF 2008, session Can Mathematics 

help Medicine? Barcelona (Spain) 

http://www.esof2008.org 

1 August 

Deadline for submission of material for the September issue of 

the EMS Newsletter 

Vicente Muñoz: vicente.munoz@imaff.cfmac.csic.es 

3–9 August 

Junior Mathematical Congress 8, Jena (Germany) 

www.jmc2008.org 

16–31 August 

EMS-SMI Summer School at Cortona (Italy) 

Mathematical and numerical methods for the cardiovascular 

system 

dipartimento@matapp.unimib.it 

8–19 September 

EMS Summer School at Montecatini (Italy) 

Mathematical models in the manufacturing of glass, polymers 

and textiles 

web.math.unifi .it/users/cime/ 

28 September – 8 October 

EMS Summer School at Będlewo (Poland) 

Risk theory and related topics 

www.impan.gov.pl/EMSsummerSchool/ 

7–9 November 

EMS Executive Committee Meeting, Valencia (Spain) 

Stephen Huggett: s.huggett@plymouth.ac.uk 

2009 

5–8 February 

European Student Conference in Mathematics EUROMATH 

2009, Cyprus 

www.euromath.org

Editorial 

Martin Raussen (Aalborg, Denmark) 

Dear Newsletter readers, 

I have had the privilege to be in charge 

of the Newsletter of the European Mathematical 

Society for almost fi ve years, 

since the autumn of 2003. Time has now 

come for me to hand over to my successor 

Vicente Muñoz from Madrid, Spain. He and I are 

jointly producing this issue and the next and he will 

take over responsibility after that. I plan to stay on the 

Newsletter’s editorial board and do bits of work here 

and then. Although I have been very fond of my job, 

of encouraging and fi nding interesting articles and assembling 

them in the Newsletter, I must admit that I 

am now looking forward to a period with more time for 

my primary occupation of mathematical research and 

teaching. 

The production of a newsletter like this one is a collaborative 

task. It could not have been done without a 

helpful and encouraging editorial board. You can fi nd 

the list of editors on page 2, and I have to thank all members 

for their contributions, whether they were produced 

from their own hands or by their intervention. Some of 

the editors have been on board much longer than I have 

and I hope that many of them will continue their good 

work in the future. This is not to say that the Newsletter 

could not use “fresh blood”, and I would like to encourage 

volunteers to contact Vicente Muñoz who will 

compose a new editorial board at the beginning of next 

year. 

Since 2005, the Newsletter has appeared under the 

EMS Publishing House – in a new format, both in print 

and electronically. Cooperation with director Thomas 

Hintermann has been very friendly and effi cient. I am 

aware of the fact that the money the EMS pays for the 

production of the Newsletter just pays the hard costs 

and I am therefore particularly grateful for the support 

from the publishing house. By the way, have a look at 

the impressive development of the publishing house 

both in terms of journals and of book publications at 

www.ems-ph.org! Cooperation with our Swiss layout experts, 

Sylvia and Micha Lotrovsky, has been swift and 

friendly; likewise the handling of technical and thus 

TeXnical articles in the hands of Christoph Eyrich in 

Berlin. Thanks to all of them! 

The main idea with this Newsletter is to give the personal 

members of the EMS immediate “value” for their 

membership dues. The concept of the Newsletter has developed 

over the years; nowadays, it contains one or two 

feature articles usually surveying a newer mathematical 

development, one article of a mainly historical character, 

at least one interview with a well-known mathematician, 

and a presentation of a national mathematical society 

Editorial 

or of a mathematical research centre within Europe, and 

quite often also an article on news from Mathematical 

Education. Moreover, there are columns with news from 

the EMS, other interesting (mainly European) mathematical 

news, a featured book review, a conference listing 

carefully produced in Romania, the section “Recent 

Books” from the hands of our Czech colleagues, and fi - 

nally (biannually) “Solved and Unsolved Problems” with 

comments and solutions, and a Personal Column. 

It has not always been easy for this editor to fi nd volunteers 

willing to compose interesting feature articles. 

I am therefore extremely grateful for the support from 

editors of several membership journals of national mathematical 

societies who have allowed me to “reuse” some 

of their articles and thus to make them visible to more 

mathematicians throughout Europe. Particular thanks 

go to the French gazette des mathématiciens, the German 

Mitteilungen der DMV and (my personal favourite) the 

Dutch Nieuw Archief voor Wiskunde. 

It has been my privilege as editor to be invited, two or 

three times per year, to the weekend meetings of the executive 

committee of the EMS. These take place in various 

European cities, usually by invitation of a national 

mathematical society. The meetings are pleasant because 

the participants work hard together for a common goal: a 

better infrastructure and a higher recognition in politics 

and in the public for European mathematics and mathematicians. 

Even more could certainly be done if the EMS 

had more active support (bottom-up) from its members 

– and from more members! 

I do not know whether a leaving editor has the right 

to pronounce a wish. Anyway, I would like to address 

three wishes to the readers of this Newsletter issue 

– which this time is distributed in a higher circulation 

due to the European Congress in Amsterdam and to 

a campaign directed to all European departments of 

mathematics: 

- Please inform the new editor-in-chief or one of the 

other editors about articles that you fi nd interesting 

for mathematicians at large, be they from your own or 

from somebody else’s pen. Every (email) alert will be 

gratefully acknowledged. 

- The EMS would be able to work more effi ciently if 

it had more personal members – this would also increase 

the circulation of the Newsletter. Could you 

not try to convince one or more of your offi ce neighbours? 

Please have a look at the new EMS web page 

http://www.euro-math-soc.eu (still under construction). 

This page already offers many services to the community, 

e.g. announcements of jobs and conferences in addition 

to news. Under Membership you will fi nd many 

more reasons to join the EMS; in addition, you can now 

become an EMS member online. 

- If you fi nd this Newsletter worth reading, why not ask 

your department or your library for a subscription? 

Refer to http://www.ems-ph.org/journals/subscription. 

php?jrn=news . 

I am grateful for your support in these matters. 


Joint mathematical weekend in 

Copenhagen 

Martin Raussen (Aalborg, Denmark) 

Following the tradition of joint mathematical weekends 

that was launched in Lisbon (2003) and continued in Prague 

(2004), Barcelona (2005) and Nantes (2006), the Danish 

Mathematical Society (DMF) organised a mathematical 

weekend in Copenhagen earlier this year starting on Friday 

29 February and lasting until Sunday 2 March. More than 

160 participants, many of them from abroad, listened to 

more than 50 lectures, making this the biggest mathematical 

event held in Denmark for years. To start with, the mayor 

of Copenhagen had invited the participants for a reception 

buffet at the city’s impressive town hall. 

The meeting had been planned by an executive committee 

consisting of four mathematicians from the Copenhagen 

area, among them DMF president Søren Eilers, and 

moreover EMS vice-president Helge Holden. Time was 

provided for four plenary talks (by Xavier Buff, Toulouse; 

Nigel Higson, Pennsylvania State; Frank Merle, Cergy- 

Pontoise; and Stefan Schwede, Bonn) and six sessions: 

- Algebraic topology (organizers: Jesper Grodal, Ib 

Madsen) 

- Coding theory (Olav Geil, Tom Høholdt) 

- Non-commutative geometry/operator algebra (Ryszard 

Nest, Mikael Rørdam) 

- Dynamical systems (Carsten Lunde Petersen, Jörg 

Schmeling) 

- Algebra and representation theory (Jørn Børling 

Olsson , Henning Haahr Andersen) 

- Partial differential equations (Gerd Grubb, Helge 

Holden) 

A detailed program can be found at http://www.math. 

ku.dk/english/research/conferences/emsweekend/. 

Group photo at the end of the conference 

Concentrated audience 

EMS News 

Support for the meeting had been granted by the 

Danish science research council, the Danish and the European 

Mathematical Societies and the Department of 

Mathematics at Copenhagen University. A very enthusiastic 

group of young mathematicians from this department 

must be thanked for their effi cient help with all the 

practical aspects of the meeting. For example, session 

talks were synchronized in order to allow participants to 

switch at the sound of a horn, which was impossible to 

miss by even the most enthusiastic speaker! 

In summary, this was a very nice meeting that was 

useful for the whole audience, which ranged widely, not 

only with respect to their mathematical interests but also 

to their age; the attendees included not only many young 

PhD students but also several emeritus professors. 

The mathematical weekend has established itself as a 

good framework for a general medium sized mathematics 

conference. Who is going to organise the next one? 


EMS News 

Arthur Besse donates royalties to 

EMS-Committee 

Mireille Martin-Deschamps (Versailles Saint-Quentin, France) 

The Friends of Arthur Besse are a group of mathematicians 

who work in the fi eld of Riemannian geometry. 

Here is their story. In 1975, Marcel Berger convinced 

his students to organise a workshop about one of his favourite 

problems: understanding manifolds all of whose 

geodesics are closed. The workshop took place in Besseen-Chandesse, 

a very pleasant village in the centre of 

France, and it turned out to be so successful that the decision 

was made to write a book about the topic. Arthur 

Besse was born. 

The experience was so pleasant and enjoyable that 

Arthur did not stop there and settled down to write another 

book. This second book “Einstein manifolds” was 

eventually published in 1987. 

Years have passed since then. Arthur’s friends have 

scattered to various places. As they say “for Arthur himself, 

who never aimed to immortality, it may be time for 

retirement”. 

Nevertheless, a new version of Einstein Manifolds has 

recently been published and the royalties have been offered 

to the EMS with the suggestion that this money 

could be used to help young people from developing 

countries. Of course this offer was gratefully accepted by 

the Executive Committee of the EMS, who decided to 

put this money at the disposal of the Developing Countries 

Committee. 

To give a rough estimate, in the long run these duties 

could provide 9000 euros. 

6 EMS Newsletter June 2008

EMS News 

EMS Executive Committee meeting 

Copenhagen, Denmark, 3 March 2008 

Vasile Berinde, EMS Publicity Officer 

Copenhagen town hall 

Organized as a continuation of the successful Joint Mathematical 

Weekend EMS – Danish Mathematical Society 

(February 29 th –March 2nd), the fi rst EC meeting of 2008 

was held at the Institute for Mathematical Sciences, University 

of Copenhagen, on Monday, 3rd of March. This 

was an unusual working day, for, by tradition, the EC 

usually only meets at week-ends. 

Except for a last minute cancellation by Carlo Sbordone, 

all members of the Executive Committee were there: Ari 

Laptev (President, in the Chair), Pavel Exner and Helge 

Holden (Vice-Presidents), Stephen Huggett (Secretary), 

Jouko Vaananen (Treasurer), Olga Gil-Medrano, Mireille 

Martin-Deschamps, Victor Buchstaber, and Klaus Schmidt 

(members), together with Riitta Ulmanen, our quiet and 

effi cient executive secretary, Martin Raussen, the current 

Editor-in-Chief of the Newsletter, his successor (starting 

with the September issue), Vicente Muñoz, who thus made 

his début at an EC meeting, Mario Primicerio (representing 

applied mathematics within EC), and myself. 

After a healthy and refreshing half hour walk trip 

from Cabinn Scandinavia to the meeting venue, people 

felt ready to run over the agenda of the day. Our President, 

imperturbably relaxed, chaired an extremely dense 

marathon EC meeting, that started at 9.00 a.m. and lasted 

until almost 6.00 p.m., only broken for coffee and a quick 

lunch. 

From the business of what was a very pleasant and 

effi cient meeting, I’ll mention just a few matters: 

- President’s report (the effect of his letter to the Presidents 

of national mathematical societies; his very successful 

visits to Strasbourg and Brussels; his valuable 

visit to Serbia, the ISE matters) 

- Treasurer’s report (on the fi nancial year 2007) 

- Publicity Ofi cer’s report (the database of European 

departments of mathematics, the EMS poster) 

- Membership matters (new applications from Montenegro, 

Serbia, and Turkey; the outline of a fee structure; 

the new individual members formally accepted by the 

EC) 

- EMS Web Site (the EC thanked Helge Holden by acclaim 

for the progress with the web site that he demonstrated 

by taking the EC through the various pages; 

various suggestions for improvement were also made; 

our main web site will be freely maintained by University 

of Bremen, while the fees payment pages would 

be hosted in Helsinki, and linked to the membership 

database) 

- 5 th European Congress of Mathematics (the EMS 

would have a booth at the 5ECM, shared with the EMS 

publishing house) 

- 6 th European Congress of Mathematics (all three bidders 

– Krakow, Prague and Vienna – would be able to 

see all three revised bids; at the Utrecht Council, each 

bid would be given 15 minutes for their presentation, 

plus some time for questions) 

- Reports by standing committees 

- Publishing (the EC confi rmed the next Editor of the 

Newsletter, Vicente Muñoz, following an informal 

meeting the previous day, where he gave a report on 

his editorial views) 

- Closing matters (The President expressed the gratitude 

of all present to Søren Eilers and to Martin Raussen 

for the excellent arrangements which they had made 

for our meeting in Copenhagen). 

The next EC meeting, in Utrecht, preceding the EMS 

Council, would start at 14.00 on the 11th of July, and 

would continue from 09.00 to 11.00 on the 12th, the day 

the EMS Council meeting starts. The last EC meeting of 

2008 will be in Valencia, 7 th –9 th of November. 


News 

What happened to our friend and 

colleague Ibni Oumar Mahamat Saleh 

from Chad? 

Aline Bonami (Orléans, France) and Marie-Françoise Roy (Rennes, France) 

Mr. President of the Republic of Chad, 

Mr. President of the French Republic 

The French learned societies 

of mathematics: SMF 

(Société Mathématique de 

France), SMAI (Société 

de Mathématiques Appliquées 

et Industrielles) and 

SFdS (Société Française 

de Statistiques) launched 

the following petition on 

10 March 2008; it can be 

found at http://smf.emath. 

fr/PetitionSaleh/. 

We want to know the truth concerning our colleague, the 

mathematician Ibni Oumar Mahamet Saleh, a Chadian 

politician and former minister. He was abducted from his 

home on 3 February 2008 and we have had no news from 

him since then. 

Ibni Oumar Mahamat Saleh is, beyond his political 

activities, an active member of the mathematical community. 

He completed his mathematical education at the University 

of Orléans, where he defended his PhD thesis in 

1978. He was appointed as a professor at the University 

of N’Djamena in 1985. His numerous positions at the 

university included: 

- Chair of the Department of Mathematics (1985), 

- Director of the Center of Scientifi c Research (1986), 

- Rectorship (1990–1991). 

In spite of his heavy administrative and ministerial duties, 

Ibni Oumar Mahamet Saleh always succeeded in 

maintaining a high level of teaching standards. In order 

to improve the scientifi c level of teachers at the 

University of N’Djamena, he negotiated in 1991 a collaboration 

between the University of Orléans and the 

University of N’Djamena, in association with INSA in 

Lyon and the University of Avignon. This agreement is 

still in force and has been very successful. In addition to 

its goal of training, it has allowed Chadian teachers to 

establish stable and fruitful contacts with European and 

African Universities. Even when called to other responsibilities, 

Ibni Oumar Mahamet Saleh was instrumental 

in the success of the project. Several times he visited the 

department of mathematics in Orléans as part of this 

framework. 

Since 3 February, contradictory information has circulated 

about Ibni Oumar Mahamat Saleh. The two other 

opponents, who were arrested the same day, have been 

released. One of them thinks that our colleague died, 

while his family thinks that he is still alive. 

We want to know the truth. » 

The petition has received over 2450 signatures from 

the mathematical community around the world, particularly 

from many mathematicians in Africa who knew Ibni 

as a colleague or as a professor. 

The petition, the list of signatories, messages of information 

to the signatories and several documents about 

the case can be found at http://smf.emath.fr/Petition- 

Saleh/. 

A blog of information has been set up by his family 

and friends at http://prisonniers-politiques.over-blog. 

com/. 

So far we have received no answer from the Chadian 

and French authorities and we fear that we may not to 

see our friend and colleague again. 

We invite you to sign the petition. 

Aline Bonami [Aline.Bonami@univ-orleans.fr] and 

Marie-Françoise Roy [marie-francoise.roy@univrennes1.fr] 


The Centre de Recerca Matemàtica (CRM, Bellaterra, 

Spain) organizes, jointly with the EMS, a scientifi c session 

at ESOF2008. 

The title of the session is Can Mathematics help Medicine? 

It is part of the activities on the theme “Engineering 

the Body”. 

The programme is as follows: 

Monday, July 21, 8:30–10:00 

Alfi o Quarteroni (Politecnico di Milano, Italy and Ecole 

Polytechnique Fedérale Lausanne, Switzerland) 

Mathematical models for the cardiovascular system 

SIMAI2008 

International conference in Rome (Italy) 

organized in cooperation with SIAM 

The Italian Society for Applied 

and Industrial Mathematics 

(SIMAI) will hold its 9th Congress 

in Rome, Italy from September 15th to 19th . This 

international event takes place every two years. This 

time it is being organized in cooperation with SIAM. 

The hosting environment will be the beautiful downtown 

of Rome. 

European Student 

Conference in Mathematics 

EUROMATH–2009 

5–8 February 2009, Cyprus; www.euromath.org 

This is a conference organized by the 

Cyprus Mathematical Society in cooperation 

with the European Mathematical 

Society, the Ministry of Education 

of Cyprus, the University of 

Cyprus and the Thales Foundation. 

The conference will consist of several workshops/symposiums 

covering many themes. Suggestions for workshops, 

symposiums, sessions and exhibitions for the conference 

are welcome. More information can be found at www.euromath.org; 

alternatively, please contact the chair of the 

organizing committee at makrides.g@ucy.ac.cy 

News 

Dominique Chapelle (INRIA-Rocquencourt, France) 

Modeling and estimation of the cardiac electromechanical 

activity 

Peter Deufl hard (Zuse Institute Berlin, Germany) 

The challenge of electrocardiology: model hierarchy and 

multiscale simulation. 

Moreover, ESOF2008 hosts the following plenary talks: 

Marcus Du Sautoy (University of Oxford, UK) 

Mathematics: creative art or useful science? 

Eva Bayer-Flückiger (Ecole Polytechnique Fédérale 

Lausanne, Switzerland) 

The Science of Communication: number theory and 

coding 

For more information, please consult 

http://www.esof2008.org . 

The main (invited) speakers will be Antonio Ambrosetti 

(SISSA, Trieste), Douglas N. Arnold (University of Minnesota, 

USA), Nicola Bellomo (Politecnico di Torino), Giovanni 

Ciccotti (Università di Roma ‚‘La Sapienza’’), Nicholas 

J. Higham (University of Manchester, UK), and Alfi o 

Quarteroni (Politecnico di Milano & EPFL Lausanne). 

Besides, there will be a number of minisymposia (usually 

several dozens), round tables, prizes to young mathematicians, 

and interactions with industrial representatives. 

More information (in progress) can be found at the 

SIMAI site http://www.iac.rm.cnr.it/simai/simai2008/ . 

Note that all associates to a member society of EMS 

are entitled to the reduced conference fee as the members 

of SIMAI. 

The themes of interest are: applications of mathematics, 

mathematics and sciences, mathematics and life, 

mathematics and technology, mathematics and social 

sciences, mathematics and space, fractals and geometry, 

mathematics and economy, mathematics and literature, 

mathematics and music, mathematics and law, mathematics 

and statistics, the history of mathematics, mathematics 

and society, mathematics and Europe, mathematics and 

philosophy, mathematics and computer science, and famous 

numbers. 

Students aged between 12 and 18 from any European 

or international school who may be interested in attending 

should send a message with their full address, fax and 

email to the organizing committee. Registration should 

be completed by 30 November 2008. The deadline for 

submission of abstracts is 17 October 2008. Presented 

papers will be reviewed and invited for publication in the 

proceedings of the conference, which will be published 

after the event. 

The registration fee is 75 euros for students and 150 

euros for teachers. 


News 

Abel Prize 2008 

John Griggs Thompson, Jacques Tits 

(Photo: University of Florida and Jean- 

François Dars/CNRS Images) 

The Norwegian Academy 

of Science 

and Letters has 

decided to award 

the Abel Prize for 

2008 to John Griggs 

Thompson (Graduate 

Research Professor, 

University of 

Florida) and Jacques 

Tits (Professor Emeritus, 

Collège de 

France, Paris) 

“for their profound achievements in algebra and in particular 

for shaping modern group theory”. 

Modern algebra grew out of two ancient traditions in 

mathematics, the art of solving equations, and the use of 

symmetry as for example in the patterns of the tiles of 

the Alhambra. The two came together in late eighteenth 

century, when it was fi rst conceived that the key to understanding 

even the simplest equations lies in the symmetries 

of their solutions. This vision was brilliantly realised 

by two young mathematicians, Niels Henrik Abel and 

Evariste Galois, in the early nineteenth century. Eventually 

it led to the notion of a group, the most powerful way 

to capture the idea of symmetry. In the twentieth century, 

the group theoretical approach was a crucial ingredient 

in the development of modern physics, from the understanding 

of crystalline symmetries to the formulation of 

models for fundamental particles and forces. 

In mathematics, the idea of a group proved enormously 

fertile. Groups have striking properties that unite 

many phenomena in different areas. The most important 

groups are fi nite groups, arising for example in the study 

of permutations, and linear groups, which are made up 

of symmetries that preserve an underlying geometry. The 

work of the two laureates has been complementary: John 

Thompson concentrated on fi nite groups, while Jacques 

Tits worked predominantly with linear groups. 

Thompson revolutionised the theory of fi nite groups 

by proving extraordinarily deep theorems that laid the 

foundation for the complete classifi cation of fi nite simple 

groups, one of the greatest achievements of twentieth century 

mathematics. Simple groups are atoms from which 

all fi nite groups are built. In a major breakthrough, Feit 

and Thompson proved that every non-elementary simple 

group has an even number of elements. Later Thompson 

extended this result to establish a classifi cation of an important 

kind of fi nite simple group called an N-group. At 

this point, the classifi cation project came within reach and 

was carried to completion by others. Its almost incredible 

conclusion is that all fi nite simple groups belong to certain 

standard families, except for 26 sporadic groups. Thompson 

and his students played a major role in understanding 

the fascinating properties of these sporadic groups, 

including the largest, the so-called Monster. 

Tits created a new and highly infl uential vision of groups 

as geometric objects. He introduced what is now known as 

a Tits building, which encodes in geometric terms the algebraic 

structure of linear groups. The theory of buildings is 

a central unifying principle with an amazing range of applications, 

for example to the classifi cation of algebraic and 

Lie groups as well as fi nite simple groups, to Kac-Moody 

groups (used by theoretical physicists), to combinatorial 

geometry (used in computer science), and to the study 

of rigidity phenomena in negatively curved spaces. Tits’s 

geometric approach was essential in the study and realisation 

of the sporadic groups, including the Monster. He also 

established the celebrated “Tits alternative”: every fi nitely 

generated linear group is either virtually solvable or contains 

a copy of the free group on two generators. This result 

has inspired numerous variations and applications. 

The achievements of John Thompson and of Jacques 

Tits are of extraordinary depth and infl uence. They complement 

each other and together form the backbone of 

modern group theory. 

This citation is taken from www.abelprisen.no 

Visit the Abel web site: www.abelprisen.no/en/abel. 

Under the auspices of the Abel prize and with EMS vice-president Helge Holden (NTNU, Trondheim) as the driving 

force, a comprehensive web site collecting together the available historical material about Niels Henrik Abel 

(1802–1829) has been established. It contains: 

- The only known painted portrait of Abel. 

- A personal biography (by Arild Stubhaug) and a scientifi c biography (by Christian Houzel). 

- Abel’s collected works and related material, scanned and available in pdf format. 

- Even several of Abel’s original handwritten manuscripts. 

Moreover, the web site contains biographical and mathematical literature on Niels Henrik 

Abel and his work (most of it scanned) and photos of Abel memorabilia (monuments, stamps, 

banknotes, medals and so on). 

Have a look and enjoy! 


Mathematics Year 2008 in Germany 

Günter M. Ziegler (Berlin) 

2008 was offi cially declared a “Mathematics Year” in 

Germany. This created an unprecedented opportunity 

to work on the public’s view of the subject. Although 

Mathematics Year 2008 is primarily “a German affair”, 

I believe that a number of the lessons we have learnt in 

preparing the Year and in promoting it to the media may 

be of interest to the international readership of the EMS 

newsletter. 

Since 2000, which was a “Year of Physics” in Germany, 

the German Federal Ministry of Science and Education 

has dedicated each year to one particular science. Topics 

have been the natural sciences like biology, chemistry, 

geology and computer science, and also humanities in 

2007. The Science Years in Germany have now acquired 

a number of well-tested and successful components: 

- Big events, among them the opening and closing gala 

and the week-long “Science Summer”. 

- Exhibitions, including a large one on the “Science Ship” 

that travels on the Rhine, Danube and Elbe rivers all 

summer. 

- A major media and public relations campaign. 

A fourth component is new for 2008; much more than 

in previous science years we are working to reach the 

schools (teachers, parents and thus students). 

The motto for Mathematics Year 2008 is “Mathematics. 

Everything that counts!”. The posters and activities 

for the schools declare: “You’re better at math than you 

think!”. 

Four partners carry Mathematics Year 2008: besides 

the Federal Ministry of Science and the “Science in Dialogue” 

agency (which represents the German Science 

Foundation and the major research organizations such 

as the Max Planck Society), there are the Deutsche Telekom 

Foundation and the German Mathematical Society 

(DMV). I am acting as the DMV president; it is important 

to have an offi cial function when communicating 

with politicians. We have a budget of roughly 7.5 million 

euro for the year; that sounds like a lot of money but the 

money is soon gone when you get into professional public 

relations, organizing big events, etc. 

News 

A professional event and advertising agency in Berlin has 

designed the look of the Year (logo, print and web design), 

organized the major events and run the editorial (and 

campaign) offi ce. However, in contrast to the approach of 

previous years, we insisted on having an additional “contents 

offi ce” for the media work. This is where we take 

advantage of expertise from the community, make sure 

it is represented in the Year, and make sure that there’s 

“math inside” the publications (and that the mathematics 

is correct). The hope is that when the Year is over, the 

mathematics content offi ce will keep running as an active 

platform for promoting mathematics to the public. 

The four major partners that run the year have identifi 

ed several aims that they would like to attain or that 

they intend to communicate. One is that mathematics is 

multi-faceted. It does include “learning to calculate” but 

also much more – mathematics is high-tech, it is art, it is 

puzzles, etc. Our main message for the year is: “There is 

lots to discover!”. People who think they don’t like mathematics 

haven’t seen much of it. Therefore we try to show 

people sides of mathematics they have not seen yet – or 

show them aspects that they had not identifi ed as being 

connected to mathematics. 

About one hundred large and small exhibitions all 

over Germany present different aspects of mathematics: 

mathematics institutes show historic objects and science 

centres have developed hands-on objects for children of 

different age groups (some exhibitions focus on numbers, 

i.e. where they come from, what they are good for, numbers 

in nature, numbers in everyday life, lucky numbers 

and so on). The Mathematical Research Institute Oberwolfach 

has developed and promoted an exposition “Imaginary”, 

showing singularities of algebraic surfaces and 

Opening session, from left to right: Klaus Kinkel, president of the 

German Telekom Foundation, Federal Minister of Education and 

Research Annette Schavan, DMV president Günter M. Ziegler, and 

Gerold Wefer, president of Science in Dialogue 


News 

other visually-striking geometric objects, mostly in three 

dimensions and partly rendered as colourful plots on 

plexiglass. With the relevant software – free to download 

– every visitor has the chance to create objects themselves. 

A major German weekly newspaper (DIE ZEIT) 

has launched a competition for the most beautiful object 

created on its web site using the Imaginary software. 

We frankly and actively admit that doing mathematics is 

diffi cult. Don’t try to say it’s all easy – it is not true and people 

will not believe you. We rather argue that mathematics 

is diffi cult but you need it. Since mathematics is diffi cult it 

is interesting for the brightest. We state that mathematics is 

everywhere – it’s just that one often doesn’t notice it. This 

has turned out to be an aspect that the media are very interested 

in. There are various columns in newspapers and 

magazines that focus on mathematics in everyday life. One 

point we stress is that every one of us uses mathematics 

daily without thinking: when comparing prices of goods, 

handing over the right amount of cash to a salesperson, 

knowing how long to park or to travel here and there, etc. 

The other point we make is that mathematics is hidden in 

many objects in daily use like iPods (compression techniques), 

automobiles (aerodynamics, navigation, optimization 

of electronics), communication, medicine, drug design 

and even sports (tactical manoeuvres, scoring systems). 

We also think it is important to show that not only 

doing mathematics but also watching mathematics can 

be fun – for instance as part of fi lm plots. An active group 

of mathematicians connected to my colleague Konrad 

Polthier from Free University in Berlin have organized a 

Mathematics Film Festival with international help. They 

created a database of mathematics fi lms ranging from 

famous blockbusters to local low budget productions at 

universities. They negotiated contracts with fi lm distribution 

companies under which some major movies have 

become available for free public viewing occasions. Some 

fi fty people and institutions in Germany have now registered 

on the database and have consequently announced 

local fi lm festivals consisting of mathematics fi lms from 

this database. 

Even if only a third of the year is over at the time of 

writing, there are several lessons that we have learned already: 

1. Don’t try to teach. There’s no hope that people will 

know more mathematics at the end of the year. If 

many people think of mathematics as something interesting 

at the end of the year, we will have been 

very successful. 

2. Images, colours, graphics and photographs are important. 

Several mathematics calendars were produced 

for the year, with some great images, and they immediately 

sold out! 

3. Faces are also important. A subject is “abstract” for 

the media as long as they don’t have people to talk to 

and individuals to write about. For all the press materials, 

we are presenting and portraying mathematicians 

“to talk to”. 

4. Talk to the press. Press releases are one thing but you 

also have to talk to the key editors about topics that 

Swimming science center with a math exhibition 

you can present especially for them. My experience is 

that they are interested! 

5. Use professionals. For us, the year is an opportunity 

to get help and learn from an advertising agency but 

also from all the other major players. For example, 

the Deutsche Telekom Foundation has been funding 

mathematics education projects for years and they 

are also sponsoring new programs for the education 

and development of mathematics teachers. 

6. Make it a community effort. We are working hard 

to get hundreds of people from all over Germany 

involved, inviting people to become “math makers” 

for the Year. This is the only way to have activities 

all over the country. A “top down” campaign cannot 

have a broad effect. 560 people had already been registered 

as “math makers” by the end of April. 

7. Use the opportunities. The physics community (represented 

by the German Physics Society) profi ted a 

lot from the Year of Physics 2000 and used it to build 

infrastructure, enlarge their membership base and 

professionalize their web, print and media appearance. 

Mathematicians all over Germany are working 

hard to grab the opportunities. 

This is my personal, preliminary collection of lessons 

learned. It might be revised in the course of the year. Still, 

many different people and groups and most of the media 

are stating that the Year of Mathematics in Germany is 

already a success. It is worth the effort and I would like 

to encourage mathematicians in Europe to start similar 

projects. If such opportunities arise for greater, national 

visibility for mathematics projects then grab them. 

A huge campaign like the German Mathematics Year 

of course takes a lot of effort but it defi nitely seems to be 

worth that effort. We’ve made a major try and have got 

some major aspects right but we are also eager to learn in 

the process. Thus, your comments are very welcome! 

Günter M. Ziegler [ziegler@math.TU-Berlin.DE] is a 

professor of mathematics at Technische Universität Berlin 

and the president of the German Mathematical Society 

(DMV). He acknowledges the kind assistance with this 

article of Thomas Vogt [vogt@jahr-der-mathematik.de] 

from the Mathematics Year Contents Offi ce situated at 

Technische Universität Berlin. 


Combinatorial Algebraic Topology 

Dmitry N. Kozlov (Bremen, Germany) 

1 Introduction 

Combinatorial Algebraic Topology is a contemporary mathematical 

discipline which transcends several classical branches 

of study. Discrete mathematics and algebraic topology should 

perhaps be mentioned in the first batch, but further connections 

exist to such areas as algebra and category theory. Outside 

of classical mathematics there are important applications 

as well – we mention here connections to theoretical computer 

science, via complexity theory, and to vision recognition, 

via point cloud data and persistent homology. 

The dramatis personae of Combinatorial Algebraic Topology 

are cell complexes which are combinatorial both locally 

and globally. Being combinatorial locally is a well-studied 

feature in algebraic topology. It means that the cell attachment 

maps can be described by discrete data rather than by 

arbitrary continuous maps. Often one simply considers simplicial 

complexes, but also more intricate constructions exist. 

Being combinatorial globally is a feature more intrinsic 

to Combinatorial Algebraic Topology. Expressed succinctly, 

it means that the cells are indexed by combinatorial objects 

and that the cell attachments are encoded by some combinatorial 

rules, or more generally by algorithms, applied to the 

combinatorial objects which index the involved cells. 

The main aspiration of the subject is then to understand 

how the algebraic invariants of such combinatorial cell complexes, 

e.g., their Betti numbers, depend on the discrete data. 

At best one may be able to completely describe these algebraic 

structures in the language of combinatorics; either by 

using existing gadgets, such as matroids, or by inventing new 

ones. The mechanisms of computation themselves are sometimes 

purely combinatorial – such as clever machinations with 

labellings, orderings, and the like. When these fail, adaptations 

of central methods of computation used in algebraic 

topology, such as spectral sequences, can be tried. 

Rather than risking to confuse the reader with too many 

general words, we have chosen to present our subject by means 

of three examples. In the first one we show how discrete methods 

can be used in existing topological situations. In the second 

one the tables are turned: Topological constructions are 

used to gain knowledge about discrete objects. The last example 

highlights connections between algebraic topology and 

complexity theory. 

2 Goresky-MacPherson formula in 

arrangement theory 

Combinatorial Algebraic Topology can help to analyze topological 

spaces which are important elsewhere if the invariants 

of the spaces which we would like to compute are completely 

determined by the combinatorial data. When they are, we often 

end up having a meaningful and deep connection between 

topology and combinatorics. 

