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ContentsEditorial TeamEditor-in-ChiefLucia Di VizioUniversité de Versailles-St QuentinLaboratoire de Mathématiques45 avenue des États-Unis78035 Versailles cedex, Francee-mail: divizio@math.cnrs.frCopy EditorChris Nunn119 St Michaels Road,Aldershot, GU12 4JW, UKe-mail: nunn2quick@qmail.comEditorsJean-Paul Allouche(Book Reviews)CNRS, Math., Équipe Combinatoireet OptimisationUniversité Pierre et Marie Curie4, Place Jussieu, case 18975252 Paris Cedex 05, Francee-mail: allouche@math.jussieu.frMariolina Bartolini Bussi(Math. Education)DESU – Universitá di Modena eReggio EmiliaVia Allegri, 9I-42121 Reggio Emilia, Italybartolini@unimore.itJorge Buescu(Societies)Dep. Matemática, Faculdadede Ciências, Edifício C6,Piso 2 Campo Grande1749-006 Lisboa, Portugale-mail: jbuescu@ptmat.fc.ul.ptEva-Maria Feichtner(Research Centres)Department of MathematicsUniversity of Bremen28359 Bremen, Germanye-mail: emf@math.uni-bremen.deEva MirandaDepartament de MatemàticaAplicada I, EPSEB, Edifici PUniversitat Politècnicade CatalunyaAv. del Dr Marañon 44–5008028 Barcelona, Spaine-mail: eva.miranda@upc.eduMădălina Păcurar(Personal Column)Department of Statistics,Forecast and MathematicsBabeș-Bolyai UniversityT. Mihaili St. 58–60400591 Cluj-Napoca, Romaniae-mail: madalina.pacurar@econ.ubbcluj.roUlf PerssonMatematiska VetenskaperChalmers tekniska högskolaS-412 96 Göteborg, Swedene-mail: ulfp@math.chalmers.seZdzisław PogodaInstitute of MathematicsJagiellonian Universityul. prof. StanisławaŁojasiewicza30-348 Kraków, Polande-mail: zdzislaw.pogoda@uj.edu.plThemistocles M. Rassias(Problem Corner)Department of MathematicsNational Technical Universityof Athens, Zografou CampusGR-15780 Athens, Greecee-mail: trassias@math.ntua.grVolker R. Remmert(History of Mathematics)IZWT, Wuppertal UniversityD-42119 Wuppertal, Germanye-mail: remmert@uni-wuppertal.deDierk SchleicherSchool of Engineering andScienceP.O. Box 750 561University Bremen28725 Bremen, Germanydierk@jacobs-university.deOlaf Teschke(Zentralblatt Column)FIZ KarlsruheFranklinstraße 1110587 Berlin, Germanye-mail: teschke@zentralblatt-math.orgJaap TopUniversity of GroningenDepartment of MathematicsP.O. Box 4079700 AK Groningen,The Netherlandse-mail: j.top@rug.nlScan the QR code to go to theNewsletter web page:http://www.euro-math-soc.eu/node/4230EuropeanMathematicalSocietyNewsletter No. 92, June 2014Editorial: IMAGINARY – Mathematics Communication for the21st Century – G.-M. Greuel, A. D. Matt & A. S. J. S. Mey ......... 3EMS Executive Committee Meeting in London – S. Huggett ......... 7New Editorial Board of the Journal of the EuropeanMathematical Society ............................................................. 8The 1st Caucasian Mathematics Conference – R. Duduchava ..... 9EMS Monograph Award ............................................................ 11The Ferran Sunyer i Balaguer Prize 2014 ................................... 12The Abel Prize Laureate 2014 ................................................... 12On Some Conjectures in Additive Number Theory – P. Mihăilescu.. 13Ideas That Will Outlast Us – L. Vainerman .................................. 16Mathematical Knowledge Management: Transcending theOne-Brain-Barrier with Theory Graphs – M. Kohlhase .............. 22Is a Graviton Detectable? – F. Dyson .......................................... 29Bill Thurston (1946–2012) – J. H. Hubbard ................................ 36Who Are the Invited Speakers at ICM 2014? – M. Andler ............ 38Interview with Alois Kufner on the Occasion of his80th Birthday – L. Pick ......................................................... 45Buffon: Did He Actually Throw Sticks? – E. Behrends & J. Buescu.. 47ICMI Column – M. Bartolini Bussi & and J.-L. Dorier ................... 50ERME Column, May 2014 – V. Durand-Guerrier ......................... 52Guarding your Searches: Data Protection at zbMATH –J. Holzkämper & O. Teschke .................................................. 54Book Reviews ........................................................................... 56Letter to the Editor – M. Ferri .................................................... 59Personal Column ...................................................................... 60The views expressed in this Newsletter are those of theauthors and do not necessarily represent those of theEMS or the Editorial Team.ISSN 1027-488X© 2014 European Mathematical SocietyPublished by theEMS Publishing HouseETH-Zentrum SEW A27CH-8092 Zürich, Switzerland.homepage: www.ems-ph.orgFor advertisements and reprint permission requestscontact: newsletter@ems-ph.orgEMS Newsletter June 2014 1

Editorial: IMAGINARY – MathematicsCommunication for the 21st CenturyEditorialGert-Martin W. Greuel (University of Kaiserslautern, Germany), Andreas Daniel Matt (Mathematisches ForschungsinstitutOberwolfach, Germany) and Antonia S. J. S. Mey (Free University Berlin, Germany)IMAGINARY is the name of a collaborative mathematicsoutreach project that aims to improve the image andunderstanding of mathematics and in this way awakean interest and fuel passion for the subject in childrenand adults. This goal is achieved in different ways: on theone hand by showing the beauty and art in mathematicsand on the other hand through surprising applications.To best understand the project we have to go back to itsbeginning.IMAGINARY was born at the Mathematische ForschungsinstitutOberwolfach (MFO) in conjunction withthe Year of Mathematics in 2008 in Germany. It startedwith the travelling exhibition “IMAGINARY – throughthe eyes of mathematics” shown in 12 German cities. Dueto its tremendous success, follow up exhibitions weresoon organised in Austria, Switzerland, Spain, UK andUkraine. The program SURFER, developed for IMAGI-NARY, became a centrepiece of the exhibition. It teachesin a playful way the connection between formula andform, between algebra and geometry through beautiful3D surfaces. In this way, it bridges the gap between artand mathematics. An example of such a surface, Citrus,is seen on the front cover of this issue. The visitors of theexhibition get the chance to alter the algebraic equations,see the effects on the displayed surfaces in real time andeven get to take a printout back home.What made the exhibition unique from the beginningis its highly interactive and intuitive nature and its openaccess and open source philosophy. This is also reflectedin the many positive comments left in the guest book:- This already beautiful exhibition is obtaining a specialliveliness by excellent leadership.- Mathematics makes happy.- Super, especially that you can also use the program inthe school.- A wonderful exhibition. I have spent much time hereand met many beautiful things, it had to take placemore often and actually as a permanent event!- Thank you and keep it up!- It is a fantastically beautiful exhibition.- The magic world of mathematics is not easy to understand.But you can bring them closer.- We were again there, because it was so fascinating.- I should have perhaps studied math…- Simply gorgeous, cool programs.IMAGINARY exhibitions and events have been organised in morethan 120 cities.IMAGINARY exhibition at the Leibniz University, Hannover, 2009.Since 2008, the IMAGINARY exhibition has beenshown in over 60 cities in Germany alone but has alsotravelled further afield to 4 continents, 29 countries andover 120 cities with more than 1 million visitors in total.In Europe, IMAGINARY has been presented in 17countries with talks, workshops, media activities and, inmost cases, exhibitions.However, the original exhibition was not enough; it focusedon a very beautiful yet small part of mathematics.The project needed to grow further and the Mathematicsof Planet Earth Year 2013 (MPE) presented a good opportunityto do so. A competition for virtual exhibitionmodules themed around MPE was announced and IM-AGINARY offered to provide the required web infrastructurein order to make the modules of the competitionavailable online. At the launch of the MPE year inEurope at the UNESCO in Paris, the web interface toIMAGINARY (imaginary.org) went live, displaying entriesfor the competition and, of course, the winners.The web platform provides an online resource formathematics outreach. One might think of it as a “pickEMS Newsletter June 2014 3

Editorialand mix science stand”, like a pick and mix sweets standbut instead of sweets one gets to choose between differentmathematical tools, all for free. Everything should beeasy to digest but also awake an interest to learn moreabout the non-trivial mathematics behind it.The tools are grouped into different categories,such as exhibitions, galleries, films and hands-on exhibitswith instructions on how to recreate them at home,school or university. Currently imaginary.org hosts twofull exhibitions: the “IMAGINARY – through the eyesof mathematics” original travelling exhibition is availableand free to download; at the same time, a completeMPE exhibition is also available, consisting of a series ofmodules with a more applied mathematics focus, such asa program that calculates the displacement of volcanicash clouds (Dune Ash) or a film discussing how mathematicalmodelling of glacial movement works in orderto predict the future behaviour of glaciers. Of course, exhibitsfrom both exhibitions may be mixed. Furthermore,IMAGINARY also ventured into school education. InDecember 2013 the so-called ENTDECKERBOX (discoverybox) was launched. It is primarily aimed at usein the classroom and provides resources for teachers inorder to make mathematics lessons more interactive andinteresting for the pupils.Interactive station Dune Ash at the UNESCO in Paris, 2013, simulatingvolcanic ash dispersion in real timeNine programs and films included in the IMAGINARY discovery boxfor schools.The Björling minimal surface by Ulrich Pinkall – part of the“IMAGINARY – through the eyes of mathematics” exhibition.Who is behind IMAGINARY nowadays? IMAGINARYis a project by the MFO, accounted by its director GerhardHuisken, with funding from the Klaus Tschira Stiftung.It is maintained by a committed core team (mathematicians,software engineers, graphic designers, etc.),who run the project, develop the internet platform andgive advice on how to coordinate exhibitions, but alsodream up new ventures of where IMAGINARY will goin the future. At the same time, and most importantly, itis community driven. This means that anyone who has aninteresting piece of software, film or other type of interactivematerial can upload this to the website and makeit available to the rest of the community. In this way, thecommunity becomes an integral part in the communicationprocess by not only experiencing but also creatingcontent and thus advancing mathematics communicationto the 21st century. Of course, anyone can just use thematerial and create a mathematics event, exhibition orworkshop. Due to this community driven aspect of thewebsite, the German news outlet “Spiegel ONLINE”called it the “YouTube of Mathematics”.As a result of its success in general, and particularlyin the last two years, IMAGINARY has seen a lot of mediacoverage across Europe for its numerous successfulexhibitions. Gert-Martin Greuel, the former director ofMFO and scientific advisor of IMAGINARY, and AndreasDaniel Matt, the curator and project manager ofIMAGINARY, both also initiators of the project, wereawarded the German Media Prize for Mathematics inNovember 2013. In the first half of 2014, a new exhibitionwas launched at the Technische Museum in Berlin, showingthree MPE related pieces until 24 June 2014. Furthermore,IMAGINARY has started a collaboration with theAfrican Institute for Mathematical Sciences (AIMS)and, in association with AIMS, an interactive IMAGI-NARY event was organised for the first time in Africa4 EMS Newsletter June 2014

EMS NewsSURFER installation at the IMAGINARY exhibition in Belgrade, 2012. Cedric Villani inaugurating the IMAGINARY exhibition in Paris, 2010.at the 10th anniversary pi-day celebrations in Dar esSalaam, Tanzania. The event was attended by over 2000schoolchildren and their teachers from three universities,33 secondary schools and 14 primary schools. In November2014, a workshop and exhibition will be organisedin Cape Town to plan future mathematics communicationactivities with partners on the African continent. Atthe same time, IMAGINARY’s travelling exhibitions arecurrently on tour or planned for the coming months inGermany, Russia, Spain, Norway, Portugal and Hungary.But most certainly, the planned highlight is an IMAGI-NARY exhibition organised by the NIMS institute at theICM in Seoul, South Korea, in August, which will be thelargest IMAGINARY exhibition yet, featuring all modulesof previous years, as well as new software, images,films and sculptures. Following the IMAGINARY philosophy,we invite all mathematicians to send us their latestimages, films or software programs to be included atthe ICM exhibition.The IMAGINARY pick and mix science stand hasbecome a franchise (non-commercial and open), whereyou can taste new maths and bring your own ideas. Ithas grown from a single exhibition into a collaborativeframework and community driven movement withIMAGINARY teams in several countries. The RoyalSpanish Mathematical Society (RSME), for example,has already organised more than 15 exhibitions andseveral museum installations. The question now is howdoes IMAGINARY continue? A new project is to connectmodern mathematics and current research to outreach.Mathematicians visiting the MFO are asked towrite about their current work but for a general public.These so-called “snapshots of modern mathematics”are then reviewed and edited and distributed throughthe project. Another new idea is to increase networkingbetween mathematics communicators. To kick startThe Lorenz attractor by Jos Leys, an image submitted at the MPEcompetition. Young IMAGINARY fans, 2008.EMS Newsletter June 2014 5

EMS Newsthis, an open map of all mathematics museums on ourplanet was prepared – yes, there are many more thanone might think!IMAGINARY is constantly evolving and always welcomesexciting new contributions and ideas (for examplethe game http://2048.imaginary.org).The Newsletter thanks IMAGINARY, and in particularAndreas Daniel Matt, for providing the cover illustration.Gert-Martin Greuel is a professor at theUniversity of Kaiserslautern. His researchinterests are algebraic geometry, singularitytheory and computer algebra. He is one ofthe authors of the computer algebra systemSingular. Since January 2012, he has beenEditor-in-Chief of Zentralblatt MATH. UntilMarch 2013, he was the Director of theMathematisches Forschungsinstitut Oberwolfach (MFO)and at present he is the scientific adviser of IMAGINARY.Andreas Daniel Matt is a mathematicscommunicator at the Mathematisches ForschungsinstitutOberwolfach. He studiedmathematics and computer science and didhis PhD in mathematics in machine learningat the University of Innsbruck and theUniversity of Buenos Aires. Since 2007, hehas coordinated the project IMAGINARY.He also curates the MiMa Museum for Minerals andMathematics, Oberwolfach.Antonia S. J. S.Mey is a postdoctoral researcherin the Mathematics and ComputerScience Department at the Free University,Berlin. She completed her PhD at the Universityof Nottingham in theoretical physics.Her research interests lie in appliedmathematics and focus on stochastic estimatorsand Markov state models and theirapplications to biological problems. Since 2014, she hasbeen a member of the IMAGINARY team.AMS-EMS-SPM InternationalMeeting 2015,10–13 June 2015, Porto, PortugalThe meeting will bring together the American Mathematical Society (AMS), the European Mathematical Society(EMS) and the hosting Portuguese Mathematical Society (SPM) in the UNESCO world heritage city ofPorto.The scientific scope of this meeting ranges from plenary talks of general interest to special sessions focusing oncurrent research in specific areas. There will be distinguished Invited Addresses, an evening Public Lecture anda variety of Special Sessions.List of invited speakers:Marcus du Sautoy (evening Public Lecture), University of Oxford, UKRui Loja Fernandes, University of Illinois, USAIrene Fonseca, Carnegie Mellon University, USAAnnette Huber-Klawitter, Albert-Ludwigs-Universität Freiburg, GermanyMikhail Khovanov, Columbia University, USAAndré Neves, Imperial College London, UKSylvia Serfaty, Université Pierre et Marie Curie Paris 6, FranceGigliola Staffilani, MIT, USAMarcelo Viana, Instituto de Matemática Pura e Aplicada, Rio de Janeiro, BrazilFor more information see http://aep-math2015.spm.pt/6 EMS Newsletter June 2014

EMS NewsEMS Executive Committee Meeting inLondon from the 28th of February tothe 2nd of March 2014Stephen Huggett (University of Plymouth, UK)MembershipThe Treasurer reported that at the Council meeting thisyear several Societies (hence, Full Members) would beapplying to move up a class. The Executive Committeewas also pleased to agree to elect the European SetTheory Society to Associate Membership of the EMS,and to approve the list of about 80 new individual members.The Publicity Officer Dmitry Feichtner-Kozlov reportedthat, as planned, a selection of institutions in theUK, France, and Germany had been sent a letter and aposter, inviting them to consider joining the EMS. ThePresident reported that she had done the same in Spain.This process would continue with Italy and Holland.The webThe consultant, Robert Carr, presented his proposal fora redesign of the EMS web site. There would be a prototypeready soon, and he requested feedback on it. Thefinished site would be ready to show Council. The ExecutiveCommittee approved this process, and thankedRobert.Lucia Di Vizio presented a paper giving detailed usagestatistics for our Newsletter presence on both Facebookand Twitter. Her report was welcomed by the ExecutiveCommittee.Scientific MeetingsVolker Mehrmann reported that in addition to the usualmathematical programme, the 7th European Congressof Mathematics would include a mathematical film festival,a public lecture on mathematics and architecture,a student conference, and historical lectures and walks.It was agreed that the membership of the NeugebauerPrize Committee can be made public, as can the Chair ofthe Klein Prize Committee. The Treasurer reported thatFoundation Compositio Mathematica is happy to fundabout half of the cost of the EMS prizes. The membershipof the EMS Prize Committee was discussed. Finally,it was agreed that Michael Atiyah and Jean-Pierre Bourguignonbe suggested as the EMS representatives on theHirzebruch Lecture panel.The Executive Committee discussed the First JointInternational Meeting RSME–SCM–SEMA–SIMAI–UMI, to be held in Bilbao from June 30th to July 4th 2014,which looks very strong. Jouko Väänänen suggested thatthe EMS organise a lecture about (and visit to) the RichardSerra sculptures in the Guggenheim Museum.Armen Sergeev reported on the Caucasian MathematicsConference, to be held in Tbilisi, Georgia, on the5th and 6th of September 2014 under the auspices of theEuropean Mathematical Society and the cooperationof the Armenian, Azerbaijan, Georgian, Iranian, Russianand Turkish Mathematical Societies.The six invitedspeakers, one from each of the six participating countries,had now been finalised. Funds from the EMS SolidarityCommittee would be used to support some of the invitedspeakers and two young mathematicians from eachcountry.EMS Council 2014The President reported that local preparations for the2014 meeting of Council in Donostia/San Sebastián werevery good, and that details were almost ready to be publishedon the Council web site. The Executive Committeethen considered whether to propose to Council anyamendments to the Statutes. It was agreed that the By-Laws 10–12 might be reviewed, in order to ensure thatdelegates were members of the EMS. After a detaileddiscussion of the membership fees it was agreed to proposesmall increases for both corporate and individualmembers. The draft agenda for Council was agreed inoutline, except that it should include a presentation ofthe web site, which would form part of the report fromthe Executive Committee.Standing CommitteesJosé Antonio Carrillo presented the report of the AppliedMathematics Committee, noting in particular thatthe Bedlewo Summer School is now confirmed. ThenVolker Mehrmann reported on the EU-MATHS-INproject, which is applying for support under COST.Alice Fialowski presented the report from the Chairof the Committee for Developing Countries, noting thatnone of the new Emerging Regional Centres of Excellenceunder consideration is from Africa.The Treasurer introduced the draft position paper onOpen Access from the Publications Committee. It wasagreed that this was very important, but the ExecutiveCommittee would only comment in detail once a morenearly final draft had been received. Gert-Martin Greuelobserved that it was unique to this paper that the role oflibraries was analysed.Gert-Martin Greuel reported that IMAGINARYwould be having an exhibition at the ICM in Seoul andat the next ECM in Berlin, and the platform www.im-EMS Newsletter June 2014 7

EMS Newsaginary.org is now available in German and Spanish (besidesEnglish); translations into French and Catalan areon the way.PublishingIt was noted that the Proceedings of the 6th EuropeanCongress of Mathematics had now been published. TheExecutive Committee agreed the details of the renewalof the Editorial Board of Journal of the European MathematicalSociety.The President reported that the committee awardingthe EMS Monograph Prize was unanimous in deciding tomake a joint award to two of the submissions:- Foundations of Garside Theory, by Patrick Dehornoywith François Digne, Eddy Godelle, Daan Krammer,and Jean Michel- Topics in Elliptic PDEs and Measure Theory, by AugustoPonceClosingThe next meeting will be a short one just before the Councilin Donostia/San Sebastián. The autumn meeting willbe in Barcelona from the 21st to the 23rd of November.The President expressed the gratitude of the wholeExecutive Committee to the London Mathematical Societyfor its hospitality.New Editorial Board of the Journal ofthe European Mathematical SocietyThe Executive Committee of the EMS has appointedthe new editorial board of its flagship journal JEMS. Thecomposition of the new editorial board is as follows:Editor-in-Chief:François Loeser, Université Pierre et Marie Curie, Paris,FranceEditors:Kari Astala, University of Helsinki, FinlandAnnalisa Buffa, CRN Pavia, ItalyTom Bridgeland, University of Oxford, UKXavier Cabré, ICREA and Universitat Politècnica deCatalunya, Barcelona, SpainHélène Esnault, Freie Universität Berlin, GermanyMichael Harris, Columbia University, USA, andUniversité Paris Diderot, FranceJános Kollár, Princeton University, USAKarl Kunisch, University of Graz, AustriaMichel Ledoux, Université de Toulouse, FrancePhilip Maini, University of Oxford, UKGünter Malle, University of Kaiserslautern, GermanyPhilippe Michel, École Polytechnique Féderale,Lausanne, SwitzerlandJaroslav Nešetřil, Charles University, Prague, CzechRepublicLaure Saint Raymond, École Normale Supérieure deParis, FrancePaul Seidel, Massachusetts Institute of Technology, USASylvia Serfaty, Université Pierre et Marie Curie, Paris,FranceStanislav Smirnov, University of Geneva, SwitzerlandJan Philip Solovej, University of Copenhagen, DenmarkBertrand Toën, Université de Montpellier, FranceXavier Tolsa, ICREA and Universitat Autònoma deBarcelona, SpainGiuseppe Toscani, University of Pavia, ItalySara van de Geer, ETH, SwitzerlandMarcelo Viana, IMPA, BrazilAlex Wilkie, University of Manchester, UKBurkhard Wilking, University of Münster, GermanyUmberto Zannier, Scuola Normale Superiore, Pisa, ItalyTamar Ziegler, The Hebrew University of Jerusalemand Technion, Haifa, IsraelThe Editor-in-Chief and the editors have an initial termof four and three years, respectively, with the possibilityof an extension for a second term. From 1 June 2014,authors submitting articles to the journal will be directedto an electronic submission system on the journal’shomepage and the new editorial board will handle thesearticles.Created by the EMS in 1998, the first volume of JEMS,published by Springer-Verlag, appeared in January 1999,with Jürgen Jost as Editor-in-Chief. Since January 2003,the Editor-in-Chief has been Haim Brézis. With the creationof the EMS Publishing House, the publication ofJEMS was moved from Springer to the EMS publisherin 2004.Today, JEMS is one of the strongest mathematicaljournals, thanks to the skill and dedication of the firsttwo editorial boards. In this transition period, the EMSwould like to fully recognise and acknowledge the extraordinarywork of the departing editorial board:Haim Brezis (Editor-in-Chief), Université Pierre etMarie Curie, Paris, France, Rutgers University, USA,and Technion, Haifa, IsraelAntonio Ambrosetti, SISSA, Trieste, ItalyEnrico Arbarello, Università di Roma “La Sapienza”,ItalyRobert John Aumann, The Hebrew University ofJerusalem, IsraelHenri Berestycki, EHESS, Paris, France8 EMS Newsletter June 2014

EMS NewsFabrice Bethuel, Université Pierre et Marie Curie, Paris,FranceJean Bourgain, Institute for Advanced Study, Princeton,USAJean-Michel Coron, Université Pierre et Marie Curie,Paris, FranceJesús Ildefonso Díaz, Real Academia de Ciencias, SpainCorrado De Concini, Università di Roma “LaSapienza”, ItalySimon Donaldson, Imperial College, London, UKYasha Eliashberg, Stanford University, USAGeoffrey Grimmett, Cambridge University, UKMarius Iosifescu, Romanian Academy, Bucharest,RomaniaSergiu Klainerman, Princeton University, USAEduard Looijenga, Utrecht University, the NetherlandsAlex Lubotzky, The Hebrew University of Jerusalem,IsraelAngus Macintyre, Queen Mary, University of London,UKIb Madsen, University of Copenhagen, DenmarkJean Mawhin, Université Catholique de Louvain,BelgiumStefan Müller, University of Bonn, GermanyBart de Smit, Universiteit Leiden, the NetherlandsH. Mete Soner, ETH Zürich, SwitzerlandAlain Sznitman, ETH Zürich, SwitzerlandMina Teicher, Bar-Ilan University, Ramat-Gan, IsraelClaire Voisin, IHES, Bures sur Yvette, FranceThis board is responsible for the handling of papers submittedto JEMS before 1 June 2014 and will terminate itsfunctions by September 2015.The EMS is greatly indebted to the departing editorialboard, which under the leadership of Professor Brézishas raised the journal to its current high rank. At the sametime, the EMS expresses its gratitude to the new editorialboard for having accepted this important responsibilityand warmly welcomes all its members.EMS Executive CommitteeMay 2014The First Caucasian MathematicsConference (CMC I)Roland Duduchava (Ivane javakhishvili Tbilisi State University, Georgia)with EMS-EC member Armen Sergeev. An organisationalmeeting in January 2014 in Istanbul gave the first shape tothe idea. The three initiators mentioned were joined by therep resentatives of the participating countries and institutions:How much can mathematicians contribute to peace andharmony in the world? At a time when the region is surroundedby old and new tensions, the first CaucasianMathematics Conference (CMC I) is being organisedin Tbilisi, Georgia, 5–6 September 2014, under the auspicesof the European Mathematical Society and withthe cooperation of the Armenian, Azerbaijan, Georgian,Iranian, Russian and Turkish Mathematical Societies. Althoughit is welcoming mathematicians from all over theworld, the first aim of the Caucasian Mathematical Conferenceis to bring together mathematicians from Caucasianand neighbouring countries annually in one of thesecountries.The idea of this gathering was expressed by Betül Tanbayat the EMS Presidents Meeting in Aarhus in 2013and, encouraged by Marta Sanz-Solé, the presidents of theGeorgian and Turkish Mathematical Societies Roland Duduchavaand Betül Tanbay started the preliminary workCarles Casacuberta (ex-officio; Chair of the EMSCommittee for European Solidarity),Mohammed Ali Dehghan (President of the IranianMathematical Society),Tigran Harutyunyan (President of the ArmenianMathematical Union),Misir Jumayil oglu Mardanov (Representative of theAzerbaijan Mathematical Union) andMarta Sanz-Solé (ex-officio; President of the EuropeanMathematical Society),forming together a Steering Committee responsible forthe scientific programme of the conference. The committeehas selected the invited spea kers:Maria Esteban (C.N.R.S. at the University Paris-Dauphine),Mohammad Sal Moslehian (Ferdowsi University ofMashhad),Garib N. Murshudov (MRC Laboratory of MolecularBiology, Cambridge),Dmitri Orlov (Steklov Institute, Moscow),EMS Newsletter June 2014 9

EMS NewsSamson Shatashvili (Trinity College, Dublin),Leon Takhtajan (University of Stony Brook) andCem Yalçin Yildirim (Bo ğaziçi University, Istanbul).A list of invited young mathematicians (under 40) will beselected by the National Mathematical Societies and approvedby the Steering Committee. Roland Duduchavawas appointed the Chairman of the CMC I and the LocalOrganising Committee consists of Buchukuri Tengiz,Davitashvili Tinatin (Scientific Secretary), Eliashvili Merab,Natroshvili David (Vice Chairman) and Sigua Levan.The programme consists of invited speakers’ talks andparallel sessions. Participants wishing to organise a parallelsession are invited to send applications before 15 June2014. The organiser of such a session should present togetherwith its title a preliminary list of participants with atleast five speakers.The conference fee is 20 USD. The organisers will applyfor financial support from different foundations and,in case of success, will try to pay accommodation for theselected young partici pants.The online registration page and all information aboutthe conference can be found at euro-math-soc.eu/cmc/ orhttp://www.gmu.ge/cmc and contact email addresses arecmc.tbilisi@gmail.com and cmc.tbilisi@gmu.ge.Citizens of many countries do not need a visa for Georgia(EU countries, USA, Japan, Israel, Turkey, Canada, Armenia,Azerbaijan, Mol do va, Russia, Ukraine, etc.) or elsemany can get a visa when entering Georgia.Tbilisi can be reached by aeroplane from most of thebig airports of Europe, Turkey and the former Soviet republics(see: http://www.tbilisiairport.com/en-EN/).Immediately after the CMC I, the fifth InternationalConference of the Georgian Mathematical Union willbe held in Batumi, Georgia, 8–12 September 2014 (see:http://www.gmu.ge/Batumi2014).New Service by EU-MATHS-INThe association EU-MATHS-IN (promoted by the EMS and the ECMI) has launched a new service: a websitefor advertising jobs for mathematicians in companies or institutions working on industrial contracts. The scopeof the website is advertising jobs in industrial mathematics and related fields.Please circulate this to all interested people, industrial contacts and young mathematicians looking for jobs.Job announcements can be found or deposited athttp://www.eu-maths-in.eu/jobs10 EMS Newsletter June 2014

