Of the European Mathematical Society

NEWSLETTER OF THE EUROPEAN MATHEMATICAL SOCIETY

S E European March 2017 M M Mathematical Issue 103 E S Society ISSN 1027-488X

Features History Spectral Synthesis for Claude Shannon: His Work Operators and Systems and Its Legacy Diffusion, Optimal Transport Discussion and Ricci Curvature for Metric Mathematics: Art and Science Measure Spaces

Bratislava, venue of the EMS Executive Committee Meeting, 17–19 March 2017 New journals published by the

Editor-in-Chief: Mark Sapir, Vanderbilt University, Nashville, USA

Editors: Goulnara Arzhantseva (University of Vienna, Austria) Frédéric Chapoton (CNRS and Université de Strasbourg, France) Pavel Etingof (Massachussetts Institute of Technology, Cambridge, USA) Harald Andrés Helfgott (Georg-August Universität Göttingen, Germany and Centre National de la Recherche Scientifique (Paris VI/VII), France) Ivan Losev (Northeastern University, Boston, USA) Volodymyr Nekrashevych (Texas A&M University, College Station, USA) Henry K. Schenck (University of Illinois, Urbana, USA) Efim Zelmanov (University of California at San Diego, USA)

ISSN print 2415-6302 Aims and Scope ISSN online 2415-6310 The Journal of Combinatorial Algebra is devoted to the publication of research articles of 2017. Vol. 1. 4 issues the highest level. Its domain is the rich and deep area of interplay between combinatorics Approx. 400 pages. 17.0 x 24.0 cm and algebra. Its scope includes combinatorial aspects of group, semigroup and ring theory, Price of subscription: representation theory, commutative algebra, algebraic geometry and dynamical systems. 198¤ online only / 238 ¤ print+online Exceptionally strong research papers from all parts of mathematics related to these fields are also welcome.

Editors: Luc Devroye (McGill University, Montreal, Canada) Gabor Lugosi (UPF Barcelona, Spain) Shahar Mendelson (Technion, Haifa, Israel and Australian National University, Canberra, Australia) Elchanan Mossel (MIT, Cambridge, USA) J. Michael Steele (University of Pennsylvania, Philadelphia, USA) Alexandre Tsybakov (CREST, Malakoff, France) Roman Vershynin (University of Michigan, Ann Arbor, USA)

Associate Editors: Sebastien Bubeck (Microsoft Research, Redmond, USA) Sara van de Geer (ETH Zurich, Switzerland) Ramon van Handel (Princeton University, USA) ISSN print 2520-2316 Andrea Montanari (Stanford University, USA) ISSN online 2520-2324 Jelani Nelson (Harvard University, Cambridge, USA) 2018. Vol. 1. 4 issues Philippe Rigollet (MIT, Cambridge, USA) Approx. 400 pages. 17.0 x 24.0 cm Rachel Ward (University of Texas, Austin, USA) Price of subscription: 198¤ online only / 238 ¤ print+online Aims and Scope Mathematical Statistics and Learning will be devoted to the publication of original and high- quality peer-reviewed research articles on mathematical aspects of statistics, including fields such as machine learning, theoretical computer science and signal processing or other areas involving significant statistical questions requiring cutting-edge mathematics.

European Mathematical Society Publishing House Scheuchzerstrasse 70 [email protected] Seminar for Applied Mathematics, ETH-Zentrum SEW A21 CH-8092 Zürich, Switzerland www.ems-ph.org Contents Editorial Team Editor-in-Chief Vladimir R. Kostic European (Social Media) Valentin Zagrebnov Department of Mathematics Institut de Mathématiques de and Informatics Marseille (UMR 7373) – CMI University of Novi Sad Mathematical Technopôle Château-Gombert 21000 Novi Sad, Serbia 39, rue F. Joliot Curie e-mail: [email protected] 13453 Marseille Cedex 13, France Eva Miranda Society e-mail: [email protected] (Research Centres) Department of Mathematics Copy Editor EPSEB, Edifici P Universitat Politècnica Newsletter No. 103, March 2017 Chris Nunn de Catalunya 119 St Michaels Road, Av. del Dr Maran˜on 44–50 Aldershot, GU12 4JW, UK 08028 Barcelona, Spain Editorial – Message from the President – P. Exner...... 3 e-mail: [email protected] e-mail: [email protected] New Editors Appointed...... 4 Editors Vladimir L. Popov New Members of the EC of the EMS...... 5 (Features and Discussions) Report from the Executive Committee Meeting – R. Elwes & Ramla Abdellatif Steklov Mathematical Institute (Contacts with SMF) Russian Academy of Sciences S. Verduyn Lunel...... 7 LAMFA – UPJV Gubkina 8 Spectral Synthesis for Operators and Systems - 80039 Amiens Cedex 1, France 119991 Moscow, Russia e-mail: [email protected] e-mail: [email protected] A. Baranov & Y. Belov...... 11 Diffusion, Optimal Transport and Ricci Curvature for Metric Jean-Paul Allouche Michael Th. Rassias (Book Reviews) (Problem Corner) Measure Spaces - L. Ambrosio, N. Gigli & G. Savaré...... 19 IMJ-PRG, UPMC Institute of Mathematics Claude Shannon: His Work and Its Legacy - 4, Place Jussieu, Case 247 University of Zurich 75252 Paris Cedex 05, France Winterthurerstrasse 190 M. Effros & H.V. Poor...... 29 e-mail: [email protected] 8057 Zürich Visions for Mathematical Learning: The Inspirational Legacy e-mail: [email protected] Fernando Pestana da Costa of Seymour Papert (1928–2016) - C. Hoyles & R. Noss...... 34 (Societies) Volker R. Remmert Do We Create Mathematics or Do We Gradually Discover Depto de Ciências e Tecnolo- (History of Mathematics) gia, Secção de Matemática, IZWT, Wuppertal University Theories Which Exist Somewhere Independently of Us? Universidade Aberta, D-42119 Wuppertal, Germany V.L. Popov...... 37 Rua da Escola Politécnica, nº e-mail: [email protected] 141-147, 1269-001 Lisboa, Mathematics: Art and Science - A. Borel ...... 37 Portugal Vladimir Salnikov Results of a Worldwide Survey of Mathematicians on e-mail: [email protected] (Young Mathematicians’ Column) University of Luxembourg Journal Reform - C. Neylon, D.M. Roberts & M.C. Wilson...... 46 Jean-Luc Dorier Mathematics Research Unit M&MoCS – International Research Center on Mathematics and (Math. Education) Campus Kirchberg FPSE – Université de Genève 6, rue Richard Coudenhove- Mechanics of Complex Systems - F. dell’Isola, L. Placidi & Bd du pont d’Arve, 40 Kalergi E. Barchiesi...... 50 1211 Genève 4, Switzerland L-1359 Luxembourg [email protected] [email protected] Join the London Mathematical Society (LMS) - E. Fisher...... 53 ICMI Column - J.-L. Dorier...... 56 Javier Fresán Dierk Schleicher (Young Mathematicians’ Column) (Features and Discussions) Book Reviews...... 58 Departement Mathematik Research I Solved and Unsolved Problems - M.T. Rassias...... 61 ETH Zürich Jacobs University Bremen 8092 Zürich, Switzerland Postfach 750 561 e-mail: [email protected] 28725 Bremen, Germany The views expressed in this Newsletter are those of the [email protected] authors and do not necessarily represent those of the EMS or the Editorial Team. Olaf Teschke (Zentralblatt Column) ISSN 1027-488X FIZ Karlsruhe © 2017 European Mathematical Society Franklinstraße 11 Published by the 10587 Berlin, Germany EMS Publishing House e-mail: [email protected] ETH-Zentrum SEW A27 CH-8092 Zürich, Switzerland. homepage: www.ems-ph.org Scan the QR code to go to the For advertisements and reprint permission requests Newsletter web page: contact: [email protected] http://euro-math-soc.eu/newsletter

EMS Newsletter March 2017 1 EMS Agenda EMS Executive Committee EMS Agenda

President Prof. Stefan Jackowski 2017 (2017–2020) Prof. Pavel Exner Institute of Mathematics 7 – 8 April (2015–2018) University of Warsaw EMS Committee for Developing Countries (CDC) Meeting, Doppler Institute Banacha 2 Rome, Italy Czech Technical University 02-097 Warszawa Brˇehová 7 Poland 28 – 29 April 11519 Prague 1 e-mail: [email protected] European Research Centres on Mathematics (ERCOM) Czech Republic Prof. Vicente Muñoz Meeting, Linz, Austria e-mail: [email protected] (2017–2020) Departamento de Geometria y Vice-Presidents Topologia Facultad de Matematicas EMS Scientific Events Universidad Complutense de Prof. Volker Mehrmann 2017 (2017–2018) Madrid Plaza de Ciencias 3 Institut für Mathematik 29 May – 2 June TU Berlin MA 4–5 28040 Madrid Spain Conference “Representation Theory at the Crossroads of Strasse des 17. Juni 136 4 e-mail: [email protected] Modern Mathematics” in honor of Alexandre Kirillov for his 3 th 10623 Berlin birthday. Reims, France Germany Prof. Beatrice Pelloni http://reims.math.cnrs.fr/pevzner/aak81.html e-mail: [email protected] (2017–2020) Prof. Armen Sergeev School of Mathematical & 12 – 15 June (2017–2020) Computer Sciences Meeting of the Catalan, Spanish and Swedish Mathematical Steklov Mathematical Institute Heriot-Watt University Societies (CAT-SP-SW-MATH), Umeå, Sweden Russian Academy of Sciences Edinburgh EH14 4AS EMS Distinguished Speaker: Kathryn Hess Gubkina str. 8 UK http://liu.se/mai/catspsw.math?l=en 119991 Moscow e-mail: [email protected] Russia Prof. Betül Tanbay 26 – 30 June e-mail: [email protected] (2017–2020) EMS-ESMTB Summer School 2017: Mathematical Modeling Department of Mathematics in Neuroscience, Copenhagen, Denmark Bogazici University http://dsin.ku.dk/calendar/ems-esmtb-summer-school/ Secretary Bebek 34342 Istanbul Prof. Sjoerd Verduyn Lunel Turkey 26 – 30 June (2015–2018) e-mail: [email protected] EMS summer school: Interactions between Dynamical Department of Mathematics Systems and Partial Differential Equations, Barcelona, Spain Utrecht University EMS Secretariat Budapestlaan 6 10 – 19 July 3584 CD Utrecht Ms Elvira Hyvönen Foundations of Computational Mathematics (FoCM’17), The Netherlands Department of Mathematics Barcelona, Spain e-mail: [email protected] and Statistics EMS Distinguished Speaker: Mireille Bousquet-Mélou P. O. Box 68 http://www.ub.edu/focm2017/ Treasurer (Gustaf Hällströmin katu 2b) 00014 University of Helsinki 17 – 21 July Finland Summer School: Between Geometry and Relativity, Prof. Mats Gyllenberg Tel: (+358) 2941 51141 ESI Vienna, Austria (2015–2018) e-mail: [email protected] http://www.univie.ac.at/AGESI_2017/school/ Department of Mathematics Web site: http://www.euro-math-soc.eu and Statistics 24 – 28 July University of Helsinki EMS Publicity Officer 31st European Meeting of Statisticians, Helsinki, Finland P. O. Box 68 EMS-Bernoulli Society Joint Lecture: Alexander Holevo 00014 University of Helsinki Dr. Richard H. Elwes http://ems2017.helsinki.fi Finland School of Mathematics e-mail: [email protected] University of Leeds 3 – 9 September Leeds, LS2 9JT EMS Summer School “Rationality, stable rationality and bira- Ordinary Members UK tionally rigidity of complex algebraic varieties”, Udine, Italy e-mail: [email protected] http://rational.dimi.uniud.it Prof. Nicola Fusco (2017–2020) 24 – 29 September Dip. di Matematica e Applicazioni 5th Heidelberg Laureate Forum Complesso Universitario di http://www.heidelberg-laureate-forum.org/ Monte Sant’ Angelo Via Cintia 80126 Napoli 15 – 19 November Italy ASTUCON – The 2nd Academic University Student Conference (Science, Technology Engineering, Mathematics), e-mail: [email protected] Larnaca, Cyprus

2 EMS Newsletter March 2017 Editorial Editorial – Message from the President Pavel Exner, President of the EMS

Dear EMS Members, Dear Friends,

Any society’s life contains both climactic and anticli- mactic periods. The year just concluded was full of major events but while the New Year promises many things we look forward to, it is day-to-day work that will provide the leitmotiv of the coming months. This includes the implementation of the decisions taken at our council meeting in Berlin last Summer. To begin with, the society’s leadership has been substantially renewed. With deep gratitude for their devoted work, we part with Franco Brezzi and Martin Raussen and we wish success to Volker Mehrmann and Armen Sergeev who replace them as the society’s vice-presidents. We also thank the other departing members of the Executive Commit- tee: Alice Fialowski, Gert-Martin Greuel and Laurence Halpern. We welcome the new members: Nicola Fusco, Stefan Jackowski, Vicente Muñoz, Beatrice Pelloni and Betül Tanbay. Equally important is the renewal of our stand- intensively and we are certain of an attractive meeting, ing committees, which form the backbone of the EMS. which will do much for the standing of mathematics in Around half the chairs and a number of the members this part of Europe. have reached the end of their tenure. Their replacements Even if it is still a long time ahead, I encourage you were decided at the recent meeting of the Executive to contemplate possible candidates for the EMS prizes Committee in Tbilisi. Let me express our gratitude to in 2020. Our main award is highly renowned – recent all of them, with personal thanks to follow separately. confirmation can be seen from two of its latest laureates, In some committees, the changes run particularly deep. Hugo Duminil-Copin and Geordie Williamson, winning This is especially true for the Education Committee, the 2017 New Horizons in Mathematics Prize just a few which will see a majority of new members. We wish them days ago. It is in all our interests to keep the flag flying success and hope that the committee’s scope will broad- high. en and include some hands-on activities. A New Year message generally strikes an optimistic Another big change concerns the Publication and tone. However, I hope it won’t do any harm to add a few Electronic Publication Committees. As technology words about our worries. Some of them, frankly, are of advances, this separation has become gradually less our own making; if I were to characterise their common justifiable and we have invited their members to dis- root, I would suggest a lack of loyalty to the mathemati- cuss the formation of a unified committee within the cal community. To give a few examples, numerous col- next few months. Before leaving the topic of commit- leagues registered for the Berlin congress but did not tees, let me also mention the generous support the EMS then pay, causing a financial headache for the organis- has received from the Simons Foundation, targeted at ers (and we know that at least some such individuals mathematics in Africa. Our Committee for Developing did indeed attend). On the other hand, far from every Countries has worked hard to create appropriate grant member of the organising committee of the congress schemes and its five year programme is now underway. opted to attend the meeting whose programme they had While none of the largest mathematical meetings will designed! occur this year, some are already looming on the hori- You may also have noticed the council amending zon. On a global scale, we look forward to ICM 2018 the society’s By-Laws (Rule 23) to state that commit- in Rio and are delighted that the following meeting in tee members must be individual EMS members “in good 2022 will return, after 16 years, to Europe: either Paris standing”. While this requirement should be self-evident, or Saint Petersburg. The EMS, as a society represent- we have spotted committee members (and even chairs) ing the entire European mathematical community, will ignoring it. (I add that this has happened despite our express no preference between the two bids but we are membership dues being far lower than those of math- confident that both offer the prospect of a wonderful ematical societies on other continents.) A few of our cor- meeting. At the European scale, the Berlin Council has porate members are also perpetually in arrears. We are, decided to hold the 8th European Congress in Portorož of course, conscious of difficult economic situations in in 2020. Our Slovenian colleagues have started working parts of our continent and will never introduce the spirit

EMS Newsletter March 2017 3 Editorial of “juste retour” to the EMS. Nevertheless, we need at Then, we come to a still wider political scene, in least to see sincere efforts to deal with this matter. which our ability to influence things is close to zero. These are problems we can resolve ourselves, with We received a harsh reminder of this on the eve of the the will to do so. That is less true of difficulties in our Berlin congress, when an attempted coup (or whatever relations to the “outside world”, including European we should call it) prevented our colleagues in Turkey funding schemes. In last year’s message, I spoke about from attending. An immediate consequence was that the the ERC, a very valuable instrument that covers, how- second Caucasian Conference (planned with EMS sup- ever, only a limited segment of mathematical activities. port) had to be postponed; subsequent events in Turkey During 2016, the European Commission led an open have thrown its new date into further doubt. In other consultation on the role of mathematics in Horizon countries, we see processes unfolding that may not be as 2020. This was a useful exercise in which many of us par- violent but that signify deep instabilities in the political ticipated but it would be overly optimistic to expect an climate. In such a situation, it is useful to keep in mind enduring effect. It is a task for each of us to seek out a double inclusion: geographical Europe is wider than opportunities within the funding system and I would political Europe and mathematical Europe is wider than like to praise members of the EU-MATHS-IN initiative geographical Europe. We can and must hold together, and other colleagues who have devoted their energies even as we sail through rough waters. to such activities. This matter regularly features at the Let me end on an optimistic note after all: these annual Meeting of the Presidents of the EMS Member political tumults are temporary but – as we all know – Societies and no doubt will be raised again in April in mathematics is eternal. Happy New Year! Lisbon.

Farewells within the Editorial Board of the EMS Newsletter

With the December 2016 issue, Lucia Di Vizio, Jorge Buescu and Jaap Top ended their editorship of the Newslet- ter. We express our deep gratitude for all the work they have carried out with great enthusiasm and competence, and thank them for contributing to a friendly and productive atmosphere. Two new members have rejoined the Editorial Board in January 2017. It is a pleasure to welcome Fernando P. da Costa and Michael Th. Rassias, introduced below.

New Editors Appointed

Fernando Pestana da Costa is an analysis and differential equations, with an emphasis on associate professor at the Depart- aspects of qualitative theory. He graduated from Técnico ment of Sciences and Technology in chemical engineering and then turned to mathemat- of Universidade Aberta, Lisbon, ics, completing his PhD in mathematics at Heriot-Watt and is a researcher at the Cen- University, Edinburgh, under the supervision of Jack tre for Mathematical Analy- Carr. He was Vice-President (2012–4) and then President sis, Geometry and Dynamical (2014–6) of the Portuguese Mathematical Society and he Systems, Técnico, University of is now (2016–8) President of its General Assembly. Lisbon. His research areas are

4 EMS Newsletter March 2017 EMS News

Michael Th. Rassias is a postdoc- year 2014-2015, he was a Postdoctoral researcher at the toral researcher in Mathematics Department of Mathematics of Princeton University at the University of Zürich and and the Department of Mathematics of ETH-Zürich, a visiting researcher at the Pro- conducting research at Princeton. While at Princeton, gram in Interdisciplinary Stud- he collaborated with Professor John F. Nash, Jr. for the ies of the Institute for Advanced preparation of the volume entitled “Open Problems in Study, Princeton, since Septem- Mathematics”, Springer, 2016. He has been awarded ber 2015. He holds a Diploma with two Gold medals in National Mathematical Olym- from the School of Electrical piads in Greece, a Silver medal in the International and Computer Engineering of Mathematical Olympiad of 2003 in Tokyo, as well as the National Technical Univer- with the Notara Prize of the Academy of Athens, 2014. sity of Athens (2010), a Master He has authored two problem-solving books in Num- of Advanced Study in Mathematics from the Univer- ber Theory and Euclidean Geometry, respectively, as sity of Cambridge (2011), and a PhD in Mathematics well as a book on Goldbach’s Ternary Conjecture and from ETH-Zürich (2014). His doctoral thesis was on has edited four books, by Springer. He has published the “Analytic investigation of cotangent sums related to several research papers in Mathematical Analysis and the Riemann zeta function”, written under the super- Analytic Number Theory. His homepage is http://www. vision of Professor E. Kowalski. During the academic mthrassias.com/ .

New Members of the EC of the EMS

Nicola Fusco It is a great honour that life is not easy for talented young mathematicians: for me to be elected to the Execu- there is little money for mobility, research funds are tive Committee (EC) of the EMS. scarce and there are inadequate job opportunities. There- In my professional career as fore, I would like to make a contribution to all the actions a mathematician, I have had a that the new EC will promote to support and defend (in lot of scientific and personal con- all countries and European institutions) the fundamen- tact with colleagues from several tal role played by mathematics in the scientific and tech- European countries. On all these nological enhancement of our society. For these reasons, occasions, I have experienced, I will also be happy to develop, together with my distin- beyond any obvious national guished colleagues of the EC, actions aimed at spread- differences, that we all face the ing, throughout the European public, the knowledge of same problems and the same mathematics and the awareness of its great achievements challenges. One of them is the funding of both pure and and its important role in so many aspects of everybody’s applied mathematical research, which is often severely life. In the hard times we are nowadays facing, I would cut in favour of other areas with more immediate impact. also stress how mathematics can promote peaceful coop- Another common and critical issue in many countries is eration among people of different cultures and countries.

Stefan Jackowski received all his Related Structures” and is a former editor of “Algebraic degrees from the University of and Geometric Topology”. Warsaw, where he is currently a In the years 2005-2013, he was President of the Polish professor. He has spent several Mathematical Society and he chaired the Local Organi- research stays abroad, includ- sation Committee of 6ECM in Kraków (2012). He is ing at the ETH Zürich, Aarhus currently a member of the Committee for Evaluation of University, the University of [Polish] Scientific Research Institutions. Oxford, the University of Chica- The goals he would like to work for while on the go, IHES, the Fields Institute and Executive Committee include promotion of collabora- Max-Planck Institute Bonn. His tion and exchange of ideas between national societies, research interests are in algebraic facilitation of national and joint initiatives under the topology, including transforma- auspices of the EMS and, last but not least, strengthen- tion groups, homotopy theory, group theory, homological ing European identity and promoting understanding algebra and their interactions. He is an editor of “Fun- between the countries represented in the EMS. damenta mathematicae” and “Journal of Homotopy and

EMS Newsletter March 2017 5 EMS News

Vicente Muñoz received his concept of s-formality on rational homotopy theory and PhD in 1996 at the University of the construction of a symplectic simply-connected non- Oxford (UK) under the supervi- formal 8-manifold (with Marisa Fernández), the realisa- sion of Simon Donaldson. After tion of real homology classes via embedded laminations this, he had positions in Univer- (with Ricardo Pérez-Marco), a Bogomolov inequal- sidad de Málaga, Universidad ity associated to Spin(7)-instantons on 8-dimensional Autónoma de Madrid and CSIC tori and a topological field theory method for comput- (Spain). Since 2009, he has been ing Deligne-Hodge polynomials of character varieties a full professor at the Univesidad of surfaces (with Peter Newstead, Marina Logares and Complutense de Madrid. He has Javier Martínez). He has supervised four doctoral stu- had visiting fellowships at IAS dents and is currently supervising two students. He has Princeton (USA) in 2007 and published more than 90 research papers and the popu- Université Paris 13 (France) in 2015 and he has been lar book “Distorting Shapes” (editorial RBA), which a member of ICMAT (Spain), 2013-2016. His research has been translated into six languages. He was an edi- interests lie in differential geometry, algebraic geometry tor of the EMS Newsletter from 2004 to 2008 and then and algebraic topology and, more specifically, gauge the- became Editor-in-Chief from 2008 to 2012. He is an edi- ory, moduli spaces, symplectic geometry, complex geom- tor of Revista Matemática Complutense (Springer) and etry and rational homotopy theory. His main results are Revista Matemática Iberoamericana (EMS) and he is a a proof of the Atiyah-Floer conjecture, the finite type member of the Governing Board of the Real Sociedad condition for Donaldson invariants of 4-manifolds, the Matemática Española.

Beatrice Pelloni I am honoured I agreed to take on the Head of School role, giving me to start serving on the Execu- the opportunity of supporting these values in the eve- tive Committee of the EMS this ryday reality within a large school with a high volume year, after serving for four years of research activity. It is also the reason I am attracted on the Council of the London by the opportunity to participate and assist the strate- Mathematical Society (LMS). I gic aims of the EMS. My research interests are at the have also served a 5-year term boundary between pure and applied mathematics. My as a member of the Women in contributions relate to the study of analytical techniques Mathematics Committee of the for nonlinear partial differential equations arising in LMS and one year as Chair of the mathematical physics. I am active in two different parts Women and Mathematics Committee of the EMS. After of this wide area of research: boundary value problems many years at the University of Reading, in April 2016 I for integrable partial differential equations and the rig- moved to Heriot-Watt University in Edinburgh, as Head orous analysis of nonlinear partial differential equations of the School of Mathematical and Computer Sciences. modelling atmospheric flows. For my work in the former I am strongly committed to promoting fairness, equality area, I was awarded the Olga Taussky-Todd prize lecture and transparency in academic life, while maintaining the at ICIAM 2011. highest standards of scientific integrity. This is the reason

Betül Tanbay completed her ences, which she directed from 2006 to 2012. She was undergraduate degree at the the first woman president of theTurkish Mathematical University of Strasbourg (ULP) Society, from 2010 to 2016, where she also became an and her graduate degrees at the Honorary Member. She has represented the society at University of California, Berke- the General Assemblies of the IMU in Germany, Spain ley (UCB). She has been a fac- and South Korea and at the Councils of the EMS in Hol- ulty member at the University land and Poland, and she was elected as a member to of Bog˘aziçi, Istanbul, since 1989. the Executive Committee at the Council in Berlin in Her research area is operator 2016. Previously, she worked as a member of the EMS algebras, mainly focusing on the Raising Public Awareness and Ethics Committees and is Kadison-Singer problem. She has held visiting positions currently EC-liaison of the latter committee and also a at the University of California, Berkeley and Santa Bar- member of the Committee for Women in Mathematics bara, the Université de Bordeaux, the Institut de Mathé- of the IMU. Betül Tanbay is fluent in Turkish, English, matiques de Jussieu, Paris VI, the University of Kansas French, German and Spanish. She has a daughter and and Pennsylvania State University. Betül Tanbay is the a son both studying mathematics in Turkish public uni- founder of the Istanbul Center for Mathematical Sci- versities.