Feature 

One representative case is determining some of the most 

important invariants, i.e., the cohomology groups of certain 

spaces related to arrangements. To start with recall that a subspace 

arrangement is a collection A = {A1,...,An} of finitely 

many linear subspaces in k d ,wherek = R or k = C, such that 

Ai �⊇ A j for i �= j. One of the spaces related to such an arrangement 

is its complement MA := k d \ �n i=1 Ai whose algebraic 

invariants are of interest in various applications in such areas 

as algebraic geometry and singularity theory. 

A classical example is obtained by taking the special hyperplane 

arrangement Ad−1, called the braid arrangement, 

consisting of all hyperplanes xi = x j, for1≤i < j ≤ d. The 

complement of the braid arrangement Ad−1 in Rd consists of 

d! contractible pieces, indexed by all possible orderings of the 

coordinates. 

The combinatorial data which is standardly associated to 

an arbitrary subspace arrangement A is the so-called intersection 

lattice LA , which is the set 

� 

KI = � 

� 

Ai, for I ⊆{1,...,n} ∪{k d } 

EMS Newsletter June 2008 13 

i∈I 

with the order given by reversing inclusions: x ≤L y if and 

A 

only if x ⊇ y. In particular, the minimal element of LA is kd , 

denoted ˆ0, and the maximal element is � 

KI∈A KI. Forexample, 

in the case of the braid arrangement, the intersection lattice 

is the so-called partition lattice Πd. This is the lattice 

whose elements are indexed by set partitions of {1,...,n}, 

with the partial order given by refinement. 

It was discovered by Goresky and MacPherson, as a corollary 

of their Stratified Morse theory, see [GoM88], that the cohomology 

groups of the complement of a subspace arrangement 

A are determined by the combinatorial data LA : 

Theorem 2.1. (Goresky-MacPherson) 

For an arbitrary subspace arrangement A we have 

�H i (MA ) ∼ = � 

�H codimR(x)−i−2(∆(ˆ0,x)), 

x∈L ≥ˆ0 

A 

where ∆(ˆ0,x) denotes the nerve of the interval in LA viewed 

as a category. 

The last words of Theorem 2.1 need a little explanation. As 

defined, LA is a partially ordered set, hence so is the interval 

(ˆ0,x). The latter consists of all elements y of LA , such that 

ˆ0 < y < x, equipped with the partial order induced from the 

one on LA . Any poset P can be seen as a category whose 

elements are the elements of P, with a unique morphism from 

x to y, forx,y ∈ P, whenever x ≥ y. The transitivity of the 

partial order guarantees the existence of the composition and 

the fulfilment of the associativity axiom. 

There is a standard way to associate a topological space, 

in fact, even a structure of a simplicial set, to any category, 

which is called the nerve of that category. In the special case

Feature 

of a poset we get a simplicial complex, the so-called order 

complex, whose vertices are the elements of the poset, and 

whose simplices are all chains (fully ordered subsets) of P. 

There are many combinatorial tools to determine cohomology 

groups of an order complex of a poset. Thus, in particular, 

Theorem 2.1 means that the cohomology groups of the complements 

of specific, combinatorially defined arrangements 

can often be found by purely discrete methods. 

One can turn a real subspace arrangement A into a complex 

one A C by taking the linear subspaces in Cd defined by 

the same equations as those in A . Certainly, this new complexified 

arrangement has the same intersection lattice. We 

therefore obtain a curious corollary: 

Proposition 2.2. For any real subspace arrangement, the sum 

of Betti numbers of its complement is equal to the sum of Betti 

numbers of the complement of its complexification. 

As an example, the complement of an arbitrary real hyperplane 

arrangement is just the union of the pieces into which 

the hyperplanes cut the Euclidean space, each of these is contractible. 

Proposition 2.2 implies that the sum of the Betti 

numbers of the complement of the complexification of a real 

hyperplane arrangement is equal to the number of these pieces. 

For instance, as explained above , the sum of the Betti numbers 

of the complement of the real braid arrangement is equal 

to d!, and hence, Proposition 2.2 implies that the sum of the 

Betti numbers of the complement of the complex braid arrangement 

is equal to d!. The latter is an interesting and important 

topological space studied in particular by Arnold, 

[Ar69]. 

Of course the relation of arrangement theory to combinatorics 

is by no means exhausted by the Goresky-MacPherson 

formula. For example, a famous result of Orlik and Solomon 

connects the cohomology algebra of the complement of a hyperplane 

arrangement to the theory of matroids, see [OS80]. 

Also the study by DeConcini and Procesi, see [DP95], of resolutions 

of singularities for a collection of divisors, whose intersections 

are locally arrangements, leads to subtle and interesting 

combinatorics of nested sets, see [Fei05, FK04, FK05]. 

One other notable situation is that of toric varieties where 

similiar ideas provide connections between algebraic and polyhedral 

geometry, see [Fu93, MS05]. 

3 Graph homomorphisms and characteristic 

classes 

Let us now describe one of the many situations in which algebraic 

topology can be of use for questions in discrete mathematics. 

A central notion in combinatorics is that of a graph. For 

two graphs T and G, agraph homomorphism from T to G is 

a set map between vertex sets ϕ : V(T ) → V(G), such that if 

x,y ∈ V(T ) are connected by an edge, then ϕ(x) and ϕ(y) are 

also connected by an edge. An important special case is provided 

by considering a graph homomorphism to a complete 

graph ϕ : G → Kn, which is the same as a vertex coloring of 

G with n colors. In particular, the chromatic number of G,denoted 

χ(G), is the minimal n, such that there exists a graph 

homomorphism ϕ : G → Kn. It is an important, but difficult 

task to compute or to estimate chromatic numbers of graphs. 

We mention the notorious 4-coloring theorem as an example. 

The topological approach to graph colorings was pioneered 

by László Lovász (who is the current IMU President 

and has turned 60 this year). In 1978 he used homotopy theory 

to tackle the problem of determining chromatic numbers 

of Kneser graphs, see [Lov78] for details, and also [dL03] for 

a nice historical account. Recall that the Kneser graphs are 

the graphs defined as follows: for fixed k and n, vertices of 

the corresponding Kneser graph are indexed by all k-element 

subsets of [n], and two vertices are connected by an edge if 

the indexing subsets are disjoint. 

We note that if ϕ : T → G and ψ : G → H are graph homomorphisms, 

then the composition ψ ◦ ϕ : T → H is again a 

graph homomorphism, and, since identity maps are also graph 

homomorphisms, it follows that graphs together with graph 

homomorphisms form a category called Graphs. 

More recently it was realized that the set of all graph homomorphisms 

between graphs T and G can be enriched to 

form a topological space Hom(T,G), see [BK06] for details. 

The construction can be rephrased as follows: let ∆ V(G) be 

a simplex whose set of vertices is V(G), andletC(T,G) denote 

the direct product ∏x∈V(T ) ∆ V(G) , i.e., the copies of ∆ V(G) 

are indexed by vertices of T . 

Definition 3.1. The complex Hom(T,G) is the subcomplex of 

C(T,G) defined by the following condition: σ = ∏x∈V (T) σx ∈ 

Hom(T,G) if and only if for any x,y ∈ V(T ), if(x,y) ∈ E(T), 

then (σx,σy) is a complete bipartite subgraph of G. 

Here a complete bipartite subgraph of a graph G is a pair 

(A,B) of subsets (not necessarily disjoint) of the set of vertices 

of G, such that any vertex from A is connected by an edge 

to any vertex from B. An example of the cell complex of all 3colorings 

of a 6-cycle is given in Figure 1. The reader should 

note that several quadrangles have been omitted for the sake 

of transparency. There should be three maximal quadrangles 

attached to the grey vertex (only the two shaded ones are 

shown); the same is the case at the other five vertices at which 

the maximal cubes touch each other. In total, there are 18 

quadrangles which are not part of higher-dimensional cells. 

The topology of Hom(T,G) is inherited from the product 

topology of C(T,G), in particular, the cells of Hom(T,G) are 

products of simplices. It turned out that when T is an odd 

cycle (a polygon) C2r+1, the topology of this space is related 

to the chromatic number of G. This is the content of the next 

theorem, which was proved in [BK04]. 

Theorem 3.2. (Lovász Conjecture). 

For an arbitrary graph G, and any integers r and k, such that 

r ≥ 1,k≥−1, we have the following implication: 

if the complex Hom(C2r+1,G) is k-connected, then 

χ(G) ≥ k + 4. 

Perhaps a useful way of thinking about this and similar results 

is to view it as a test: the (higher) connectivity of a certain 

topological space is being tested, and if that quantity is 

possible to compute (which of course depends heavily on the 

space) it provides some information on the chromatic number. 

Whenever a loopfree graph T possesses an involution 

which flips an edge, the topological space Hom(T,G) has a 


2 

2 

1 23 

13 

1 2 

1 

2 

2 

23 

1 

1 

1 2 

1 

2 

13 2 

13 

2 13 3 12 

13 

Figure 1. (Essential parts of) the complex Hom(C6,K3) 

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12 

free involution. This allows us to talk about the Stiefel–Whitney 

characteristic classes in the cohomology of that space, see 

[Ko05a]. It is an intriguing fact that the chromatic number of 

G is in fact related to the vanishing of the powers of this characteristic 

class, and the Lovász Conjecture is one of the indications 

of that. These connections can be analysed directly, 

or by using spectral sequences, we refer to [Ko07, Ko05b] for 

further details. 

4 A topological approach to evasiveness 

In the last section we would like to illustrate ties which exist 

between Combinatorial Algebraic Topology and theoretical 

computer science, more specifically – complexity theory. 

Let us fix n and consider the set of all graphs with n vertices. 

A particular graph is then characterized by the set E(G) 

considered as a subset of the set of ordered pairs {(i, j) | 1 ≤ 

i < j ≤ n}. Recall that a graph property is called monotone if 

the set of graphs which satisfy this property is closed under 

removal of edges; examples include planarity and the property 

of being disconnected. Given such a monotone property 

G , we can construct a simplicial complex ∆(G ) by taking ordered 

pairs (i, j), for1≤ i < j ≤ n, as vertices, and saying 

that a set of such pairs forms a simplex in ∆(G ) if the corresponding 

graph has the property G . 

Let us now consider the following algorithmic situation. 

We are given a certain graph property G , which we know, 

and a certain graph G, which we do not know. However, we 

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3 

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EMS Newsletter June 2008 15 

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Hom(C6,K3) 

3 

12 

1 

3 

23 1 

23 

12 

13 

1 

3 

23 

2 13 

2 

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can ask the oracle questions of the type: Is the edge (i, j) in 

G? Our task is to decide whether the graph G satisfies the 

property G or not. We would like to ask as few questions as 

possible. More precisely, we are interested in knowing the socalled 

worst-case complexity, that is: how many questions do 

we have to ask in the worst case scenario? For example, if the 

graph property is trivial (meaning that either no graphs or all 

graphs satisfy it), then we do not need to ask any questions 

at all. On the other hand, if the property is being a complete 

graph, then we might be forced into asking �n� 2 questions: this 

would happen if the answers to the first � � n 

2 − 1 questions are 

all positive. 

Definition 4.1. A graph property G of graphs on n vertices 

is called evasive, if we need to ask �n� 2 questions in the worst 

case in the course of the algorithm described above; otherwise, 

we say that G is nonevasive. 

Certainly not all non-trivial properties are evasive. Consider 

for example the property of being a so-called scorpion graph. 

Here a graph G on n vertices is called a scorpion graph if it 

has 3 vertices s, t,andb, such that 

(1) the vertex s, called the sting, is only connected to vertex t, 

(2) the vertex t, called the tail, is only connected to vertices s 

and b, 

(3) the vertex b, called the body, is connected to all vertices, 

but s. 

This notion is illustrated on Figure 2. It is a nice exercise 

in complexity theory to verify that this property is actually 

nonevasive. 

1 

2 

3

Feature 

Figure 2. An example of a scorpion graph 

Clearly, if an edge is removed from a scorpion graph, the 

obtained graph does not have to be a scorpion graph anymore: 

being a scorpion graph is not a monotone graph property. In 

fact, the following is perhaps the most important open question 

pertaining to evasiveness of graphs. 

Conjecture 4.2. (Evasiveness Conjecture, also known as Karp 

Conjecture) 

Every nontrivial monotone graph property for graphs on n 

vertices is evasive. 

It is mind-boggling that so far, the Evasiveness Conjecture 

has been verified in the case when n is a prime power, and, 

additionally, when n = 6, using topological techniques only; 

see [KSS84]. The proof in these cases relies on the interplay 

of algebraic invariants of the complex ∆(G ) mentioned above 

and the action of the symmetry group Sn and its subgroups. 

It is difficult to imagine that whether n is a prime power or 

not is essential for the validity of the Evasiveness Conjecture, 

and it is still not understood what topology has to do with it. 

Acknowledgments 

The author would like to thank Martin Raussen for many 

comments which helped to improve the presentation substantially. 

Bibliography 

[Ar69] V.I. Arnold, The cohomology ring of the group of dyed 

braids, Mat. Zametki 5, (1969), 227–231. (Russian) 

[BK04] E. Babson, D.N. Kozlov, Proof of the Lovász Conjecture, 

Annals of Mathematics (2) 165 (2007), no. 3, 965–1007. 

[BK06] E. Babson, D.N. Kozlov, Complexes of graph homomorphisms, 

Israel J. Math., 152 (2006), 285–312. 

[DP95] C. De Concini, C. Procesi, Wonderful models of subspace 

arrangements, Selecta Math. (N.S.) 1 (1995), no. 

3, 459–494. 

[Fei05] E.-M. Feichtner, De Concini-Procesi wonderful arrangement 

models: a discrete geometer’s point of view, in: 

Combinatorial and computational geometry, Math. Sci. 

Res. Inst. Publ. 52, Cambridge Univ. Press, Cambridge, 

2005, pp. 333–360. 

[FK04] E.-M. Feichtner, D.N. Kozlov, Incidence combinatorics 

of resolutions, Selecta Math. (N.S.) 10 (2004), no. 1, pp. 

37–60. 

[FK05] E.-M. Feichtner, D.N. Kozlov, A desingularization of 

real differentiable actions of finite groups, Int. Math. Res. 

Not. 2005, no. 15, pp. 881–898. 

s 

t 

b 

[Fu93] W. Fulton, Introduction to toric varieties, Annals of 

Mathematics Studies 131, The William H. Roever Lectures 

in Geometry, Princeton University Press, Princeton, 

NJ, 1993, xii+157 pp. 

[GoM88] M. Goresky, R. MacPherson, Stratified Morse Theory, 

Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 

14, Springer-Verlag, Berlin Heidelberg New York, 1988. 

[KSS84] J. Kahn, M. Saks, D. Sturtevant, A topological approach 

to evasiveness, Combinatorica 4 (1984), 297–306. 

[Ko05a] D.N. Kozlov, Chromatic numbers, morphism complexes, 

and Stiefel–Whitney characteristic classes,in:Geometric 

Combinatorics (eds. E. Miller, V. Reiner, B. Sturmfels), 

IAS/Park City Mathematics Series 14, American Mathematical 

Society, Providence, RI; Institute for Advanced 

Study (IAS), Princeton, NJ, 2007; pp. 249–316. 

[Ko05b] D.N. Kozlov, Cohomology of colorings of cycles, American 

J. Math., in press. arXiv:math.AT/0507117 

[Ko07] D.N. Kozlov, Combinatorial Algebraic Topology, Algorithms 

and Computation in Mathematics 21, Springer- 

Verlag Berlin Heidelberg, 2008, xx+390 pp., 115 illus. 

[dL03] M. de Longueville, 25 Jahre Beweis der Kneservermutung. 

Der Beginn der topologischen Kombinatorik, (German) 

[25th anniversary of the proof of the Kneser conjecture. 

The start of topological combinatorics], Mitt. 

Deutsch. Math.-Ver. 2003, no. 4, pp. 8–11. 

[Lov78] L. Lovász, Kneser’s conjecture, chromatic number, and 

homotopy, J. Combin. Theory Ser. A 25, (1978), no. 3, 

319–324. 

[McC01] J. McCleary, A user’s guide to spectral sequences, Second 

edition, Cambridge Studies in advanced mathematics 

58, Cambridge University Press, Cambridge, 2001. 

[MS05] E. Miller, B. Sturmfels, Combinatorial Commutative Algebra, 

Graduate Texts in Mathematics 227, Springer- 

Verlag, New York, 2005, xiv+417 pp. 

[OS80] P. Orlik, L. Solomon, Combinatorics and Topology of 

complements of hyperplanes, Invent. Math. 56 (1980), 

167–189. 

Dmitry N. Kozlov [dfk@math.uni-bremen.de] 

The author is recipient of the Wallenberg 

Prize of the Swedish Mathematics Society 

(2003), the Gustafsson Prize of the Göran 

Gustafsson Foundation (2004), and the European 

Prize in Combinatorics (2005). He has 

also authored a book on the topic which has 

recently appeared in Springer-Verlag. His research lies at the 

interface of Discrete Mathematics, Algebraic Topology, and 

Theoretical Computer Science. 

He has obtained his doctorate from the Royal Institute of 

Technology, Stockholm in 1996. After longer stays at the 

Mathematical Sciences Research Institute at Berkeley, the 

Massachusetts Institute of Technology, the Institute for Advanced 

Study at Princeton, the University of Washington, 

Seattle, and Bern University, he has been a Senior Lecturer 

at the Royal Institute of Technology, Stockholm, and an Assistant 

Professor at ETH Zurich. Currently he holds the Chair of 

Algebra and Geometry at the University of Bremen, Germany. 


On Platonism 

Reuben Hersh (University of New Mexico, USA) 

“Platonism” can mean a lot of different 

things. Even “Platonism in mathematics” 

can mean a lot of different 

things. In my writings on the philosophy 

of mathematics, I have been concerned 

about the philosophical stance 

or preconceptions of practicing mathematicians, 

whether explicitly formulated 

or not. As I have written, this 

usually involves some choice or combination or alternation 

of “formalism” and “Platonism”, both of them in 

rough-and-ready, naïve versions. Their “Platonism” says 

mathematical objects exist independently of our knowledge 

or activity, and mathematical truth is objective, 

with the same status as scientifi c truth about the physical 

world. This may be boiled down to the phrase “out 

there.” That’s where mathematical entities are, meaning, 

not “in here.” 

As for “formalism”, that is taken to mean that math is 

no more than logical deduction from formally stated “axioms,” 

so that the “facts” about mathematical objects are 

nothing more than statements that one formal sentence 

follows logically from some other formal sentence; there 

is no “meaning” or “reference,” just logical manipulation 

of logical formulas. 

These two points of view are incompatible, yet it 

seems that most mathematicians oscillate between the 

two. My purpose in writing about this has been to argue 

that both notions are false. Formalism is wildly incompatible 

with the actual practice of mathematics, whether in 

research or in teaching. Platonism is incompatible with 

the standard view of the nature of reality, as held by the 

majority of scientists and mathematicians: there is only 

one universe, one real world, which is physical reality, including 

its elaboration into the realm of living things, and 

elaborated from there into the realm of humankind with 

its social, cultural, and psychological aspects. I have argued 

that these social, cultural and psychological aspects 

of humankind are real, not illusory or negligible, and that 

the nature of mathematical truth and existence is to be 

understood in that realm, rather than in some other independent 

“abstract” reality. 

My view of Platonism – always referring to the common, 

every-day Platonism of the typical working mathematician 

– is that it expresses a correct recognition that 

there are mathematical facts and entities, that these are 

not subject to the will or whim of the individual mathematician 

but are forced on him as objective facts and entities 

which he must learn about and whose independent 

existence and qualities he seeks to recognize and discover. 

The fallacy of Platonism is in the misinterpretation of 

this objective reality, putting it outside of human culture 

and consciousness. Like many other cultural realities, it is 

external, objective, 

Feature 

from the viewpoint of any individual, but internal, historical, 

socially conditioned, from the viewpoint of the society 

or the culture as a whole. 

(One might note, in passing, that other socio-cultural 

phenomena, like the divine right of kings, the innate natural 

inferiority of slaves, etc etc have also been regarded for 

centuries as part of the objective order of the Universe.) 

The recent article by Davies 1 declares death to Platonism, 

on the grounds of neurophysiological discoveries 

about arithmetical and geometrical thinking. Researchers 

are now able to observe blood fl ow into specifi c sections 

of the brain associated with various mental processes, including 

mathematical ones. These discoveries add a lot of 

empirical weight to our previous general belief, that thinking 

is something that happens mostly in the brain. What 

new philosophical insights do we gain? Well, Platonism 

requires some mental faculty by which the mathematician 

can connect to the abstract external non-physical realm 

where the objects he is studying are supposed to exist. 

This faculty can be labeled “intuition,” but such a label is 

not an explanation. In fact, the dualism that is forced by 

Platonism – the division of reality into two realms, one 

physical, including the brain of the mathematician, and 

the other “abstract,” “out there,” neither physical nor social 

nor psychological, just “abstract” and “out there” – is 

a fatal fl aw, much the same as the fl aw in Descartes’ dualism 

of mind and matter. It seems that Davies regards the 

evidence that thinking takes place in the brain as proof 

that there is no such “intuition”, in the sense of a special 

mental faculty for connecting to “out there.” But with 

or without neurophysiological evidence, it is pretty clear 

that the posited “intuition” is an ad hoc artifact, lacking 

any specifi city or clear description, let alone empirical 

evidence. It is nothing more than something that, for 

the Platonist, “has to” be available to the mathematician, 

since he/she evidently is able to discover mathematical 

facts, and these facts are “out there,” not “in here.” 

If by Platonism one simply means the rejection of formalism 

– the conviction that the things we are studying 

are real, and that they have objective properties which 

we can discover – then there is no doubt that this conviction 

is supportive, indeed perhaps even necessary, for 

mathematical research. While one is proving a theorem 

or fi nding a counterexample, one is not in the mode of 

considering the statement of the theorem or counterexample 

meaningless! But it is not necessary, either logically 

or psychologically, to locate this meaning in some 

eternal, inhuman, abstract reality. 

I once took a vote in a talk at New Mexico State University 

in Las Cruces. The question was, “Was the spectral 

theorem on self-adjoint operators in Hilbert space true 

before the Big Bang, before there was a universe?” The 

vote was yes, by a margin of 75 to 25. But there were no 

self-adjoint operators, no Hilbert space, before the twentieth 

century! So no statement about them could be true or 

false or meaningful (it could not refer to anything) before 

these concepts had been established in the consciousness 

of mathematicians. However, it is true that mathematical 

1 Let Platonism die, EMS-Newsletter 64 (2007), 24–25. 


Feature 

theories like the theory of self-adjoint operators in Hilbert 

space, while they come into existence historically (and can 

even go out of existence historically) are time-independent, 

in the sense that they do not refer to or in their content 

include or make note of any non-mathematical, external 

data, including the particular century of their birth. We 

construct mathematics without reference to non-mathematical 

data, including spatial (in what country was the 

mathematician who proved the theorem?) and temporal. 

The spectral theorem’s content or reference excludes any 

spatial-temporal data of ordinary daily life. 

I have compared this to a movie, which we know is 

just shadows and colors projected on a screen. This factual 

knowledge in no way confl icts with seeing the movie 

as it is intended to be seen, as a (fi ctional) reality which 

we fully understand, participate in, and can reason about. 

There are objective facts relating to the content of the 

movie, without relegating it to an abstract realm “out 

there.” At the same time, the movie has a physical reality 

in terms of lights and shadows, and cannot exist except on 

the basis of this or some other comparable physical reality. 

Similarly, the content of mathematical theorems and 

concepts is real, we do have reliable knowledge of it. That 

content is part of human culture and consciousness, and 

is contingent on the existence and activity of the species 

homo sapiens. With this understanding, full acceptance 

and acknowledgment of the reality and meaningfulness 

of mathematics does not contradict or confl ict with the 

ordinary scientifi c view of reality. The mental, social and 

cultural, including the mathematical, are grounded in the 

physical – the fl esh and blood of past and present humans, 

especially mathematicians. We can recognize this, 

even while the detailed nature of this grounding – just 

how our thoughts are carried out by our brains – may 

never be completely understood. 

The puzzle for us to resolve is the universality of 

mathematics. Unlike other cultural realms, there is a 

universality or commonality of mathematical facts – the 

“laws” of arithmetic, for instance – which is strikingly different 

from the great variations in other cultural realms 

– language, religion, music, family structure, etc, in different 

cultures around the world. Why is this? Well, to begin 

with, the facts of elementary arithmetic, the small natural 

numbers, are physical facts, they describe the behavior of 

collections of stable, non-interacting, discrete objects like 

coins or buttons. These are as independent of culture as 

other parts of physics, like the fact that a stone will fall to 

the ground. The ability to develop logical consequences 

also seems to be to a great extent culture – independent, 

which suggests a possible basis in our brain structures, just 

as linguists have postulated some basis in our brain structures 

for the universal features that all human languages 

seem to share. Certainly, along with physical reality, we 

must consider the special features of the human brain, if 

we want to understand more deeply the nature of mathematics. 

This is a major project for empirical science – not 

for philosophy! From this point of view, the facts presented 

by Davies are relevant and interesting. They do not 

refute Platonism, they are part of the scientifi c program 

that one focuses on after rejecting Platonism. 

To put the matter as simply as possible: everyone 

agrees that mathematics is a historically derived cultural 

activity. I say, there is no need to give it any other “metaphysical” 

description. As a cultural activity it has certain 

puzzling features: its seeming “certainty,” and “universality.” 

Platonists account for this puzzle by postulating a 

special existential realm, with matching mental faculty 

(“intuition”). They simply disregard the strange incompatibility 

of such a realm and faculty with our common 

understandings of both philosophy of science and philosophy 

of mind. I consider it more promising to regard 

the nature of mathematics as a problem for a multi-disciplinary 

empirical investigation, analogous, for example, 

to the nature of language, where an empirical science, 

linguistics, is thriving. The best example of such investigation 

is the fascinating work of Stanislas Dehaene [D]. 

My recent edited book was intended as a contribution to 

this enterprise [H1]. 

My most persistent opponent has been Martin Gardner, 

who is a theist, a believer in the effi cacy of prayer, 

and who has declared that I am on the “slippery slope” 

to solipsism (in one review) and to Stalinism (in another 

review). Others have seen a resemblance to Jung’s “collective 

unconscious”. I see nothing Jungian here, but 

Jungianism is not so bad, compared to solipsism and 

Stalinism! I have also been honored by a critique from 

William Tait, who is described on the jacket of his book 

as “one of the most eminent philosophers of mathematics 

of his generation” [T]. He fi nds my description of 

Platonism rather ridiculous, and seems not to know the 

difference between my saying what some other people 

seem to think and saying what I myself think. (He proposes 

a distinction between “realism” and what he calls 

“hyperrealism.” “Realism” is OK, says William Tait. The 

absurdities of Platonism, he explains, are attributes of 

“hyperrealism.”) 

References 

[D] Dehaene, S. The Number Sense, Oxford University Press 1997 

[H1] Hersh, R. What is Mathematics, Really? Oxford University Press 

1997 

[H2] Hersh, R. (Ed.) 18 Unconventional essays on the nature of mathematics, 

Springer, 2006 

[T] Tait, W. The Provenance of Pure Reason, Oxford University Press, 

2005 

Reuben Hersh [rhersh@math.unm.edu] 

has published research on partial differential 

equations, random evolutions, 

and probability. He lives in Santa Fe, 

New Mexico, in the USA. He was a student 

of Peter Lax, and a co-author with 

Phil Davis, Martin Davis, Paul Cohen, Richard J. Griego, 

Jim Donaldson, Tosio Kato, George Papanicolaou, Mark 

Pinsky, Priscilla Greenwood, and others. With Vera John- 

Steiner, he has a forthcoming book entitled “Loving and 

Hating Mathematics: Inside Mathematical Life.” 



Platonism and 

its Opposites 

Barry Mazur (Harvard University, Cambridge MA, USA) 

We had the sky up there, all speckled with stars, and we 

used to lay on our backs and look up at them, and discuss 

about whether they was made or only just happened – Jim 

he allowed they was made, but I allowed they happened; I 

judged it would have took too long to make so many. 

mused Huckleberry Finn. The analogous query that 

mathematicians continually fi nd themselves confronted 

with when discussing their art with people who are not 

mathematicians is: 

Is mathematics discovered or invented? 

I will refer to this as The Question, acknowledging 

that this fi ve-word sentence, ending in a question mark 

– and phrased in far less contemplative language than 

that used by Huck and Jim – may open conversations, 

but is hardly more than a token, standing for puzzlement 

regarding the status of mathematics. 

If you engage in mathematics long enough, you bump 

into The Question, and it won’t just go away. 1 If we wish 

to pay homage to the passionate felt experience that 

makes it so wonderful to think mathematics, we had better 

pay attention to it. 

Some intellectual disciplines are marked, even scarred, 

by analogous concerns. Anthropology, for example has a 

vast, and dolefully introspective, literature dealing with 

the conundrum of whether we can ever avoid – wittingly 

or unwittingly – clamping the templates of our own culture 

onto whatever it is we think we are studying: how 

much are we discovering, how much inventing? 

Such a discovered/invented perplexity may or may 

not be a burning issue for other intellectual pursuits, but 

it burns exceedingly bright for mathematics, and with a 

strangeness that isn’t quite matched when it pops up in 

other fi elds. For example, if you were to say “Priestley 

discovered oxygen but Lavoisier invented it” I think I 

know roughly what you mean by that utterance, without 

our having to synchronize our private vocabularies terribly 

much. But to intelligently comprehend each other’s 

possibly differing attitudes towards circles, triangles, and 

numbers, we would also have to come to some – albeit 

ever-so-sketchy – understanding of how we each view, 

and talk about, a lot more than mathematics. 2 

For me, at least, the anchor of any conversation 

about these matters is the experience of doing mathematics, 

and of groping for mathematical ideas. When I 

read literature that is ostensibly about The Question, I 

ask myself whether or not it connects in any way with 

my felt experience, and even better, whether it reveals 

something about it. I’m often – perhaps always – disap- 

Feature 

pointed. The bizarre aspect of the mathematical experience 

– and this is what gives such fi erce energy to The 

Question – is that one feels (I feel) that mathematical 

ideas can be hunted down, and in a way that is essentially 

different from, say, the way I am currently hunting 

the next word to write to fi nish this sentence. One 

can be a hunter and gatherer of mathematical concepts, 

but one has no ready words for the location of the hunting 

grounds. Of course we humans are beset with illusions, 

and the feeling just described could be yet another. 

There may be no location. 

There are at least two standard ways of – if not exactly 

answering, at least – fi elding The Question by offering 

a vocabulary of location. The colloquial tags for these 

locations are In Here and Out There (which seems to me 

to cover the fi eld). 

The fi rst of these standard attitudes, the one with the 

logo In Here – which is sometimes called the Kantian 

(poor Kant!) – would place the source of mathematics 

squarely within our faculties of understanding. Of course 

faculties (Vermögen) and understanding (Verstand) are 

loaded eighteenth century words and it would be good 

– in this discussion at least – to disburden ourselves of 

their baggage as much as possible. But if this camp had to 

choose between discovery and invention, those two toobrittle 

words, it would opt for invention. 

The “Out There” stance regarding the discovery/invention 

question whose heraldic symbol is Plato (poor 

Plato!) is to make the claim, starkly, that mathematics is 

the account we give of the timeless architecture of the 

cosmos. The essential mission, then, of mathematics is the 

accurate description, and exfoliation, of this architecture. 

This approach to the question would surely pick discovery 

over invention. 

Strange things tend to happen when you think hard 

about either of these preferences. 

For example, if we adopt what I labelled the Kantian 

position we should keep an eye on the stealth word “our” 

in the description of it that I gave, hidden as it is among 

behemoths of vocabulary (Vermögen, Verstand). Exactly 

whose faculties are being described? Who is the we? Is 

the we meant to be each and every one of us, given our 

separate and perhaps differing and often faulty faculties? 

If you feel this to be the case, then you are committed to 

viewing the mathematical enterprise to be as variable as 

humankind. Or are you envisioning some sort of distillate 

of all actual faculties, a more transcendental faculty, 

possessed by a kind of universal or ideal we, in which 

1 Garrison Keillor, a wonderful radio raconteur has in his repertoire 

a fi ctional character, Guy Noir, who tangles indefatigably 

with “life’s persistent questions.” This is all to the good. 

We should pay particular honour to the category of persistent 

questions even though – or, especially because – those are 

the chestnuts that we’ll never crack. 

2 For a start: you and I turn adjectives into nouns (red cows → 

red; fi ve cows → fi ve) with only the barest fl ick of a thought. 

What is that fl ick? Understanding the differences in our 

sense of what is happening here may tell us lots about our 

differences regarding matters that can only be discussed with 

much more mathematical vocabulary. 


Feature 

case the Kantian view would seem to merge with the Platonic. 

3 

If we adopt the Platonic view that mathematics is discovered, 

we are suddenly in surprising territory, for this 

is a full-fl edged theistic position. Not that it necessarily 

posits a god, but rather that its stance is such that the only 

way one can adequately express one’s faith in it, the only 

way one can hope to persuade others of its truth, is by 

abandoning the arsenal of rationality, and relying on the 

resources of the prophets. 

Of course, professional philosophers are in the business 

of formulating anti-metaphysical or metaphysical 

positions, decorticating them, defending them, and refuting 

them. 4 Mathematicians, though, may have another 

– or at least a prior – duty in dealing with The Question. 

That is, to be meticulous participant/observers, faithful 

to the one aspect of The Question to which they have 

sole proprietary rights: their own imaginative experience. 

What, precisely, describes our inner experience when we 

(and here the we is you and me) grope for mathematical 

ideas? We should ask this question open-eyed, allowing 

for the possibility that whatever it is we experience may 

delude us into fabricating ideas about some larger framework, 

ideas that have no basis. 5 

I suspect that many mathematicians are as unsatisfi ed 

by much of the existent literature about The Question as 

I am. To be helpful here, I’ve compiled a list of Do’s and 

Don’t’s for future writers promoting the Platonic or the 

Anti-Platonic persuasions. 

For the Platonists 

One crucial consequence of the Platonic position is that 

it views mathematics as a project akin to physics, Platonic 

mathematicians being – as physicists certainly are 

– describers or possibly predictors – not, of course, of 

the physical world, but of some other more noetic 6 entity. 

Mathematics – from the Platonic perspective – aims, 

among other things, to come up with the most faithful 

description of that entity. 