EMS NewsEMS Monograph AwardThe EMS Monograph Award is assigned every two years to the author(s) of a monograph in any area of mathematics thatis judged by the selection committee to be an outstanding contribution to its field. The prize is endowed with 10,000 Euroand the winning monograph will be published by the EMS Publishing House in the series “EMS Tracts in Mathematics”.The first EMS Monograph Award (2014) has been assigned jointly toPatrick Dehornoy (Université de Caen, France)with François Digne (Université de Picardie Jules-Verne, Amiens, France), Eddy Godelle (Université de Caen, France),Daan Krammer (University of Warwick, Coventry, UK) and Jean Michel (Université Denis Diderot Paris 7, France)for their workFoundations of Garside TheoryandAugusto C. Ponce (Université Catholique de Louvain, Belgium)for his workElliptic PDEs, Measures and Capacities – From the Poisson Equation to Nonlinear Thomas–Fermi ProblemsScientific CommitteeJohn Coates, Pierre Degond, Carlos Kenig, Jaroslav Nesetril, Michael Roeckner, Vladimir TuraevSubmissionThe second award will be announced in 2016, the deadline for submissions is 30 June 2015.The monograph must be original and unpublished, written in English and should not be submitted elsewhere until aneditorial decision is rendered on the submission. Monographs should preferably be typeset in TeX. Authors should senda pdf file of the manuscript by email and a hard copy together with a letter to:European Mathematical Society Publishing HouseETH-Zentrum SEW A27, Scheuchzerstrasse 70, CH-8092 Zürich, SwitzerlandE-mail: info@ems-ph.orgEMS Tracts in MathematicsEditorial Board:Carlos E. Kenig (University of Chicago, USA)Andrew Ranicki (University of Edinburgh, UK)Michael Röckner (Universität Bielefeld, Germany, and Purdue University, USA)Vladimir Turaev (Indiana University, Bloomington, USA)Alexander Varchenko (University of North Carolina at Chapel Hill, USA)This series includes advanced texts and monographs covering all fields in pure and applied mathematics.Tracts will give a reliable introduction and reference to special fields of current research.The books in the series will in most cases be authored monographs, although edited volumes maybe published if appropriate. They are addressed to graduate students seeking access to researchtopics as well as to the experts in the field working at the frontier of research.Most recent titles:Vol. 21 Kaspar Nipp and Daniel Stoffer: Invariant Manifolds in Discrete and Continuous Dynamical Systems (see also page 68)978-3-03719-124-8. 2013. 225 pages. 58.00 EuroVol. 20 Hans Triebel: Local Function Spaces, Heat and Navier–Stokes Equations978-3-03719-123-1. 2013. 241 pages. 64.00 EuroVol. 19 Bogdan Bojarski, Vladimir Gutlyanskii, Olli Martio and Vladimir Ryazanov: Infinitesimal Geometry of Quasiconformal andBi-Lipschitz Mappings in the PlaneISBN 978-3-03719-122-4. 2013. 214 pages. 58.00 EuroEMS Newsletter June 2014 11

NewsThe Abel Prize Laureate 2014The Norwegian Academy of Science and Letters has awarded the Abel Prize for 2014 toYakov G. Sinaiof Princeton University, USA, and the Landau Institute for Theoretical Physics, RussianAcademy of Sciences,“for his fundamental contributions to dynamical systems, ergodic theory and mathematicalphysics”.Photo: Archives of the Mathematisches Forschungsinstitut OberwolfachThe Ferran Sunyer i Balaguer Prize 2014The Ferran Sunyer i Balaguer Prize 2014 winnerswere:Professors Véronique Fischer and MichaelRuzhansky (Imperial College London), for theworkQuantization on Nilpotent Lie GroupsAbstract: The theory of quantisation originated at theend of the 1920s in mathematical physics. The mathematicalaspect of this theory concerned with theanalysis of operators has been intensively developedover the past 50 years. Nowadays, it is widely acceptedand used as a powerful tool to study different questionsin the theory of partial differential equations.However, the settings of these studies have beenmainly Euclidean and commutative in nature. Thismonograph presents the recent advances of the theoriesof quantisation and pseudo-differential operatorson compact and nilpotent Lie groups. It also containsapplications to non-commutative global analysis andto partial differential equations, in particular on theHeisenberg group.This monograph will be published by Springer Baselin their Birkhäuser series Progress in Mathematics.Call for theFerran Sunyer iBalaguer Prize 2015The prize will be awarded for a mathematicalmonograph of an expositorynature presenting the latest developmentsin an active area of research inmathematics.The prize consists of 15,000 Euros andthe winning monograph will be publishedin Springer Basel’s Birkhäuserseries “Progress in Mathematics”.DEADLINE FOR SUBMISSION:2 December 2014http://ffsb.iec.cat12 EMS Newsletter June 2014

On Some some Conjectures inAdditive in Additive Number Number Theory TheoryPreda Mihăilescu (Universität Göttingen, Germany)FeatureNumber theory is noted for problems that are easy to formulatebut not as easy to solve. It is also a fact that mathematicianslike to make progress. So, if involved in this field, ratherthan collapse under the weight of simple questions hundredsof years old, they will typically either develop interesting theorieswhich gain a life of their own and eventually pay offwith some solution of the old problems – at least partial, if notcomplete, as in the case of Fermat’s Last Theorem – or theyimagine variants. In this short presentation, these processeswill be illustrated with a family of number theoretical problemsconcerned with the distribution of primes and primetuples,an area which has experienced dramatic progress inthe last decade.1 The Cramér Model and gaps between primesThe distribution of prime numbers is one of the most relevantquestions of analytic number theory, and it can be a specialisationof arbitrary depth. We fix a notation and write P ⊂ Nfor the set of rational primes and let their sequence be numeratedby p 1 = 2, p 2 = 3,...,p n ,...∈ P. The counting functionis π(x) = |P ≤x | and the prime number theorem states thatlimn→∞π(x)x/ log(x) = 1,which suggests that typically we should havep n+1 − p n ∼ O(log(n)).To try to make the meanings of the terms precise and to explainwhy one expects a certain average behaviour, Cramérused the following model, which has become a reference eversince. To quote its description from Soundararajan’s presentation[9]: The indicator function for the set of primes (thatis, the function whose value at n is 1 or 0 depending onwhether n is prime or not) behaves roughly like a sequence ofindependent, Bernoulli random variables X(n) with parameters1/ log n(n ≥ 3). In other words, for n ≥ 3, the randomvariable X(n) takes the value 1 (n is prime) with probability1/ log n, and X(n) takes the value 0 (n is composite) withprobability 1 − 1log n. For completeness, let us set X(1) = 0 andX(2) = 1. This probabilistic approach awakens some reasonablereserve: numbers are not almost prime or almost composite;they are either one or the other. However, it turned outthat the model leads to some astonishing insights and suggestionsconcerning the behaviour of primes at large. One of theimmediate consequences is related to the interval among consecutiveprimes. The following conjecture is deduced fromthe assumption underlying Cramér’s model:Conjecture 1. Given an interval 0 ≤ α c log(2) (p n ) · log (4) (p n )log (3) (p n ) ,2for infinitely many values of n > 1; here, the upper index ofthe (natural) logarithm indicates the iteration count, thuslog (2) (p n ) = log(log(p n )).This is hardly the largest size of gap that one would expect;on the contrary, the limitlim supn→∞δ(n)log(p n ) = 1is suggested by the model. However, neither empirical computationsnor rigorous proofs have been made so far to providelarger lower bounds than Rankin’s. Erdős offered a prizeof $10, 000 for the first person to provide an improvement ofRankin’s results but the prize has not been challenged to date.As already mentioned, the Cramér model cannot be expectedto provide a completely consistent picture of the distributionof primes. According to the model, this distributionshould be Poisson – however, Helmut Maier proved [7]in 1985 a theorem which reflects, in an impressively simplemanner, the deviation of the set of primes from the predictedPoisson distribution:Theorem 1. For any λ>1, we defineg(λ, x) = π x + (log(x)) λ − π(x).(log(x)) λ−1Thenlim inf g(λ, x) < 1 < lim sup g(λ, x).x→∞x→∞Since within the Cramér model the limit should exist andbe 1, the theorem makes the limitations of the model concrete.EMS Newsletter June 2014 13

FeatureSmall gapsMoving on to small gaps, we encounter one of the fascinatingmathematical thrillers of the last decade. Note first that theprime gaps arep n+1 − p n ≥ 2,the value 2 corresponding to the prime twins. Thus provingthe expected fact that there exist an infinity of prime twinswould be solving one of the important conjectures concerningsmall prime gaps. But evenlim inf δ(n) = 0n→∞was not known in 2002 when Goldston announced, on behalfof himself and Yildirim, an amazing result: he claimed theproof thatδ(n)/ log(n) 1/16 < 1for infinitely many primes and thus a sharpening of the vanishingof the limes inferior, a sharpening which is muchstronger than the corresponding result of Rankin for largegaps. The proof was unfortunately erroneous and was forgottenfor several years, until the same authors, together with J.Pintz, made the breakthrough in 2005, giving a correct proofoflim inf δ(n) = 0.n→∞While the three authors joined forces with Graham, consideringvarious related problems, the next major breakthroughfollowed in 2013 and was achieved by Y. Zhang, who refinedthe method of Goldson, Pintz and Yildirim, thus proving afixed upper boundlim inf (p n+1 − p n ) < 7 · 10 7 .n→∞The upper bound has since been successively improved overthe whole year and, at the time of writing, had dropped below5 · 10 5 ; see [1] for a tabular presentation of the detailedevolution of the proven bounds.The repartition of primes can be considered on a more detailedbasis by asking the question about repartition of primesin arithmetic sequences; concretely, given two coprime integersm, a, one wishes to gather information about the functionπ(x; m, a) = |{p ∈ P ≤x : p ≡ a mod m}|.Dirichlet had already established the asymptotic behaviourϕ(m) · π(x; m, a)lim= 1x→∞ π(x)but more subtle questions exist concerning, for instance, theasymptotics of the difference|π(x; m, a) − π(x; m, b)|.An important conjecture due to Elliott and Halberstam refersto these oscillations; more precisely, considering the maximaldeviationE(x; m) = maxπ(x) (a,m)=1π(x; m, a) −ϕ(m) ,Elliott and Halberstam stipulate the following.Conjecture 2. For every θ with 0 < θ < 1 and for everyA > 0 there exists a constant C > 0 such thatCxE(x; m) ≤log A (x) .1≤m≤x θThe conjecture has been proven for all θ

Featureist arbitrarily long arithmetic progressions consisting only ofprimes. As one would expect, the method leads to new questions– one of them is related to the case when k polynomialsp i (x) ∈ Z[X], i = 1, 2,...,k, are given and one wishes to knowif there are infinitely many integers x, m such that x + p i (m)are all simultaneously prime. The result was established in2006 by Tao and Ziegler and it obviously implies the one onarithmetic progressions.The problem is more intricate if one wants to count thefrequency of arithmetic sequences or, more generally, of givenpatterns of primes. In this context, an accurate analysis basedon the Cramér model leads to the famous conjecture of Hardyand Littlewood: for integer sets H = {h 1 , ..., h k } they first defineda constant Σ(H) ∈ R ≥0 , which vanishes if there is no(k + 1)-tuple of primes with mutual distances h i ; otherwise itis a constant which is deduced from the Poisson distributionof gaps (see [9] for more details), such that the following isconsistent with the Cramér model:Conjecture 3. Let H = {h 1 , ..., h k } be a set of positive integerssuch that Σ(H) 0. Thenx|{n ≤ x : n + h 1 , ..., n + h k ∈ P}| ∼ Σ(H)(log x) . kThis is what one would expect and it suggests not onlythat there should be infinitely many Sophie-Germain pairsbut that their occurence is quite frequent, about one in everylog(x) 2 , on average. A stronger version of this conjecture isdue to Bateman and Horn and it replaces the linear polynomialsf i (x) = x + h i , for which the Hardy Littlewood conjecturerequires that f i (n) ∈ P for all i, by some predetermined polynomialsf i (x) of arbitrary degree. The claim of the conjectureis then similar, with the same distribution function dependingonly on k and a structural constant that replaces Σ(H).Particular problems and conjecturesA variety of problems related to prime gaps or to distributionsof primes have been proposed, sometimes in the hopethat these might represent a simpler, more accessible problem.The hope did not often come true.The Hypothesis H of Schinzel [10] states:Conjecture 4. Consider s polynomials f i (x) ∈ Z[X]; i =1, 2,...,s with positive leading coefficients and such that theproductsF(X) = f i (x)is not divisible, as a polynomial, by any integer different from±1. Then there is at least one integer x for which all the polynomialsf i (x) take prime values.Apparently, this is only a more accessible version of theBateman-Horn conjecture.Dorin Andrică [2] proposed during his years of study thefollowing conjecture, which was sufficiently supported by numericalevidence:Conjecture 5. Let p n ∈ P denote as usual the n−th prime.Then, for all n ∈ N, the distance√pn+1 − √ p n < 1.i=1Equivalently, the gapg n = p n+1 − p n < 2 √ p n + 1.This appears as if it was an easy exercise, given that we“know" from the Cramér model that the typical gap has lengthlog(p n ) and we even expect that no larger gaps than log 2 (p n )exist, with little to say about exponential gaps like O( √ p n ).But not only does the estimate lay in the order of magnitudeof what one can prove by assuming the Riemann Hypothesisbut it appears that proving the one is essentially equivalent toproving the other.An apparently even simpler conjecture was formulated byMichael Th. Rassias [8], at the age of 14, while preparingfor the International Mathematical Olympiad that was heldin Tokyo in 2003. Rassias’ numerically supported conjectureclaims:Conjecture 6. For any prime p > 2 there are two otherprimes p 1 < p 2 such thatp = p 1 + p 2 + 1.p 1We have here a surprising feature of presenting a prime asa quotient; however, after some algebraic manipulation, werecover the expression(p − 1)p 1 = p 2 + 1,thus obtaining an interesting combination of a generalised SophieGermain twin problemp 2 = 2ap 1 − 1,strengthened by the additional condition that 2a+1 be a primenumber too. We have seen that such questions are caught bythe Hardy–Littlewood conjecture. One may ask if Rassias’conjecture is to some extent simpler than the general Hardy–Littlewood conjecture or its special case concerning distributionof generalised Sophie-Germain pairs p, 2ap+1 ∈ P? Thisis not likely to be true.A further related problem, much appreciated by amateurnumber theorists in search of interesting computations, is relatedto Cunningham chains, i.e. sequences of primesp i+1 = mp i + n, i = 1, 2,...,k − 1,for fixed coprime m, n ∈ N >1 .As for the large arithmetic progressions of primes, there arecomputing competitions for the longest Cunningham chain orfor the one built up of the largest primes – but unlike thebreakthrough of Green and Tao, there is no general resultknown on large Cunningham chains to date. Rassias’ conjecturecan also be stated in terms of Cunningham chains,namely: there exist Cunningham chains with parameters 2a, −1for a such that 2a − 1 = p is prime.References[1] http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_primes#World_records.[2] T. Andreescu and D. Andrica: Number Theory, Birkhäuser,Boston, Basel, Berlin (2009), p. 12.[3] H. Dubner: Large Sophie-Germain primes, Math. Comp.65(1996), pp. 393–396.EMS Newsletter June 2014 15

Featurecernant les nombres premiers, Acta Arith. 4 (1958), pp. 185–208.[11] Y. Zhang: Bounded gaps between primes, Arxive.[4] A. Granville: Bounded Gaps between Primes, http://www.dms.umontreal.ca/~andrew/CurrentEventsArticle.pdf.[5] B. Green an T. Tao: The primes contain arbitrarily long arithmeticprogressions, Ann. Math., Vol 167 (2008), pp. 481–547.[6] E. Klarreich: Together and Alone, closing the Prime Gap,QUANTA Magazine,https://www.simonsfoundation.org/quanta/20130519-unheralded-mathematician-bridges-the-prime-gap/.[7] H. Maier, Primes in short intervals, The Michigan MathematicalJournal 32(2)(1985), pp. 221–225.[8] M. Th. Rassias: Problem Solving and Selected Topics in NumberTheory, Springer (2011).[9] K. Soundararajan: Small Gaps between Prime Numbers: theWork of Goldston-Pintz-Yildirim, Bull. Am. Math. Soc. 44, Nr.1, (2007) pp. 1–18.[10] A. Schinzel and W. Sierpiński: Sur certaines hypothèses concernantles nombres premiers, Acta Arith. 4 (1958), pp. 185–208.[11] Y. Zhang: Bounded gaps between primes, Arxive.Preda Mihailescu [preda@unimath.gwdg.de]was born in Bucharest,1955. He studied mathematics and computerscience in Zürich, receiving his PhDfrom ETH-Zürich. He was active during15 years in the industry, as a numericalanalyst and cryptography specialist. In2002, Mihailescu proved Catalan’s Conjecture. This numbertheoretical conjecture, formulated by the French mathematicianE. C. Catalan in 1844, had stood unresolved for over acentury. The result is known as Mihailescu’s Theorem. He iscurrently a professor at the Institute of Mathematics of theUniversity of Göttingen.Preda Mihailescu [preda@unimath.gwdg.de]was born in Bucharest,1955. He studied mathematics and computerscience in Zürich, receiving his PhDfrom ETH-Zürich. He was active duringTo the Memory of George Kac (Georgiy Isaakovich Kac)15 years in the industry, as a numericalanalyst and cryptography specialist. InLeonid Vainerman (University of Caen, France)2002, Mihailescu proved Catalan’s Conjecture. This numbertheoreticalTranslatedconjecture,into Englishformulatedby Nataliyaby theMarkovaFrench mathematicianE. C. Catalan in 1844, had stood unresolved for over acentury. The result is known as Mihailescu’s Theorem. He isIdeas That Will Outlast UsPrefacecurrently a professor at the Institute of Mathematics of theHere is a story about a person whose work and destinyUniversity of Göttingen.were closely related to two important advances in 20thcentury mathematics: supermathematics and quantumgroups. The first serious step towards supermathematics(the introduction of Lie superalgebras and formal Liesupergroups) was made by F. A. Berezin and G. I. Kac in[3]. Interesting memories have been recorded about FelixAlexandrovich Berezin but almost nothing has beenwritten about Georgiy Isaakovich Kac (in fact, only [2]).Discussing the impact of [3] on mathematics and mathematicalphysics, Yu. A. Neretin wrote: “I have heard verylittle about the second author of this paper, G. I. Kac. Itis worth noting that Kac’s papers on ‘ring groups’ haveto a great extent set the stage for another ‘explosion’,namely, the works on ‘quantum groups’.” See also V. G.Drinfeld’s discussion in [4].******I knew George Kac (a remarkable mathematician and aremarkable person) between 1968 and 1978. He died ofa heart attack on 20 May 1978, in the prime of his talentand vitality. His friend B. I. Khatset wrote that his modestyand generosity had earned him the nickname PierreBezukhov (one of the principal characters of Leo Tolstoy’sWar and Peace) and, as you will see below, he waslike this in mathematics, too. After G. K.’s death, whilstspending considerable time at research and academiccentres where follow-up work was taking place, I ob-served the strong development of his ideas. That is whyyou will find here not just personal memories but alsoreflections on his mathematical ideas, their genesis, theirevolution and their impact on other researchers. This isnot a scientific paper so it does not claim to be rigorousor exhaustive. Nevertheless, it will be more understandableto those who are to a certain extent familiar withalgebra and analysis. I am writing mainly about eventsthat I have witnessed.Kac algebrasIn the autumn of 1968 I started attending G. K.’s seminaron operator rings. He was a remarkable speaker and hismanner of setting out the material was clear and rigorousand, moreover, lively and accessible. Nothing I hadheard before could match it and long after his death onlya handful of speakers left a comparable impression.We came to know each other more closely after mygraduation from Kiev University, when I was hired bythe Kiev Aviation Engineering Military College, whereKac served as a professor of mathematics. Even thoughI graduated from the university with honours, in thoseyears, at the height of anti-Semitism, it was impossiblefor me to formally enrol in a PhD programme. For thesame reason G. K. could not work either at the universityor at the academy of sciences. Happily, Yu. M. Berezanskihelped him set up his seminar at the institute ofmathematics. Over the following years, I was fortunate16 EMS Newsletter June 2014

Featureto work with G. K., which not only guided my scientificcareer but also determined my future life path. I haveto mention that at the same period I was doing my PhDthesis under the informal supervision of M. L. Gorbachuk,to whom I am deeply grateful.G. K. offered me the choice of two research topics. Thefirst was to further develop his joint work with Berezin[3], one of the pioneering works in the area now calledsupermathematics and dedicated to the study of mathematicalstructures (superalgebras, Lie superalgebras,supermanifolds, etc.) graded, in the simplest case, by thegroup Z/2Z. The motivation was provided by quantummechanics, where two kinds of particles, fermions andbosons, are governed by totally different statistical laws.Both Kac and Berezin had a strong background in theoreticalphysics, so no wonder they became pioneers inthis new area. Kac’s PhD thesis, defended in 1950 (withN. N. Bogolyubov as an advisor), was dedicated to thecorrelation theory of electron gas. Let us also mentionthe superanalogue of Frobenius’ theorem, obtained byG. K. and A. I. Koronkevich shortly after [3].Nevertheless, I missed my chance to become a “supermathematician”and chose another topic, ring groups,introduced by G. K. around 1960. Below is a reminder ofthe sources of this theory.Let G be a commutative, locally compact group andG its dual, i.e. the group of its unitary continuous characters(which is also commutative and locally compact).Pontryagin’s duality says that the dual to G is isomorphicto G. However, this beautiful theory breaks down if G isnon-commutative, even if it is finite, since the charactersof such a group are too few and do not contain all theinformation about the group. To save the duality theoryone could replace the characters with the irreducibleunitary representations (in the commutative case theyare just characters). Indeed, T. Tannaka showed in 1938that a compact group can be restored up to isomorphismfrom a collection of its representations, while in1949 M. G. Krein gave an axiomatic description of such adual of a compact group. Nevertheless, the fact that thisdual, a block-algebra, is not a group breaks the symmetryof the duality. Later on, such a non-symmetric dualitytheory was developed by W. F. Stinespring (1959) forunimodular groups and by N. Tatsuuma (1965–66) forgeneral locally compact groups.Translating Stinespring’s paper into Russian for thecollection “Matematika”, G. K. came up with the idea toconstruct a new category of ring groups that would containboth groups and their duals with a symmetric dualityacting within (i.e. an object and its dual must havethe same mathematical structure, like in Potryagin’s duality).A ring group is a collection (A, G, S, m), where Ais an operator algebra, an invariant measure m is a traceon it (i.e. a positive central linear form which can be unbounded),a coproduct G is an algebra homomorphismfrom A to its tensor square, and an antipode S is an involutiveanti-isomorphism of A, satisfying certain axioms.When A is commutative, it can be realised as the algebraof essentially bounded measurable functions on aunimodular group G with respect to the invariant measureof G, Gf(x) = f(xy), Sf(x) = f(x –1 ) (where x, y are elementsof G) and m is the integral over the invariantmeasure on G. So, unimodular groups are included inthe category of ring groups. Their duals are exactly cocommutativering groups (i.e. G is stable under the permutationof factors in the tensor product). In this case,the algebra A is generated by the translation operatorsL(x) or, equivalently, by the convolution operators L(f)with f a continuous integrable function on G. G mapsL(x) to its tensor square, S(L(x)) = L(x –1 ) and m(L(f)) =f(e), where e is the unit of G.In purely algebraic terms, ring groups are nothingother than Hopf algebras studied earlier in topology.G. K. was not aware of the existence of Hopf algebrasand reinvented them when he introduced ring groups.Lastly, G. K. gave a construction of a dual in this category.Applying this construction twice, one has an objectisomorphic to the original one, like in Pontryagin’s duality.These results were first announced in Soviet MathematicalDoklady in 1961 and later published in detail in[7]. That work used the techniques from I. Segal’s paperon traces on operator algebras, also translated into Russianby G. K. for the “Matematika” collection.The research topic I chose was the open problem formulatedin G. K.’s habilitation thesis (Moscow University,1963): to extend ring group theory in a way to coverall locally compact groups.Fairly soon, I identified the technical tools to be prepared.Firstly, the above mentioned traces had to bereplaced by possibly unbounded positive forms on operatoralgebras that are not necessarily central, calledweights. Secondly, G. K. made systematic use of closeties between traces and so-called Hilbert algebras, soone had, passing from traces to weights, to develop anappropriate generalisation of Hilbert algebras. The ideaappealed to G. K. and we started working on it with vigour.Shortly afterwards, the papers by M. Takesaki andF. Combes appeared, containing the needed techniques;they were clear to us since we had already gone halfway.Now the coveted target was within reach but we hadto hurry, since we were not alone in our pursuit. By thattime, Takesaki had already written a paper on generalisationof ring groups; in addition, he had mastered all thenecessary techniques. It remains a mystery to me whyhe did not come first in the race, being a leading specialistwith a number of brilliant results to his credit. G. K.was also convinced that J. Dixmier saw the same objective.He said: “We must hurry. I’m sure that Dixmier hassomeone tackling this problem.” As it turned out, he wasabsolutely right. But even in that stressful situation hisintegrity did not fail him: since he considered that it wasme who had suggested the basic ideas, he decided to allowme the opportunity to complete the solution by myself,thus becoming the sole author, while it was he whohad set the problem and had put much effort into adequatelypreparing me for solving it. He liked me to cometo the lecture room at the end of his lectures to discussand to walk with him from Uritski Square, where theMilitary College was situated, past the railway stationEMS Newsletter June 2014 17