6 EMS Newsletter March 2017 EMS News Report from the Executive Committee Meeting in Tbilisi, 4–6 November 2016

Richard Elwes, EMS Publicity Officer and Sjoerd Verduyn Lunel, EMS Secretary

The Executive Committee of the EMS met in Tbilisi, in Berlin in July. The committee considered a report from 4–6 November 2016, at Ivane Javakhishvili Tbilisi State local organiser Volker Mehrmann. It was agreed that, on University, on the kind invitation of the Georgian Math- most metrics, the meeting was a great success and the ematical Society. As well as the committee’s current committee reiterated its profound thanks to all involved. incumbents, several incoming members for 2017 also However, there were also problems, notably a surpris- attended by invitation, making this event something of ing disparity between the number of people registering a handover. On Friday evening, the gathered assembly and those paying the registration fee. Consequently, the enjoyed a warm welcome from Roland Duduchava, meeting ended up in financial deficit. The organisers of Otar Chkadua and Tinatin Davitashvili (President and ECM8 must be aware of this danger. Vice-Presidents of the Georgian Mathematical Society), The committee opted to amend the profile for mem- as well as Mikhail Chkhenkeli, Vice-Rector of Tbilisi bers of future ECM Scientific Committees, who should State University. The oldest university in the Caucasus, it not only be top mathematicians but also people with was founded in 1918 by the historian Ivane Javakhishvili, broad mathematical interests willing to attend the event with, amongst others, the mathematician Andrea Raz- in person. madze, after whom the mathematics institute is named. This was also the first Executive Committee meeting since the venue of the 8th European Congress Officers’ Reports and Membership was settled at July’s council meeting: it will be held in President Pavel Exner welcomed all members and Portorož in Slovenia, 5–11 July 2020. Local organiser guests and opened the meeting by relating his recent Tomaz Pisanski delivered a presentation on prepara- activities. The treasurer, Mats Gyllenberg, then present- tions so far, including some thoughts on the philosophy ed his report on the society’s 2016 income and expendi- of the congress. In the ensuing discussion, the commit- ture, recording healthy results. Thus, the EMS can afford tee contemplated the mission of the ECM in our chang- to maintain its recently elevated expenditure on scien- ing mathematical world. How can we use such events tific projects. He then presented the society’s budget for to strengthen our community? How can we help young 2017. mathematicians feel welcome? Some initial ideas were The committee was pleased to approve a list of 54 aired and it was agreed that this was an important topic new individual members. The office has received no new that will be revisited. applications for corporate membership, although there have been enquiries that may develop. Too many indi- Other Scientific Meetings vidual members remain in arrears, although several have The committee discussed reports from the 2016 EMS belatedly paid their dues after a reminder. The office will Summer Schools and agreed that these were scientifi- continue to chase outstanding payments. The conversa- cally successful and a worthwhile use of EMS resourc- tion then turned to the thornier problem of member es. Considering recommendations from the Meetings societies that are perpetually in arrears. The society is Committee, support for four 2017 Summer Schools was mindful of the financial difficulties that national mathe- approved. It was decided to require applicants for Sum- matical societies can experience. Nevertheless, the com- mer School funding to provide more details on topics mittee previously agreed the principle that membership such as geographic, age and gender distributions of par- will lapse for societies that neither pay their dues nor ticipants. The remit for the Meetings Committee was respond to letters from the president. In early 2016, then adapted to encourage a more proactive approach the president wrote to all corporate members badly in to attracting proposals and thus hopefully increasing arrears. The committee thus resolved that he will now competition. It may also be useful to publicise recent write final reminders to such members, warning them of successful Summer Schools. the committee’s intention to propose the termination of Committee member Alice Fialowski reported from their membership at the next council meeting. the EMS Distinguished Speaker talk, delivered by Ernest Vinberg at the 50th Seminar Sophus Lie. It was agreed ECMs 7&8 to increase the level of ceremony at EMS Distinguished This was the first meeting of the Executive Committee Speaker events to reflect the honour of the award. EMS since the 7th European Congress of Mathematics (ECM) Distinguished Speakers for 2017 were decided: Mireille

EMS Newsletter March 2017 7 EMS News

Bousquet-Mélou at the Conference on Foundations of lence programme and plans for the grant scheme for Computational Mathematics (Barcelona, 10–19 July researchers from Africa (made possible by generous 2017) and Kathryn Hess at the Meeting of the Catalan, funding from the Simons Foundation for Africa). The Spanish and Swedish Math Societies (Umeå, 12–15 June, Executive Committee thanked her for the CDC’s excel- 2017). lent work and discussed ways to increase its visibility. The committee agreed that the EMS will endorse EMS members should be reminded of the option of the following events: the 9th European Student Confer- donating to the CDC when paying their EMS member- ence in Mathematics, EUROMATH-2017 (Bucharest, ship fees (the amount raised by this route has decreased 29 March – 2 April 2017); the 2nd Academic University in recent years). Student Conference 2017 (15–19 November 2017, Lar- Reports from the chairs of other standing commit- naca, Cyprus); and the 11th European Conference on tees were also discussed, along with reports from the Mathematical and Theoretical Biology (Lisbon, 23–27 Publicity Officer and Editor-in-Chief of the Newsletter, July 2018). and reports on other EMS-affiliated projects including Meanwhile, the EMS-Bernoulli Society Joint Lecture the European Digital Mathematical Library, Zentral- for 2017 will be given by Alexander Holevo at the 31st blatt MATH and EU-MATHS-IN (the European Ser- European Meeting of Statisticians (Helsinki, 24–28 July vice Network of Mathematics for Industry and Inno- 2017). vation). The latter has had a proposal (“Mathematical The committee approved separate funding for the Modelling, Simulation and Optimization for Societal Applied Mathematics Committee for scientific activi- Challenges with Scientific Computing”) funded under ties, to the tune of 5,000 euros in 2017 and 10,000 euros the Horizon 2020 programme on user-driven e-infra- in 2018 (bearing in mind that 2018 is the Year of Math- structure innovation. EU-MATHS-IN is also involved ematical Biology). in making mathematics more visible in H2020 generally.

Society Meetings Funding, Political and Scientific Organisations The Executive Committee discussed the president’s The president reported on recent developments in Hori- report from the council meeting from July and contem- zon2020 and the European Research Council. He also plated ways to make it a more interactive event. This discussed the new legal status of the Initiative for Sci- conversation will be continued at future meetings. The ence in Europe (ISE), reminding the committee that the committee was delighted to accept an invitation from fee for ISE membership for the EMS is set to double the Czech Mathematical Society to hold the 2018 Coun- to 3,000 euros. Thus, after two years, the benefits of ISE cil in Prague. membership will be re-evaluated. The president gave a The next Executive Committee meeting will be held short report about ESOF (EuroScience Open Forum) on 17–19 March in Bratislava and the next meeting of in Manchester in July 2016, noting that the Committee the Presidents of Member Societies will be held on 1–2 for Raising Public Awareness had run a successful panel April 2017 in Lisbon. session on “The Myth of Turing”. There are live bids from both Paris and St. Petersberg Standing Committees & Projects for the ICM in 2022. The EMS is supporting both. The With numerous upcoming vacancies across the society’s committee also discussed EMS ongoing business with 11 standing committees (excluding the Executive Com- other mathematical organisations, including ICIAM, mittee), the committee expressed its sincere appreciation the Abel Prize, CIMPA, the Fermat Prize and various to all outgoing members for their efforts in carrying out research facilities around Europe. the society’s work. It was then pleased to fill these plac- es, including appointing Jürg Kramer as Chair and Tine Conclusion Kjedsen as Vice-Chair of the Education Committee, San- The meeting was closed with enthusiastic thanks to our dra di Rocco as Chair and Stanislaw Janeczko as Vice- hosts, the Georgian Mathematical Society (particularly Chair of the European Solidarity Committee, Michael its President Roland Duduchava) and Tbilisi State Uni- Drmota as Chair and Ciro Ciliberto as Vice-Chair of the versity for excellent organisation and generous hospital- Meetings Committee, and Alessandra Celletti as Chair of ity. the Women in Mathematics Committee. The society’s standing committees will soon reduce to 10, following a decision to merge the Publishing and Electronic Publishing Committees. The details, and further appointments to the combined committee, will await the outcome of a discussion between the two. The Executive Committee then approved an updated remit for the Ethics Committee. The Chair of the Committee for Developing Coun- tries (CDC) Giulia Di Nunno, in attendance by invitation, then delivered a presentation on the CDC’s work, including on the Emerging Regional Centres of Excel-

8 EMS Newsletter March 2017 News Call for Nominations for the Fermat Prize 2017

The new edition of the Fermat Prize for Mathematics Research opened in October 2016.

The call for nominations of candidates will be open until 30 June 2017 and the results will be announced in Decem- ber 2017.

The Fermat Prize rewards the research of one or more mathematicians in fields where the contributions of Pierre de Fermat have been decisive: statements of variational principles or, more generally, partial differential equations; foundations of probability and analytical geometry; and number theory.

More information about the Fermat Prize, in particular concerning the application process, are available at

http://www.math.univ-toulouse.fr/FermatPrize.

Call for Nominations for the Ostrowski Prize 2017

The aim of the Ostrowski Foundation is to promote the mathematical sciences. Every second year it provides a prize for recent outstanding achievements in pure mathematics and in the foundations of numerical mathematics. The value of the prize for 2017 is 100,000 Swiss francs.

The prize has been awarded every two years since 1989. The most recent winners are Ben Green and Terence Tao in 2005, Oded Schramm in 2007, Sorin Popa in 2009, Ib Madsen, David Preiss and Kannan Soundararajan in 2011, Yitang Zhang in 2013, and Peter Scholze in 2015. See

https://www.ostrowski.ch/index_e.php

for the complete list and further details.

The jury invites nominations for candidates for the 2017 Ostrowski Prize. Nominations should include a CV of the candidate, a letter of nomination and 2–3 letters of reference.

The Chair of the jury for 2017 is Gil Kalai of the Hebrew University of Jerusalem, Israel. Nominations should be sent to [email protected] by May 15, 2017.

EMS Newsletter March 2017 9 News Call for Nominations for the 2019 ICIAM Prizes

The ICIAM Prize Committee for 2019 is calling for nomi- Nominations should be made electronically through the nations for the five ICIAM Prizes to be awarded in 2019 website https://iciamprizes.org/. The deadline for nomi- (the Collatz Prize, the Lagrange Prize, the Maxwell Prize, nations is 15 July 2017. the Pioneer Prize and the Su Buchin Prize). Each ICIAM Please contact [email protected] if you have any Prize has its own special character but each one is truly questions regarding the nomination procedure. international in character. Nominations are therefore welcome from every part of the world. A nomination should ICIAM Prize Committee for 2019: take into account the specifications of each particular Committee Chair: Maria J. Esteban prize (see below and at http://www.iciam.org/iciam-prizes) Zdenek Strakos (Chair of Collatz Prize Subcommittee) and should contain the following information: Alexandre Chorin (Chair of Lagrange Prize Subcommit- tee) - Full name and address of the person nominated. Alexander Mielke (Chair of Maxwell Prize Subcommit- - Web homepage if any. tee) - Name of particular ICIAM Prize. Denis Talay (Chair of Pioneer Prize Subcommittee) - Justification for nomination (cite nominator’s reason Zuowei Shen (Chair of Su Buchin Prize Subcommittee) for considering the candidate to be deserving of the Margaret H. Wright prize, including explanations of the scientific and practical influence of the candidate’s work and publications). The International Council for Industrial and Applied - Proposed citation (concise statement about the out- Mathematics (ICIAM) is the world organisation for standing contribution in fewer than 250 words). applied and industrial mathematics. - CV of the nominee. Its members are mathematical societies based in - 2–3 letters of support from experts in the field and/or 2–3 more than 30 countries. names of experts to be consulted by the Prize Committee. For more information, see the council’s webpage at - Name and contact details of the proposer. http://www.iciam.org/. Simon Donaldson Awarded Doctor Honoris Causa

Sir Simon Kirwan Donaldson FRS, professor in pure Professor Simon Donaldson is a renowned geom- mathematics at Imperial College London (UK) and per- eter who has received numerous prizes, including the manent member of the Simons Center for Geometry and Fields Medal, the Shaw Prize and the Breakthrough Physics at Stony Brook University (US), received the Prize. The ceremony can be viewed at www.youtube.com/ honorary degree Doctor Honoris Causa by Universidad watch?v=xoC53Smg-WI. A report (in Spanish) can be Complutense de Madrid (Spain) on 20 January 2017. found at tribuna.ucm.es/43/art2605.php. The ceremony was presided over by the Rector, Carlos Andradas, who is also a professor in algebra at UCM. Vicente Muñoz, a professor in geometry and topology at UCM and a member of the Executive Com- mittee of the EMS, was in charge of reading the Lau- datio. There were 150 people in attendance, including the President of the Spanish Mathematical Society and the President of the Spanish Academy of Sciences. The ceremony took place in the Faculty of Mathematics of UCM. This is the first time that such an event has been hosted in this relatively new building. This allowed for an affluence of undergraduate and postgraduate students and served to commemorate the 25th anniversary Simon Donaldson (left) and Carlos Andradas (right) during the cer- of the faculty building. emony (photo: Jesús de Miguel/UCM)

10 EMS Newsletter March 2017 Feature Spectral Synthesis Synthesis for for Operators Operators and Systems Systems Anton Baranov (St. Petersburg State University, St. Petersburg, Russia) and Yurii Belov (St. Petersburg State University,Anton Baranov St. Petersburg, (St. Petersburg Russia) State University, St. Petersburg, Russia) and Yurii Belov (St. Petersburg State University, St. Petersburg, Russia)

Spectral synthesis is the reconstruction of the whole lattice any invariant subspace E of A, we have of invariant subspaces of a linear operator from generalised E = Span x E : x Ker (A λI)n . eigenvectors. A closely related problem is the reconstruction ∈ ∈∪λ,n − of a vector in a Hilbert space from its Fourier series with re- Equivalently, this means that the restriction A has a com- |E spect to some complete and minimal system. This article dis- plete set of generalised eigenvectors. cusses the spectral synthesis problem in the context of operator and function theory and presents several recent advances The notion of spectral synthesis for a general operator in this area. Among them is the solution of the spectral syn- goes back to J. Wermer (1952). In the special context of thesis problem for systems of exponentials in L2( π, π). translation invariant subspaces in spaces of continuous or − A more detailed account of these problems can be found smooth functions, similar problems were studied by J. Del- in the survey [1], to appear in the proceedings of 7ECM. sarte (1935), L. Schwartz (1947) and J.-P. Kahane (1953). Note that, in this case, the generalised eigenvectors are exponentials and exponential monomials. Wermer showed, in particular, that any compact nor- 1 Introduction mal operator in a Hilbert space admits spectral synthesis. Eigenfunction expansions play a central role in analysis and However, both of these conditions are essential: there ex- its applications. We discuss several questions concerning such ist non-synthesable compact operators and there exist non- expansions for special systems of functions, e.g. exponen- synthesable normal operators. For a normal operator A with λ tials in L2 on an interval and in weighted spaces, phase-space simple eigenvalues n, Wermer showed that A Synt if and only if the set λ carries a complex measure orthog- shifts of the Gaussian function in L2(R) and reproducing ker- { n} nels in de Branges spaces (which include certain families of onal to polynomials, i.e. there exists a nontrivial sequence µ 1 such that µ λk = 0, k N . Existence of Bessel or Airy type functions). We present solutions of some { n}∈ n n n ∈ 0 such measures follows from Wolff’s classical example of a problems in the area (including the longstanding spectral synthesis problem for systems of exponentials in L2( π, π)) and Cauchy transform vanishing outside of the disc: there exist − λ D = z C : z < 1 and µ 1 such that mention several open questions, such as the Newman–Shapiro n ∈ { ∈ | | } { n}∈ problem about synthesis in Bargmann–Fock space. µn 0, z > 1. λ ≡ | | n z n Spectral synthesis for operators − One of the basic ideas of operator theory is to consider a lin- At the same time, there exist compact operators that do not ad- ear operator as a “sum” of its simple parts, e.g. its restrictions mit spectral synthesis. Curiously, the first example of such a onto invariant subspaces. In the finite-dimensional case, the situation was implicitly given by H. Hamburger in 1951 (even possibility of such decomposition is guaranteed by the Jor- before Wermer’s paper). Further results were obtained in the dan normal form. Moreover, any invariant subspace coincides 1970s by N. Nikolski and A. Markus. For example, Nikol- with the span of the generalised eigenvectors it contains (re- ski [20] proved that any Volterra operator can be a part of a call that x is said to be a generalised eigenvector or a root complete compact operator (recall that a Volterra operator is vector of a linear operator A if x Ker (A λI)n for some a compact operator whose spectrum is 0 ). ∈ − { } λ C and n N). ∈ ∈ Theorem 2 (Nikolski). For any Volterra operator V in a However, the situation in the infinite-dimensional case is Hilbert space H, there exists a complete compact operator A much more complex. Assume that A is a bounded linear op- on a larger Hilbert space H H such that H is A-invariant erator in a separable Hilbert space H that has a complete set ⊕ and A H = V. In particular, A Synt. of generalised eigenvectors (in this case we say that A is com- | plete). Is it true that any A-invariant subspace is spectral, that A. Markus [16] found a relation between spectral synthe- is, it coincides with the closed linear span of the generalised sis for a compact operator and the geometric properties of eigenvectors it contains? In general, the answer is negative. the eigenvectors. We now introduce the required “strong com- Therefore, it is natural to introduce the following notion. pleteness” property.

Deﬁnition 1. A continuous linear operator A in a separable Hereditarily complete systems Hilbert (or Banach, or Frechét) space H is said to admit spec- Let xn n N be a system of vectors in a separable Hilbert space { } ∈ tral synthesis or to be synthesable (we write A Synt) if, for H that is both complete (i.e. its linear span is dense in H) and ∈

EMS Newsletter March 2017 11 Feature minimal (meaning that it fails to be complete when we remove Indeed, assume that the system of eigenvectors x is not { n} any of its vectors). Let yn n N be its (unique) biorthogonal hereditarily complete and, for some partition N = N1 N2, { } ∈ ∪ system, i.e. the system such that (xm, yn) = δmn, where δmn the mixed system xn n N yn n N is not complete. The { } ∈ 1 ∪{ } ∈ 2 is the Kronecker delta. Note that a system coincides with its biorthogonal system yn consists of eigenvectors of the adjoint biorthogonal system if and only if it is an orthonormal basis. operator A . Then, E = Span y : n N is A-invariant and ∗ { n ∈ 2}⊥ With any x H, we associate its (formal) Fourier series x : n N E but E Span x : n N . ∈ { n ∈ 1}⊂ { n ∈ 1} with respect to the biorthogonal pair x , y : Note that hereditary completeness includes the require- { n} { n} ment that the biorthogonal system y is complete in H, { n} x H (x, y )x . (1) which is by no means automatic. In fact, it is very easy to ∈ ∼ n n n N construct a complete and minimal system whose biorthogo- ∈ nal is not complete. It is one of the basic problems of analysis to find conditions on the system xn that ensure the convergence of the Fourier Example 5. Let en n N be an orthonormal basis in H. Set { } { } ∈ series to x in some sense. In applications, the system x is xn = e1 + en, n 2. Then, it is easy to see that xn is com- { n} ≥ { } often given as the system of eigenvectors of some operator. plete and minimal, while its biorthogonal is clearly given by The choice of the coefficients in the expansion (1) is nat- yn = en, n 2. Taking direct sums of such examples, one ≥ ural: note that if the series n cn xn converges to x in H then, obtains biorthogonal systems with arbitrary finite or infinite necessarily, cn = (x, yn). There are many ways to understand codimension. / convergence reconstruction: It is not so trivial to construct a complete and minimal The simplest case (with the exception of orthonormal system x with a complete biorthogonal system y that • { n} { n} bases, of course) is a Riesz basis: a system xn is a Riesz is not hereditarily complete (i.e. the mixed system (2) fails = { } basis if, for any x, we have x n cn xn (the series con- to be complete for some partition of the index set). A first verges in the norm) and A cn 2 x B cn 2 for explicit construction was given by Markus (1970). Further , { } ≤≤={ } some positive constants A B. Equivalently, xn Ten for an results about the structure of nonhereditarily complete sys- orthonormal basis e and some bounded invertible opera- { n} tems were obtained by N. Nikolski, L. Dovbysh and V. Su- tor T. dakov (1977). For an extensive survey of spectral synthesis Bases with brackets: there exists a sequence nk such that • n and hereditary completeness, the reader is referred to [13]. k (x, y )x converges to x as k . n=1 n n →∞ Interesting examples of hereditarily (in)complete systems Existence of a linear (matrix) summation method • also appear in papers by D. Larson and W. Wogen (1990), (e.g. Cesàro, Abel–Poisson, etc.): this means that there E. Azoff and H. Shehada (1993) and A. Katavolos, M. Lam- exists a doubly infinite matrix (Am,n) such that brou and M. Papadakis (1993) in connection with reflexive x = limm n am,nS n(x), i.e. some means of the partial →∞ operator algebras. sums S n(x) of the series (1) converge to x. The following property, known as hereditary complete- 2 Spectral synthesis for exponential systems ness, may be understood as the weakest form of the recon- iλt struction of a vector x from its Fourier series (x, y )x . Let eλ(t) = e be a complex exponential. For Λ= λn C, n N n n { }⊂ ∈ we consider Definition 3. A complete and minimal system xn n N in a { } ∈ Hilbert space H is said to be hereditarily complete if, for any (Λ) = eλ λ Λ E { } ∈ x H, we have 2 ∈ as a system in L ( a, a). The series cneλ are often referred − n n x Span (x, yn)xn . to as nonharmonic Fourier series, in contrast to “harmonic” ∈ orthogonal series. A good introduction to the subject can be It is easy to see that hereditary completeness is equivalent found in [26]. to the following property: for any partition N = N1 N2, ∪ Exponential systems play a most prominent role in anal- N1 N2 = , of the index set N, the mixed system ∩ ∅ ysis and its applications. Geometric properties of exponential 2 x y (2) systems in L ( a, a) were one of the major themes of 20th n n N1 n n N2 − { } ∈ ∪{ } ∈ century harmonic analysis. Let us briefly mention some of is complete in H. Clearly, hereditary completeness is neces- the milestones of the theory. sary for the existence of a linear summation method for the (i) Completeness of exponential systems. series (1) (otherwise, there exists a vector x orthogonal to all This basic problem was studied in the 1930–1940s by partial sums of (1)). N. Levinson and B. Ya. Levin. One of the most important con- Hereditarily complete systems are also known as strong tributions is the famous result of A. Beurling and P. Malliavin Markushevich bases. We will also say, in this case, that the (1967), who gave an explicit formula for the radius of com- system admits spectral synthesis motivated by the following pleteness of a system (Λ) in terms of a certain density of Λ. E theorem of Markus [16]. By the radius of completeness of (Λ), we mean E Theorem 4 (Markus). Let A be a complete compact operator r(Λ) = sup a > 0: (Λ) is complete in L2( a, a) . with generalised eigenvectors x and trivial kernel. Then, E − { n} A Synt if and only if the system xn is hereditarily com- A new approach to these (and related) problems and their far- ∈ { } plete. reaching extensions can be found in [9, 10, 17, 18].