This attitude has the curious effect of reducing some 

of the urgency of that staple of mathematical life: rigorous 

proof. Some mathematicians think of mathematical 

proof as the certifi cate guaranteeing trustworthiness of, 

and formulating the nature of, the building-blocks of the 

edifi ces that comprise our constructions. Without proof: 

no building-blocks, no edifi ce. Our step-by-step articulated 

arguments are the devices that some mathematicians 

feel are responsible for bringing into being the theories 

we work in. This can’t quite be so for the ardent Platonist, 

or at least it can’t be so in the same way that it might 

be for the non-Platonist. Mathematicians often wonder 

about – sometimes lament – the laxity of proof in the 

physics literature. But I believe this kind of lamentation 

is based on a misconception, namely the misunderstanding 

of the fundamental function of proof in physics. Proof 

has principally (as it should have, in physics) a rhetorical 

role: to convince others that your description holds 

together, that your model is a faithful re-production, and 

possibly to persuade yourself of that as well. It seems to 

me that, in the hands of a mathematician who is a de- 

termined Platonist, proof could very well serve primarily 

this kind of rhetorical function – making sure that the description 

is on track – and not (or at least: not necessarily) 

have the rigorous theory-building function it is often 

conceived as fulfi lling. 

My feeling, when I read a Platonist’s account of his 

or her view of mathematics, is that unless such issues regarding 

the nature of proof are addressed and conscientiously 

examined, I am getting a superfi cial account of 

the philosophical position, and I lose interest in what I 

am reading. 

But the main task of the Platonist who wishes to persuade 

non-believers is to learn the trade, from prophets 

and lyrical poets, of how to communicate an experience 

that transcends the language available to describe it. If 

all you are going to do is to chant credos synonymous 

with “the mathematical forms are out there,” – which 

some proud essays about mathematical Platonism content 

themselves to do – well, that will not persuade. 

For the Anti-Platonists 

Here there are many pitfalls. A common claim, which is 

meant to undermine Platonic leanings, is to introduce 

into the discussion the theme of mathematics as a human, 

and culturally dependent pursuit and to think that one 

is actually conversing about the topic at hand. Consider 

this, though: If the pursuit were writing a description of 

the Grand Canyon and if a Navajo, an Irishman, and a 

Zoroastrian were each to set about writing their descriptions, 

you can bet that these descriptions will be culturally-dependent, 

and even dependent upon the moods 

and education and the language of the three describers. 

But my having just recited all this relativism regarding 

the three descriptions does not undermine our fi rm faith 

in the existence of the Grand Canyon, their common focus. 

Similarly, one can be the most ethno-mathematically 

3 A more general lurking question is exactly how we are to 

view the various ghosts in the machine of Kantian idealism – 

for example, who exactly is that little-described player haunting 

the elegant concept of universally subjective judgments 

and going under a variety of aliases: the sensus communis or 

the allgemeine Stimme? 

4 A very useful – and to my mind, fi ne – text that does exactly 

this type of lepidoptery is Mark Balaguer’s Platonism and 

Anti-Platonism in Mathematics, Oxford Univ. Press (1998). 

5 When I’m working I sometimes have the sense – possibly the 

illusion – of gazing on the bare platonic beauty of structure 

or of mathematical objects, and at other times I’m a happy 

Kantian, marvelling at the generative power of the intuitions 

for setting what an Aristotelian might call the formal 

conditions of an object. And sometimes I seem to straddle 

these camps (and this represents no contradiction to me). I 

feel that the intensity of this experience, the vertiginous imaginings, 

the leaps of intuition, the breathlessness that results 

from “seeing” but where the sights are of entities abiding in 

some realm of ideas, and the passion of it all, is what makes 

mathematics so supremely important for me. Of course, the 

realm might be illusion. But the experience? 

6 “inner knowing”, a kind of intuitive consciousness – direct 

and immediate access to knowledge beyond what is available 

to our normal senses and the power of reason. 


conscious mathematician on the globe, claiming that all 

our mathematical scribing is as contingent on ephemeral 

circumstance as this morning’s rain, and still one can be 

the most devout of mathematical Platonists. 

Now this pitfall that I have just described is harmless. 

If I ever encounter this type of mathematics-is-a-human-activity 

argument when I read an essay purporting 

to defuse, or dispirit, mathematical Platonism I think to 

myself: human activity! what else could it be? I take this 

part of the essay as being irrelevant to The Question. 

A second theme that seems to have captured the imagination 

of some anti-Platonists is recent neurophysiological 

work – a study of blood fl ow into specifi c sections 

of the brain – as if this gives an insider’s view of things. 7 

Well, who knows? Neuro-anatomy and chemistry have 

been helpful in some discussions, and useless in others . To 

show this theme to be relevant would require a precisely 

argued explanation of exactly how blood fl ow patterns 

can refute, or substantiate, a Platonist – or any – disposition. 

A satisfying argument of that sort would be quite a 

marvel! But just slapping the words blood fl ow – as if it 

were a poker-hand – onto a page doesn’t really work. 

Sometimes the mathematical anti-Platonist believes 

that headway is made by showing Platonism to be unsupportable 

by rational means, and that it is an incoherent 

position to take when formulated in a propositional vocabulary. 

It is easy enough to throw together propositional sentences. 

But it is a good deal more diffi cult to capture a Platonic 

disposition in a propositional formulation that is a 

full and honest expression of some fl esh-and-blood mathematician’s 

view of things. There is, of course, no harm in 

7 like the old Woody Allen movie Everything you wanted to 

know about sex but were afraid to ask 

EMS Tracts in Mathematics Vol. 5 

Feature 

trying – and maybe a good exercise. But even if we cleverly 

came up with a proposition that is up to the task of expressing 

Platonism formally, the mere fact that the proposition 

cannot be demonstrated to be true won’t necessarily 

make it vanish. There are many things – some true, some 

false – unsupportable by rational means. For example, if 

you challenge me to support – by rational means – my 

claim that I dreamt of Waikiki last night, I couldn’t. 

So, when is there harm? It is when the essayist becomes 

a leveller. Often this happens when the author writes extremely 

well, super coherently, slowly withering away the 

Platonist position by – well – the brilliant subterfuge of 

making the whole discussion boring, until I, the reader, 

become convinced – albeit momentarily, within the framework 

of my reading the essay – that there is no “big deal” 

here: the mathematical enterprise is precisely like any other 

cultural construct, and there is a fallacy lurking in any 

claim that it is otherwise. The Question is a non-question. 

But someone who is not in love won’t manage to defi 

nitively convince someone in love of the non-existence 

of eros; so this mood never overtakes me for long. Happily 

I soon snap out of it, and remember again the remarkable 

sense of independence – autonomy even – of 

mathematical concepts, and the transcendental quality, 

the uniqueness – and the passion – of doing mathematics. 

I resolve then that (Plato or Anti-Plato) whatever I come 

to believe about The Question, my belief must thoroughly 

respect and not ignore all this. 

Barry Mazur [mazur@math.harvard. 

edu] is the Gerhard Gade University 

Professor at Harvard University. 

Gennadiy Feldman (National Academy of Sciences, Kharkov, Ukraine) 

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013750x

The meeting at Wernigerode 

Arild Stubhaug (Hyggen, Norway) 

Karl Weierstrass (1815–1897) 

© Institut Mittag-Leffl er 

Gösta Mittag-Leffl er (1846–1927) 


Sonja Kovalevsky (1850–1891) 


Vito Volterra (1860–1940) 

The following is an excerpt from Arild Stubhaug’s biography 

of the Swedish mathematician Mittag-Leffl er 

(1846–1927) chosen by the author himself and translated 

by Ulf Persson, who is one of the editors of the EMS 

Newsletter. The biography (to be reviewed elsewhere in 

this issue) has appeared in its original Norwegian and 

as a Swedish translation (by K.O. Widman) last year. An 

English edition in preparation with Springer-Verlag and 

due 2009 (independent of the excerpt below). 

The excerpt takes place in August 1888 in the Harz 

Mountains in Northern Germany, still a popular ‘Spa-type’ 

destination. Wernigerode is a small, touristy town situated 

in the north, sporting medieval half-timbered houses and 

a castle with a wide view of the surroundings. The visitor 

will fi nd here an ample offering of both culture and nature 

but I doubt any references to its mathematical history. 

Since the beginning of July 1888, Weierstrass had been 

staying in Wernigerode in the Harz. He rented a number 

of rooms at Müllers Hotel for himself and his two sisters. 

Sonja Kovalevsky had also come here from London and 

Paris. She was putting the fi nal touches to the treatise 

that would eventually get her the Bordin prize and she 

History 

hoped that by being around Weierstrass she would be 

able to conclude her work. 

On 3 August Mittag-Leffl er arrived in Wernigerode, 

or rather at a station one hour away by train from it, 

where he was met and welcomed by Weierstrass and 

Sonja. Weierstrass looked well and Sonja ‘smashing’, he 

noted in his diary. For Mittag-Leffl er, the main purpose of 

meeting Weierstrass was to discuss and assess the twelve 

papers submitted for the mathematical prize of Oscar II. 

But he also entertained plans of asking Weierstrass to 

present his latest results so that they would not be lost 

to posterity should Weierstrass not fi nd the time to edit 

them. 

When he told Sonja about his plan the fi rst evening, she 

implored him to abstain. She badly needed Weierstrass’ 

time and attention in order to complete her own work. 

Besides the old master had solemnly asked her to come 

so he could relay to her what he no longer felt able to 

edit himself. 

The next day Mittag-Leffl er noted in his diary: ‘Mathematical 

discussions [with Weierstrass and Sonja] of high 

interest’. In the evening Volterra also arrived at Wernigerode 

and for the next few days, Weierstrass divided his 

time between his three former students. In the morning 

he listened to and commented on Volterra’s generalizations 

of Abel’s theorem for multiple integrals; in the afternoon 

he expounded at length on the three-body problem 

to the other two. In addition they went on shorter 

or longer walks, sometimes all together and sometimes 

two and two. Weierstrass expressed to Mittag-Leffl er his 

concern of whether Sonja would get her work completed 

– so much still remained in order to bring it into shape. 

When Volterra left Wernigerode after three days, it 

was cold and raining. Mittag-Leffl er took some short 

walks, fi rst with Weierstrass and then with Sonja. Sonja 

told him about her relationship with Maxim Kovalevsky 

– that she did not want to marry him but would rather 

live with him clandestinely. In this way she hoped to have 

a hold on him and make him remain the ‘devoted lover’ 

she had always wanted to have. Maxim on the other hand 

had always claimed that anything but marriage would 

denigrate her. But such a view, she thought, was just 

brought about by his vanity; the real reason he wanted 

her as a wife was just to be able to keep mistresses on the 

side. On top of everything he would rather be the most 

famous of the two. She continued to talk about wanting 

to be a mistress, not a wife, and thus it was sad to realize 

that men only respected her in such a way that the 

mere thought of her as a mistress appeared impossible. 

And when she started to suggest that in the vacations she 

would not mind having one, or better still several intellectually 

well-endowed men as her lovers, while during 

the semesters she would rather prefer to be left alone 


History 

Georg Cantor (1845–1918) 


Adolf Hurwitz (1859–1919) 


and work, Mittag-Leffl er could not help but remark that 

‘this is not how it usually comes across during the actual 

semesters’ and he added that unfortunately for her 

she lacked those physical attributes a man demands of 

a mistress. On the other hand she was probably richly 

equipped with those attributes a man would want in a 

wife. ‘But unhappy the man who had her for his wife,’ 

he added because the egotism Sonja had developed to 

such a high degree and her indifference, concealed under 

a lively and engaging temperament, would rather soon 

bring a husband to the end of his wits. ‘With her, everything 

is head and imagination but her heart has not yet 

vibrated’, he wrote, and thought of her terrible temper 

and the scenes she was capable of creating, as well as 

her total absence of any sense of duty and responsibility, 

which she so openly admitted to. He concluded that 

Sonja was probably completely in the right. It would be 

best for her not to get married at all. 

About a week after Mittag-Leffl er’s arrival in Wernigerode, 

Cantor came as well. In the meantime both professor 

Tietgen from Berlin and professor Hettner, who 

did not live too far away, had dropped by. 

Cantor turned out to prove himself an eager companion 

and Mittag-Leffl er went on many a walk together 

with him and Sonja. One day they climbed de Brocken, 

the highest mountain in the Harz and stayed overnight 

there. The next day, when they returned to Möllers Hotel, 

they found out that Adolf Hurwitz had arrived. The next 

few days were fi lled with what Mittag-Leffl er referred to 

as talks and conferences – mostly with Weierstrass – as 

well as walks with Cantor, Sonja and Hurwitz. Mittag- 

Leffl er found Hurwitz to be a modest young man fi lled 

with new and good ideas, with a genuine love for his science. 

‘Sonja has mobilized her full battery of coquettish 

charms aimed at his simple innocence,’ he added. 

At the dinner table on 16 August, Sonja started a discussion 

concerning the liberation of women, something 

which would turn out to be very unpleasant for Mittag- 

Leffl er. Present at the table, in addition to Sonja and Mittag-Leffl 

er, were Hurwitz, Cantor and the two sisters of 

Weierstrass; Weierstrass himself never had his dinners 

there. Mittag-Leffl er, according to his diary notes, did 

nothing but develop his own ideas that women should 

also have the right to vote, when Sonja suddenly interrupted 

him and started to talk about how much more than 

him she did and achieved 

in Stockholm. She lectured 

twice as much and had written 

several treatises; she had 

never had a leave of absence 

and had never been sick. 

Mittag-Leffl er responded 

calmly that he did not want 

to defend himself but noted 

in his diary that he had been 

on the verge of remarking 

that included among his duties 

in Stockholm was all the 

work involved in maintaining 

the position for his female colleague. Cantor on the 

other hand reacted to Sonja’s outburst by pointing out 

that she had forgotten all the work Mittag-Leffl er did 

with Acta Mathematica. But this was sprightly dismissed 

by Sonja, who claimed that the work with the journal did 

not make more demands on his time than did her own 

domestic chores on hers, and besides he had a secretary. 

A bit later in the day he realized that Sonja had also 

spoken disparagingly of him to Weierstrass. And from 

Weierstrass he learned that Schwarz, who was due in 

Wernigerode two days later, thought of Mittag-Leffl er as 

his adversary. The reason was supposedly that everyone 

believed that Mittag-Leffl er was fi shing for a position in 

Berlin, an ambition, he insisted in his diary, that had not 

been admitted to his thoughts, although it certainly would 

be worth thinking over. His most fervent desire for the 

coming winter was only to get enough energy to work 

with mathematics, he noted, and concluded discouraged 

at the end of that day: ‘But it will not be so. Things are 

just getting worse and worse and it will probably soon 

end as it did for my father. 1 

Herman Schwarz (1843–1921) 

Through daily meetings with Weierstrass the assessment 

of the submitted treatises progressed. The work of 

Poincaré was in particular discussed and Mittag-Leffl er 

was continually struck by the penetrating acuity of 

Weierstrass’ mind, his unusual powers of recollection 

and how easily he quenched even the most diffi cult of 

questions. 

Schwarz’ presence in Wernigerode led to several unpleasant 

scenes. Both Sonja and Weierstrass had written 

to Schwarz and asked him to come and take part in 

constructing a model for the body of rotation with which 

she was working for her prize-winning essay. In addition 

Weierstrass had asked Schwartz to come in order 

to try to bring about reconciliation between him and 

Mittag-Leffl er. At the fi rst meeting at the dinner table, 

Schwarz bowed to him several times and Mittag-Leffl er 

thought that he had looked very embarrassed. Later in 

the evening, however, after Schwarz had retired and just 

Sonja and Cantor remained, Mittag-Leffl er was for the 

fi rst time told about the details of the rumours Schwarz 

had spread about him during the last few years and he 

1 Mittag-Leffl er’s father had a mental breakdown and was 

committed for the rest of his life to an institution, while Mittag-Leffl 

er was still in his youth [translator’s note] 


started to grasp what the persistent strain in his relations 

with Schwarz was really founded on. Everywhere in Germany, 

France and Italy, Schwarz had disseminated the rumour 

that Mittag-Leffl er was a lecher, a Wüstling, leading 

a life of debauchery – so morally depraved that he had 

married Signe while he was still suffering from syphilis. 

That was why she had not been able to give birth and 

that was why she had consulted doctors all over Europe. 

To substantiate his claims he had bragged that he had 

sought out Mittag-Leffl er’s doctor in Helsinki. Cantor 

confessed that after having heard this he had refused to 

shake hands with Schwarz and that he would rather not 

have anything at all to do with such a person. And when 

Schwarz had spread the same rumours in Paris, Poincaré 

had reacted very strongly and forbidden anyone to badmouth 

his friend Mittag-Leffl er and that the rumours, 

even if there were some substance to them, were the sole 

business of Mittag-Leffl er himself and no one else. Both 

Poincaré and Picard had threatened to have him evicted 

out of the country if he would not give up his slanderous 

allegations. 

Cantor continued the next day to reveal more about 

Schwarz and how the story about the syphilis had been 

expanded upon. Cantor was curious as to what steps Mittag-Leffl 

er would take. Cantor was also able to relate that 

when Schwarz was newly engaged to Kummer’s daughter, 

he had asked for advice about whether he should tell 

his future father-in-law that as a soldier in 1866 he had 

had a severe episode of syphilis. 

‘Did not sleep during the night,’ Mittag-Leffl er noted 

in his diary. He had decided to confront Schwarz with 

what he had learned. At the dinner table on 20 August 

he asked Schwarz for a meeting between four eyes and 

just after dinner he escorted Schwarz to his room. Mittag-Leffl 

er started by remarking that he had expected an 

unconditional apology. When that did not come about, 

he minced no words in telling him that he could expect 

something very unpleasant. In Germany such defamations 

would be punished and so many witnesses could be 

brought up from Germany, France and Italy that there 

should be no doubt as to the outcome. He would be 

condemned to at least six months in some correctional 

institution, in addition to being saddled with paying indemnities. 

According to Mittag-Leffl er, Schwarz only 

stuttered something about having come to Wernigerode 

only because Weierstrass had asked him to and that Mittag-Leffl 

er was to do what he thought just. Much more 

was not said as Schwarz was then asked to leave. 

Mittag-Leffl er went right away to Weierstrass to tell 

him what had transpired but as he found him asleep, he 

instead sought out Sonja. She showed indignation as to 

the behaviour of Schwarz but thought nevertheless that 

Mittag-Leffl er should not have brought about such a 

scene. She meant that Mittag-Leffl er had only thought of 

himself, not on how the affair would affect Weierstrass. 

He countered that he had not thought of himself but of 

his wife Signe. Had it not been for the lies spread about 

her, he would have done nothing. Sonja was nevertheless 

not fully satisfi ed, something Mittag-Leffl er related 

to her need of Schwarz for help in her own work. 

History 

When Weierstrass was later fi lled in on what had 

happened, he thought that Mittag-Leffl er had acted in 

the right way, even if he found the whole thing very embarrassing 

and painful. He may not have believed that 

Mittag-Leffl er would be satisfi ed with what had already 

been said and he took it upon himself to force Schwarz 

to give a full apology. In his diary Mittag-Leffl er noted 

that since a Swede could not duel – and since he had now 

upbraided Schwarz so strongly, was it not rather Schwarz 

who could demand satisfaction? – there remained nothing 

but a legal process. But he did not look forward to 

bringing about such a scandal. 

The same evening Weierstrass succeeded in extracting 

an apology from Schwarz and his word of honour that he 

had not spread such stories during the last few years and 

that he would never ever do it again. Mittag-Leffl er who 

desired to accommodate Weierstrass decided to let matters 

rest for his sake and declared that he accepted the 

apology. Cantor, on the other hand, thought that Mittag- 

Leffl er ought to have demanded an apology in writing, 

something Mittag-Leffl er admitted he would have done 

had it not been for Weierstrass – anyway he would ‘treat 

Schwarz as air’ until he came around and personally 

asked for forgiveness. 

The next day Mittag-Leffl er noted in his diary that he 

slept extremely well during the night. During the afternoon 

Weierstrass told him that he had had another talk 

with Schwarz, who in the presence of Weierstrass would 

like to ask for forgiveness. Schwarz once again swore that 

during the last year, he – after Weierstrass had admonished 

him during the Easter of 1887 – had not told anyone 

the story, whose veracity he no longer believed in. 

But he had heard the story from the Finn Neovius, who 

had relayed the contents of a Finnish short story in which 

this had been told and had been informed that Mittag- 

Leffl er had been the model for it. 

In the evening Weierstrass took the two of them for a 

walk but when, after the evening meal, he tried to bring 

about a reconciliation, Schwarz excused himself saying 

he was tired and wanted to retire to bed. Not until the 

next afternoon did Schwarz arrive along with Weierstrass 

and present his apology for what had happened. Mittag- 

Leffl er promised to forget all about it and hoped that in 

the future they could work without hostility and with a 

mutual desire toward common scientifi c goals. He expressed 

a certain understanding that because of the 

strong competition between mathematicians in Germany, 

there could arise such back-stabbing but that he, who 

was above all such things, ought to be allowed to live in 

peace with everybody else. Schwarz started then to claim 

that he did not begrudge anyone nor compete with anyone 

but was stopped by Weierstrass who said he should 

not talk about things that were not true. 

During a walk in which everybody took part Schwarz 

expressed his intention to give a talk, which he later 

wanted to write down and send to Acta as a proof for all 

the world to see of their reconciliation. Next afternoon 

Schwarz did give a lecture but it seems never to have 

been sent to Stockholm. 

The next days in Wernigerode were devoted to further 


History 

discussions of the manuscripts for the prize of King Oscar 

II and both Schwarz and Hettner participated actively in 

these. In addition Mittag-Leffl er and Weierstrass tried to 

help Sonja in integrating a differential equation that had 

arisen in connection with her problem. Weierstrass was 

also working on the stability of the n-body problem. 

A kind of serene sense of consensus seemed to have 

descended on them those last days at the Müllers Hotel. 

When Schwarz left on 15 August, he showered Mittag- 

Leffl er with declarations of friendship. The only things 

that created some discord and kept confl icts alive were 

the machinations of Sonja. And Mittag-Leffl er thought 

that the reason that Sonja had been so unpleasant was 

her jealousy of the kindness Weierstrass had shown him. 

Many times their old teacher had turned to Mittag-Leffl 

er before he had addressed himself to her. Now she did 

not want to hear anything about Mittag-Leffl er’s plans to 

invite him to Sweden next summer. As she was not going 

to be in Sweden at the time, there was no reason why 

Weierstrass should be either. 

Weierstrass told Mittag-Leffl er that he strongly disapproved 

of Sonja’s plans to get a position in Paris and he 

resented her lack of tact in that she exploited every opportunity 

to speak disparagingly of Stockholm. He had 

always reminded her that no other country would have 

done as much for her as Sweden. On 26 August, Mittag- 

Leffl er concludes his diary with the following. 

My decision is made; I will try to bring it to the point 

that she [Sonja] will be if not personally indifferent to 

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me, at least far more distant than up to now. Only then 

can I endure all those pains that she infl icts on me. So 

long as she is so personally close to me, as she has been 

up to now, she tortures the life out of me. I can put up 

with much from those whom I am indifferent to but 

nothing at all from the few who are really close to me. 

On one of the last evenings, Mittag-Leffl er and Cantor 

went down to the centre of Wernigerode, where a large 

crowd of people had congregated and where festivities 

had been arranged in connection with the silver jubilee 

of the wedding of the local prince Graf Otto zu Stolberg- 

Wernigerode. The bridal couple, accompanied by servants, 

were driven through the town in elegant wagons 

towards the castle. ‘The whole made a tragicomic impression,’ 

Mittag-Leffl er wrote in his diary. On 28 August, he 

left Wernigerode. 

Arild Stubhaug is an established literary 

writer in his native Norway. Among 

mathematicians, he is known world-wide 

for his biographies of Niels-Henrik Abel 

and Sophus Lie. Both books have been translated to English, 

French, German and Japanese; the fi rst metioned one, 

moreover, to Russian and Chinese. 

ESI Lectures in Mathematics and Physics 

Recent Developments in 

Pseudo-Riemannian Geometry 

D. V. Alekseevsky (University of Edinburgh, UK) 

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Germany), Editors 

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This book provides an introduction to and survey of recent developments 

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Interview with Preda Mihăilescu 

Göttingen (Germany) 

from left to right: Samuel J. Patterson, Preda V. Mihăilescu, 

Axel Munk and Hanno Ehrler 

Preda V. Mihăilescu, born in Bucharest, Romania, is best 

known for his proof of Catalan’s conjecture. After leaving 

Romania in 1973, he settled in Switzerland. He studied 

mathematics and informatics in Zürich and then pursued 

a career as a mathematician in the Swiss banking sector. 

While still working in industry, he received his PhD from 

ETH Zürich in 1997. His thesis, titled Cyclotomy of rings 

and primality testing, was written under the direction of 

Erwin Engeler and Hendrik Lenstra. 

Mihăilescu started to work in academia as a research 

professor at the University of Paderborn, Germany. Since 

2005 he has been a professor at the Georg-August University 

of Göttingen (Germany). 

This interview took place on 3 December 2007 at the 

Mathematisches Institut of the University of Göttingen. It 

was prepared by Axel Munk (Mu) and Samuel Patterson 

(Pa), both from the University of Göttingen, who also participated; 

it was conducted by Hanno Ehrler from Deutschlandfunk, 

a German public broadcasting channel. 

Childhood and education, Romania and Switzerland 

First of all, I would like to ask you to tell us a little 

about your curriculum vitae, starting with your childhood. 

I read in an article that you had a talent for 

mathematics and numbers very early on. 

The game with numbers indeed started about the age of 

four, and before I was fi ve I had the reputation of the 

child who multiplies three digit numbers. I liked the 

game and could not understand what others found so astonishing 

about it – a little bit like listening to music that 

not many appear to notice. There were variations; friends 

of the family who were engineers visited and verifi ed my 

237 × 523, say, with a slide rule – and I found it quite awkward 

when I looked at “the competition”. I saw all these 

lines at varying distances on the slide ruler and realized 

that it could never yield the precise 6 digits of the result, 

since the maximum accuracy was 5 digits. So I knew that 

the verifi cation was only approximate – yet I knew that 

my result was precise. It was different on the playground; 

Interview 

only much later did I learn that the older people whom 

I did not know and who came making their challenges 

– sometimes they even tried to teach me to perform multiplications 

on paper – had actually made bets with their 

friends. 

Of course this gave a predisposition for mathematics, 

but it was not cultivated in any particular way for quite a 

while. Humanities came fi rst in my early life. 

What other important recollections do you have from 

your childhood? 

My childhood was relatively protected at a time of intense 

dictatorship. My parents did not hide from us all of 

what was happening. Thus we knew that an uncle was in a 

camp and that others had no job or worked as street dog 

catchers (it happened), having been poets and philosophers 

before that. We knew that twice a year my grandfather 

visited his former professor and master, who had 

recently fi nished his camp sentence, taking him a large 

dried sausage, which made the old man cry. Then they 

had several hours of discussions, which brought them 

back into the spiritual fi elds of their youth; I sensed a 

particular intensity in my grandfather when he was coming 

back from these visits, like a veil was raised for a brief 

period of time over a world I would never encounter in 

my own life. 

We knew that we had to be careful about what we 

said and we feared the power – but in fact I had little direct 

contact with “those guys” as a child. The parades, the 

songs and the doctrines were visible but not so tangible 

as the family circle that was offering comfort. Eventually, 

when going to school, a third category emerged: the 

school teachers, some of whom were really nice – yet they 

were employed by “them” so one could not be sure. For 

us, Stalinization came to an end towards the mid-60s, although 

it formally ended in 1956. In the 60s there was, on 

the one hand, an amnesty and many people came out of 

camps, and on the other hand … there was “Strawberry 

fi elds” and “Penny Lane”. If I recall correctly, we heard 

the fi rst Beatles songs even on public radio. By the time 

I was able to consciously ask the question: ‘why did one 

grandfather die under untold conditions while the other 

one lives with us to the greater joy of all, why did uncles 

disappear, and whether or not we had been close to such 

a destiny?’ a sort of a warmer wind was blowing. 

And then, when I became a teenager I was elected, 

as a good pupil, to some leading communist youth role 

in the school. I thought I was choosing a smart way out 

by basically saying at the meetings that one should try to 

be oneself even within this organization because according 

to what they said, they actually expected this from 

the youth – and not, like everybody knew, frightful obedience 

and indoctrination. The words were not exactly 

these but the meaning was quite clear, and after several 


Interview 

calls to order in offi ces of higher ranking “comrades”, I 

came to understand that I did not have all one needs for 

civil disobedience. This may be one of the personal experiences 

that made me choose, four years later, to seek 

asylum in Switzerland. 

And there you studied mathematics? 

And there I studied mathematics and entered a new life 

that was challenging in all directions. 

There were two things that led me to the choice of 

doing applied mathematics: one was the pragmatic estimate 

of survival odds for the inexperienced refugee that 

I was. But this could never suffi ce, not without the feeling 

of being in a world where one could do things, a lack of 

limitations that had to be experienced. So I went to do 

something that none of the males of my family had done, 

at least in the last three generations – become practical. 

They had all ended up teaching in one way or another; I 

was going to work in some productive area – quite ironic, 

looking back now, isn’t it? It is probably from my mother 

that I have learned a love for very practical, well done 

things. She was a gynaecologist and a surgeon. Delivering 

babies under the most diffi cult conditions, both medical 

and physical, was an unspoken, penetrating passion for 

her. From her I think I received the love for things with a 

practical purpose – developing a project like delivering a 

child, bringing something new to life… 

Industrial career 

So it is quite natural that you studied applied mathematics 

and then went into industry. What did you do 

there? What kind of work did you have to do? 

There are two periods of my industrial career. After 

the second “Vordiplom” in mathematics until the end 

of my computer science Masters degree, I worked, with 

interruptions, in a company in the machine industry. I 

applied numerical analysis and developed methods and 

programs for designing models of turbine blades so that 

the gas fl ow around these blades could be computed 

exactly. It is a typical application meant to provide reference 

data for testing larger software packages. From 

the desire for a visual interaction with these models, I 

was led to several visualisation applications; in the end, 

I even developed a visual interface for software that 

computes load fl ow and short-cut simulations in large 

electrical networks. It was an effective sales argument 

for that specialized, professional software and my fi rst 

application that had an impact on the market. That was 

also in the very early beginnings of graphical interfaces 

so I had to invent many of the tools I needed pretty 

much from scratch. 

Then, after completing my computer science studies, 

I switched to information security. I worked in that domain 

fi rst in banks and later in a consulting and development 

company. I would say that in the productive phases 

of my industry life, I was peacefully conceiving projects 

and bringing them to life, and there was nothing spectacular 

about it. The security of online ATM system in Switzerland, 

for example, which has been running since 1990 

with some natural renewals and no essential changes to 

the core, has withstood the test of time and represents 

the silent success behind that work. 

What does “information security” mean? Were you 

concerned with security systems? 

Yes – security of information. That means protecting data 

on the net against intrusion or modifi cation, protecting 

identities – in the sense that the identity of a person may 

be associated with documents or “electronic signatures” 

in an inalterable and irrevocable way – and variations of 

these basic problems of confi dentiality and identity. The 

methods used belong to the fi eld of cryptography, which 

uses number theory, and this is why a mathematician was 

preferred. The mathematical background was certainly 

useful so that the cryptographic issues were easily accessible. 

From my point of view – and probably not only 

mine – the challenge of the job was to supervise a large 

system so that no security gaps arose. Gaps due to organizational 

reasons rather than purely cryptographic 

ones often played the most important role; cryptography 

is well under control due to the theory. 

It was also the beginning of public key cryptography 

and I had a lot to do helping potential users be a bit more 

sensible. Primarily, security made the life of the normal 

programmer or computer user slightly harder, not only 

because of the numerous password protected applications 

one had to sign into sometimes dozens of times a 

day but also because a secured application could, at that 

time, be logically slower. One had to convince co-workers 

to accept such inconveniences to enforce the security 

of the internal network. 

At one stage, I was employed in the most important 

bank in Switzerland, which had the ambition of being 

ahead of its time in IT; it could also afford to pursue this 

goal. At that time the term “single sign-on” was starting 

to be coined as a desirable alternative to the multiple 

applications asking for a password on the desk of every 

employee. The technology answering the expectations of 

this time was only developed over fi ve years later. 

I was asked if I “could develop a single sign-on system” 

for the company’s network. What I was doing 

practically at that time was cryptographically securing 

the communications in the bank. This was very useful 

and could be done with the manpower available. I was 

unsuccessful in using a detailed technical argument to 

explain that “single sign-on” was not at that time within 

reach. On that occasion, I learned that sometimes a 

negative proof is not accepted as a basis for a decision. 

Maybe that experience made me wish for an environment 

where a negative proof is well regarded; anyhow, 

years later I was able to present a negative proof, which 

confi rmed Catalan’s conjecture, and that one was very 

well accepted. 

Pure mathematics during spare time… 

So we are back to pure mathematics. As I know, you 

also had time for studying pure mathematics; after all, 

you solved Catalan’s conjecture. How was it possible to 


have this job in the bank and the work in industry and 

at the same time be concerned with pure mathematics? 

Is it some kind of hobby or an inner drive? 

The inner drive was there and mathematics was also helpful 

for my mental balance. For instance, I used to work at 

the weekends or in the evenings at or around my PhD. 

And the PhD was about… 

The PhD was about primality testing and was thus in applied 

number theory with an explicit application to cryptography 

too. It was completed during my time at the 

bank. I then liked designing effi cient algorithms. Some 

of the ideas from that time came to fruition later, for instance 

recently in the fastest general primality test for 

large numbers. 

…and as a main occupation 

But at some time you decided to move from your industry 

job into research. Why did you make this decision? 

Probably the feeling had accumulated that I wanted to 

do more mathematics. It happened that I was offered a 

position at the RSA labs, the research group of a major 

cryptography company in the USA, and, at the same 

time, at a university. Doors were open in both directions; 

in fact, I fi rst went to RSA for several months and after 

that I accepted the university position, at Paderborn University 

in Germany. I guess my inner voice told me that 

I wanted to do academic research and teaching too. In 

Boston, I was already working at Catalan in the evenings 

– one may speculate whether this had an impact on my 

decision too … it is probably just speculation! 