Featureto his home on Bolshaya Podvalnaya Street (the streetnames are of the period). Sometimes these discussionscontinued in his apartment.Working on my own involved some risk because Ihad not yet fully mastered the necessary techniques andneeded more time. To make matters worse, I was deeplyupset by the death of my father in August 1971. Finally,seeing that I was not making much progress, G. K. understoodthat we could well lose the race and took over.He promptly got through a couple of issues that had baffledme and with his support I started to advance muchquicker. In a joint effort, we had fairly rapidly completeda draft version. Even then G. K. remained true to himself;he suggested that I first publish part of the solution onmy own, and only after that we publish the entire solution,which we did – the note [16] was submitted earlierthan both papers [17].Then we were in for an ordeal. Takesaki’s paper [14]appeared, in which he took one more step towards thegeneralisation of ring groups. It was unavailable in Kiev,while the review on it in “Реферативный Журнал” (theSoviet analogue of Mathematical Reviews) indicatedthe construction of a duality theory that generalisedthe one of ring groups and covered all locally compactgroups. Given Takesaki’s reputation and the title of thepaper, there was little doubt that we had lost. G. K., upset,dropped all his work and rushed to Moscow to readit firsthand (usually he visited Moscow several times ayear in order to keep track of publications unavailablein Kiev and to socialise with colleagues, such as M. A.Naimark, F. A. Berezin, A. A. Kirillov and others). Backin Kiev with a photocopy of Takesaki’s paper in hand,G. K. said that the reviewer was wrong and that the paper’sresults had not achieved the goal.We completed and published our papers. But aboutthe same time, papers came out by M. Enock and J.-M.Schwartz (who worked under Dixmier’s supervision)containing equivalent results, although using a somewhatdifferent technique. As Michel Enock later toldme, Dixmier had also urged them to hurry, explainingthat besides Takesaki, there had to be someone in Kievworking on the same problem with Kac. Our French colleaguesused to send us their preprints and papers, whilewe could not respond to them, since we worked at a MilitaryCollege and, bound by secrecy regulations, were notallowed to communicate with foreigners. For example,in 1975 we were invited to participate in a conferencein Marseille dedicated to the subject of our studies andI rashly showed my invitation at the so-called First Departmentof the College. I was lucky to get away with it;the KGB men told me that if I wanted to continue mywork there, I must throw the invitation away.In view of the fundamental role of G. K. in the discoveryof the new mathematical objects, the suggestionof Enock and Schwartz to name them Kac algebras washighly appropriate. I learned about it from G. K. himself.He had very expressive eyes and it was apparent howmuch he enjoyed the news. Of course, being a man ofgreat modesty, he never actually pronounced this name.Today, an entire book [5] is dedicated to this subject.Kac algebras were invented not just for the sake ofbeauty but with a view to applying them to the solutionof various problems. According to G. K., ring groups hadto be regarded in the same way as the ordinary groupsthat they generalise, while their application areas mightbe wider, which later proved to be the case. Nevertheless,bringing ring groups into effective use required a deeperunderstanding of their structure, examples and properties.Back in the early 1960s, G. K. had distinguished andstarted to explore special classes of ring groups: compact,discrete and finite.G. K.’s habilitation thesis contained a list of openproblems – some of them have been solved since (likethe one examined above). One of these open problemsis the fact that the existence of an invariant measure isan axiom of a Kac algebra and is not derived from otheraxioms, as in the case of ordinary groups (the uniquenessof this measure is easy to prove). This is justified by Weil’sand Haar’s theorems, which state that the existence ofan invariant measure is equivalent to the existence of alocally compact topology on a group. However, it wouldbe natural to define Kac algebras only in algebraic andtopological terms and to prove the existence of an invariantmeasure, thereby generalising the Haar theoremfor ordinary groups. For the above special classes of Kacalgebras this was done in [11, 13].In particular, finite Kac algebras [11] are finite-dimensionalsemisimple Hopf *-algebras over the fieldof complex numbers. Their axiomatics contains a counitthat is the analogue of the unit in an ordinary group. Theexistence of an invariant measure is a theorem.In his habilitation thesis and in [9], G. K. extended anumber of classical results on finite groups to finite Kacalgebras. In particular, he obtained the analogue of theLagrange theorem stating that the order of a subgroupdivides the order of a group. Later on, a stronger statementwas established by V. D. Nichols and M. B. Zoellerfor any finite-dimensional Hopf algebra. As for finitegroups, the only Kac algebra of prime dimension is thecyclic group. This result has been generalised many timesin recent works by various authors. G. K. also provedthat for any irreducible representation of a finite Kacalgebra, there is a basis in which its matrix elements arealgebraic integers.We saw above that commutative and co-commutativeKac algebras correspond to ordinary groups andtheir duals respectively. Of special interest are examplesof nontrivial Kac algebras that do not belong to thesetwo classes. The first such examples were built by G. I.Kac and V. G. Paljutkin [8, 10, 11]. As noted by V. G. Drinfeld[4], they were the first known examples of quantumgroups.Ordinary groups can be realised as set transformationgroups and Kac algebras can also act, but on algebrasinstead of sets. G. K. repeatedly voiced the opinion,which is now generally accepted, that algebras are noncommutativecounterparts of sets and ring groups arenoncommutative counterparts of groups. He suggested adefinition of a ring group action on an algebra and a constructionof the crossed product of these objects. Later18 EMS Newsletter June 2014

Featureon, Enock and Schwartz dedicated a series of papers tothat type of construction and related results.Discussing the choice of new problems in the mid-1970s, G. K. strongly advocated an in-depth study of finitering groups and their representations in the vein ofhis works. Another option was the search for new examplesof nontrivial Kac algebras but that turned out to bean uphill task, and only 20 years later did I understandhow to systematically construct them. However, in thosedays analytical aspects of the theory seemed closer tome than algebraic ones.I decided to explore such generalisations of Kac algebras,which would cover so-called generalised shiftoperators or hypergroups, studied earlier by J. Delsarte,Yu. M. Berezanski, S. G. Krein and B. M. Levitan. This lineof research offered challenging analytical problems andinteresting applications. But, despite the fact that thisproblem was mentioned in his habilitation thesis and in[8], G. K. thought that such a theory would be somewhatdeficient because the coproduct must be just a positivemap and not a homomorphism of operator algebras likein the Kac algebra theory, so that certain essential propertiesof Kac algebras would be lost.Still, the fact that due to weaker constraints thetheory became far richer in applications seemed of importanceto me. In the 1980s and 1990s, encouraged byYu. M. Berezanski, I worked in this direction. But in themid-1970s, to my great sorrow, our active collaborationwith G. K. was practically over, even though we continuedto see each other and discuss mathematical and nonmathematicaltopics. Since then, I had always kept trackof Kac algebra theory, my first love in mathematics, but itwas not until the mid-1990s that I returned to it.Quantum GroupsLet us now turn to the evolution of G. K.’s ideas after hisdeath. Kac algebras solved the problem of duality buttheir range of applications was not wide enough (whichwas the reason for me to take up a generalisation).However, the genuine breakthrough in understandingthe approach to extend this circle of ideas occurred inthe mid-1980s, when V. G. Drinfeld and others discoveredquantum groups [4]. In purely algebraic terms, thesquared antipode in a quantum group is not necessarilytrivial (it is trivial in a Kac algebra) – a seemingly smallmatter but one that makes the fundamental difference.Many important examples and applications appeared,in particular in theoretical physics and topology. AsAlain Connes put it in the preface to [5], Kac algebrashave proved “not sufficiently non-unimodular” to covernew applications and therefore the necessity arose forquantum groups. Note that Kac algebras are “non-unimodular”generalisations of ring groups in their ownright!As for quantum group theories in operator algebraicframework, S. L. Woronowicz [18] built the theory ofcompact quantum groups that contained the theoremof existence of invariant measure (for compact Kac algebrasthis theorem had been proved earlier [13]) andstudied their irreducible representations. Constructingconcrete examples of quantum groups, he overcamenumerous functional analytical difficulties which highlightedthe challenges in extending Kac algebra theoryto capture all interesting examples while keeping itsbeauty and symmetry.In the 1950s, Stinespring noted the essential roleplayed in the duality of unimodular groups by the unitarysending f(x,y) to f(x, xy). G. K. had built an analogueof this operator for arbitrary ring groups and discoveredits fundamental property, the so-called pentagonal relation.Later on, G. K. and I, as well as Enock and Schwartz,made full use of this observation passing to non-unimodularKac algebras. But S. Baaj and G. Skandalis [1]took it even further, pointing out that a multiplicativeunitary (i.e. satisfying the pentagonal relation) undercertain regularity conditions already allows one to buildtwo operator algebras in duality, each of them carrying amore general structure than that of a Kac algebra.After the end of the Soviet era, Michel Enock and Iexchanged all our past publications. He could not knowthat I had already received everything he had sent mebefore, since not a single response came back from behindthe Iron Curtain (the envelopes I often receivedhad traces of being open and re-sealed, certainly by theKGB, but they contained only mathematical articles).We started to correspond and soon Michel came to Kiev.One of the sentimental events of his stay was our visit toG. K.’s tomb at Gostomelskoye cemetery.When I came to Paris in the spring of 1992, many colleaguessaid that they were happy to see a representativeof George Kac’s school. They did not know thatG. K. could never have formal graduate students, andhis “school” eventually consisted of just V. G. Paljutkin,me, A. I. Koronkevich, and V. Zhuk. It was so unfair thatG. K. had not enjoyed even a small part of the recognitionthat he so much deserved!Besides the above mentioned difficulties of his life,his scientific results had not been properly appreciatedby some leading specialists, e.g. by L. S. Pontryagin, whoseduality principle had been so brilliantly extended byG. K., and by I. M. Gel’fand, who had been rather criticalof G. K.’s works. In 1983, Gel’fand tried to persuade meto take up a different subject, saying: “Do not bury yourtalent as your teacher Kac did!” This was just before thediscovery of quantum groups, for which V. G. Drinfeldreceived the Fields Medal and which had Kac algebratheory as a direct predecessor (see [4])!Gradually the idea arose to get back to constructingnon-trivial examples of Kac algebras and quantumgroups. When Woronowicz and others built examples ofquantum groups, each case was unique and presentedspecific difficulties. It was desirable to have constructionsgenerating a number of different examples in a unifiedway. For example, Drinfeld proposed a purely algebraicway to deform the coproduct and antipode withoutchanging the algebra of a given quantum group in orderto get a new one. Applying this construction, called twisting,to the dual of an ordinary group one could hope toEMS Newsletter June 2014 19

Featurecome up with interesting examples of quantum groups. Istarted to work on analytical aspects of twisting.Enock came to Kiev once again in May 1994 on theoccasion of G. K.’s 70th anniversary. At the meeting ofthe Kiev Mathematical Society, Enock and I gave presentationsabout different facets of G. K.’s activities and,in particular, discussed the problem of examples of Kacalgebras. Afterwards, we continued our discussions andin the spring of 1995 the paper [6] was finished. Later on,I extended and reinforced these results. In particular, weconstructed a series of new “quantisations” of the Heisenberggroup, which is popular with physicists. All ofthese objects turned out to be not only Kac algebras buteven unimodular ring groups in the sense of G. K.’s veryfirst definition.The finite-dimensional aspect of twisting was discussedin Kiev with D. Nikshych, who built deformationsof classical series of finite groups – symmetric, dihedral,quasiquaternionic and alternated. The latter gave thefirst known examples of simple Kac algebras (i.e. withoutproper normal Kac subalgebras).In the late 1990s, there was progress in the longawaitedgeneralisation of Kac algebra theory. Baajand Skandalis, on the one hand, and Woronowicz, onthe other hand, better understood the conditions to beplaced on a multiplicative unitary so that it generates apair of quantum groups in duality. A version of such atheory was proposed by T. Masuda, I. Nakagami and S. L.Woronowicz. In Leuven (Belgium), A. Van Daele discussedat his seminar new examples of quantum groupsand approaches to the construction of a general theory.Among other things, he distinguished a class of quantumgroups defined in a purely algebraic way such that theirtopological properties could be derived.Finally, a satisfactory theory of locally compact quantumgroups was proposed in 1999 by J. Kustermans andS. Vaes [17]. It was as beautiful and symmetric as Kacalgebra theory was; without much exaggeration onecan say that it had been modelled on the latter. A locallycompact quantum group is a collection (A, G, m,n,), where A, G, m are the same as in Kac algebra theory,and n is a right invariant weight on A. The axioms donot mention antipode explicitly but imply its existenceand properties and not one but two weights are present– here it is: the second non-unimodularity! It resemblesan ordinary locally compact group with two invariantmeasures: left and right.In 1999–2000, I spent a few months in Leuven andcould familiarise myself with these things firsthand,whereupon I got an idea to generalise the theory of extensions[8], which G. K. used for getting non-trivial examplesof Kac algebras [10, 11]. Given a commutativeKac algebra K 1, which is an algebra of functions on anordinary group G 1, and a co-commutative K 2, which is adual of an ordinary group G 2, the question is whether itis possible to build an extension of K 1by K 2– a new Kacalgebra such that K 1would be its normal Kac subalgebraand K 2would be the corresponding quotient. G. K.showed that such an extension exists if and only if thegroups G 1and G 2act on each other as on sets and theseactions are compatible in a special way. He had describedall these extensions in terms of what is referred to as bicrossedproduct construction. Later on, this constructionwas rediscovered by M. Takeuchi and S. Majid.Now that we had a far wider category of locally compactquantum groups in hand, one might expect to makethe most of the construction. I shared these ideas withStefaan Vaes and in a few months the paper [15] was finished.Taking various G 1and G 2, one could get plenty ofconcrete examples of Kac algebras and quantum groups.In our later work, we classified extensions with G 1andG 2low-dimensional Lie groups; another quantum groupwith surprising regularity properties was built by Baaj,Skandalis and Vaes with G 1and G 2coming from numbertheory; in yet another case, my former PhD studentPierre Fima used similar groups to construct examplesof quantum groups with prescribed types of their vonNeumann algebras.Baaj, Skandalis and Vaes also extended to locallycompact quantum groups another brilliant idea of G. K.,who had built in [8] what was later (when the importanceof this sequence in various problems became evident)called the Kac exact sequence.I will conclude my notes with a story about Kac algebras“in action”. In the postscript to [5], A. Ocneanuexplained that Kac algebras must arise as non-commutativeanalogues of group symmetries in the subfactortheory founded by V. Jones. Indeed, in 1994, W. Szymanskiand R. Longo independently of one another provedthat if N is a finite index subfactor of a factor M thenunder certain conditions there is necessarily a Kac algebraK acting on M such that N is a subalgebra of fixedpoints with respect to this action. Vice versa, given a Kacalgebra and its action on a factor, one can build a finiteindex subfactor. This is a far-reaching extension of classicalGalois theory, where N and M are fields, and K is aGalois group. Later on, M. Enock and R. Nest came upwith a similar result for subfactors of infinite index, inwhich case a Kac algebra had to be replaced by a locallycompact quantum group.PostscriptThe purpose of these notes was to show, apart from personalmemories, the powerful influence of Georgiy Isaakovich’sworks on the progress of a wide area of mathematics.He has been gone for more than 35 years but inscores of recently published works, you will effortlesslyfind clear evidence of their foundation in his ideas. Indeed,as Pushkin put it: “I have built a monument notwrought by hands…”References1. S. Baaj, G. Skandalis, Unitaires multiplicatifs et dualité pour lesproduits croisés de C*-algèbres, Ann. Sci. Ecole Normale Sup., Ser.4, 26 (1993), 425–488.20 EMS Newsletter June 2014

Feature2. Yu. M. Berezanski, F. A. Berezin, N. N. Bogolyubov, L. I. Vainerman,Yu. L. Daletski, A. A. Kirillov, V. G. Paljutkin, B. I. Khatset and S. D.Eidelman, Georjiy Isaakovich Kac (Obitiary), Soviet Math. Uspekhi34, n.2 (1979).3. F. A. Berezin and G. I. Kac, Lie groups with commuting and anticommutingparameters, Math. USSR Sbornik 11(1970), 311–325.4. V. G. Drinfel’d, Quantum groups. Proceedings of the InternationalCongress of Mathematicians, Berkeley, vol. 1, 1986, 798–820.5. M. Enock and J.-M. Schwartz, Kac Algebras and Duality of LocallyCompact Groups, Springer, 1992.6. M. Enock and L. Vainerman, Deformation of a Kac algebra by anAbelian subgroup, Comm. Math. Phys., 178 (1996), 571–596.7. G. I. Kac, Ring groups and the principle of duality, I, II, Trans. MoscowMath. Soc. 12 (1963), 291–339; 13 (1965), 94–126.8. G. I. Kac, Extensions of groups to ring groups, Math. USSR Sbornik5 (1968), 451–474.9. G. I. Kac, Certain arithmetic properties of ring groups, Funct. Anal.Appl. 6 (1972), 158–160.10. G. I. Kac and V. G. Paljutkin, Пример кольцевой группы,порождённой группами Ли, Укр. матем. ж. 16 (1964), 99–105 (inRussian).11. G. I. Kac and V. G. Paljutkin, Finite ring groups, Trans. MoscowMath. Soc. 15 (1966), 251–294.12. J. Kustermans and S. Vaes, A simple definition for locally compactquantum groups, Comptes Rendus Acad. Sci. Paris. Ser. I 328 (10)(1999), 871–876; Locally compact quantum groups, Ann. Sci. EcoleNormale Sup., Ser. 4, 33 (2000), 837–934.13. V. G. Paljutkin, Invariant measure of a compact ring group, Am.Math. Soc. Transl. 84 (1969), 89–102.14. M. Takesaki, Duality and von Neumann algebras, Lecture Notes inMath. 24 (1972), 665–785.15. S. Vaes and L. Vainerman, Extensions of locally compact quantumgroups and the bicrossed product construction, Advances in Mathematics,175, n.1(2003), 1–101.16. L. I. Vainerman, Characterization of objects dual to locally compactgroups, Funct. Anal. Appl. 8, (1974), 66–67.17. L. I. Vainerman and G. I. Kac, Nonunimodular ring groups andHopf-von Neumann algebras, Soviet Math. Dokl. 14 (1974), 1144–1148; Math. USSR Sbornik 23 (1974), 185–214.18. S. L. Woronowicz, Compact matrix pseudogroups, Comm. Math.Phys., 111 (1987), 613–665.Currently Leonid Vainerman is a professorat the University of Caen, France.He is working in the domain of quantumgroups and groupoids and their applicationsto operator algebras. It was hiscollaboration with George I. Kac thataroused his interest in this domain. Hereceived his PhD from the Institute of Mathematics of theUkrainian National Academy of Sciences.2014Clay Research Conference and Workshops28 September – 3 OctoberMathematical Institute | Andrew Wiles Building | University of Oxford, UKClay Research Conference1 OctoberSpeakers:Ben Green (Oxford)Jonathan Pila (Oxford)Paul Seidel (MIT)Scott Sheffield (MIT)Conference Workshops28 September – 2 OctoberAdvances in Probability:Integrability, Universality andBeyondOrganizers:Ivan Corwin (Columbia, IHP, CMI)Martin Hairer (Warwick)29 September – 3 OctoberAnalytic Number TheoryOrganizers:Ben Green (Oxford)Roger Heath-Brown (Oxford)29 September – 3 OctoberFunctional Transcendencearound Ax–SchanuelOrganizers:Jonathan Pila (Oxford)Alex Wilkie (Manchester)29 September – 3 OctoberSymplectic TopologyOrganizers:Dominic Joyce (Oxford)Alexander Ritter (Oxford)Ivan Smith (Cambridge)RegistrationRegistration for the Clay Research Conference is free but required. Participationin the workshops is by invitation; a limited number of additional places isavailable. To register for the Conference and to register interest in a workshop,email Naomi Kraker at admin@claymath.org.For full details, including the schedule, titles and abstractswhen they become available, see www.claymath.orgEMS Newsletter June 2014 21

FeatureMathematical KnowledgeManagement: Transcending theOne-Brain-BarrierOne-Brain-Barrier withwithTheoryTheoryGraphsGraphsMichael Kohlhase (Jacobs University Bremen, Germany)Michael Kohlhase (Jacobs University Bremen, Germany)We present the emerging discipline of Mathematical KnowledgeManagement (MKM), which studies the possibility ofcomputer-supporting and even automating the representation,cataloguing, retrieval, refactoring, plausibilization, changepropagation and in some cases even application of mathematicalknowledge.We focus on theory graph technology here, which supportsmodular and thus space-efficient representations ofmathematical knowledge and allows MKM systems to achievea limited mathematical literacy that is necessary to complementthe abilities of human mathematicians and thus to enhancetheir productivity.1 IntroductionComputers and Humans have complementary strengths. Whilethe former can handle large computations and data volumesflawlessly at enormous speeds, humans can sense the environment,react to unforeseen circumstances and use their intuitionsto guide them through only partially understood situations.In mathematics we make use of this complementarityby letting humans explore mathematical theories and come upwith novel proofs, while delegating symbolic/numeric computationand typesetting of documents to computers.There is, however, one area where the specific strengthsof computers are not commonly exploited in mathematics: themanagement of existing mathematical knowledge. In contrastto the creation of new knowledge, which requires (human) insightsand intuition, the management (cataloguing, retrieval,refactoring, plausibilization, change propagation and in somecases even application) of mathematical knowledge can besupported by machines and can even be automated in the nearfuture given suitable representation formats and algorithms.With more than 100 thousand articles published annually,the times where a human could have an overview over – letalone have working knowledge in – all of mathematics arelong gone. Even web-based information systems with globalreach like the Cornell e-print arXiv [1] and reviewing serviceslike the AMS reviews [16] or Zentralblatt Math [22] cannotprevent duplication of work and missed opportunities for theapplication of mathematical results.Even if they make mathematical documents available atthe click of a mouse, they cannot transcend a fundamentalinvariant: to do mathematics with any of that knowledge, itmust pass through a human brain, which has a very limitedcapacity – compared to the volume, breadth and diversity ofmathematical knowledge currently accessible in documents.We will call this the one-brain-barrier (OBB) of mathematics.Of course the OBB applies to all areas of human knowledgebut the highly networked nature and rigorous presentationof mathematical knowledge make it a particularly appealingsubject for studying the possibilities of machine supportin knowledge management.The effect of the OBB is particularly noticeable for themathematical practice of framing, i.e. establishing that an objecta (of class A) can be viewed as an incarnation of a class Bof objects (via an interpretation mapping ι : B → A); see [7]for an introduction. Framing is exceedingly successful in establishingseemingly new mathematical knowledge in bulk:proving a single theorem T about objects b in B yields thetheorem ι(T) about objects ι(b) in A “for free”. Influential interpretations(where the theory of B is particularly rich) areoften referred to as “representation theorems” as they crossfertilisebetween mathematical areas by bridging represenationaldifferences. But the method of framing is useful alsoin the small: we use it every time we apply a theorem, onlythat we usually do not state necessary interpretations explicitlybut leave their reconstruction to the reader. The ability toto this without conscious thought is part of every mathematicaleducation. The utility and ubiquity of framing lets somemathematicians consider it a defining characteristic of mathematicalpractice.Note that the proof that any A can be interpreted to be a Bmay be highly non-trivial; but the problems with finding interpretationsand representation theorems begin way before that:a prerequisite is that working knowledge about both As andBs is co-located in one brain. There are two kinds of situationwhere interpretations are needed:1. We have established a new result about a class B of objectsand want to apply it to all known objects a that can beinterpreted to be of class B.2. We study object b (of class B) and want to see whetherthere are classes A it can be interpreted as, so that we cantake advantage of the knowledge about A.Both induce a search problem for an interpretation ι : B → A,where A ranges over “all mathematical knowledge”. So, evenif we assume that the knowledge about B is already available– a questionable assumption in 2 – the OBB hits us with fullforce. Note that even though the proof of the representationtheorem for ι usually requires (human) insight and intuition,the application of the frame (the transport of the knowledgefrom B to A) is a largely syntactical process, which can bemachine-supported.In this article, we want to survey the field of MathematicalKnowledge Management, its basic ideas and methodologicalbasis in general (next section), then focus on a subset of meth-22 EMS Newsletter June 2014

Featureods that put the notion of framing at the centre (Section 3) andshow how they can be used to transcend the one-brain barrierin Section 4. Section 5 concludes the article.2 Mathematical Knowledge ManagementMathematical Knowledge Management (see [15, 3]) is ayoung research area at the intersection of mathematics, artificialintelligence, computer science, library science and scientificpublishing. The objective of MKM is to develop new andbetter ways of managing sophisticated mathematical knowledgebased on innovative technology of computer science, theinternet and intelligent knowledge processing. MKM is expectedto serve mathematicians, scientists and engineers whoproduce and use mathematical knowledge; educators and studentswho teach and learn mathematics; publishers who offermathematical textbooks and disseminate new mathematicalresults; and librarians who catalogue and organise mathematicalknowledge.Even though MKM focuses on (MK)M (the “managementof mathematical knowledge”), its methods touch on M(KM)(the “mathematics of knowledge management”). It is a fundamentaltenet of MKM that important aspects of knowledgecan be formalised into a state where their management can besupported by machines. Conversely, it is assumed that only bymaking such aspects of knowledge formal – e.g. by semanticsextraction – will they become amenable to automation.As a consequence, semantisation is an important concern forMKM.As the “client list” of MKM above already suggests, thenotion of knowledge management is rather inclusive. Correspondingly,the aspects of knowledge studied by MKM andthe “depth of formalisation” are varied, ranging from full formalisationof the mathematical content in a logical system[5, 21] to machine-readable versions of the Mathematics SubjectClassification [12].It has always been an aim of the MKM community tobuild a digital mathematical library – envisioned to be universalin [3]. Based on this, sophisticated mathematical softwaresystems could help humans articulate, organise, disseminateand access mathematical knowledge to allow them to concentrateon those parts of doing mathematics that need humancognitive facilities – creating new mathematical knowledgeand appreciating the beauty of the existing canon.In this paper we focus on ways to structure a digitalmathematical library into modular theory graphs and explainsalient mathematical practices in this setting. Our expositionhere follows [20], which introduces OMDoc/MMT, currentlythe strongest formulation of theory graph ideas, which wereinitially introduced in [4] and extended to a comprehensiverepresentation format for mathematical knowledge and documents(OMDoc: Open Mathematical Documents [8]).3 Theory Graphs & Computer-SupportedFramingThe main idea of the theory graphs paradigm in MKM is totake the idea of interpretations seriously and use it as a structuringprinciple in a modular representation methodology formathematical knowledge. Neither the modularity principlenor the use of interpretations are particularly new in mathematics– they have been the main structural mechanism behindBourbaki’s systematic redevelopment of mathematics.But the MKM community is developing ways of formalisingthem and is making use of these structures for machinesupport. We will now explore the basic notions and show howthese relate to the mathematical practice of framing.Theory graphsbuild on two central notions: theories (the nodes of the graph)and theory morphisms (the edges). The former are just collectionsof symbols denoting indivisible mathematical objectsand axioms stating the assumptions this theory makes aboutthem.Theory morphisms are mappings from symbols in thesource theory to expressions in the target theory, such that (*)all axioms are mapped to theorems of the target theory.Figure 1 shows a simple example of a theory graph. It consistsof a theory monoid of monoids, which has the symbolsop for the binary composition operation and unit for its neutralelement. Symbols and axioms (which we omit in Figure 1 forsimplicity) have tripartite global names of the form g?t?c inOMDoc/MMT, where g is the URL of the document that containsthe theory t and c is the name of a symbol in t. Note thatsuch tripartite names (MMT URIs) are valid uniform resourceidentifiers, which is a crucial prerequisite for being used inweb-based MKM systems. In Figure 1, we have assumed thatall theories are in the same document and we can thereforeomit the URL g throughout.InheritanceNow, a theory cgp of Abelian groups can be obtained from amonoid by adding an inverse operation inv. In OMDoc/MMTwe can avoid duplication of representations by utilising an inclusionmorphism (visualised by an arrow ֒→) that “copies”all symbols and axioms from the source theory monoid tocgp. Note that inclusion morphisms are trivially theory morphisms,since the target is defined to make them so. OM-Doc/MMT treats incoming inclusions like symbols and namesthem in their target theory, so that the inclusion of the monoidin Figure 1 is globally accessible to MKM systems via theMMT URI g?cgp?mon. The inherited symbols are availablein cgp via the name of the inclusion that supplied them, e.g.composition as mon/op or globally as g?cgp?mon/op. Axiomsare inherited similarly.In the next step we can obtain a theory ring via two inclusions:ring?add and ring?mul (and adding a distributivity axiom).Note that ring has two distinct binary operations: the additivering?add/mon/op and the multiplicative ring?mul/op,ringadd, mulring?mulFigure 1. A simple Theory Graphring?addcgp?moncgpmon, invmonoidop, unitEMS Newsletter June 2014 23