12 EMS Newsletter March 2017 Feature

(ii) Riesz bases of exponentials. Thus, in general, there exists no linear summation method The first results about Riesz bases of exponentials, which go for nonharmonic Fourier series λ Λ( f, e˜λ)eλ associated to a back to R. Paley and N. Wiener (1930s), were of perturbative complete and minimal exponential∈ system. nature. Assume that the frequencies λn are small perturbations It is worth mentioning that “bad” sequences Λ can be reg- of integers, ularly distributed, e.g. be a bounded perturbation of integers: in Theorem 7 one can choose a uniformly separated sequence sup λn n < δ. Λ so that λ n < 1, n Z. n Z | − | n ∈ | − | ∈ Paley and Wiener showed that (Λ) is a Riesz basis in Theorem 8. If the system of exponentials eλ λ Λ is complete E 2 { } ∈ L2( π, π) if δ = π 2. It was a longstanding problem to find and minimal in L ( a, a) then, for any partition Λ=Λ1 Λ2, − − − ∪ δ Λ1 Λ2 = , the corresponding mixed system has defect at the best possible ; finally, M. Kadets (1965) showed that the ∩ ∅ sharp bound is 1/4. However, it was clear that one cannot most 1, that is, describe all Riesz bases in terms of “individual” perturbations. A complete description of exponential bases in terms dim eλ λ Λ e˜λ λ Λ ⊥ 1. { } ∈ 1 ∪{ } ∈ 2 ≤ of the Muckenhoupt (or Helson–Szegö) condition was given by B. S. Pavlov (1979) and was further extended by S. V. Hru- It turns out that incomplete mixed systems are always schev, N. K. Nikolski and B. S. Pavlov (see [11] for a detailed highly asymmetric. Given a complete and minimal system account of the problem). Yu. Lyubarskii and K. Seip (1997) (Λ) in L2( π, π), it is natural to expect that “in the main” p E − extended this description to the L -setting. These results re- Λ is similar to Z. This is indeed the case. As a very rough vealed the connection of the problem to the theory of singular consequence of more delicate results (such as the Cartwright– integrals. Levinson theorem), let us mention that Λ always has den- (iii) Exponential frames (sampling sequences). sity 1: A system xn in a Hilbert space H is said to be a frame if n (Λ) { } lim r = 1, there exist positive constants A, B such that r →∞ 2r 2 2 2 Λ Λ A (x, xn) x B (x, xn) , where nr( ) is the usual counting function, nr( ) | | ≤ ≤ | | = # λ Λ, λ r . Analogously, one can define the upper n n { ∈ | |≤ } density D+(Λ): i.e. a generalised Parceval identity holds. Here, we omit the requirement of minimality to gain in “stability” of the recon- nr(Λ) struction; there is a canonical choice of coefficients so that the D+(Λ) = lim sup . r 2r series c x converge to x. If a frame x is minimal then →∞ n n n { n} it is a Riesz basis. Theorem 9. Let Λ C, let the system (Λ) be complete and Exponential frames were introduced by R. Duffin and 2 ⊂ E minimal in L ( a, a) and let the partition Λ=Λ1 Λ2 satisfy A. C. Schaeffer (1952), while their complete description was − ∪ D+(Λ2) > 0. Then, the mixed system eλ λ Λ e˜λ λ Λ is { } ∈ 1 ∪{ } ∈ 2 obtained relatively recently by J. Ortega–Cerdà and K. Seip complete in L2( a, a). [22]; this solution involves the theory of de Branges spaces of − entire functions, which is to be discussed below. For an exten- Theorem 9 shows that there is a strong asymmetry be- sive review on exponential frames on disconnected sets, see a tween the systems of reproducing kernels and their biorthog- recent monograph by A. Olevskii and A. Ulanovskii [21]. onal systems. The completeness of a mixed system may fail only when we take a sparse (but infinite!) subsequence Λ1. We conclude this subsection with one open problem. Synthesis up to codimension 1 The spectral synthesis (or hereditary completeness) problem Problem 10. Given a hereditarily complete system of expo- for exponential systems was also a longstanding problem in nentials, does there exist a linear summation method for the nonharmonic Fourier analysis. Let (Λ) be a complete and corresponding nonharmonic Fourier series? E minimal system of exponentials in L2( a, a) and let e˜ be − { λ} the biorthogonal system. It was shown by R. Young (1981) Translation to the entire functions setting that the biorthogonal system e˜λ is always complete. A classical approach to a completeness problem is to translate { } it (via a certain integral transform) to a uniqueness problem in Problem 6. Is it true that any complete and minimal system 2 some space of analytic functions. In the case of exponentials of exponentials eλ λ Λ in L ( a, a) is hereditarily complete, { } ∈ − on an interval, the role of such a transform is played by the i.e. any function f L2( a, a) belongs to the closed linear ∈ − standard Fourier transform , span of its “harmonics” ( f, e˜λ)eλ? F This question was answered in the negative by the authors a 1 izt jointly with Alexander Borichev [2]. Surprisingly, it turned ( f )(z) = f (t)e− dt. F 2π a out, at the same time, that spectral synthesis for exponential − systems always holds up to at most one-dimensional defect. By the classical Paley–Wiener theorem, maps L2( π, π) F − Theorem 7. There exists a complete and minimal system of unitarily onto the space 2 exponentials eλ λ Λ, Λ R, in L ( π, π) that is not heredi- { } ∈ ⊂ − 2 a z tarily complete. W = F entire, F L (R), F(z) Ce | | . P a { − ∈ | |≤ }

EMS Newsletter March 2017 13 Feature

Paley–Wiener space W (also known as the space of ban- 3 Spectral synthesis in de Branges spaces and P a dlimited functions of bandwidth 2a) plays a remarkably im- applications portant role in signal processing. The Fourier transform maps exponentials in L2( a, a) to Preliminaries on de Branges spaces − the cardinal sine functions: The theory of Hilbert spaces of entire functions was created by L. de Branges at the end of the 1950s and the beginning sin a(z λ) of the 1960s. It was the main tool in his famous solution of ( eλ¯ )(z) = kλ(z) = − . F π(z λ) the direct and inverse spectral problems for two-dimensional − canonical systems. These are second order ODEs that include, Note that the functions kλ are the reproducing kernels in as particular cases, the Schrödinger equation on an interval, Paley–Wiener space W , i.e. the function k generates the the Dirac equation and Krein’s string equation. For the gen- P a λ evaluation functional at the point λ: eral theory of de Branges spaces, we refer to the original monograph by de Branges [8]; for information relating to in- F(λ) = (F, k ), F W . verse problems, see [23, 24]. λ ∈P a De Branges spaces proved to be highly nontrivial and are int In particular, the orthogonal expansion f = n Z cne in interesting objects from the point of view of function theory. ∈ L2( π, π) becomes Surprisingly, they appear to be unavoidable in substantially − different branches of analysis, for example: sin π(z n) Polynomial approximations on the real line. F(z) = cn − , cn = F(n), (3) • n Z π(z n) Orthogonal polynomials and random matrix theory (see, ∈ − • for example, [15]). the classical Shannon–Kotelnikov–Whittacker sampling for- Model (backward shift invariant) subspaces of Hardy space: mula. • K = H2 ΘH2, where H2 is Hardy space and Θ is an inner Moreover, this translation makes it possible to find an ex- Θ function (for a discussion of this relation, see, for example, plicit form of the biorthogonal system, which is not possible 2 [9]). when staying inside L ( a, a). Let kλ λ Λ be a complete and − { } ∈ Functional models for different classes of linear operators minimal system in W . Its biorthogonal system may then be • P a [5, 12]. obtained from one function G known as the generating func- Λ and even tion of Λ. This is an entire function with the zero set Λ, which Analytic number theory, the Riemann Hypothesis and can be defined by the formula • Dirichlet L-functions [14]. z There are equivalent ways to introduce de Branges spaces: GΛ(z) = lim 1 , an axiomatic approach or a definition in terms of the gen- R − λ →∞ λ Λ, λ

dim kλ λ Λ1 gλ λ Λ2 ⊥ 1 (5) { } ∈ ∪{ } ∈ ≤ Example 11. If T = Z, µn = 1 and A(z) = sin πz then but the defect 1 is possible. = Wπ. The corresponding representation of the elements H P

14 EMS Newsletter March 2017 Feature of coincides with the Schannon–Kotelnikov–Whittaker Now we turn to the problem of the size of the defect (i.e. H sampling formula (3). the dimension of the complement to a mixed system). It turns out that one can construct examples of systems of reproducing An important feature of de Branges spaces is that they kernels with large or even infinite defect. have the so-called division property: if a function f in a de Branges space vanishes at some point w C then the func- Theorem 14. For any increasing sequence T = tn with H ∈ { } tion f (z)/(z w) also belongs to . Another characteristic t , n , and for any N N , there exists − H | n|→∞ | |→∞ ∈ ∪ {∞} property of a de Branges space is existence of orthogonal a measure µ such that, in the de Branges space with spectral bases of reproducing kernels. It is clear from the definition data (T,µ), there exists a complete and minimal system of re- that the functions A(z)/(z tn) form an orthogonal basis in producing kernels kλ λ Λ whose biorthogonal system gλ λ Λ − H { } ∈ { } ∈ and are reproducing kernels of up to normalisation. is also complete but, for some partition Λ=Λ Λ , H 1 ∪ 2 Spectral synthesis problem and its solution dim ( gλ λ Λ1 kλ λ Λ2 )⊥ = N. Let k be a complete and minimal system of reproducing { } ∈ ∪{ } ∈ { λ} kernels in a de Branges space . As in the Paley–Wiener H The key role in this construction plays the balance be- space, its biorthogonal system is given by formula (4) for tween the “spectrum” tn and the masses µn . If n µn = , some appropriate generating function GΛ. However, in con- { } { } 2N 2µ ∞< but there exists a subsequence tnk of T such that k tnk − nk trast to the Paley–Wiener case, the biorthogonal system in , then, in the corresponding de Branges space, one has ∞ general need not be complete (Baranov, Belov, 2011). mixed systems with any defect up to N. Conversely, the esti- We will say that a de Branges space has the spectral syn- mate µ t M for some M > 0 and all n implies an estimate n ≥|n|− thesis property if any complete and minimal system of repro- from the above in terms of M for the defect. ducing kernels with the complete biorthogonal system (this assumption is included) is also hereditarily complete, i.e. all Spectral theory of rank one perturbations of compact mixed systems are complete. In [3], the following problem is self-adjoint operators addressed. Let A be a compact self-adjoint operator in a separable Hilbert Problem 12. Which de Branges spaces have the spectral syn- space H with simple point spectrum s and trivial kernel. { n} thesis property? If a space does not have the spectral synthesis In other words, A is the simplest infinite-dimensional opera- property, what is the possible size of the defect for a mixed tor one can imagine, diagonalisable by the classical Hilbert– system? Schmidt theorem. Surprisingly, the spectral theory of rank one Why is hereditary completeness of reproducing kernels in perturbations of such operators is already highly nontrivial. For a, b H, consider the rank one perturbation L of A, de Branges spaces an interesting and significant topic? There ∈ are several motivations for that: Relation to exponential systems and nonharmonic Fourier L = A + a b, Lf = Af + ( f, b)a, f H. • ⊗ ∈ series as discussed above. N. Nikolski’s question: whether there exist nonhereditarily For example, one may obtain examples of rank one pertur- • complete systems of reproducing kernels in model spaces bations changing one boundary condition in a second order 2 2 KΘ = H ΘH ? Note that de Branges spaces form an differential equation. important special subclass of model spaces. In [5], the spectral properties of rank one perturbations Spectral synthesis for rank one perturbations of self-adjoint are studied via a functional model. Several similar functional • operators (see Subsection 3). models for rank one perturbations (or close classes of oper- In [3], a complete description of de Branges spaces with ators) have been developed, e.g. by V. Kapustin (1996) and the spectral synthesis property was obtained. To state it, we G. Gubreev and A. Tarasenko (2010). Let us present the idea need one more definition. An increasing sequence T = tn is of this model without going into technicalities. { } said to be lacunary (or Hadamard lacunary) if For t = s 1, consider a de Branges space with spectral n n− H data (T,µ), where µ is some measure supported by T. Let G tn+1 tn lim inf > 1, lim inf | | > 1, be an entire function such that G but G(z)/(z w) tn tn tn tn+1 H − ∈ →∞ →−∞ | | whenever G(w) = 0. This means that the function G has H i.e. the moduli of tn tend to infinity at least exponentially. growth just slightly larger than is possible for the elements of | | . Assume also that G(0) = 1. Consider the linear operator Theorem 13. Let be a de Branges space with spectral data H H (T,µ). Then, has the spectral synthesis property if and only H F(z) F(0)G(z) if one of the following conditions holds: (MF)(z) = − , F . (8) z ∈H (i) µ < . n n ∞ (ii) The sequence tn is lacunary and, for some C > 0 and It is easily seen that M is a rank one perturbation of a compact { } 1 any n, self-adjoint operator with spectrum tn− = sn. The functional µ model theorem from [5] proves that any rank one perturbation µ + t2 k Cµ . (7) k n 2 ≤ n of A is unitary equivalent to a model operator M of the form t t t > t tk | k|≤| n| | k| | n| (8) for some de Branges space and function G. Therefore, H Note that condition (3) implies that the sequence of while the spectrum T = t is fixed, the masses µ and the { n} n masses µn also grows at least exponentially. function G are free parameters of the model.

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It is easy to see that the eigenfunctions of M are of the where m stands for the area Lebesgue measure. This space (as form G(z)/(z λ) for λ Z , where Z(G) denotes the zero set well as its multi-dimensional analogues) plays a most promi- − ∈ G of G. The point spectrum of M is thus given by λ 1 : λ Z . nent role in theoretical physics, serving as a model of the { − ∈ G} Multiple zeros of G correspond to generalised eigenvectors phase space of a particle in quantum mechanics. It also ap- but we assume, for simplicity, that G has simple zeros only. pears naturally in time-frequency analysis and Gabor frame Thus, the system of eigenfunctions of the rank one perturba- theory. There is a canonical unitary map from L2(R) to F tion L is unitary equivalent to a system of the form gλ λ Λ, (the Bargmann transform), which plays a role similar to that { } ∈ as in (4), while eigenfunctions of L∗ (which is also a rank one of the Fourier transform in the Paley–Wiener space setting. perturbation of A) correspond to a system kλ λ Λ of repro- Note that, clearly, all functions in are of order at most 2 { } ∈ F ducing kernels in . and satisfy the estimate F(z) C exp(π z 2/2) (which can be H | |≤ | | Thus, we relate the spectral properties of rank one per- slightly refined). turbations to geometric properties of systems of reproducing The reproducing kernels of are the usual complex ex- πλ¯z F kernels (in view of the symmetry, we interchange the roles of ponentials, kλ(z) = e . Moreover, the Bargmann transform L and L∗): of the phase-space shift of the Gaussian, i.e. of the func- 2 Completeness of L completeness of a system of repro- tion e2πiηte π(t ξ) , coincides up to normalisation with eπλz, • ⇐⇒ − − ducing kernels k in (E). λ = ξ+iη. Thus, geometric properties (e.g. spectral synthesis) { λ} H Completeness of L completeness of the system bi- of the phase-space shifts of the Gaussian are equivalent to the • ∗ ⇐⇒ orthogonal to the system of reproducing kernels. corresponding properties of the exponentials in Fock space. Spectral synthesis for L hereditary completeness of • ⇐⇒ kλ λ Λ, i.e. for any partition Λ=Λ1 Λ2, the system Radial Fock-type spaces { } ∈ ∪ kλ λ Λ gλ λ Λ is complete in . Considering radial weights differing from the Gaussian weight, { } ∈ 1 ∪{ } ∈ 2 H The results of Subsection 3 lead to a number of striking one obtains a wide class of Hilbert spaces of entire functions. examples for rank one perturbations of compact self-adjoint Namely, for a continuous function ϕ : [0, ) (0, ), we ∞ → ∞ operators. These examples show that the spectral theory of define the radial Fock-type space as such perturbations is a rich and complicated subject that is far 2 1 2 ϕ( z ) from being completely understood. = = − | | < . ϕ F entire : F ϕ : F(z) e dm(z) F F π C | | ∞ Theorem 15 (Baranov, Yakubovich). For any compact self- adjoint operator A with simple spectrum, there exists its rank We always assume that log r = o(ϕ(r)), r , to exclude →∞ one perturbation L = A + a b such that: finite-dimensional spaces. ⊗ Any Fock-type space is a reproducing kernel Hilbert (i) Ker L = Ker L∗ = 0 and L is complete but the eigenvec- space. It was shown by K. Seip [25] that in classical Fock tors of L∗ span a subspace with infinite codimension. space there are no Riesz bases of reproducing kernels. Re- (ii) L and L∗ are complete but L admits no spectral synthe- cently, A. Borichev and Yu. Lyubarskii [7] showed that Fock- sis with infinite defect (i.e. there exists an L-invariant type spaces with slowly growing weights ϕ(r) = (log r)γ, γ ∈ subspace E such that the generalised vectors of L that (1, 2], have Riesz bases of reproducing kernels corresponding belong to E have infinite codimension in E). to real points and, thus, can be realised as de Branges spaces with equivalence of norms (this is clear from the representa- For more results about completeness and spectral synthe- tion of de Branges spaces via their spectral data). Moreover, sis of rank one perturbations, see [5]. One may also ask for ϕ(r) = (log r)2 is in a sense the sharp bound for this phe- which compact self-adjoint operators A there exists a rank nomenon. Namely, it is shown in [7] that if (log r)2 = o(ϕ(r)), one perturbation that is a Volterra operator (i.e. the spectrum r , and ϕ has a certain regularity then has no Riesz can be destroyed by a rank one perturbation). This problem →∞ Fϕ bases of reproducing kernels. was solved in [4], where it was shown that the spectrum In view of the examples above, one may ask which de s is “destructible” if and only if t = s 1 form the zero { n} n n− Branges spaces can be realised as radial Fock-type spaces, set of an entire function of some special class introduced by that is, there is an area integral norm that is equivalent to M. G. Krein (1947). the initial de Branges space norm. Surprisingly, it turns out that this class of de Branges spaces exactly coincides with 4 Spectral synthesis in Fock-type spaces the class of de Branges spaces (ii) with the spectral synthesis Classical Bargmann–Fock space property in Theorem 13. Fock-type spaces form another important class of Hilbert Theorem 16 (Baranov, Belov, Borichev). Let be a de H spaces of entire functions. In contrast to de Branges spaces, Branges space with spectral data (T,µ). Then, the following where the norm is defined as an integral over the real axis statements are equivalent: (with respect to some continuous or discrete measure), in (i) There exists a Fock-type space such that = . Fock-type spaces the norm is defined as an area integral. Fϕ H Fϕ Classical Fock space (also known as Bargmann, Segal– (ii) is rotation invariant, that is, the operator R : f (z) F H θ → Bargmann or Bargmann–Fock space) is defined as the set of f (eiθz) is a bounded invertible operator in for some H all entire functions F for which (all) θ (0,π). ∈ 2 1 2 π z 2 F := F(z) e− | | dm(z) < , π | | ∞ (iii) The sequence T is lacunary and (7) holds. F C

16 EMS Newsletter March 2017 Feature

Thus, in the space with ϕ(r) = (log r)γ, γ (1, 2], Problem 19. Let G be such that ewzG for any w C. Fϕ ∈ ∈F ∈F ∈ any complete and minimal system of reproducing kernels is Is it true that hereditarily complete. We also mention that Riesz bases in wz some de Branges spaces with lacunary spectral data have been Span e G : w C = G ? described by Yu. Belov, T. Mengestie and K. Seip [6]. { ∈ } R Newman and Shapiro showed that the equality holds in Synthesis in Fock space and the Newman–Shapiro the case where G is a linear combination of exponential problem monomials. In our example in Theorem 18, however, the Now we turn to the case of classical Fock space . Though F function G admits multiplication by polynomials in Fock it has no Riesz bases of reproducing kernels, there exist many space but not multiplication by the exponents. Thus, the spec- complete and minimal systems of reproducing kernels. The tral synthesis problem of Newman and Shapiro remains open. two-dimensional lattice Z+iZ plays for Fock space a role similar to the role of the lattice Z for Paley–Wiener space Wπ. πλz P Bibliography In particular, if Λ=(Z + iZ) 0 then kλ λ Λ = e λ Λ is a \{ } { } ∈ { } ∈ complete and minimal system, whose generating function is [1] A. Baranov, Yu. Belov, Spectral synthesis in Hilbert spaces of the Weierstrass sigma-function (up to the factor z). The sec- entire functions, to appear in Proc. of 7th European Congress ond author proved (2015) the following Young-type theorem of Mathematics (Berlin). for Fock space. [2] A. Baranov, Y. Belov, A. Borichev, Hereditary completeness for systems of exponentials and reproducing kernels, Adv. Theorem 17 (Belov). For any complete and minimal system Math. 235 (2013), 525–554. of reproducing kernels (i.e. exponentials) in , its biorthogo- [3] A. Baranov, Y. Belov, A. Borichev, Spectral synthesis in de F nal system is also complete. Branges spaces, Geom. Funct. Anal. (GAFA) 25 (2015), 2, 417–452. On the other hand, the first author recently proved that [4] A. Baranov, D. Yakubovich, One-dimensional perturbations classical Fock space has no spectral synthesis property. Equiv- of unbounded selfadjoint operators with empty spectrum, J. alently, this means that there exist nonhereditarily complete Math. Anal. Appl. 424 (2015), 2, 1404–1424. [5] A. Baranov, D. Yakubovich, Completeness and spectral syn- systems of phase-space shifts of the Gaussian in L2(R). At thesis of nonselfadjoint one-dimensional perturbations of self- the same time, there are good reasons to believe that there ex- adjoint operators, Adv. Math. 302 (2016), 740–798. ists a universal upper bound for the defects of mixed systems. [6] Yu. Belov, T. Mengestie, K. Seip, Discrete Hilbert transforms The proofs of these results are to appear elsewhere. on sparse sequences, Proc. Lond. Math. Soc. 103 (2011) 3, 73–105. Theorem 18 (Baranov). There exist complete and minimal [7] A. Borichev, Yu. Lyubarskii, Riesz bases of reproducing ker- πλz systems e λ Λ of reproducing kernels in that are not nels in Fock type spaces, J. Inst. Math. Jussieu 9 (2010), 449– { } ∈ F hereditarily complete, that is, for some partition Λ=Λ1 Λ2, 461. πλz ∪ the mixed system e λ Λ gλ λ Λ is not complete in . [8] L. de Branges, Hilbert Spaces of Entire Functions, Prentice– { } ∈ 1 ∪{ } ∈ 2 F Hall, Englewood Cliffs, 1968. Moreover, this example of a nonhereditarily complete sys- [9] V. P. Havin, J. Mashreghi, Admissible majorants for model tem of reproducing kernels in admits the following refor- subspaces of H2. Part I: slow winding of the generating inner F mulation. Given a function G , let us denote by the function, Can. J. Math. 55, 6 (2003), 1231–1263; Part II: fast ∈F RG subspace of defined as winding of the generating inner function, Can. J. Math. 55,6 F (2003), 1264-1301. [10] V. P. Havin, J. Mashreghi, F. Nazarov, Beurling–Malliavin G = GF : GF , F – entire . multiplier theorem: the 7th proof, St. Petersburg Math. J. R { ∈F } 17 (2006), 5, 699–744. Thus, is the (closed) subspace in that consists of func- [11] S. V. Hruschev, N. K. Nikolskii, B. S. Pavlov, Unconditional RG F bases of exponentials and of reproducing kernels, Lecture tions in that vanish at the zeros of G with appropriate F Notes in Math. 864 (1981), 214–335. multiplicities. The example of Theorem 18 shows that there [12] G. Gubreev, A. Tarasenko, Spectral decomposition of model n exists G such that z G for any n 1 and operators in de Branges spaces, Sb. Math. 201 (2010), 11, n ∈F ∈F ≥ Span z G : n Z+ . 1599–1634. { ∈ } RG We stated this result to compare it with a longstanding [13] S. V. Khruschev, N. Nikolski, A function model and some problem in Fock space that has a similar form. This problem problems in spectral function theory, Proc. Steklov. Inst. Math. was posed in the 1960s by D. J. Newman and H. S. Shapiro 3 (1988), 101–214. [19], who were motivated by an old paper of E. Fisher (1917) [14] J. Lagarias, Hilbert spaces of entire functions and Dirichlet L- functions, Frontiers in number theory, physics, and geometry. on differential operators. Assume that a function G from is F I, 365–377, Springer, Berlin, 2006. such that ewzG for any w C, that is, its growth is smaller ∈F ∈ [15] D. S. Lubinsky, Universality limits for random matrices and de than the critical one. One may define on the linear span of all Branges spaces of entire functions, J. Funct. Anal. 256 (2009), exponentials the (unbounded) multiplication operator MG F = 11, 3688–3729. GF. It is natural to expect that the adjoint of MG will then be [16] A. Markus, The problem of spectral synthesis for operators ∂ with point spectrum, Math. USSR-Izv. (1970) 3, 670–696. given by an infinite order differential operator G∗ , where 4 ∂z [17] N. Makarov, A. Poltoratski, Meromorphic inner functions, z = G z G∗( ) ( ). Newman and Shapiro showed that the positive Toeplitz kernels and the uncertainty principle, Perspectives in answer to this question is equivalent to the positive solution Analysis, Math. Phys. Stud. 27, Springer, Berlin, 2005, 185– of the following problem. 252.

EMS Newsletter March 2017 17 Feature

[18][18] N. N. Makarov, Makarov, A. A. Poltoratski, Poltoratski, Beurling–Malliavin Beurling–Malliavin theory theory for for Anton Baranov [anton.d.baranov@gmail. Toeplitz kernels, Invent. Math. , 3 (2010), 443–480. Anton Baranov [anton.d.baranov@gmail. Toeplitz kernels, Invent. Math. 180, 3 (2010), 443–480. com] is a professor of mathematics at St. [19] D. J. Newman, H. S. Shapiro, Certain Hilbert spaces of entire com] is a professor of mathematics at St. [19] D. J. Newman, H. S. Shapiro, Certain Hilbert spaces of entire Petersburg State University. He works in functions,functions, Bull. Amer. Math. Soc. 72 (1966),(1966), 971–977. 971–977. Petersburg State University. He works in 72 harmonic analysis, function theory and op- [20][20] N. N. Nikolski, Nikolski, Complete Complete extensions extensions of of Volterra Volterra operators, operators, Math. harmonic analysis, function theory and op- USSR-Izv. 3 (1969)(1969) 6, 6, 1271–1276. 1271–1276. erator theory, and is particularly inter- [21][21] A. A. M. M. Olevskii, Olevskii, A. A. Ulanovskii, Ulanovskii, Functions with Disconnected ested in the interaction between these ar- Spectrum: Sampling, Interpolation, Translates, AMS, Provi- Spectrum: Sampling, Interpolation, Translates, AMS, Provi- eas, such as the theory of holomorphic dence, RI, 2016. function spaces and functional models. He has held invited [22] J. Ortega-Cerdà, K. Seip, Fourier frames, Ann. of Math. (2), function spaces and functional models. He has held invited [22] J. Ortega-Cerdà, K. Seip, Fourier frames, Ann. of Math. (2), professor positions at Aix-Marseille University and Univer- 155 (2002), 3, 789–806. professor positions at Aix-Marseille University and Univer- 155 (2002), 3, 789–806. sity Paris-Est. He received the Young Mathematician Prize of [23][23] C. C. Remling, Remling, Schrödinger Schrödinger operators operators and and de de Branges Branges spaces, spaces, J. sity Paris-Est. He received the Young Mathematician Prize of the Saint Petersburg Mathematical Society and was an invited Funct. Anal. 196 (2002),(2002), 2, 2, 323–394. 323–394. the Saint Petersburg Mathematical Society and was an invited [24][24] R. R. Romanov, Romanov, Canonical Canonical systems systems and and de de Branges Branges spaces, spaces, speaker at 7ECM in Berlin. arXiv:1408.6022. Yurii Belov [[email protected]] is an [25] K. Seip, Density theorems for sampling and interpolation Yurii Belov [[email protected]] is an [25] K. Seip, Density theorems for sampling and interpolation associate professor of mathematics at St. in the Bargmann–Fock space, I, J. Reine Angew. Math. 429 associate professor of mathematics at St. in the Bargmann–Fock space, I, J. Reine Angew. Math. 429 Petersburg State University. He works in (1992),(1992), 91–106. 91–106. Petersburg State University. He works in complex and harmonic analysis and func- [26][26] R. R. M. M. Young, Young, An Introduction to Nonharmonic Fourier Series,, complex and harmonic analysis and func- Academic Press, New-York, 1980. tiontion theory, theory, and and is is particularly particularly interested interested inin structural structural properties properties of of Hilbert Hilbert spaces spaces of of entire functions. He has held an invited pro- fessorfessor position position at at University University Paris-Est. Paris-Est. He He received received the the Young Young Mathematician Prize of the Saint Petersburg Mathematical Society and the Euler Award from the Government of Saint Petersburg. Monograph Award Call for Submissions – Deadline to expire soon! N The EMS Monograph Award is assigned every year to the author(s) of a

monograph, in any area of mathematics, that is judged by the selection M committee to be an outstanding contribution to its field. The prize is endowed with 10,000 Euro and the winning monograph will be published by the EMS Publishing House in the series “EMS Tracts in Mathematics”.