Pa: Did the university come to you or did you apply for 

a university position at this time? 

I corresponded with Professor von zur Gathen (Paderborn 

University, Germany), who had invited me before 

to give some seminar lectures on the results of my PhD. 

We kept in touch after that and in fact he offered me a 

position that was also connected to some cryptographic 

applications, yes. 

Catalan’s conjecture 1 

Let’s speak shortly about Catalan’s conjecture. How 

did it come that you were concerned with this particular 

problem and how did you solve it? 

This is more than one question… 

Maybe you could tell us how you got involved with the 

problem and on which scientifi c shoulders your solution 

was built upon. 

Let’s try to break this down into several questions. Firstly, 

why did I get concerned with Catalan? I came across it at 

a conference in Rome during an interesting talk by Guillaume 

Hanrot. Returning to Paris where I was spending 

a summer doing research, I found myself alone and I 

worked several days to understand more about Hanrot’s 

methods. I did not stop until I had more than a proof of 

that conjecture! It happened that within a week or two, I 

Interview 

could do a step that was then considered to be non-negligible; 

I proved the so-called “double Wieferich conditions”. 

That was encouraging and may have strengthened 

my tenacity. Then I was in Paderborn, where I had the 

peace of mind to think over a longer period about this 

problem – it was different from the evenings of mathematics 

after work. 

You were asking about the “shoulders” on which I 

was standing. I would at fi rst answer that I was jumping 

from shoulders to shoulders. At the beginning it was the 

shoulders of various people who had investigated Catalan’s 

conjecture in the last 30 years; there’s a long list. 

After Bugeaud and Hanrot, who gave me a taste for the 

question, I got acquainted with Maurice Mignotte, who 

was very active in the area. I had to understand the ideas 

of Inkeri and Schwarz, and Bennet, Glass and Steiner, 

Tijdeman, and, of course, the consequences of the work 

of Baker. At the base of all practical results there was 

an analytic simplifi cation made by Cassels in the 1950s 

– everybody needed that result as a starting point. I certainly 

forget half of the important names… However, 

today I would claim that the shoulders that still hold 

the present proof are those of Kummer, Leopoldt and 

Thaine, who gave the fundamental facts from cyclotomy 

that are used. 

Perspectives of number theory 

Catalan’s conjecture is in the fi eld of number theory. 

What are the perspectives of number theory today? 

Pa: I’m glad this question has been asked of you. 

(smiles) 

Mi: Glorious as ever … I am sorry; on the one hand, I 

do not retain the authority to expand on this question 

– there are much more competent people to do that. And 

I can imagine they would also be very careful with prognostics. 

But since I stand here as one who has worked 

applying mathematics and who has also done research, 

I would like to recall one impressive example. Algebraic 

geometry in positive characteristic was born with Weil 

and Grothendieck (pursuing work done by Dedekind 

and Weber, and Kronecker and Artin) as I see it – being 

the amateur expert that I am – in the second half of the 

last century. It took some time to be established among 

geometers themselves. In the 80s however, with Schoof’s 

algorithm for fast counting of points on elliptic curves 

defi ned over (large) fi nite fi elds, the topic became in a 

short time a domain of intensive research in algorithmic 

number theory with practical applications in cryptography. 

It is hard to say where such phenomena happen or 

predict how they will happen again in the future, but they 

certainly will! 

1 Mihăilescu’s theorem (formerly Catalan’s conjecture) was 

conjectured by the mathematician Eugène Charles Catalan 

in 1844 and proved in 2002 by Preda Mihăilescu. It states that 

the only solution of the equation x a – y b = 1 for x, a, y, b > 1 

is x = 3, a = 2, y = 2, b = 3. The proof appeared under the title 

Primary cyclotomic units and a proof of Catalan’s conjecture, 

J. Reine Angew. Math. 572 (2004), 167–195. 


Interview 

The Göttingen tradition 

We are here at the University of Göttingen and, as Professor 

Munk told me, there’s a tradition at this university 

of combining pure and applied mathematics. Professor 

Patterson, could you tell us a little bit about the 

tradition of this university and about the state of art 

today. 

Pa: Well, the origins of mathematics in Göttingen really 

have to be traced back to Carl Friedrich Gauss, who was 

the director of the Sternwarte (the observatory here) for 

a very long period of time. He wasn’t actually a professor 

of mathematics – he was a professor of astronomy – but 

he was regarded as the leading mathematician in Europe 

at the same time. So at this point you have someone who 

was, for example, doing very practical work during the triangulation 

of the Kingdom of Hannover, and on the other 

hand developing, using this experience, the abstract area 

of differential geometry. And the tradition of Gauss has 

essentially inspired everybody in Göttingen since then, in 

one way or another. One of the people who took this up 

most strongly was Felix Klein, who was very, very much 

involved in the connections between mathematics and in- 

Mathematisches Institut Göttingen at the times of Courant and 

Hilbert and today 

New buildings under construction for Institut für mathematische 

Stochastik 

dustry. He didn’t get much gratitude. Such questions were 

hotly debated about 110 years ago. Klein was involved in 

a whole series of public arguments. This was part of the 

development of the Prussian state and industry. 

And at present? Is there some kind of tradition that 

leads to the present in the university here? 

Pa: Well, we are like most universities. We are quite small 

as a university, as far as mathematics is concerned, but 

there are three mathematical institutes inside the faculty. 

One is called, for historical reasons, the Mathematisches 

Institut, which is merely pure mathematics now, and then 

there are the two applied mathematics departments: the 

Institutes for Numerical and Applied Mathematics and 

the Institute for Mathematical Stochastics. If I may allow 

myself a personal opinion here, I would like to see, within 

a reasonable amount of time, all of these under one roof 

and working together with one another. At the moment 

this is not about to happen and the present planning is to 

move to a new building in about nine years. 

The perception of mathematics in industry 

In your biography you had both: on the one hand, you 

worked in applied mathematics in industry and on 

the other hand, you were working in research and now 

in academia. I would be interested to hear a little bit 

about how industry views mathematics? 

Let me give some partial answers and perspectives. First, 

there are those domains that are traditional “customers” 

of mathematics: mechanical and electrical engineering, 

fi nancial and insurance mathematics and a few more. 

There, mathematics is probably supposed to be an important 

tool for the precision and accuracy of the required 

application and the image of what one can expect from 

mathematics generally has precise contours. This is simply 

due to tradition. 

And outside these traditional branches? 

There have been many more recent applications born 

with the development of the computer. From statistics 

and information theory to the very dynamical branches 

of computer science – they are encountered almost everywhere, 

from the pharmaceutical industry to transportation 

scheduling, from the food industry to, say, fl ood and 

earthquake prediction. 

And what is the reception of mathematics here? 

Frankly, I would not know the answer. An educated guess 

may be that every domain reaches a degree of complexity 

beyond which the immediate solutions are not suffi cient 

and some mathematical understanding is called for. 

And this would then have a positive impact on the image 

of mathematics!? 

In principle, yes, depending on how widespread the 

awareness is that a change was brought about by mathematics. 

After all, maybe to a large extent, people in industry 

care less about the distinctions between various 

scientifi c disciplines, as long as things work… Therefore, 


there is a difference between the image of mathematics 

and the image of mathematicians. Maybe the latter 

is more distinct and I believe they are, in general, well 

regarded for their ability to structure complex problems, 

identify possible solutions and modify targets so that 

they become affordable. 

You mean that mathematicians are asked for in industry? 

I mean they are well regarded. There is often the principle 

of learning by doing in industry. You may fi nd a physicist 

solving integer programming problems or a number theorist 

becoming a software manager or a bank manager. 

So, mathematicians have a good reputation for coming 

around to problems on which they are trained and those 

for which they are not. They soon have to discover that 

they belong to a micro-culture to which absence of proof 

is close to illusion. This is a positive contribution to the 

collective, especially when they do not expect that the 

others adhere to the same standards of formal consistency. 

It is a good habit to explain convictions, even obtained 

by some proof reasoning, in terms closer to common language. 

Most people may resent an excess of precision 

and accuracy of expression as a kind of snobbery. 

How did you experience the change from industry to 

academia? 

The time frames are certainly different in industry and 

it is sometimes hard to pursue an important idea over 

a longer period of time. My taste and my own history 

make me long for places where the two meet or it is foreseeable 

that they could meet. To an extent, I was offered 

this opportunity in Göttingen for this reason. Here it is 

possible to pursue both theoretical research in number 

theory at the Mathematical Institute and practical research 

in biometrics at the Institute for Mathematical 

Stochastics. 

Biometrics 

Mr. Munk, in your institute you work jointly with Mr. 

Mihăilescu on biometrics. Are there many groups in 

mathematics that are working on biometrics? 

Mu: No – interestingly, this is not the case. Worldwide, 

there are many groups, either in academia or in industry, 

who work on that issue, of course, but usually their background 

is from computer science, in particular pattern 

recognition or electrical engineering. The group here is, 

as far as I know, the only larger group in a mathematical 

department that deals extensively with those issues 

– and that certainly makes it unique. Indeed, in our group 

we bring together experts from several areas. Preda 

Mihăilescu has a background in cryptography, on the one 

hand, and on the other hand, in biometrical identifi cation 

analysis. We combine his expertise with my statistical 

knowledge and our knowledge in pattern recognition, 

image processing and enhancement and specifi c aspects 

of geometry. Actually, our research process itself gradually 

increased the insight that many interesting mathematical 

questions are inherent to biometric issues. 

Mi: There is a nice historical analogue to what we are 

Interview 

doing. At the beginnings of data transmission with modems, 

the data were transmitted in clear text and the 

transmission rate was low. There is a fundamental law of 

information theory, Shannon’s law, that states that one 

cannot transmit data over a channel at a rate beyond the 

so called “channel capacity”, which is essentially limited 

by the physical channel used: wires, radio, etc. That physical 

limitation was hard to relax, so nobody knew how to 

reach better transmission rates for modems. Some people 

believed this was not possible without improving 

the physical channels – and that belief was scientifi cally 

backed up by Shannon’s law. Yet, Liv and Zempel looked 

at the problem from a different perspective and found 

that one does not have to send more text at a time in 

order to increase the rate. It suffi ces to increase the information 

that the text contains – and for this, one can compress 

the text before sending it on the physical channel 

and then decompress it, since in fact most of the data sent 

over the net have a high rate of redundancy! By using 

compression, the transmission rate suddenly increased, 

without changing the physical channels and without contradicting 

Shannon’s law. 

Fingerprints 

Mi: In fi ngerprint security we are in a somewhat similar 

situation now. Our group has proved that the security 

based on the currently used fi ngerprint data is bounded 

and in fact insuffi cient. Using presently extracted data, 

one can – by state of the art methods – never reach the 

degree of security required for internet transactions. The 

proof is based on statistical evidence from the literature, 

evidence we want to back up by our own extended fi eld 

research. But the order of magnitude is quite reliable. 

So, by changing the perspective, as Ziv and Lempel did, 

one is led to the observation that the fi nger contains a 

wealth of information that is currently not used in AFIS 

(Automated Fingerprint Identifi cation Systems). The human 

operator uses this passive information, so she can 

sort out evidence in cases in which the machine does not. 

Obviously, we need new algorithms capable of extracting 

and structuring such information; this will also increase 

the security rate. This was one major goal for our group 

from the start. We fi rst thought only of improving the 

identifi cation rate – now it shows that the same work can 

help improve the security systems based on biometry. 

I think it would be interesting for the readers to describe 

the different credentials that are present in the 

Institute for Mathematical Stochastics; moreover, what 

are people actually working on? 

Mu: In fact, we combine the expertise of various people 

with different backgrounds. As I said before, Preda 

Mihăilescu has an expertise in cryptography and practical 

biometrics. Then we have somebody whose background 

is from differential geometry, obviously an area of pure 

mathematics, that turns out to play a very important role 

in the modern analysis of fi ngerprints. My own background 

is statistics; we have people with a degree in computer 

science and even a degree in both computer science and 


Interview 

Screenshot from fi ngerprint development and analysis software 

mathematics. We collaborate with people from physics 

and from biology. This makes such an area more attractive 

for me personally; it is interdisciplinary in the true sense. 

At the end of the day, I think mathematical modelling 

plays a very important role, and this can and has to 

be done by mathematicians. This is probably the particular 

strength of our group: we tackle various applications 

from the perspective of mathematics. Our major focus is 

mathematical modelling and the understanding of mathematical 

structures that we use for fi ngerprint identifi - 

cation, for example. This is then combined with statistical 

issues, e.g. the investigation of the distribution of the 

main characteristics of a fi ngerprint on the fi nger. Just 

view your own fi nger tip and you experience a nice fl ow 

fi eld of ridges and bifurcations with fascinating random 

patterns – a living geometry! 

Human vs. machine expertise 

Talking about biometrics, where do you see the major 

challenges in the near future where mathematical ideas 

will make the difference? 

Mu: An important issue, which I think will play a major 

role in the next few years in the whole area, is the intersection 

between biometric identifi cation systems and 

security issues, i.e. security issues in the sense of cryptography. 

Here, Preda will certainly play a prominent role 

in our group. Furthermore, statistical ideas are required. 

Traditional cryptography uses reproducible information 

– any password or secret key is a unique chain of characters 

from an alphabet. In contrast, fi ngerprints have an 

inherent variability that has to be taken into account and 

cannot be avoided. This leads to a paradoxical situation: 

security under variability of the key. This seriously limits 

the potential security of biometrical systems! It appears 

that the state of the art approaches are unaware 

of this limitation in various applications, in particular in 

‘soft biometrics’, where time and money often imposes 

restrictions to security. However, I believe that a combination 

of thorough statistical understanding of these 

limitations and innovative use of additional fi ngerprint 

information, which so far has been neglected as well, will 

improve both reliability and security of such systems – a 

great challenge for the future! It is by thorough understanding 

of the nature of limitations that one can, like in 

the case of Liv and Zempel, fi nd a way to bypass them. 

Mi: I’d like to add that this is a domain of an intellectual 

nature where there is some unknown mathematics. It is 

interesting that computer scientists who have worked in 

“expert systems” have repeatedly faced the problem of 

learning from human experts. Some simple general rules 

can be easily understood and transformed into algorithms 

but then running “man against machine” on some 

specifi c decision problem, one fi nds that human experts 

tend to be better, yet they cannot explain to the computer 

scientist the choices behind their decision processes. This 

is a complex problem and certainly the human has a wide 

capacity to integrate – let me use the word holistically 

– complex, even apparently irrelevant data, and therefore 

improve the decision. We are thus trying to improve 

the algorithmic integration of the layers of information 

by learning from humans. If we make some step ahead 

in this direction, certainly the immense processing capacity 

of the computer will then make it possible to use the 

machine for double-checking expert decisions in critical 

contexts; experts do make errors too! 

Mathematics has the capacity of modelling. Situations 

like this one teach us a certain kind of humility, 

since the major, powerful tools of mathematics don’t get 

to the facts. There is a need for some interaction with 

the object that approaches a mathematical and machine 

understanding of what man was doing before. We are in 

this process, and it brings up new notions and new ideas, 

which then offer a feedback to the possibility of bringing 

more well-known – or possibly new – domains of mathematics 

into the game. We are at this turning point. 

Can you give an example of what man did before the 

machine? 

Yes, certainly. If you look at fi ngerprints, it was defi ned in 

a somehow conventional way that the so called minutiae 

identify the fi ngerprint and thus the person. Minutiae are, 

for instance, line endings and line bifurcations on the fi nger. 

Matching suffi ciently many of them (12–18, say, according 

to domestic laws of various countries) identifi es 

a person in court. However, an expert looks at the whole 

image of the fi nger and he uses everything he sees there 

for the orientation and thus for the matching of minutiae. 

The machine was (essentially) only taught to use the actual 

location of minutiae. So the integrative process will 

probably need to take ridge connections, curvatures and 

further visual impressions into account, which the expert 

may use in case of uncertainty. 

Mu: But there is another aspect, and maybe this clarifi es 

why we require this broad range of expertise: from statistics, 

imaging, pattern recognition and cryptography. If 

you, for example, simply raise the question of how secure 

a fi ngerprint identifi cation system can potentially be, 

various issues have to be considered. One is the pattern 

recognition part, which means how accurate the informa- 


tion is that the algorithms extract in order to optimize 

the identifi cation rate. The other issue, and no machine 

can answer this question, is how good the quality of fi ngerprints 

are in nature in order to make these things 

work, i.e. what are the inherent natural limitations of the 

achievable error rates in biometrics. This is not very well 

understood, albeit of great practical relevance. 

For example, we have recently started a project with 

the Bundeskriminalamt – the German Criminology Agency 

– where we investigate statistically how fi ngerprints 

transform over a lifetime. There appears to be a problem in 

quality with prints of elderly people and very young people. 

Fingerprints change during relatively short time periods 

in these age groups and this is not understood at all. 

This causes diffi culties for the minutiae – based approach, 

which is used in AFIS nowadays, because there 

is some physical distortion of the fi nger, especially with 

teenagers. It is not easy to identify the same person over 

a gap of two to three years because of fi nger growth. But 

you can do it if you use the background information that 

we are extracting, since the line connections are still the 

same; it is just the rectangular reference grid of the minutiae 

that is distorted, like one of those popular paintings 

of Vasarely. 

Pa: I’m just wondering here – fi ngerprints have been used 

for 100 years. Was this not a problem before now? 

Mi: As I explained, the human expert had less of this 

problem. On the other hand, the human can never process 

the huge amount of data that a machine can. We wish 

to teach the machine to use more information in uncertain 

cases and thus approach the differentiated attitude of 

the human, while keeping its specifi c high performance. 

Mu: And there is another issue that is very important. At 

the end of the day, the target of your identifi cation system 

is to guarantee certain error rates, irrespective of the 

particular method used or even of the fact that human 

experts are involved or not. However, the requirements 

set for these error rates logically depend on the application. 

Thirty years ago, fi ngerprints were mainly used in 

forensics but nowadays we are talking about fi ngerprint 

identifi cation in a variety of applications, including commercial 

systems like access control to personal computers 

or cell phones, where the security does not need to 

reach forensic standards. In contrast, other commercial 

applications involving fi nancial transactions require very 

low error rates and are only just developing. 

Security issues 

A particular problem of your research is probably to 

estimate the security of biometrical systems? 

Mu: How secure is the method? Certainly, we are investigating 

the limitations of technologies. To be more constructive, 

we believe that we can occasionally bring some 

new mathematical ideas into the business, which really 

can help to improve the identifi cation systems – and 

which to some extent also explore the possible limits of 

technologies. 

Interview 

Mi: As I said, some leading methods that are currently 

proposed have insuffi cient security. I explained why we 

are optimistic about the possibilities of improvement. It 

belongs to the purpose of research that one tries to prove 

lower bounds to the security that one can offer. This requires 

more work (in statistical modelling too). It is a diffi 

cult task that we take upon us. 

But we cannot provide the certainty that a deployed 

system doesn’t have gaps in the security conception and 

guidelines. The conception of the system is very important 

and it has to reduce the risks in a hierarchical structure. I 

know from earlier experience in the industry that security 

is not only a matter of the cryptography used but also of 

the risk hierarchies. The fi rst project I developed was the 

security of online ATM systems in Switzerland. There you 

have a hierarchy of keys and the ultimate keys are, by design, 

unknown to anybody. They are born, kept and used 

in a tamperproof security box. This is a computer whose 

memory is instantly deleted when you shake it or physically 

tamper with it. The next hierarchy of keys are known 

to a very few well-trusted people, etc. Actually I believe 

that the management did not completely trust this technology, 

so they also printed out the core key, making it accessible 

to a designated bank director; maybe this is better. 

However, the most important keys should be known 

to a minimal number of people. This is a matter of design; 

it’s not a matter of cryptography or of biometrics. 

There is a very strong connection between this research 

and the applied aspect, the aspect of concrete applications 

in industry, in bank systems. You told us that the 

mathematical aspect is very important in biometrics. 

Are there concepts or aspects that you can take from 

pure mathematics and then apply in your research? 

Could you describe some of these? 

Mu: In our institute, we are of course not only concerned 

with biometrics. To give you an example, we have recently 

started to work on growth modelling of biological 

objects such as trees and leafs. This is supported by the 

German Science Foundation and we collaborate closely 

with colleagues from the forest science department, who 

perform fi eld experiments. This practical problem has 

initiated fundamental research towards principal component 

analysis on manifolds, which we call geodesic 

principal component analysis. Here we benefi t a lot from 

discussions with colleagues from differential geometry 

and optimization. 

Mathematics and Applications 

Pa: There’s one comment I’d like to make on this: the notions 

of pure mathematics and applied mathematics are 

not terribly fi rm or precise. When I was a student, applied 

mathematics was very much associated with physics but 

what is now stated as applied mathematics, for example 

in cryptography, is very, very much what in those days 

was considered as pure mathematics; it was number theory. 

Number theory was considered, let us say, in the late 

1960s as one of the purest areas of mathematics; nowadays 

it is one of the most applied areas of mathematics. 


Interview 

Mi: Exactly, for instance the algebraic geometry over fi - 

nite fi elds that I mentioned. 

Conversely, do you think that there is also some payback 

from applications of mathematics, fostering new 

theoretical research? 

Mi: Mathematics progresses both by the impulse from 

questions raised in physics – and nowadays from a much 

wider domain of applications – and from questions arising 

in the mathematical research itself. An important aspect 

is that the period it takes for some new fundamental 

mathematical insight to fi nd an application – a refl ection 

in the sensible world, I would say – is long, usually very 

much longer than, for instance, the time it takes for a 

discovery in experimental physics to be translated into a 

revolutionary technology. 

Sometimes, this process is iterative. I am far from being 

an expert but here I think for instance of Riemannian 

geometry used in relativity theory, whose further 

development has continuous impacts on mathematics 

and physics, as I understand it. 

Pa: I think it actually seems to occur on a rather slow, 

almost geological, time scale. What happens is that you 

fi nd a student coming up, and she or he learns two or 

more different areas either from different teachers or 

from books or something like that and then brings them 

together in her or his person. And then the development 

takes place. Each generation of mathematicians has got 

this essentially dialectic aspect that brings together new 

connections and this is exactly what keeps mathematics 

alive. Just as a single line, it would die out. 

Public awareness – in Germany 

2008 is the year of mathematics in Germany. How do 

you think the awareness of mathematics is in the real 

world, in the public? 

Mu: Personally, I would say that a major challenge for 

the mathematical community is to focus the attention 

of society on the benefi ts it draws from mathematical 

research. When something in mathematics is invented, 

typically it takes a very long time until it is recognized 

in society as a value or a contribution. Moreover, when 

a mathematical result is at the heart of a practical invention 

it is not recognized as such anymore. In other disciplines 

this goes faster and this process is much more 

direct. Inventions in molecular biology or in medicine, 

let’s say, are fi rst of all recognized as inventions of these 

disciplines. At the beginning of the 20 th century Radon 

developed what nowadays is called the Radon transform, 

and about 60 or 70 years later people used it for tomography. 

Of course, nobody in the public related these two 

things anymore. And there are many other examples: the 

mathematics of fi nancial markets that is used in every 

investment bank nowadays is founded on the famous 

Ito-calculus from the early 50s, which itself is based on 

Brownian motion developed by Einstein and Wiener in 

the fi rst quarter of the last century. This has been very 

successfully applied in the 70s to option pricing, and the 

Noble prize in economy was awarded for this development. 

These very long periods of time prevent the public 

from appreciating the value of mathematics and how it 

really matters to us. 

Pa: Can I say something about my experience here? 

Much of modern mathematics started in Germany during 

the 19 th century and it came to me as a huge surprise 

to discover how negative many people in Germany are 

about mathematics. It is a very strange experience: when 

one says one is a mathematician here, one usually gets the 

answer, ‘I was never any good at mathematics at school,’ 

or something of this nature. In fact, the attitude is quite 

different, let’s say, in the UK or in France where there are 

many programs in the media about mathematics, on the 

BBC or in other places. We’ve just been involved here in 

a program that is being made by Marcus du Sautoy for 

the BBC. One might hope, though it’s a rather weak hope, 

that in the course of the Year of Mathematics one might 

actually change this a little; it seems to be a specifi cally 

German attitude that you do not fi nd in other countries. 

May I come to my last question – do you think that the 

special work you’re doing in biometrics, which combines 

some aspects of academics and some aspects of 

industry and applicability may have an effect on the 

recognition of mathematics in the public? 

Mi: If it succeeds in reaching the goals that we have set… 

Our goals are to improve technological systems, on the 

one hand, and to develop some theoretical models that 

can prove something positively in areas that are poorly 

understood, on the other hand. Maybe the best hope is 

that solutions found for this specifi c problem may call 

for some new mathematics, if we develop concepts that 

can be used in similar problems. This is the best I can 

hope for, and then let the waves go the way the waves go. 

It’s always a matter of who hears at the other end of the 

channel. It’s never the whole public. 

Hanno Ehrler [hanno.ehrler@gmx.de] studied musicology, 

art history and ethnology in Mainz. He works in part 

as a freelance journalist for the Frankfurter Allgemeine 

Zeitung (FAZ) and the public German TV station ARD 

(subjects: contemporary music, science fi ction, modern 

technologies). 

Axel Munk [munk@math.uni-goettingen.de] is a professor 

at the Institut für Mathematische Stochastik at Göttingen 

University. His research in statistics focuses on nonparametric 

regression, medical statistics, statistical inverse 

problems and imaging, statistical modelling and shape 

analysis. Within pattern recognition, he is mainly interested 

in the analysis of fi ngerprints. 

Samuel J. Patterson [sjp@math.uni-goettingen.de] is a 

professor at the Mathematisches Institut, Göttingen University. 

His main research areas comprise analysis on and 

around Kleinian groups, generalized theta functions and 

metaplectic groups, and analytic number theory in general. 


Personal column 

Please send information on mathematical awards and 

deaths to the editor. 

Awards 

The 2007 Oberwolfach Prize for Excellent Achievements in 

Algebra and Number Theory was awarded to Ngô Bao Châu 

(Paris-Sud/Orsay). 

The 2008 Clay Research Awards has been given to Cliff Taubes 

(Harvard University, Cambridge MA, USA) for his proof of 

the Weinstein conjecture in dimension three, and to Claire Voisin 

(CNRS, IHES, and the Institut Mathématique de Jussieu, 

Paris, France) for her disproof of the Kodaira conjecture. 

Gitta Kutyniok (Giessen University, Germany) has been selected 

to receive this year’s von Kaven Prize in Mathematics for 

her outstanding work in the fi eld of applied harmonic analysis. 

This prize is awarded by the von Kaven Foundation, which is 

administered by the Deutsche Forschungsgemeinschaft (DFG, 

German Research Foundation). 

Andreas Neuenkirch (University of Frankfurt, Germany) has 

been awarded the 2007 Information-Based Complexity Award 

for Young Researchers. 

Newsletter editor Themistocles Rassias (National Technical 

University of Athens, Greece) has been awarded a Doctor 

Honoris Causa from the University of Alba Iulia (Romania). 

Congratulations! 

László Lovász (Eötvös Loránd University, Budapest, Hungary 

and current president of the International Mathematical Union 

IMU) has been awarded the Bolyai Prize, one of the highest 

honours in Hungarian scientifi c life. 

The 2008 Steele Prize for a Seminal Contribution to Mathematical 

Research has been awarded to Endre Szemerédi (Budapest, 

Hungary, and Rutgers, USA) for the paper “On sets of 

integers containing no k elements in arithmetic progression”, 

Acta Arithmetica XXVII (1975). 

One of the 2008 Maxime Bôcher Memorial Prizes has been 

awarded to Alberto Bressan (Italy and State College, PA, USA) 

for his fundamental works on hyperbolic conservation laws. 

The 2008 Doob Prizes have been given to Enrico Bombieri (Italy 

and IAS Princeton, USA) and Walter Gubler (Switzerland and 

Humboldt University, Berlin, Germany) for their book “Heights 

in Diophantine Geometry” (Cambridge University Press, 2006). 

The Autumn Prize of the Mathematical Society of Japan (MSJ) 

for 2007 has been awarded to Tadahisa Funaki (University of 

Tokyo, Japan) for his outstanding contribution to stochastic 

analysis on large scale interacting systems. 

The 2007 MSJ Geometry Prizes have been awarded to Shigeyuji 

Morita and Kenichi Yoshikawa (both from the University of 

Tokyo, Japan). The award to S. Morita has been made in recognition 

of his fundamental research on mapping class groups; 

Personal column 

the award to K. Yoshikawa has been made for his outstanding 

research on Ray-Singer analytic torsion and its behaviour on 

various moduli spaces. 

The MSJ Analysis Prizes have been awarded to Shigeki Aida 

(Osaka University, Japan) for his contributions to stochastic 

analysis in infi nite dimensional spaces, to Toshiaki Hishida 

(Niigata University, Japan) for his contributions to new developments 

in Fujita-Kato theory for the Navier-Stokes equations, 

and Takeshi Hirai (Kyoto University, Japan) for his contributions 

to representation theory of infi nite symmetric groups. 

Zbigniew Błocki (Kraków, Poland) was awarded the Zaremba 

Great Prize of the Polish Mathematical Society for his papers 

on complex analysis. 

Grzegorz Świątek (Warsaw, Poland and Pennsylvania) was 

awarded the Banach Great Prize of the Polish Mathematical 

Society for his papers on dynamical systems. 

Teresa Ledwina (Wrocław, Poland) was awarded the Steinhaus 

Great Prize of the Polish Mathematical Society for her papers 

on mathematical statistics. 

Adam Skalski (Łódź, Poland) was awarded the Kuratowski Prize. 

Radosław Adamczak (Warsaw, Poland) was awarded the Prize 

of the Polish Mathematical Society for young mathematicians. 

Luis Barreira (Instituto Superior Técnico, Lisbon, Portugal) 

was the 2008 winner of the Ferran Sunyer i Balaguer Prize. He 

was acknowledged for his book “Dimension and Recurrence in 

Hyperbolic Dynamics”. 

George Lusztig (Romania and MIT, Boston, USA) has been 

awarded the 2008 Steele Prize for Lifetime Achievement. 

The Adams Prize of Cambridge University has been awarded 

jointly to Tom Bridgeland (University of Sheffi eld, UK) and 

David Tong (University of Cambridge, UK). 

The 2007 SASTRA Ramanujan Prize has been awarded to Ben 

Green (University of Cambridge, UK). This annual prize, which 

was established in 2005, is for outstanding contributions to areas 

of mathematics infl uenced by Srinivasa Ramanujan. 

Ulrike Tillmann (Oxford University, UK) has been selected as 

the recipient of a Friedrich Wilhelm Bessel Research Award. 

This award is conferred in recognition of lifetime achievements 

in research. 

Deaths 

We regret to announce the deaths of: 

Karl-Heinz Diener (Germany, 18 September 2007) 

Henry R. Dowson (UK, 28 January 2008) 

Anders Frankild (Denmark, 10 June 2007) 

Helmuth Gericke (Germany, 15 August 2007) 

Karl Gruenberg (UK, 10 October 2007) 

Samuel Karlin (UK, 18 December 2007) 

David Kendall (UK, 23 October 2007) 

Richard Lewis (UK, 26 July 2007) 

Farkas Miklos (Hungary, 28 August 2007) 

Georg Nöbeling (Germany, 16 February 2008) 

Alex Rosenberg (Germany and USA, 27 October 2007) 

Jean-Luc Verley (France, 25 October 2007) 


Journal of the European Mathematical Society 

Aims and Scope 

Journal of the European Mathematical Society (JEMS) is the official journal of the EMS. The Society, founded in 1990, works at promoting joint scientific 

efforts between the many different structures that characterize European mathematics. JEMS will publish research articles in all active areas of pure 

and applied mathematics. These will be selected by a distinguished, international board of editors for their outstanding quality and interest, according 

to the highest international standards. Occasionally, substantial survey papers on topics of exceptional interest will also be published. Starting in 1999, 

the Journal has been published by Springer-Verlag until the end of 2003. Since 2004 it is published by the EMS Publishing House. The first Editor-in- 

Chief of the Journal was J. Jost, succeeded by H. Brezis in 2003. 

Editorial Board 

Haïm Brezis (Editor-in-Chief), Université Pierre et Marie Curie, Paris, France, 

Rutgers University, USA and Technion, Haifa, Israel 

Antonio Ambrosetti, SISSA, Trieste, Italy 

Luigi Ambrosio, Scuola Normale Superiore Pisa, Italy 

Enrico Arbarello, Università di Roma “La Sapienza”, Italy 

Robert John Aumann, The Hebrew University of Jerusalem, Israel 

Ole E. Barndorff-Nielsen, Aarhus University, Denmark 

Henri Berestycki, EHESS, Paris, France 

Joseph Bernstein, Tel Aviv University, Israel 

Fabrice Bethuel, Université Pierre et Marie Curie, Paris, France 

Jean Bourgain, Institute for Advanced Study, Princeton, USA 

Jean-Michel Coron, Université Pierre et Marie Curie, Paris, France 

Ildefonso Diaz, Instituto de España, Madrid, Spain 

Corrado De Concini, Università di Roma “La Sapienza”, Italy 

Simon Donaldson, Imperial College, London, UK 

Yasha Eliashberg, Stanford University, USA 

Geoffrey Grimmett, Cambridge University, UK 

Gerhard Huisken, Max-Planck-Institute for Gravitational Physics, Golm, Germany 

Marius Iosifescu, Romanian Academy, Bucharest, Romania 

Wilfrid S. Kendall, University of Warwick, Coventry, UK 

Sergiu Klainerman, Princeton University, USA 

Hendrik Lenstra, University of Leiden, The Netherlands 

Eduard Looijenga, Utrecht University, The Netherlands 

Angus Macintyre, Queen Mary, University of London, UK 

Ib Madsen, Aarhus University, Denmark 

Jean Mawhin, Université Catholique de Louvain, Belgium 

Stefan Müller, Max-Planck Institute for Mathematics in the Sciences, Leipzig, Germany 

Sergey Novikov, University of Maryland, College Park, USA, 

and Russian Academy of Sciences, Moscow, Russia 

Lambertus Peletier, University of Leiden, The Netherlands 

Alfio Quarteroni, EPFL Lausanne, Switzerland 

Mete Soner, Sabanci University, Istanbul, Turkey 

Alain Sznitman, ETH Zürich, Switzerland 

Mina Teicher, Bar-Ilan University, Ramat-Gan, Israel 

Claire Voisin, IHES, Bures sur Yvette, France 

Efim Zelmanov, University of California, San Diego, USA 

Günter M. Ziegler, Technische Universität Berlin, Germany 

Subscription information 

Journal of the European Mathematical Society is published in one volume per annum, four issues per volume. Print ISSN: 1435-9855, online ISSN: 1435-9863 

The annual subscription rate is 320 Euro (suggested retail price, add 30 Euro for postage and handling). Other subscriptions on request. 