Featuretwo neutral elements: ring?add/mon/unit and ring?mul/unit,and the additive inverse ring?add/inv. Note that the respectivecopies of the axioms of mon and cgp are inherited as well.ring inherits eight axioms: two closure, two associativity andtwo neutrality axioms, plus additive inverse and commutativity.We see that OMDoc/MMT can mimic the efficiency ofmathematical vernacular in this respect.NotationBefore we go on with our exploration of Figure 1, a clarificationof the level of representation is in order. We have seenthat the use of inclusion morphisms gives us theory structuresthat systematically supply URIs for the objects we expect.We should think of OMDoc/MMT as a kind of “machinelanguage” for representing mathematical knowledge.In particular, URIs as names are good for communicationof knowledge over the internet and its manipulation by machinesbut they are woefully inadequate for communicationwith humans, who expect to see, for example, ◦ instead ofMMT URI g?cgp?mon/op in Abelian groups and may use +for ring?add/mon/op and · for ring?mul/op in rings. To accountfor this, OMDoc/MMT offers a system of notation definitionsthat allow us to associate human-oriented notationswith symbols and regain the familiar and helpful notations ofmathematical discourse. Indeed, making notation definitionsfirst-class citizens of the representation format allows us tomodel them and their effects in modular theory graphs: theyare inherited along inclusions and thus available wherever theinherited symbols are visible. In MKM systems that communicatewith humans, we use notation definitions to generateformula presentations for the user but keep the symbol URIsto render additional services, e.g. definition lookup.ViewsWe have seen that inclusions can be used for a modular developmentof, for example, the elementary algebraic hierarchy.But the real power of theory graphs comes from another kindof theory morphisms: views. In contrast to inclusions they linktwo pre-existing theories and thus we have to prove that all thesource axioms are theorems of the target theory to establishthe theory morphism condition (*). Moreover, views usuallyhave non-trivial symbol mappings. In Figure 2 we extend thetheory graph from Figure 1 with a theory of integers and twoviews. v 1 maps the monoid operation to integer addition andthe neutral element to the integer number zero. To establishringadd, mulviewinclusionring?mulv 4 : x op y → y op xv 3ring?addv 2 :mon/op → +mon/unit → 0inv → −cgp?moncgpmon, invmonoidop, unitFigure 2. A Theory Graph with Views/Examplesv 2or mon → v 1integers+, 0, −v 1 :op → +unit → 0this mapping as a view we have to establish (*) by provingclosure of Z under +, associativity of +, and x + 0 = x for allx ∈ Z (the proof obligations induced by v 1 ) from the axiomsof integers.The view v 2 is similar, only that it additionally maps theinverse operation to − and has proof obligations for the inverseaxiom and commutativity. Note that v 2 can be mademodular by mapping the inclusion mon to v 1 instead of thedirect mapping of (inherited) symbols. This also allows us toinherit/reuse the proof obligations of v 1 in v 2 . A view v 3 fromring to integers can be established similarly.Theory Morphisms Transport Proofs and TheoremsBut more important than the mechanics of theory graphs is theσrealisation that in the presence of a theory morphism S −→T by (*) any S -theorem t becomes a T-theorem σ(t) as theS -proof π of t can be translated along σ: any appeal to anS -axiom in π becomes an appeal to a T-theorem in σ(π) by(*). In particular, σ(t) can be referred to as T?[v]/t, just likeinduced symbols, only that v is the MMT URI of the view σ.In this context, note that the pragmatics of structures (inclusionswith non-trivial mappings) and views are complementary.Structures are constitutive to a theory and restrictthe “applicability” of a theory, since they introduce axiomsthat must be fulfilled by proof obligations, while views are(logically) redundant but allow us to “apply” a theory elsewhere.The subgraph induced by structures must be acyclic,while views may induce cycles. Indeed, theory isomorphisms(pairs of inverse theory morphisms) are a good way to representequivalent theories, alternative definitions, etc.Views at Work in MathematicsIn our example in Figure 2, this means that all group theoremsapply to (Z, +) (and all theories that are reachable from it viatheory morphisms). This already explains one mathematicalpractice: proving a conjecture in the greatest possible generality.In theory graph terms, this means establishing a theoremas low as possible (in terms of the pre-order induced by theorymorphisms) in the graph, since the “cone of influence”gets bigger this way.Another mathematical practice is to abbreviate proofs bymeta-arguments, e.g. “(iv) the proof of this case follows fromcase (iii) by symmetry”. In most cases, such proofs by metaargumentscan be seen as making the reader aware of a theoryendomorphismlike v 4 that can be used to transport the proofof (iii) into one of (iv). Finally, mathematical examples canbe interpreted as theory morphisms in theory graphs (anotherconsequence of (*)). In our example, (Z, +, 0) is an exampleof a monoid by v 1 ,(Z, +, 0, −) is an example of an Abeliangroup via v 2 and (Z, +, 0, −, ∗, 1) is an example of a ring viav 3 . An example becomes “non-obvious” if it has at least oneview component.FramingIn conclusion, we note that theory morphisms are very naturalcandidates for representing the structures underlying themathematical practice of framing highlighted in the introduction.Indeed, we can see the condition (*) as being the essenceof framing. We are currently exploring the logical and cogni-24 EMS Newsletter June 2014

Featuretive consequences of this idea. It seems that theory morphismsdirectly account for representation theorems like Stone’s theoremor Ken Ribet’s link from the Taniyama-Shimura conjectureto Fermat’s last theorem, which made Wiles’ proofpossible. But framing also seems to be at work in less obviousplaces, e.g. in mathematical synonyms. The concepts of“nodes” and “edges” in a graph are often called “points” and“lines”, borrowing terminology from geometry. I believe thatthis metaphoric usage of names is licensed by a partial viewfrom (linear, ordered) geometry to theory graphs. A problemin the study of framing is that it is a cultural skill of mathematicsand that mathematical literacy requires mastering framingto a level where it becomes almost subconscious, and thereforehard to observe via introspection.MKM for Theory GraphsEven our simple example in Figure 2 shows that theory graphsare very information-rich objects whose structure can be usedto explain many mathematical practices. But the full power oftheory graphs only comes out when they are implemented in asoftware system that supports these practices. OMDoc/MMThas been implemented in the MMT API [19], which supportsservices including notation-definition-based presentation (seeabove), incremental flattening (i.e. computation of the symbols,axioms and theorems of a theory S induced by the theorygraph), as well as type- and proof-checking (an importantaspect of MKM we have neglected in this article).Such systems allow us to make use of the space efficiencyafforded by the modular representation in theory graphs: weonly need to represent (and store) a small subset of the knowledgeavailable to an experienced mathematician. In our experience,the induced knowledge outweighs the represented bya factor of 50/1 (if it is finite at all) even for relatively smalltheory graphs. Note that the quotient of the size of the inducedknowledge over the represented knowledge is a good measurenot only for the space efficiency of the representation systembut also for the mathematical literacy of a human. Experiencedmathematicians induce much more (implicit) knowledgefrom what they read than inexperienced ones – becausethe former, it is conjectured, have more densely connectedmental theory graphs at their disposal. OMDoc/MMT providesstandardised identifiers for all induced knowledge itemsand the MMT API can compute their values on demand andreason with them. In this sense, the MMT API can be consideredto possess a certain amount of “mathematics literacy”that allows it to render higher-level mathematical services.4 Overcoming the One-Brain-BarrierWe have seen above that with theory graph technology, MKMsystems can achieve a certain level of mathematical literacyeven without the ability for automated reasoning – anotherimportant aspect of MKM we will not cover in this article. Butin the foreseeable future it seems clear that no machine willeven come close to a human mathematician’s creative abilityof inventing new mathematics. But we can make use ofthe high storage capacity and cheap computation that MKMoffers. It is claimed that even with limited mathematical literacy,machines can (i) complement human mathematicians and(ii) even uncover – rather than discover – novel mathematicsby overcoming the OBB, if we have theory graphs that containmore knowledge than the average human mathematician.In this section, we will assume such theory graphs will becomeavailable and sketch one example for (i) and two for (ii)– which may provide motivation for organising a communityeffort to build such graphs. Starting points could be theoremprover libraries like the Mizar library [14], which has morethan 1000 theories which together contain more than 50,000formal theorems and definitions (see Figure 4 for the inheritancegraph – the Mizar theory language only supports inclusions).We are currently in the process of exporting half-adozentheorem prover libraries into the OMDoc/MMT formatto create a large theory graph (the Open Archive of Formalizations;OAF) to experiment on.Complementing Humans by Searching the InducedKnowledge SpaceThe idea is very simple: we use a formula search engine likeMathWebSearch (see [10, 11]) and instead of indexing allthe represented knowledge items with their respective URIs,we extend this to the set of induced items and their MMTURIs (which the MMT API can compute by flattening). Considerthe situation in Figure 5, where we are searching for theassociativity formula X+(Y+Z) = (X+Y)+Z, and MathWeb-Search returns the MMT URI http://latin.omdoc.org/math?IntAryth?assoc, which – together with the underlying theoryinducedrepresentedFigure 3. Knowledge Addressable in MMTFigure 4. The Mizar Inheritance GraphEMS Newsletter June 2014 25

Featuretheorems – after all, the HOL Light and Isabelle Librariescontain Tom Hales’ proof of the Kepler Conjecture and theCoq library contains the proof of Feit-Thomson’s Odd-OrderTheorem.Note that once a set of views has been found by the viewfinder,we can generate the induced theorems in all accessibletheories and iterate the view finding process, which might findmore views with the induced theorems.Figure 5. Searching for Induced Knowledge Itemsgraph – contains enough information to generate an explanationof the reason + is associative on Z. In essence, thisexperimental search engine searches the space of (induced)mathematical knowledge rather than just the space of mathematicaldocuments.Making Bourbaki AccessibleIdeas like the ones above can solve one of the problems withthe Bourbaki book series, which is written in such a very conciseand modular manner that it can only be understood if onehas all the previous parts in memory. We have extracted thetheory graph underlying the first 30 pages of Algebra I [2].It contains 51 theories, 94 inclusions and 10 views. The theoriescontain 82 Symbols, 38 axioms, 30 theorems and 17definitions [13]. For knowledge items higher up in the graph,there is no (single) place in the book which states all theiraxioms or properties. With the MMT API, we can generateflattened descriptions for reference and with FlatSearch wecan search for their properties. We conjecture that the simpleexplanation feature from Figure 5 can be extended into a“course generator” that generates a self-contained – modulothe reader’s prerequisite knowledge – document that explainsall aspects necessary for understanding the search hits. Withtheory graph technology, Bourbaki’s Elements can be read byneed, as a foundational treatise should be, instead of beingrestricted to beginning-to-end reading.Uncovering TheoremsWe can now come back to the discussion on the OBB from theintroduction. One way to overcome the problem of missinginterpretations is to systematically search for them – after all,machine computation is comparatively cheap and the numberof potential theory morphisms is bounded by the squareof the number of theories times the number of symbols intheories. In particular, the MathWebSearch formula searchengine [10] can efficiently search for substitutions (which areessentially the same as the symbol-to-expression mappingsof theory morphisms). We have explored this idea before thetheory graph technology was fully developed, and the Theo-Scrutor system [17] found a considerable number of simpleviews in the Mizar library. While these were relatively syntacticand obvious – after all, for a view finder to work, theproofs for the proof obligations have to be part of the libraryalready – they had not previously been noticed because of theOBB, even though the Mizar project has a “library committee”tasked with finding such linkages. We expect that oncethe OAF is sufficiently stable, a renewed experiment will yieldmany more, and more interesting views and possibly novelRefining Theory GraphsFinally, even partial views – which should be much more numerousthan total ones in a theory graph – can be utilised. Saywe have the situation below with two theories S and T, a partialtheory morphism S −→ σT with domain D and codomainC, and its partial inverse δ.SDσδTCThen we can pass to the following more modular theorygraph, where S ′ := S \D and T ′ := T\C. In this case wethink of the equivalent theories D and C as the intersection oftheories S and T along σ and δ. Note that any views out of Sand T now have to be studied, if they can be pulled back to Cand D.S ′ T ′δσσDCδWe have observed that many of the lesser known algebraicstructures in Bourbaki naturally arise as theory intersectionsbetween better known structures. We hope to explain the remainingones via other category-theory-inspired theory graphtransformation operations.Note that operations like theory intersections apply theorygraph technology to the problem of theory graph maintenance,which is itself a problem greatly hampered by the OBBwithout MKM techniques.5 ConclusionIn this paper we have explored opportunities to lift the onebrainbarrier in mathematics, which limits the application ofmathematical knowledge both inside the mathematical domainas well as in other disciplines. We propose that the wayforward is to employ computer systems that can systematicallyexplore immense knowledge spaces, if these are representedin sufficiently content-oriented formats. Together withthe creation, curation and application of digital mathematicallibraries (DMLs), this is one of the central concerns of thenew field of Mathematical Knowledge Management (MKM).We have presented the theory graphs approach as a representationparadigm for mathematical knowledge that allowsus to make its modular and highly networked structure explicitand therefore machine-actionable. We have seen thattheory graphs following the “little theories” approach containthe information structures necessary to explain – and thus ultimatelysupport by computer – many mathematical practices26 EMS Newsletter June 2014

Featureand culture skills. This has the potential to significantly extendthe “mathematical literacy” of mathematical knowledgemanagement systems and consequently make them more suitableas tools that complement human skills.We have explored three exemplary mathematical applicationsof theory graph technologies and one MKM-internal togive an intuition of what services we can expect if we embarkon the enterprise of representing large bodies of mathematicalknowledge and its network structures in machine-actionableformats. The availability of such DMLs is currently the largestbottleneck for overcoming the OBB in mathematics. We arecurrently experimenting with establishing an open archive offormal mathematics (the OAF project [18]) by integrating theoremprover libraries. But formalisation often poses a highburden on the author and forces decisions about logical foundationsthat are irrelevant mathematically. Therefore, we arecurrently researching ways the theory graph methods presentedhere can be extended to representations of mathematicalknowledge in which the degree of formalisation is flexible.Flexiformal representations – see [9] for a discussion –are much closer to mathematical vernacular, which mixes informalparts (natural language) with formal parts (e.g. formulaeand functional markup for mathematical statements) andare therefore easier to obtain in practice. But mathematicalliteracy may be limited by the availability of formal/machineactionableparts; therefore, we are additionally investigatingmethods for automated semantics-extraction from mathematicaldocuments, which would greatly enhance the reach of themethods described in this paper.AcknowledgementsWork on the concepts presented here has been partially supportedby the Leibniz association under grant SAW-2012-FIZ_KA-2 and the German Research Foundation (DFG) undergrants KO 2428/9-1 and KO 2428/13-1.Bibliography[1] arxiv.org e-Print archive. URL: http://www.arxiv.org.[2] N. Bourbaki. Algebra I. Elements of Mathematics. SpringerVerlag, 1974.[3] W. M. Farmer. Mathematical Knowledge Management. InD. Schwartz and D. Te’eni, editors, Encyclopedia of KnowledgeManagement, pages 1082–1089. Idea Group Reference,2 edition, 2011.[4] W. M. Farmer, J. Guttman, and X. Thayer. Little theories.In D. Kapur, editor, Proceedings of the 11 th Conference onAutomated Deduction, number 607 in LNCS, pages 467–581,Saratoga Springs, NY, USA, 1992. Springer Verlag.[5] J. Harrison. Formal Proof – Theory and Practice. Noticesof the AMS, 55(11):1395–1406, 2008. URL: http://www.ams.org/notices/200811/tx081101395p.pdf.[6] J. Jeuring, J. A. Campbell, J. Carette, G. Dos Reis, P. Sojka,M. Wenzel, and V. Sorge, editors. Intelligent Computer Mathematics,number 7362 in LNAI. Springer Verlag, 2012.[7] A. Kohlhase and M. Kohlhase. Spreadsheet interaction withframes: Exploring a mathematical practice. In J. Carette,L. Dixon, C. Sacerdoti Coen, and S. M. Watt, editors, MKM/-Calculemus Proceedings, number 5625 in LNAI, pages 341–356. Springer Verlag, July 2009. URL: http://kwarc.info/kohlhase/papers/mkm09-framing.pdf.[8] M. Kohlhase. OMDoc – An open markup format for mathematicaldocuments [Version 1.2]. Number 4180 in LNAI.Springer Verlag, Aug. 2006. URL: http://omdoc.org/pubs/omdoc1.2.pdf.[9] M. Kohlhase. The flexiformalist manifesto. In A. Voronkov,V. Negru, T. Ida, T. Jebelean, D. Petcu, S. M. Watt, andD. Zaharie, editors, 14th International Workshop on Symbolicand Numeric Algorithms for Scientific Computing (SYNASC2012), pages 30–36, Timisoara, Romania, 2013. IEEE Press.URL: http://kwarc.info/kohlhase/papers/synasc13.pdf.[10] M. Kohlhase, B. A. Matican, and C. C. Prodescu. MathWeb-Search 0.5 – Scaling an Open Formula Search Engine. InJeuring et al. [6], pages 342–357. URL: http://kwarc.info/kohlhase/papers/aisc12-mws.pdf.[11] M. Kohlhase, H. Mihaljevic-Brandt, W. Sperber, andO. Teschke. Zentralblatt column: Mathematical formulasearch. EMS Newsletter, pages 56–57, Sept. 2013. URL: http://www.ems-ph.org/journals/newsletter/pdf/2013-09-89.pdf.[12] C. Lange, P. Ion, A. Dimou, C. Bratsas, J. Corneli,W. Sperber, M. Kohlhase, and I. Antoniou. Reimplementingthe mathematics subject classification (MSC) as alinked open dataset. In Jeuring et al. [6], pages 458–462.http://arxiv.org/abs/1204.5086 arXiv:1204.5086.[13] B. Laubner. Using theory graphs to map mathematics: A casestudy and a prototype. Master’s thesis, Jacobs University, Bremen,Aug. 2007. URL: https://svn.eecs.jacobs-university.de/svn/eecs/archive/msc-2007/blaubner.pdf.[14] Mizar. URL: http://www.mizar.org.[15] International Conference on Mathematic Knowledge Management.URL: http://www.mkm-ig.org/.[16] Mathematical reviews. URL: http://www.ams.org/mr-database.[17] I. Normann. Automated Theory Interpretation. PhD thesis,Jacobs University, Bremen, Germany, 2008. URL:https://svn.eecs.jacobs-university.de/svn/eecs/archive/phd-2008/normann.pdf.[18] The oaf project. URL: http://mathhub.info/OAF-project.[19] F. Rabe. The MMT API: A Generic MKM System. InJ. Carette, D. Aspinall, C. Lange, P. Sojka, and W. Windsteiger,editors, Intelligent Computer Mathematics, number7961 in Lecture Notes in Computer Science, pages 339–343.Springer, 2013. http://dx.doi.org/10.1007/978-3-642-39320-4doi:10.1007/978-3-642-39320-4.[20] F. Rabe and M. Kohlhase. Information & Computation,0(230):1–54, 2013. URL: http://kwarc.info/frabe/Research/mmt.pdf.[21] F. Wiedijk. Formal proof – Getting Started. Notices of theAMS, 55(11):1408–1414, 2008. URL: http://www.ams.org/notices/200811/tx081101408p.pdf.[22] Zentralblatt MATH. URL: http://www.zentralblatt-math.org/zbmath/.Michael Kohlhase [m.kohlhase@jacobsuniversity.de]is a full professor for computerscience at Jacobs University Bremenand an associate adjunct professorat Carnegie Mellon University. He studiedpure mathematics at the Universities ofTübingen and Bonn (1983–1989) and continuedwith computer science, in particularhigher-order unification and automated theorem proving(PhD 1994, Saarland University). His current researchinterests include knowledge representation for mathematics,inference-based techniques for natural language processingand computer-supported education.EMS Newsletter June 2014 27

WorldWorld ScientificConnecting Great MindsNOTABLE BOOKS BYFREEMAN DYSONFreeman John Dyson FRS, born December15, 1923, is an eminent English-bornAmerican physicist, mathematician, andfuturist. He is famous for his work inquantum electrodynamics, solid-statephysics, mathematics, astronomy andnuclear engineering, as well as a renownedand best-selling author. He has spent most of his life as aprofessor of physics at the Institute for Advanced Studyin Princeton, taking time off to advise the US governmentand write books for the public. He has won numerousnotable awards including the Enrico Fermi Award,Templeton Prize, Wolf Prize, Pomeranchuk Prize, andHenri Poincaré Prize.PROCEEDINGS OF THE CONFERENCEIN HONOUR OF THE 90TH BIRTHDAYOF FREEMAN DYSONNanyang Technological University, Singapore,26 – 29 August 2013edited by K K Phua & L C Kwek (Nanyang TechnologicalUniversity), N P Chang (City College of New York), & A H Chan(National University of Singapore)Professor Freeman Dyson, a greatphysicist, thinker and futurist, has beenvery active in scientific, literary andpublic policy activities throughout hiscareer. As a tribute to him on theoccasion of his 90th birthday and tocelebrate his lifelong contributions inphysics, mathematics, astronomy,nuclear engineering and globalwarming, a conference covering awide range of topics was held inSingapore, August 2013.This memorable volume collects an interesting lecture byProfessor Dyson, Is a Graviton Detectable?, and contributionsby many other speakers, including C DeWitt-Morette, M-L Ge,K Huang, T Kinoshita, J L Lebowitz, E H Lieb, M Longuet-Higgins,B H J McKellar and P F Schewe.Readership: Academics and students interested in high energyphysics, astrophysics, cosmology, and condensed matter physics.500pp April 2014978-981-4590-10-5 US$85 £56978-981-4590-70-9(pbk) US$44 £29BIRDS AND FROGSSelected Papers, 1990–2014by Freeman J Dyson (IAS, Princeton)This book, a sequel to the volume, SelectedPapers of Freeman Dyson with Commentary,published by the American MathematicalSociety in 1996, that contains a selection ofDyson’s papers up to 1990, is a collectionof the most interesting writings of FreemanDyson from the period 1990–2014, selectedby himself.It comprises five sections. The first introduces how the various itemscame to be written, while the second section carries Talks aboutScience. The most substantial is a lecture to the American MathematicalSociety entitled Birds and Frogs, describing two kinds of mathematician,with examples from real life. The next section has an important tributeto Yang Chen Ning written for his retirement banquet at Stony BrookUniversity, and is followed by a section on Politics and History. Thelast and final section carries a recent article, Is a Graviton Detectable?discussing the question whether any conceivable experiment coulddetect single gravitons, to provide direct evidence of the quantizationof gravity.Readership: Students of physics and mathematics and all members ofthe general public interested in science.320pp Dec 2014978-981-4602-85-3 US$58 £38978-981-4602-86-0(pbk) US$28 £18ADVANCED QUANTUM MECHANICSSecond Editionby Freeman Dyson (IAS, Princeton),translated and transcribed by David Derbes (University of Chicago)Reviews of the First Edition“The fame of these lectures as well as ofthis author, together with the enduringinterest in their contents attested bythis transcription, obviously classify thebook as of great interest to students andresearchers willing to hear a presentationof quantum electrodynamics from one ofthe founding fathers.”Zentralblatt MATHReadership: Physicists, graduate andundergraduate students.316pp Nov 2011978-981-4383-40-0 US$98 £65978-981-4383-41-7(pbk) US$42 £28AD/PK/05/14/03/JLOrder your copies @ www.worldscientific.com

Is a Graviton Detectable?Freeman Dyson (Institute for Advanced Study, Princeton, USA)Freeman Dyson (Institute for Advanced Study, Princeton, USA)Feature1 IntroductionI am enormously grateful to Dr. K. K. Phua, and to everyoneelse who had a hand in organizing this conference, for invitingme to visit Singapore. I am also grateful to my old and newfriends who came to Singapore to help me celebrate my birthday.As a former Brit, I am delighted to see this sparkling newcountry, which has prospered by giving free play to Chineseenterprise while still driving on the left side of the road.Now I come to the technical substance of my talk. It isgenerally agreed that a gravitational field exists, satisfyingEinstein’s equations of general relativity, and that gravitationalwaves traveling at the speed of light also exist. The observedorbital shrinkage of the double pulsar [1] provides directevidence that the pulsar is emitting gravitational waves atthe rate predicted by the theory. The LIGO experiment now inoperation is designed to detect kilohertz gravitational wavesfrom astronomical sources. Nobody doubts that gravitationalwaves are in principle detectable.This talk is concerned with a different question, whetherit is in principle possible to detect individual gravitons, orin other words, whether it is possible to detect the quantizationof the gravitational field. The words “in principle” areambiguous. The meaning of “in principle” depends on therules of the game that we are playing. If we assert that detectionof a graviton is in principle impossible, this may havethree meanings. Meaning (a): We can prove a theorem assertingthat detection of a graviton would contradict the laws ofphysics. Meaning (b): We have examined a class of possiblegraviton detectors and demonstrated that they cannot work.Meaning (c): We have examined a class of graviton detectorsand demonstrated that they cannot work in the environmentprovided by the real universe. We do not claim to have answeredthe question of “in principle” detectability accordingto meaning (a). In Sec. 3 we look at detectors with the LIGOdesign, detecting gravitational waves by measuring their effectson the geometry of space–time, and conclude that theycannot detect gravitons according to meaning (b). In Secs. 4and 5 we look at a different class of detectors, observing theinteractions of gravitons with individual atoms, and concludethat they cannot detect gravitons according to meaning (c). InSecs. 6 and 7 we look at a third class of detectors, observingthe coherent transitions between graviton and photon statesinduced by an extended classical magnetic field, and find thatthey also fail according to meaning (c).In Sec. 2 we look at a historic argument used by NielsBohr and Leon Rosenfeld to demonstrate the quantum behaviorof the electromagnetic field, and explain why this argumentdoes not apply to the gravitational field. In Sec. 8 webriefly examine the possibility of observing primordial gravitonsat the beginning of the universe by measuring the polarizationof the cosmic background radiation today. No definiteconclusions are reached. This talk is a report of workin progress, not a finished product. It raises the question ofthe observability of gravitons but does not answer it. There ismuch work still to do.2 The Bohr–Rosenfeld ArgumentBefore looking in detail at graviton detectors, I want to discussa general theoretical question. In 1933 a famous paperby Niels Bohr and Leon Rosenfeld, [2] was published in theproceedings of the Danish Academy of Sciences with the title,“On the Question of the Measurability of the ElectromagneticField Strengths.” An English translation by Brycede Witt, dated 1960, is in the Institute library in Princeton,bound in an elegant hard cover. This paper was a historic displayof Bohr’s way of thinking, expounded in long and convolutedGerman sentences. Rosenfeld was almost driven crazy,writing and rewriting 14 drafts before Bohr was finally satisfiedwith it. The paper demonstrates, by a careful and detailedstudy of imaginary experiments, that the electric andmagnetic fields must be quantum fields with the commutationrelations dictated by the theory of quantum electrodynamics.The field-strengths are assumed to be measured by observingthe motion of massive objects carrying charges and currentswith which the fields interact. The massive objects are subjectto the rules of ordinary quantum mechanics which setlimits to the accuracy of simultaneous measurement of positionsand velocities of the objects. Bohr and Rosenfeld showthat the quantum-mechanical limitation of measurement ofthe motion of the masses implies precisely the limitation ofmeasurement of the field-strengths imposed by quantum electrodynamics.In other words, it is mathematically inconsistentto have a classical electromagnetic field interacting witha quantum-mechanical measuring apparatus.A typical result of the Bohr–Rosenfeld analysis is theirequation (58),∆E x (1)∆E x (2) ∼ |A(1, 2) − A(2, 1)| . (1)Here the left side is the product of the uncertainties of measurementof two averages of the x-component of the electricfield, averaged over two space–time regions (1) and (2). Onthe right side, A(1, 2) is the double average over regions (1)and (2) of the retarded electric field produced in (2) by aunit dipole charge in (1). They deduce (1) from the standardHeisenberg uncertainty relation obeyed by the measuring apparatus.The result (1) is precisely the uncertainty relation impliedby the commutation rules of quantum electrodynamics.Similar results are found for other components of the electricand magnetic fields.The question that I am asking is whether the argument ofBohr and Rosenfeld applies also to the gravitational field. Ifthe same argument applies, then the gravitational field mustbe a quantum field and its quantum nature is in principle observable.However, a close inspection of the Bohr–Rosenfeldargument reveals a crucial feature of their measurement apparatusthat makes it inapplicable to gravitational fields. In theEMS Newsletter June 2014 29