Previous prize winners were: Patrick Dehornoy et al., Foundations of Garside Theory Augusto C. Ponce, Elliptic PDEs, Measures and Capacities. From the Poisson Equation to Nonlinear Thomas– Fermi Problems Vincent Guedj and Ahmed Zeriahi, Degenerate Complex Monge–Ampère Equations Yves de Cornulier and Pierre de la Harpe, Metric Geometry of Locally Compact Groups The deadline for the next award, to be announced in 2018, is 30 June 2017. Submission of manuscripts The monograph must be original and unpublished, written in English and should not be submitted elsewhere until an editorial decision is rendered on the submission. Monographs should preferably be typeset in TeX. Authors should send a pdf file of the manuscript to:[email protected] Scientific Committee John Coates (University of Cambridge, UK) Pierre Degond (Université Paul Sabatier, Toulouse, France) Carlos Kenig (University of Chicago, USA) Jaroslav Nesˇetrˇil (Charles University, Prague, Czech Republic) Michael Röckner (Universität Bielefeld, Germany, and Purdue University, USA) Vladimir Turaev (Indiana University, Bloomington, USA)

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18 EMS Newsletter March 2017 Feature Diffusion, Optimal Transport and Ricci Diffusion, Optimal Transport and Ricci Curvature for for Metric Metric Measure Measure Spaces Spaces Luigi AmbrosioAmbrosio (Scuola (Scuola Normale Normale Superiore, Superiore, Pisa, Pisa, Italy), Italy), Nicola Nicola Gigli Gigli (SISSA, (SISSA, Trieste, Trieste, Italy) Italy) and Giuseppe Savaré (Universitàand Giuseppe di Pavia, Savaré Italy) (Università di Pavia, Italy)

Lower Ricci curvature bounds play a crucial role in several and to the second order differential operator L =∆ V, g − ∇ ·g deep geometric and functional inequalities in Riemannian ge- satisfying the integration-by-parts formula ometry and diffusion processes. Bakry–Émery [8] introduced an elegant and powerful technique, based on commutator estimates for differential operators and so-called Γ-calculus, to ( f, g) = f Lg dm. E − M derive many sharp results. Their curvature-dimension condition has been further developed by many authors, mainly in L reads in local coordinates as the framework of Markov diffusion modelled on weighted Rie- mannian manifolds, with relevant applications to infinite di- W W ij 1 ij L f = e ∂ e− g ∂ f , W = V + log det(g ), mensional problems. i j 2 A new synthetic approach relying on entropy and opti- i, j mal transport has been more recently introduced by Lott, Sturm and Villani. It relies structurally on the notions of dis- and generates a semigroup (Pt)t 0 through the solution ft = ≥ tance and measure and can therefore be used to extend the Pt f of the diffusion equation curvature-dimension condition to the general nonsmooth setting of metric measure spaces (X, d, m). Among its many beau- ∂t ft = L ft in [0, ) M, f0 = f. (2) tiful properties, the synthetic approach is stable with respect ∞ × to measured Gromov convergence. Whenever f C (M) is smooth with compact support, f is a The equivalence of the two points of view can be directly 0 ∈ c∞ classical C solution to (2) and one can extend L from C (M) proved in a smooth differential setting but it is a difficult task ∞ c∞ to a self-adjoint operator in L2(X, m) generating a Markov in a general metric framework, when explicit calculations in semigroup. local charts are hard (if not impossible) to justify. We will try to give a brief and informal introduction to the Example 1. Choosing M = Rd with the Euclidean metric and two approaches and show how the Otto variational interpre- V 0, one gets the Laplace operator and the corresponding ≡ tation of the Fokker-Planck equation and the theory of metric heat flow. More general choices of V yield the drift-diffusion gradient flows has provided a unifying point of view, which al- operator and the weighted Dirichlet form lows one to prove their equivalence for arbitrary metric measure spaces. As a byproduct, by combining Γ-calculus and op- 2 V timal transport techniques, an impressive list of deep results L f =∆f V f, ( f ):= f e− dx; −∇ ·∇ E d |∇ | in Riemannian geometry and smooth diffusion have a natural R counterpart in the nonsmooth metric measure framework. the Gaussian space (Rd, , m ) corresponds to the Gaussian |·| G measure m associated to V (x):= 1 x 2 + d ln(2π) and to the 1 Ricci curvature and Γ-calculus in a smooth G G 2 | | 2 Ornstein-Uhlenbeck operator L f =∆f x f . setting − ·∇ The usual elliptic second order operators in divergence form In order to introduce and explain the basic notions we will deal with, we consider a smooth, complete and connected d- ij ij L f = ∂i g ∂ j f , ( f ) = g ∂i f ∂ j f dx, M, E d dimensional Riemannian manifold ( g) endowed with its i, j R i, j V Riemannian distance dg and a reference measure m = e− Vol g that is absolutely continuous with respect to the Riemannian ij volume form Vol ; its density is associated to the smooth po- are associated to the uniformly elliptic metric tensor g and g to the potential V := 1 log det(gij) by the expression of the tential V : M R. The metric tensor g induces a norm f 2 → |∇ |g − of the gradient of a smooth function f : M R, given in volume measure in local coordinates. → ∆ local coordinates by f 2 = gij∂ f ∂ f . The Laplace-Beltrami operator g and the energy form |∇ |g i, j i j The combination with the reference measure m gives rise to the quadratic energy form L f =∆ f, ( f ) = f 2 dVol , g E |∇ |g g M ( f ):= f 2 dm, ( f, g):= f, g dm, (1) E |∇ |g E ∇ ∇ g in a Riemannian manifold corresponds to the choice V 0. M M ≡

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It is interesting to note that the operator L encodes the in- if, for every smooth function f with compact support and for formation concerning the metric tensor, which can be recon- every t 0, ≥ structed by the commutation identity yielding the Γ-tensor 2 e2Kt P f 2 + E (t) LP f 2 P f 2, (6) 1 |∇ t |g N 2K | t | ≤ t|∇ |g Γ( f, g):= L( fg) (L f ) g f Lg , Γ( f ):=Γ( f, f ), 2 − − where since one easily gets t 1 λt λs λ− e 1 if λ 0, Eλ(t):= e ds = − 2 0 t if λ = 0. Γ( f, g) = f, g g, Γ( f ) = f g.  ∇ ∇ |∇ |   In the flat Euclidean caseM = Rd, V 0, (6) with N = Γ also characterises the class of the Lipschitz functions (and ≡ ∞ thus the Riemannian distance), since simply follows by Jensen’s inequality and the commutation property of the heat equation P f = P ( f ). In the gen- ∇ t t ∇ Γ( f ) L2 f is L-Lipschitz. (3) eral case, (6) reflects the commutator bounds coded in (5); the ≤ ⇒ simplest situation is provided by the BE(K, ) case, when (6) ∞ follows (at least formally – see, for example, [9, Sec. 3.2.3]) Bakry–Émery [8, 9] introduced a further geometric tensor, by the monotonicity property of the quantity called Γ2, obtained by an iterated commutation: Λ = Γ , , 1 (s): Ps (Pt s f ) 0 s t Γ ( f ):= LΓ( f ) Γ( f, L f ). (4) − ≤ ≤ 2 2 − which satisfies the differential inequality Thanks to the Bochner-Lichnerowicz identity, the Γ tensor 2 d can be expressed by the following remarkable formula, in- Λ(s) = 2Ps Γ2(Pt s f ) 2KPs Γ(Pt s f ) = 2K Λ(s). volving, in a crucial way, Ricci curvature and the Hessian of ds − ≥ − V: Brunn–Minkowski and Prékopa–Leindler inequalities 2 A second instance of application of curvature bounds con- Γ ( f ) = 2 f + Ric ( f, f ) + 2V( f, f ). 2 ∇ g g ∇ ∇ ∇g ∇ ∇ cerns the curved version of the celebrated Brunn–Minkowski inequality According to Barky-Émery, the weighted Riemannian man- 1 ϑ ϑ ifold (M, d , m) satisfies the curvature-dimension condition Vol((1 ϑ)A + ϑB) Vol ( A) − Vol ( B) ,ϑ[0, 1], g − ≥ ∈ BE(K, N), K R, N [1, ] if, for every smooth function f , ∈ ∈ ∞ for an arbitrary couple of Borel sets A, B Rd; here, (1 ⊂ − 1 2 ϑ)A + ϑB = (1 ϑ)a + ϑb : a A, b B . Γ ( f ) K Γ( f ) + L f . (5) { − ∈ ∈ } 2 ≥ N In order to state it in a Riemannian manifold, it is conve- nient to denote by Zϑ(a, b), a, b M, the set of interpolating ∈ Example 2. When V 0, the BE(K, N) condition is equiva- points x M satisfying ≡ ∈ lent to Ric Kg and d N. In particular, Euclidean space g ≥ ≤ Rd satisfies the BE(0, d) condition; the d-dimensional unit d (a, x) = (1 ϑ)d (a, b), d (x, b) = ϑd (a, b),ϑ[0, 1]. g − g g g ∈ sphere Sd (resp. hyperbolic space Hd) is the reference model for BE(d 1, d) (resp. BE( (d 1), d)). Theorem 4 ([17]). The weighted manifold (M, d , m) satis- − − − g When a general potential V is involved, (5) also reflects the fies the curvature-dimension condition BE(K, ) if and only ∞ convexity of V: in the simplest Euclidean case of Rd and if, for every ϑ [0, 1] and Borel functions f, f , f : M ∈ 0 1 → N = ,(5) is equivalent to 2V KI. In particular, Gaussian [0, ) satisfying ∞ ∇ ≥ ∞ space (Rd, , m ) satisfies the BE(1, ) condition. |·| G ∞ K 2 1 ϑ ϑ f (x) exp ϑ(1 ϑ)dg(a, b) f0(a) − f1(b) (7a) 2 Two equivalent formulations of the ≥ − 2 − curvature-dimension condition whenever a, b M and x Z (a, b), it holds that ∈ ∈ ϑ The BE(K, N) condition has many deep and beautiful func- 1 ϑ ϑ − tional and geometric consequences (some of them are listed f dm f dm f dm . (7b) ≥ 0 1 at the end of this paper in Section 7). Here we focus on two M M M relevant (and seemingly far-reaching) aspects. In particular, if K 0, we have ≥ Pointwise gradient estimates for Markov diffusion 1 ϑ ϑ m Z (A, B) m(A) − m(B) , (8) A first application concerns the behaviour of the semigroup ϑ ≥ (Pt)t 0 associated to (2). ≥ where Theorem 3 ([8, 45]). The weighted manifold (M, dg, m) satisfies the curvature-dimension condition BE(K, N) if and only Z (A, B):= x Z (a, b) for some a A, b B . ϑ ∈ ϑ ∈ ∈ 20 EMS Newsletter March 2017 Feature

(8) admits the refined N-dimensional version, The dynamical formulation of Benamou–Brenier The Wasserstein distance Wd enjoys two other important dy- 1/N 1 ϑ 1/N ϑ 1/N namical characterisations, which play a crucial role in the ge- m Zϑ(A, B) τ − (δ)m(A) + τ (δ)m(B) , ≥ K,N K,N ometric formulation of the properties discussed in Section 2. According to the first one, which is due to Benamou– where τK,N ( ) are suitable distortion coefficients only depend- Brenier [10], the Wasserstein distance between µ ,µ · 0 1 ∈ ing on K and N and δ is the minimal (resp. maximal) distance (M) in the Riemannian manifold M can be evaluated by P2 between the points of A and B if K 0 (resp. K < 0). minimising the action ≥ We shall see that the optimal transport point of view provides a nice unifying interpretation of (6) and (7a,b), which 1 W2(µ ,µ ) = min v 2 dµ dt (10a) will only depend on the metric dg and on the reference mea- d 0 1 g t 0 M | | sure m defined on M, without referring to its differential structure. The basic idea is to lift the geometric properties of M to among all the weakly continuous solutions (µ, v) of the con- the space of Borel probability measures (M): (2) can be in- P tinuity equation terpreted as a gradient flow in (M) and an interpolation of P sets as in (8) becomes a geodesic in (M). In both cases, the P ∂tµt + divg(µtvt) = 0, t [0, 1], (10b) curvature-dimension condition can be characterised by the be- ∈ haviour of the entropy functional along these two classes of connecting µ0 to µ1. Equation (10b) has to be intended in du- curves. ality with smooth test functions, i.e.

3 Optimal transport and the geometry of 1 probability measures ∂tζ(x, t) + ζ(x, t), v(x, t) g dµt dt = 0, 0 M ∇ Optimal transport provides a natural way to introduce a geo- for every ζ C∞(M (0, 1)). metric distance between probability measures, which reflects ∈ c × The Benamou–Brenier representation (10a,b) can be fur- the properties of d in M. We introduce it in the more general g ther extended to Lipschitz (or even absolutely continuous) framework of a complete and separable metric space (X, d). curves (µt)t [0,1] in 2(M): they can be characterised as so- We call (X) the space of Borel probability measures ∈ P P2 lutions to the continuity equation (10b) with a Borel vector with finite quadratic moment: every µ (X) satisfies ∈P2 field v satisfying

2 2 W (µt,µt+h) d (x, xo)dµ(x) < 2 d vt(x) g dµt(x) = lim (11) X ∞ | | h 0 h2 X → for some (and thus any) reference point x X. for a.e. t (0, 1). The limit on the right side of equation o ∈ ∈ For a given couple of measures µ0,µ1 2(X), we con- (11) has a natural interpretation as the squared metric velocity ∈P 2 sider the collection Plan(µ ,µ ) of all the transport plans or µ˙t of µ at the time t, whose integral 0 1 | |Wd couplings between µ0,µ1, i.e. measures µ (X X) with ∈P × 1/2 marginals µ0,µ1, thus satisfying 1 1 2 µ˙t dt = vt dµt dt | |Wd | |g 0 0 M µ(A X) = µ0(A), µ(X B) = µ1(B) × × expresses the length of the curve in (X). P2 for every Borel subset A, B X. The squared Kantorovich- ⊂ Duality and Hamilton–Jacobi equations Rubinstein-Wasserstein distance W (µ ,µ ) (Wasserstein dis- d 0 1 The second characterisation of W is a dynamical version of tance, for short) is then deﬁned as d the dual Kantorovich formulation

2 2 1 2 W (µ0,µ1):= min d (x0, x1)dµ(x0, x1). (9) Wd (µ0,µ1) = sup Q1φ(y)dµ1(x) φ(x)dµ0(x) d µ ,µ µ Plan( 0 1) X X 2 φ Cb(X) X − X ∈ × ∈

Equation (9) is an important example of the class of opti- shared with all optimal transport problems. Here, Q1 denotes mal transport problems, where the squared distance function the inf-convolution 2 d (x0, x1)in(9) is replaced by a general cost c : X X R. × → 1 2 Wd is a distance on 2(X) inducing the topology of weak Q1φ(y):= inf d (x, y) + φ(x) P x X 2 convergence with quadratic moments, i.e. convergence of all ∈ the integrals µ φ dµ whenever φ : X R is continuous → X → and it is the value at time t = 1 of the Hopf-Lax evolution with at most quadratic growth. ( 2(X), Wd) is a complete and P (Qt)t 0, deﬁned by separable metric space and it inherits other useful properties ≥ from (X, d) such as compactness or existence of geodesics. We 1 2 Qtφ(y):= inf d (x, y) + φ(x),φCb(X). refer the interested reader to a number of books [3, 44, 39]. x X ∈ 2t ∈

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When (X, d)isalength space, i.e. for every couple of points if and only if P∗ is a L-Wasserstein contraction, i.e. x, y X and ε>0, there exists an ε-midpoint zε X satisfy- ∈ ∈ Wd(P∗µ, P∗ν) L Wd(µ, ν) for every µ, ν 2(X). ing ≤ ∈P In the case of the Markov semigroup (Pt)t 0 introduced in ≥ max d(x, zε), d(zε, y) (1/2 + ε) d(x, y), Section 1 in a Riemannian setting, one gets the following the- ≤ orem [38, 42]. then (Qt)t 0 is a semigroup in Lip b(X), the space of bounded ≥ and Lipschitz real functions defined in X. In the Euclidean Theorem 5. The semigroup (Pt)t 0 satisfies the pointwise ≥ case X = Rd, φ ( ):= Q φ is the unique viscosity solution of bound (6) of Theorem 3 with K R, N = , if and only t · t ∈ ∞ the Hamilton–Jacobi equation if the semigroup (Pt∗)t 0, defined on m-absolutely continuous ≥ measures µ = m by Pt∗µ := (Pt)m, admits a unique exten- 1 2 sion to (X) satisfying the contraction property ∂tφt(x) + φt(x) = 0, lim φt(x) = φ(x), P2 2|∇ | t 0 Kt ↓ W (P∗µ, P∗ν) e− W (µ, ν) for every µ, ν (X). (15) d t t ≤ d ∈P2 and an analogous property holds in the Riemannian setting. Even more refined estimates hold in the case N < . ∞ In general metric spaces, one can give a metric interpretation Notice that one of the advantages of (15) with respect to to the quantity φ by considering the local slope or local |∇ | (6) and to the differential formulation of the BE(K, ) con- Lipschitz constant of a map φ : X R, ∞ → dition given by (5) relies on the weaker regularity requirement of its formulation: it involves probability measures and φ(y) φ(x) φ (x):= lim sup | − |. (12) the Markov semigroup (Pt)t 0 but avoids local differentiabil- |∇ | y x d(y, x) ≥ → ity structures. This is one of the recurrent themes of pushing the geometric information coded into lower Ricci curvature It is possible to prove [4] that, for every x X, φ ( ):= Q φ ∈ t · t bounds toward general metric measure spaces. satisfies 1 Prékopa–Leindler inequality and convexity of the entropy ∂ φ (x) + φ 2(x) = 0 (13) t t 2|∇ t| A second crucial interpretation provides a link between the for every t (0, ) with at most countably many exceptions. weighted Prékopa–Leindler inequality (7a,b), the geometric ∈ ∞ A further regularisation in time [2] shows that notion of geodesics in metric spaces and the entropy functional. 1 2 Geodesics in a metric space ( , d ) are length-minimising Wd (µ0,µ1) = sup φ1(y)dµ1(y) φ0(x)dµ0(x), (14a) X X 2 − curves (xϑ)ϑ [0,1] in satisfying X X ∈ X d (xϑ , xϑ ) = (ϑ1 ϑ0)d (x0, x1), 0 ϑ0 ϑ1 1. where the supremum runs among all the regular subsolutions X 0 1 − X ≤ ≤ ≤ φ C1([0, 1]; Lip (X)) of (13), i.e. solving A real functional Φ : D(Φ) R is called geodesically ∈ b ⊂X→ K-convex if every couple of points x0, x1 D(Φ) can be con- 1 ∈ ∂ φ + φ 2(x) 0 in X [0, 1]. (14b) nected by a geodesic (xϑ)ϑ [0,1] along which t t 2|∇ t| ≤ × ∈ K 2 Φ(xϑ) (1 ϑ)Φ(x ) + ϑΦ(x ) ϑ(1 ϑ)d (x , x ). (16) ≤ − 0 1 − 2 − 0 1 4 Wasserstein distance and lower curvature Since ( (X), W ) is a metric space, we can also consider P2 d bounds in smooth Riemannian manifolds geodesics at the level of probability measures: these are curves (µϑ)ϑ [0,1] in 2(X) satisfying The two equivalent formulations of the curvature-dimension ∈ P condition BE(K, ) in Section 2 have nice counterparts in W (µ ,µ ) = (ϑ ϑ )W (µ ,µ ), 0 ϑ ϑ 1. ∞ d ϑ0 ϑ1 1 − 0 d 0 1 ≤ 0 ≤ 1 ≤ terms of the Kantorovich-Rubinstein-Wasserstein distance. In (17) order to keep the exposition simpler, we just focus on the case In a pioneering paper, McCann [36] pointed out the role and N = . the interest of geodesic convexity of suitable integral func- ∞ tionals in 2(X) as the relative entropy, Pointwise gradient estimates and Wasserstein contraction P log dm if µ = m,log L1(X, m), The first formulation relies on the Kuwada duality result Ent(µ):= X | |∈ [34], which exploits the dual dynamic representation formula  + otherwise. (14a,b) and deals with a couple of dual maps: P :C(X)  ∞ b → As a beautiful example, Theorem 4 admits a nice reformula- C (X) linear and continuous and P : (X) (X) satisfy-  b ∗ P →P tion in terms of its geodesic K-convexity. ing Theorem 6 ([37, 18]). The Prékopa–Leindler inequality (7a, b) holds in the weighted Riemannian manifold (M, dg, m) if Pφ dµ = φ d(P∗µ) for every ϕ Cb(X),µ (X). X X ∈ ∈P and only if the functional Ent is geodesically K-convex in 2(M). In this case, for every (µϑ)ϑ [0,1] satisfying (17), we P ∈ Kuwada’s duality states that in length metric spaces and un- have der minimal assumptions, P preserves Lipschitz functions and satisfies the pointwise bound Ent(µ ) (1 ϑ)Ent(µ ) + ϑEnt(µ ) ϑ ≤ − 0 1 K Pφ L P φ for every φ Lip (X) ϑ(1 ϑ)Wd(µ0,µ1). (18) |∇ |≤ |∇ | ∈ b − 2 −

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Gradient flows in (X): convexity and contraction The minimising movement approach. P2 The two remarkable properties highlighted by Theorems 5 A second point of view (and, in fact, the original one adopted and 6 can be better understood by the Jordan-Otto interpre- by Jordan–Kinderlehrer–Otto [31]) concerns a variational tation of the semigroup P∗ as the gradient flow of the en- scheme that can be used to construct gradient flows in gen- tropy functional in Wasserstein space (M). There are (at eral metric spaces [20, 3]. It is strongly related to the implicit P2 least) three different ways to justify this interpretation. The Euler scheme for (19), which suggests approximating the val- first one combines the Benamou–Brenier result with the De ues of the solution x at the discrete points kτ, k N,ofa ∈ Giorgi notion of curves of maximal slope [3], the second one uniform grid of step τ>0 by the solutions Xk x(kτ) of the τ ≈ is related to the so-called JKO/minimising movement varia- recursive discrete scheme tional scheme [20, 31, 3] and the last one captures the energy- 1 k k 1 k distance interaction of convex functions in Euclidean space. X X − + Φ(X ) = 0, k = 1, 2, , (24) τ τ − τ ∇ τ ··· Curves of maximal slope. starting from an approximation X0 x(0). One can select One starts from the basic remark that solutions in Rd to the τ ≈ solutions to (24) by choosing Xk among the minimisers of gradient flow equation τ

1 k 1 2 X X Xτ− +Φ(X). x(t) = Φ(x(t)) (19) → 2τ| − | −∇

d At the level of measures, replacing the Euclidean distance in for a smooth function Φ : R R can be characterised by d → R by the Wasserstein distance in 2(X) and the function Φ the maximal rate decay of Φ, in the sense that along a general P by the relative entropy, one eventually obtains the following smooth curve y, we have scheme.