Contents Volume 10 (2008) 

Issue 1: 

J. Bourgain and W.-M. Wang, Quasi-periodic solutions of nonlinear random Schrödinger equations 

Eugenio Montefusco, Benedetta Pellacci and Marco Squassina, Semiclassical states for weakly coupled nonlinear Schrödinger systems 

C. Bandle, J. v. Below and W. Reichel, Positivity and anti-maximum principles for elliptic operators with mixed boundary conditions 

Marek Fila and Michael Winkler, Single-point blow-up on the boundary where the zero Dirichlet boundary condition is imposed 

Itai Benjamini and Alain-Sol Sznitman, Giant component and vacant set for random walk on a discrete torus 

Koen Thas and Don Zagier, Finite projective planes, Fermat curves, and Gaussian periods 

Hoai-Minh Nguyen, Further characterizations of Sobolev spaces 

Gabriele Mondello, A remark on the homotopical dimension of some moduli spaces of stable Riemann surfaces 

M. Farber and D. Schütz, Homological category weights and estimates for cat1 (X, ξ) 

Issue 2: 

Dong Li and Ya. G. Sinai, Blow ups of complex solutions of the 3D Navier–Stokes system and renormalization group method 

Ursula Hamenstädt, Bounded cohomology and isometry groups of hyperbolic spaces 

Michael Larsen and Alexander Lubotzky, Representation growth of linear groups 

I. P. Shestakov and E. Zelmanov, Some examples of nil Lie algebras 

Y.-P. Lee, Invariance of tautological equations I: conjectures and applications 

Andreas Čap, Infinitesimal automorphisms and deformations of parabolic geometries 

Jean Van Schaftingen and Michel Willem, Symmetry of solutions of semilinear elliptic problems 

Margarida Mendes Lopes and Rita Pardini, Numerical Campedelli surfaces with fundamental group of order 9 

Sergiu Klainerman and Igor Rodnianski, Sharp L1 estimates for singular transport equations 

M. Burak Erdogˇ an, Michael Goldberg and Wilhelm Schlag, Strichartz and smoothing estimates for Schrödinger operators with large magnetic potentials in �3 Leonid Makar-Limanov and Jie-Tai Yu, Degree estimate for subalgebras generated by two elements 

J.-P. Francoise, N. Roytvarf and Y. Yomdin, Analytic continuation and fixed points of the Poincaré mapping for a polynomial Abel equation 

To appear: 

Bjorn Poonen,The moduli space of commutative algebras of finite rank 

Jean Bourgain and Alex Gamburd, Expansion and random walks in SL d (�=p n �): I 

L. Escauriaza, C.E. Kenig, G. Ponce and L. Vega, Hardy’s uncertainty principle, convexity and Schrödinger Evolutions 

Tomek Szankowski, Three-space problems for the approximation property 

Eitan Tadmor and Dongming Wei, On the global regularity of sub-critical Euler-Poisson equations with pressure 

Ioan Bejenaru and Daniel Tataru, Large data local solutions for the derivative NLS equation 

Radu Ignat and Felix Otto, A compactness result in thin-film micromagnetics and the optimality of the Néel wall 

Sergio Albeverio, Alexei Daletskii and Alexander Kalyuzhnyi, Random Witten Laplacians: Traces of Semigroups, L 2 -Betti Numbers and Index 

J. Krieger and W. Schlag, Non-generic blow-up solutions for the critical focusing NLS in 1-d 

Jean Van Schaftingen, Estimates for L 1 -vector fields under higher-order differential conditions 

European Mathematical Society Publishing House / Seminar for Applied Mathematics / ETH-Zentrum FLI C4 

CH-8092 Zürich, Switzerland / www.ems-ph.org / subscriptions@ems-ph.org

Interview with JEMS editor-in-chief 

Haïm Brezis 

Conducted by Thomas Hintermann (EMS Publishing House, Zurich, Switzerland) 

What publishing experience did you have before you 

started working for JEMS? 

Shortly after I got my PhD, in the early seventies, I was 

invited to serve on the board of several journals, including 

the Journal of Functional Analysis, the Archive for 

Rational Mechanics and Analysis, and others. I learned 

much from the chief editors of those journals, particularly 

Jacques-Louis Lions, Paul Malliavin, Irving Segal and 

Jim Serrin. I am currently on the board of about 25 journals, 

but of course with various levels of involvement. I 

initiated Communications in Contemporary Mathematics 

ten years ago together with Xiao-Song Lin. Moreover, 

I am a chief editor for two book series published by 

Birkhäuser and CRC Press. 

How long have you been managing JEMS? 

I was asked in 2003 by Rolf Jeltsch, then President of 

the EMS, to take over from Jürgen Jost, the fi rst chief 

editor of JEMS, who had done a very good job but was 

eager to step down. This was around the time when the 

Interview 

EMS had decided to transfer JEMS from Springer to the 

newly created EMS Publishing House. At fi rst, I was reluctant 

to accept, being already engaged with many other 

projects. I discussed the matter with some colleagues who 

were closely involved with the EMS, among them Mina 

Teicher, Doina Cioranescu and Jean-Pierre Bourguignon. 

They convinced me that it was important to carry on this 

project, which was originated by the fi rst president of the 

EMS, F. Hirzebruch. The change of publisher offered a 

unique opportunity to shape the journal and build on the 

good reputation the journal had already acquired before 

my time. Your own encouragement, Thomas, was another 

factor in my decision to accept the challenge. 

Please describe your cooperation with the other editors 

and the refereeing process for JEMS. 

It is my principle to allow individual editors as much 

freedom in their activities as possible. In my experience, 

the more responsibility an editor of a journal is given, the 

more active and reliable he becomes. Usually, the editors 

of JEMS receive or solicit manuscripts and contact reviewers 

entirely on their own, and then forward them to 

me with a well-documented recommendation. Of course, 

the ultimate decision lies in my hands but it is very rare 

that I challenge the advice of an editor. The editors are 

aware of this policy and they act with utmost care because 

they feel responsible for the quality of JEMS. This 

is not always the policy for other journals; I know examples 

where the chief editor has a much tighter hand 

on the process, often soliciting additional opinions that 

might confl ict with the original advice. As a result, members 

of the board become less involved and sometimes 

are even reluctant to communicate great papers out of 

fear that their enthusiasm might be challenged. 

When I took over, I reshaped the board. My primary 

concern was to bring in leading fi gures, at the peak 

of their creativity and connected to young people – the 

most natural source of papers! From my past experience 

with other journals I knew who was taking seriously his/ 

her mission of editor; I avoided distinguished mathematicians 

who do not reply to email! 

Since JEMS is a general mathematical journal, it was 

important to have a good distribution in terms of topics. 

Geography also played a role: one of our goals was to 

increase the visibility of the European Mathematical Society 

within the international mathematical community. 

Most of the members of the board are based in Europe 

but we also have many European “expatriates” working 

in other countries, notably the United States. Everyone 

on the board understands that the excellent ranking of 

JEMS – among the top ten – is a direct result of his/her 


Interview 

personal activity. Moreover, my policy is to encourage 

our editors to publish some of their own work in JEMS. 

I know that this is considered “politically incorrect” in 

some editorial circles. But our journal is young and I feel 

that this is the best way to send a strong signal to authors 

about the quality of JEMS. 

You are a prolifi c mathematical author (nonlinear 

functional analysis and PDEs and more) yourself. How 

do you fi nd the time necessary to manage the JEMS enterprise? 

Finding the time is indeed a big problem and often requires 

compromise. On the other hand, I view this task as 

an integral part of my scientifi c activity. It also provides 

an excellent opportunity to become acquainted with new 

developments, particularly those outside my immediate 

fi eld of research. It is a unique observatory; the information 

I receive at JEMS becomes very useful when I take 

part in committees awarding prizes, e.g. at the Académie 

des Sciences. 

Are journals still as important as in the past, in times 

where preprints are freely circulating on the Internet 

and in preprint archives? Please comment on the added 

value of refereeing! 

With all the preprints circulating via email, it is really hard 

to say where we are heading. Publishing in a respectable 

journal, and particularly in a top journal like JEMS, endows 

any paper with a quality stamp most authors fi nd 

very desirable. For young people it is often necessary on 

their CV. It is the ultimate sign that a paper has been 

found correct and worthy of attention by the mathematical 

community. One should not underestimate the psychological 

impact on some talented people who work in 

isolation and sometimes under diffi cult conditions. 

JEMS is a general mathematical journal, accepting 

papers from all areas. However, are there particularly 

strongly represented fi elds? 

My specialty is nonlinear PDEs and it is inevitable that 

my own fi eld is well represented. For example, we decided 

to publish a special issue dedicated to Antonio 

Ambrosetti, one of our board members, who is a leading 

fi gure in nonlinear PDEs. This tilted the balance towards 

analysis. However, we took care to readjust matters in 

later issues of the journal. I am closely acquainted with 

several editors from other fi elds and I encourage them 

to compensate my personal bias towards analysis. As it 

stands now, I think we can say that JEMS is a truly interdisciplinary 

journal within mathematics. 

Do you think that JEMS is representing the EMS in the 

eyes of mathematicians, or is it perceived as just a top 

journal disconnected from the society? 

It is the fate of most mathematical journals to be disconnected 

from their “owners”. The journals of the AMS are 

not especially linked with the society in the eyes of the 

public. The same can be said of the CRAS or the Annales 

de l’ENS. Communications on Pure and Applied Mathematics, 

which is an NYU journal, is an exception. 

In the case of JEMS the situation is slightly different. 

The EMS is a rather young society closely connected with 

the emergence of Europe as a major scientifi c power and 

it is natural for the EMS to increase its visibility through 

a leading journal. I am pleased with the wide geographical 

distributions of authors (about two thirds from Europe 

and one third from the rest of the world). One of the 

goals of JEMS is to become the “fl agship” of the EMS. 

In the last few years, JEMS has more than doubled the 

number of published papers, and the development is 

likely to continue. What makes JEMS so attractive for 

authors? What are, in your experience, the most important 

factors for authors when deciding on the journal 

for their article? The EMS is a not-for-profi t organization; 

do you think that this is an important detail for 

authors and editors? 

I suppose that the excellent reputation of the members 

of the board (as leading scientists) and their effi cient 

personalities plays a role. Also, many active young European 

mathematicians have enjoyed the support of European 

exchange networks in their formative years. They 

had the opportunity to meet and collaborate with colleagues 

from other European countries. Not surprisingly, 

they have sympathy for a European journal. In addition, 

there is increasing discomfort among authors and editors 

toward the profi t-making giants in the publication industry; 

this is also a factor in favour of JEMS. 

Any fi nal comments? 

I derive much pleasure from my activities at JEMS and 

the friendly collaboration of all the members of our 

board. The working relationship with the whole editorial 

team of the EMS has been superb, especially when a major 

overfl ow of high-class papers required a substantial 

increase in the size of the journal. I am very grateful to 

you, Thomas, for your continued enthusiasm during these 

“pioneering times” for JEMS. 

Haïm Brezis [brezis@math.rutgers.edu] 

is professor emeritus at the Université 

Pierre et Marie Curie (Paris VI), visiting 

distinguished professor at Rutgers 

University, USA, and at the Technion – Israel Institute of 

Technology. He is the editor-in-chief of the Journal of the 

European Mathematical Society. 

Thomas Hintermann [hintermann@ 

ems-ph.org] is the director of the EMS 

Publishing House. 


The 100 th 

anniversary of ICMI 

Symposium, Rome, 5–8 March 2008 

The fi rst century of the International Commission on 

Mathematical Instruction (1908–2008) 

Refl ecting and shaping the world of mathematics 

education 

Maria G. (Mariolina) Bartolini Bussi (Modena, Italy) 

The logo of the symposium reproduces the fl oor of 

the Capitol Square in Rome, which was designed by 

Michelangelo. 

At the beginning of March, in Rome, the centennial of 

the ICMI was celebrated. In this short paper, rather than 

covering all of the scientifi c components of the symposium, 

I shall try to convey the emotion of being there. 

The website of the symposium (http://www.unige.ch/ 

math/EnsMath/Rome2008/) contains a lot of information 

about the program and also photos of the scientifi c 

and the social events. 

The symposium was held in two historical buildings: the 

Corsini Palace, which dates back to the 16 th century and hosts 

the Academy of Lincei; and the Mattei Palace (again 16 th 

century), which hosts the Institute of Enciclopedia Italiana. 

The Academy of Lincei is the most ancient learned society 

in the world. It was established as an international society 

in 1603 by Federico Cesi and others (a Dutch scientist 

among them). Galileo Galilei added his name and fame to 

the society a few years later (in 1611) and the number of 

academicians increased steadily with the addition of Italians 

and non-Italians from the worlds of science, poetry, 

law and philology. In front of the Corsini Palace there is 

a villa called “The Farnesina”, built between the 15 th and 

the 16 th centuries; it has a beautiful garden and has famous 

Raffaello’s frescoes inside. The villa belongs to the academy, 

which holds occasional formal celebrations there. 

Both the Corsini and Mattei Palaces are in the city centre, 

within walking distance of the Vatican and other famous 

sites in Rome. In particular, Corsini Palace is in Trastevere, 

a well-known district along the river Tevere and an area 

beloved by tourists; most of the district is for pedestrians 

only and visitors can enjoy the narrow streets with low, old 

houses and famous restaurants on the ground fl oor. 

The ICMI was established in 1908, during the International 

Congress of Mathematicians held in Rome, with the 

aim of supporting and expanding the interest of mathematicians 

in teaching in schools. Its fi rst president was Felix 

Klein. Something similar was attempted in many different 

subjects but only in mathematics was there success in obtaining 

widespread international collaboration, in order 

to face problems relating to the social image of mathematics, 

to diffi culties in learning and to links with research 

and applications. Some years ago the idea of celebrating 

Mathematics Education 

the centennial in the same place where the commission 

had been established was launched. In spite of the many 

diffi culties of hosting a large congress in a city crowded 

with tourists all through the year, Ferdinando Arzarello 

and Marta Menghini accepted the challenge and designed 

a celebration that aimed to evoke the original event as 

much as possible: only a few days separated the true birthday 

from the dates of the symposium; the palace was the 

same (the Palace of the Academy of Lincei); and even the 

social program was the same, with a beautiful banquet and 

excursion to the famous villas of Tivoli (Villa Adriana and 

Villa d’Este). The scientifi c committee (chaired by Arzarello) 

was composed of researchers from every continent 

who were well-known fi gures in the fi eld of the didactics 

of mathematics, both for the research that they had carried 

out and for the institutional positions they held. The 

local organising committee was made up of professors 

from Italian Departments of Mathematics. 

Everything worked in a wonderful way. It was not easy 

in Rome to leave the beautiful surroundings to go into the 

Corsini Palace to take part in the symposium. Yet the program 

was so interesting that the large room of the meeting 

was always crowded. 

When a society turns one hundred, the memories are 

usually to be reconstructed by historians. Yet the organizers 

succeeded in interviewing (and in most cases also inviting 

to Rome) some of the most relevant mathematicians and 

mathematics educators who bear witness to this long history. 

The culmination of effort for the event has produced 

a website that will constitute an extraordinary source for 

the future (http://www.icmihistory.unito.it/). In the section 

‘Interviews and fi lm clips’ many eyewitnesses utter their 

thoughts, e.g. Emma Castelnuovo, Trevor Fletcher, Maurice 

Glaymann, Geoffrey Howson, Jean-Pierre Kahane, 

Heinz Kunle, André Revuz and Bryan Thwaites. Others 

have been evoked by the invited lecturers. Hyman Bass 

(the ex-president of the ICMI) opened the symposium 

with a speech on ‘Moments in the history of the ICMI’. 

The closing plenary was given by Michèle Artigue, the 

president of the ICMI, on the theme ‘One century at the 

interface between mathematics and mathematics education 

– refl ections and perspectives’. Between them the following 

themes were addressed in plenary speechs: The development 

of mathematics education as an academic fi eld; Intuition 

and rigour in mathematics education; Perspectives 

on the balance between application & modelling and ‘pure’ 

mathematics in the teaching and learning of mathematics; 

The relationship between research and practice in mathematics 

education – international examples of good practice; 

The origins and early incarnations of the ICMI; the ICMI 

Renaissance – the emergence of new issues in mathematics 

education; and Centres and peripheries in mathematics 

education, the ICMI’s challenges and future. 

This list of titles conveys the idea that the celebration 

was not just reminiscing in history but was open to the 

future directions of research in mathematics education 

and the possible action to be taken to improve the level 

of scientifi c culture in various countries. Other scientifi c 

activities were the working groups and short talks. Details 

may be found on the website mentioned above. 


Mathematics Education 

More than 180 invited participants were present in 

Rome from all over the world. Besides the representatives 

of the Italian institutions that supported the event and some 

of the past offi cers of the ICMI, it is worthwhile to note 

the attendance of the president Lazlo Lovasz and the vicepresident 

Claudio Procesi of the IMU and of the president 

of the ICTP (International Centre for Theoretical Physics) 

Ramadas Ramakrishnan, on behalf of UNESCO. Most 

participants were accompanied by relatives and friends, as 

Rome is always appealing for a spring holiday. 

The proceedings are in progress and will be ready in a 

few months. There is no doubt that they will constitute an 

indefeasible source for all the mathematics educators who 

feel the need to know about the roots of their academic 

fi eld and also for the organizers of the next centennial 

symposium in Rome in 2108! 

ICMI AWARDS 

In 2000, the ICMI decided to create two awards for research 

in mathematics education: the Felix Klein Medal, 

from the name of the fi rst president of the ICMI, and the 

Hans Freudenthal Medal, from the name of its eighth president. 

The fi rst two medals were announced in 2003 and 

were awarded to Guy Brousseau, France (Klein medal) and 

Celia Hoyles, UK (Freudenthal medal). The citations are 

available at http://www.icme10.dk/pages/081icmi-award. 

htm. The medallists for 2005 were Ubiratan D’Ambrosio, 

Brazil (Klein medal) and Paul Cobb, USA (Freudenthal 

medal). The citations are available at http://www.univ-paris-diderot.fr/2006/04-icmi.pdf. 

The medallists for 2007 are 

Jeremy Kilpatrick, USA (Klein medal) and Anna Sfard, 

Israel (Freudenthal medal). At ICME11 (Monterrey, Mexico, 

5-13 July 2008) the last four medals (2005-2007) will be 

presented. Short citations of the 2007 medals follow: 

Jeremy Kilpatrick, University of Georgia, 

Athens, GA, USA 

It is with great pleasure that the ICMI 

Awards Committee hereby announces 

that the Felix Klein Medal for 2007 is 

given to Professor Jeremy Kilpatrick, University 

of Georgia, Athens, GA, USA, in 

recognition of his more than forty years of sustained and 

distinguished lifetime achievement in mathematics education 

research and development. Jeremy Kilpatrick’s numerous 

contributions and services to mathematics education 

as a fi eld of theory and practice, as he prefers to call it, 

are centred around his extraordinary ability to refl ect on, 

critically analyse and unify essential aspects of our fi eld as 

it has developed since the early 20 th century, while always 

insisting on the need for reconciliation and balance among 

the points of view taken, the approaches undertaken and 

the methodologies adopted for research. It is a characteristic 

feature of Jeremy Kilpatrick that he has always embraced 

a very cosmopolitan perspective on mathematics 

education. Thus he has worked in Brazil, Colombia, El Salvador, 

Italy, New Zealand, Singapore, South Africa, Spain, 

Sweden and Thailand, in addition to being, of course, extraordinarily 

knowledgeable about the international literature. 

Throughout his academic career, Jeremy Kilpatrick 

has published groundbreaking papers, book chapters and 

books – many of which are now standard references in the 

literature – on problem solving, on the history of research 

in mathematics education, on teachers’ profi ciency, on curriculum 

change and its history, and on assessment. 

Anna Sfard, University of Haifa, Israel, 

and the Institute of Education, University 

of London, UK (also affi liated to Michigan 

State University). 

It is with great pleasure that the ICMI 

Awards Committee hereby announces 

that the Hans Freudenthal Medal for 2007 

is given to Professor Anna Sfard, University of Haifa, Israel, 

and the University of London, UK, in recognition of 

her highly signifi cant and scientifi cally deep accomplishments 

within a consistent, long-term research programme 

focused on objectifi cation and discourse in mathematics 

education, which has had a major impact on many strands 

of research in mathematics education and on numerous 

young researchers. In addition to publications related to 

the above-mentioned research programme, Anna Sfard 

has published numerous other papers and book chapters 

within a broad range of topics. It is a characteristic feature 

of Anna Sfard’s scientifi c achievements that they are 

always very thorough, original and intellectually sharp. 

She often uncovers the tacit if not hidden assumptions 

behind notions, approaches and conventional wisdom, 

and by turning things upside-down she usually succeeds 

in generating new fundamental and striking insights into 

complex issues and problems. 

Mogens Niss, Chair, The ICMI Awards Committee, mn@ 

ruc.dk. 

People who are interested in being informed about ICMI 

activities are invited to subscribe to the ICMI News (free). 

There are two ways of subscribing to ICMI News: 

1. Click on http://www.mathunion.org/ICMI/Mailinglist 

with a Web browser and go to the “Subscribe” button to 

subscribe to ICMI News online. 

2. Send an email to icmi-news-request@mathunion.org with 

the subject-line: Subject: subscribe. 

Previous issues can be seen at http://www.mathunion.org/ 

pipermail/icmi-news. 

In the issue 67 of this Newsletter (p. 18) the ICMI study 

n. 19 on Proof and Proving in Mathematics Education was 

announced. The address of the website of the study has 

been changed and is now: 

http://www.icmi19.com 

The deadline for submissions is still June 30, 2008. 

Mariolina Bartolini Bussi [bartolini@ 

unimo.it] is the Newsletter Editor within 

Mathematics Education. A short biography 

can be found in issue 55, page 4. 


ERCOM: Mathematisches 

Forschungsinstitut Oberwolfach 

The Mathematisches Forschungsinstitut Oberwolfach 

(MFO) was founded in 1944 and has developed over the 

years into an international research centre. In mathematical 

research, interchange of ideas plays a central role. The 

high degree of abstraction of mathematics and the compact 

way it is presented necessitate direct personal communication. 

Although most new results are nowadays 

quickly made available to the mathematical community 

via electronic media, this cannot replace personal contact 

among scientists. The importance of personal contact increases 

with the constant increase of specialization. 

Therefore, the institute concentrates on cooperative 

research activities of larger (workshop programme) or 

smaller (mini-workshop programme and Research in 

Pairs) groups. Leading representatives of particularly 

relevant research areas from all over the world are invited 

to Oberwolfach (about 30% coming from Germany 

and 40% from elsewhere in Europe) offering them the 

opportunity to pursue their research activities. The institute’s 

premises offer free accommodation (full board) for 

about 55–60 invited visitors. In all activities, participation 

of promising young scientists plays an important role. 

Henri Cartan at the original Oberwolfach building 

Scientifi c Programmes 

The MFO has two large, central tasks: the weekly workshop 

programme and the Research in Pairs programme 

for longer-term research stays. Additionally, there are further 

activities of the MFO that address young researchers. 

The scientifi c programme is operated annually over 

50 weeks and covers all areas in mathematics, including 

applications. 

1. The Workshop Programme. The main scientifi c programme 

consists of about 40 week-long workshops per 

year, each with about 50 participants. Alternatively, there 

ERCOM 

can be two parallel workshops of half that size (about 

25 participants). The workshops are organised by internationally 

leading experts of the respective fi elds. The 

participants are personally invited by the director after 

recommendations by the organizers. A special characteristic 

feature of the Oberwolfach Workshops is a research 

orientation. Very often the guest researchers appreciate 

the stimulating atmosphere. Many signifi cant research 

projects owe their origins to the realisation of a workshop 

in Oberwolfach. 

2. The Mini-Workshop Programme. This programme offers 

12 week-long mini-workshops per year, each with about 

15 participants. These mini-workshops are aimed especially 

at junior researchers who have a focused subject, and 

allow them to react on recent developments since the subjects 

are fi xed only half a year before the mini-workshop 

takes place. 

3. The Oberwolfach Arbeitsgemeinschaft. The idea of the 

Arbeitsgemeinschaft for young as well as for senior researchers 

is to learn about a new active topic by giving 

a lecture, guided by leading international specialists. The 

Arbeitsgemeinschaft meets twice a year for one week and 

is organized by Professor Christopher Deninger and Professor 

Gerd Faltings. 

4. The Oberwolfach Seminars. The Oberwolfach Seminars 

are week-long events taking place six times a year. They 

are organised by leading experts in the fi eld and address 

post-docs and PhD students from all over the world. The 

aim is to introduce 25 participants to a particularly hot 

development. 

5. The Research in Pairs Programme. The Research in Pairs 

(RiP) Programme aims at small groups of 2–4 researchers 

from different places working together at the Mathematisches 

Forschungsinstitut Oberwolfach for anything from 

2 weeks to 3 months on a specifi c project. 

6. Oberwolfach Leibniz Fellows. The focus of this new 

postdoctoral programme, which has been set up for the 

fi rst time from 2007 to 2010, is to support excellent young 

researchers in an important period of their scientifi c career 

by providing ideal working conditions in an international 

atmosphere. Outstanding young researchers can apply 

to carry out a research project, individually or in small 

groups, for a period from two to six months. They may 

propose for co-workers to visit Oberwolfach for shorter 

periods. Oberwolfach Leibniz Fellows should be involved 

in an active research group at a university or another research 

institute. It is also possible to make an application 

to the European Post-Doctoral Institute (EPDI). 


Just published / Coming soon 

www.degruyter.com 

Victor Zvyagin / Dmitry Vorotnikov 

■ Topological Approximation Methods for Evolutionary 

Problems of Nonlinear Hydrodynamics 

DE GRUYTER 

June 2008. XII, 230 pages. Hardcover. 

RRP € [D] 98.00 / * US$ 128.00. 

ISBN 978-3-11-020222-9 

de Gruyter Series in Nonlinear Analysis and Applications 12 

The authors present functional analytical methods for solving a class of partial differential equations. The results 

have important applications to the numerical treatment of rheology (specific examples are the behaviour 

of blood or print colours) and to other applications in fluid mechanics. 

A class of methods for solving problems in hydrodynamics is presented. 

Barbara Kaltenbacher / Andreas Neubauer / Otmar Scherzer 

■ Iterative Regularization Methods for Nonlinear Ill-Posed 

Problems 

May 2008. Approx. VIII, 196 pages. Hardcover. 

RRP € [D] 78.00 / * US$ 115.00. 

ISBN 978-3-11-020420-9 

Radon Series on Computational and Applied Mathematics 6 

Nonlinear inverse problems appear in many applications, and typically they lead to mathematical models that 

are ill-posed, i.e., they are unstable under data perturbations. Those problems require a regularization, i.e., a 

special numerical treatment. This book presents regularization schemes which are based on iteration methods, 

e.g., nonlinear Landweber iteration, level set methods, multilevel methods and Newton type methods. 

Ulrich Kulisch 

■ Computer Arithmetic and Validity 

Theory, Implementation, and Applications 

May 2008. Approx. XVIII, 410 pages. Hardcover. 

RRP € [D] 78.00 / * US$ 108.00. 

ISBN 978-3-11-020318-9 

de Gruyter Studies in Mathematics 33 

The present book deals with the theory of computer arithmetic, its implementation on digital computers and 

applications in applied mathematics to compute highly accurate and mathematically verified results. The aim 

is to improve the accuracy of numerical computing (by implementing advanced computer arithmetic) and to 

control the quality of the computed results (validity). The book can be useful as high-level undergraduate textbook 

but also as reference work for scientists researching computer arithmetic and applied mathematics. 

Hans-Otto Georgii 

■ Stochastics 

Introduction to Probability and Statistics 

Transl. by Marcel Ortgiese / Ellen Baake / Hans-Otto Georgii 

February 2008. IX, 370 pages. Paperback. 

RRP € [D] 39.95 / * US$ 49.00. 

ISBN 978-3-11-019145-5 

de Gruyter Textbook 

This book is a translation of the third edition of the well accepted German textbook ‘Stochastik’, which presents 

the fundamental ideas and results of both probability theory and statistics, and comprises the material of a 

one-year course. The stochastic concepts, models and methods are motivated by examples and problems and 

then developed and analysed systematically. 

*For orders placed in North America. 

Prices are subject to change. 

Prices do not include postage and handling.

Publications and Further Activities 

1. The Oberwolfach Reports (OWR) began in 2004 as a 

new series of publications of the institute in collaboration 

with the EMS Publishing House. The four issues comprise 

more than 3000 pages per year. The OWR are formed of 

the offi cial reports of every workshop, containing extended 

abstracts of the given talks, from one to three pages per 

talk, including references. The aim is to report periodically 

upon the state of mathematical research, and to make 

these reports available to the mathematical community. 

2. The book series Oberwolfach Seminars (OWS) is published 

in cooperation with Birkhäuser. In this series, the 

material of the Oberwolfach seminars is made available 

to an even larger audience. 

3. The Oberwolfach Preprints (OWP) mainly contain research 

results related to a longer stay in Oberwolfach. In 

particular, this concerns the Research in Pairs Programme 

and the Oberwolfach Leibniz Fellows. 

4. Once every three years the Oberwolfach Prize is awarded 

by the Gesellschaft für mathematische Forschung e.V. 

and by the Oberwolfach Stiftung to young mathematicians. 

The prize is awarded for excellent achievements in 

changing fi elds of mathematics. 

5. The John Todd Fellowship is awarded every three years 

by the Oberwolfach Foundation and the Mathematisches 

Forschungsinstitut Oberwolfach to young excellent mathematicians 

working in numerical analysis. 

6. On a two year rotation, a training week for school teachers 

(and librarians on the alternate years) of the State of 

Baden-Württemberg takes place. 

7. The institute also hosts the annual fi nal training week 

for especially gifted German pupils to prepare for the International 

Mathematical Olympiad. 

Buildings 

The buildings, which were mainly fi nanced by the 

Volkswagen-Stiftung, do not only offer accommodation 

facilities for visitors but also form an excellent and stimulating 

frame for research activities at the institute. Besides 

the well-adapted scientifi c infrastructure it is also 

the institute’s remote location, and the excellent service 

with board and lodging in the guest house close to the 

conference and library building, that guarantees effi cient 

and concentrated working conditions for the guests. 

The library building is located immediately downhill 

from the guest house. It has replaced the former building 

(the old castle) and was inaugurated in 1975. Comprising 

the library, two lecture halls, several small discussion 

rooms and numerous computer stations, it offers 

excellent working conditions for scientifi c research. The 

offi ces of the scientifi c administration are also part of 

ERCOM 

this building. The library, the discussion rooms and the 

cafeteria are open day and night and offer scientists the 

possibility to work at any time, an option that is eagerly 

taken up. 

The guest house was inaugurated in 1967. Each scientist is 

housed in a single room with its own bathroom. In addition, 

eight apartments and fi ve bungalows enable a longer 

stay at the MFO within the Research in Pairs and the 

Oberwolfach Leibniz Fellows programmes. Due to their 

age, the maintenance of the buildings is of greatest importance. 

The refurbishment of the guest house and the bungalows 

is going on until 2010. 

Infrastructure 

The basis for the successful realisation of the research 

programmes described above is an excellent infrastructure. 

Traditionally in mathematics, the library plays the 

main role. The Oberwolfach Library, which some years 

ago was ranked by the AMS as the best mathematical 

library outside the USA, contains about 45,000 issues of 

books, proceedings, etc. and subscribes to approximately 

500 print journals and 3,000 electronic journals. The library 

can be used by the guest researchers 24 hours a day. 

By an enlargement in 2007, which was fi nanced by the 

Klaus Tschira Foundation and the Volkswagen Foundation, 

the MFO is able to provide new capacity for about 

the next 20 years. 

During the last few years the computer pool has been 

improved signifi cantly and is now up-to-date. 

The MFO owns a large collection of more than 9,000 

photographs of mathematicians, which are also available 

online. Professor Konrad Jacobs, Erlangen, is one of 

those who have contributed to this photo collection. Special 

thanks go to Springer-Verlag Heidelberg for helping 

organize the digitalization of the photos. 

During the last twenty years, mathematical software 

has developed into an established tool for mathematical 

research and education. In some fi elds, its importance is 

comparable to that of mathematical literature. However, 

collections of information about mathematical software 

so far only exist in a rudimentary form. The intention of 

the Oberwolfach References on Mathematical Software 

(ORMS) project is to fi ll this gap. This includes a web-interfaced 

collection of detailed information and links and 

a classifi cation scheme for mathematical software eventually 

aiming to cover all thematic aspects of mathematical 

software. 

Institutional Structure and Statutes of the MFO 

Since 2005, the MFO has been registered as a non-profi t 

corporation (gemeinnützige GmbH). The MFO is headed 

by a director supported by a vice-director. The sole associate 

of the MFO is the Gesellschaft für Mathematische Forschung 

e.V. (GMF), represented by its board. The GMF is 

also a non-profi t society and was founded in Freiburg, 1959, 

in order to set up legal representation and scientifi c back- 


Book review 

ing for the institute. Financing of the MFO is shared by the 

Federal Republic of Germany and the Federal States with 

emphasis on the local state of Baden-Württemberg. Being 

a member of the Leibniz-Gemeinschaft is a prerequisite 

for the common fi nancing. The fi nancial partners are represented 

in the Administrative Council of the MFO, which 

in its function as most important supervisory panel decides 

on the medium-term and long-term fi nance and budget 

planning. The institute and the administrative council are 

supported by the Scientifi c Advisory Board, which is composed 

of 6-8 internationally renowned mathematicians. 