Featurelast paragraph of Sec. 3 of the Bohr–Rosenfeld paper, theywrite: “In order to disturb the electromagnetic field to be measuredas little as possible during the presence of the test bodysystem, we shall imagine placed beside each electric or magneticcomponent particle another exactly oppositely chargedneutralizing particle.” The neutralizing particles have the followingfunction. Suppose we have a mass carrying a chargeor current J whose movement is observed in order to measurethe local electric or magnetic field. The movement of thecharge or current J produces an additional electromagneticfield that interferes with the field that we are trying to measure.So we must compensate the additional field by addinga second mass, carrying the charge or current −J and occupyingthe same volume as the first mass. The second mass isconstrained by a system of mechanical linkages and springsto follow the movement of the first mass and cancels the fieldsgenerated by the first mass. This cancellation is an essentialpart of the Bohr–Rosenfeld strategy. It is then immediatelyobvious that the strategy fails for measurement of the gravitationalfield. The test-objects for measuring the gravitationalfield are masses rather than charges, and there exist no negativemasses that could compensate the fields produced bypositive masses.The conclusion of this argument is that the Bohr–Rosenfeldanalysis does not apply to the gravitational field. This does notmean that the gravitational field cannot be quantized. It meansonly that the quantization of the gravitational field is not alogical consequence of the quantum behavior of the measuringapparatus. The fact that the electromagnetic field must bequantized does not imply that the gravitational field must bequantized.3 Can LIGO Detect a Graviton?In the LIGO experiment, if it is successful, we shall detecta classical gravitational wave, not an individual quantum ofgravity. A classical wave may be considered to be a coherentsuperposition of a large number of gravitons. LIGO is supposedto detect a wave with a strain amplitude f of the orderof 10 −21 . According to Landau and Lifshitz [3], page 370, theenergy density of this wave isc 2 E = ω 2 f 2 , (2)32πGwhere G is Newton’s constant of gravitation and ω is the angularfrequency. For a wave with angular frequency 1 Kilohertzand amplitude 10 −21 , Eq. (2) gives an energy density ofroughly 10 −10 ergs per cubic centimeter. A single graviton of agiven angular frequency ω cannot be confined within a regionwith linear dimension smaller than the reduced wave-length(c/ω). Therefore the energy density of a single graviton ofthis frequency is at most equal to the energy of the gravitondivided by the cube of its reduced wave-length, namely ω4 E s = . (3)c 3For an angular frequency of 1 Kilohertz, the single gravitonenergy density is at most 3 × 10 −47 ergs per cubic centimeter.So any gravitational wave detectable by LIGO must contain atleast 3×10 37 gravitons. This wave would be barely detectableby the existing LIGO. For a LIGO apparatus to detect a singlegraviton, its sensitivity would have to be improved by a factorof the order of 3 × 10 37 . Even this vast improvement of sensitivitywould probably not be sufficient, because the detectionof weak signals is usually limited not only by the sensitivityof the apparatus but also by the presence of background noise.But to see whether detection of single gravitons is possible inprinciple, we disregard the problem of background noise andanalyze the structure and operation of a super-sensitive LIGOdetector.For a rough estimate of the sensitivity of a LIGO apparatusrequired to detect a single graviton, we equate (2) with(3). This gives the strain f to be detected by the apparatus,f = (32π) 1/2 Lp ω, (4)cwhere L p is the Planck length G 1/2L p =c 3 = 1.4 × 10 −33 cm . (5)The strain is derived from a measurement of the variation ofdistance between two mirrors separated by a distance D. Thevariation of the measured distance is equal to fD, so longas D does not exceed the reduced wave-length (c/ω) of thegraviton. For optimum detectability we take D equal to (c/ω).Then the variation of distance is by (4)δ = (32π) 1/2 L p . (6)Up to a factor of order unity, the required precision of measurementof the separation between the two mirrors is equalto the Planck length, and is independent of the frequency ofthe graviton.Is it possible in principle for a LIGO apparatus to measuredistances between macroscopic objects to Planck-lengthaccuracy? The following simple arguments give a negative answerto this question. First consider the case in which the objectsare floating freely in space. The Heisenberg uncertaintyrelation between position and momentum of freely floatingobjects gives the lower boundMδ 2 ≥ T , (7)for the variation of distance δ, where M is the mass of eachobject and T is the duration of the measurement. Now T mustbe greater than the time (D/c) required to communicate betweenthe two mirrors. If δ is equal to the Planck length, (5)and (7) imply GMD ≤ . (8)c 2So the separation between the two mirrors is less than theSchwarzschild radius of each of them, the negative gravitationalpotential pulling them together is greater than Mc 2 , andthey are bound to collapse into a black hole before the measurementcan be completed.We next consider the situation that arises when the twomirrors are clamped in position by a rigid structure. In thiscase the precision of measurement of the distance betweenthe two mirrors is limited by quantum fluctuations of the rigidstructure. We use a simple dimensional argument to estimatethe magnitude of the fluctuations. Let s be the velocity ofsound in the structure, let D be the separation between themirrors, and let M be the mass of the structure. There will beat least one mode of sound-vibration of the structure which30 EMS Newsletter June 2014

Feature5 Thermal Graviton GeneratorsWe have a splendid natural generator of thermal gravitonswith energies in the kilovolt range, producing far more gravitonsthan any artificial source. It is called the sun. StephenWeinberg long ago calculated [5] the graviton luminosity ofthe sun, caused by gravitational bremsstrahlung in collisionsof electrons and ions in the sun’s core. A later calculation[6] corrected a mistake in Weinberg’s paper but does not substantiallychange the result. For an electron–ion collision withenergy E, the differential cross-section p(ω) for producing agraviton of energy ω is divergent at low energies, so thatthe total cross-section has no meaning. The physically meaningfulquantity is the integral of the differential cross-sectionmultiplied by the energy of the graviton, 320p(ω)ω dω = Z 2 α 2 L 29pE , (22)where α is the electromagnetic fine-structure constant and Zis the charge of the ion. Including a similar contribution fromelectron–electron collisions, (22) gives a total graviton luminosityof the sunL g = 79 Megawatts , (23)or about 10 24 gravitons per second with energy in the kilovoltrange. This gives a flux at the earth ofF g = 4 × 10 −4 gravitons per cm 2 per second . (24)If we imagine the whole mass of the earth to be availableas raw material for the manufacture of graviton detectors,with the cross-section (21) per electron and the flux (24), thecounting-rate is 2.4×10 −17 per second. If the experiment continuesfor the lifetime of the sun, which is 5 billion years, theexpected total number of gravitons detected will be 4. The experimentbarely succeeds, but in principle it can detect gravitons.According to Gould, [6] there exist in the universe sourcesof thermal gravitons which are stronger than the sun, namelyhot white dwarfs at the beginning of their lives, and hot neutronstars. Gould estimates the graviton luminosities of a typicalwhite dwarf and a typical neutron star to be respectively10 4 and 10 10 times solar. Their luminosities are roughly proportionalto their central densities. But the lifetimes duringwhich the stars remain hot are shorter than the lifetime of thesun, being of the order of tens of millions of years for thewhite dwarf and tens of thousands of years for the neutronstar. The lifetime output of gravitons will therefore be respectively100 and 10 5 times solar. To stretch the theoretical possibilitiesof detection to the limit, we may suppose the detectorto have mass equal to the sun and to be orbiting aroundthe source of gravitons at a distance of 0.01 astronomical unitwith an orbital period of 8 hours. Then the expected numberof gravitons detected will be of the order of 10 13 for thewhite dwarf and 10 16 for the neutron star. The detection rateis roughly one per minute for the white dwarf and 3 × 10 4 persecond for the neutron star. The conclusion of this calculationis that graviton detection is in principle possible, if we disregardthe problem of discriminating the graviton signal frombackground noise.The most important source of background noise is probablythe neutrinos emitted by the sun or the white dwarf orthe neutron star as the case may be. These neutrinos canmimic graviton absorption events by ejecting electrons fromatoms as a result of neutrino–electron scattering. The neutrinoshave higher energy than the gravitons, but only a smallfraction of the neutrino energy may be transferred to the electron.From the sun, about 10 14 neutrinos are emitted for eachgraviton, and the cross-section for neutrino–electron scatteringis about 10 20 times the cross-section for graviton absorption(see Ref. [7]). Therefore there will be about 10 34 neutrinobackground events for each graviton absorption event.For white-dwarfs and neutron-stars the ratio of backgroundto signal is even larger, since neutrino productionand scattering cross-sections increase with temperature morerapidly than graviton production and absorption cross-sections.Without performing detailed calculations, we can assert thatfor all thermal sources of gravitons the ratio of neutrino backgroundto graviton signal will be of the order of 10 34 orgreater. In all cases, the total number of detected gravitonevents is vastly smaller than the square-root of the numberof background events. The graviton signal will be swampedby the statistical scatter of the background noise.Before jumping to conclusions about the detectability ofgravitons, we must explore possible ways in which the neutrinobackground events might be excluded. The first possibleway is to surround the detector with a shield thick enough tostop neutrinos but let gravitons pass. If the shield is made ofmatter of ordinary density, its thickness must be of the order10 10 kilometers, and its mass is so large that it will collapseinto a black hole. The second possible way is to surround thegraviton detector with neutrino detectors in anticoincidence,to catch the outgoing neutrino after each scattering event. Thisway fails for the same reason as the shield. The neutrino detectorswould need to be at least as massive as the shield. Thethird possible way is to build a shield or a set of anticoincidencedetectors out of some mythical material with superhighdensity. The known laws of physics give us no clue as to howthis might be done. We conclude that, if we are using knownmaterials and known physical processes in a noisy universe,detection of thermal gravitons appears to be impossible.6 Nonthermal GravitonsIt is possible to imagine various ways in which energeticobjects such as pulsars may emit nonthermal gravitons ofhigh energy. One such way is a process first identified byGertsenshtein[8], the coherent mixing of photon and gravitonstates in the presence of an extended classical magnetic field.The graviton emission from various celestial objects resultingfrom the Gertsenshtein process was calculated by Papiniand Valluri [9]. Some interestingly high graviton luminositieswere predicted.The Gertsenshtein process results from the interaction energy 8πGh ij T ij , (25)c 4between the gravitational field h ij and the energy–momentumtensor T ij of the electromagnetic field. This interaction expressesthe fact that electromagnetic fields have weight, justlike other forms of energy. Now suppose that h ij is the field of32 EMS Newsletter June 2014

Featurea graviton traveling in the z direction and 1T ij = (B i + b i )(B j + b j ) , (26)4πis the energy–momentum of the photon magnetic field b i superimposedon a fixed classical magnetic field B i . Then theinteraction (25) contains the term 4GI = hc 4 xy B x b y , (27)bilinear in the graviton and photon fields. The effect of this bilinearterm is to mix the photon and graviton fields, so that aparticle that is created as a photon may be transformed into agraviton and vice versa. There is an oscillation between gravitonand photon states, just like the oscillation between neutrinostates that causes neutrinos to change their flavors whiletraveling between the sun and the earth. If a photon travels adistance D though a uniform transverse magnetic field B, itwill emerge as a graviton with probability GP = sin 2 1/2 BD D2c 2 = sin 2 , (28)Lwith the mixing-length 2c2 L =(29)G 1/2 Bindependent of wave-length. In all practical situations, D willbe small compared with L, so that GB 2 D 2 P = . (30)4c 4The quadratic dependence of P on D makes this process interestingas a possible astrophysical source of gravitons. Thenumerical value of L according to (29) is roughly 1025 L = , (31)Bwhen L is measured in centimeters and B in Gauss.We may also consider the Gertsenshtein process as the basisof a graviton detector consisting of a hollow pipe of lengthD filled with a transverse magnetic field B. The tube must beaccurately pointed at a putative source of gravitons in the sky.At the far end of the tube is a shield to block incident photons,and at the near end is a detector of photons resulting from theconversion of gravitons on their way through the tube. If D isone astronomical unit (10 13 cm), then (30) givesP = 10 −24 B 2 . (32)The field B must be very strong to obtain a reasonable rate ofconversion of gravitons to photons. A detector with the samedesign has been used in a real experiment to detect axionsthat might be created by thermal processes in the core of thesun [10]. The axion field is supposed to interact with the electromagneticfield with an interaction energy similar to (27),but with a much larger coupling constant. The experimentersat CERN in Switzerland are using a surplus magnet from theLarge Hadron Collider project as an axion-detector, pointingit at the sun and looking for kilovolt photons resulting fromconversion of axions into photons. The length of the magnetis 9 meters and the magnetic field is 9 × 10 4 Gauss. They havenot yet detected any axions.The Gertsenshtein process does not require the classicalmagnetic field to be uniform. For a nonuniform field, the conversionof photons to gravitons still occurs with probabilitygiven by (28), if we replace the product BD by the integralof the transverse component of B along the trajectory of thephotons. Likewise, the conversion will not be disturbed by abackground gravitational field, even when the field is strongenough to curve the photon trajectory, because the gravitationalfield acts in the same way on photons and gravitons.In a curved space–time, the photons and the gravitons followthe same geodesic paths, and the photon and graviton wavesremain coherent.7 Nonlinear ElectrodynamicsHowever, there is an important disturbing factor which wasneglected in previous discussions of the Gertsenshtein process.The disturbing factor is the nonlinearity of the electromagneticfield caused by quantum fluctuations of electron–positron pairs in the vacuum [11, 12]. The fourth-order termin the electromagnetic field energy density is (Ref. [12],page 190),α (E 2 − H 2 ) 2 + 7(E · H) 2 , (33)360π 2 Hc2where α is the fine-structure constant and m 2 c 3 H c = = 5 × 10 13 Gauss (34)eis the critical magnetic field at which electron–positron pairfluctuations become noticeable.When the field in (33) is divided into classical and photoncomponents as in (26), there is a term quadratic in both theclassical and photon fields,α360π 2 Hc2 (4(B · b) 2 + 7(B · e) 2 ) , (35)where b and e are the magnetic and electric fields of the photon.From (35) it follows that the photon velocity v is not equalto c but is reduced by a fraction vkαBg = 1 − =c2 360π 2 Hc2 . (36)The coefficient k is equal to 4 or 7 for a photon polarizedwith its magnetic field or its electric field parallel to B. Weconsider the case k = 4, since that case is more favorableto the Gertsenshtein process. Since the graviton field is notaffected by the nonlinear electromagnetic interaction (33), thegraviton velocity is precisely c, and the photon and gravitonwaves will lose coherence after traveling for a distance c 90π 2 cH 2 c 1043 L c = == . (37)gω αB 2 ω B 2 ωIf the propagation distance D is larger than L c , the Gertsenshteinprocess fails and the formula (30) for the photon–graviton conversion probability is incorrect. A necessary conditionfor the Gertsenshtein process to operate isDB 2 ω ≤ 10 43 . (38)Furthermore, even when the Gertsenshtein process is operating,the probability of photon–graviton conversion accordingto (30) and (38) is 1036 P ≤ . (39)B 2 ω 2EMS Newsletter June 2014 33

FeatureWe are interested in detecting astrophysical sources of gravitonswith energies up to 100 kilovolts, which means frequenciesup to 10 20 . With ω = 10 20 , (38) and (39) becomeD ≤ 1023B 2 , P ≤ 10−4B 2. (40)We consider two situations in which (40) has important consequences.First, with typical values for the magnetic fieldand linear dimension of a pulsar, B = 10 12 and D = 10 6 ,(40) shows that the Gertsenshtein process fails by a wide margin.The calculations of the graviton luminosity of pulsars inRef. [9] assume that the Gertsenshtein process is producinghigh-energy gravitons. These calculations, and the high luminositiesthat they predict, are therefore incorrect. Second,in the hollow pipe graviton detector which we consideredearlier, (40) shows that the Gertsenshtein process can operatewith a modest field, B = 10 5 Gauss, and a pipe lengthD = 10 13 cm, but the probability of detection of each gravitontraveling through the pipe is only 10 −14 . If the field is madestronger, the length of the pipe must be shorter according to(40), and the probability of detecting a graviton becomes evensmaller. The detector can work in principle, but fails for practicalreasons in the real universe.8 ConclusionsWe have examined three possible kinds of graviton detectorwith increasingly uncertain results. First, the LIGO detectorfor low-energy gravitons, which we prove ineffective as a consequenceof the laws of physics. Second, the gravitoelectricdetector for kilovolt gravitons, which we prove ineffective asa consequence of the background noise caused by neutrinoprocesses in the real universe. Third, the coherent gravitonconversiondetector for high-energy gravitons, is ineffectiveonly because of practical limits to the size of magnetic detectors.In addition to these three kinds of detector, there isa fourth kind which actually exists, the Planck space telescope,detecting polarization of the microwave backgroundradiation. According to Alan Guth [13], the polarization ofthe background radiation in an inflationary universe couldprovide direct evidence of the existence of single gravitonsin the primordial universe before inflation. The results of thePlanck polarization measurements are not yet published, andit remains to be seen whether the observations are able to distinguishbetween primordial gravitons and other gravitationaleffects of primordial matter. The question, whether a detectorof present-day microwave radiation is in principle able todetect primordial gravitons, remains open.Many papers have been published, for example Eppleyand Hannah [14] and Page and Geilker[15], claiming todemonstrate that the gravitational field must be quantized.What these papers demonstrate is that a particular theorywith a classical gravitational field interacting with quantummechanicalmatter is inconsistent. Page and Geilker assumethat the classical gravitational field is generated by the expectationvalue of the energy–momentum tensor of the matter inwhichever quantum state the matter happens to be. They performedan ingenious experiment to verify that this assumptiongives the wrong answer for a measurement of the gravitationalfield in a real situation.In this talk I am not advocating any particular theory of aclassical gravitational field existing in an otherwise quantummechanicalworld. I am raising three separate questions. I amasking whether either one of three theoretical hypotheses maybe experimentally testable. One hypothesis is that gravity isa quantum field and gravitons exist as free particles. A secondhypothesis is that gravity is a quantum field but gravitonsexist only as confined particles, like quarks, hidden insidecomposite structures which we observe as classical gravitationalfields. The third hypothesis is that gravity is a statisticalconcept like entropy or temperature, only defined for gravitationaleffects of matter in bulk and not for effects of individualelementary particles. If the third hypothesis is true, then thegravitational field is not a local field like the electromagneticfield. The third hypothesis implies that the gravitational fieldat a point in space–time does not exist, either as a classical oras a quantum field.I conclude that the first hypothesis may be experimentallytestable, but the second and third may not. Analysis of theproperties of graviton-detectors, following the methods of thispaper, cannot distinguish between the second and third hypotheses.Three outcomes are logically possible. If a gravitondetector is possible and succeeds in detecting gravitons, thenthe first hypothesis is true. If graviton detectors are possibleand fail to detect gravitons, then the first hypothesis is falseand the second and third are open. If a graviton detector is inprinciple impossible, then all three hypotheses remain open.Even if their existence is not experimentally testable, gravitonsmay still exist.The conclusion of the analysis is that we are still a longway from settling the question whether gravitons exist. Butthe question whether gravitons are in principle detectable isalso interesting and may be easier to decide.In conclusion, I wish to thank Tony Rothman and StevenBoughn, [16] for helpful conversations and for sharing theirthoughts with me before their paper was published.Bibliography[1] J. M. Weisberg and J. H. Taylor, in Binary Radio Pulsars, AstronomicalSociety of the Pacific Conference Series, Vol. 328,eds. F. Rasio and L. H. Stairs (Astronomical Society of thePacific, 2005).[2] N. Bohr and L. Rosenfeld, Kgl. Danske Vidensk. Selsk., Mat.-Fys. Med. 12, 8 (1933) [English translation: Bryce DeWitt, Onthe question of the measurability of the electromagnetic fieldstrengths, preserves the Germanic style of the original].[3] L. D. Landau and E. M. Lifshitz, The Classical Theory ofFields, Course of Theoretical Physics, Vol. 2, fourth revisedEnglish edition, translated by M. Hamermesh (Elsevier, Oxford,1975).[4] A. Einstein, Ann. der Phys. 17, 132 (1905).[5] S. Weinberg, Phys. Rev. B 140, 516 (1965).[6] R. J. Gould, Astrophys. J. 288, 789 (1985).[7] M. Fukugita and T. Yanagida, Physics of Neutrinos and Applicationsto Astrophysics (Springer-Verlag, Berlin, 2003).[8] M. E. Gertsenshtein, Zh. Eksp. Teor. Fiz. 41, 113 (1961) [Sov.Phys. JETP 14, 84 (1962)].[9] G. Papini and S.-R. Valluri, Astron. Astrophys. 208, 345(1989), and earlier papers there cited.[10] K. Zioutas et al., First results from the CERN axion solar telescope(CAST), arXiv:hep-ex/0411033v2 [for this reference Iam indebted to E. Witten].34 EMS Newsletter June 2014

ObituaryBill Thurston (1946–2012)John H. Hubbard (Cornell University, Ithaca, US)Photo courtesy of theCornell Math DepartmentWilliam Thurston (30 October1946–21 August 2012) wasa geometer: he taught a wholegeneration a new way to thinkabout geometry. In all the fieldsin which he worked, the “afterThurston’’ landscape is nearlyunrecognisable from what itwas “before Thurston’’.His influence still dominatesthese fields, through his ownresults and through his manystudents: he supervised 33 thesesand many of these former students are now majormathematicians in their own right. At last count (April2014), he had 177 students, grand-students and greatgrand-students.Brief biographyThurston received his bachelor’s degree from New College(now New College of Florida) in 1967. For his undergraduatethesis he developed an intuitionist foundationfor topology. Following this, he earned a doctorate inmathematics from the University of California, Berkeley,in 1972. His PhD advisor was Morris W. Hirsch and hisdissertation was on Foliations of Three-Manifolds whichare Circle Bundles.After completing his PhD, he spent a year at the Institutefor Advanced Study (where he worked with JohnMilnor and started Milnor’s interest in dynamical systems),then another year at MIT as an assistant professor.In 1974, he was appointed as a professor of mathematicsat Princeton University. In 1991, he returned toUC Berkeley as a professor of mathematics and in 1993became Director of the Mathematical Sciences ResearchInstitute. In 1996, he moved to University of California,Davis. In 2003, he became a professor of mathematics atCornell University.He did revolutionary work in foliation theory, lowdimensionaltopology, dynamics and geometric grouptheory.Foliation TheoryHis early work, in the early 1970s, was mainly about foliations.He solved so many problems that the field was“tsunamied’’: graduate students were discouraged fromits study because it appeared that Thurston would solveall the problems. Only now is the field catching up withhis contributions.At the time, finding a codimension one foliation of amanifold was a major result in its own right. He solvedthe problem completely: a compact manifold has a codi-mension one foliation if and only if its Euler characteristicis 0.He also found that any real number could be a Godbillon-Veyinvariant, a major open problem at the time.With John Mather, he showed that the cohomology ofthe group of homeomorphisms of a manifold is the samewhether one gives the group the discrete topology or thetopology of uniform convergence on compact sets. This resultwas prescient: at the time no one thought about suchquestions, but today they are central to a whole field.Results involving complex analysisIn the late 1970s and early 1980s, Thurston found a collectionof results that appear to be unrelated but areall closely connected; the connecting thread was Teichmüllertheory and complex analysis. This led Lars Ahlfors,a professor at Harvard, arguably the greatest complexanalyst of the 20th century and one of the two firstField’s Medallists, to write as his complete NSF proposalin the early 1980s: “I will continue to study the work ofBill Thurston.’’ Ahlfors was indeed awarded the grant.And Al Marden, of Minnesota, another great complexanalyst, claimed that “everyone saw that Thurston wasthe best complex analyst in the world’’, and created theGeometry Center as a playground for Thurston’s ideas.Many great things came out of the Geometry Center, inparticular the two videos Not Knot and Inside out, thatset a whole new standard for mathematical videos.The results that so electrified the mathematical communitywere the following.1. Homeomorphisms of surfaces.Every homeomorphism of a compact surface is eitherof finite order, reducible or pseudo-Anosov.2. Hyperbolization of 3-manifolds that fiber over thecircle.A compact 3-manifold that fibers over the circle hasa hyperbolic structure if and only if its monodromy ispseudo-Anosov.3. Hyperbolization of Haken manifolds.A compact Haken 3-manifold admits a hyperbolicstructure if and only if it contains no incompressibletori.4. Topological characterization of rational functions.A post-critically finite branched map of the 2-sphereto itself is equivalent to a rational function if and onlyif it admits no Thurston obstructions.Proving just one of these results requires an entire graduatecourse; each involves a multitude of totally new ideasand techniques. There are international conferences everyyear about just aspects of each of them.36 EMS Newsletter June 2014

Obituarythings. It has inspired literally hundreds of further papers;it has inspired a whole generation of mathematicians.Photo courtesy of Thurston 60th Birthday Conference, PrincetonUniversity 2007Further, the two hyperbolization theorems inspiredThurston to make the geometrization conjecture: everyirreducible compact 3-manifold has a natural geometry,which belongs to one of eight types. This conjecture wassolved by Perelman in 2005; the mathematical communityis still coming to terms with his proof.As an interesting sideline, the famous Japanese couturierIssey Miyake and his clothes designer Dai Fujiwaradesigned their 2010 collection after Thurston’s “eightgeometries’’. Bill was invited to the presentation of thecollection at Fashion Week in Paris; further, he arrangedthat I was also invited to the show. Bill really got into theswing of things: the set was suggestive of the fundamentaldomain for a Kleinian group and Bill wandered around itwinding giant ropes around Dai and himself, all the whilespeaking about how this reflected various aspects of thegeometry of knots. The press corps were mystified butmuch entertained, and had a field day.This was actually far more in keeping with Bill’s generaloutlook on mathematics (and life in general): he keptin his office and at home an extensive collection of toysand games that could be given mathematical meaning;he always thought that mathematics should be a playfulactivity.Dynamical systemsItem 4 on the list above is part of a bigger picture. Allof Bill’s thinking was influenced by dynamical systems,which are somehow the essence of “infinite processes’’.All of Bill’s major results are concerned with thequestion: what is the right geometry underlying thisproblem? He would create the geometry by some infiniteprocess, starting with some inappropriate geometryand applying a transformation coming from his problemover and over, until it converged to the right geometry.So in some sense all his work concerned dynamical systems.But more specifically, Milnor and Thurston wrotea paper about iteration in one dimension in the 1980s.This paper contains an enormous wealth of ideas involvingkneading sequences, zeta-functions and many otherGeometric group theoryAround 1980, Bill broadened his focus from groups thatarise in specific geometric problems to groups in general.I well remember feeling that this was an unfortunatedevelopment: I loved the immediacy of the geometricproblems and found the new direction too general andabstract.Not so, said Bill: “A group is a very geometric object.”It took me a long time to see that he was right. The bookWord processing in Groups was a massive attempt towrite in one place the many insights that Bill brought tothe subject. It took the combined efforts of Epstein, Paterson,Cannon, Holt, Levy and Thurston.The entire mathematical community will miss his deepinsights. I miss him on a more personal level: Bill was agood friend. He had a wonderfully sunny outlook and hekept his good temper through the incredible hardships ofhis final illness.John Hamal Hubbard was born on 6 or7 October 1945 (the actual date is unknown).He is a professor at CornellUniversity in Ithaca, New York, and atthe Université de Provence in Marseille,France. He is well known for mathematicalcontributions to the fields of complexdynamics and Teichmüller theory.Hubbard graduated with a Doctorat d’État from Universitéde Paris-Sud in 1973 under the direction of AdrienDouady. He has written many influential papers on complexdynamics and he has written several books. In 2006,he completed the first volume of a series devoted to Teichmüllertheory and applications to four revolutionary theoremsof Bill Thurston. In his free time, he enjoys long bikerides in Ithaca and he likes collecting various mushroomsfound across the Cornell campus; he can easily give thebinomial nomenclature for each species.Photo courtesy of Mathematisches ForschungsinstitutOberwolfach.EMS Newsletter June 2014 37

DiscussionTable 1a: Country of birth – World. Number of ICM speakers per country or region.Table 1b: Country of birth – Europe. Number of ICM speakers per country in Europe.ity of education in universities will decline, or fail to rise;hence the training of mathematics teachers will not besatisfactory; hence…Giving some numbers may illustrate the economic aspectof the discussion. Following the OCDE publicationEducation at a Glance 2013, Tables B1.3a and B1.3b, thetotal average expenditure for primary/secondary educationin the USA is $142,000 in purchasing power parity,and for undergraduate education it is $81,000. Hiringa foreign trained mathematician represents a saving of$223,000.As we have briefly discussed, brain drain is indeedan intricate matter. All four approaches to the problemhave a strong claim to the truth. I will not argue further– everyone should weigh the different aspects and considerhow they might be reconciled. Hopefully these datacan help both policymakers and our community to weighthe various arguments for the further development (andspread) of mathematics.II. The Results1. The median age of ICM lecturers is 46; the averageis slightly higher, at 46.66 ± 1. For plenary speakers, themedian is 52 and the average 51.66 ± 1, whereas for sectionalspeakers the median and the average are both 46.Among the speakers, there are 27 women.2. The average time difference between PhD year andyear of first degree is 5 years (for the 150 people forwhom I have reliable data). Because the notion of firstdegree varies considerably from country to country (seeitem III.2 below) and because some students do not goEMS Newsletter June 2014 39