0 k d Given M := µ = m, find M 2(X) by minimising Φ(y(t)) = Φ(y(t)) y(t) Φ(y(t)) y(t) , (20) τ τ dt −∇ · ≥ −|∇ || | ∈P 1 2 k 1 M W (M, M − ) + Ent(M), k = 1, 2, . → 2τ d τ ··· whereas along any solution of (19), we have precisely Defining Mτ(t) as the piecewise constant interpolant of the k d 2 2 values Mτ in each interval ((k 1)τ, kτ], one can prove, under Φ(x(t)) = Φ(x(t) x(t) = Φ(x(t)) = x(t) , − dt −|∇ || | −|∇ | −| | very general assumptions [31, 3, 4], that the family Mτ con- (21) verges locally uniformly in (X) to the solution µ = P µ = P2 t t∗ an identity that is, in fact, equivalent to (19). (Pt)m associated to (2). In order to mimic the above argument at the Wasserstein level, we first observe that, along any smooth solution µt = Gradient flows and evolution variational inequalities. tm of the continuity equation (10b), it is possible to compute In the Euclidean framework, one can evaluate the derivative the time derivative of the entropy by an integration by parts, of the squared distance function of a solution x of (19) from obtaining any given point y Rd, obtaining ∈ d d 1 Ent(µ ) = log , v dµ x(t) y 2 = Φ(x(t), y x(t) , dt t ∇ t t t dt 2| − | ∇ − M 1/2 1/2 2 2 and then use the K-convexity inequality log t g dµt vt g dµt . ≥− M |∇ | M | | K Φ(x), y x Φ(y) Φ(x) x y 2 Thus, the rate of decay of Ent is bounded from below by the ∇ − ≤ − − 2 | − | product of the Wasserstein velocity of the curve (11) and the to obtain the Evolution Variational Inequality (EVIK) square root of the Fisher information functional

d 1 2 K 2 d 2 x(t) y Φ(y) Φ(x(t)) x(t) y y R . (25) g dt 2| − | ≤ − − 2 | − | ∀ ∈ F(µ) = log 2 dµ = |∇ | dm |∇ |g t >0 >0 (22) It is not hard to see that (25), in fact, characterises the solu- 2 tions to (19). Since (25) just involves the ambient distance and = 4 √ g dm,µ= m, M |∇ | the driving functional Φ, one is tempted to use it for a possible definition of gradient flows in arbitrary metric spaces [3]. which plays the same role as Φ in (20). On the other hand, |∇ | Definition 7. Let ( , d ) be a metric space and let Φ : along the solution µt = Pt∗µ = (Pt)m induced by (2), one has X X D(Φ) R be a given l.s.c. functional. A semigroup ⊂X→ (St)t 0 in D(Φ) is an EVIK-flow of Φ if, for every x D(Φ), d 2 ≥ ∈ Ent(µt) = F(µt) = µ˙t , (23) the curve xt := St x is locally Lipschitz in (0, ) and solves dt − −| |Wd ∞

d 1 2 K 2 which corresponds to (21). d (x(t), y) Φ(y) Φ(x(t)) d (x(t), y) y . dt 2 X ≤ − − 2 X ∀ ∈X

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In general metric spaces, the existence of an EVI -flow where n N and ϕ C (Rn n). K ∈ ∈ b × is a much stronger requirement than the simple energy- It is clear that if two metric measure spaces (Xi, di, mi), i = dissipation identity (21): it encodes both the K-convexity of 1, 2, are isomorphic, i.e. there exists an isometry ι : X X 1 → 2 Φ and a sort of infinitesimal Riemannian behaviour of the dis- preserving distances and volumes tance d [19]. X 1 d2(ι(x),ι(y)) = d1(x, y), m1(ι− (A)) = m2(A) Theorem 8. Suppose that (St)t 0 is an EVIK flow for Φ on ≥ ( , d ). Then, S is a K-exponential contraction, i.e. X X for every couple of points x, y X1 and every Borel set ∈ Kt A X2, then ϕ (X1, d1, m1) = ϕ (X1, d2, m2) for every d (St x, Sty) e− d (x, y). ⊂ X ≤ X cylindrical functional ϕ as in (28). The Gromov reconstruc- If, moreover, any couple of points x, y D(Φ) can be joined tion theorem guarantees the converse property: if two nor- ∈ by a geodesic in D(Φ) then Φ is strongly K-geodesically con- malised metric measure spaces are indistinguishable by all vex. the cylindrical functionals then they are isomorphic. It jus- tifies the following definition [30, 28] (see [40, 44, 27] for Taking the above result and Otto heuristics concerning other equivalent approaches and for the relation with mea- the Riemannian character of the Wasserstein distance into ac- sured Gromov-Hausdorff convergence). count, it is not completely surprising that Theorems 5 and 6 can be obtained as a consequence of the EVIK character- Definition 10 (Measured Gromov convergence). A sequence isation of the Markov semigroup P in (M). In fact, the of normalised metric measure spaces (Xh, dh, mh), h N, con- ∗ P2 ∈ weighted Riemannian manifold (M, dg, m) satisfies the curva- verges to a limit metric measure space (X, d, m) if, for every ture-dimension BE(K, ) condition if and only if the semi- cylindrical functional ϕ as in (28), we have ∞ group (Pt∗)t 0 is the EVIK flow of the entropy functional in ≥ h h h 2(X): for every µ D(Ent), the curve µt = P∗µ satisfies lim ϕ (X , d , m ) = ϕ (X, d, m) . t h P ∈ →∞ d 1 2 K 2 Wd (µt,ν) Ent(ν) Ent(µt) Wd (µt,ν) (26) In view of Theorem 9, it is natural to look for a synthetic dt 2 ≤ − − 2 definition of lower Ricci curvature bounds for general metric for every ν D(Ent). measure spaces that is stable under measured Gromov conver- ∈ gence, a programme that has been outlined in [14, Appendix 5 Metric measure spaces, Gromov 2]. Such a definition has been independently introduced by convergence and the Lott–Sturm–Villani Sturm [40, 41] and Lott–Villani [35], starting from the smooth curvature-dimension condition characterisation given by Theorem 6. Definition 11 (The Lott–Sturm–Villani CD(K, ) condition). Let us now address the question of how to extend the previ- ∞ A metric measure space (X, d, m) satisfies the CD(K, ) con- ous results to a general metric measure space (X, d, m), i.e. a ∞ dition if the entropy functional Ent is geodesically K-convex complete and separable metric space (X, d) endowed with a in ( (X), W ): every couple µ ,µ D(Ent) can be con- non-negative Borel measure m, which we assume here to be P2 d 0 1 ∈ nected by a geodesic (µϑ)ϑ [0,1] satisfying (17) along which with full support, satisfying the growth condition ∈ (18) holds. 2 m(B (x )) a ebr , a, b R . (27) r o ≤ ∈ + A similar but technically more complicated notion can be introduced in the case of a finite dimension upper bound N < This general class of spaces naturally arises when lower Ricci (and we do not distinguish here between CD, CD or CDe curvature bounds are considered, thanks to the following re- ∞ ∗ classes of spaces). Besides its intrinsic geometric structure, markable result of Gromov (see, for example, [30]) and to the just involving the notion of distance (through Wasserstein deep contributions by Cheeger-Colding [14, 15, 16]. geodesics) and measure (through the entropy functional), a Theorem 9 (Gromov compactness). Let (Mh, dh, mh), h N, crucial feature of the above definition is its stability with re- g ∈ be a sequence of weighted Riemannian manifolds with uni- spect to measured Gromov convergence: if (Xh, dh, mh), h ∈ formly bounded diameter and satisfying a uniform BE(K, N) N, is a sequence of CD(K, N) metric measure spaces con- condition, for some K, N R independent of h. Then, there verging to (X, d, m) in the measured Gromov topology then ∈ exist a limit metric measure space (X, d, m) and a subsequence (X, d, m) is a CD(K, N) metric measure space. h(n) n h(n) such that (Mh(n), d , mh(n)) converges to (X, d, m) It is possible to prove that CD(K, N) metric measure → g under the measured Gromov convergence. spaces enjoy many of the geometric properties that are a consequence of the curvature-dimension condition in the smooth Perhaps the simplest way to introduce measured Gromov Riemannian setting (see the final remarks below). convergence for normalised (i.e. m(X) = 1) metric measure Definition 11, however, captures only part of the informa- spaces is to resort to another beautiful theorem of Gromov tion coded in the Riemannian formalism, since geodesic K- [30], characterising the equivalence class of metric measure convexity of the entropy functional is also shared by Finsler spaces up to measure-preserving isometries. It relies on the (non-Riemannian) geometries: perhaps the simplest example notion of cylindrical metric functionals of the form is given by the space (Rd, , Vol), where the distance is in- · duced by an arbitrary norm (not necessarily Hilbertian) n n · ϕ (X, d, m) := ϕ d(xi, x j)i, j=1 dm⊗ (x1, , xn), (28) n ··· and the measure is the usual Lebesgue one. X 24 EMS Newsletter March 2017 Feature

On the other hand, in many examples where a specific of a nonnegative measure µ = m as in (22): Dirichlet energy form and a corresponding Markov semi- 2 group can be constructed, one may adopt the Bakry–Émery w F(µ) = 8 Ch( √ ) = |∇ | dm,µ= m. (30) point of view to characterise a curvature-dimension condi- >0 tion. In this direction, Gromov wrote [29, page 85]: “There is another option for the abstract theory of Ricci 0, where Even if Ch is not a quadratic form, it is still possible to use ≥ instead of the metric one emphasizes the heat flow (diffusion), convex analysis to define the nonlinear Laplacian ∆X f as the − but at this stage it is unclear whether the two approaches are element of minimal L2-norm of its L2-subdifferential, consist- equivalent and if not which one is better for applications.” ing of all the functions ξ L2(X, m) satisfying the variational ∈ A relevant question is to single out a stronger condition for inequality general metric measure spaces that is still stable under measured Gromov convergence and is capable of reproducing the ξ(g f )dm Ch(g) Ch( f ) for every g D(Ch). − ≤ − ∈ pointwise gradient estimates (6) along a suitably adapted ver- X sion of the heat flow. This would have the great advantage of It is a remarkable result of the theory of gradient flows in combining both the tools from Γ-calculus and optimal trans- Hilbert spaces that, for every f L2(X, m), there exists a port and hopefully extending to the non-smooth framework ∈ unique locally Lipschitz curve ( f ) solving many deep results and techniques available in the Rieman- t t>0 nian setting. As explained in Section 4, the Wasserstein gra- d dient flow of the entropy functional provides a unifying point ft =∆X ft for a.e. t > 0, lim ft = f. (31) dt t 0 of view for both the approaches and keeps the basic feature of ↓ a pure geometric formulation in terms of distance and mea- The map Pt : f ft defines a continuous semigroup of con- 2 → sure. tractions in L (X, m); by the specific property of Ch, (Pt)t 0 ≥ can also be extended to a semigroup of contractions in every p 6 RCD(K, ) spaces and the link between BE L (X, m), preserving positivity, mass and constants. ∞ and CD It is then possible to prove, in many cases, that the semigroup (Pt)t 0 coincides with the Wasserstein gradient flow ≥ The Cheeger energy and its L2-gradient flow of the entropy functional (as a curve of maximal slope and A first step in the direction of extending the Bakry–Émery as a limit of the minimising movement variational scheme): approach to the setting of metric measure spaces concerns this important identification holds, in particular, for the whole the construction of a canonical energy form, the so called class of CD(K, ) metric measure spaces [24, 4]. ∞ Cheeger energy [13], and of the related evolution semigroup. Theorem 13. If (X, d, m) is a CD(K, ) space then, for every The Cheeger energy can be obtained by a relaxation pro- ∞ µ = m (X) with Ent(µ) < , the curve µ = (P )m is cedure from the functional f 2(x)dm(x), initially defined ∈P2 ∞ t t X |∇ | locally Lipschitz in 2(X) and the map t Ent(µt) is locally on bounded Lipschitz functions and involving the local slope P → Lipschitz and satisfies (23); it is, moreover, the limit of the introduced in (12). minimising movement scheme (30). Conversely, any locally 2 µ = m Definition 12 (The Cheeger energy). For every f L (X, m), Lipschitz curve t t , t 0, in 2(X) solving (23) satisfies ∈ ≥ P we define t = Pt0. Quadratic Cheeger energies and a metric setting for the 1 2 Ch( f ):= inf lim inf fn dm : n 2 |∇ | Bakry–Émery approach →∞ X (29) L2 If one looks for a good metric framework where the Bakry– f Lip (X), f f , n ∈ b n → Émery approach can be applied, there are at least two essential properties: the linearity of Pt and the link between distance with proper domain D(Ch) := f L2(X, m) : Ch( f ) < . and energy. Since Pt is originally defined as the gradient flow ∈ ∞ of the Cheeger energy in Hilbert space L2(X, m), it is not sur- It is possible to prove that Ch is a convex, 2-homogeneous, prising that the linearity of Pt is related to the quadraticity lower semicontinuous functional in L2(X, m) with a dense do- of Ch; as a byproduct, it induces a nice connection between main. For every f D(Ch), there exists at least one opti- weak gradients and Γ-calculus. ∈ mal sequence ( f ) Lip (X) converging to f in L2(X, m) n n ⊂ b Theorem 14 (Cheeger energy, Dirichlet forms and Γ). The and realising the infimum in (29): the corresponding slopes 2 semigroup (Pt)t 0 is linear if and only if the Cheeger energy f converge strongly in L (X, m) to a unique limit that is ≥ |∇ n| is quadratic, i.e. for every f, g D(Ch), called the weak gradient of f and is denoted by f w. The ∈ |∇ | map f f is 1-homogeneous and subadditive, enjoys → |∇ |w Ch( f + g) + Ch( f g) = 2 Ch( f ) + 2 Ch(g). (32) some natural calculus rules [4, 26] and represents Ch by the − formula In this case, ( f, g):= Ch( f +g) Ch( f ) Ch(g) is a strongly E − − 1 local Dirichlet form, whose Γ-tensor coincides with the weak Ch( f ) = f 2 dm, 2 |∇ |w gradient and whose generator L coincides with ∆X: X which can also be useful to define the Fisher information F(µ) Γ( f ) = f 2 if f D(Ch), L f =∆ f if f D(∆ ). |∇ |w ∈ X ∈ X

EMS Newsletter March 2017 25 Feature

In the metric setting, a nice collection of smooth functions Theorem 17. If (Xh, dh, mh), h N, is a sequence of ∈ where the pointwise diﬀerential formulation of the Bakry– RCD(K, N) metric measure spaces converging to (X, d, m) in Émery condition (5) can be stated is lacking; however, it is the measured Gromov topology then (X, d, m) is an RCD(K, N) possible to give a suitable weak formulation that is still equiv- metric measure space. Moreover, if the diameters of Xh are alent to the pointwise gradient estimate (6): e.g. the BE(K, ) uniformly bounded and λ (Lh), k N, are the ordered se- ∞ k ∈ condition is equivalent to asking that the map quences of eigenvalues of the compact operator Lh, we have

h 2Ks lim λk(L ) = λk(L) for every k N. s e− Γ(Pt s f )Psφ dm h ∈ → − →∞ X is increasing in (0, t) for every f D(Ch) and every nonneg- 7 Applications ∈ ative φ L∞(X, m). ∈ ffi Concerning the link with distance, the very definition of It is really di cult to give even a partial account of the ongo- Cheeger energy shows that every bounded L-Lipschitz func- ing and striking developments of the metric theory of CD and ffi tion f satisfies RCD spaces. Both are su ciently flexible and strong to guarantee a series of structural geometric results: among them, we 2 quote the tensorisation property, the global-to-local and local- Γ( f ) = f w L m-a.e. (33) |∇ | ≤ to-global characterisations of the CD/RCD conditions and the In order to infer geometric properties on (X, d) from the en- development of a nice first and second order calculus [26]. ergy form, it is natural to ask that every function f D(Ch) We now recall some of the most important geometric and ∈ satisfying (33) admits an L-Lipschitz representative. functional analytic estimates (often stated in particular exem- plifying cases) that can be derived for a general metric mea- The RCD(K, ) condition and the entropic EVI -flow sure space (X, d, m). We start from the properties valid for ∞ K Summarising the discussion above for a general metric mea- all CD(K, N) spaces, where the recent results of Cavalletti- sure space with a quadratic Cheeger energy, it is possible to Mondino [11, 12] solve a series of important open problems ask for the Lott–Sturm–Villani CD(K, ) condition or for the and show the power of the optimal transport approach (in ∞ Bakry–Émery condition BE(K, ). It turns out that these are, the RCD framework, they can also be deduced by Γ-calculus ∞ in fact, equivalent and can be unified by the notion of EVIK tools – see [9]). flow [4, 5, 6]. Bishop–Gromov inequality: For x X, the map 0 ∈ X, d, m Theorem 15. For a general metric measure space ( ), m(B (x )) the following properties are equivalent: r r 0 is nonincreasing, → r K,N 0 s (t)dt (1) The Cheeger energy is quadratic according to (32) (and K,N thus (Pt)t 0 is linear) and (X, d, m) is a CD(K, ) space. where s is the function providing the measure of the ≥ ∞ spheres in the model space of Ricci curvature K and dimen- (2) The Cheeger energy is quadratic according to (32), ev- sion N [44]. ery function satisfying (33) is L-Lipschitz and the Bakry– Émery condition holds (in a suitably weak formulation). Bonnet–Myers diameter estimate: If K > 0 then the diameter of X is bounded by π √(N 1)/K. (3) The entropy functional Ent admits a EVIK flow according − to Definition (7). Spectral gap and Poincaré inequality: If K > 0 then

This result leads to the following definition. ¯ 2 N 1 2 ¯ ( f f ) dm − f w dm, f = f dm, X − ≤ NK X |∇ | X Definition 16 (The Riemannian curvature-dimension condition). A metric measure space (X, d, m) satisfies the and a sharp inequality also holds for Lp with p 2[12]. RCD(K, ) condition if one of the equivalent properties of ∞ Log–Sobolev and Talagrand inequalities: If K > 0 and Theorem 15 is satisfied. m(X) = 1 then [12] The theorem above has been remarkably extended to the KN N 1 case of the finite dimension condition by Erbar-Kuwada- W2(µ, m) Ent(µ) − F(µ). 2(N 1) d ≤ ≤ 2KN Sturm [21] by introducing a suitable notion of EVIK,N flow − for the entropy power functional Sharp Sobolev inequalities: If K > 0, N > 2, 2 < p 2 := ≤ 1 2N/(N 2) then [12] H (µ):= exp Ent(µ) . − N − N 2 2 (p 2)(N 1) 2 f Lp f L2 + − − f w dm. A different approach, using Rény entropies in the original for- ≤ KN X |∇ | mulation of the Lott–Sturm–Villani condition, has also been developed by [1]. A crucial result due to the formulation in Levy–Gromov inequality: If m(X) = 1, diam(X) = D and terms of entropy and Wasserstein distance is the following A X with perimeter P(A) < then stability property with respect to measured Gromov conver- ⊂ ∞ gence [6, 27]. P(A) (m(A)), ≥IK,N,D

26 EMS Newsletter March 2017 Feature where is a suitably defined model isoperimetric profile for [8] D. Bakry and M. Émery. Diffusions hypercontractives. In I the parameters K, N, D (such as the N-dimensional sphere, Séminaire de probabilités, XIX, 1983/84, volume 1123, pages when N is an integer and K > 0). The case N = holds 177–206. Springer, Berlin, 1985. ∞ [9] D. Bakry, I. Gentil, and M. Ledoux. On Harnack inequalities in RCD spaces [9, Cor. 8.5.5], [7]. and optimal transportation. Ann. Sc. Norm. Super. Pisa Cl. Sci. Let us now consider the specific case of RCD(K, N) spaces, (5), 14(3):705–727, 2015. where (Pt)t 0 is a linear Markov semigroup associated to a [10] J.-D. Benamou and Y. Brenier. A computational fluid mechan- ≥ Markov process and second order calculus tools can also be ics solution to the Monge-Kantorovich mass transfer problem. developed [22]. Numer. Math., 84(3):375–393, 2000. [11] F. Cavalletti and A. Mondino. Sharp and rigid isoperimetric Li-Yau and Harnack inequalities: If K 0 and N < then inequalities in metric-measure spaces with lower Ricci curva- ≥ ∞ [9, Cor. 6.7.6] ture bounds. To appear in Invent. Math., arXiv:1502.06465, 2015. N L(log Pt f ) t > 0, [12] F. Cavalletti and A. Mondino. Sharp geometric and func- ≥−2t tional inequalities in metric measure spaces with lower Ricci N/2 t + s d2(x,y)/2 curvature bounds. To appear in Geom. Topol., ArXiv e- P f (x) P + f (y) e . t ≤ t s t prints:1505.02061, 2015. [13] J. Cheeger. Differentiability of Lipschitz functions on metric The splitting theorem [25]: If K 0, N [2, ) and X measure spaces. Geom. Funct. Anal., 9(3):428–517, 1999. ≥ ∈ ∞ contains a line, i.e. there exists a map γ : R X such that [14] J. Cheeger and T. H. Colding. On the structure of spaces → d(γ(s),γ(t)) = t s for every s, t R, then (X, d, m) is iso- with Ricci curvature bounded below. I. J. Differential Geom., | − | ∈ morphic to the product of R (with Euclidean distance and the 46(3):406–480, 1997. [15] J. Cheeger and T. H. Colding. On the structure of spaces usual Lebesgue measure) and a RCD(0, N 1) space. − with Ricci curvature bounded below. II. J. Differential Geom., The maximal diameter theorem [32]: If (X, d, m) satisfies 54(1):13–35, 2000. the RCD(N, N + 1) condition with N > 0 and there exists [16] J. Cheeger and T. H. Colding. On the structure of spaces with points x, y X such that d(x, y) = π then (X, d, m) is isomor- Ricci curvature bounded below. III. J. Differential Geom., ∈ phic to the spherical product of [0,π] and a RCD(N 1, N) 54(1):37–74, 2000. − space with diameter less than π. [17] D. Cordero-Erausquin, R. J. McCann, and M. Schmucken- schläger. A Riemannian interpolation inequality à la Borell, Volume-to-metric cones [23]: If K = 0, there exists xo X Brascamp and Lieb. Invent. Math., 146(2):219–257, 2001. N ∈ such that m(Br(xo)) = (R/r) m(Br(xo)) for some R > r > 0 [18] D. Cordero-Erausquin, R. J. McCann, and M. Schmucken- and the sphere centred at x0 of radius R/2 contains at least schläger. Prékopa-Leindler type inequalities on Riemannian three points then the ball BR(xo) is locally isometric to the manifolds, Jacobi fields, and optimal transport. Ann. Fac. Sci. ball B (y ) of the cone Y built over an RCD(N 2, N 1) Toulouse Math. (6), 15(4):613–635, 2006. R o − − space. [19] S. Daneri and G. Savaré. Eulerian calculus for the displace- ment convexity in the Wasserstein distance. SIAM J. Math. We conclude this brief review by noting that there have been Anal., 40(3):1104–1122, 2008. some recent striking applications to time-dependent metric [20] E. De Giorgi. New problems on minimizing movements. In measure spaces and Ricci flows [43, 33]. C. Baiocchi and J. L. Lions, editors, Boundary Value Problems for PDE and Applications, pages 81–98. Masson, 1993. [21] M. Erbar, K. Kuwada, and K.-T. Sturm. On the equivalence of Bibliography the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces. Invent. Math., 201(3):993– ff [1] L. Ambrosio, A. Mondino, and G. Savaré. Nonlinear di usion 1071, 2015. equations and curvature conditions in metric measure spaces. [22] N. Gigli. Nonsmooth differential geometry – An approach ArXiv e-prints, 1509.07273, 2015. tailored for spaces with Ricci curvature bounded from below. [2] L. Ambrosio, M. Erbar, and G. Savaré. Optimal transport, To Mem. Amer. Math. Soc., ArXiv:1407.0809, 2015. Cheeger energies and contractivity of dynamic transport distances in extended spaces. Nonlinear Anal., 137:77–134, [23] N. Gigli and G. de Philippis. From volume cone to metric cone Geom. Funct. Anal. 2016. in the nonsmooth setting. , 26(6):1526– [3] L. Ambrosio, N. Gigli, and G. Savaré. Gradient flows in metric 1587, 2016. spaces and in the space of probability measures. Lectures in [24] N. Gigli, K. Kuwada, and S. Ohta. Heat flow on Alexandrov Mathematics ETH Zürich. Birkhäuser Verlag, Basel, second spaces. Comm. Pure Appl. Math., 66:307–331, 2013. edition, 2008. [25] N. Gigli. An overview of the proof of the splitting theorem in [4] L. Ambrosio, N. Gigli, and G. Savaré. Calculus and heat flow spaces with non-negative Ricci curvature. Anal. Geom. Metr. in metric measure spaces and applications to spaces with Ricci Spaces, 2:169–213, 2014. bounds from below. Invent. Math., 195(2):289–391, 2014. [26] N. Gigli. On the differential structure of metric measure spaces [5] L. Ambrosio, N. Gigli, and G. Savaré. Metric measure spaces and applications. Mem. Amer. Math. Soc., 236(1113):vi+91, with Riemannian Ricci curvature bounded from below. Duke 2015. Math. J., 163(7):1405–1490, 2014. [27] N. Gigli, A. Mondino, and G. Savaré. Convergence of pointed [6] L. Ambrosio, N. Gigli, and G. Savaré. 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[29] M. Gromov. Sign and geometric meaning of curvature. Rend. [43] K. Sturm. Super-Ricci flows for metric measure spaces. I. Sem. Mat. Fis. Milano, 61:9–123 (1994), 1991. arXiv:1603.02193, 2016. [30] M. Gromov. Metric structures for Riemannian and non- [44] C. Villani. Optimal transport. Old and new, Volume 338 of Riemannian spaces. Modern Birkhäuser Classics. Birkhäuser, Grundlehren der Mathematischen Wissenschaften. Springer- Boston, 2007. Verlag, Berlin, 2009. [31] R. Jordan, D. Kinderlehrer, and F. Otto. The variational for- [45] F.-Y. Wang. Equivalent semigroup properties for the mulation of the Fokker–Planck equation. SIAM J. Math. Anal., curvature-dimension condition. Bull. Sci. Math., 135(6– 29(1):1–17, 1998. 7):803–815, 2011. [32] C. Ketterer. Cones over metric measure spaces and the maximal diameter theorem. J. Math. Pures Appl. (9), 103(5):1228– 1275, 2015. [33] E. Kopfer and K.-T. Sturm. Heat flows on time-dependent Luigi Ambrosio [[email protected]] is metric measure spaces and super-Ricci flows. ArXiv e-prints, a professor of mathematics at Scuola Nor- 2016. male Superiore, Pisa. His research inter- [34] K. Kuwada. Duality on gradient estimates and Wasserstein ests include geometric measure theory, cal- controls. J. Funct. Anal., 258(11):3758–3774, 2010. culus of variations and optimal transport. [35] J. Lott and C. Villani. Ricci curvature for metric-measure spaces via optimal transport. Ann. of Math. (2), 169(3):903– 991, 2009. [36] R. J. McCann. A convexity principle for interacting gases. Nicola Gigli [[email protected]] is a profes- Adv. Math., 128(1):153–179, 1997. sor of mathematics at SISSA, Trieste. He [37] F. Otto and C. Villani. Generalization of an inequality by Ta- lagrand and links with the logarithmic Sobolev inequality. J. is currently interested in geometric and Funct. Anal., 173(2):361–400, 2000. non-smooth analysis, with a particular fo- [38] F. Otto and M. Westdickenberg. Eulerian calculus for the con- cus on properties of spaces with curvature traction in the Wasserstein distance. SIAM J. Math. Anal., bounded from below. 37(4):1227–1255 (electronic), 2005. [39] F. Santambrogio. Optimal transport for applied mathemati- Giuseppe Savaré [giuseppe.savare@unipv. cians, volume 87 of Progress in Nonlinear Differential Equa- it] is a professor of mathematics at Pavia tions and their Applications. Birkhäuser/Springer, Cham, 2015. Calculus of variations, PDEs, and modeling. University. His current research interests [40] K.-T. Sturm. On the geometry of metric measure spaces. I. involve optimal entropy transport prob- Acta Math., 196(1):65–131, 2006. lems and variational methods for gradi- [41] K.-T. Sturm. On the geometry of metric measure spaces. II. ent flows and rate-independent evolutions. Acta Math., 196(1):133–177, 2006. This article follows his invited talk at the [42] K.-T. Sturm and M.-K. von Renesse. Transport inequalities, 7th European Congress of Mathematics in gradient estimates, entropy, and Ricci curvature. Comm. Pure Berlin, 2016. Appl. Math., 58(7):923–940, 2005.