The statutes of the MFO and of the GMF contain 

the following aims, to be achieved with an international 

scope: 

- The promotion of research in mathematics. 

- The intensifi cation of scientifi c collaboration. 

- The intensifi cation of education and training in mathematics 

and related areas. 

- The promotion of young scientists. 

The director of the institute decides on the scientifi c programme 

in cooperation with the scientifi c committee of 

the GMF. For the scientifi c programme, this is the most 

important panel of the institute. It is based on the honorary 

work of about 20 mathematicians, covering all areas 

of mathematics. The scientifi c committee examines all 

scientifi c events at the institute prior to their approval. 

The programme is fi xed in a competitive procedure according 

to strictly scientifi c criteria. 

For the planning and realization of the scientifi c programme 

of the MFO approximately 20 staff positions are 

Book review 

Arne Jensen (Aalborg, Denmark) 

Att våga sitt tärningskast 

Gösta Mittag-Leffl er 1846–1927 

Arild Stubhaug 

Translated from Norwegian by 

Kjell-Ove Widman 

Atlantis 2007 

ISBN 978-91-7353-185-6 

This book is a biography of Gösta Mittag-Leffl er comprising 

more than 750 pages. It was written by Arild 

Stubhaug, who is also the author of biographies of 

Niels Henrik Abel and Sophus Lie and originally published 

in Norwegian under the title “Med viten og vilje” 

Participants of the 2004 mini workshop Geometry and Duality in 

String Theory 

provided in various divisions, such as scientifi c and administration 

management, library, IT-service, guest service 

and housekeeping. 

The Förderverein (Friends of Oberwolfach) of the 

MFO has more than 700 members and provides additional 

fi nancial support for the MFO by its membership 

fees. The Oberwolfach Foundation, a foundation of public 

utility within the Förderverein, provides long-term 

fi nancial support by building up an endowment. Within 

the Oberwolfach Stiftung the Horst Tietz Fund plays an 

important role by providing special funds. 

Please consult the web side of the MFO (www.mfo. 

de) for further, more detailed information concerning 

the scientifi c programme and the various committees 

and boards. A short guide for applications is online at 

http://www.mfo.de/programme/ProposalGuide2008.pdf . 

(Aschehoug 2007); an English edition is in preparation 

with Springer-Verlag and due in 2009. Arild Stubhaug 

has a background both in mathematics and literature, 

making him ideally suited to embark on the monumental 

task of writing a biography of such a complex and at 

times controversial person as Gösta Mittag-Leffl er. 

The book starts in the middle of events, with a description 

of a visit to Egypt by Gösta Mittag-Leffl er 

and his wife Signe. They were also accompanied by a 

personal physician. Mittag-Leffl er was 53 years old and 

his wife was 38. The trip lasted from the end of 1899 

until Easter 1900. The main reason for the trip was 

Mittag-Leffl er’s health problems (in particular stomach 

problems). This is a recurring theme throughout 

the biography. The dry desert air seems to have had a 

benefi cial infl uence. 

As can be inferred from this short summary of the 

introductory chapter, the biography brings us very close 

to Mittag-Leffl er and his daily life, routines and cares. 

The biography is based on the wealth of material that 

Mittag-Leffl er left behind. Some documents are kept in 

the National Library of Sweden (Kungliga biblioteket), 

Stockholm, while other documents are at the Mittag- 

Leffl er Institute in the suburb of Djursholm. The docu- 


ments include diaries, letters, newspaper clippings, etc. 

Arild Stubhaug has carefully researched this wealth of 

material and distilled out of it a fascinating account of a 

complex personality. 

Mittag-Leffl er was born in 1846, the son of Johan Olof 

Leffl er and Gustava Wilhelmina, née Mittag. He had one 

sister Anne Charlotte (born 1849) and two brothers Fritz 

(born 1847) and Arthur (born 1854). His father developed 

a mental disorder and was committed to care from 

1870 until his death. This illness might be one of the reasons 

why Gösta Mittag-Leffl er had no children. He was 

very close to his mother and wrote her many letters. From 

these letters (and the replies) Arild Stubhaug has gained 

access to some of his personal thoughts during formative 

periods of his life. This has given a view of him that would 

otherwise not be available. 

Mittag-Leffl er received his doctorate from Uppsala 

University in 1872. His thesis was on some results in 

complex analysis. One characteristic of this biography is 

that it does not provide any details on his mathematical 

achievements. As a mathematician one may regret this. 

However, it is not for his mathematical achievements 

that he deserves a well researched biography. 

Probably one of the most important events in his life 

was the award of a travel grant (Bysantinska resestipendiet) 

in 1873, allowing him to visit mathematicians in Paris 

and Berlin. In Paris he visited Hermite and also became 

acquainted with other French mathematicians. These 

contacts were to play an important role in his career, for 

example leading to a close acquaintance with Hermite’s 

student Poincaré. After a semester in Paris he moved to 

Berlin and met Weierstrass. This connection was to determine 

the direction of many of his activities in the future. 

In particular, he heard about Sonja Kovalevsky for the 

fi rst time. 

Mittag-Leffl er was called to a professorship at the 

newly established Stockholms Högskola (later Stockholm 

University) in 1881 and from that time his base was 

Stockholm. However he kept up contacts with a large 

number of European mathematicians during the rest of 

his career. He often travelled and reports on his travels 

take up a large part of the biography. See the accompanying 

translation of a selected chapter in this newsletter 

issue. 

Many of the high points of his career after returning 

to Stockholm are probably known to many mathematicians. 

He managed to get Sonja Kovalevsky appointed 

to a professorship at Stockholms Högskola, he founded 

the journal Acta Mathematica, and he educated and 

promoted a number of brilliant Swedish mathematicians, 

including I. Fredholm. All this, and much more, is 

meticulously related. Many details are added that most 

of us probably never knew. For example that the economy 

of Acta Mathematica was precarious for many years 

and that Mittag-Leffl er paid part of the expenses from 

his own funds. 

1 A report on the present Mittag-Leffl er Institute appeared in 

the Newsletter issue 56 (June 2005), pp. 29–30. 

Book review 

Apart from mathematics Mittag-Leffl er also had a 

career as a business man. He invested in many businesses 

and made and lost money. This is a fascinating 

part of his life that is exposed in detail for the fi rst time. 

Again the wealth of written material makes this possible. 

At some points in his career he was very wealthy. Towards 

the end of his life he had very little left. In 1916 

on his 70th birthday he and his wife Signe announced 

in a testament the formation of Makarna Mittag-Leffl ers 

Matematiska Stiftelse under the Royal Academy. The 

purpose was the formation of a mathematical research 

institute. 1 The donation included his library and his villa 

in Djursholm. Extracts of the testament were published 

in Acta Mathematica. 

Downturns in his fi nancial situation and other events 

meant that the vision was not realized for many years. It 

is fortunate that the donation in 1916 of the library and 

the villa put those out of reach of his creditors. It was 

only in the late 1960s that Lennart Carleson managed to 

secure funding for turning the villa in Djursholm into a 

world class mathematical research institute. 

The impression one gets after reading this comprehensive 

account is of a very complex personality. So many 

facets are revealed that it is clear he must have been at 

times both very charming and able to get close to people 

of very different personalities, for example Hermite, 

Painlevé and Weierstrass, and at other times not a person 

one would like to go up against. 

I will not say anything about Gösta Mittag-Leffl er 

and Alfred Nobel. To fi nd out about this you will have to 

read the book. The account given changed the picture I 

had of the two and their relations. 

The reader is also rewarded with a lively picture of 

the life and social activities of the upper segment of 

Swedish society during that period. It is important to see 

the mathematics and the mathematicians in their contemporary 

context. 

I enjoyed reading this book very much and hope you 

will do the same. 

Arne Jensen [matarne@math.aau.dk] got his PhD from 

the University of Aarhus in 1979. He has been a professor 

of mathematics at Aalborg University since 1988. 

He served as acting director of the Mittag-Leffl er Institute 

from 1993 to the beginning of 1995. In 2000–01 he 

was a visiting professor at the University of Tokyo. His 

research interests are spectral and scattering theory for 

Schrödinger operators. 


Conferences 

Forthcoming conferences 

compiled by Mădălina Păcurar (Cluj-Napoca, Romania) 

Please e-mail announcements of European conferences, workshops 

and mathematical meetings of interest to EMS members, to 

one of the addresses madalina.pacurar@econ.ubbcluj.ro or madalina_ 

pacurar@yahoo.com. Announcements should be written 

in a style similar to those here, and sent as Microsoft Word fi les 

or as text fi les (but not as TeX input fi les). 

June 2008 

1–7: Applications of Ultrafi lters and Ultraproducts in 

Mathematics (ULTRAMATH 2008), Pisa, Italy 

Information: ultramath@dm.unipi.it; 

http://www.dm.unipi.it/~ultramath 

2–6: International Conference on Random Matrices 

(ICRAM), Sousse, Tunisia 

Information: abdelhamid.hassairi@fss.rnu.tn; 

http://www.tunss.net/accueil.php?id=ICRAM 

2–6: Thompson’s groups: new developments and interfaces, 

CIRM Luminy, Marseille, France 

Information: colloque@cirm.univ-mrs.fr; 

http://www.cirm.univ-mrs.fr 

3–6: Chaotic Modeling and Simulation International Conference 

(CHAOS2008), Chania, Crete, Greece 

Information: skiadas@asmda.net; http://www.asmda.net/chaos2008 

5–6: 12 th Galway Topology Colloquium, Galway, Ireland 

Information: aisling.mccluskey@nuigalway.ie; 

http://www.maths.nuigalway.ie/conferences/topology08.html 

6–11: Tenth International Conference on Geometry, Integrability 

and Quantization, Sts. Constantine and Elena resort, 

Varna, Bulgaria 

Information: mladenov@bio21.bas.bg; 

http://www.bio21.bas.bg/conference/ 

8–14: Mathematical Inequalities and Applications 2008, 

Trogir – Split, Croatia 

Information: mia2008@math.hr; http://mia2008.ele-math.com/ 

8–14: 34 th International Conference “Applications of Mathematics 

in Engineering and Economics”, Resort of Sozopol, 

Bulgaria 

Information: mtod@tu-sofi a.bg; 

http://www.tu-sofi a.bg/fpmi/amee/index.html 

9–11: International Conference “Modelling and Computation 

on Complex Networks and Related Topics” (Net- 

Works 2008), Pamplona, Spain 

Information: networks2008@unav.es; 

http://www.fi sica.unav.es/networks2008/default.html 

9–13: Geometric Applications of Microlocal Analysis, 




9–13: Conference on Algebraic and Geometric Topology, 

Gdańsk, Poland 

Information: cagt@math.univ.gda.pl; 

http://math.univ.gda.pl/cagt/ 

9–13: Educational Week on Noncommutative Integration, 

Leiden, Netherlands 

Information: mdejeu@math.leidenuniv.nl; 

http://www.math.leidenuniv.nl/~mdejeu/NoncomIntWeek_ 

2008 

9–14: II International Summer School on Geometry, Mechanics 

and Control, La Palma (Canary Islands), Spain 

Information: gmcnet@ull.es; 

http://webpages.ull.es/users/gmcnet/Summer-School08/home1. 

htm 

9–14: Colloquium of Non Commutative Algebra, Sherbrooke, 

Québec, Canada 

Information: Ibrahim.Assem@USherbrooke.ca; 

http://www.crm.umontreal.ca/~paradis/Sherbrooke/index_e. 

shtml 

9–19: Advances in Set-Theoretic Topology: Conference 

in Honour of Tsugunori Nogura on his 60 th Birthday, Erice, 

Sicily, Italy 

Information: erice@dmitri.math.sci.ehime-u.ac.jp; 

http://www.math.sci.ehime-u.ac.jp/erice/ 

9–20: GTEM/TUBITAK Summer School: Geometry and 

Arithmetic of Moduli Spaces of Coverings, Istanbul, Turkey 

Information: gamsc.school@gmail.com; 

http://math.gsu.edu.tr/GAMSC/index.htm 

15–22: ESF-MSHE-PAN Conference on Operator Theory, 

Analysis and Mathematical Physics, Będlewo, Poland 

Information: ablondeel@esf.org; 

http://www.esf.org/conferences/08279 

15–24: CIMPA-School Nonlinear analysis and Geometric 

PDE, Tsaghkadzor, Armenia 

Information: imprs@mis.mpg.de; 

http://www.imprs-mis.mpg.de/schools.html 

16–17: Fifth European PKI Workshop, Trondheim, Norway 

Information: sjouke.mauw@uni.lu; 

http://www.item.ntnu.no/europki08/ 

16–19: 2nd International Conference on Mathematics & 

Statistics, Athens, Greece 

Information: atiner@atiner.gr; 

http://www.atiner.gr/docs/Mathematics.htm 

16–20: Workshop on population dynamics and mathematical 

biology, CIRM Luminy, Marseille, France 



16–20: Homotopical Group Theory and Homological Algebraic 

Geometry Workshop, Copenhagen, Denmark 

Information: http://www.math.ku.dk/~jg/homotopical2008/ 


16–20: Fourth Conference on Numerical Analysis and Applications, 

Lozenetz, Bulgaria 

Information: http://www.ru.acad.bg/naa08/ 

16–20: Conference on vector bundles in honour of S. Ramanan 

(on the occasion of his 70th birthday), Mirafl ores de 

la Sierra (Madrid), Spain 

Information: oscar.garcia-prada@uam.es; 

http://www.mat.csic.es/webpages/moduli2008/ramanan/ 

16–20: 15-th Conference of the International Linear Algebra 

Society (ILAS 2008), Cancún, Mexico 

Information: ilas08@star.izt.uam.mx; http://star.izt.uam.mx/ 

ILAS08/ 

16–20: The 11 th Rhine Workshop on Computer Algebra, 

Levico Terme, Trento, Italy 

Information: michelet@science.unitn.it; 

http://science.unitn.it/~degraaf/rwca.html 

16–21: International Scientifi c Conference „Differential 

Equations, Theory of Functions and their Applications“ 

dedicated to 70 th birthday of academician of NAS of 

Ukraine A.M.Samoilenko, Melitopol, Ukraine 

Information: conf2008@imath.kiev.ua; 

http://www.imath.kiev.ua/conf2008/ 

17–20: Structural Dynamical Systems: Computational Aspects, 

Capitolo-Monopoli, Bari, Italy 

Information: sds08@dm.uniba.it; 

http://www.dm.uniba.it/~delbuono/sds2008.htm 

17–22: International conference “Differential Equations 

and Topology” dedicated to the Centennial Anniversary 

of Lev Semenovich Pontryagin, Moscow, Russia 

Information: pont2008@cs.msu.ru; http://pont2008.cs.msu.ru 

22–27: CR Geometry and PDE’s – III, Levico Terme, Trento, 

Italy 


http://www.science.unitn.it/cirm/AnnCR2008.html 

22–28: Combinatorics 2008, Costermano (VR), Italy 

Information: combinatorics@ing.unibs.it; 

http://combinatorics.ing.unibs.it 

23–27: Homotopical Group Theory and Topological Algebraic 

Geometry, Max Planck Institute for Mathematics Bonn, 

Germany 

Information: admin@mpim-bonn.mpg.de; 

http://www.ruhr-uni-bochum.de/topologie/conf08/ 

23–27: Hermitian symmetric spaces, Jordan algebras and 

related problems, CIRM Luminy, Marseille, France 



23–27: Conference on Differential and Difference Equations 

and Applications 2008 (CDDEA 2008), Strečno, Slovak 

Republic 

Information: cddea@fpv.uniza.sk; http://www.fpv.uniza.sk/cddea/ 

23–27: Workshop on Geometric Analysis, Elasticity and 

PDEs, on the 60 th Birthday of John Ball, Heriot Watt University, 

Edinburgh, UK 

Information: morag.burton@icms.org.uk; 

http://www.icms.org.uk/workshops/pde 

Conferences 

23–27: First Iberoamerican Meeting on Geometry, Mechanics, 

and Control, Santiago de Compostela, Spain 

Information: gmcnet@ull.es; 

http://www.gmcnetwork.org/eia08 

23–27: Phenomena in High Dimensions, 4 th Annual Conference, 

Seville, Spain 

Information: phd@us.es; 

http://www.congreso.us.es/phd/ 

24–26: Current Geometry, the IX Edition of the International 

Conference on problems and trends of contemporary 

geometry, Naples, Italy 

Information: http://school.diffi ety.org/page82/page82.html 

25–28: VII Iberoamerican Conference on Topology and its 

Applications, Valencia, Spain 

Information: cita@mat.upv.es; http://cita.webs.upv.es 

30–July 3: Analysis, PDEs and Applications, Roma, Italy 

Information: mazya08@mat.uniroma1.it; 

http://www.mat.uniroma1.it/~mazya08/ 

28–July 1: 3rd Small Workshop on Operator Theory, Krakow, 

Poland 

Information: swot08@ar.krakow.pl; 

http://www.zzm.ar.krakow.pl/swot08/ 

30–July 3: Analysis, PDEs and Applications. On the occasion 

of the 70 th birthday of V. Maz’ya, Roma, Italy 

Information: mazya08@mat.uniroma1.it; 

http://www.mat.uniroma1.it/~mazya08/ 

30–July 4: Joint ICMI/IASE Study; Teaching Statistics in 

School Mathematics. Challenges for Teaching and Teacher 

Education, Monterrey, Mexico 

Information: batanero@ugr.es; 

http://www.ugr.es/~icmi/iase_study/ 

30–July 4: Geometry of complex manifolds, CIRM Luminy, 

Marseille, France 



30–July 4: The European Consortium for Mathematics in 

Industry (ECMI2008), University College, London, UK 

Information: lucy.nye@ima.org.uk; http://www.ecmi2008.org/ 

30–July 5: 9 th Conference on Geometry and Topology of 

Manifolds, Krakow, Poland 

Information: robert.wolak@im.uj.edu.pl; 

http://www.im.uj.edu.pl/gtm2008/ 

July 2008 

1–4: VIth Geometry Symposium, Bursa, Turkey 

Information: arslan@uludag.edu.tr; 

http://www20.uludag.edu.tr/~geomsymp/index.htm 

2–4: The 2008 International Conference of Applied and 

Engineering Mathematics, London, UK 

Information: wce@iaeng.org; 

http://www.iaeng.org/WCE2008/ICAEM2008.html 

2–11: Soria Summer School on Computational Mathematics: 

“Algebraic Coding Theory” (S3CM), Universidad de 

Valladolid, Soria, Spain 

Information: edgar@maf.uva.es; http://www.ma.uva.es/~s3cm/ 


Conferences 

3–8: 22nd International Conference on Operator Theory, 

West University of Timisoara, Timisoara, Romania 

Information: ot@theta.ro; http://www.imar.ro/~ot/ 

7–10: The Tenth International Conference on Integral 

Methods in Science and Engineering (IMSE 2008), University 

of Cantabria, Santander, Spain 

Information: imse08@unican.es, meperez@unican.es; 

http://www.imse08.unican.es/ 

7–10: International Workshop on Applied Probability 

(IWAP 2008), Compiègne, France 

Information: nikolaos.limnios@utc.fr,joseph.glaz@uconn.edu; 

http://www.lmac.utc.fr/IWAP2008/ 

7–11: VIII International Colloquium on Differential Geometry 

(E. Vidal Abascal Centennial Congress), Santiago de 

Compostela, Spain 

Information: icdg2008@usc.es; http://xtsunxet.usc.es/icdg2008 

7–11: Spectral and Scattering Theory for Quantum Magnetic 

Systems, CIRM Luminy, Marseille, France 



7–11: Algebraic Topological Methods in Computer Science 

(ATMCS) III, Paris, France 

Information: atmcs08@lix.polytechnique.fr; 

http://www.lix.polytechnique.fr/~sanjeevi/atmcs 

7–11: Set Theory, Topology and Banach Spaces, Kielce, 

Poland 

Information: topoconf@pu.kielce.pl; 

http://www.pu.kielce.pl/~topoconf/ 

7–12: New Horizons in Toric Topology, Manchester, UK 

Information: Helen.Kirkbright@manchester.ac.uk; 

http://www.mims.manchester.ac.uk/events/workshops/NHTT08/ 

7–12: International Conference on Modules and Representation 

Theory, Cluj-Napoca, Romania 

Information: aga_team@math.ubbcluj.ro, aga.team.cluj@gmail. 

com; 

http://math.ubbcluj.ro/~aga_team/AlgebraConferenceCluj 

2008.html 

13: Joint EWM/EMS Workshop, Amsterdam (The Netherlands) 

Information: http://womenandmath.wordpress.com/joint-ewm 

ems-worskhop-amsterdam-july-13th-2007/ 

14–17: International Conference on Differential and Difference 

Equations, Veszprem, Hungary 

Information: ddea2008@szt.uni-pannon.hu; 

http://www.szt.uni-pannon.hu/~ddea2008 

14–18: Fifth European Congress of Mathematics (5ECM), 

Amsterdam, Netherlands 

Information: http://www.5ecm.nl 

14–18: Effi cient Monte Carlo: From Variance Reduction 

to Combinatorial Optimization. A Conference on the Occasion 

of R.Y. Rubinstein’s 70 th Birthday, Sandbjerg Estate, 

Sønderborg, Denmark 

Information: oddbjorg@imf.au.dk; 

http://www.thiele.au.dk/Rubinstein/ 

15–18: Mathematics of program construction, CIRM Luminy, 




15–19: The 5 th World Congress of the Bachelier Finance 

Society, London, UK 

Information: mark@chartfi eld.org; http://www.bfs2008.com 

17–19: 7 th International Conference on Retrial Queues (7 th 

WRQ), Athens, Greece 

Information: aeconom@math.uoa.gr; 

http://users.uoa.gr/~aeconom/7thWRQ_Initial.html 

17–August 1: XI Edition of the Italian Diffi ety School, Santo 

Stefano del Sole (Avellino), Italy 

Information: http://school.diffi ety.org/page3/page0/page64/ 

page64.html 

18–22 : European Open Forum ESOF 2008: Science for a 

better life, Barcelona, Spain 

Information: http://www.esof2008.org 

20–23: International Symposium on Symbolic and Algebraic 

Computation, Hagenberg, Austria 

Information: franz.winkler@risc.uni-linz.ac.at; 

http://www.risc.uni-linz.ac.at/about/conferences/issac2008/ 

21–24: SIAM Conference on Nonlinear Waves and Coherent 

Structures, Rome, Italy 

Information: meetings@siam.org; 

http://www.siam.org/meetings/nw08/ 

21–25: Operator Structures and Dynamical Systems, Leiden, 

the Netherlands 

Information: mdejeu@math.leidenuniv.nl; 

http://www.lorentzcenter.nl/lc/web/2008/288/info.php3?wsid 

=288 

21–25: Summer School “PDE from Geometry”, Cologne, 

Germany 

Information: gk-admin@math.uni-koeln.de; 

http://www.mi.uni-koeln.de/~gk/school08 

21–August 29: CEMRACS, CIRM Luminy, Marseille, France 



24–26: Workshop on Current Trends and Challenges in 

Model Selection and Related Areas, University of Vienna, 

Vienna, Austria 

Information: hannes.leeb@yale.edu; 

http://www.univie.ac.at/workshop_modelselection/ 

August 2008 

3–9: Junior Mathematical Congress, Jena, Germany 

Information: info@jmc2008.org; http://www.jmc2008.org/ 

13–19: XXVII International Colloquium on Group Theoretical 

Methods in Physics (Group27), Yerevan, Armenia 

Information: pogosyan@ysu.am; http://theor.jinr.ru/~group27 

14–19: Complex Analysis and Related Topics (The XI th Romanian-Finnish 

Seminar), Alba Iulia, Romania 

Information: rofi nsem@gmail.com; 

http://www.imar.ro/~purice/conferences/Ro-Fin-11/rofin11 

sem.html 


16–31: EMS-SMI Summer School: Mathematical and numerical 

methods for the cardiovascular system, Cortona, 

Italy 

Information: dipartimento@matapp.unimib.it 

18–22: International conference on ring and module theory, 

Ankara, Turkey 

Information: http://www.algebra2008.hacettepe.edu.tr 

19–22: Duality and Involutions in Representation Theory, 

National University of Ireland, Maynooth, Ireland 

Information: involutions@maths.nuim.ie; 

http://www.maths.nuim.ie/conference/ 

21–23: International Congress of 20 th Jangjeon Mathematical 

Society, Bursa, Turkey 

Information: cangul@uludag.edu.tr, hozden@uludag.edu.tr, 

inam@uludag.edu.tr; http://www20.uludag.edu.tr/~icjms20/ 

25–28: The 14 th general meeting of European Women in 

Mathematics (EWM), Novi Sad, Serbia 

Information: ewm2009@im.ns.ac.yu; 

http://ewm2009.wordpress.com/ 

25–29: Function spaces, Differential Operators and Nonlinear 

Analysis, Helsinki, Finland 

Information: ljp@rni.helsinki.fi ; http://mathstat.helsinki.fi /fsdona 

27–29: Journées MAS de la SMAI: Modélisation et Statistiques 

des Réseaux, Rennes, France 

Information: jian-feng.yao@univ-rennes1.fr; 

http://mas2008.univ-rennes1.fr/ 

28–September 2: 9ème Colloque Franco-Roumain de 

Mathématiques Appliquées, Brașov, Romania 

Information: colloque2008@unitbv.ro; 

http://cs.unitbv.ro/colloque2008/site/ 

29–September 2: The International Conference of Differential 

Geometry and Dynamical Systems (DGDS-2008) 

and The V-th International Colloquium of Mathematics in 

Engineering and Numerical Physics (MENP-5, mathematics 

sections), Mangalia, Romania 

Information: dept@mathem.pub.ro; vbalan@mathem.pub.ro 

http://www.mathem.pub.ro 

31–September 5: The 22nd Summer Conference on Real 

Functions Theory, Stara Lesna, Slovakia 

Information: http://www.saske.sk/MI/confer/lsrf08.html 

31–September 6: Summer School on General Algebra and 

Ordered Sets, Trest, Czech Republic 

Information: http://www.karlin.mff.cuni.cz/~ssaos 

September 2008 

1–5: Representation of surface groups, CIRM Luminy, 




1–5: Conference on Numerical Analysis (NumAn 2008), 

Kalamata, Greece 

Information: numan2008@math.upatras.gr; 

http://www.math.upatras.gr/numan2008/ 

1–6: School (and Workshop) on the Geometry of Algebraic 

Stacks, Trento, Italy 


http://www.science.unitn.it/cirm/ 

Conferences 

1–12: School on Algebraic Topics of Automata, Lisbon, 

Portugal 

Information: patricia@cii.fc.ul.pt; 

http://caul.cii.fc.ul.pt/SATA2008/ 

2–5: X Spanish Meeting on Cryptology and Information 

Security, Salamanca, Spain 

Information: delrey@usal.es; 

http://www.usal.es/xrecsi/english/main.htm 

2–7: International Conference on Geometry, Dynamics, 

Integrable Systems, Belgrade, Serbia 

Information: gdis08@mi.sanu.ac.yu; 

http://www.mi.sanu.ac.yu/~gdis08/ 

6–14: First European Summer School in Mathematical Finance, 

Dourdan near Paris, France 

Information: euroschoolmathfi @cmap.polytechnique.fr; 

http://www.ceremade.dauphine.fr/~bouchard/ESCMF/ 

8–12: Chinese-French meeting in probability and analysis, 




8–10: Calculus of Variations and its Applications From 

Engineering to Economy, Universidade Nova de Lisboa, Caparica, 

Portugal 

Information: cva2008@fct.unl.pt; 

http://ferrari.dmat.fct.unl.pt/cva2008/ 

8–12: International Workshop on Orthogonal Polynomials 

and Approximation Theory 2008. Conference in honor 

of professor Guillermo López Lagomasino’s 60 th birthday, 

Universidad Carlos III de Madrid, Leganés, Spain 

Information: iwopa08@gmail.com; 

http://www.uc3m.es/iwopa08 

8–19: EMS Summer School: Mathematical models in the 

manufacturing of glass, polymers and textiles, Montecatini, 

Italy 

Information: cime@math.unifi .it; 

http://web.math.unifi .it/users/cime// 

9–12: INDAM Workshop on Holomorphic Iteration, Semigroups 

and Loewner Chains, Rome, Italy 

Information: iterates@mat.uniroma2.it; 

http://www.congreso.us.es/holomorphic/ 

10–12: Nonlinear Differential Equations (A Tribute to the 

work of Patrick Habets and Jean Mawhin on the occasion 

of their 65 th birthdays), Brussels, Belgium 

Information: node2008@uclouvain.be; 

http://www.uclouvain.be/node2008.html 

12–15: Mathematical Analysis, Differential Equations and 

Their Applications, Famagusta, North Cyprus 

Information: fabdul@mersin.edu.tr; 

http://madd2008.emu.edu.tr/contact.php 

14–18: 7 th Euromech Fluid Mechanics Conference, Manchester, 

UK 

Information: http://www.mims.manchester.ac.uk/events/workshops/EFMC7/ 


Conferences 

15–19: Geometry and Integrability in Mathematical Physics, 




15–19: International Conference on K-Theory and Homotopy 

Theory, Santiago de Compostela, Spain 

Information: regaca@usc.es; 

http://www.usc.es/regaca/ktht/ 

15–19: 9 th SIMAI Congress, Rome, Italy 

Information: simai2008@iac.rm.cnr.it; http://www.simai.eu/ 

16–20: International Conference of Numerical Analysis 

and Applied Mathematics 2008 (ICNAAM 2008), Psalidi, 

Kos, Greece 

Information: tsimos@mail.ariadne-t.gr; http://www.icnaam.org/ 

18–21: 6 th International Conference on Applied Mathematics 

(ICAM6), Baia Mare, Romania 

Information: vberinde@ubm.ro; http://www.icam.ubm.ro 

18–29: Crimean Autumn Mathematical School-Symposium, 

Batiliman, Ukraine 

Information: pavelstarkov@list.ru 

19–26: International Conference on Harmonic Analysis 

and Approximations IV, Tsaghkadzor, Armenia 

Information: mathconf@ysu.am; 

http://math.sci.am/conference/sept2008/conf.html 

21–24: The 8 th International FLINS Conference on Computational 

Intelligence in Decision and Control (FLINS 2008), 

Madrid, Spain 

Information: fl ins2008@mat.ucm.es; 

http://www.mat.ucm.es/congresos/fl ins2008 

22–25: Symposium on Trends in Applications of Mathematics 

to Mechanics (STAMM 2008), Levico, Italy 

Information: stamm08@gmail.com; 

http://mate.unipv.it/pier/stamm08.html 

22–26: 10 th International workshop in set theory, CIRM Luminy, 




22–28: 4 th International Kyiv Conference on Analytic 

Number Theory and Spatial Tessellations, Jointly with 5 th 

Annual International Conference on Voronoi Diagrams in 

Science and Engineering (dedicated to the centenary of 

Georgiy Voronoi), Kyiv, Ukraine 

Information: voronoi@imath.kiev.ua; 

http://www.imath.kiev.ua/~voronoi 

29–October 3: Commutative algebra and its interactions 

with algebraic geometry, CIRM Luminy, Marseille, France 



29–October 8: EMS Summer School: Risk theory and related 

topics, Będlewo, Poland 

Information: stettner@iman.gov.pl; 

www.impan.gov.pl/EMSsummerSchool/ 

October 2008 

3–5: II Iberian Mathematical Meeting, Badajoz, Spain 

Information: imm2@unex.es; http://imm2.unex.es 

5–12: International Conference “Differential Equations. 

Function Spaces. Approximation Theory” dedicated to 

the 100 th anniversary of the birthday of S.L. Sobolev, Novosibirsk, 

Russia 

Information: sobolev100@math.nsc.ru; 

http://www.math.nsc.ru/conference/sobolev100/english 

6–10: Partial differential equations and differential Galois 

theory, CIRM Luminy, Marseille, France 



9–11: Algebra, Geometry and Mathematical Physics, Tartu, 

Estonia 

Information: agmf@astralgo.eu; http://www.agmf.astralgo.eu/ 

tartu08/ 

9–12: The XVI- th Conference on Applied and Industrial 

Mathematics (CAIM 2008), Oradea, Romania 

Information: serban_e_vlad@yahoo.com; http://www.romai.ro 

13–17: Hecke algebras, groups and geometry, CIRM Luminy, 




20–24: Symbolic computation days, CIRM Luminy, Marseille, 

France 



22–24: International Conference on Modeling, Simulation 

and Control 2008, San Francisco, USA 

Information: wcecs@iaeng.org; 

http://www.iaeng.org/WCECS2008/ICMSC2008.html 

26–28: 10 th WSEAS Int. Conf. on Mathematical Methods 

and Computational Techniques in Electrical Engineering 

(MMACTEE 8), Corfu, Greece 

Information: info@wseas.org; 

http://www.wseas.org/conferences/2008/corfu/mmactee/ 

26–28: 7 th WSEAS Int. Conf. on Non-Linear Analysis, Non- 

Linear Systems and Chaos (NOLASC 8), Corfu, Greece 

Information: info@wseas.org; 

http://www.wseas.org/conferences/2008/corfu/nolasc/ 

26–31: New trends for modeling laser-matter interaction, 




27–31: New trends for modeling laser-matter interaction, 




November 2008 

3–7: Harmonic analysis, operator algebras and representations, 




5–7: Fractional Differentiation and its Applications, Ankara, 

Turkey 

Information: dumitru@cankaya.edu.tr; 

http://www.cankaya.edu.tr/fda08/ 


5–7: Modern Problems of Differential Geometry and General 

Algebra, Saratov City, Russia 

Information: vagner2008@bk.ru; 

http://mexmat.sgu.ru/vagner2008.php?lang=en 

10–14: The 6 th Euro-Maghreb workshop on semigroup 

theory, evolution equations and applications, CIRM Luminy, 




17–21: Geometry and topology in low dimension, CIRM 

Luminy, Marseille, France 



21–22: International Conference on Nolinear Analysis and 

Applied Mathematics, Targoviste, Romania 

Information: dteodorescu2003@yahoo.com; 

http://icnaam.valahia.ro/ 

24–28: Approximation, geometric modelling and applications, 




December 2008 

1–5: Homology of algebra: structures and applications, 


information: colloque@cirm.univ-mrs.fr; http://www.cirm.univmrs.fr 

8–12: Latent variables and mixture models, CIRM Luminy, 



15–19: Meeting on mathematical statistics, CIRM Luminy, 



16–18: Eighth IMA International Conference on Mathematics 

in Signal Processing, Cirencester, UK 

Information: pam.bye@ima.org.uk; 

http://www.ima.org.uk/Conferences/signal_processing/signal_ 

processing08.html 

February 2009 

5-8 : European Student Conference EUROMATH 2009, Cyprus 

Information: www.euromath.org 

March 2009 

15-20: ALGORITMY 2009 – Conference on Scientifi c Computing, 

High Tatra Mountains, Podbanske, Slovakia 

Information: algoritm@math.sk; http://www.math.sk/alg2009 

July 2009 

6–10: 26 th Journées arithmétiques, Saint-Etienne, France 

Information: ja2009@univ-st-etienne.fr; 

http://ja2009.univ-st-etienne.fr 

Recent Books 

Recent books 

edited by Ivan Netuka and Vladimír Souček (Prague) 

Books submitted for review should be sent to: Ivan Netuka, 

MÚUK, Sokolovská, 83, 186 75 Praha 8, Czech Republic. 