DiscussionWestern Europe France 32Germany 11Italy 9UK 16Switzerland 3Rest of Western Europe 14.5Eastern EuropeUSSR (Russia, Belarus,Ukraine, Georgia,22.5Lithuania)Hungary 5Romania, Czech Republic,Serbia, Poland, Bulgaria8North America Canada 4USA 36South America Argentina 3Brazil 2Near East Asia Israel 9Rest of Near-Eastern Asia(Iran, Lebanon, Turkey)2East and South Asia China 9Japan 6Korea 6Rest of East and SouthAsia (India, Taiwan)3AfricaEgypt, Morocco, SouthAfrica1Oceania Australia 2Table 2: Country of undergraduate studies. Number of ICM speakersper country of undergraduate studies.directly from undergraduate to PhD studies, this numbershould be taken with caution, and overestimates slightlythe actual duration of PhD studies. However, it does indicatethat the path to high level international recognitiondoes not require finishing a PhD very quickly.3. Most of the details for the geography of ICM speakersare given below. However, one can summarise themain conclusions in a few lines.a. The countries giving birth to the largest number ofICM speakers are: France (30), USSR (27), USA (26),UK and Germany (12), Italy and China (9), Hungary(8), Israel (7) and Japan and Korea (6). Europe (notincluding the USSR) has altogether 95.b. Geographic mobility is marginal between countryof birth and country of secondary education, whichis expressed, for a mathematically alert audience, bythe fact that the transition matrix is almost diagonal.Between birth and undergraduate studies, this is stillthe case. Apart from mobility inside the USSR, theonly significant flow is to the USA and to the UK:10 persons in our sample left their country to studyin the USA at an undergraduate level, the numberbeing 3 for the UK. It would be very interesting tosee how the numbers change in the next 20 years, inview of the aggressive recruiting policies of a numberof universities in the US and the UK, for instancegeared to International Mathematics Olympiad participants.c. The situation is, expectedly, very different at the graduatelevel. The main beneficiary of the flow of studentsis the USA (which attracted 58.5 PhD foreignbornstudents, representing two thirds of the total of84.5 ICM speakers who obtained their PhD in theUSA); the UK is a distant second, with 4.5 foreignbornout of 16.5 PhD students, and Israel third, with 3foreign-born out of 7 students. The countries that losethe most at that level are: USSR (with 10 departuresout of 27 students), Korea (6 out of 6), Hungary (6 outof 8) and Israel (3 out of 7).Interestingly, greater Europe, not including the formerUSSR, loses 23 individuals (who obtained their PhDin the US) out of a total of 95.d. Finally, where do they work?- The USA has the lion’s share, with 73 speakers workingthere, before France (35), the UK (19.5), Canada(10.5), Germany (8), Switzerland (7), Italy (6.5) andJapan and Korea (6).- In terms of mobility, the main fact emerging from thetables is that 30 individuals who obtained their PhDin the US leave the US at some point after their PhDto work in another country (often their home country),whereas 20 who obtained their PhD outside ofthe US work there.III. Some remarks on the methodologyThe data that I gathered and analysed are the following:- Gender.- Date of birth.- Country of birth.- Country of primary-secondary education (K-12).- Country and institution of undergraduate education;date of undergraduate degree.- Country, institution and adviser of PhD training.- Main positions held before present.- Present position if different from the one on the officialICM programme.My aim was to study the national origins of the 206speakers and to describe their mobility. Obviously, notall of the information is statistically interesting: if thereare 1 or 2 speakers born in country C, where they studyor work does not teach us much.I chose to restrict the study to the 206 mathematicianswho are either plenary speakers or sectionalspeakers in one of the mathematics sessions (Sections1 to 17), at the exclusion of the history, popularisationand education sessions – not that I think that these activitiesare less worthy (in view of the fact that a substantialpart of my present activity revolves preciselyaround history of mathematics, popularisation of mathematicsand mathematics education, it would be somewhatparadoxical).40 EMS Newsletter June 2014

DiscussionCountry of PhD – World. Number of ICM speakers per country or region of PhD studies.Table 3b: Country of PhD – Europe. Number of ICM speakers per country of PhD studies in Europe.I started by searching the internet for publicly availableinformation on each of the 206 speakers. I foundmore than 80 CVs on their webpages, plus other piecesof information. Of course, the mathematics genealogyproject (http://genealogy.math.ndsu.nodak.edu/) provedto be extremely useful. I wrote directly to 60 of them forwhom my information was lacking, asking them to fillin a short questionnaire, guaranteeing that I would useit only for statistical purposes. I received 37 responses.Further research, and some educated guesses, providedme with reasonably reliable information (again, only forstatistical purposes) for all but two speakers.The information that proved to be the most difficultto gather was birth years. Therefore I do not attempt tocompute interesting information like the average age ofPhD graduation (see also the remark below). However,the available data does not indicate a strong pattern ofextremely quick studies, particularly at the PhD level.There are significant national differences in the organisationof studies in the present sample. Indeed, theEMS Newsletter June 2014 41

DiscussionTable 4a: Country of work – World. Number of ICM speakers per country or region of employment.Table 4b: Country of work – Europe. Number of ICM speakers per country of employment in Europe.42 EMS Newsletter June 2014

Discussion4+4 US system (4 years of undergraduate study andtheoretically 4 years of PhD) is not present everywhere.Many European countries did not, until recently, offera degree below the Master’s level – corresponding to 5years of university studies. In several countries, there aresystems with two doctorates: PhD and habilitation or, forpeople roughly over 50 in France, the thèse de 3ème cycleand the thèse d’Etat. In Italy, the PhD system is fairlyrecent; older colleagues do not hold degrees beyond thelaurea – a Master’s level degree. For those, I have chosento classify them as holding an Italian PhD.For most mathematicians in our sample from thecountries belonging to the former Soviet Union, theUSSR classification seems to be more relevant, sincemost of them studied before 1989. For this reason, I haveregrouped those mathematicians in the ‘USSR’ category,even though it may seem dubious in terms of the politicalsituation today. For similar reasons, I have groupedtogether in the tables former Eastern European countriesbelonging to the Eastern Bloc – of course todaymost of them belong to the European Union or plan tojoin it.As may be expected, there is no significant mobilitybetween birth and country of primary/secondary schooling.Therefore the tables I give omit primary/secondaryschooling data altogether.Several speakers have dual appointments: I have chosento count them as 0.5 for each country.It would certainly be very interesting to make a similarstudy for past and future ICMs.The tables below may reveal obvious mistakes in caseswhere the numbers are small. Please feel free to writeto me so that I can correct the data in my tables and possiblypost corrected tables.Martin Andler is a professor at the Universityof Versailles. Before that, he held aCNRS position, first at the Université Paris7 and then at Ecole Normale Spérieure.He has held visiting positions at MIT, RutgersUniversity and IAS Princeton. Hisresearch concerns representation theory ofLie groups and history of mathematics. He is heavily involvedin outreach activities as the President of Animath(www.animath.fr).FranceGermanyItalyUKSwitzerlandRest ofWestern EuropeUSSRHungaryRest ofEastern EuropeCanadaUSAArgentinaBrazilRest ofSouth AmericaIsraelRest of NearEastern AsiaFrance 27 1 2Germany 10 1 1Italy 0.5 7.5 1UK 1 11Switzerland 3Rest WesternEurope1.5 7.5 8USSR 1 17 6 3Hungary 1 1 2 4Rest EasternEurope1 2 5Canada 5USA 26Argentina 1 2Brazil 0.5 1.5Rest SouthAmerica2Israel 1 2 4Rest NearEastern Asia1 1 1China 1 5 2 1Japan 6Korea 6SingapureRest East andSouth Asia5Africa 1 1 1Oceania 2ChinaJapanKoreaSingapureRest of East andSouth AsiaOceaniaTable 5: Birth to PhD. The number on line i and column j is the number of speakers born in country i, who obtained their PhD in country j.EMS Newsletter June 2014 43

DiscussionFranceGermanyItalyUKSwitzerlandRest ofWestern EuropeUSSRHungaryRest ofEastern EuropeCanadaFrance 27.5 1 0.5 1Germany 6 3 2 1 1Italy 1 6.5 0.5 1UK 1 8 3Switzerland 0.5 1 1 0.5Rest WesternEurope3.5 1 4.5 0.5 6 1USSR 2 1 1 1.5 3.5 1 13 2 2Hungary 2 1 1 1 3Rest EasternEurope2 6Canada 2 3USA 1 0.5 2 22.5Argentina 1 2Brazil 2Rest SouthAmerica1 1Israel 4 3Rest NearEastern Asia1 1 1China 0.5 4 2 2 0.5Japan 6Korea 6Rest East andSouth Asia2 1 2Africa 1 1Oceania 1 1USAArgentinaBrazilRest ofSouth AmericaIsraelRest of NearEastern AsiaChinaJapanKoreaSingapureRest of East andSouth AsiaAfricaOceaniaTable 6: Birth to work. The number on line i and column j is the number of speakers born in country i, who work in country j.FranceGermanyItalyUKSwitzerlandRest ofWestern EuropeUSSRHungaryCanadaFrance 26.5 1 1 0.5 1.5Germany 7 3 2Italy 1 5.5 0.5 0.5UK 10.5 4 1 1Switzerland 1.5 1 1 0.5Rest Western3.5 0.5 1.5 1EuropeUSSR 2 1 1.5 2.5 1 7 2Hungary 1 1Rest Eastern1 2EuropeCanada 2 1 1USA 4 1 1 1 1 1 6.5 53 0.5 2 6 2 2.5Argentina 2Brazil 1.5Israel 4 3China 1 1Japan 6USAArgentinaBrazilIsraelRest of NearEastern AsiaChinaJapanKoreaSingapureRest of East andSouth AsiaAfricaOceaniaTable 7: PhD to work. The number on line i and column j is the number of speakers who obtained their PhD in country i, who work in country j.44 EMS Newsletter June 2014

Interview with Alois Kufner on theOccasion of his 80th BirthdayInterviewLuboš Pick (Charles University in Prague, Czech Republic)Alois Kufner (born 1 February1934 in Pilsen, CzechRepublic) is a professor ofmathematics at the MathematicalInstitute of theAcademy of Sciences of theCzech Republic in Prague.He was the director of thisinstitute from 1979 to 1990.He is the author of manybooks. His research interestsinclude mainly the theory offunction spaces and its applications.For many years he worked in the central bodiesof the Union of Czech Mathematicians and Physicists. Hehas spent a lot of effort working for the benefit of talentedyoung mathematicians.Do you still remember when and why you decided tobecome a mathematician?That happened against my own will. I wanted to becomea high-school teacher of mathematics and physics. HenceI applied as an undergraduate to the Faculty of Mathematicsand Physics at Charles University in Prague. Butexactly in that very year, the study programme for teacherswas carried over to a different faculty, specifically thePedagogical Faculty of the same university. In order tofollow my plan, I would have to change my application.But my teacher at the secondary school talked me out ofit and I ended up studying mathematics.The supervisor of your PhD thesis was Jindřich Nečas.Was collaboration with him exciting? Who else had influenceon your choice of research area?Let me first slightly correct you. My diploma thesis wason singular integral equations and my supervisor was JanPotoček. He had two diploma students and only one PhDposition. He offered that Ivo Babuška could take one ofthem over to his department of PDEs in the academy,and Babuška chose me. This resulted in my very first job,and I did the doctorate simultaneously with it. Babuška’sdepartment was quite impressive and the work was interesting.At that time, Sobolev spaces and functionalanalyticsolutions of PDEs were the cutting edge of researchin analysis, and I happily joined in. Nečas was amember of that department and worked with me in aninformal way. He pointed me towards applications ofweighted spaces, which has been my principal researchtopic ever since. I also collaborated with several studentsfrom Charles University, where Nečas had a part-timejob. In this way, I encountered nonlinear equations. Col-laboration with Nečas definitely had significant impact onme. Aside from him and Potoček, I consider also SergeyMikhailovich Nikolskii and Sergey L’vovich Sobolev myequally important (though rather remote) teachers.You authored an impressive array of books and monographs.Do you still remember the first one of them?Which of your books is your favourite one?I prefer books with an impact on the general public ratherthan monographs specialised in a narrow field. That’swhy I wrote three little books for the “School of YoungMathematicians”. My first book was a compendium onFourier series, written jointly with Jan Kadlec. Oddlyenough, it is one of my most cited ones. It is difficult tosay which of my books I like most: perhaps “Geometryof Hilbert Space” (in Czech), in which my main goal wasto explain to a wide range of readers (including studentsand applied people) the basics of functional analysis.You founded the “Prague school of function spaces”,now quite well known in the world. How did it allstart?I first studied Sobolev spaces in connection with PDEs.When Nečas moved to the USA in 1968, we began, togetherwith his students Oldřich John and SvatoplukFučík, studying function spaces systematically. We establishedan informal seminar. Later it moved from the universityto the academy and that was exactly the momentwhen the Prague school began. Its first generation wasformed by our diploma students.You rebuilt a once purely service Department of Mathematicsat the University of Pilsen into one of the bestmathematics departments in the country. How did youmanage it?I did not really build it – that is exaggerated. Severalquite able people had been there from the very beginning.I got there by accident. In 1972, during my militaryservice in Pilsen, I visited Jiří Holenda, a schoolmate ofmine from university. He persuaded me to invest timeand effort into reorganising their department, which thenseemed rather incoherent. I organised seminars on PDEsthere and I tried to attract some dedicated young people.Then the process started to run on its own. The expertiseand organising abilities of other people such as StanislavMíka, Pavel Drábek and Jiří Holenda significantly contributedto the rise and glory of the Pilsen department.During your directorship of the Mathematical Instituteof the Czechoslovak Academy of Sciences, in the middleof a politically difficult period, you kept some peopleEMS Newsletter June 2014 45

Interviewemployed who would otherwise have had problems findinga job. How did you manage it?First of all, I never had the ambition of being a directorand I got into this position by sheer coincidence of circumstances.I took it as a service to my “maternal” institution.The times were not easy indeed. My principal goalwas to keep a quiet atmosphere in the institute, rathercalm down than provoke conflicts. Somehow, it worked.You spent a lot of effort and energy for the benefit ofthe Union of Czech Mathematicians and Physicists andalso of the EMS. What are your impressions about theorigins and the current state of the EMS?I was a deputy of the union at the first informal meetingdedicated to establishing a European organisationof mathematicians, held in 1978 in Helsinki. Hospitalityand engagement of Finnish mathematicians were veryimportant for the beginning of the EMS. The EMS alsoprobably wouldn’t exist without the effort of Sir MichaelAtiyah and later of Professor Friedrich Hirzebruch. TheCzech union contributed quite significantly, too. I was intensivelyworking in the Extended Preliminary Committeeof the EMS till 1992. The current publication activityof the EMS is formidable. My overall impression is thatthe original goal of establishing an analogue of the AMShas not been met yet.You had a decisive impact on the fruitful long-termcollaboration of Czech mathematicians with expertsfrom abroad. Did your personal friendship with foreignmathematicians play a role here?Personal contacts have always been very important. Anothersignificant parameter is the location. For example,the Friedrich Schiller University in Jena was easy to travelto, and moreover it is not far from Prague. Collaborationwith the Soviet Union was also possible throughvarious mutual agreements. Contacts with the Royal Societyin the UK were more difficult to maintain, but possible.After 1990, everything became easy. I always madesome original contacts and my younger colleagues couldthen take advantage of them. An important factor wasthe organisation of conferences in our country since newcontacts were built there.her maximum support. I believe that she also has understandingfor my mathematics. I appreciate that she accompaniesme during my travels if circumstances allow.She is however not always delighted by me leaving heron her own during conference talks.What do you like most and what do you like least aboutbeing a mathematician?I like the life in general and the life of a mathematicianin particular. What I do not like is the sometimes negativeattitude of the public towards mathematics. Too oftenwe hear celebrities emphasising that they never likedmathematics, that they have never been capable of understandingit. Mathematics deserves better!Luboš Pick is a professor of mathematicalanalysis at Charles University in Prague,Faculty of Mathematics and Physics. Hestudied at Charles University in Pragueand obtained a PhD at the MathematicalInstitute of the Czechoslovak Academy ofSciences in 1990. In the period 1991–1994he was employed by the University of Wales in Cardiff,UK. He has written one monograph and over 60 researchpapers in the theory of function spaces. In his free timehe translates books from English to Czech and plays bassguitar in several rock and folk bands.The first Spring School on Nonlinear Analysis, FunctionSpaces and Applications (NAFSA) was organised in1978; the 10th one is just coming. How was the organisationin times of no email, internet or cell phones?NAFSA schools were an offspring of the Pilsen seminars.Conditions for the organising work were of course verydifferent to the current ones. But we were able to manageit even with quite modest means such as the usualmail. And hard conditions had a good impact on the integrityof our team.Your wife is a quite well-known translator from Germanand other languages. What is family life like fortwo professionals so deeply engaged in their work?We have been together for over 55 years. I respect hertranslating job very much and I have always tried to give46 EMS Newsletter June 2014

Raising Public AwarenessBuffon: Did He Actually Throw Sticks?Ehrhard Behrends (Freie Universität Berlin, Germany) and Jorge Buescu (Universidade de Lisboa, Portugal)George-Louis Leclerc, Comte de Buffon (1707–1788) isfamous for the following “experiment”:Find a room with a plank floor, and denote the widthof the planks by a. You will need a stick or some similarlyshaped object, whose length 2r must be less thana. This condition guarantees that the stick will crossthe edge of at most one plank when it is tossed onthe floor.today in the fact that in the fifth arrondissement, at thesouthern border of the Jardin des Plantes, near the JussieuCampus, there is to be found a Rue Buffon.La Rue Buffon.Buffon and the “needle experiment.”The probability P that such an event occurs (thatis, that the stick does not end up entirely on a singleplank) is then 4r/(pa). This formula contains themathematical constant p, and we have, therefore, theopportunity of computing this number “experimentally.”It will, to be sure, be necessary to throw thestick “very often,” a number of times that we shall denoteby n. If the stick lies across two boards k timesout of the n throws, then according to the law of largenumbers, k/n should be a good approximation of P,and by solving the equation P = 4r/(p a) for p, we willhave found a good approximation of p.(Of course, instead of planks and sticks, one could simplyuse paper, along with needles or matches; the onlyrequirement is that the distance between the lines on thepaper be large enough.)It is generally accepted that Buffon’s experiment representsthe first Monte Carlo experiment in the history ofmathematics, namely an experiment in which a randommethod is used to solve a problem at least approximately.Such methods can be used, for example, to compute definiteintegrals over multidimensional domains of definitionand solve numerous counting problems.Buffon’s family became wealthy through an inheritance,and so in his youth, Buffon became financially independent.Like many of his contemporaries, he becamefascinated by the rapidly developing natural sciences. Indeed,he was interested in everything. His studies led toa planned fifty-volume encyclopaedia Histoire naturelle,générale et particulière, thirty-six volumes of which werepublished. From 1739 on, he served as the administratorof the royal garden in Paris (“Jardin Royal,” today“Jardin des Plantes”). An echo of his service can be seenThere were plans for this street to play an important rolein November 2013 as part of a campaign for the popularizationof mathematics. The rpa committee (rpa = “raisingpublic awareness”) of the European Mathematical Society(EMS) met that month in Paris, and someone cameup with the idea of “reconstructing” the Buffon experimentin the Rue Buffon. The conditions could not havebeen more propitious, for the relevant mathematics canbe easily understood by a lay audience (while at the sametime being not at all trivial), and with a bit of luck andprofessional preparation, one could hope for good coveragein the media and to score a triumph for the popularizationof mathematics.But the project was cancelled. When some committeemembers got in touch with the Parisian historian ofmathematics Bernard Bru to learn more about the historicaldetails, they learned that there exists no historicalevidence that Buffon saw any relationship between histheoretical deliberations and a calculation to approximatethe value of p, nor whether his famous “experiment”was in fact ever carried out. We have here, then,an interesting example of the fact that historical truthand “generally accepted knowledge” need have little relationshipto each other. Let us now look more closely atthis discrepancy between fact and supposition.We begin with facts that are not in dispute.1. In 1733, Buffon submitted an article to the Royal Academyof Sciences (of which he became a member in 1734)in which he correctly calculated the probability that arandomly tossed stick of length 2r would intersect one ofseveral parallel lines an equal distance a apart (providedthat 2r < a):Sur un plancher qui n’est formé que de plancheségales & parallèles, on jette une Baguette d’une certainelongueur, & qu’on suppose sans largeur. Quandtombera-t-elle sur une seule planche? 1 (See [2])1Freely translated: “One tosses a stick of a certain length and ofnegligible thickness onto a floor consisting of parallel boardsof equal width. When will it fall on only a single board?EMS Newsletter June 2014 47

Raising Public AwarenessHe points out that one can use his formula to determinethe value of a for which the probability of landingon a single board is fifty percent: “Il y a donc une certainelargeur de la planche qui rendroit le pari ou le jeuégal.” 22. This research was published – in a more extensive version– in 1777, in his Histoire naturelle. It will now becomeeven clearer that the principal motivation for hisinvestigations was in calculating odds for gamblers:Je suppose que, dans une chambre dont le parquet estsimplement divisé par des points parallèles, on jette enl’air une baguette, et que l’un des joueurs parie que labaguette ne croisera aucune des parallèles du parquet,et que l’autre au contraire parie que la baguette croiseraquelques-unes des ces parallèles; on demande lesort de ces deux joueurs. (On peut jouer ce jeu sur undamier avec une aiguille à coudre ou une épingle sanstête.) 33. Buffon later carried out experiments related to theSt. Petersburg paradox. This involves a game in which afair coin is tossed repeatedly until it first lands showingheads. If this occurs at the kth toss, the player wins 2 kducats. Since the expected value of the player’s winningsis infinite, it makes sense that the cost to play the gameshould also be infinite.Buffon describes the game in [3], beginning on page394, and on page 399, one learns that he has made somerelevant experiments:J’ai donc fait deux mille quarante-huit expériences surcette question, c’est-à-dire j’ai joué deux mille quarante-huitfois ce jeu, en faisant jeter la pièce par unenfant. 44. Laplace took up Buffon’s needle problem and statedexplicitly that one could use the theoretical calculationsto determine an experimental approximation of p. Afterdetermining the probability that a line would be crossed,he writes:Si l’on projette un grand nombre de fois ce cylindre,le rapport du nombre de fois où le cylindre rencontreral’une des divisions du plan au nombre total desprojections sera, à très peu près, la valeur de 4r/(a p),2Freely translated: “There is, therefore, a certain width ofboard for which the wager – that is, the game – is fair.”3Freely translated: “I assume that within a room in which theparquet is simply divided by parallel points, one tosses a stickinto the air, and one player wagers that the the stick will notcross any of the parallels in the parquet, while the other playerwagers that it will cross one of the parallels. One asks forthe odds for each player. (One could also play this game on acheckerboard with knitting needles or headless pins.)” (See[3], p. 411ff.)4 Freely translated: I have carried out 2048 experiments withrespect to this question, that is, I have played this game 2048times, making use of a child to toss the coin.”ce qui fera connaître la valeur de la circonférence2p. 5 [9]5. Buffon’s needle problem seems to have sparked interestin actual experiments from the mid-19th centuryonwards. Apparently the first documented one was performedin 1850 by Rudolf Wolf [13], then a professorat the University of Bern. Augustus de Morgan refersin 1859 ([11], pp. 283–4) that a certain Ambrose Smithperformed the experiment in 1855 with 3204 trials and astudent of his with 600 trials.How reasonable are such experiments? The theory saysthat only results of the following type are to be expected.If one tosses the stick n times and hits a line k times, usingk/n in calculating an approximation to p, then with someprobability p, the result that one obtains is within somevalue e of p. Here one may choose a value of p (close to1) and a value of e (small) and one can then determinea suitable value of n, for example using Chebyshev’s inequality.Unfortunately, for moderately large p and notvery small e, the required n is astronomically large andconvergence exceedingly slow. Buffon’s method is thereforenot well suited to obtaining information about thedigits of the number p.It is worth noting in this connection the experimentsreported in 1901 of one Lazzerini, who maintained thatafter throwing 3408 sticks, he had obtained a value of paccurate to six decimal places ([10], p. 120). Papers byGridgeman [5] and Badger [1] refute this claim as extremelyunlikely and probably due to data manipulation.In notable contrast to the historically verified evidence,there remains the impression that Buffon explicitly hadin mind a determination of an approximate value of pby means of an “experiment” and that he in fact carriedit out:1. The Buffon problem is treated regularly in textbookson stochastic theory. In books on this subject in bothEnglish and German, we have found not a single one inwhich the least doubt is cast on the statement that “Buffonwished to calculate an approximation to p with hisexperiment”. (This is true, alas, of the textbook ElementareStochastik of the first author.)2. These textbook authors find themselves in good company,for even in books on the history of the theory ofprobability, it is maintained, without citing any sources,that Buffon performed such experiments. Here are twoexamples:- “It was originally performed with a needle” ([7], p.75).5Freely translated: If one tosses this cylinder with great frequency,then the quotient of the total number of throws forwhich the stick lands on one of the divisions of the plane andthe total number of throws will have approximately the value4r = (a p) which will suffice to determine the value of the circumference2p.”48 EMS Newsletter June 2014

Raising Public Awareness- Many investigators (including Buffon) used this resultfor the experimental determination of p” ([10], p. 120).(Other history books at least leave open the connectionof the approximation of p.) According to Bernard Bru,this situation is the result of a misunderstanding: the experimentson the St. Petersburg paradox were at sometime or other extrapolated to encompass experiments onp and then the “facts” were simply repeated without verificationfrom original sources.3. The internet is not a big help. At the website Mac TutorHistory of Mathematics (http://turnbull.mcs.st-and.ac.uk/history/), which we visit frequently and value highly, thefollowing is said about Buffon: “His most notable contributionto mathematics was a probability experimentwhich he carried out calculating p by throwing sticks overhis shoulder onto a tiled floor and counting the numberof times the sticks fell across the lines between the tiles.”(Many years ago, there was even more nonsense servedup. Instead of “sticks” being tossed, it was “white loavesof bread”. One does not have to look far to see how thaterror arose. The word “baguettes” in Buffon’s originaltext (see above) was mistranslated as the familiar stickshapedloaf of French bread (la baguette de pain). But “labaguette” has a number of meanings in French, includingsimply “stick”, which makes considerably more sensein this context. At the time, the first author notified themanager of the website to check the translation and soonthereafter the offending word was changed.It is also worth noting that Buffon’s “experiment”lends itself to a number of interesting generalisations.Here are a few examples:- What happens if the stick is longer than the distancebetween the planks? (This question was answered byLaplace. For more recent treatments, see [4] by P. Diaconis.)- Can one replace sticks with some two-dimensional surface,such as a coaster? (One can, of course, obtain formulasfor the probability but it depends on the shapeof the surface whether the number p will appear in theformula. Thus, for example, square coasters are suitablefor calculating approximations to p, while circularones are not.)- How does the situation change if one replaces the stickwith a curved segment in the plane? (See in this regard“Buffon’s Noodles” [12].)Buffon’s needle problem is the first in the then unknownterritory of geometric probability, and opened up a wholenew area of mathematical thinking. Klain and Rota statethat it is “(…) the theorem leading into the heart of GeometricProbability” ([8], p. 3).For this reason alone Buffon would rightfully deservea place in the history of mathematics. Regarding thequestions raised here, on the other hand, it is unlikelythat new documents will surface that will provide conclusiveinformation. Therefore, we recommend to all authorsof future books on stochastic or probability theorynot to involve Monsieur Buffon in any throwing of sticks,needles, loaves of bread or similar articles.In closing, we would like to thank Bernard Bru (Paris)and Eberhard Knobloch (Berlin) for their help in elucidatingthe problems described in this work.References[1] L. Badger, Lazzarini’s Lucky Approximation of p. Math. Mag. 67(1994), 2, pp. 83–91.[2] Leclerc de Buffon, Buffon’s presentation to l’Academie royal dessciences. From the bulletin of the academy, 1733, from p. 43: http://gallica.bnf.fr/ark:/12148/bpt6k3530m/f51.image.[3] Leclerc de Buffon, Essai d’Arithmetique morale. 1777.[4] P. Diaconis, Buffon’s Problem with a Long Needle. J. of AppliedProbability 13, 1976, pp. 614–618.[5] N. T. Gridgeman, Geometric probability and the number p. ScriptaMath. 25 (1960), 183–95.[6] Ph. Holgate, Buffon’s Cycloid. Biometrika 68, 1981, pp. 712–716.[7] A. C. King und C. B. Read, Pathways to Probability. 1963.[8] D. Klain, G.-C. Rota, Introduction to Geometric Probability. LezioneLincee, Cambridge University Press, Cambridge, 1997.[9] P.-S. Laplace, Theorie Analytique des Probabilites. 1812 (see p. 366in volume VII of the collected works).[10] L.E. Maistrov, Probability Theory: A historical sketch. 1974, p.120.[11] Augustus de Morgan, A budget of paradoxes. Cosimo Classics, NY,2007; 1st edition 1872.[12] J. F. Ramaley, Buffon’s Noodle Problem. American MathematicalMonthly 76, 1969, pp. 916–918.[13] H. Riedwyl, Rudolf Wolf’s contribution to the Buffon needleproblem (an early Monte Carlo experiment) and application ofleast squares. The American Statistician 44 (1990) 44, 2, 138–139.The present paper is an extended version of an articlethat was originally published in German by the first authorin the proceedings of the German Mathematical Society(March 2014; this version was translated to Englishby David Kramer).Ehrhard Behrends was a professorfor mathematics at FreieUniversität Berlin until March2014. For many years he was thesecretary of the German MathematicalSociety; at present heis the Chair of the rpa committeeof the EMS. He is responsible for the development ofwww.mathematics-in-europe.eu, a popular mathematicalwebpage under the auspices of the EMS.Jorge Buescu is a professor ofmathematics at the University ofLisbon. Besides his mathematicalactivity, he is highly engagedin the popularisation of mathematics,being the author of severalbooks. He is a member ofthe EMS rpa committee and an editor of the EMS Newsletter.EMS Newsletter June 2014 49