European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum SEW A21, CH-8092 Zürich, Switzerland [email protected] / www.ems-ph.org

Geometry, Analysis and Dynamics on sub-Riemannian Manifolds, Volume I and Volume II (EMS Series of Lectures in Mathematics) Davide Barilari (Université Paris 7 Denis Diderot, Paris, France), Ugo Boscain (École Polytechnique, Palaiseau, France) and Mario Sigalotti (École Polytechnique, Palaiseau, France), Editors Vol. I: ISBN 978-3-03719-162-0. 2016. 332 pages. Softcover. 17 x 24 cm. 44.00 Euro Vol. II ISBN 978-3-03719-163-7. 2016. Approx. 306 pages. Softcover. 17 x 24 cm. 44 Euro Sub-Riemannian manifolds model media with constrained dynamics: motion at any point is only allowed along a limited set of directions, which are prescribed by the physical problem. From the theoretical point of view, sub- Riemannian geometry is the geometry underlying the theory of hypoelliptic operators and degenerate diffusions on manifolds. In the last twenty years, sub-Riemannian geometry has emerged as an independent research domain, with extremely rich motivations and ramifications in several parts of pure and applied mathematics, such as geometric analysis, geometric measure theory, stochastic calculus and evolution equations together with applications in mechanics, optimal control and biology. The aim of the lectures collected here is to present sub-Riemannian structures for the use of both researchers and graduate students.

28 EMS Newsletter March 2017 History Claude Shannon: His Work and Its Legacy1

Michelle Effros (California Institute of Technology, USA) and H. Vincent Poor (Princeton University, USA)

The year 2016 marked the centennial of the birth of Claude Elwood Shannon, that singular genius whose fer- tile mind gave birth to the field of information theory. In addition to providing a source of elegant and intriguing mathematical problems, this field has also had a profound impact on other fields of science and engineering, notably communications and computing, among many others. While the life of this remarkable man has been recounted elsewhere, in this article we seek to provide an overview of his major scientific contributions and their legacy in today’s world. This is both an enviable and an unenviable task. It is enviable, of course, because it is a wonderful story; it is unenviable because it would take volumes to give this subject its due. Nevertheless, in the hope of providing the reader with an appreciation of the extent and impact of Shannon’s major works, we shall try. To approach this task, we have divided Shannon’s work into 10 topical areas:

- Channel capacity - Channel coding - Multiuser channels - Network coding - Source coding - Detection and hypothesis testing - Learning and big data - Complexity and combinatorics - Secrecy - Applications

We will describe each one briefly, both in terms of Shan- non’s own contribution and in terms of how the concepts initiated by Shannon have influenced work in the inter- communication is possible even in the face of noise and vening decades. By necessity, we will take a minimalist the demonstration that a channel has an inherent maxi- approach in this discussion. We offer apologies for the mal rate at which it can reliably deliver information are, many topics and aspects of these problems that we must arguably, Shannon’s most important contributions to the necessarily omit. field. These concepts were first expounded in his foun- dational 1948 paper. He developed these ideas further Channel capacity throughout the 1950s and even into the 1960s, examining By Shannon’s own characterisation: “The fundamental the capacity of particular channels, looking at the effects problem of communication is that of reproducing at one of feedback and other features of existing communica- point either exactly or approximately a message select- tion networks and also, because capacity in his vision is ed at another point.” The channel is the medium – wire, an asymptotic quantity, looking at ways in which that cable, air, water, etc. – through which that communication asymptote is achieved. occurs. Often, the channel transmits information in a way Since Shannon’s original work, the notion of capacity that is noisy or imperfect. The notion that truly reliable has evolved in several directions. For example, the traditional notion of capacity has been generalised to remove 1 This paper was adapted from a talk presented at The Bell many of Shannon’s original simplifying assumptions. In Labs Shannon Conference on the Future of the Information addition, the notion of capacity has been expanded to Age, Murray Hill, NJ, 28–29 April 2016, celebrating the occa- capture other notions of communication. For example, sion of the Shannon centennial. identification capacity was introduced to measure the

EMS Newsletter March 2017 29 History capacity when one is only interested in knowing when a data transmission. This was followed quickly by another message is present, not necessarily what is in the message, revolution, namely space-time coding. These ideas have computation capacity was introduced to measure how driven a lot of what has happened in practice since that many computations are possible in certain circumstanc- time, including the revival of the near-capacity-achieving es, and so on. Capacity has also been applied to many low-density parity-check codes and the introduction of types of channels that have emerged since Shannon’s multiple-input multiple-output (MIMO) systems. These day. Examples include quantum channels, which include advances have enabled modern high-capacity data com- both quantum as well as classical notions of transmission munication systems. And, of course, there have been and noise, fading channels, which model signal attenua- many other key developments, including fountain and tion in wireless transmissions, and, most famously, mul- Raptor codes, polar codes, etc. In recent times, we have tiple-antenna channels, which form the basis of modern also seen a resurgence of some of the earlier ideas relat- wireless broadband communications. Even more recent- ed to areas such as cloud storage and other distributed ly, Shannon’s asymptotic concept of capacity, which relies storage applications. So, channel coding provides a fur- on the ability to use a channel an unlimited number of ther example of a very early idea of Shannon’s that has times, has been examined in a finite-blocklength setting, played a critical role in driving what is happening in tech- where only a limited number of channel uses is consid- nology today. ered; the finite-blocklength constraint is relevant to modern, delay-constrained applications such as multimedia Multiuser channels communications. Shannon introduced the notions of channel coding Channel capacity has been an enduring concept. Even and capacity in a very simple communication setting in today, almost seven decades later, we are still using the which a single transmitter sends information to a single notion of capacity to think about how communication receiver. The techniques that he used to analyse chan- channels behave. We have every expectation that it will nels in this setting are applicable well beyond this simple continue to be an important concept well into the future. “point-to-point” communication model. Multiuser channel models generalise point-to-point channel models by Channel coding incorporating multiple transmitters, multiple receivers, In his 1948 paper, Shannon showed that, for any com- multi-directional flow of information or some combina- munication rate less than capacity, one can communicate tion of these features. with arbitrarily small error probabilities. In Shannon’s The generalisation from point-to-point channels to paradigm, reliability is achieved through channel coding: multiuser channels shows up in Shannon’s own work as transmitters protect signals against errors by introducing early as the 1950s. In his 1956 paper, Shannon general- redundancy into each message before transmission, and ised his network model from the point-to-point scenario receivers apply their knowledge of the type of redun- to channels incorporating feedback; the goal in that work dancy employed to improve their probability of correct- was to understand when feedback from the receiver to ly determining the intended message from the channel the transmitter increases the rate at which the transmit- output. The idea of adding redundancy to a signal was ter can send to the receiver. That work employed two not new but, prior to Shannon, many communications notions of capacity: the capacity achievable with an engineers thought that achieving arbitrarily small error asymptotic notion of reliability and the capacity achiev- required more and more redundancy, therefore neces- able with perfect reliability. In the former, information sarily forcing the rate of transmission to zero. The idea delivery is considered reliable if the probability of error that an arbitrarily small probability of error could be can be made arbitrarily small. In the latter, information achieved with some constant rate of transmission there- delivery is considered reliable only if the probability of fore flew in the face of conventional wisdom at the time error can be made to equal zero for a sufficiently large of its introduction. number of channel uses. Shannon’s notion of channel coding initiated a tre- In 1960, Shannon generalised the network further by mendous amount of research and spawned entire sub- considering two-way channels. Two-way channels differ fields within the field of information theory. In particu- from point-to-point channels with feedback in that the lar, a significant amount of fundamental work went on point-to-point channel with feedback has only a single in the 1950s through to the 1980s, when some of the message travelling from the transmitter to the receiver very basic codes and decoding algorithms that we still while the two-way channel has messages travelling from use today were developed. Notable examples include each node to the other. The 1960 paper also mentions a algebraic codes, such as the Reed-Solomon family of channel in which a pair of transmitters sends information channel codes that form the basis of codes used in mod- through a shared medium to a single receiver; that chan- ern storage media, and the Viterbi sequential decod- nel would today be called a “multiple access channel”. ing algorithm, which has found an astonishing array of The 1960 paper mentions future work to appear on this applications, including its use in essentially every mobile topic; while no such paper is found in the literature, it phone in use today. The developments of more recent is clear that Shannon was thinking about generalisations times have been no less impressive. In the 1990s, turbo beyond two-communicator models. codes were discovered, which, together with correspond- Starting in the late 1960s, multiuser channels became an ing iterative decoding ideas, revolutionised the field of important area for information theory research. Research

30 EMS Newsletter March 2017 History on feedback investigated the improved trade-offs between shown that if one could solve all index coding networks rate and error probability achievable through feedback. then that would provide a means of solving all network Research on two-way channels yielded improved upper coding networks as well. That is, any network coding and lower bounds on achievable rate regions. A wide instance can be represented by an index coding instance array of new channel models were developed, including whose solution would give you a solution to the original multiple access channels (in which multiple transmitters network coding problem. send information to a single receiver), broadcast channels In addition to work on network coding capacity, there (in which a single transmitter sends possibly distinct infor- has also been quite a bit of work on network code design, mation to multiple receivers), relay channels (in which a as well as work on the relationship between networks single transmitter sends information to a single receiver of capacitated links and the corresponding networks of with the aid of a relay that can both transmit and receive noisy channels that they are intended to model. Results information but has no messages of its own to transmit) in this domain demonstrate that the capacity of a net- and interference channels (in which the transmissions of work of noisy channels is exactly equal to the capacity multiple transmitters interfere at the multiple receivers of the network coding network achieved by replacing with which they are trying to communicate). each channel by a noiseless, capacitated link of the same Generalisations of Shannon’s channel model are not capacity. Thus, Shannon’s channel capacity fully charac- limited to increasing the number of transmitters or receiv- terises the behaviour of noisy, memoryless channels at ers in the networks. Other generalisations include com- least insofar as they affect the capacity of the networks pound channels, which capture channels with unknown in which they are employed. or varying statistics, wiretap channels, which model chan- Other questions considered in the domain of network nels with eavesdroppers, and arbitrarily varying channels, coding include network error correction, secure network which capture channels under jamming. Joint source- coding, network coding in the presence of eavesdroppers, channel coding has also been a major topic in the mul- network coding techniques for distributed storage and tiuser communication literature. While the optimality of network coding for wireless applications with unreliable separation between source and channel coding holds for packet reception. the point-to-point scenario studied by Shannon, it does not hold in general and a good deal of work has gone into Source coding understanding when such separation is optimal and how Source coding, also called data compression, refers to the to achieve optimal performance when it is not. efficient representation of information. Shannon’s work While interest in multiuser channels waxes and wanes introduces two types of source coding to the literature: over time due to the difficulty of the problems, the mas- lossless source coding, in which the data can be recon- sive size and huge importance of modern communication structed from its description either perfectly or with a networks makes multiuser information theory an impor- probability of error approaching zero, and lossy source tant area for continuing research. coding, in which greater efficiency in data representation is obtained through the allowance of some level of Network coding inaccuracy or “distortion” in the resulting reproduction. The examples given above of multiuser channels are While Shannon’s 1948 paper famously discusses both typically used to model wireless communication environ- source coding and channel coding and is often described ments. But wireless networks are not the only multiuser as the origination point for both ideas, Shannon first communication networks. After all, Shannon’s work was posed the lossy source coding problem in an earlier com- itself originally inspired by communication networks munication. like the wireline phone and telegraph networks of his Shannon’s 1948 paper sets a lot of highly influen- day, each of which connected vast numbers of users over tial precedents for the field of lossless source coding. It massive networks of wires. The modern field of network introduces the now-classical approach to deriving upper coding studies such networks of point-to-point chan- bounds on the rates needed to reliably describe a source nels. Typically, the point-to-point channels in these mod- and gives a strong converse to prove that no better rates els are assumed to be noiseless, capacitated links. The can be achieved. It also includes both the ideas of fixed- field of network coding began with questions about the and variable-length codes, that is, codes that give the capacity of network coding networks. In some scenarios, same description length to all symbols, and codes that notably the case of multicast network coding, the capac- give different description lengths to different symbols. ity is known, and efficient algorithms are available for Arithmetic codes, which remain ubiquitous to this day, achieving those bounds in practice. However, for most have their roots in this paper. The notions of entropy, networks, the network coding capacity remains incom- entropy rate, typical sequences and many others also pletely understood. come from the 1948 paper. Given the difficulty of solving the general network The 1948 paper also looks at lossy source coding, coding problem, a variety of special cases have been con- describing the optimal trade-off between rate and distor- sidered. One of these is the family of index coding net- tion in lossy source description and intuitively explain- works. Unlike general network coding networks, index ing its derivation. In a 1959 paper, Shannon revisits the coding networks are networks in which only one node trade-off between rate and distortion in lossy source in the network has an opportunity to code. It has been coding, giving more details of the proof, coining the term

EMS Newsletter March 2017 31 History

“rate distortion function” to describe that bound and These ideas have motivated quite a bit of work in presenting more examples of solutions of the rate distor- subsequent years and to the present day. For example, tion function for different sources. channel decoding is, in essence, hypothesis testing with Since then, there has been a lot of work in both loss- large numbers of hypotheses, and some famous results less and lossy source coding. Much of the work in the from this area have been developed within the context of 1950s through the 1970s looked at detailed proofs and sequence detection, including the Viterbi algorithm, not- extensions of the original ideas. Extensions include mod- ed above, and Forney’s maximum likelihood sequence el generalisations to allow sources with memory, non- detector. Related to these developments is multiuser ergodic sources, and so on. Advances were also made in detection, which is also motivated by data detection in practical code design for both lossless and lossy source multiple-access communications, and the closely related coding. Huffman developed his famous source coding problem of data detection in MIMO systems. Distributed algorithm, which is still in use. Tunstall codes looked at detection, which is a problem motivated by wireless sen- coding from variable-length blocks of source symbols to sor networking, is also a successor to these ideas. And, fixed-length descriptions. Arithmetic codes were further returning to the sampling theorem, one of the major developed for speed and performance. Algorithms for trends today in signal processing is compressed sensing, designing fixed and variable-rate vector quantizers were which exploits signal sparsity to go well beyond Nyquist- also introduced. Shannon sampling to capture the essence of a signal with In the years that followed, a lot of work was done on far fewer samples. So, again, although we might not think universal source coding and multi-terminal source cod- of Shannon as being a progenitor of this field, these con- ing. Universal source codes are data compression algo- nections show that his work has had a major influence rithms that achieve the asymptotic limits promised by either directly or as a motivator. Shannon without requiring a priori knowledge of the distribution from which the source samples will be drawn. Machine learning Results on universal source coding include code designs Another topic that is very much in evidence today is that for both lossless and lossy universal source coding and of machine learning and its role in big data applications. analyses of code performance measures such as the rate Shannon was an early actor in the application of machine at which a code’s achievable rate (and, in the case of lossy learning ideas – in 1950, he wrote one of the earliest coding, distortion) approaches the optimal bound. Like chess-playing computer programs and, in 1952, he devel- multiuser channel codes, multi-terminal source codes are oped “Theseus”, the famous maze-solving mouse. Of data compression algorithms for networks with multiple course, we have come a very long way in machine learn- transmitters of information, multiple receivers of infor- ing since those early contributions, driven by ever more mation or both. Examples include the work of Ahlswede powerful computers. For example, many games have and Körner on source coding with coded side informa- been conquered: checkers in the 1950s, chess with Deep tion and the work of Slepian and Wolf on distributed Blue in the 1990s, Jeopardy with Watson in 2011 and go source coding networks, where source coded descriptions with AlphaGo in 2016. And, of course, there have been are sent by independent encoders to a single decoder. many fundamental developments in learning and related In addition to advances in the theory of optimal tasks, such as neural networks and decision trees, and source codes and their performance, there has been also graphical models, which have played a major role in much research and development aimed at building and channel decoding. These developments are behind con- standardising lossless and lossy source codes for a vari- temporary developments such as deep learning and self- ety of communication applications. These algorithms are driving cars. So, again, Shannon was an early pioneer of critical parts of many of the data-rich applications that a field that has turned out to be a very important part of are becoming increasingly ubiquitous in our world. modern technology and science.

Detection and hypothesis testing Complexity and combinatorics Another field in which Shannon’s influence has been felt Quite a bit of Shannon’s work related to and influenced has been that of signal detection and hypothesis testing. the fields of complexity and combinatorics. Shannon’s Although one might not normally think of Shannon in Master’s thesis, perhaps his most famous work next to the this context, he worked directly on signal detection in 1948 paper, drew a relationship between switching cir- some of his very early work in 1944, in which he explored cuits and Boolean algebra. His 1948 paper also introduced the problem of the best detection of pulses, deriving the many tools that continue to be useful to combinatorial optimal maximum a posteriori probability (MAP) proce- applications. Shannon’s 1956 paper on zero-error capacity dure for signal detection; his work was one of the earli- revisits the capacity problem – shifting the approach from est expositions of the so-called “matched filter” principle. a probabilistic perspective with asymptotic guarantees of Also, by revealing the advantages of digital transmission reliability to a combinatoric perspective in which reliabili- in communications, he showed the importance of these ty requires the guaranteed accurate reproduction of every fields to communication theory in general. And further, possible message that can be sent by the transmitter. he expounded the idea that there is an optimal sampling Over time, information theory has been and contin- rate for digitising signals through the famous Nyquist- ues to be used for a variety of applications in the com- Shannon sampling theorem. plexity and combinatorics literature. Results include

32 EMS Newsletter March 2017 History generalisations of tools originally developed by Shan- the broadcast channel with confidential messages, which non in the probabilistic framework to their combinatoric is a model that has driven considerable research since, alternatives. An example is Shannon’s typical set, which particularly in the recent development of wireless physi- captures a small set of sequences of approximately equal cal layer security, which makes use of radio physics to individual probability that together capture a fraction of provide a degree of security in wireless transmission. The the total probability approaching 1. This set generalises 1990s notion of common randomness as a source of dis- to more combinatoric alternatives such as that used in tilling secret keys for use in cipher systems also has its the method of types. Fano’s inequality, entropy space roots in information theory and is another basis for wire- characterisations, and a variety of other tools from infor- less physical layer security. mation theory likewise play a role in the combinatorics So, again, we see another very important field of con- and complexity literature. temporary technology development influenced by Shan- The field of communication complexity also draws non’s work. upon information theory tools. Concentration inequalities are another example of areas that sit at that bound- Applications ary between combinatorics and information theory, While Shannon originally developed information theory bringing in tools from both of these communities to solve as a means of studying the problems of information com- important problems. One can also find many examples in munication and storage, ideas from his work were very the literature involving bounding various counting argu- quickly taken up by other fields. In 1956, Shannon wrote ments using information theory tools. about this phenomenon in an article titled “The Band- wagon,” where he warned of “an element of danger” in Cyber security the widespread adoption of information theory tools and Cyber security is another extremely important aspect of terminology. In that article, he noted his personal belief modern technology that has its roots, at least in terms that “information theory will prove useful in these other of its fundamentals, in Shannon’s work. In particular, he fields” but also argued that “establishing of such applica- established an information theoretic basis for this field in tions is not a trivial matter of translating words to a new his 1949 paper (in turn based on earlier classified work), domain, but rather the slow tedious process of hypoth- in which he addressed the question of when a cipher sys- esis and experimental verification”. tem is perfectly secure in an information theoretic sense. Today, information theory is used in a wide variety In this context, he showed the very fundamental result of fields. Biology and finance are two major examples that cipher systems can only be secure if the key – that is, of fields where people are starting to apply information the secret key that is shared by sender and receiver and theoretic tools: in one case to study how biological sys- used to create an enciphered message – has at least the tems transmit and store information and in the other to same entropy as the source message to be transmitted. model long-term behaviour of markets and strategies Or, in other words, he showed that only one-time pads for maximising performance in such markets. Applica- are perfectly secure in an information theoretic context. tions also exist in fields like linguistics, computer science, Shannon’s work was allegedly motivated by the SIG- mathematics, probability, statistical inference and so on. SALY system, which was used between Churchill and Roosevelt to communicate by radio telephone in World Concluding remarks War II and which made use of one-time pads provided While one can barely skim the surface of Shannon’s through physical transport of recordings of keys from work and legacy in an article such as this, it should be Virginia to London. Most cyber security systems today, of clear that his genius has benefitted modern science and course, do not use one-time pads. In fact, almost none do. engineering, and thereby society, in countless ways. We Rather, they use smaller bits of randomness, expand that hope that this very brief overview will inspire continuing into a key and use computational difficulty to provide interest in Shannon and his work and continuing interac- security. Nevertheless, the fundamental thinking comes tion across the boundaries of the many distinct fields that from Shannon. Public key cryptosystems, of course, were share tools, philosophies, and interests with the field of not invented by Shannon but they are basically part of information theory. the legacy of looking at cyber security, or secret communications, from a fundamental point of view. Selected bibliography Another major advance in information theoretic Shannon, C. E. (1938). A symbolic analysis of relay and switching cir- characterisations of security was Wyner’s introduction cuits. Transactions of the AIEE 57(12): 713–723. of the wire-tap channel in 1975, which gets away from Shannon, C. E. (1944). The best detection of pulses. Bell Laboratories a shared secret and uses the difference in the physical memorandum dated 22 June 1944. (In Claude Elwood Shannon: Collected Papers channels, from the transmitter to a legitimate receiver , N. J. A. Sloane and A. D. Wyner, Eds. Piscataway, NJ: IEEE Press, 1993, pp. 148–150.) and to an eavesdropper, to provide data confidentiality. Shannon, C. E. (1948). A mathematical theory of communication. Bell This setting introduces the notion of secrecy capacity, System Technical Journal 27(3): 379–423. which is defined as the maximum rate at which a message Shannon, C. E. (1949). Communication theory of secrecy systems. Bell can be transmitted reliably to the legitimate receiver System Technical Journal 28(4): 656–715. while being kept perfectly secret from the eavesdropper. Shannon, C. E. (1949). Communications in the presence of noise. Pro- Wyner’s work was extended by Csiszár and Körner to ceedings of the IRE 37(1): 10–21.

EMS Newsletter March 2017 33 Obituary

“Claude Shannon Demonstrates Machine Learning”, AT&T Archives. Poor, H. V., Schaefer, R. (2017). Wireless physical layer security. Proceed- (Online at http://techchannel.att.com/play-video.cfm/2010/3/16/ ings of the National Academy of Sciences of the USA 114(1): 19–26. In-Their-Own-Words-Claude-Shannon-Demonstrates-Machine- Learning.) Shannon, C. E. (1956). The bandwagon. IRE Transactions on Informa- Michelle Effros [[email protected]] is the tion Theory 2(1): 3. George van Osdol Professor of Electrical Shannon, C. E. (1959). Coding theorems for a discrete source with a fidelity criterion. IRE Conv. Rec. 7:142–163. Engineering at the California Institute of Berlekamp, E. (1974). Key Papers in the Development of Coding Theo- Technology. Her research focuses primarily ry. IEEE Press, New York. on information theory, with a particular in- Slepian, D. (1974). Key Papers in the Development of Information The- terest in information theoretic tools for the ory. IEEE Press, New York. analysis and design of very large networks. Sloane, N. J. A., Wyner, A. D., Eds. (1993). Claude Elwood Shannon: Col- She currently serves as Senior Past President of the IEEE lected Papers. IEEE Press, Piscataway, NJ. Information Theory Society. Verdú, S., McLaughlin, S., Eds. (2000). Information Theory: 50 Years of Discovery. IEEE Press, Piscataway, NJ. H. Vincent Poor [[email protected]] is MacKay, D. J. C. (2003). Information Theory, Inference and Learning Al- the Michael Henry Strater University Pro- gorithms. Cambridge University Press, Cambridge, UK. fessor of Electrical Engineering at Princeton Cover, T. M., Thomas, J. A. (2006). Elements of Information Theory – Second Edition. John Wiley & Sons, New York. University. His research interests include in- Richardson, T., Urbanke, R. (2008). Modern Coding Theory. Cambridge formation theory and signal processing and University Press, Cambridge, UK. their applications in various fields. He has El Gamal, A., Kim, Y.-H. (2011). Network Information Theory. Cam- served as President of the IEEE Informa- bridge University Press, Cambridge, UK. tion Theory Society and as Editor-in-Chief of the IEEE Elder, Y. C. (2015). Sampling Theory: Beyond Bandlimited Systems. Transactions on Information Theory. Cambridge University Press, Cambridge, UK.

Visions for Mathematical Learning: The Inspirational Legacy of Seymour Papert (1928–2016)

Celia Hoyles and Richard Noss (UCL Knowledge Lab, University College, London, UK)

Seymour Papert, who died on 31 July, was a mathemati- 1980: Keynote in ICME Berkeley, USA cian with two PhDs in pure mathematics, from the Uni- Seymour gave one of the four plenaries at ICME 1980. versity of Witwatersrand, South Africa, and the Univer- Sadly, as far as we can tell, there was no transcript pro- sity of Cambridge, UK. He was a founder of artificial duced of Seymour’s remarks. We are, however, grateful intelligence with Marvin Minsky at MIT, a psychologist to Jeremy Kilpatrick (who attended the talk) for pointing working alongside Jean Piaget, a political activist against us to a 1980 book edited by Lynn Steen and Don Albers, apartheid and, on a personal level, a wonderful cook and which includes a 4-page synopsis of Seymour’s talk.1 loyal friend. Since his death, the web has been awash with Apparently, Seymour was inspirational. From the reminiscences and detailed accounts of his intellectual abstract, we know that he began: contributions, not only to the fundamental subjects in which he was the undisputed leader but also to the field of “We are at the beginning of what is the decade of math- education, to a scholar who believed and showed that the ematics education. Not just in how children learn, but computer, or at least the very carefully crafted use of the what they learn: we will see dramatic changes in what computer, could introduce young and old alike to the joys children learn; we will see subject matters that for- and power of mathematics and mathematical thinking. merly seemed inaccessible or difficult even at college In this short article, we have selected four pieces of level learned by young children; we will see changes in work that directly impacted on the mathematics educa- where learning takes place, and in the process of learn- tion field and community. Significantly, these are among ing itself.” his less well-known lectures and papers and we hope that, by airing them, the realisation of Papert’s vision of a new 1 https://books.google.cz/books?id=zcq9BwAAQBAJ&pg=P kind of learnable mathematics may be one step closer. A12&lpg=PA12&dq=%22.