C. D. Aliprantis, R. Tourky: Cones and Duality, Graduate 

Studies in Mathematics, vol. 84, American Mathematical Society, 

Providence, 2007, 279 pp., USD 55, ISBN 978-0-8218-4146-4 

Ordered vector spaces and cones were introduced in mathematics 

at the beginning of the 20th century and were developed 

in parallel with functional analysis and operator theory. Since 

cones are employed to solve optimization problems, the theory 

of ordered vector spaces is an indispensable tool for solving a 

variety of applied problems appearing in areas such as engineering, 

econometrics and the social sciences. 

The aim of this book is to present the theory of ordered 

vector spaces from a contemporary perspective, which has 

been infl uenced by a study of ordered vector spaces in economics 

as well as other recent applications. The material is spread 

out in eight chapters. The fi rst chapters start with fundamental 

properties of wedges and cones and illustrate a variety of 

remarkable results from the connection between the topology 

and the order. The role of the Riesz decomposition property 

and normal cones is pointed out. The next section studies in 

detail cones in fi nite dimensional spaces and polyhedral cones. 

The authors proceed with a presentation of the fi xed points and 

eigenvalues of Krein operators. Further chapters contain material 

on K-lattices, Riesz-Kantorovich functionals and piecewise 

affi ne functions, which is a topic that has not been included in 

any monograph before. The last chapter serves as an appendix 

on linear topologies and their basic properties. 

At the end of each section, there is a list of exercises of 

varying degrees of diffi culty designed to help the reader to 

better understand the material. This aim is further supported 

by the provision of hints to selected exercises. Since the topics 

discussed in the book have their origins in problems from 

economics and fi nance, the book will be valuable not only for 

students and researchers in mathematics but also for those interested 

in economics, fi nance and engineering. (jsp) 

J. Appell, M. Väth: Elemente der Funktionalanalysis, Vieweg, 

Wiesbaden, 2005, 349 pp., EUR 24,90, ISBN 3-528-03222-7 

This book provides an elementary introduction to both linear 

and nonlinear functional analysis. The authors emphasize the 

variability of infi nite dimensional spaces and highlight the role of 

compactness. The linear part of the book covers classical means 

of doing functional analysis (normed linear spaces and operators, 

the criteria of compactness in varied spaces and the Riesz- 

Schauder theory of compact operators). The second part of the 

book concentrates mainly on fi xed point theorems (Banach, 

Brouwer, Schauder, Darbo, Borsuk). The appendix includes the 

Baire category theorem, bases in Banach spaces, the Weierstrass 

approximation theorem and the Tietze-Urysohn and Dugundji 

extension theorems. 

Each chapter ends with many exercises and problems of 

varying diffi culty, which give further applications and exten- 


Recent books 

sions of the theory. The book is easily understandable and it can 

be warmly recommended to graduate students in mathematics, 

physics, biology, chemistry and engineering as a neat introduction 

to functional analysis. (jl) 

G. Balkerma, P. Embrechts: High Risk Scenarios and Extremes, 

Zurich Lectures in Advanced Mathematics, European 

Mathematical Society, Zurich, 2007, 375 pp., EUR 48, ISBN 978- 

3-03719-035-7 

The problem of multivariate extremes and their fi nancial application 

is of central interest in the fi eld of quantitative risk management 

and it is the main topic of this monograph. The book is 

divided into fi ve parts and twenty chapters. After a description 

of motivations, the fi rst part contains an introduction to the theory 

of point processes and their applications in extreme value 

theory. The second part covers the classical univariate and multivariate 

extreme value theories. The coordinate-wise approach 

and max-stable distributions are discussed here. 

The main part of the book comprises parts III and IV. In the 

third part a modern geometric (coordinate free) approach to 

multivariate extremes is broadly studied. Particular interest is 

paid to heavy tailed distributions and to the generalized Pareto 

distribution (candidate multivariate extension of the GPD). 

Extreme value theory is often based on threshold exceedance, 

which is studied in the fourth part of the book. There is no 

unique threshold in the multivariate case and the authors study 

two classes (horizontal and elliptic) of thresholds. In the last 

part of the book, some open problems in the theory and statistical 

applications of multivariate extremes are summarized. 

The book is written with a broad audience of theoretical 

and applied mathematicians in mind. Problems are mostly 

motivated by examples from insurance and fi nance (portfolio 

theory) but their solution is purely mathematical. The level of 

rigour in the book is very high and to understand all the techniques 

of modern extreme value theory, a solid mathematical 

background is required. On the other hand, if one just needs 

to apply the results, the proofs and technical sections may be 

skipped. The book can also be recommended as the basis for an 

advanced course on multivariate extreme value theory. (dh) 

W. Byers: How Mathematicians Think. Using Ambiguity, 

Contradiction, and Paradox to Create Mathematics, Princeton 

University Press, Princeton, 2007, 415 pp., USD 35, ISBN 978- 

0-691-12738-5 

A lot has been said and written about the philosophy of mathematics, 

yet not enough. There are too many unanswered questions. 

For example, is mathematics discovered or created? There 

still seems to be room for a good book in this fi eld, and this one 

is wonderful, a must-read for everyone interested in mathematics, 

philosophy and/or history. One of the most pervasive myths 

about mathematics is that it is a dull technocratic discipline carried 

out by grim computer-like minds that have no feelings and 

who work without intuition, ambiguity or doubt and produce 

strictly formal algorithms and theorems free of contradictions, 

confl icts and paradoxes. Such myths, unfortunately still rather 

common among ‘outsiders’, call for such a book. 

The author, a great mathematician and philosopher, and 

also a practitioner of Zen-Buddhism, shows how essential nonlogical 

qualities are in mathematical research and creativity 

and that the secret of successful mathematics is not in its logical 

structure, or at least not only there. Excellent discussions are 

presented about ambiguity, contradiction, paradox and their 

central role in the world of mathematical discovery. (lp) 

A. Carbery et al., Eds.: Complex and Harmonic Analysis, Proceedings, 

May 25-27, 2006, Thessaloniki, DEStech Publications, 

Lancaster, 2007, 327 pp., USD 89,50, ISBN 978-1-932078-73-2 

This book represents the proceedings of the international conference 

on complex and harmonic analysis, which was held in 

May 2006 at the Aristotle University of Thessaloniki. The volume 

contains 23 articles from a broad range of topics including 

geometric function theory, complex iteration, function spaces, 

composition operators, extremal problems, potential theory, 

maximal functions, analysis on manifolds and others. The conference 

was dedicated to the memory of Nikos Danikas (a 

member of Thessaloniki University) who died in 2004. The introductory 

paper on the mathematical work of Nikos Danikas 

is written by W.K. Hayman and it is followed by an article on 

Danikas measures written by V. Nestoridis. The book can be 

recommended to students and researchers in the area. (jl) 

A. D. D. Craik: Mr Hopkins’ Men. Cambridge Reform and 

British Mathematics in the 19th Century, Springer, Berlin, 

2007, 405 pp., 78 fi g., EUR 99.95, ISBN 978-1-84628-790-9 

This book gives a fascinating view of Cambridge University 

during the Victorian era. It is divided into two parts. The fi rst 

part “Educating the Elite” describes the system of education in 

Cambridge, its dramatic changes in the 19th century and its role 

in the development of the mathematical sciences in Britain. The 

author describes the evolution of curricula and reforms leading 

to Cambridge’s dominance of British higher education. The 

Cambridge reforms are analysed in a comprehensive context 

(the role of political decisions, the infl uence of the church and 

parliament, the development of the town and colleges in Cambridge, 

life at the university, social interest in higher education 

and scientifi c work, and so on). A detailed biography of William 

Hopkins, the most remarkable tutor in Cambridge in the mid- 

19th century, and an appreciation of his mathematical achievements 

have been created with the help of a study of archival 

sources. Brief biographies of Hopkins’ top students from 1829 

up to 1854 are added at the end of the fi rst part. Pencil and watercolour 

portraits of top students from Hopkins’ own collection 

(attributed to the artist T. C. Wageman, who created them 

between 1829 and 1852) are published here for the fi rst time. 

The second part “Careers of the Wrangles” is devoted to 

a description of the careers of some top students and graduates 

(so-called Wrangles) including many famous scientists and 

mathematicians, professors at English and Scottish universities 

and colleges, fi rst professors in Australia, offi cial educators 

overseas and throughout the colonies, tutors of prominent people 

(for example an Indian maharajah), churchmen, etc. Detailed 

biographies of four excellent scientists (G. Green, J. C. 

Adams, G. G. Stokes and H. Goodwin) are included. At the end 

of the second part, the author describes the transformation of 

Cambridge from an unimportant institution into a world centre 

for mathematical and physical sciences and education. Various 

topics are discussed (growth of a research community, the birth 

of journals, the development of scientifi c institutions, the analysis 

of achievements in mathematical sciences before 1830 and 

then from 1830 up to 1880). 


The book can be recommended to people who are interested 

in the history of Victorian Britain in general and in the history 

of Cambridge University, mathematical education, mathematics, 

and scientifi c life and work, as well as the connections 

of science and religious belief, politics, etc. (mbec) 

P. Daskalopoulos, C.E. Kenig: Degenerate Diffusions. Initial 

Value Problems and Local Regularity Theory, Tracts in Mathematics, 

vol. 1, European Mathematical Society, Zürich, 2007, 

198 pp., EUR 48, ISBN 978-3-03719-033-3 

This book is devoted to the study of nonnegative solutions of 

the Cauchy initial value problem or the Dirichlet boundary value 

problem for a class of nonlinear evolution differential equations 

∂u/∂t = ∆ ϕ(u) on R n x (0,T) or on a cylinder B x (t1,t2), 

where the nonlinearity ϕ is assumed to be continuous and increasing 

with ϕ (0) = 0, and is assumed to satisfy the growth 

condition a ≤ [u ϕ ´(u)]/[ϕ (u)] ≤ 1/a for all u > 0 for a in (0,1) 

and the normalization condition ϕ (1) = 1. This class forms a 

natural generalization of the power case ϕ(u) = u m and has a 

wide range of applications both in physics and in geometry. In 

1940, D. Widder characterized the set of all nonnegative weak 

solutions of the Cauchy problem for the heat equation on the 

strip R n x (0,T) by fi ve fundamental properties. 

The authors are interested in analogues of these properties 

for the nonlinearities ϕ described above. The results are divided 

into two groups according to the growth of ϕ. The fi rst group 

concerns the case of “slow diffusion” and it is characterized 

in the power case by the condition m > 1. The second group 

includes the supercritical “fast diffusion” case, which generalizes 

the case ϕ (u) = u m for (n-2)/n < m < 1. Besides the items 

mentioned above, the book collects together local regularity 

results in chapter 1 (a priori L ∞ bounds, the Harnack inequality 

and equicontinuity of solutions to the slow diffusion case) and 

a proof of continuity of weak solutions to the porous medium 

equation in the fi nal (fi fth) chapter. 

The book gives an up-to-date, clear and concise overview of 

results concerning degenerate diffusion together with powerful 

methods and useful techniques for studying existence and 

qualitative properties of solutions. A number of comments and 

discussions of various topics, a brief summary of further known 

results and some open problems are listed in the last section of 

each chapter and an up-to-date bibliography will be appreciated 

by both researchers and graduate students with a background 

in analysis and partial differential equations. (jsta) 

P. Drábek, J. Milota: Methods of Nonlinear Analysis. Applications 

to Differential Equations, Birkhäuser Advanced Texts, 

Birkhäuser, Basel, 2007, 568 pp., EUR 74.79, ISBN 978-3-7643- 

8146-2 

In many cases, existence problems in the theory of differential 

equations and in the calculus of variations refer to results of 

nonlinear analysis in abstract infi nite-dimensional spaces. For 

example, the problem of fi nding stationary points of a functional 

in the calculus of variations may lead to an abstract saddle 

point theorem. To prove existence of a solution of a problem 

in the theory of partial differential equations, fi xed points and 

other topological tools of nonlinear operator theory are often 

useful. The book shows methods of how to apply abstract nonlinear 

analysis to specifi c problems. Thus both nonlinear functional 

analysis and application sections are well developed. 

Recent books 

The list of all the topics would be too long so we will only 

mention the main themes. The exposition of differential calculus 

in normed linear spaces discusses the inverse function 

theorem and the implicit function theorem. Some further topics 

are specifi c to fi nite dimensions; they include the rank theorem, 

fi nite-dimensional bifurcation theorems, integration of 

differential forms on manifolds with Stokes’ theorem and the 

Brouwer degree. The topological methods in nonlinear analysis 

are based on infi nite-dimensional generalizations of the degree. 

The Leray-Schauder degree is available for compact perturbations 

of the identity operator. Another extension of the concept 

of degree can be used for monotone operators and their generalizations. 

Applications include fi xed point theorems, global 

bifurcation theorems and existence theorems based on subsolutions 

and supersolutions. 

The exposition of variational methods starts with a classical 

analysis of extrema on infi nite dimensional spaces and direct 

methods of the calculus of variations. The analysis of stationary 

points covers bifurcation of potential operators, the mountain 

pass lemma, the Lusternik-Schnirelmann method and other 

tools for searching saddle points. Finally, the last chapter is devoted 

to a systematic treatment of classical and weak solutions 

of partial differential equations, demonstrating techniques 

developed in the preceding text. The authors reveal the fruits 

of their long and rich teaching experience. The presentation is 

self-contained and covers all the fundamental tools of nonlinear 

analysis. On the other hand, they also organize advanced 

excursions, mostly as appendices, to topics of interest and some 

parts are developed up to recent research level. Consequently, 

this volume can be used as a reference book or a textbook, especially 

as an important source of knowledge and inspiration 

for university students and teachers of all levels. (jama) 

H. Dym: Linear Algebra in Action, Graduate Studies in Mathematics, 

vol. 78, American Mathematical Society, Providence, 

2007, 541 pp., USD 79, ISBN 978-0-8218-3813-6 

This is a book on linear algebra, playing an important role in 

pure and applied mathematics, computer science, physics and 

engineering. The book is divided into 23 chapters and 2 appendices. 

The fi rst six chapters and some selected parts from chapters 

7-9 are based on classical linear algebra topics. The reader 

will fi nd here many interesting principles and results concerning 

vector spaces, Gaussian elimination and its applications, 

determinants, eigenvalues and eigenvectors, Jordan forms and 

their calculations, normed linear spaces, inner product spaces 

and orthogonality, and symmetric, Hermitian and normal matrices. 

The next three chapters are devoted to singular values 

and related inequalities, pseudoinverses and triangular factorization 

and positive defi nite matrices. 

Chapter 13 treats difference equations, differential equations 

and their systems. Chapters 14-16 contain applications to 

vector valued functions, the implicit function theorem and extremal 

problems. The subsequent chapters deal with matrix valued 

holomorphic functions, matrix equations, realization theory, 

eigenvalue location problems, zero location problems, convexity 

and matrices with nonnegative entries. Two appendices describe 

useful facts from complex function theory. The book offers basic 

and advanced techniques of linear algebra from the point of 

view of analysis. Each technique is illustrated by a wide sample 

of applications and it is accompanied by many exercises of 


Recent books 

varying diffi culty, which give further extensions of the theory. 

The book can be recommended as a general text for a variety of 

courses on linear algebra and its applications, as well as a selfstudy 

aid for graduate and undergraduate students. (mbec) 

H.-D. Ebbinghaus: Ernst Zermelo – An Approach to His Life 

and Work, Springer, Berlin, 2007, 356 pp., EUR 49.95, ISBN 978- 

3-540-49551-2 

This book gives the fi rst English, detailed scientifi c biography 

of Ernst Zermelo (1871–1953), a German mathematician best 

known for a formulation of the axiom of choice and for an axiomatization 

of set theory. He is considered to be one of the 

greatest logicians in the fi rst half of the 20th century. In fi ve 

chapters, the author describes Zermelo’s life and his scientifi c 

achievements from his youth through his studies, works, teaching 

and scientifi c activities. The author also analyses Zermelo’s 

conditions for scientifi c study and research before, during and 

after World War II, his isolation during the war and his activities 

after the war. Zermelo’s major scientifi c contributions (in 

set theory, the calculus of variations, the theory of games, the 

theory of rating systems, applied mathematics) are discussed 

without assuming a detailed mathematical knowledge of the 

fi eld in question. 

The presentation of Zermelo’s works explores his motivations, 

aims, acceptance and infl uence on the development of 

mathematics. The book is based on a study of unknown and 

unpublished sources, archival materials, private letters, family 

documents, previously unpublished notes, interviews and memoirs. 

The main text is followed by a schematic curriculum vitae 

of Ernst Zermelo summarizing the most important events of 

his life. The book concludes with an appendix containing a German 

version of Zermelo’s unpublished ideas and parts of his 

works (for example selected proofs), his original letters, notes 

and samples of his literary activities. The book contains more 

than 40 photos and facsimiles, a large list of references and an 

index. It gives new light to all facets of Zermelo’s life and mathematical 

achievements and it can be recommended to a wide 

audience. The book is suitable for mathematicians, historians of 

mathematics and science, students and teachers. (mbec) 

J. Friberg: A Remarkable Collection of Babylonian Mathematical 

Texts, Sources and Studies in the History of Mathematics 

and Physical Sciences, Springer, Berlin, 2007, 533 pp., EUR 

99.95, ISBN 978-0-387-34543-7 

This fascinating book presents 121 unpublished mathematical 

clay tablets from the Norwegian Schøyen Collection including 

many interesting tablets from the Early Dynastic III period 

(circa 2600–2350 BC) up to the Late Kassite period (circa 

14th–13th century BC). Based on descriptions, transliterations, 

translations, interpretations, analysis and synthesis of the mathematical 

cuneiform texts and their comparison with previously 

published mathematical clay tablets, the author has made numerous 

amazing discoveries and has created a new comprehensive 

treatment of Old Babylonian mathematical texts. The book 

is divided into 12 chapters, 10 appendices, a vocabulary for MS 

texts, an index of subjects, an index of texts and a large list of 

references. 

The author starts with an introduction explaining how to 

better understand mathematical cuneiform texts. He explains 

mathematical terminology, cuneiform systems of notations for 

numbers and measures, operations with sexadecimal numbers 

(addition, subtraction, multiplication, division, and an ingenious 

factorization method for the computation of reciprocals 

and square roots). Then he explains the use of the Old Babylonian 

standard tablets of squares, square roots, cube roots and 

reciprocals, as well as the non-standard tablets of “quasi-cube 

roots”. The next chapters are devoted to a discussion of Old 

Babylonian metrological systems (standard types for volumes, 

area, length and weight are described). 

A large part of the book analyses some typical Babylonian 

geometrical problems (computation of the area of a triangle, a 

trapezoid or a circle; computation of the diagonals of the sides 

of a square; the problem of how to inscribe and circumscribe 

circles, triangles and squares, how to divide fi gures into other 

fi gures with given special properties, and how to change fi gures 

to other fi gures, while retaining the area; computation of the 

volume of prisms, cubes and cylinders; construction of labyrinths, 

mazes, decorative patterns and ground plans of palaces), 

some typical algebraic problems (arithmetic and geometric 

progressions, solutions of linear and quadratic equations and 

their systems) and some problems from daily life (construction 

of walls, building of a ramp, dividing works, money and so on). 

Many pictures, drawings and coloured photos of the most interesting 

tablets are also included. The book opens up Babylonian 

mathematics to a new generation of mathematicians, historians 

of science and mathematics, teachers and students. It can therefore 

be recommended to a wide audience. (mbec) 

B. Fristedt, N. Jain, N. Krylov: Filtering and Prediction – A 

Primer, Student Mathematical Library, vol. 38, American Mathematical 

Society, Providence, 2007, 252 pp., USD 39, ISBN 978- 

0-8218-4333-8 

This book is mainly intended as an introduction to the problem 

of fi ltering and prediction in time homogeneous Markov 

chains and the Wiener process. The problem of how to estimate 

an unobservable signal based on an available observation of response 

(a signal corrupted by noise) is known as fi ltering. Prediction 

means estimating the future of a random process based 

on its history. 

The fi rst two chapters of the book are introductory and deal 

with basic probability concepts and discrete Markov chains. In 

the third chapter, the discrete Markov chains are used as the 

simplest model for an introduction of the fi ltering problem. In 

chapter 4, the conditional expectation is introduced as a necessary 

tool for continuous (in space and/or time) processes. The 

continuous space Markov chains are then discussed in chapter 

5; the famous Kalman fi lter is described and the chapter closes 

with linear fi ltering. Chapter 6 covers fi ltering for the Wiener 

process and the continuous time Kalman fi lter is introduced in 

a classical setting. The last two chapters deal with the prediction 

problem in stationary random sequences. 

The book is written in an elementary way (no special 

knowledge of probability and statistics is needed to read the 

book) but it is still mathematically rigorous. The book can be 

recommended to all students interested in stochastic models. 

Since the book contains many problems and remarks, it can 

also be recommended as the basis of a one semester introductory 

course; almost all theorems are proved and problems may 

be used as homework assignments although some of the problems 

are more challenging than others. (dh) 


M. Giaquinta, G. Modica: Mathematical Analysis – Linear 

and Metric Structures and Continuity, Birkhäuser, Boston, 

2007, 465 pp., EUR 119, ISBN 978-0-8176-4374-4 

This book provides a comprehensive explanation of the background 

of functional analysis and its origins. It is divided into 

three parts. It begins with a careful and inspiring exposition 

of basic ideas on linear and metric structures. The second part 

is dedicated to the role of general topological notions in the 

framework of metric spaces. These two parts help the reader 

to prepare for the main principles of functional analysis, which 

are then explained in the third part of the book. At each step, 

theory is accompanied by important examples and applications. 

Carefully chosen exercises stimulate the reader to work actively 

with the text. On the whole, this self-contained and up-to-date 

book can be very useful not only to the advanced undergraduate 

or graduate student but also to anybody who wants to systemize 

their knowledge on the elements of the subject. (oj) 

J. J. Gray, K. H. Parshall, Eds.: Episodes in the History of 

Modern Algebra (1800–1950), History of Mathematics, vol. 32, 

American Mathematical Society, Providence, 2007, 336 pp., USD 

69, ISBN 978-0-8218-4343-7 

This book offers new light on the development and history of 

modern algebra. The book brings together suitably revised, 

chapter-length versions of twelve lectures that were given at 

the workshop on the history of 19th and 20th century algebra 

held at the Mathematical Sciences Research Institute in Berkeley 

in 2003. 

The introduction and chapters written by prominent mathematicians 

and historians of mathematics (J. J. Gray, K. H. 

Parshall, E. L. Ortiz, S. E. Despeaux, O. Neumann, H. M. Edwards, 

G. Frei, J. Schwermer, D. D. Fenster, Ch. W. Curtis, 

L. Corry, N. Schappacher, S. Slembek, C. McLarty) provide 

complex and detailed overviews of the evolution of modern 

algebra from the early 19th century work of Ch. Babbage on 

function equations through to the description of the development 

of calculus operations (British research done before and 

documented within the Cambridge Mathematical Journal), 

the analysis of divisibility theories in the early history of commutative 

algebra (contributions by C. F. Gauss, E. E. Kummer, 

E. I. Zolotarev, L. Kronecker , D. Hilbert, E. Noether, etc.), a 

presentation of Kronecker’s fundamental theorem on general 

arithmetic, a description of advances in the theory of algebras 

and algebraic number theory (works by J. H. M. Wedderburn, 

A. Hurwitz, R. Brauer, H. Hasse, E. Noether, H. Minkowski, 

K. Hensel, L. Dickson, A. A. Albert, etc.) and in the analysis of 

the foundations of algebraic geometry and its arithmetization 

and development (results of H. Hasse, E. Noether, F. Severi, 

A. Grothendieck, etc.). 

The topics discussed represent the long and diffi cult process 

of the changing organization of the main subjects, the changing 

algebraic themes, ideas and concepts, as well as the results 

of mathematical communication and collaboration within and 

across national boundaries. A comprehensive list of references 

as well as many detailed notes are added at the end of each 

chapter. The book can be recommended to readers who are interested 

in the history of modern mathematics in general and 

modern algebra in particular. It is suitable for mathematicians, 

historians of mathematics and science, teachers and students. 

(mbec) 

Recent books 

M. Greenacre: Correspondence Analysis in Practice, 2nd edition, 

Chapman and Hall/CRC, Boca Raton, 2007, USD 79,95, 

ISBN 1-58488-616-1 

The previous edition, published in 1993, has been totally revised 

and extended. Like the fi rst edition, this book is intended 

to be practically oriented and didactic. The author’s wish is to 

represent in each of the 25 chapter (and fi ve appendices) a 

fi xed amount of material. As a result, each chapter is exactly 

eight pages in length to offer a fi xed amount of material both 

for reading and teaching. However, this does not always have 

pleasant consequences in some chapters, where more information 

would be useful for the reader. A short summary of the 

main points in the form of a list concludes each chapter. 

As in the fi rst edition, the book’s main thrust is toward the 

practice of correspondence analysis, so that most technical issues 

and mathematical aspects are gathered in appendix A. The 

theoretical part here is more extensive than that of the fi rst edition, 

including additional theory on new topics such as canonical 

correspondence analysis, transition and regression relationships, 

stacked tables, subset correspondence analysis and the 

analysis of square tables. Concerning computational issues, a 

very strong feature of this edition is the use of the R program 

for all computations. The lengthy appendix B (45 pages) summarizes 

(on examples) possibilities of relating macros from R. 

In addition to that, three different technologies are described 

enabling the creation of the graphical displays presented in the 

book. No references at all are given in the 25 chapters; a relatively 

brief bibliography in appendix C is given to point readers 

toward further reading containing a more complete literature 

guide. A glossary of the most important terms and an epilogue 

with some fi nal thoughts conclude the book. 

Summarizing, this is a nice book for all those who wish to 

acquaint themselves with a versatile methodology of correspondence 

analysis and the way it can be used for the analysis 

and visualization of data arriving typically from the fi elds of 

social, environmental and health sciences, marketing and economics. 

Numerous examples provide a real fl avour of the possibilities 

of the method. (jant) 

D. D. Haroske, H. Triebel: Distributions, Sobolev Spaces, Elliptic 

Equations, EMS Textbooks in Mathematics, European 

Mathematical Society, Zürich, 2007, 294 pp., EUR 48, ISBN 978- 

3-03719-042-5 

Many problems in mathematical physics reduce to elliptic differential 

equations of second order and their boundary value 

problems, and also to the spectral theory of such operators. 

Central role are played by the Dirichlet problem, the Neumann 

problem and the eigenvalue and eigenfunction problem. The 

main aim of the book is to develop the L 2 theory of the listed 

problems on bounded domains with smooth boundaries in Euclidean 

space. This is done in a reader-friendly way at a moderate 

level of diffi culty, aiming at graduate students and their 

teachers. 

Among the topics covered, we fi nd the classical theory of 

the Laplace-Poisson equation, the theory of distributions, the 

theory of Sobolev spaces on domains, abstract spectral theory 

in Hilbert and Banach spaces and compact embeddings. One 

of the principal assets of the book is a very good, friendly and 

accessible introduction to various aspects of function space theory 

and their applications in the theory of partial differential 


Recent books 

equations. The book is richly furnished with interesting exercises 

and a thorough set of notes is attached to each chapter. This 

is an ideal book for both students and teachers of a modern 

graduate course on partial differential equations. (lp) 

M. Hata: Problems and Solutions in Real Analysis, Series on 

Number Theory and Its Applications, vol. 4, World Scientifi c, 

New Jersey, 2007, 292 pp., USD 42, ISBN 978-981-277-949-6 

As is well-known, we learn mathematics by trying to solve 

problems. A fascinating thing about mathematics is that even 

when we do not succeed in actually solving a given problem, 

we can still gain something by trying to solve it, for example by 

learning or developing new techniques. 

The author presents over 150 very interesting mathematical 

problems and their detailed solutions. The majority of the 

questions belong to real analysis, while some of them are taken 

from analytic number theory (such as those concerning the 

uniform distribution or a proof of the prime number theorem, 

which a reader accesses in an elementary way through several 

exercises). The book is divided into 18 chapters, each of which 

starts with a list of some basic knowledge in the fi eld followed 

by problems and their solutions. Topics include sequences and 

limits, infi nite series, continuous functions, differentiation, integration, 

improper integrals, series of functions, approximation 

by polynomials, convex functions, the formula evaluating the 

Riemann zeta function at 2, functions of several variables, the 

uniform distribution, Rademacher functions, Legendre polynomials, 

Chebyshev polynomials, Gamma functions and the prime 

number theorem. 

The standard of problems varies from easy, aimed at undergraduate 

students of calculus and linear algebra, to deep 

and complex, which could challenge experts. The text contains 

many useful and interesting historical comments on various 

signifi cant mathematical results in real analysis, and is accompanied 

with corresponding references. (lp) 

L. Hathout: Crimes and Mathdemeanors, A.K. Peters, Wellesley, 

2007, 196 pp., USD 14.95, ISBN 978-1-56881-260-1 

As the title suggests, this book is something of a detective story 

with a mathematical slant. Fourteen chapters contain fourteen 

cases with more or less criminal plots, solved by a fourteen year 

old Ravi, a brilliant high school student and mathematics genius 

who is able to reduce tangled criminal cases that baffl e police 

and lawyers to mathematics, logic and probability, and to 

solve them. Amazingly, the author is a high school student himself. 

He has managed to hide intriguing math problems within 

cleverly written criminal mysteries including murder and fraud. 

Readers are invited to challenge their own wits and try to solve 

the mysteries themselves. Each case is followed by an analysis, 

in which the mathematical or logical background of the crime 

is clearly formulated. Ravi’s brilliant solution comes next, followed 

by an extension to more general or more complicated 

mathematical questions. 

Only high school mathematics (combinatorics, fi nite probability, 

elementary inequalities, geometry, algebra, etc.) and 

physics is needed for solving all the problems, except one for 

which a bit of integration is required. Therefore the book will 

bring pleasure to a very broad (not necessarily mathematical) 

public of interested readers who like mysteries and solving conundrums. 

(lp) 

P. M. Higgins: Nets, Puzzles, and Postmen. An Exploration of 

Mathematical Connections, Oxford University Press, Oxford, 

2007, 247 pp., GBP 15.99, ISBN 978-0-19-921842-4 

In this lovely and fascinating book, the author describes the 

world and the main functions of networks, which are all around 

us (from age-old examples, such as family trees, to the modern 

phenomena of the Internet and the World Wide Web). 

Beginning with structures of chemical isomers, family trees, 

simple mathematical puzzles (for example tic-tac-toe, familiar 

logic games, exotic squares, Sudoku) and famous mathematical 

problems (the bridges of Königsberg, the hand-shaking 

problem, the Hamilton cycle, the party problem), the author 

explains how networks are represented, how they work, and 

how they can be described, deciphered and understood. After 

some puzzles and games, the author introduces examples and 

applications of networks from natural sciences, social sciences, 

technology, economics, transportation sciences and genetics. 

Various topics are discussed (the four-colour map problem, 

guarding the museum, the Brouwer fi xed point theorem, the 

Chinese postman problem, nets as machines, automata, planning 

routes, maximizing profi ts, the quick route, spanning 

networks, secret codes, RNA reconstructions, labyrinths and 

mazes and so on). 

Understanding the book does not require a deep mathematical 

knowledge. For the reader wanting to study these topics 

from a mathematical point of view, the fi nal chapter “For 

Connoisseurs” can be recommended. The book will open the 

eyes of the reader to hidden networks, hence it can be recommended 

to people wanting to discover a remarkable new view 

of our world. (mbec) 

A. Iosevich: A View from the Top. Analysis, Combinatorics 

and Number Theory, Student Mathematical Library, vol. 39, 

American Mathematical Society, Providence, 2007, 136 pp., USD 

29, ISBN 978-0-8218-4397-0 

The author has based this nice little book on a capstone course 

that he has taught to upper division undergraduate students, 

the main objective of which was to explain, in a visual way, a 

unity of mathematics and interactions between its fundamental 

areas such as analysis, algebra, number theory, topology, basic 

counting and probability theory. Such courses are unfortunately 

taught rather rarely; the book will thus be a tremendous asset 

and an endless source of inspiration to anyone who has a 

course like that in mind. 

Starting with basic inequalities of Cauchy-Schwarz and 

Hölder, the author proceeds by pursuing their applications in 

geometry, then turns attention to the Besicovitch-Kakeya conjecture 

on a connection of the size of a set with a number of 

contained line segments with different slopes, a central problem 

in contemporary harmonic analysis, and then presents 

some brilliant ideas concerning basic counting (combinatorics, 

the binomial theorem, expected values, random walks, etc.) and 

probability reasoning, and fi nishes with oscillatory integrals, 

trigonometric sums and Fourier analysis with applications to 

geometry and number theory. The basic idea is to get the reader 

enthusiastic about mathematical research by observing the evolution 

of ideas. Most of the book requires no prerequisites (just 

high school mathematics) but for some chapters the reader is 

supposed to have certain knowledge about functions of several 

variables. (lp) 


M. Jarnicki, P. Pfl ug: First Steps in Several Complex Variables 

– Reinhardt Domains, EMS Textbooks in Mathematics, European 

Mathematical Society, Zürich, 2008, 359 pp., EUR 58, ISBN 

978-3-03719-049-4 

This book gives a nice introduction to some parts of the theory of 

several complex variables. It concentrates on a study of the properties 

of a special class of domains called Reinhardt domains. 