Mathematics EducationICMI ColumnMariolina Bartolini Bussi (Università di Modena e Reggio Emilia, Italy) and Jean-Luc Dorier (Université de Genève,Switzerland)The 23rd ICMI Study: Primary MathematicsStudy on Whole NumbersCo-Chairs: Mariolina Bartolini Bussi & Sun XuhuaA new study will be conducted by the International Commissionon Mathematical Instruction. This study, the 23rdled by the ICMI, addresses for the first time mathematicsteaching and learning in primary school (and pre-school),taking into account inclusive international perspectives,socio-cultural diversity and institutional constraints. Thebroad area of Whole Number Arithmetic (WNA), includingoperations and relations and arithmetic word problems,forms the core content of all primary mathematicscurricula. The study of this core content area is oftenregarded as foundational for later mathematics learning.However, the principles and main goals of instruction inthe foundational concepts and skills in WNA are far fromuniversally agreed upon, and practice varies substantiallyfrom country to country. An ICMI study that provides ameta-level analysis and synthesis of what is known aboutWNA would provide a useful base from which to gaugegaps and silences and an opportunity to learn from thepractice of different countries and contexts.Whole numbers are part of everyday language in mostcultures but there are different views on the most appropriateage at which to introduce whole numbers in theschool context. Whole numbers, in some countries, areintroduced in pre-school, where the majority of childrenattend before the age of 6. In some countries, primaryschooling comprises Grades 1–6; in others it comprisesGrades 1–5. Thus the entrance age of students for primaryschool may vary from country to country. For thesereasons, this study addresses teaching and learning WNAfrom the early grades, i.e. the periods in which WNA issystematically approached in formal schooling – in somecontexts this includes pre-school.Primary schooling is compulsory in most countries(in all Western countries), although there is considerablevariation in the facilities, resources and opportunities forstudents. This is the uneven context where mathematicsteaching and learning takes place. Mathematics is a centralfeature of early education and the content, quality anddelivery of the curriculum is of critical importance in viewof the kinds of citizens each country seeks to produce.In the international literature, there are many contributionsabout primary school mathematics. In manycases, especially in the West, early processes of mathematicalthinking, often observed in early childhood (i.e.3–8 year-old children), are also investigated by cognitiveand developmental psychologists. They sometimes studythe emergence of these processes in clinical settings,where children are stimulated by suitable models so asto observe the emergence of aspects such as one-to-onecorrespondences, counting, measuring and other proc-esses. In several countries, Piaget’s theory has been veryinfluential despite criticism. Neuroscientists have alsobeen studying for some years the emergence of “numbersense”. However, recent perspectives highlight that whatis still needed is serious and deep interdisciplinary workwith experts in mathematics education.The ICMI has acknowledged that it is timely to launch,for the first time in its history, an international study thatspecifically focuses on early mathematics education thatis both basic and fundamental. When foundational processesare concerned, a strong epistemological basis isneeded. This is where the involvement of the ICMI addsvalue with respect to analyses carried out in other fields.Such epistemological analysis was part of the classicalworks of professional mathematicians (e.g. Klein, Smithand Freudenthal) who played a major role in the historyof the ICMI and considered mathematics teaching as awhole.The ICMI study will be organised around five themesthat provide complementary perspectives on approachesto early WNA in mathematics teaching and learning. Contributionsto the separate themes will be distinguishedby the theme’s specific foci and questions, although itis expected that interconnections between themes willemerge and warrant attention.The five themes are:1. The why and what of WNA.2. Whole number thinking, learning and development.3. Aspects that affect whole number learning.4. How to teach and assess WNA.5. Whole numbers and connections with other parts ofmathematics.Themes 1 and 2 address foundational aspects from thecultural-historic-epistemological perspective and fromthe (neuro)cognitive perspective. What is especiallyneeded are reports about the impact of foundational aspectson practices (both at the micro-level of students andclassrooms and at the macro-level of curricular choices).Themes 3 and 4 address learning and teaching, respectively,although it is quite difficult, sometimes, to separatethe two aspects; for example, in some languages and cultures(e.g. Chinese, Japanese and Russian) the two wordscollapse into only one.Theme 5 addresses the usefulness (or the need) toconsider WNA in connection with (or as the basis for)the transition to other kinds of numbers (e.g. rationalnumbers) or with other areas of mathematics traditionallyseparated from arithmetic (e.g. algebra, geometryand modelling).ICMI Study 23 is designed to enable teachers, teachereducators, researchers and policymakers around the50 EMS Newsletter June 2014

Mathematics Educationworld to share research, practices, projects and analyses.Although reports will form part of the programme, substantialtime will also be allocated for collective work onsignificant problems in the field, which will eventuallyform part of the study volume. As in every ICMI study,ICMI Study 23 is built around an international conferenceand directed towards the preparation of a publishedvolume. The study conference will take place in Macau,China, and will be hosted by the University of Macau(3–7 June 2015).As is usual practice for ICMI studies, participation inthe study conference will be by invitation only for theauthors of submitted contributions that are accepted.Contributions have to be submitted before 15 September2014; they will be reviewed and a selection will be madeaccording to the quality of the work and the potential tocontribute to the advancement of the study, with explicitlinks to the themes contained in the discussion documentand the need to ensure diversity among the perspectives.The number of invited participants will be limited to approximately100 people.The first product of ICMI Study 23 is an electronicvolume of the proceedings, to be made available first onthe conference website and later on the ICMI website. Itwill contain all the accepted papers as reviewed papers inthe conference proceedings (with an ISBN number).The second product is a gallery of commented videoclipsabout practices in WNA, to be hosted on the conferencewebsite and later, possibly, on the ICMI website.The third product is the ICMI study volume. The volumewill be informed by the papers, the video-clips andthe discussions at the study conference as well as its outcomes.The International Programme Committee for ICMIStudy 23 invites submissions of contributions of severalkinds: theoretical or cultural-historic-epistemologicalessays(with deep connections to classroom practice,curricula or teacher education programmes); positionpapers discussing policy and practice issues; discussionpapers related to curriculum issues; reports on empiricalstudies; and video-clips on explicit classroom or teachereducation practice. To ensure rich and varied discussion,participation from countries with different economic capacityor with different cultural heritage and practices isencouraged.The ICMI Study 23 website is open at the address:http://www.umac.mo/fed/ICMI23/.The website contains a longer version of this discussiondocument and the detailed information about deadlinesfor submission; it will be regularly updated andused for sharing the contributions of those invited to theconference in the form of conference pre-proceedings.Further information may be requested at the followingaddress:icmiStudy23@gmail.com.The members of the International Programme Committeeare: Maria G. Bartolini Bussi (University of Modenaand Reggio Emilia, Italy), Xuhua Sun (University ofMacau, China), Berinderjeet Kaur (National Institute ofEducation, Singapore), Hamsa Venkatakrishnan (Universityof the Witwatersrand, Johannesburg, South Africa),Joanne Mulligan (Macquarie University, Sydney,Australia), Jarmila Novotna (Charles University, Prague,Czech Republic), Lieven Verschaffel (KU Leuven University,Belgium), Maitree Inpasitha (Khon Kaen University,Thailand), Sarah Gonzalez de Lora (PUC Madrey Maestra, Dominican Republic), Sybilla Beckmann(University of Georgia, Athens, GA USA), Roger E.Howe, ICMI Liaison (Yale University, New Haven, CT,USA), Abraham Arcavi, ex-officio, ICMI Secretary General(The Weizman Institute of Science, Rehovot, Israel)and Ferdinando Arzarello, ex-officio, President of theICMI (University of Turin, Italy).The ICMI Emma Castelnuovo Award for Excellencein the Practice of Mathematics EducationThe International Commission on Mathematical Instruction(ICMI) is committed to the “development ofmathematical education at all levels” and its aims are“to promote the reflection, collaboration, exchange anddissemination of ideas on the teaching and learning ofmathematics, from primary to university level. The workof the ICMI stimulates the growth, synthesis and disseminationof new knowledge (research) and of resources forinstruction (curricular materials, pedagogical methods,uses of technology, etc.)”.The ICMI has decided in the past to create two awardsto recognise outstanding achievements in mathematicseducation research: the Felix Klein Award, honouring alifetime achievement, and the Hans Freudenthal Award,recognising a major cumulative programme of research.In order to reflect a main aspect of the ICMI (as statedabove) not yet recognised in the form of an award, theICMI has decided to create a third award to recogniseoutstanding achievements in the practice of mathematicseducation. This award will be named after Emma Castelnuovo,an Italian mathematics educator born in 1913, tocelebrate her 100th birthday and honour her pioneeringwork. Further details are available at http://www.mathunion.org/icmi/activities/awards/emma-castelnuovoaward/.While preparing this column, we received the verysad news that, on 13 April, Emma Castelnuovo passedaway in her sleep. In the discussion lists of Italian mathematicsteachers, dozens of condolence and memorymessages have been posted. An interview with Emmacan be downloaded from http://www.icmihistory.unito.it/clips.php.EMF2015, Tipaza (Alger), Algeria,10–15 October 2015The scientific meetings of the Espace MathématiqueFrancophone have been organised every third year since2000 and are acknowledged as a regional conference bythe ICMI. The next one will take place in Tipaza (Algeria)in 2015 (10–15 October). The theme will be: ‘Culturalplurality and universality of mathematics: challenges andprospects for their teaching and learning’. A very shortexcerpt from the general presentation follows.EMS Newsletter June 2014 51

Mathematics EducationOne of the strengths of mathematics lies in the universalnature of the results. However, before reachingthe status of universality, each mathematical conceptfirst emerges in a particular cultural context and is enrichedby the contributions of various civilisations thathave contributed to its development and diffusion. Thus,depending on the period, various civilisations have hada leading role in the mathematical discoveries and disseminationof older concepts. The Maghreb (especiallythe territory that is now Algeria) was, at a turning pointin history, one of the important places in the developmentand dissemination of mathematics throughout theMediterranean region. The most famous examples arethe popularisation of the positional decimal system inthe 9th century and its circulation through the Maghreband Europe and also the development of symbolisationfor scoring fractions and writing equations and the developmentof combinatorics.This movement between the plurality of cultural rootsand universality of mathematics is found in many aspectsin teaching and learning mathematics. On the one hand,in the construction of knowledge the student meets theconcept in specific situations; then a necessary contextualisation/depersonalisationleads to the institutionalisationof knowledge. On the other hand, universal knowledgeis not taught directly but undergoes changes atdifferent levels of the chain of didactic transposition. Towhat extent are cultural roots being taken into accountin these transformations? Is it necessary to consider thiscultural dimension and how? To what extent would thecontextualisation of mathematical concepts in their culturaldimension possibly promote student learning? TheInternational Programme Committee of this conferenceis chaired by Laurent Theis (Université de Sherbrooke,Canada) with Ahmed Djebbar (Université de Lille,France) as vice-chair and Rachid Bebbouchi (UniversityHouari Boumediene, Alger) at the head of the localorganisation. Further information is available from thewebsite http://www.mathunion.org/ICMI > news or fromJean-Luc Dorier (Jean-Luc.Dorier@unige.ch), who is amember of the International Programme Committee.Educating the educators: Conference oninternational approaches to scaling-upprofessional development in maths and scienceeducation15–16 December 2014 in Essen, GermanyThe European project mascil (Mathematics and Sciencefor life!) and the German Centre for Mathematics TeacherEducation (DZLM) are pleased to host a conferenceat the University of Duisburg-Essen in the northwest ofGermany.This is the first international conference specificallydevoted to the topic of educating the educators (such asteachers, teacher educators and the educators of teachereducators as well as multipliers and institutions engagedin teacher professional development) in particular in relationto disseminating innovative teaching approaches,such as inquiry-based learning. It will serve as a lever andplatform for international exchange about concepts andexperiences concerning questions such as: ‘What are thefeatures of successful concepts and professional development?What are the needs and experiences of the differenttarget groups? Which pitfalls have to be avoided?’This international conference will connect researchersand practitioners engaged in the fields of maths andscience education in order to discuss concepts of scaling-upteacher professional development. Please visithttp://educating-the-educators.ph-freiburg.de for regularlyupdated information about the conference. We arelooking forward to receiving your contributions. Furtherinformation can also be obtained from Günter Törner(guenter.toerner@uni-due.de), Chair of the EMS-Committeeof Education.ERME Column, May 2014Viviane Durand-Guerrier (ERME President, Université de Montpellier 2, France)CERME 8-proceedingsThe 8th Congress of European Research in MathematicsEducation (CERME 8) took place in Antalya, Turkey,6-12 February 2013. The publication of the proceedingswas carried out by Maria Alessandra Mariotti (Chair ofthe CERME8-IPC), Behiye Ubuz (Chair of the CERM8-Loc) and Çiğdem Haser, with the collaboration of theWorking Group leaders and co-leaders. The proceedingshave been published by Middle East Technical University,Ankara, Turkey, on behalf of the ERME. They areavailable on the website of the ERME alongside the proceedingsof all previous CERMEs: http://www.mathematik.uni-dortmund.de/~erme/.News from CERME 9The 9th Congress of European Research in MathematicsEducation (CERME 9) will take place in Prague (CzechRepublic), 4–8 February 2015.Konrad Krainer (Austria) is the Chair of the IPC(International Programme Committee), which is composedof Jorryt Van Bommel (Sweden), Marianna Bosch(Spain), Jason Cooper (Israel), Andreas Eichler (Germany),Ghislaine Gueudet (France), Marja van der Heuvel-Panhuizen (the Netherlands), Uffe Jankvist (Denmark,Co-Chair), Maria-Alessandra Mariotti (Italy), DespinaPotari (Greece), Ewa Swoboda (Poland), Nad’a Vondrová(Czech Republic) and Carl Winsløw (Denmark).52 EMS Newsletter June 2014

Mathematics EducationNad’a Vondrová (Chair) and Jarmila Novotná (Co-Chair) are in charge of the local organisation.The chief aims of the ERME are to promote communication,cooperation and collaboration in researchin mathematics education in Europe, in order to knowmore about research and research interests in differentEuropean countries and to create opportunities forinter-European cooperation between researchers incollaborative projects. This conference is designed as astarting point in promoting these aims in a communicativespirit.In order to achieve these goals, CERME deliberatelyand distinctively moves away from research presentationsby individuals towards collaborative group work.Its main feature is the Thematic Working Group (TWG),whose members work together on a common researchdomain. TWGs will have about 12 hours over four daysin which to meet and progress their work. Each ThematicWorking Group (TWG) should aim to provide a goodscientific debate with the purpose of deepening mutualknowledge about the problems and methods of researchin the field. Conference participants are expected to workwithin just one group.There will be about 20 Thematic Working Groupsat CERME 9: details about the focus of each group canbe accessed from the list of the TWGs on the CERME9 website. Researchers wishing to have their work discussedat the conference should submit a paper or a posterproposal to one of these TWGs. CERME papers andposters must be about research (empirical, theoretical ordevelopmental). In addition, there will be plenary activitiesand a poster session.The deadline for submission of papers will be 15 September;the deadline for poster proposals will be 1 October.All the deadline details are available at http://www.cerme9.org/about-cerme-9/deadlines/.In order to support researchers who might not be ableto attend the CERME without financial assistance, theERME has developed the Graham Litter Fund. A limitedamount of financial support is available to researchers.Applications should be sent before 15 September2014. Information is available at http://www.cerme9.org/about-cerme-9/financial-support/.The ERME is deeply interested in the active participationin CERME of young researchers in mathematicseducation. From CERME 9 onwards, the InternationalProgramme Committee (IPC) will comprise two electedrepresentatives of Young Researchers, and the inclusionof Young Researchers in the co-chairing of the ThematicWorking Groups is considered.I invite you to visit the website of CERME 9: http://www.cerme9.org, which will be regularly updated.Chow Rings,Decomposition ofthe Diagonal, and theTopology of FamiliesClaire VoisinIn this book, Claire Voisin provides an introduction to algebraic cycleson complex algebraic varieties, to the major conjectures relating them tocohomology, and even more precisely to Hodge structures on cohomology.The volume is intended for both students and researchers, and not onlypresents a survey of the geometric methods developed in the last thirty yearsto understand the famous Bloch-Beilinson conjectures, but also examinesrecent work by Voisin.Annals of Mathematics Studies, 187Phillip A. Griffiths, John N. Mather, and Elias M. Stein, Series EditorsPaper $75.00 978-0-691-16051-1Cloth $1 65.00 978-0-691-16050-4Hangzhou Lectureson Eigenfunctions ofthe LaplacianChristopher D. SoggeBased on lectures given at Zhejiang University in Hangzhou, China,and Johns Hopkins University, this book introduces eigenfunctions onRiemannian manifolds. Christopher Sogge gives a proof of the sharp Weylformula for the distribution of eigenvalues of Laplace-Beltrami operators,as well as an improved version of the Weyl formula, the Duistermaat-Guillemin theorem under natural assumptions on the geodesic flow. Soggeshows that there is quantum ergodicity of eigenfunctions if the geodesicflow is ergodic.Annals of Mathematics Studies, 188Phillip A. Griffiths, John N. Mather, and Elias M. Stein, Series EditorsPaper $75.00 978-0-691-16078-8Cloth $165.00 978-0-691-16075-7See our E-Books atpress.princeton.eduEMS Newsletter June 2014 53

ZentralblattGuarding your Searches:Data Protection at zbMATHJens Holzkämper (FIZ Karlsruhe, Germany) and Olaf Teschke (FIZ Karlsruhe, Germany)This year, as in the quadrennial cycles before, the mathematicalcommunity eagerly awaits the announcementof the Fields Medallists during the opening ceremony ofthe ICM on 13 August. Though there are as usual manyspeculations and educated guesses, we can be pretty optimisticthat the names of the prize winners will onceagain remain secret up till then; indeed, discretion hasworked well so far and this has been a great part of thefun.Of course, perfect secrecy is hard to maintain and isprobably harder than ever nowadays due to technicaldevelopments. It is not necessary to kidnap and interrogatethe Chair of the Fields Committee; surveillanceof communication of the people involved would do (includingnot just possible committee members but, atleast at a later stage, press officers of mainstream media),since at least some of them would most likely notrefrain from using insecure channels. Alternatively, theanalysis of search queries as employed in trend miningcould evolve into a pretty clear picture of the ongoingdiscussions if restricted to a sufficiently adapteduser pool – which might be intricate for Google or Bing(where mathematics is just a tiny noise in the large datastream) but would surely be applicable for MathSciNetor zbMATH requests.Fortunately, this Orwellian scenario is unlikely, butonly since it appears that no one capable of doing sowould be interested in spoiling the party at the ICM.However, there is much more sensitive informationaround. Hiring decisions are often connected to an evaluationthrough scientific databases. On a general level,knowing in advance what mathematical research is goingon is definitely of interest both inside and outside ofmathematics. Frequently, allegations of plagiarism havebeen made, involving claims that new results might havebeen copied from a colleague and (pre-)published firstby someone else; on the other hand, knowing developmentsin many applications ahead of publication couldlead to significant advantages. Cryptography is a wellknownexample, which directly pertains to organisationsthat are capable of large-scale surveillance. However,mathematical results are part of so many applicationsthat it wouldn’t make sense to restrict to this (or extendto network algorithms, data mining, pattern recognition,etc.) as it is not certain whether more impact might comefrom theoretical foundations of number theory or fastermatrix multiplication. Mathematics is interesting as awhole, which is actually pursued on a rather transparentlevel by the NSA (which publicly spends a lot of moneyon grants, and even more on recruiting mathematicians).No doubt similar activities occur, if less prominently, inother services and regions 1 . With the information thatbecame public from the Snowden files, one would besurprised if there were no algorithms that keep track ofresearch activities and persons in areas like, for example,cryptography; and the uneasy feeling that these algorithmsmay raise alarm due to connections unknowneven to the researchers themselves, or just due to falsepositive signals, is certainly not a good environment forindependent research.While the solution of this dilemma obviously requiresefforts of the society as a whole, we can try toimprove things in our small area. More than 20 millionsearch queries every year in zbMATH are only a verytiny fraction of the world’s web traffic but possibly largeenough to derive sensitive information in our subject,especially when queries could be personalised (whichhappens, for example, when EMS member accounts areused). This confronts us with the task of taking measuresfor data protection – at least as far as can be donefrom the zbMATH supplier side. Concerning the dataconnection, an SSL certificate (on a non-Heartbleed 2 -affected server) has been set up over the last month toprotect zbMATH queries (soon to be upgraded furtherwith software that allows perfect forward secrecy 3 ), encryptingall data exchange between your browser andour servers.While this closes the most obvious vector of attack,the question of handling the information on our serversremains. The level of possible access by the secretservices to user data stored by providers has been a centraltopic in recent discussions. While connection dataare elusive, the only secure way to protect search logs atsupply servers is their permanent deletion. On the otherhand, erasure is in conflict with requests from librarians,and also possibly cripples functions of the interface.Hence there are decisions to be taken, which will be outlinedbelow.From the librarians’ side, there is an ongoing demandfor access data. At the moment, the most common standardis described by the COUNTER Code of Practice fore-Resources 4 . While the desire to evaluate the usage ofthe resources is highly understandable, the 90 pages (in-1While there has been a discussion about the impact of mathematicalfinance to the banking crisis, it seems that the ethicaldimension of mathematicians’ contribution to global surveillanceinfrastructure is yet unexplored.2The crypto-apocalypse of April 2014: http://heartbleed.com/,https://xkcd.com/1354/.3https://en.wikipedia.org/wiki/Forward_secrecy.4http://www.projectcounter.org/code_practice.html.54 EMS Newsletter June 2014

Zentralblattcluding appendices) of the recent 4th release indicatean increasing demand for the granularity of informationabout the usage of journals and databases – informationwhich can only be generated from access logs. Details invarious library requests from the past included, for example,differentiation between searches, clicks, long andshort views, and sessions, together with overall accessnumbers. The need for each single number makes senseon its own (since, for example, the total number of viewsor downloads for articles, reviews or profiles is not alwaysmeaningful) but there is a prospective danger thatalong with the legitimate wish to evaluate the relevanceof a resource, a framework evolves around quantitativemeasures that monitors user behaviour in too detailed away. What is lost is the awareness that such statistics simplycannot grasp fundamental aspects from the side ofmathematical content. This might be best illustrated withan example. At the climax of the El Naschie scandal, oneof the authors asked a librarian whether they had succeededin eliminating Chaos, Solitons and Fractals fromtheir Elsevier bundle. The surprising answer was: “Whyshould we? The access numbers skyrocketed over thelast few months – this journal is obviously of highest importancefor our mathematicians!” The conclusion is notjust the old insight that ideally mathematicians shouldhave the final decision on which resources they need butalso that they should not overly rely on possibly treacherousstatistics. What we would like to add is that thecreation of detailed statistics may evolve into a privacyproblem itself. This problem reaches even beyond librarylicensing – in the area of Open Access, the trendto justify relevance by download figures is even morepronounced; on the other hand, when such statistics areused for ranking purposes (“most popular article”, etc.),the threat of manipulation is immanent. Detection andlevelling of manipulation attempts would again requirean overhead of user surveillance which doesn’t seem desirable– hence, the preferable alternative seems to refrainfrom an overuse of quantitative data.This also concerns the second issue: availability offunctions based on usage data. Nowadays, we are accustomedfrom shopping platforms to seeing options like“most popular items” or “users interested in this alsoviewed…”. Wouldn’t it make sense to implement somethingsimilar in scientific databases? The barrier wouldbe, again, the willingness to exploit data from users at alarge scale. Though not personalised at the level of suchapplications, sensitive data may become available implicitly.As a very basic example, it is known (and plausible)that researchers often search for themselves to ensurethe correctness of their data. Therefore, publishing“popular searches” is not fully independent of informationon who uses the database to what extent, somethingwhich is certainly not of public interest. On a more sophisticatedlevel, let us consider the example of hiringmentioned above. Institutions that have special hiringseasons tend to have significantly larger access numbersto zbMATH in these months, which may indicate thatthe evaluation by profiles and reviews contributes muchto the database usage. A seemingly innocent functionlike “people searching for this person looked also for…”would be prone to inadvertently revealing competingapplications in the hiring process, which is certainly notdesirable. Again, even the attempt would require longtermstorage and data mining of usage data, with all theknown (and, most likely, many yet unknown) problemsinvolved.Our approach to the topic is rather straightforward.Traditionally, FIZ Karlsruhe (as provider of zbMATH)has very high standards for data protection 5 , which alsoderives from the history of supplying not only informationon scientific progress but also, for example, on patents,where usage data would reveal business-relevantdevelopment strategies. Beyond these standards (whichforbid, for example, monitoring individual actions) and,of course, the requirements of German data protectionlaw (which is generally considered to meet the highestlevels in international comparison), we would assess thevalue of user data protection much higher than the potentialbenefit of applications derived from them. Therefore,our decision is to delete user logs permanently afterthe finalisation of the general access statistics requiredby the libraries and to refrain from further analysis. Thedeletion currently takes places at the latest a year afteraccess.We hope that this policy finds the acceptance of thezbMATH user community.5Privacy policy available at http://www.fiz-karlsruhe.de/fiz_privacy_policy.html?&L=1EMS Newsletter June 2014 55