Some basic notions in the theory (domains of holomorphy, 

holomorphic convexity, pseudoconvexity) can be nicely introduced 

and studied in the setting of Reinhardt domains without 

too many technical details. In the fi rst chapter of the book, 

the authors present such an introduction to problems arising 

in connection to the holomorphic continuation of holomorphic 

functions in several complex variables. Biholomorphisms of 

domains and the Cartan theory in the setting of Reinhardt domains 

are treated in the second chapter. A short third chapter 

contains a discussion of S-domains of holomorphy for various 

special classes S of holomorphic functions. The last chapter is 

devoted to a study of holomorphically contractible families on 

Reinhardt domains and it includes a lot of explicit calculations 

in specifi c cases. Many results presented in the book are based 

on research by both authors. (vs) 

N. L. Johnson, V. Jha, M. Biliotti: Handbook of Finite Translation 

Planes, Pure and Applied Mathematics, vol. 289, Chapman 

& Hall/CRC, Boca Raton, 2007, 861 pp., USD 99.95, ISBN 

978-1-58488-605-1 

This book is something of a compendium of examples, processes, 

construction techniques and models of fi nite translation 

planes. A translation plane is an affi ne plane that admits a 

group of translations acting transitively on its points. The guiding 

principle of the authors is to present an atlas of translation 

planes. The tremendous variety of translation planes makes any 

attempt at classifi cation rather diffi cult. So the authors provide 

a combination of descriptions and methods of construction to 

give an explanation of various classes of planes. Since interest 

in certain translation planes arises primarily from their place 

within certain classifi cation results, the authors also provide 

comprehensive sketches of the major classifi cation theorems 

together with outlines of their proofs. The bulk of the book is 

a comprehensive treatment of known examples of the planes 

in more than 800 pages and 105 chapters, culminating with the 

atlas of planes and construction processes. (jtu) 

I. Kononenko, M. Kukar: Machine Learning and Data Mining. 

Introduction to Principles and Algorithms, Horwood Publishing, 

Chichester, 2007, 454 pp., GBP 50, ISBN 978-1-904275-21-3 

Recent advances in machine learning have had a strong impact on 

rapid developments in related areas like data analysis, knowledge 

discovery and computational learning theory. This (already wellestablished) 

research discipline has found many applications in 

expert and business systems, data mining, databases, game playing, 

text, speech and image processing, etc. However, due to the 

rather interdisciplinary character of this new fi eld, many books 

on machine learning currently appearing are quite specialised to 

the considered discipline. In that sense, this book differs from several 

other excellent books on machine learning and data analysis, 

as its authors have succeeded in providing a detailed, yet comprehensive, 

overview of the fi eld. The entire textbook consists of 

fourteen chapters and can be divided into fi ve parts. 

Recent books 

The introductory part of the book encompasses the fi rst 

two chapters, which provide an historical overview of the fi eld 

and outline philosophical issues related to machine and human 

learning, intelligence and consciousness. The following four 

chapters deal with the basic principles of machine learning, 

representation of knowledge, basic search algorithms used to 

search hypothesis space, and attribute quality measures used to 

guide the search. The next two chapters set practical guidelines 

for preparing, cleansing and transforming the processed data. 

Chapters 9 – 12 are devoted to a description of the respective 

learning algorithms: symbolic and statistical learning, artifi cial 

neural networks and cluster analysis. The last two chapters introduce 

formal approaches to machine learning: the problem of 

identifi cation in the limit and computational learning theory. 

The attached appendix reviews some theoretical concepts 

used in the book. Although oriented primarily towards advanced 

undergraduate and postgraduate students, each section 

of the textbook is relatively self-contained and only requires 

a background knowledge in calculus. The discussed topics are 

explained in an interesting way with a lot of illustrative fi gures 

and supporting graphs. Each chapter is accompanied with a 

summary of the concepts taught. For these reasons, this book 

represents both a valuable teaching resource for students and a 

good reference source of applicable ideas for a wide audience 

including researchers and application specialists interested in 

machine learning paradigms. (im) 

E. Maor: The Pythagorean Theorem. A 4,000-Year History, 

Princeton University Press, Princeton, 2007, 259 pp., USD 24.95, 

ISBN 978-0-691-12526-8 

This interesting book is devoted to the Pythagorean theorem, 

which is the most frequently used theorem in all branches of 

mathematics and which is learnt in geometry at school. The author 

shows the long route the Pythagorean theorem has taken 

through cultural history and he analyses its role in the development 

of mathematical thinking, research and teaching. 

The book starts with a description of the earliest evidence 

of knowledge of the theorem, which was known to the Babylonians 

over 4000 years ago. It continues with an analysis of the 

work of the Greek mathematician Euclid, which immortalized 

this theorem as Proposition 47 in the fi rst book of his famous 

Elements. Then the book describes the role of the theorem in 

Greek mathematics (Archimedes, Apollonius, etc.) and its positions 

in pure and applied mathematics, as well as its infl uence 

on the arts, poetry, music and culture through the Arabic world, 

the Middle Ages, the Renaissance and the New Age in Europe 

up to Einstein’s theory of relativity. The most beautiful proofs 

of the theorem are shown and explained. The book contains 

an index and many interesting photos and pictures. It can be 

recommended to readers who want to learn about mathematics 

and its history, who want to be inspired and who want to understand 

important mathematical ideas more deeply. (mbec) 

E. Menzler-Trott: Logic’s Lost Genius – The Life of Gerhard 

Gentzen, History of Mathematics, vol. 33, American Mathematical 

Society, Providence, 2007, 440 pp., USD 89, ISBN 978-0- 

8218-2550-0 

This book gives (for the fi rst time in English) a detailed scientific 

biography of Gerhard Gentzen (1909-1945), a German mathematician 

who was one of the founders of modern structural 


Recent books 

proof theory and who is considered to be one of the greatest 

logicians from the fi rst half of the 20th century. In six chapters, 

the author describes Gentzen’s life and his scientifi c achievements 

from his youth, through his studies, works, teaching and 

scientifi c activities, and his military service as well as his political 

ideas, until his arrest and tragic death in Prague in 1945. The 

author also analyses Gentzen’s conditions for scientifi c study 

and research in National Socialist Germany before and during 

World War II, his fi ghts for “German logic”, and his battles 

with colleagues and the political system. The book is based on a 

study of unknown and unpublished sources, archive materials, 

private letters, family documents and memoirs. 

The book ends with four interesting appendices. The fi rst 

and second appendix (written by C. Smoryński – Gentzen and 

Geometry, Hilbert’s Programme) contain a short essay on 

Gentzen’s results in geometry and a deeper analysis of Hilbert’s 

programme showing relations to ideas and mathematical results 

of Hilbert, Brouwer, Weyl, Gödel and Gentzen. In the third appendix, 

there are (for the fi rst time in English) three lectures 

by G. Gentzen (The Concept of Infi nity in Mathematics, The 

Concept of Infi nity and the Consistency of Mathematics, and 

The Current Situation in Research in the Foundation of Mathematics), 

which were presented by G. Gentzen to a wide mathematical 

public in Münster (1936), at the Descartes Congress in 

Paris (1937) and in Bad Kreuznach (1937). The fourth appendix 

(written by John von Plato) explains in detail the Gentzen 

mathematical program, his ordinal proof theory, his work on 

natural deduction and his calculus, and it gives a general survey 

of later developments in structural proof theory. 

The book can be recommended to a wide audience; it is 

suitable for mathematicians, historians of science, students and 

teachers. (mbec) 

R. E. Moore, M.J. Cloud: Computational Functional Analysis, 

second edition, Horwood Publishing, Chichester, 2007, 180 

pp., GBP 27.50, ISBN 978-1-904275-24-4 

This book is devoted to a brief survey of basic structures and 

methods of functional analysis used in computational mathematics 

and numerical analysis. It is an introductory text intended 

for students at the fi rst year graduate level; despite its 

occasional lack of depth and precision, it provides a good opportunity 

to get acquainted with the rudiments of this powerful 

discipline. The main emphasis is on numerical methods for operator 

equations – in particular, on analysis of approximation 

error in various methods for obtaining approximate solutions 

to equations and system of equations. 

The book could be loosely divided into two parts; the former 

concentrates on linear operator equations, the latter on nonlinear 

operator equations. After introducing the basic framework 

used in functional analysis (such as linear, topological, metric, 

Banach and Hilbert spaces), the authors move on to linear 

functionals and operators and several types of convergence in 

Banach spaces. Special chapters are devoted to reproducing 

kernel Hilbert spaces and order relations in function spaces. 

The fi rst part of the book fi nishes with basic elements of 

the Fredholm theory of compact operators on Hilbert spaces 

and with approximation methods for linear operator equations. 

The second part (concentrating on nonlinear equations) starts 

with interval methods for operator equations and basic fi xed 

point problems. After introducing Fréchet derivatives in Ba- 

nach spaces and its elementary properties, their applications in 

Newton’s method and its variant in infi nite dimensional spaces 

are presented. The last chapter is devoted to a particular example 

of a use of the theory in a “real-world” problem; the authors 

choose a hybrid method for a free boundary problem. The 

topics are mostly discussed without proofs but each chapter is 

accompanied by a series of exercises that are designed to help 

students to learn how to discover mathematics for themselves. 

Hence the book serves as a readable introduction to functional 

analytical tools involved in computation. (jsp) 

A. Neeman: Algebraic and Analytic Geometry, London Mathematical 

Society Lecture Note Series 345, Cambridge University 

Press, Cambridge, 2007, 420 pp., GBP 40, ISBN 978-0-521- 

70983-5 

To explain basic principles of modern algebraic geometry to 

undergraduate students with a standard education is a very 

diffi cult task. Nevertheless, Amnon Neeman has succeeded in 

this book in showing that it is possible. The book is based on 

his own teaching experience and the basic idea is simple. He 

has chosen to present a formulation and the proof of Serre’s 

famous theorem on ‘géométrie algébriques and géométrie analytique’ 

(GAGA) as the ultimate aim of the book. On the way 

to this goal, he introduces a lot of notions and tools of modern 

algebraic geometry (e.g. the spectrum of a ring, Zariski topology, 

schemes, algebraic and analytic coherent sheaves, localizations, 

sheaf cohomology). The whole book shows to the reader 

a beautiful mixture of analysis, algebra, geometry and topology, 

all used together in the modern language of algebraic geometry 

and, at the same time, nicely illustrating its power. 

The book is written in a very understandable way, with a 

lot of details and with many remarks and comments helping to 

develop intuition for the fi eld. It is an extraordinary book and 

it can be very useful for teachers as well as for students or nonexperts 

from other fi elds. (vs) 

R. Niedermeier: Invitation to Fixed-Parameter Algorithms, 

Oxford Lecture Series in Mathematics and its Applications 31, 

Oxford University Press, Oxford, 2006, 300 pp., GBP 55, ISBN 

0-19-856607-7 

Theoretical computer science as the borderline between mathematics 

and computer science is a wonderland of fast evolving 

theories, challenging open problems and plentiful algorithmic 

and complexity results, all backed up by practical motivation 

and applications in computing. One of the recently opened and 

intensively studied directions is the so-called Fixed-Parameter 

Complexity, designed and developed in the 1990s by Downey 

and Fellows. Peppered with practical questions of solving problems 

with small parameter values, problems that are computationally 

hard in general, this is a surprisingly rich theory, where 

computational classes are defi ned based on precise logic constructions. 

It has immediately become popular among researchers; 

every major computer science conference gets a number 

of presentations from this area, and specialized workshops are 

now regularly organized. And yet, only one monograph devoted 

to this topic was available until recently; the book under review 

organically complements this previously published monograph 

(Downey-Fellows, Parametrized Complexity, 1999). 

Downey and Fellows, the gurus of FPT (Fixed Parameter 

Tractability), introduce the theory by concentrating more on 


structural defi nitions and properties whilst Nidermeier comes 

in with numerous samples of FPT algorithms for particular 

problems. Moreover, a lot has happened since the Downey-Fellows 

monograph was published. The book under review is an 

excellent survey of new results, focused primarily on algorithmic 

issues. 

The book is divided into three chapters. The fi rst chapter 

provides necessary defi nitions accompanied by motivating examples 

and a broad exposition of the philosophy of Fixed Parameter 

Complexity. Don’t skip this chapter if you are a novice 

in this area or if you want to enjoy learning new views on it. The 

second chapter provides a thorough catalogue of the methodology 

of fi xed parameter algorithms. This is an encyclopaedia 

of methods such as kernelization, depth-bounded search trees 

and graph decompositions, all thoroughly explained with many 

examples. The third chapter gives examples of hardness reductions 

in Fixed Parameter Complexity Theory and describes the 

W[t] classes, with most attention paid to W[1] and W[2], which 

is where most of the natural problems lie. 

The last section, called “Selected Case Studies”, is a Madame 

Tussauds Museum of FPT. The difference is, you do not fi nd 

any wax problems here; all of them are alive and developing at 

the time you read about them. The exposition here is infl uenced 

by the famous NP-completeness bible of Garey and Johnson: 

“Computers and Intractability – A guide to the Theory of NPcompleteness” 

(1979), and the problems are described as carefully 

from the FPT perspective as Garey and Johnson did from 

the P-NP one. And not only because of this section is the book 

becoming as useful as Garey-Johnson’s monograph. 

This is an advanced textbook intended and well-suited both 

for researchers and graduate students in theoretical computer 

science. You will enjoy reading it whether you want to do research 

in the area or just want to learn about it. And while merrily 

intending the latter, you are most likely going to end up 

doing the former when you fi nish reading it. (jkrat) 

V. V. Prasolov: Elements of Homology Theory, Graduate Studies 

in Mathematics, vol. 81, American Mathematical Society, 

Providence, 2007, 418 pp., USD 69, ISBN 978-0-8218-3812-9 

This book is a translation of the Russian original (published 

in 2005). It is a continuation of the author’s earlier book “Elements 

of Combinatorial and Differential Topology”. It is convenient 

to have the companion volume at hand because the 

book refers to it for basic defi nitions and concepts. In the fi rst 

half of the book, the author builds the theory of simplicial homology 

and cohomology and describes some of its applications 

(most of the space being devoted to characteristic classes in the 

simplicial setting). It is followed by a description of singular 

homology. The penultimate chapter defi nes Čech cohomology 

and de Rham cohomology and it proves the de Rham theorem 

and its simplicial analogue. The last chapter “Miscellany” contains 

material on invariants of links, embeddings and immersions 

of manifolds, and a section on cohomology of Lie Groups 

and H-Spaces. 

The book contains 136 problems (mostly in the chapters on 

simplicial homology and cohomology and their applications). 

For the majority of problems, there are solutions or hints in 

a 37-page appendix. Together with its companion volume, the 

book can be recommended as a basis for an introductory course 

in algebraic topology. (dsm) 

Recent books 

G. Royer: An Initiation to Logarithmic Sobolev Inequalities, 

SMF/AMS Texts and Monographs, vol. 14, American Mathematical 

Society, Providence, 2007, 119 pp., USD 39, ISBN 978- 

0-8218-4401-4 

Classical Sobolev inequalities constitute an extremely important 

part of functional analysis with a surprisingly wide fi eld of applications. 

The study of certain specifi c topics such as quantum fi elds 

and hypercontractivity semigroups requires an extension of a 

classical Sobolev inequality to the setting of infi nitely many variables. 

This is a diffi cult problem because the Lebesgue measure 

in infi nitely many variables is meaningless. A solution was discovered 

by Leonard Gross in his famous pioneering paper from 

1975, where he replaced the Lebesgue measure with the Gauss 

measure. In the end, the power integrability gain known from 

classical inequalites is replaced by a more delicate logarithmic 

growth – hence the name. Gross’ logarithmic Sobolev inequalities 

have found, again, an impressive range of applications. 

The book under review provides an introduction to logarithmic 

Sobolev inequalities and to one of its specifi c applications 

in the fi eld of mathematical statistical physics, more precisely 

to the example concerning real spin models with weak 

interactions on a lattice. A proof of the uniqueness of the Gibbs 

measure is given based on the exponential stabilization of the 

stochastic evolution of an infi nite-dimensional diffusion process, 

a generalization of the Ising model. The author begins with 

the background material on self-adjoint operators and semigroups, 

then proceeds to logarithmic Sobolev inequalities with 

applications to Kolmogorov diffusion processes, and fi nishes 

with Gibbs measures and the Ising models. The text is complemented 

with exercises and appendices that extend the material 

to related areas such as Markov chains. (lp) 

A. A. Samarskii, P. N. Vabishchevich: Numerical Methods for 

Solving Inverse Problems of Mathematical Physics, Inverse 

and Ill-Posed Problems Series, Walter de Gruyter, Berlin, 2007, 

438 pp., EUR 148, ISBN 978-3-11-019666-5 

In this monograph, the authors consider the main classes of 

inverse problems in mathematical physics and their numerical 

treatment. They include in the book many numerical illustrations 

and codes for their realization. They give a rather complete 

treatment of basic diffi culties for approximate solutions 

of inverse problems. They use minimal mathematical apparatus 

(in particular, basic properties of operators in fi nite-dimensional 

spaces). Russian mathematicians contributed in a substantial 

way to solutions of theoretical and practical questions studied 

in inverse problems in mathematical physics (pioneering work 

in this direction was done by Andrei Nikolaevich Tikhonov). 

The main topics treated in the book are boundary value 

problems for ordinary differential equations and elliptic and 

parabolic partial differential equations, methods of solutions 

for ill-posed problems and evolutionary inverse problems. The 

book is intended for graduate students and scientists interested 

in applied mathematics, computational mathematics and mathematical 

modelling. (knaj) 

C. G. Small: Functional Equations and How to Solve Them, 

Problem Books in Mathematics, Springer, Berlin, 2007, 129 pp., 

EUR 32.95, ISBN 978-0-387-34539-0 

This book is devoted to functional equations of a special type, 

namely to those appearing in competitions like the Interna- 


Recent books 

tional Mathematical Olympiad for high school students or in 

the William Lowell Putnam Competition for undergraduates. 

Its aim is to present methods of solving functional equations 

and related problems and to provide basic information on its 

history. The intention limits the generality of the treatment; 

more complicated cases are omitted to keep the exposition 

understandable for secondary school students. To give a feeling 

of how it is done I will describe the content of the fi rst 

two chapters. In the introductory part, contributions by Nicole 

Oresme, Gregory of Saint Vincent, Cauchy, d’Alembert, Babbage 

and Ramanujan are presented together with some facts 

from their lives. Also, a simultaneous solution of two equations 

is found and then the terminology is fi xed. The chapter 

ends with some simple problems testing understanding of the 

subject. 

The second chapter starts with the Cauchy equation for 

additive functions and with Jensen’s equation. Some generalizations 

like Pexider’s and Vincze’s equations are treated. 

Cauchy’s inequality for subadditive functions is studied as well 

as the Euler and d’Alembert equations. The author is able in 

such a way to show important types of equations and ways/ 

tricks necessary for dealing with problems containing them. 

The book contains many solved examples and problems at the 

end of each chapter. More than 25 pages at the end of the book 

offer hints and briefl y formulated solutions of those problems. 

Also a short appendix on Hammel basis is included. The book 

has 130 pages, 5 chapters and an appendix, a Hints/Solutions 

section, a short bibliography and an index. It has a nice and 

clear exposition and is therefore a very readable book, accessible 

without special or highly advanced mathematics. The book 

will be valuable for instructors working with young gifted students 

in problem solving seminars. (jive) 

L. Smith: Chaos – A Very Short Introduction, Oxford University 

Press, Oxford, 2007, 180 pp., GBP 6.99, ISBN 978-0-19- 

285378-3 

Chaos is everywhere. It is a behaviour that looks random but is 

not random. It is also an important area of contemporary mathematical 

research, studying situations where very small changes 

at the beginning of a certain process cause huge errors at the 

end. This is not only about weather forecasting or fi ve-year 

economy planning that has been used by communist systems. 

The idea of a small mistake having enormous consequences 

after some time is intuitive, yet it leads to new ways of understanding 

physical sciences. Chaos is a scientifi c discipline, which 

has recently been developing rapidly with interesting facts being 

discovered. It is also littered with pervasive myths such as 

the one about non-predictability of chaotic systems, which calls 

for explanation. 

Everyone interested in such an explanation as well as in 

questions, like whether the fl ap of a butterfl y’s wing can affect a 

hurricane on the other side of the world, should read this book. 

The author provides an excellent introductory explanation to 

the fascinating topic of chaos, on an accessible level and in a 

highly entertaining way. (lp) 

T. Timmermann: An invitation to quantum groups and duality. 

From Hopf algebras to multiplicative unitaries and beyond, 

EMS Textbooks in mathematics, EMS Publishing House, 

Zürich, 2008, 407 pp., EUR 58, ISBN 978-3-03719-043-2 

The Pontrjagin duality for locally compact Abelian groups is 

the best setting for a generalization of classical harmonic analysis 

and Fourier transforms. Dual objects to compact Abelian 

groups are discrete groups. To have a suitable analogue for noncommutative 

locally compact groups, it is necessary to fi nd a 

larger category containing both groups and their duals. Such a 

broader scheme was recently developed under the infl uence of 

new ideas coming from physics (quantum groups). It gives an 

interpretation of quantum groups in the setting of C*-algebras 

and von Neumann algebras. The compact case (developed by 

Woronowicz) is included as a special case. 

This book is devoted to a description of this theory. It has 

three parts. The fi rst part treats quantum groups (and their 

duality) in a purely algebraic setting. It contains a description 

of the Van Daele duality of algebraic quantum groups (giving 

a model for further generalizations) and a discussion of 

the Woronowicz compact quantum groups. In the second part, 

quantum groups are treated in the setting of C*-algebras and 

von Neumann algebras. In particular, it contains a presentation 

of the Woronowicz of C*-algebraic compact quantum groups 

and a study of multiplicative unitaries. The last part contains 

several topics. One is the cross product construction and the 

duality theorem (due to Baaj and Skandalis), the other is pseudo-multiplicative 

unitaries on Hilbert spaces. The last chapter 

contains the author’s results on pseudo-multiplicative unitaries 

on C*-modules. 

The book is nicely written and very well organized. It offers 

an excellent possibility for students and non-experts to learn 

this elegant new part of mathematics. (vs) 

K. Zhu: Spaces of Holomorphic Functions in the Unit Ball, 

Graduate Texts in Mathematics, vol. 226, Springer, Berlin, 2005, 

268 pp., EUR 69,95, ISBN 0-387-22036-4 

This book is devoted to the study of various spaces of holomorphic 

functions on the unit ball. The main tool used for their 

study is an explicit form of the Bergman and Cauchy-Szëgo 

kernels. There is a lot of different spaces of that sort and the 

author has chosen some of them for a detailed study. There is 

always one chapter of the book devoted to one type of function 

spaces. The spaces treated in the book are Bergman spaces, 

Bloch space, Hardy spaces, spaces of functions with bounded 

mean oscillation (BMO), Besov spaces and Lipschitz spaces. 

The author discusses the various characterizations, atomic decompositions, 

interpolations, and duality properties of each 

type of function space in turn. To read the book, knowledge of 

standard complex function theory is expected but the theory 

of several complex variables is not a prerequisite. Results presented 

are standard but their proofs are often new. (vs) 


Birkhäuser Verlag AG 

Viaduktstrasse 42 

4051 Basel / Switzerland 

New 

Series 

Vanishing and 

Finiteness Results in 

Geometric Analysis 

A Generalization of the Bochner 

Technique 

Pigola, S., Università dell‘Insubria, 

Como, Italia / Rigoli, M., Università di 

Milano, Italia / Setti, A.G., Università 

dell‘Insubria, Como, Italia 

2008. XIV, 282 p. Hardcover 

EUR 49.90 / CHF 89.90 

ISBN 978-3-7643-8641-2 

PM — Progress in Mathematics, Vol. 266 

Hemodynamical Flows 

Modeling, Analysis and Simulation 

Galdi, G.P., University of Pittsburgh, 

PA, USA / Rannacher, R., Universität 

Heidelberg, Germany / 

Robertson, A.M., University of 

Pittsburgh, PA, USA / Turek, S., 

Universität Dortmund, Germany 

2008. XII, 501 p. Softcover 

EUR 49.90 / CHF 65.00 

ISBN 978-3-7643-7805-9 

OWS — Oberwolfach Seminars, Vol. 37 

Studies in 

Universal Logic 

Editorial Board Members: 

H. Andréka (Hungarian Academy 

of Sciences, Hungary), M. Burgin 

(University of California, USA), 

R. Diaconescu (Romanian Academy, 

Romania), J. M. Font (University of 

Barcelona, Spain), A. Herzig (CNRS, 

France), A. Koslow (City University of 

New York, USA), J.-L. Lee (National 

Chung-Cheng University, Taiwan), 

L. Maksimova (Russian Academy 

of Sciences, Russia), G. Malinowski 

(University of Lodz, Poland), D. Sarenac 

(Stanford University, USA), 

P. Schröder-Heister (Universität 

Tübingen, Germany), V. Vasyukov 

(Academy of Sciences Moscow, Russia) 

Tel. +41 61 205 07 77 

e-mail: sales@birkhauser.ch 

www.birkhauser.ch 

This book presents very recent results involving 

an extensive use of analytical tools in the study of 

geometrical and topological properties of complete 

Riemannian manifolds. It analyzes in detail an extension 

of the Bochner technique to the non compact setting, 

yielding conditions which ensure that solutions of 

geometrically signi� cant differential equations either 

are trivial (vanishing results) or give rise to � nite 

dimensional vector spaces (� niteness results). The 

book develops a range of methods from spectral theory 

and qualitative properties of solutions of PDEs to 

comparison theorems in Riemannian geometry and 

potential theory. 

All needed tools are described in detail, often with an 

original approach. Some of the applications presented 

concern the topology at in� nity of submanifolds, Lp 

cohomology, metric rigidity of manifolds with positive 

spectrum, and structure theorems for Kähler manifolds. 

The book is essentially self-contained and supplies 

in an original presentation the necessary background 

material not easily available in book form. 

This book surveys results on the physical and mathematical 

modeling as well as the numerical simulation of 

hemodynamical � ows, i.e., of � uid and structural mechanical 

processes occurring in the human blood circuit. The topics 

treated are continuum mechanical description, choice of 

suitable liquid and wall models, mathematical analysis of 

coupled models, numerical methods for � ow simulation, 

parameter identi� cation and model calibration, � uid-solid 

interaction, mathematical analysis of piping systems, 

particle transport in channels and pipes, arti� cial boundary 

conditions, and many more. 

Hemodynamics is an area of active current research, and this 

book provides an entry into the � eld for graduate students 

and researchers. It has grown out of a series of lectures given 

by the authors at the Oberwolfach Research Institute in 

November, 2005. 

This series is devoted to the universal approach to logic and 

the development of a general theory of logics. It covers topics 

such as global set-ups for fundamental theorems of logic and 

frameworks for the study of logics, in particular logical matrices, 

Kripke structures, combination of logics, categorical logic, 

abstract proof theory, consequence operators, and algebraic 

logic. It includes also books with historical and philosophical 

discussions about the nature and scope of logic. Three types of 

books will appear in the series: graduate textbooks, research 

monographs, and volumes with contributed papers. 

Completeness Theory for Propositional Logics 

Pogorzelski, W.A., University of Bialystok, Poland / Wojtylak, P., Silesian 

University Katowice, Poland 

2008. VIII, 178 p. Softcover 

EUR 49.90 / CHF 85.00 / ISBN 978-3-7643-8517-0 

Institution-independent Model Theory 

Diaconescu, R., Simion Stoilow Romanian Academy, Romania 

2008. Approx. 390 p. Softcover 

EUR 69.90 / CHF 125.00 / ISBN 978-3-7643-8707-5 

EUR prices are net prices. CHF prices are gross prices. 

All prices are recommended and subject to change without notice.

ABCD springer.com 

New Textbooks from Springer 

Lectures on Advances in 

Combinatorics 

R. Ahlswede, University of Bielefeld, Germany; 

V. Blinovsky, Russian Academy of Sciences, 

Moscow, Russua 

The book contains a complete presentation of some 

highly sophisticated proofs following new methods 

of pushing and pulling appearing mostly for the 

first time in a book. New in a book are also several 

diametric theorems for sequence spaces, bounds 

for list codes, and even the concepts of higher level 

and dimension constrained extremal problems, and 

splitting antichains. Furthermore for the first time 

some number-theoretical and combinatorial 

extremal problems are treated in parallel and thus 

connections are made more transparent. 

2008. Approx. 310 p. (Universitext) Softcover 

ISBN 978-3-540-78601-6 � € 39,95 | £30.50 

An Introduction to Mathematical 

Cryptography 

J. Hoffstein, J. Pipher, J. Silverman, Brown University, 

Providence, RI, USA 

An Introduction to Mathematical Cryptography 

provides an introduction to public key cryptography 

and underlying mathematics that is required 

for the subject. Each of the eight chapters expands 

on a specific area of mathematical cryptography 

and provides an extensive list of exercises. 

It is a suitable text for advanced students in pure 

and applied mathematics and computer science, 

or the book may be used as a self-study. This book 

also provides a self-contained treatment of 

mathematical cryptography for the reader with 

limited mathematical background. 

2008. XVI, 524 p. 29 illus. (Undergraduate Texts in 

Mathematics) Hardcover 

ISBN 978-0-387-77993-5 � € 34,95 | £26.50 

Analysis by Its History 

G. Wanner, E. Hairer, University of Geneva, 

Switzerland 

This book presents first-year calculus roughly in the 

order in which it first was discovered. The first two 

chapters show how the ancient calculations of practical 

problems led to infinite series, differential and 

integral calculus and to differential equations. The 

establishment of mathematical rigour for these 

subjects in the 19th century for one and several 

variables is treated in chapters III and IV. The text is 

complemented by a large number of examples, 

calculations and mathematical pictures. 

2008. Approx. 390 p. 192 illus. Softcover 

ISBN 978-0-387-77031-4 � € 32,95 | £25.50 

Generalized 

Curvatures 

J. Morvan, Université 

Claude Bernard Lyon 1, 

Villeurbanne, France 

Following a historical and 

didactic approach, the book 

introduces the mathematical 

background of the subject, beginning with 

curves and surfaces, going on with convex 

subsets, smooth submanifolds, subsets of positive 

reach, polyhedra and triangulations, and ending 

with surface reconstruction. It can serve as a 

textbook to any mathematician or computer 

scientist, engineer or researcher who is interested 

in the theory of curvature measures. 

2008. IX, 266 p. 107 illus., 36 in color. (Geometry and 

Computing, Volume 2) Hardcover 

ISBN 978-3-540-73791-9 � € 79,95 | £61.50 

Braid Groups 

C. Kassel, Université Louis Pasteur - CNRS, 

Strasbourg, France; V. Turaev, Indiana University, 

Indiana, US 

In this well-written presentation, motivated by 

numerous examples and problems, the authors 

introduce the basic theory of braid groups, 

highlighting several definitions that show their 

equivalence; this is followed by a treatment of the 

relationship between braids, knots and links. 

Important results then treat the linearity and 

orderability of the subject. Relevant additional 

material is included in five large appendices. Braid 

Groups will serve graduate students and a 

number of mathematicians coming from diverse 

disciplines. 

2008. XII, 340 p. 60 illus. (Graduate Texts in 

Mathematics, Volume 247) Hardcover 

ISBN 978-0-387-33841-5 � € 42,95 | £32.50 

Boundary Integral Equations 

G. C. Hsiao, University of Delaware, Newark, DE, USA; 

W. L. Wendland, University of Stuttgart, Germany 

This book is devoted to the basic mathematical 

properties of solutions to boundary integral 

equations and presents a systematic approach to 

the variational methods for the boundary integral 

equations arising in elasticity, fluid mechanics, and 

acoustic scattering theory. It may also serve as the 

mathematical foundation of the boundary 

element methods. 

2008. Approx. 635 p. (Applied Mathematical Sciences, 

Volume 164) Hardcover 

ISBN 978-3-540-15284-2 � € 79,95 | £60.00 

Catalan’s 

Conjecture 

R. Schoof, Università di 

Roma, Italia 

Eugène Charles Catalan 

made his famous 

conjecture – that 8 and 9 

are the only two consecutive 

perfect powers of natural numbers – in 1844 

in a letter to the editor of Crelle’s mathematical 

journal. One hundred and fifty-eight years later, 

Preda Mihailescu proved it. 

Catalan’s Conjecture presents this spectacular 

result in a way that is accessible to the advanced 

undergraduate. The author dissects both 

Mihailescu’s proof and the earlier work it made use 

of, taking great care to select streamlined and 

transparent versions of the arguments and to keep 

the text self-contained. Only in the proof of 

Thaine’s theorem is a little class field theory used; 

it is hoped that this application will motivate the 

interested reader to study the theory further. 

2008. Approx. 150 p. 10 illus. (Universitext) Softcover 

ISBN 978-1-84800-184-8 � € 39,95 | £27.00 

Subdivision 

Surfaces 

J. Peters, University of 

Florida, Gainesville, FL, 

USA; U. Reif, TU Darmstadt, 

Germany 

This book summarizes the 

current knowledge on the 

subject. It contains both meanwhile classical 

results as well as brand-new, unpublished 

material, such as a new framework for 

constructing C -algorithms. The focus of the book 

2 

is on the development of a comprehensive 

mathematical theory, and less on algorithmic 

aspects. It is intended to serve researchers and 

engineers - both new to the beauty of the subject 

- as well as experts, academic teachers and 

graduate students or, in short, anybody who is 

interested in the foundations of this flourishing 

branch of applied geometry. 

2008. XVI, 204 p. 52 illus., 8 in color. (Geometry and 

Computing, Volume 3) Hardcover 

ISBN 978-3-540-76405-2 � € 49,95 | £38.50 

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