Book ReviewsBook ReviewsReviewer: Jean-Paul AlloucheAlan TuringHis Work and ImpactTS. Barry Cooper andJ. van Leeuwen, EditorsElsevier, Amsterdam, 2013xxi + 914 p.ISBN print 978-0-12-386980-7ISBN e-book 978-0-12-387012-4A hundred years after his birth, who – even without beinga mathematician or a computer scientist – has neverheard the name Alan Turing? But who can explain hiswork in detail or at least describe all the directions ofhis work? The book under review is a monumental (almost1000 pages) tribute to Alan Turing, which gatherstogether texts by Turing as well as comments and papersby 70 contributors. The “tour de force” is that it is quitepossible, and even recommended, to read the book notlinearly but by selecting pieces randomly, just for fun. Toquote the introduction: “So we have not tried to reproducethe style of an archive, rather aiming at a book to beread, to be dipped into for pure pleasure, to be enjoyedand browsed in office, on train or bus, or accompanyingthe proposer to some distant scientific meeting or placeof relaxation.” Before trying to describe the contentsmore precisely, we would like to underline, as the authorsof the book did, the enormous influence of Turingin many various fields. The regularly updated URL http://ftp.math.utah.edu/pub//bibnet/authors/t/turing-alan-mathison.htmlshows a bibliography of his work and relatedworks.The book is divided into four parts. The first of theseparts, entitled ‘How do we compute? Can we prove?’contains in particular the extraordinarily influential paper‘On Computable Numbers, with an Application tothe Entscheidungsproblem’. Almost everybody has heardthe expression ‘Turing machine’. Now the question ofwhether or not Turing is the real father of computers canbe addressed by typing the words ‘Turing machine’ intoa search engine on a … computer, and looking at the approximately2,200,000 answers. This first part of the bookends with a text by Turing ‘The reform of mathematicalnotation and phraseology’, where Turing indicates thathis statement of the “type principle” was suggested bylectures of Wittgenstein.The next part, entitled ‘Hiding and unhiding information:cryptology, complexity and number theory’, speaksof course of Turing cracking (or seriously contributingto cracking) the Enigma (the most important ciphermachine(s) used by the Germans during World War II).But this part also contains, in particular, two number-theoreticworks of Turing Some calculations of the RiemannZeta-function and (with S. Skewes) On a theorem of Littlewood,where the authors prove the existence of somereal x greater than 2 and less than a double exponentialconstant such that π (x) > li (x) (where, as usual, π (x) isthe number of primes less than x and li (x) is the logarithmicintegral of x, i.e. the principal value of ∫ 0xd t/log t).Readers interested in the question of intelligent computersor, more precisely, in the possible links between(human) intelligence and “intelligent” computers will bedelighted to read the third part ‘Building a brain: intelligentmachines, practice and theory’. This is, of course, avast field, whose most popular item is the so-called Turingtest: a test to show the ability of a machine to have(or to simulate?) intelligent human behaviour or, moreprecisely, to have behaviour that cannot be distinguishedfrom an intelligent human.The last part is entitled ‘The mathematics of emergence:the mysteries of morphogenesis’. It is devoted tothe important work of Turing in biology. The main pointsare Turing’s 1952 paper ‘The chemical basis of morphogenesis’and his work (manuscripts and lectures) on themorphogen theory of phyllotaxis.Full of enthusiasm for this book, I tried to find at leastone criticism. This was not that easy! Possibly a very minorpoint is that I dislike the word “impact” in the title(Webster: Latin impactus p.p. of impingere, to push, strikeagainst. See “Impinge”). The deep and longstanding influenceof Turing deserves a term better and more elegantthan this fashionable and somewhat hollow word.My short description of this exciting book is probablyfrustrating, and for two reasons: firstly, because it onlyscratches the surface of Turing’s work; secondly, becauseit does not (and cannot) give an idea of the extremelyinteresting contributions of the 70 contributors. We hopethat this possible frustration will urge the readers of thisbrief review to read the book. We are sure that they willdefinitely enjoy it. It is quite possible that they will alsofind there ideas or flashes of inspiration for their own research.56 EMS Newsletter June 2014

Book ReviewsDana MackenzieThe Universe in Zero WordsThe story of mathematics as toldthrough equationsPrinceton University Press, 2012224 p.ISBN print 978-0-691-15282-0ISBN e-book 978-1-400-84168-4Reviewer: Yuriy V. Rogovchenko (Kristiansand)The Newsletter thanks Zentralblatt MATH and Yuriy V.Rogovchenko for the permission to republish this review,originally appeared as Zbl 1273.00014.This nice book invites readers to explore the beauty ofthe most influential mathematical equations. Dr. DanaMackenzie, who became a full-time writer after being amathematics professor for thirteen years, writes in theintroduction: “Equations are the lifeblood of mathematicsand science. They are the brush strokes that mathematiciansuse to create their art, or the secret codethat they use to express their ideas about the universe.The rest of the world, outside of science, does not speakthe language of equations, and thus a vast cultural gaphas emerged between those who understand them andthose who do not. This book is an attempt to build abridge across that chasm. It is intended for the readerwho would like to understand mathematics on its ownterms, and who would like to appreciate mathematics asan art.”Although mathematics has numerous branches, allknown for their beauty, significance, and abundance ofimportant equations, the author decided to limit his selectiononly to four. “I consider the four main tributariesof mathematics to be algebra, geometry, applied mathematics,and analysis. All four of them mingle togetherand cooperate in a most wonderful way, and witnessingthis interaction is one of the great joys of being a mathematician.Nearly every mathematician finds himselfdrawn more to one of these tributaries than the others,but the beauty and power of the subject undoubtedlyderives from all four. For that reason, the four chaptersin this book each have a theme, or “storyline” runningthroughout, relating the evolution of the four branchesover the ages.” This book tells the story of twenty fourbeautiful and powerful equations that may be regardedas milestones in the development of mathematics, scienceand society.Recently, I had a pleasure of reviewing another nicebook by Professor Stewart [I. Stewart, Seventeen equationsthat changed the world. London: Profile Books]where the stories of some equations discussed in thisvolume can be found. It was really interesting to comparethe selection of equations by two authors. As Dr.Mackenzie explains, “the choice of equations was necessarilya matter of individual taste and preference.” Someof criteria he used “to decide what makes an equationgreat” are listed below.“1. It is surprising. […] [It] tells us something thatwe did not know before. 2. It is concise. […] It containsnothing but the essentials. 3. It is consequential. […] Theequations that make the deepest impression are the onesthat revolutionize mathematics, change our view of theworld, or change the material possibilities of our lives. 4.It is universal. […] An equation proven today will remaintrue forever.”Equations picked for this volume range from elementary,1 + 1 = 2 known to preschoolers, to inspirational Fermat’slast theorem x n + y n = z n still fascinating thousandsof amateurs, to exquisitely sophisticated Chern-Gauss-Bonnet equation ∫ Pf (Ω) 2π n χ() that allows to deduceinformation about the shape of the universe if the curvatureat each point is given. The book is written in avery transparent and elegant manner; it is both enjoyableand informative reading. The reader will absolutelylove exciting historical facts and excellent illustrations,diagrams, pictures carefully selected by the author. Thevolume concludes with a useful bibliography and a helpfulindex. A very entertaining text that appeals not onlyto mathematics enthusiasts, but also to a wide audiencewith a quite limited mathematical background.Yuriy Rogovchenko received his Ph.D.in Differential Equations from the Instituteof Mathematics, National Academyof Sciences of the Ukraine, in 1987. Heheld professor’s positions at the EasternMediterranean University, North Cyprus,University of Kalmar, Sweden and UmeåUniversity, Sweden. Since August 2012, Dr. Rogovchenkois Professor of Mathematics at the University of Agder,Norway. His research interests lie within qualitative theoryof ordinary, functional, and impulsive differential equations,perturbation methods, mathematical modeling inbiology, social sciences and economics. Dr. Rogovchenkowas nominated a distinguished reviewer of ZentralblattMATH in 2009 and received a Certificate of Recognitionfrom the President and CEO of FIZ Karlsruhe SabineBrünger-Weilandt and the President of the EuropeanMathematical Society Ari Laptev.EMS Newsletter June 2014 57

Book Reviews ReviewsMartin AignerMarkov’s Theorem and 100 Yearsof the Uniqueness ConjectureA mathematical journey fromirrational numbers to perfectmatchingsSpringer International Publishers,Cham, 2013, x + 257 pp.ISBN print: 978-3-319-00887-5ISBN e-book: 978-3-319-00888-2Reviewer: Franz Lemmermeyer (Jagstzell, Germany)The Newsletter thanks Zentralblatt MATH and Franz Lemmermeyerfor the permission to republish this review, originallyappeared as Zbl 1276.00006.Let α be a real number. Dirichlet showed that there exist infinitelymany fractions p q ∈ Q with |α− p q |≤ 1 . Now considerq 2all positive real numbers L such that |α − p q | < 1/Lq2 holdsfor infinitely many fractions p q. The supremum of all these Lis denoted by L(α); then L(α) = 0 if and only if α is rational,and L(α) ≥ 1 otherwise. The set L = {L(α) :α ∈ RQ}is called the Lagrange spectrum. A. Markoff [Math. Ann. 15,381–407 (1879); 17, 379–400 (1880)] showed that the Lagrange√spectrum below 3 consists of all numbers of the form9m2 − 4/m, where m runs through the “Markoff numbers”.These are defined as the set of all natural numbers x i occurringas solutions of the Diophantine equation x 2 1 + x2 2 + x2 3 =3x 1 x 2 x 3 . It is easy to show that every Markoff number appearsas the largest number in some Markoff triple (x 1 x 2 x 3 ), and theuniqueness conjecture predicts that each Markoff number isthe maximum of a unique Markoff triple.The investigation of the uniqueness conjecture from differentperspectives is the main goal of this book. After someintroductory chapters, the reader is introduced to Cohn matrices,the modular group SL(2, Z), free groups, graphs andtrees, and to partial results towards the uniqueness conjectureusing the arithmetic of quadratic fields. The whole discussionis very elementary, and requires no preliminaries except somefamiliarity with basic concepts of algebra and number theory.The reviewer regrets that the author has given in to thetemptation of keeping the book on a very elementary levelthroughout. Readers enjoying the section on hyperbolic geometrywill be well advised to have a look at the verynice book “Fuchsian groups” by S. Katok [Fuchsian groups.Chicago: The University of Chicago Press (1992)]. Similarly,Markoff’s original motivation for studying these questions,the theory of binary quadratic forms, is only briefly mentionedon pp. 36–38 even though quadratic forms cast theirshadows almost everywhere in this book: continued fractionsand the uni-modular group are intimately connected with Lagrangereduction of quadratic forms, and the arithmetic ofideals in quadratic number fields also is just one way of presentingGauss composition and the class group of forms. Thereaders will find a beautiful introduction to the dictionary betweenthese languages in the recent book “Algebraic theoryof quadratic numbers” by M. Trifkovič [Algebraic theory ofquadratic numbers. New York, NY: Springer (2013)].This beautiful book gives readers a chance to familiarizethemselves with a very simple and yet very difficult problemin number theory, and teaches them that it pays to look at aproblem from many different angles. I recommend it to all studentswho are already hooked to number theory, and perhapseven more to those who are not.Franz Lemmermeyer received his Ph.D. fromthe University of Heidelberg and is currentlyteaching mathematics at the gymnasium St.Gertrudis in Ellwangen. His interests includenumber theory and the history of mathematics.New journal from theNew in2014L’Enseignement MathématiqueOrgane officiel de la Commission internationale de l’enseignement mathématiqueISSN print 0013-8584 / ISSN online 2309-46722014. Vol 60, 2 issues. Approx. 450 pages. 17 x 24 cmPrice of subscription: 198 Euro (online only) / 238 Euro (print+online)European Mathematical Society Publishing HouseSeminar for Applied Mathematics, ETH-Zentrum SEW A27CH-8092 Zürich, Switzerlandsubscriptions@ems-ph.org / www.ems-ph.orgAims and Scope:The Journal was founded in 1899 by Henri Fehr (Geneva) and Charles-Ange Laisant (Paris). It is intended primarily for publicationof high-quality research and expository papers in mathematics. Approximately 60 pages each year will be devotedto book reviews.Editors:Anton Alekseev, David Cimasoni, Daniel Coray, Pierre de la Harpe, Anders Karlsson, Tatiana Smirnova-Nagnibeda, András Szenes (all Université deGenève, Switzerland), Nicolas Monod (EPFL, Switzerland), John Steinig, and Vaughan F. R. Jones (University of California at Berkeley, USA)58 EMS Newsletter June 2014

Letter to the EditorLetter to the EditorMassimo Ferri (Università di Bologna, Italy)The Role of ApplicationorientedMathematicsDear Editor,In the March Newsletter, Ciro Ciliberto, President of theItalian Mathematical Union, complained about the negativecomments on a Marie Curie fellowship proposal inmathematics, concerning a lack of interdisciplinarity andsocio-economic impact of the proposed project. I fullyagree with him on the fact that the evaluation criteriaseem not to comprehend the specificities of pure mathematicalresearch; I hope that it will be possible to givethese criteria more flexibility and also that interdisciplinaritywithin mathematics will be recognised as such.Still, it seems to me that this event stresses a point thathas already appeared in this newsletter in other forms:does mathematics “sell” itself conveniently to the scientificcommunity and society at large? I think it would benecessary to do it not only for getting funds, grants, etc.,but also as a fair attitude of mathematicians toward society.I’m not only speaking of popularisation – which isan important cultural issue anyway – but also of a bettertwo-way communication between mathematics andother scientific and technological areas. There is not just“applied” mathematics but “application-oriented” mathematics,inspired by the many facets of modern technol-ogy, which is growing fast in our departments and canprovide the link I am hoping for.Unfortunately, in my opinion, application-orientedmathematics is not given the correct appreciation withinmathematics when it comes to fund distribution and –above all – academic competition, at least in Italy. Theremay be several reasons for that: the claim that application-inspiredmathematics is of a lower level; the fact thatits correctness is more difficult to check; even a sort ofretaliation against the attitude denounced by ProfessorCiliberto (an attitude which has severe fallout in termsof funding). On the other side, mathematicians often andcorrectly “defend” on the media our discipline, pointingout its manifold applications.I think it’s high time to discuss this issue openly. I seetwo symmetric rigidities: a lack of sensitivity of somescientific environments to the specificities of pure mathematics,and a lack of sensitivity of some mathematicalenvironments to the importance of application-orientedresearch. Should we go on pretending that the problemdoes not exist? If not, how can we face it? Is this just anItalian phenomenon or a European one? I hope that theEMS Newsletter will offer its pages to a debate whichcannot be deferred any longer.Massimo FerriProfessor of Geometry at the University of BolognaNew journal from theNew in2014Annales de l’Institut Henri Poincaré DCombinatorics, Physics and their InteractionsISSN print 2308-5827 / ISSN online 2308-58352014. Vol 1, 4 issues. Approx. 400 pages. 17 x 24 cmPrice of subscription: 198 Euro (online only) / 238 Euro (print+online)European Mathematical Society Publishing HouseSeminar for Applied Mathematics, ETH-Zentrum SEW A27CH-8092 Zürich, Switzerlandsubscriptions@ems-ph.org / www.ems-ph.orgAims and Scope:The unfolding of new ideas in physics is often tied to the development of new combinatorial methods, and conversely someproblems in combinatorics have been successfully attacked using methods inspired by statistical physics or quantum fieldtheory. The journal is dedicated to publishing high-quality original research articles and survey articles in which combinatoricsand physics interact in both directions. Combinatorial papers should be motivated by potential applications to physicalphenomena or models, while physics papers should contain some interesting combinatorial development. Both rigorousmathematical proof and heuristic physical reasoning clearly labeled as such have a place in this journal.Editors-in-Chief:Gérard H. E. Duchamp (Univ. Paris XIII, France)Vincent Rivasseau (Univ. Paris XI, France)Alan Sokal (New York Univ., USA and Univ. College London, UK)Managing Editor:Adrian Tanasa (Univ. Paris XIII, France)EMS Newsletter June 2014 59

Personal ColumnPersonal ColumnPlease send information on mathematical awards anddeaths to Mădălina Păcurar [madalina.pacurar@econ.ubbcluj.ro]AwardsChristoph Aistleitner (Technische Universität Graz, Austria) hasbeen awarded the 2013 Information-Based Complexity YoungResearcher Award.Michèle Artigue (Université Paris Diderot) has been awardedthe 2013 Felix Klein Medal of the International Commission onMathematical Instruction (ICMI).Markus Bachmayr (RWTH Aachen University, Germany) hasbeen awarded the John Todd Award 2013 by the OberwolfachFoundation.Hugo Duminil-Copin (Université de Genève, Switzerland) hasbeen awarded the Oberwolfach Prize 2013 by the OberwolfachFoundation.Gerd Faltings (Max Planck Institute for Mathematics in Bonn,University of Bonn, Germany) has been awarded the 2014 KingFaisal International Prize for Science.Gert-Martin Greuel (Technische Universität Kaiserslautern,Germany) and Andreas Matt (Mathematisches ForschungsinstitutOberwolfach, Germany) have been awarded the Medien preisMathematik 2013 by the Deutsche Mathematiker Vereinigung(DMV) for their work on the project IMAGINARY.Camillo De Lellis (University of Zurich, Switzerland) and MartinHairer (Warwick University, UK) have been awarded the FermatPrize 2014 by the Institut de Mathématiques de Toulouse.John N. Mather (Princeton University) has been awarded theBrouwer Medal 2014 by the Royal Dutch Mathematical Society.Stefan Müller (University of Bonn, Germany) has been awardedthe Heinz Gumin Prize 2013 by the Carl Friedrich von SiemensStiftung.Maria Pe Pereira (Institut de Mathematiques de Jussieu, France)has been awarded the 2013 Rubio de Francia Prize of the RoyalSpanish Mathematical Society.János Pintz (Alfréd Rényi Mathematical Institute of the HungarianAcademy of Sciences, Hungary) and Cem Y. Yıldırım(Boğaziçi University, Istanbul, Turkey) have been awarded the2014 Cole Prize in Number Theory, together with Yitang Zhangand Daniel Goldston.Peter Sarnak (Institute for Advanced Study in Princeton, USA)has been awarded the 2014 Wolf Prize in Mathematics.Peter Scholze (University of Bonn, Germany) has been awardedthe 2013 SASTRA Ramanujan Prize.Yakov G. Sinai (Princeton University, USA, and the LandauInstitute for Theoretical Physics, Russian Academy of Sciences,Russia) will be awarded the 2014 Abel Prize by the NorwegianAcademy of Science and Letters.Herbert Spohn (Technical University in Munich, Germany) hasbeen awarded the Cantor Medal 2014 by the German MathematicalSociety.Cédric Villani (Université de Lyon, France) has been awardedthe 2014 Doob Prize of the American Mathematical Society(AMS).Yitang Zhang (University of New Hampshire, USA) has beenawarded the Rolf Schock Prize 2014 by the Royal SwedishAcademy of Sciences and also the Ostrowski Prize 2013 by theOstrowski Foundation.DeathsWe regret to announce the deaths of:Gerhard Becker (12 July 2013, Germany)Dan Laksov (25 October 2013, Norway)Kurt Leichtweiss (23 June 2013, Germany)Hanfried Lenz (1 June 2013, Germany)Günther Nürnberger (11 May 2013, Germany)Joel Sakarovitch (20 March 2014, France)Wolfgang Schwarz (19 July 2013, Germany)Christoph Scriba (26 July 2013, Germany)Stein Arild Strømme (31 January 1014, Norway)Piero Villaggio (4 January 2014, Italy)Marc Yor (10 January 2014, France)Jacqueline Ferrand (26 April 2014, France)60 EMS Newsletter June 2014

AMERICAN MATHEMATICAL SOCIETYASYMPTOPIAJoel Spencer, New York UniversityWith Laura Florescu, New York UniversityAsymptotics in one form or another are part of the landscape for everymathematician. The objective of this book is to present the ideas of how to approachasymptotic problems that arise in discrete mathematics, analysis of algorithms, andnumber theory. A broad range of topics is covered, including distribution of primeintegers, Erdös Magic, random graphs, Ramsey numbers, and asymptotic geometry.Student Mathematical Library, Vol. 71Jun 2014 195pp 9781470409043 Paperback €36.00ANALYSIS OF STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONSDavar Khoshnevisan, University of UtahThe stochastic PDEs that are studied in this book are similar to the familiar PDEfor heat in a thin rod, but with the additional restriction that the external forcingdensity is a two-parameter stochastic process, or what is more commonly the case,the forcing is a “random noise,” also known as a “generalized random field.” Atseveral points in the lectures, there are examples that highlight the phenomenon thatstochastic PDEs are not a subset of PDEs.CBMS Regional Conference Series in Mathematics, Vol. 119Jul 2014 116pp 9781470415471 Paperback €34.00A co-publication of the AMS and CBMSTHE GEOMETRIZATION CONJECTUREJohn Morgan, Simons Center for Geometry and Physics & Gang Tian, Princeton University andPeking UniversityProvides a complete proof of the geometrization conjecture, which describes allcompact 3-manifolds in terms of geometric pieces. The method is to understand thelimits as time goes to infinity of Ricci flow with surgery. The first half of the book isdevoted to showing that these limits divide naturally along incompressible tori intopieces on which the metric is converging smoothly to hyperbolic metrics and piecesthat are locally more and more volume collapsed. The second half of the book isdevoted to showing that the latter pieces are themselves geometric.Clay Mathematics Monographs, Vol. 5May 2014 291pp 9780821852019 Hardback €73.00A co-publication of the AMS and Clay Mathematics InstituteMATHEMATICS ACROSS THE IRON CURTAINA History of the Algebraic Theory of SemigroupsChristopher HollingsThe theory of semigroups is a relatively young branch of mathematics, with most ofthe major results having appeared after the Second World War. This book describesthe evolution of (algebraic) semigroup theory from its earliest origins to theestablishment of a full-fledged theory. Semigroup theory might be termed “Cold Warmathematics” because of the time during which it developed. There were thrivingschools on both sides of the Iron Curtain, although the two sides were not alwaysable to communicate with each other, or even gain access to the other’s publications.History of Mathematics, Vol. 41Sep 2014 449pp 9781470414931 Hardback €99.00To order AMS titles visit www.eurospanbookstore.com/amsCUSTOMER SERVICES:Tel: +44 (0)1767 604972Fax: +44 (0)1767 601640Email: eurospan@turpin-distribution.comFURTHER INFORMATION:Tel: +44 (0)20 7240 0856Fax: +44 (0)20 7379 0609Email: info@eurospangroup.comdistributed by Eurospan|group

New books published by theIndividual members of the EMS, member societiesor societies with a reciprocity agreement (such asthe American, Australian and Canadian MathematicalSocieties) are entitled to a discount of 20%on any book purchases, if ordered directly at theEMS Publishing House.Benoît Grébert (Université de Nantes, France) and Thomas Kappeler (Universität Zürich, Switzerland)The Defocusing NLS Equation and Its Normal Form (EMS Series of Lectures in Mathematics)ISBN 978-3-03719-131-6. March 2014. 176 pages. Softcover. 17 x 24 cm. 32.00 EuroThe theme of this monograph is the nonlinear Schrödinger equation. This equation models slowly varying wave envelopes in dispersive media and arises in variousphysical systems such as water waves, plasma physics, solid state physics and nonlinear optics. More specifically, this book treats the defocusing nonlinear Schrödinger(dNLS) equation on the circle with a dynamical systems viewpoint. By developing the normal form theory it is shown that this equation is an integrable partialdifferential equation in the strongest possible sense. In particular, all solutions of the dNLS equation on the circle are periodic, quasi-periodic or almost-periodic intime and Hamiltonian perturbations of this equation can be studied near solutions far away from the equilibrium.The book is not only intended for specialists working at the intersection of integrable PDEs and dynamical systems, but also for researchers farther away from thesefields as well as for graduate students. It is written in a modular fashion, each of its chapters and appendices can be read independently of each other.Emmanuel Hebey (Université de Cergy-Pontoise, France)Compactness and Stability for Nonlinear Elliptic Equations (Zurich Lectures in Advanced Mathematics)ISBN 978-3-03719-134-7. 2014. 304 pages. Softcover. 17 x 24 cm. 42.00 EuroThe book offers an expanded version of lectures given at ETH Zürich in the framework of a Nachdiplomvorlesung. Compactness and stability for nonlinear ellipticequations in the inhomogeneous context of closed Riemannian manifolds are investigated, a field presently undergoing great development. The author describesblow-up phenomena and presents the progress made over the past years on the subject, giving an up-to-date description of the new ideas, concepts, methods, andtheories in the field. Special attention is devoted to the nonlinear stationary Schrödinger equation and to its critical formulation.Intended to be as self-contained as possible, the book is accessible to a broad audience of readers, including graduate students and researchers.Robert J. Marsh (University of Leeds, UK)Lecture Notes on Cluster Algebras (Zurich Lectures in Advanced Mathematics)978-3-03719-130-9. 2013. 132 pages. Softcover. 17 x 24 cm. 28.00 EuroThe aim of these notes is to give an introduction to cluster algebras which is accessible to graduate students or researchers interested in learning more about the field,while giving a taste of the wide connections between cluster algebras and other areas of mathematics.The approach taken emphasizes combinatorial and geometric aspects of cluster algebras. Cluster algebras of finite type are classified by the Dynkin diagrams, so a shortintroduction to reflection groups is given in order to describe this and the corresponding generalized associahedra. A discussion of cluster algebra periodicity, which hasa close relationship with discrete integrable systems, is included. The book ends with a description of the cluster algebras of finite mutation type and the cluster structureof the homogeneous coordinate ring of the Grassmannian, both of which have a beautiful description in terms of combinatorial geometry.Isabelle Gallagher (Université Paris-Diderot, France), Laure Saint-Raymond (Université Pierre et Marie Curie, Paris) and Benjamin Texier (Université Paris-Diderot,France)From Newton to Boltzmann: Hard Spheres and Short-range Potentials (Zurich Lectures in Advanced Mathematics)978-3-03719-129-3. 2013. 150 pages. Softcover. 17 x 24 cm. 32.00 EuroThe question addressed in this monograph is the relationship between the time-reversible Newton dynamics for a system of particles interacting via elastic collisions,and the irreversible Boltzmann dynamics which gives a statistical description of the collision mechanism. Two types of elastic collisions are considered: hard spheres, andcompactly supported potentials.Following the steps suggested by Lanford in 1974, we describe the transition from Newton to Boltzmann by proving a rigorous convergence result in short time, as thenumber of particles tends to infinity and their size simultaneously goes to zero, in the Boltzmann-Grad scaling.This book is intended for mathematicians working in the fields of partial differential equations and mathematical physics, and is accessible to graduate students with abackground in analysis.Eduardo Casas-Alvero (Universitat de Barcelona, Spain)Analytic Projective Geometry (EMS Textbooks in Mathematics)ISBN 978-3-03719-138-5. 2014. 636 pages. Hardcover. 16.5 x 23.5 cm. 58.00 EuroThis book contains a comprehensive presentation of projective geometry, over the real and complex number fields, and its applications to affine and Euclideangeometries. It covers central topics such as linear varieties, cross ratio, duality, projective transformations, quadrics and their classifications – projective, affine andmetric –, as well as the more advanced and less usual spaces of quadrics, rational normal curves, line complexes and the classifications of collineations, pencils ofquadrics and correlations. Two appendices are devoted to the projective foundations of perspective and to the projective models of plane non-Euclidean geometries.The presentation uses modern language, is based on linear algebra and provides complete proofs. Exercises are proposed at the end of each chapter; many of themare beautiful classical results.The material in this book is suitable for courses on projective geometry for undergraduate students, with a working knowledge of a standard first course on linearalgebra. The text is a valuable guide to graduate students and researchers working in areas using or related to projective geometry.MATHEON – Mathematics for Key Technologies (EMS Series in Industrial and Applied Mathematics, Vol. 1)Peter Deuflhard, Martin Grötschel, Frank Schmidt (all Konrad-Zuse-Zentrum, Berlin, Germany), Dietmar Hömberg, Volker Mehrmann, Martin Skutella (all TechnischeUniversität Berlin, Germany), Ulrich Horst, Jürg Kramer (both Humboldt-Universität zu Berlin, Germany), Konrad Polthier, Christof Schütte (both Freie UniversitätBerlin, Germany) and Jürgen Sprekels (Weierstraß Institut für Angewandte Analysis und Stochastik, Berlin, Germany), Editors978-3-03719-137-8. 2014. 466 pages. Hardcover. 17 x 24 cm. 48.00 EuroMathematics: intellectual endeavor, production factor, key technology, key to key technologies? Mathematics is all of these! The last three of its facets have been thefocus of the research and development in the Berlin-based DFG Research Center MATHEON in the last twelve years. Through these activities MATHEON has become aninternational trademark for carrying out creative, application-driven research in mathematics and for cooperating with industrial partners in the solution of complexproblems in key technologies.Modern key technologies have become highly sophisticated, integrating aspects of engineering, computer, business and other sciences. Flexible mathematical models,as well as fast and accurate methods for numerical simulation and optimization open new possibilities to handle the indicated complexities, to react quickly, andto explore new options. Researchers in mathematical fields such as Optimization, Discrete Mathematics, Numerical Analysis, Scientific Computing, Applied Analysisand Stochastic Analysis have to work hand in hand with scientists and engineers to fully exploit this potential and to strengthen the transversal role of mathematicsin the solution of the challenging problems in key technologies. This book presents in seven chapters the highlights of the research work carried out in the MATHEONapplication areas: Life Sciences, Networks, Production, Electronic and Photonic Devices, Finance, Visualization, and Education.European Mathematical Society Publishing HouseSeminar for Applied Mathematics, ETH-Zentrum SEW A27Scheuchzerstrasse 70CH-8092 Zürich, Switzerlandorders@ems-ph.orgwww.ems-ph.org