34 EMS Newsletter March 2017 Obituary

Seymour Papert and Celia Hoyles. Seymour Papert and Richard Noss.

Even at this early stage, some 30 years before the pres- starts with the intuitions of body movement rather than ence of computers in school became commonplace, Sey- abstract points and lines can be no less rigorous but con- mour was addressing the question of epistemology, the siderably more inviting. ‘what’ of mathematics education – a theme that perme- ated his writings and speeches ever since. 1996: Launching a new journal: the International Journal of Computers for Mathematics Learning 1986: Keynote at the Tenth Conference of the (IJCML) International Group for the Psychology of In 1996, Seymour became the founding editor of a new Mathematics Education (PME 10) in London, UK journal, IJCML. In the first issue, he undertakes “An The title of Seymour’s talk was “Beyond the Cognitive: Exploration in the Space of Mathematics Educations”. the Other Face of Mathematics”.2 This talk was again This brilliant contribution begins memorably: inspirational and maybe a little controversial. He began by stating how he A mathematical metaphor frames the intentions of this paper. Imagine that we know how to construct an “shared with Piaget the heuristic value that trying as N-dimensional space, ME, in which each point repre- hard as one can to understand as much as one can of sents an alternative mathematics education – or ame children’s mathematics and mathematicians’ math- – and each dimension a feature, such as a component ematics in the same categories. Doing so can illuminate of content, a pedagogical method, a theoretical or ideo- both sides” (p. 1). logical position. Each “reform” of mathematics education introduces new points and each fundamental idea How right he was – as so many of us have now experienced a new dimension. Thus, if one considers a particular in our own work. Seymour argued for a greater impor- point (an ame) in ME, among its many “coordinates” tance to be accorded to the affective side of mathemat- are a (metaphorical) measure that runs from informal ics; remember that this keynote was 30 years ago, when to formal and another that runs from instructionist to mathematics education research was firmly grounded constructivist. In the paper I shall define seven more in the cognitive paradigm. In particular, Seymour noted such oppositional principles that have not been recog- how some people tended to identify with mathematical nized in the past as structuring choices in mathematics objects: a precursor of the hugely influential movement education. (Papert, 1996) ‘embodied mathematics’? “Do you observe the mathematical scene in your head or are you in it?” he asks The reader will not miss the daring and imaginative style (p. 2). And then the punchline that we will never forget: of this metaphor. The article focuses on how the medium he showed how the “Euclidean propositions can be seen of expression can make any specific ‘ame’ seem ‘natural’, in a different light as special cases of turtle theorems” again using elementary geometric examples as illustra- (p. 3), thus illustrating beautifully how a geometry that tion. But Seymour argues:

“…there is no doubt that in general much more can 2 http://dailypapert.com/wp-content/uploads/2015/07/Beyon- be done at an elementary level with dynamic than with dTheCognitive.pdf algebraic characterizations of curves”.

EMS Newsletter March 2017 35 Obituary

Recall that this was a decade or so before dynamic geom- Acknowledgment etry became widespread! In 2004, Seymour took up the This article was originally published in the November 2016 theme of the mediation of knowledge more generally issue of the ICMI Newsletter (http://www.mathunion.org/ in a little-known but, for us, highly significant speech to fileadmin/ICMI/files/News/ICMI_Newsletter_Novem- open the London Knowledge Lab, where we both work. ber_2016_LK_final.pdf). We gratefully acknowledge its Take a look at https://mediacentral.ucl.ac.uk/Play/3004. editors Abraham Arcavi (ICMI Secretary General) and Cheryl E. Praeger (ICMI Vice-president) for the reprint 2006: Opening keynote to ICMI 17 Study permission. Conference, Technology Revisited, Vietnam In July 2002, the ICMI Executive Committee launched References the 17th ICMI Study, called “Technology Revisited”, the Hoyles, C., & Lagrange, J.-B. (Eds) (2010) Mathematics Education and title reflecting the fact that the very first ICMI Study, held Technology – Rethinking the Terrain, Springer. in Strasbourg in 1985, focused on the influence of com- Steen, L., & Albers, D. J. (Eds) (1981) Teaching Teachers, Teaching Stu- puters and informatics on mathematics and its teaching. dents – Reflections on Mathematical Education, Springer. The Programme Committee wanted the Study Confer- Papert, S., (1996) An Exploration in the Space of Mathematics Educa- tions, Vol. 1, No. 1, pp. 95–123. ence to be opened by a scholar with vision, experience and stature in the fields of mathematics, mathematics education and technology. We chose Seymour and to our delight he accepted by return of email. The tone of his emails became more and more excited as the confer- Celia Hoyles has been a professor of math- ence approached. In his talk, Seymour spoke to the title ematics education at the Institute of Edu- ‘30 Years of Digital Technologies in Mathematics Educa- cation, University of London, since 1986. tion and the Future’, using the recently prototyped and She was a recipient of the ICMI Hans revolutionary ‘100 dollar laptop’ (renamed the ‘XO’) to Freudenthal Medal in 2004 and the Royal present his talk. He argued that, with full and easy access Society Kavli Education Medal in 2011. She to computers, we face the challenge to consider not only was President of the Institute of Mathemat- how existing knowledge can be addressed in technology- ics and its Applications in 2014–15 and a Member of the enhanced ways but also that we should reserve at least European Mathematical Society (EMS) Committee for 10% of our time and energy to consider what new types of Mathematics Education in 2008–16. mathematical knowledge and practices might emerge as a result. His accident the next day was a most terrible shock Richard Noss is a founding member of the to us and to all the participants, and the conference strug- Logo and Mathematics Education Group gled to continue after this tragedy, even as Seymour strug- inspired by Seymour. He became a profes- gled in hospital. The best tribute we could think of was to sor of mathematics education at the Institute try to keep the spirit of his ambition alive throughout the of Education in 1996. He was the founder meeting by asking for participants to consider ‘Seymour’s and director of the London Knowledge Lab 10%’ in all their sessions and their subsequent papers. – exploring the future of learning with digi- (Adapted from Hoyles, C., & Lagrange, J. B., 2010) tal technologies – for its first decade. Richard co-authored We hope that this short piece will keep Seymour’s (with Celia Hoyles) ‘Windows on Mathematical Mean- vision and struggle alive. ings: Learning Cultures and Computers’ in 1996.

36 EMS Newsletter March 2017 Discussion Do We Create Mathematics or Do We Gradually Discover Theories Which Exist Somewhere Independently of Us?

Vladimir L. Popov (Steklov Mathematical Institute, Moscow, Russia), Editor of EMS Newsletter

The headline question is a phrase from the article problem of its relationship with the natural and applied “Mathematics: Art and Science” by A. Borel, which is the sciences. The severity of the statements on these issues English translation of the text of his lecture delivered (in does not abate with time: to the citations in the reprinted German) in Munich in 1981. I remembered about this paper by A. Borel, one could add the well known view- article in October 2016 when I saw the title “Discoveries, point of V. Arnold (and see also the recent interview with not inventions – Interview with Ernest B. Vinberg” of an the Fields Medallist S. Novikov [3]). These are the items article for publication in the EMS Newsletter, No. 102, for discussion that might engage readers of the EMS December 2016 [2]. The very title clearly implies a defi- Newsletter. Note that the reprinted article by A. Borel nite answer to this question but it turns out that, in the is not his only public statement on the subject (see [1]). interview, the issue of whether mathematicians explore something that exists independently of them or some- References thing they have invented is not discussed. However, as [1] A. Borel, On the place of mathematics in culture (Faculty lecture is shown in the article by A. Borel, this issue is, in fact, given at the Institute for Advanced Study, Oct. 16, 1991), Œuvres, deeper than it may seem and contains items for discus- Vol. IV, 1983–1999, Springer, 419–440. sion that might engage readers of the EMS Newsletter. [2] A. Fialowski, J. Hilgert, B. Ørsted, V. Salnikov, Discoveries, not in- Because of this, the mentioned article by A. Borel is ventions – Interview with Ernest B. Vinberg, EMS Newsletter 102 (2016), 31–34. reprinted below. There is also another reason to do this: [3] E. Kudryavtseva, The breakup of the mandatory knowledge had in addition to a discussion of the formulated question, in happen – Interview with the Fields medalist Sergei Novikov, this article, A. Borel also discusses other principal issues 19.12.2016, http://www.kommersant.ru/doc/3169063 (in Russian). of a general nature related to mathematics, such as the

Mathematics: Art and Science

A. Borel

Published with the permission of Springer Science and schaft, Themenreihe XXXIII). We are grateful for their Business Media: extracts from pp. 9–17, The Mathematical permission to publish this English language version. Intelligencer, Vol. 5, No. 4, 1983, doi:10.1007/BF03026504. Ladies and Gentlemen, The Mathematical Intelligencer Editor’s note: Apart from some minor changes, the following article is a translation It is a great honour to be invited to address you here of the text of a lecture delivered, in German, at the Carl but one which is fraught with difficulties. First, there is Friedrich von Siemens Stiftung, Munich, on May 7, 1981, a rather natural reluctance for a practicing mathemati- and, in a slightly modified form, as the first of three “Pauli- cian to philosophise about mathematics instead of just Vorlesungen”, on February 1, 1982, at the Federal School giving a mathematical talk. As an illustration, the Eng- of Technology, Zurich. lish mathematician G. Hardy called it a “melancholy The Intelligencer requested permission from the experience”to write about mathematics rather than just author to publish a translation of the text (translated by prove theorems! However, had I not surmounted that Kevin M. Lenzen). We supplied the translation which the feeling, I wouldn’t be here, so I need not dwell on it any author checked and modified. We wish to thank him for more. More serious difficulties arise from the fact that his considerable help in improving the original translation. there are mathematicians and non-mathematicians in The German text of this lecture was published by the the audience. Whether one should conclude from this C. F. v. Siemens Stiftung (Mathematik: Kunst und Wissen- that my talk is best suited for an empty audience is a

EMS Newsletter March 2017 37 Discussion question that every one of you will have answered with- eral formulae. One can call this economy of thought or in the next hour and therefore needs no further elabo- laziness. An age-old example is the solution to a second- ration. The difficulty brought about by the presence of degree equation, say mathematicians here is that it makes me aware (almost painfully aware) that, in fact, everything about my top- x2 + 2 bx + c = 0. ic has already been said, all arguments have already been presented and pros and cons argued: mathematics Here, b and c are given real numbers. We are looking for is only an art, or only a science, the queen of sciences, a real number x that will satisfy this equation. For centu- merely a servant of science or even art and science com- ries, it has been known that x can be expressed in terms bined. The very subject of my address, in Latin Mathesis of b and c by the formula et Ars et Scientia Dicenda, appeared as the third topic in the defence of a dissertation in the year 1845. The x = –b ± b2 – c. opponent claimed it was only art but not science [1]. It has occasionally been maintained that mathematics is If b2 > c, we can take the square root and get two solu- rather trivial, almost tautological, and as such certainly tions. If b2 = c then x = –b is said to be a double solution. If unworthy of being regarded either as art or as science b2 < c, however, then we cannot take the square root and [2]. Most arguments can be supported by many refer- we maintain, at least at the beginning secondary school ences to outstanding mathematicians. It is even possible level, that there is no solution. sometimes, by selective citation, to attribute widely dif- In the 16th century, similar formulas were devised ferent opinions to one and the same mathematician. So for third- and even fourth-degree equations, such as the I would like to emphasise at the outset that the profes- equation sional mathematicians assembled here are unlikely to hear anything new. x3 + ax + b = 0. If I turn to the non-mathematicians, however, I encounter a much bigger, almost opposite problem: my I won’t write the formula out but it contains square task is to say something about the essence, the nature, of roots and cube roots – so-called radicals. An extremely mathematics. In so doing, however, I cannot assume that interesting phenomenon was discovered that came to be the object of my statements is common knowledge. Of called the casus irreducibilis. If this equation has three course, I can presuppose a certain familiarity with Greek distinct real solutions and we apply the formula, which mathematics, Euclidean geometry, for example, perhaps in principle allows one to compute them, then we meet the theory of conic sections, or even the rudiments of square roots of negative numbers; at the outset, these are algebra or analytical geometry. But they have little to do meaningless. If we ignore the fact that they don’t exist, with the object of present-day mathematical research. however, and are not afraid to compute with them then Starting from this more or less familiar ground, math- they cancel out and we get the solutions, provided we ematicians have gone on to develop ever more abstract carefully follow certain formal rules. In short, starting theories, which have less and less to do with everyday from the given real numbers a, b, we arrive at the sought experience, even when they later find important appli- for ones by using “nonreal numbers”. The square roots cations in the natural sciences. The transition from one of negative numbers were called “imaginary numbers”to level of abstraction to the next has often been very dif- distinguish them from the real numbers and controver- ficult even for the best mathematicians and it represent- sies raged as to whether it was actually legitimate to use ed, in their time, an extremely bold step. I couldn’t pos- such nonreal numbers. Descartes, for example, did not sibly give a satisfactory survey of this accumulation of want to have anything to do with them. Only around the abstractions upon abstractions and of their applications year 1800 was a satisfactory solution – satisfactory for in just a few minutes. Still, I would feel quite uncomfort- some at least – to this problem found. The real numbers able simply to philosophise about mathematics without are embedded in a bigger system consisting of the points saying anything specific on its contents. I would also like of the plane, i.e. pairs of real numbers, between which to have a small supply of examples at hand to be able one defines certain operations that have the same for- to illustrate general statements about mathematics or mal properties as the four basic operations in arithmetic. the position of mathematics with respect to art and the The real numbers are identified with the points on the natural sciences. I shall therefore attempt to describe, or horizontal axis and the square roots of negative numbers at least to give an idea of, some such steps. with those on the vertical axis. One then began to speak In doing so, I will not be able to define precisely all of complex (or imaginary) numbers. Formally, we can use my terms and I don’t expect full understanding by all. these mathematical objects almost as easily as the real But that is not essential. What I want to communicate numbers and can obtain solutions that are sometimes is really just a feeling for the nature of these transitions, real, sometimes complex. For the second-degree equa- perhaps even for their boldness and significance in the tion mentioned earlier, we can now say that there are two history of thought. And I promise not to spend any more complex solutions if b2 < c. than 20 minutes doing so. To a certain extent, this is, of course, merely a conven- A mathematician often aims for general solutions. He tion but it wasn’t easy to grant these complex numbers the enjoys solving many special problems with a few gen- same right to existence as real numbers and not to regard

38 EMS Newsletter March 2017 Discussion them as a mere tool for arriving at real numbers. There tician and a physicist were discussing the curriculum for was no strict definition of real numbers back then but physics at Princeton University around the year 1910, the the close connection between mathematics and measure- physicist said they could no doubt leave out group the- ment or practical computation gave real numbers a cer- ory, for it would never be applicable to physics [5]. Not tain reality, in spite of the difficulties with irrational and 20 years later, three books on group theory and quantum negative numbers. It wasn’t the same with complex num- mechanics appeared and, since then, groups have been bers, however. That was a step in an entirely new direc- fundamental in physics as well. tion, bringing a purely intellectual creation to the fore. The following will serve as a final example. I said ear- As mathematicians became used to this new step, they lier that we can consider complex numbers to be points began to realise that many operations performed with in the plane. An Irish mathematician, N. R. Hamilton, functions, such as polynomials, trigonometric functions, wondered whether one could define an analogue of the etc., still made sense when complex values were accepted four basic operations among the points of three-dimen- as arguments and as values. This marked the beginning of sional space, thus forming an even more comprehensive complex analysis or function theory. As early as 1811, the number system. It took him about 10 years to find the mathematician Gauss pointed out the necessity of devis- answer: it is not possible in three-dimensional space but ing such a theory for its own sake: it is in four-dimensional space. We do not need to try to imagine just what four-dimensional space is here. It The point here is not practical utility; rather, for me, is simply a figure of speech for quadruples of real num- analysis is an independent science which would lose bers instead of triples or pairs of real numbers. He called an extraordinary amount of beauty and roundness by these new numbers quaternions. He did, however, have discriminating against those fictitious quantities [3]. to do without one property of real or complex numbers, which, up until then, had been taken for granted: Apparently, even he did not foresee the practical rel- commutativity in multiplication, i.e. a × b = b × a. He also evance complex analysis was later to achieve, as in the showed that calculus with quaternions had applications theories of electricity or aerodynamics, for example. in the mathematical treatment of questions in physics But that is not the end of it. Allow me, if you will, and mechanics. Later, many other algebraic systems with to mention two further steps toward greater abstraction. a noncommutative product were defined, notably matrix Let us return to our second-degree equation. One can algebras. This also appeared to be an entirely abstract now say that it has, in general, two solutions that may form of mathematics, without connections to the outside be complex numbers. Similarly, an equation of the n-th world. In 1925, however, as Max Born was thinking about degree has n solutions if one accepts complex numbers. some new ideas of W. Heisenberg’s, he discovered that From the 16th century on, people wondered whether the most appropriate formalism for expressing them was there was also a general formula that would express the none other than matrix algebra, and this suggested that solutions of an equation of degree at least five from the physical quantities be represented by means of algebraic coefficients by means of radicals. It was finally proved objects that do not necessarily commute. This led to the to be impossible. One proof (chronologically the third) uncertainty principle and was the beginning of matrix was given by the French mathematician E. Galois within quantum mechanics and of the assignment of operators the framework of a more general theory, which was not to physical quantities, which is at the basis of quantum understood at the time and subsequently forgotten. Some mechanics [6]. 15 years later, his work was rediscovered and understood With this last example, I shall conclude my attempts only with great difficulty by a very few, so new was his to describe some mathematical topics. The examples are, viewpoint. Given an equation, Galois considered a cer- of course, extremely incomplete and not at all represent- tain set of permutations of the roots and showed that ative of all areas of mathematics. They do have two prop- certain properties of this set of permutations are deci- erties in common, however, which I would like to empha- sive. That was the beginning of an independent study of sise since they are valid in a great many cases. First of all, such sets of permutations, which later came to be known these developments lead in the direction of ever greater as Galois groups. He showed that an equation is solv- abstraction, further and further away from nature. Sec- able by means of radicals only when the groups involved ond, abstract theories developed for their own sake have belong to a certain class: namely, the solvable groups, as found important applications in the natural sciences. The they came to be called. The theorem mentioned earlier, suitability of mathematics to the needs of the natural sci- regarding equations of degree at least five, is then a con- ences is, in fact, astonishingly great (one physicist spoke sequence of the fact that the group associated to a gen- once of the “unreasonable effectiveness of mathemat- eral equation of the n-th degree is solvable only when n ics”[7]) and is worthy of a far more detailed discussion = 1, 2, 3, 4 [4]. The important properties of such groups, than I can afford to enter into here. for instance to be solvable, are actually independent of The transition to ever greater abstraction is not to be the nature of the objects to be permuted and this led to taken for granted, as you may have gathered from Gauss’ the idea of an “abstract group”and to theorems of great quotation. Mathematics was originally developed for significance, applicable in many areas of mathematics. practical purposes such as bookkeeping, measurements But, for many years, this appeared to be nothing more and mechanics; even the great discoveries of the 17th than pure and very abstract mathematics. As a mathema- century, such as infinitesimal and integral calculus, were,

EMS Newsletter March 2017 39 Discussion at first, primarily tools for solving problems in mechan- about in the dark, not knowing whether he should attempt ics, astronomy and physics. The mathematician Euler, to prove or disprove a certain proposition, and essential who was active in all areas of mathematics and its appli- ideas often occur to him quite unexpectedly, without him cations – including shipbuilding – also wrote papers on even being able to see a clear and logical path leading to pure number theory and, more than once, felt the need them from earlier considerations. Just as with composers to explain that it was as justified and important as more and artists, one should speak of inspiration [11]. practically oriented work [8]. Mathematics was, from the Other mathematicians, however, are opposed to this very beginning, of course, a kind of idealisation but, for a view and maintain that an involvement with mathemat- long time, was not as far removed from reality or, more ics without being guided by the needs of the natural sci- precisely, from our perception of reality as in the exam- ences is dangerous and almost certainly leads to theo- ples mentioned earlier. As mathematicians went further ries that may be quite subtle and may provide the mind in this direction, they became increasingly aware that a with a peculiar pleasure but which represent a kind of mathematical concept has a right to existence as soon as intellectual mirror that is completely worthless from the it has been defined in a logically consistent manner, with- standpoint of science or knowledge. For example, the out necessarily having a connection with the physical mathematician J. von Neumann wrote in 1947: world, and that they had the right to study it even when there seemed to be no practical applications at hand. In As a mathematical discipline travels far from its empir- short, this led more and more to “Pure Mathematics”or ical sources, or still more, if it is second and third gen- “Mathematics for Its Own Sake”. eration only indirectly inspired by ideas coming from But if one leaves out the controlling function of prac- “reality”, it is beset with very grave dangers. It becomes tical applicability, the question immediately arises as to more and more purely aestheticizing, more and more how one can make value judgments. Surely not all con- purely l’art pour l’art … there is a great danger that cepts and theorems are equal; as in George Orwell’s Ani- the subject will develop along the line of least resist- mal Farm, some must be more so than others. Are there ance … will separate into a multitude of insignificant then internal criteria that can lead to a more or less objec- branches… tive hierarchy? You will notice that the same basic ques- In any event … the only remedy seems to me to be the tion can be asked about painting, music or art in general. rejuvenating return to the source: the reinjection of It thus becomes a question of aesthetics. Indeed, a usual more or less directly empirical ideas [12]. answer is that mathematics is, to a great extent, an art, an art whose development has been derived from, guided Still others have taken a more intermediate stance: they by and judged according to aesthetic criteria. For the lay- fully recognise the importance of the aesthetic side of person, it is often surprising to learn that one can speak mathematics but feel that it is dangerous to push math- of aesthetic criteria in so grim a discipline as mathemat- ematics for its own sake too far. Poincaré, for example, ics. But this feeling is very strong for the mathematician, wrote: even though it is difficult to explain. What are the rules of this aesthetic? Wherein lies the beauty of a theorem, In addition to this, it provides its disciples with pleas- of a theory? Of course, there is no single answer that will ures similar to painting and music. They admire the satisfy all mathematicians but there is a surprising degree delicate harmony of the numbers and the forms; they of agreement, to a far greater extent, I think, than exists marvel when a new discovery opens up to them an in music or painting. unexpected vista; and does the joy that they feel not Without wishing to maintain that I can explain this have an aesthetic character even if the senses are not fully, I would like to attempt to say a bit more about it lat- involved at all? … er. At the moment, I shall content myself with the asser- For this reason, I do not hesitate to say that mathemat- tion that the analogy with art is one with which many ics deserves to be cultivated for its own sake, and I mathematicians agree. For example, G. H. Hardy was of mean the theories which cannot be applied to physics the opinion that if mathematics has any right to exist at just as much as the others [13]. all then it is only as art [9]. Our activity has much in common with that of an artist: a painter combines colours But a few pages further on, he returns to this comparison and forms, a musician tones, a poet words, and we com- and adds: bine ideas of a certain sort. The painter E. Degas wrote sonnets from time to time. Once, in a conversation with If I may be allowed to continue my comparison with the poet S. Mallarmé, he complained that he found writ- the fine arts then the pure mathematician who would ing difficult even though he had many ideas, indeed an forget the existence of the outside world could be lik- overabundance of ideas. Mallarmé answered that poems ened to the painter who knew how to combine colours were made of words, not ideas [10]. We, on the other and forms harmoniously but who lacked models. His hand, work primarily with ideas. creative power would soon be exhausted [14]. This feeling of art becomes even stronger when one thinks of how a researcher works and progresses. One This denial of the possibility of abstract painting strikes should not imagine that the mathematician operates me as especially noteworthy since we are in Munich, entirely logically and systematically. He often gropes where, not much later, an artist would concern himself

40 EMS Newsletter March 2017 Interview quite deeply with this question (namely, Wassily Kandin- application outside mathematics but which one cannot sky). It was sometime in the first decade of this century help viewing as great achievements. I am thinking, for that he suddenly felt, after looking at one of his own can- example, of the theory of algebraic numbers, class field vases, that the subject can be detrimental to the painting theory, automorphic functions, transfinite numbers, etc. in that it may be an obstacle to direct access to forms and Let us return to the comparison with painting once colours: that is, to the actual artistic qualities of the work again and take as “subject”the problems that are drawn itself. But, as he wrote later [15], “a frightening gap”(eine from the physical world. Then, we see that we have erschreckende Tiefe) and a mass of questions confronted painting drawn from nature as well as pure or abstract him, the most important of which was: “What should painting. replace the missing subject?”Kandinsky was fully aware This comparison is, however, not yet entirely satis- of the danger of ornamentation, of a purely decorative factory, for such a description of mathematics would art, and wanted to avoid it at all costs. Contrary to Poin- not encompass all its essential aspects, in particular its caré, however, he did not conclude that painting without coherence and unity. Indeed, mathematics displays a a real subject had to be fruitless. In fact, he even devel- coherence that I feel is much greater than in art. As a oped a theory of the “inner necessity”and “intellectual testimony to this, note that the same theorem is often content”of a painting. Since about 1910, as you know, he proved independently by mathematicians living in and other painters in increasing numbers have dedicated widely separated locations or that a considerable num- themselves to so-called abstract or pure painting, which ber of papers have two, sometimes more, authors. It can has little or nothing to do with nature. also happen that parts of mathematics that have been If one does not want to admit an analogous possibility developed completely independently of one another for mathematics, however, then one will be led to a con- suddenly demonstrate deep connections under the ception of mathematics that I would like to summarise impact of new insights. Mathematics is, to a great extent, as follows. On the one hand, it is a science because its a collective undertaking. Simplifications and unifica- main goal is to serve the natural sciences and technology. tions maintain the balance with unending development This goal is actually at the origin of mathematics and is and expansion; they display again and again a remark- constantly a wellspring of problems. On the other hand, able unity even though mathematics is far too large to it is an art because it is primarily a creation of the mind be mastered by a single individual. and progress is achieved by intellectual means, many of I think it would be difficult to account fully for this which issue from the depths of the human mind and for by appealing solely to the criteria mentioned earlier: which aesthetic criteria are the final arbiters. But this namely, subjective ones like intellectual elegance and intellectual freedom to move in a world of pure thought beauty, and consideration of the needs of natural sci- must be governed, to some extent, by possible applica- ences and technology. One is then led to ask whether tions in the natural sciences. there are criteria or guidelines other than those. In my However, this view is really too narrow; in particu- opinion, this is the case and I would now like to com- lar, the final clause is too limiting and many mathema- plete the earlier description of mathematics by looking ticians have insisted on complete freedom of activity. at it from a third standpoint and adding another essen- First of all, as has already been pointed out, many areas tial element to it. In preparation for this, I would like of mathematics that have proved important for applica- to digress, or at least apparently digress, and take up tions would not have been developed at all if one had the question: ‘Does mathematics have an existence of insisted on applicability from the beginning. In spite of its own? Do we create mathematics or do we gradu- the above quotation, von Neumann himself pointed this ally discover theories which exist somewhere indepen- out in a later lecture: dently of us?’ If this is so, where is this mathematical reality located? But still a large part of mathematics which became use- It is, of course, not absolutely clear that such a ques- ful developed with absolutely no desire to be useful, tion is really meaningful. But this feeling – that math- and in a situation where nobody could possibly know ematics somehow, somewhere, pre-exists – is widespread. in what area it would become useful: and there were no It was expressed quite sharply, for example, by G. H. general indications that it even would be so … This is Hardy: true of all science. Successes were largely due to forget- ting completely about what one ultimately wanted, or I believe that mathematical reality lies outside us, that whether one wanted anything ultimately, in refusing to our function is to discover or observe it, and that the investigate things which profit, and in relying solely on theorems which we prove, and which we describe gran- guidance by criteria of intellectual elegance… diloquently as our “creations”, are simply our notes of And I think it extremely instructive to watch the role of our observations. This view has been held, in one form science in everyday life, and to note how in this area the or another, by many philosophers of high reputation, principle of laissez faire has led to strange and won- from Plato onwards… [17].} derful results [16]. If one is a believer then one will see this pre-existent Secondly, and for me more importantly, there are are- mathematical reality in God. This was actually the belief as of pure mathematics which have found little or no of Hermite, who once said:

EMS Newsletter March 2017 41 Discussion

There exists, if I am not mistaken, an entire world Now back to mathematics. Mathematicians share an intel- which is the totality of mathematical truths, to which lectual reality: a gigantic number of mathematical ideas, we have access only with our mind, just as a world of objects whose properties are partly known and partly physical reality exists, the one like the other independ- unknown, theories, theorems, solved and unsolved prob- ent of ourselves, both of divine creation [18]. lems, which they study with mental tools. These problems and ideas are partially suggested by the physical world; It wasn’t too long ago that a colleague explained in an primarily, however, they arise from purely mathematical introductory lecture that the following question had considerations (such as groups or quaternions to go back occupied him for years: ‘Why has God created the excep- to my earlier examples). This totality, although stemming tional series?’ from the human mind, appears to us to be a natural sci- But a reference to divine origin would hardly satisfy ence in the normal sense, such as physics or biology, and the nonbeliever. Many do, however, have a vague feeling is for us just as concrete. I would actually maintain that that mathematics exists somewhere, even though, when mathematics not only has a theoretical side but also an they think about it, they cannot escape the conclusion experimental one. The former is clear: we strive for gen- that mathematics is exclusively a human creation. eral theorems, principles, proofs and methods. That is Such questions can be asked of many other concepts the theory. But, in the beginning, one often has no idea such as state, moral values, religion, etc., and would prob- of what to expect and how to continue, and one gains ably be worthy of consideration all by themselves. But understanding and intuition through experimentation, for want of time and competence, I shall have to content that is, through the study of special cases. First, one hopes myself with a short and possibly oversimplified answer to be led in this way to a sensible conjecture and, second, to this apparent dilemma by agreeing with the thesis that perhaps to stumble upon an idea that will lead to a gen- we tend to posit existence on all those things that belong eral proof. It can also happen, of course, that certain spe- to a civilization or culture in that we share them with oth- cial cases are of great interest in themselves. That is the er people and can exchange thoughts about them. Some- experimental side. The fact that we operate with intellec- thing becomes objective (as opposed to “subjective”) as tual objects more than with real objects and laboratory soon as we are convinced that it exists in the minds of equipment is actually not important. The feeling that others in the same form as it does in ours and that we can mathematics is, in this sense, an experimental science is think about it and discuss it together [19]. Because the also not new. language of mathematics is so precise, it is ideally suited Hermite, for example, wrote to L. Königsberger to defining concepts for which such a consensus exists. In around 1880: my opinion, that is sufficient to provide us with a {\it feeling} of an objective existence, of a reality of mathematics The feeling expressed at that point in your letter where similar to that mentioned by Hardy and Hermite above, you say to me: “The more I think about all these things, regardless of whether it has another origin, as Hardy and the more I come to realise that mathematics is an Hermite maintain. One could speculate forever on this experimental science like all other sciences.”This feel- last point, of course, but that is actually irrelevant to the ing, I say, is also my feeling [22]. continuation of this discussion. Before I elaborate on this, I would like to note that Traditionally, these experiments are carried out in one’s similar thoughts about our conception of physical reality head (or with pen and paper) and for this reason I have have been expressed. For example, Poincaré wrote: spoken of mental tools. I should add, however, that for about 20 years, real apparatuses, namely, electronic com- Our guarantee of the objectivity of the world in which puters, have been playing an increasing role. They have we live is the fact that we share this world with other actually given this experimental side of mathematics a sentient beings… new dimension. This has advanced to the extent that one That is therefore the first requirement of objectivity: can already see important, reciprocal and fascinating that which is objective must be common to more than interactions between computer science and pure math- one spirit and as a result be transmittable from one to ematics. the other… [20] The word “science”in my title now takes a broader meaning: it refers not only to the natural sciences, as it And Einstein: did earlier, but also – and this to a much greater extent – to the conception of mathematics itself as an experi- By the aid of speech, different individuals can, to a mental and theoretical science or, I would venture to say, certain extent, compare their experiences. In this way, as a mental natural science, as a natural science of the it is shown that certain sense perceptions of different intellect, whose objects and modes of investigations are individuals correspond to each other, while for other all creations of the mind. sense perceptions no such correspondence can be This makes it somewhat easier for me to speak of established. We are accustomed to regard as real those motivation and aesthetics. If one does not want to take sense perceptions which are common to different indi- applications in the natural sciences as a yardstick, one viduals, and which therefore are, in a measure, imper- is still not thrown back upon mere intellectual elegance. sonal [21]. There still remain almost practical criteria: namely, appli-

42 EMS Newsletter March 2017 Discussion cability in mathematics itself. The consideration of this one has still de jure the freedom to put aside apparently mathematical reality, the open problems, the structure, unsolvable, overly difficult problems and turn to other, needs and connections among various areas, already more manageable ones and maybe, in fact, follow the indicates possibly fruitful, valuable directions and allows path of least resistance, just as von Neumann had feared. the mathematician to orient himself and attach relative Wouldn’t that be a temptation for a mathematician who values to problems as well as to theories. Often a test for defines mathematics as “the art of finding problems that the value of a new theory is whether it can solve old prob- one can solve”? Interestingly enough, I heard this defini- lems. De facto, this limits the freedom of a mathematician, tion from a mathematician whose works are especially in a way which is comparable to the constraints imposed remarkable because they treated so many problems on a physicist, who after all doesn’t choose at random the which seemed quite special at the time but which later phenomena for which he wants to construct a theory or proved fundamental and whose solutions opened up new to devise experiments. Many examples show that math- paths, namely, Heinz Hopf. ematicians have often been able to foresee how certain It cannot be denied, however, that sometimes paths areas of mathematics will develop and which problems of least resistance are indeed followed, leading to trivial should be taken up and probably quickly solved. Rather or meaningless work. It can also happen that a successful often, statements about the future of mathematics have school later falls into a sterile period and then even, at proved true. Such predictions are not perfect but they are worst, exerts a harmful influence. Remarkably enough, successful enough to indicate a difference from art. Anal- however, an antidote always comes along, a reaction that ogous relatively successful forecasts about the future of eliminates these mistaken paths and fruitless directions. painting, for example, hardly exist at all. Up until now, mathematics has always been able to over- I don’t want to go too far in this. However, I suggested come such growth diseases and I am convinced that it the concept of mathematics as a mental natural science as will always do so, as long as there are so many talented one of three elements, not as the whole. On the one hand, mathematicians. It is very odd, however. Many of us have I don’t want to overlook the importance of the interac- this feeling of a unity in mathematics but it is dangerous tions between mathematics and the natural sciences. to prescribe overly precise guidelines in the name of our First, it is a common saying that all disciplines in the nat- conception of it. It is more important that freedom reigns, ural sciences must strive for a mathematical formulation despite occasional misuse. Why this is so successful can- and treatment – indeed, that a discipline achieves the sta- not be fully explained. If one thinks of Hopf, for exam- tus of a science only when this has been carried out. Thus, ple, one can, to a certain extent, see rational criteria in it is surely important that mathematicians try to help in his choice of problems: they were, for instance, often the this way. Second, it is doubtless a great achievement to first special cases of a general problem for which known formulate and treat complicated phenomena mathemati- methods of proof were not applicable. He was, of course, cally, and the new problems that are thereby introduced aware of this. But that doesn’t explain everything. He represent an enrichment for mathematics. One need only probably didn’t always foresee how influential his work think of probability. I only mean that it is simply not nec- would become; and, most likely, he did not worry about essary to put the idea of applicability in the foreground it. It is simply a part of the talent of a mathematician to in order to do valuable mathematics. The history of be drawn to “good” problems, i.e. to problems that turn mathematics shows that many outstanding achievements out to be significant later, even if it is not obvious at the came from mathematicians who weren’t thinking at all time he takes them up. The mathematician is led to this about external applications and who were led by purely partly by rational, scientific observations and partly by mathematical considerations. And as has already been sheer curiosity, instinct, intuition or purely aesthetic con- mentioned and illustrated, these contributions often siderations. Which brings me to my final subject: the aes- found important applications in the natural sciences or thetic feeling in mathematics. in engineering, often in completely unforeseen ways. I have already mentioned the idea of mathematics On the other hand, I don’t want to say that one can as an art, a poetry of ideas. With that as a starting point, foresee everything completely rationally. Actually, this one would conclude that, in order for one to appreciate isn’t the case even in the natural sciences, especially since mathematics, to enjoy it, one needs a unique feeling for one often does not know in advance which experiments intellectual elegance and beauty of ideas in a very spe- will prove interesting. Outstanding mathematicians have cial world of thought. It is not surprising that this can also been wrong and have sometimes, precisely in the hardly be shared with non-mathematicians: our poems name of applicability within mathematics, termed fruit- are written in a highly specialised language, the math- less, idle or even dangerous, new ideas that later proved ematical language; although it is expressed in many of fundamental. The freedom not to consider practical the more familiar languages, it is nevertheless unique and applications, which von Neumann demanded for science translatable into no other language; and unfortunately, as a whole, must also be demanded within mathematics. these poems can only be understood in the original. The One could object that this analogy between mathe- resemblance to an art is clear. One must also have a cer- matics and the natural sciences overlooks one essential tain education for the appreciation of music or painting, difference: in the natural sciences or in technology, one which is to say one must learn a certain language. often encounters problems that one has to solve in order I have long agreed with such opinions and analogies. to advance at all. In the world of mathematical thought, Without changing my fundamental position with regard

EMS Newsletter March 2017 43 Discussion to mathematics, I would nonetheless like to reformulate ments about mathematics. But unfortunately, or for- them somewhat in the direction of my previous state- tunately, just as in other human undertakings to which ments. I believe that our aesthetics are not always so pure many people have contributed over many centuries, and esoteric but also include a few more earthly yard- mathematics refuses to let itself be described by just a sticks such as meaning, consequences, applicability, use- few simple formulas. Almost every general statement fulness – but within the mathematical science. Our judg- about mathematics has to be qualified somehow. One ment of a theorem, a theory or a proof is also influenced exception, perhaps the only one, might be this statement by this but it is often simply equated to the aesthetic. itself. I hope I have, at least, given the impression that I would like to try to explain this using Galois’ theory mathematics is an extremely complex creation, which mentioned earlier. This theory is generally treasured as displays so many essential traits in common with art and one of the most beautiful chapters in mathematics. Why? experimental and theoretical sciences that it has to be First, it solved a very old and, at that time, most impor- regarded as all three at the same time, and thus must be tant question about equations. Second, it is an extremely differentiated from all three as well. comprehensive theory that goes far beyond the origi- I am aware that I have raised more questions than I nal question of solvability by radicals. Third, it is based have answered, treated too briefly those I have discussed on only a few principles of great elegance and simplic- and not even touched upon some important ones, such ity, which are formulated within a new framework with as the value of this creation. One can, of course, point new concepts that demonstrate the greatest originality. to innumerable applications in the natural sciences and Fourth, these new viewpoints and concepts, especially in engineering, many of which have a great influence on the concept of a group, opened new paths and had a last- our daily life, thereby establishing a social right to existing influence on the whole of mathematics. ence for mathematics. But I must confess that, as a pure You will notice that of these four points only the mathematician, I am more interested in an assessment third is a truly aesthetic judgment, and one about which of mathematics in itself. The contributions of the vari- one can have one’s own opinion only when one under- ous mathematicians meld into an enormous intellectual stands the technical details of the theory. The others construct, which, in my opinion, represents an impressive have a different character. One could make similar testimony to the power of human thinking. The math- statements about theories in any natural science. They ematician Jacobi once wrote that “the only purpose of have a greater objective content, and a mathematician science is to honour the human mind”[23]. I believe that can have his own opinion about them even if he doesn’t this creation does indeed do the human mind great hon- fully grasp the technical details of the theory. For the our. purpose of this discussion, I have separated these four The Institute for Advanced Study elements but normally I would not always do so explic- Princeton, New Jersey 08540 itly, and all four contribute to the impression of beauty. I do think that, in this respect, this example is fairly Notes typical: what we describe as aesthetic is actually often 1 The dissertation was by L. Kronecker, see Werke, 5 Vol., Teubner, a fusion of different views. For example, I would natu- Leipzig, 1895–1930, Vol. 1, p. 73. The opponent was G. Eisenstein. rally find a method of proof more beautiful if it found The source I am aware of because the name and the opinion of new and unexpected applications, although the method the opponent is a footnote by E. Lampe to a lecture by P. du Bois- itself hadn’t changed. It may have become more impor- Reymond, “Was will die Mathematik und was will der Mathema- tant but in and of itself not more beautiful. Since all this tiker?”, published posthumously by E. Lampe in Jahresbericht der takes place within mathematics itself, it will hardly help Deutschen Mathematiker-Vereinigung 19 (1910), 190–198. the non-mathematician penetrate our aesthetic world. I 2 For a discussion of a number of such opinions, see A. Pringsheim, hope, however, that it will help him find more plausible “Ueber den Wert und angeblichen Unwert der Mathematik”, the fact that our so-called aesthetic judgments display a Jahresbericht der Deutschen Mathematiker-Vereinigung 13 (1904), 357–382. greater consensus than in art, a consensus that goes far 3 Letter to F. W. Bessel, 18 November 1811. See G. F. Auwers Verlag, beyond geographical and chronological limitations. In Briefwechsel zwischen Gauss und Bessel, Leipzig 1880, p. 156. any case, I regard this as being a major factor. But once 4 Actually, the beginnings of group theory can already be traced to again, I must avoid taking this too far. It is a question some earlier work, notably by Lagrange, which was, in part, familiar of degree, not an absolute difference. An aesthetic judg- to Galois. The latter’s standpoint was, however, so general and ab- ment on the work of a composer or a painter also draws stract and, in addition, so sketchily described that it was assimilated on external factors such as influence, predecessors and only slowly. For historical information on the theory of equations the position of the work with relation to other works, and the beginnings of group theory, see, for example, N. Bourbaki, even if it is to a lesser extent. On the other hand, there Eléments d’histoire des mathématiques, Hermann ed., Paris, 1969, are differences of opinion and fluctuations in time in the third and fifth articles. evaluation of mathematical works, though not to such 5 F. J. Dyson, “Mathematics in the physical sciences”, Scientific Amer- ican 211, September (1964), 129–146. a strong degree, I would add. All these nuances need a 6 See B. L. van der Waerden’s historical introduction in “Sources in good deal of explanation, which I cannot go into here for Quantum Mechanics”, Classics of Science, Vol. 5, Dover Publica- lack of time. tions, New York, 1967, especially pp. 36–38. Also see Dirac’s re- In the limited amount of time at my disposal, it would, marks on the introduction of non-commutativity in quantum me- of course, be easier to make only sweeping short state- chanics in loc. cit. [7].

44 EMS Newsletter March 2017 Discussion

7 E. P. Wigner, “The unreasonable effectiveness of mathematics in “Finally, just a few days ago, success – but not as a result of my la- the natural sciences”, Communications on Pure and Applied Math- borious search but only by the grace of God I would say. Just as it is ematics 13 (1960), 1–14. when lightning strikes, the puzzle was solved; I myself would not be Among the many aspects of this interaction, the one that appears able to show the threads which connect that which I knew before, most remarkable to me is that the mathematical formalism some- that with which I had made my last attempt, and that by which it

times leads to basic, new and purely physical ideas. One well known succeeded.”See Gauss, Gesammelte Werke, Vol. 10I, pp. 24–25. Here example is the discovery of the positron. In 1928, P. A. M. Dirac set one must also mention H. Poincaré’s description of some of his up quantum mechanic relativistic equations for the movement of fundamental discoveries on automorphic functions. H. Poincaré, the electron. These equations also allowed a solution with the same “L’invention mathematique” in Science et Méthode, E. Flammarion mass as the electron but with the opposite electrical charge. All at- éd., Paris, 1908, Chap. III. tempts to explain these solutions satisfactorily, or to eliminate them 12 J. v. Neumann, “The mathematician” in Robert B. Heywood, The by some suitable modification of the equation, were unsuccessful. Works of the Mind, University of Chicago Press, 1947, pp. 180–187. This led Dirac eventually to conjecture the existence of a particle Collected Works, 6 Vol., Pergamon, New York, 1961, Vol. I, pp. 1–9. with the necessary properties, which was later established by An- 13 H. Poincaré, La Valeur de la Science, E. Flammarion, Paris, 1905, derson. For this, see P. A. M. Dirac, “The development of quantum Chap. 5, p. 139. Actually, this chapter is the printed version of a theory”(J. R. Oppenheimer Memorial Prize acceptance speech), lecture that Poincaré delivered at the First International Congress Gordon and Breach, New York, 1971. of Mathematicians, Zurich, 1897. A newer and even more comprehensive example would be the use 14 Loc. cit. [13], p. 147. of irreducible representations of the special unitary group SU(3) in 15 W. Kandinsky, Rückblick 1901–1913, H. Walden ed., 1913. New three complex variables, which led to the so-called “eightfold way”. printing by W. Klein Verlag, Baden–Baden, 1955. See pp. 20–21. One of the first successes of this theory was quite striking, name- 16 J. v. Neumann, “The role of mathematics in the science and in socie- ly, the discovery of the particle –: nine baryons were assigned, ty”, address to Princeton Graduate Alumni, June 1954. See Collect- through consideration of two of their characteristic quantum num- ed Works, 6 Vol., Pergamon, New York, 1961, Vol. VI, pp. 477–490. bers, to nine points of a very specific mathematical configuration 17 See G. H. Hardy, loc. cit. [9], pp. 123–124. consisting of 10 points in a plane [the 10 weights of an irreducible 18 G. Darboux, “La vie et l’Œuvre de Charles Hermite”, Revue du 10-dimensional representation of SU(3)]; this led M. Gell’man to mois, 10 January 1906, p. 46. conjecture that there should also be a particle corresponding to the 19 See L. White, “The locus of mathematical reality: An anthropologi- tenth point, which would then possess certain well-defined prop- cal footnote”, Philosophy of Science 14 (1947), 189303; also in J. R. erties. Such a particle was observed some two years later. A fur- Newman, The World of Mathematics, 4 Vol., Simon and Schuster, ther development along these lines led to the theory of “quarks”. New York, 1956, Vol. 4, pp. 2348–2364. For the beginnings of this theory, see F. J. Dyson, loc. cit. [5] and 20 H. Poincaré, loc. cit. [13], p. 262. M. Gell’man and Y. Ne’eman, The Eightfold Way, W. A. Benjamin, 21 A. Einstein, Vier Vorlesungen über Relativitätstheorie, held in May New York, 1964. 1921 at Princeton University, Fr. Vieweg und Sohn, Braunschweig, 8 See a number of papers in L. Euler’s Opera Omnia, especially I.2, 1922, p. 1. English translation in: The Meaning of Relativity, Prince- 62–63, 285, 461, 576; I.3, 5.2. I want to thank A. Weil for pointing this ton University Press, Princeton, 1945. out to me. Here is an example (translated from Latin by Weil), loc. 22 See L. Königsberger, “Die Mathematik eine Geistes- oder Natur- cit. pp. 62–63, published in 1747: wissenschaft?”, Jahresbericht der Deutschen Mathematiker-Vereini- “Nor is the author disturbed by the authority of the greatest math- gung 23 (1914), 1–12. ematicians when they sometimes pronounce that number theory is 23 In a letter of 2 July 1830 to A. M. Legendre, see C. G. J. Jacobi, Ge- altogether useless and does not deserve investigation. In the first sammelte Werke, G. Riemer, Berlin, 1881–1891, Vol. 1, pp. 453–455. place, knowledge is always good in itself, even when it seems to Since this statement is sometimes misquoted, we prefer to give here be far removed from common use. Secondly, all the aspects of the its original context: truth which are accessible to our mind are so closely related to one “Mais M. Poisson n’aurait pas dû reproduire dans son rapport une another that we dare not reject any of them as being altogether phrase peu adroite de feu M. Fourier, où ce dernier nous fait des useless. Moreover, even if the proof of some proposition does not reproches, à Abel et à moi, de ne pas nous être occupés de préfé- appear to have any present use, it usually turns out that the method rence du mouvement de la chaleur. II est vrai que M. Fourier avait by which this problem has been solved opens the way to the discov- l’opinion que le but principal des mathématiques était l’utilité pub- ery of more useful results. lique et l’explication des phénomènes naturels; mais un philosophe “Consequently, the present author considers that he has by no comme lui aurait dû savoir que le but unique de la science, c’est means wasted his time and effort in attempting to prove various l’honneur de l’esprit humain et que sous ce titre une question de theorems concerning integers and their divisors. Actually, far from nombres vaut autant qu’une question du système du monde.” being useless, this theory is of no little use even in analysis. Moreo- ver, there is little doubt that the method used here by the author will turn out to be of no small value in other investigations of greater import.” 9 G. H. Hardy, A Mathematician’s Apology, Cambridge University Press, 1940; new printing with a foreword by C. P. Snow, pp. 139–140. 10 P. Valery, Degas, danse, dessin, A. Vollard éd., Paris, 1936; Œuvres II, La Pléiade, Gallimard éd., Paris, 1966, pp. 1163–1240, especially pp. 1207–1209. 11 The following excerpt from a letter from C. F. Gauss to Olbers, written on 3 September 1805, shortly after Gauss had solved a problem (the “sign of the Gaussian Sums”) he had been working on for years, can serve as an example:

EMS Newsletter March 2017 45 Discussion Results of a Worldwide Survey of Mathematicians on Journal Reform

Cameron Neylon (Curtin University, Perth, Australia), David M. Roberts (University of Adelaide, Australia) and Mark C. Wilson (University of Auckland, New Zealand)

1 Introduction The survey cannot be taken as representing the gen- Scholarly communication is in a state of ferment. The eral opinion of mathematicians because we have no shift over the last few decades from print to digital dis- information about who responded – full anonymity was semination has set off a wide range of movements for promised to participants. However, we are confident that change, from the radical to the more modest and incre- we reached a broad cross-section of the community. Of mental. At the centre of many of these debates is the respondents, in the last three years, 33% have acted as an move toward wider access to the research literature. editor for a mathematics journal, 93% have authored a Mathematics occupies an unusual place in these debates, paper and 86% have acted as a referee. being simultaneously radical in the degree of uptake of The survey addressed general questions of desire for new approaches such as arXiv.org for rapid dissemina- change, specific issues and the association of specific fac- tion prior to peer review but also highly conservative tors with journal prestige. Questions on prestige were in terms of the move to online-only journals and wide framed in two different ways. In one set, respondents access models more generally. were asked how they personally associate specific factors Several surveys have examined the opinions of with the prestige of journals. In the second set, they were researchers generally (most recently Tenopir et al. [6] – asked how the community associate those same factors see also their literature review – Taylor & Francis [5] and with the prestige of journals. This allows us to identify Solomon [4] and issues of access to funding (Solomon consistent differences between individual (self-reported) and Björk [2], Björk & Solomon [1]) and Dallmeier- views and the assumptions those same individuals have Tiessen et al. [3] but few have focused on the views of about community views. All data, including a copy of mathematicians specifically. We sought to understand the survey itself and raw and processed responses, and how those engaged in mathematical research viewed the code used for processing, are available at https:// the importance of enhancing access to the mathematics figshare.com/projects/Survey_of_mathematical_publish- research literature and their interest in a wider range of ing/16944. innovations, including changes to peer review and publication practice. We also aimed to get feedback from the 2 Results mathematics community on specific issues they saw with We closed the survey when it reached exactly 1000 mathematical journals. responses, on 28 August 2016.2

1.1 Methodology 2.1 Demographics An online survey instrument was made available via Respondents self-reported as PhD student (10.5%), Google Forms from 12 April 2016 and submissions were postdoc (15.5%), tenure-track (7%), tenured (57%) and initially solicited through personal emails, social media and other (emeritus, librarian, etc.) (10%). Surveys in Europe research mathematics mailing lists (including DMANET, and North America of career stages of researchers give the Australian Mathematical Society and the European very different results for the distribution of career stag- Mathematical Society –– note that the American Math- es. The respondent distribution is not inconsistent with ematical Society declined to advertise it). In order to these other surveys but we cannot show that the respond- increase the number of responses, we made a second wave ents are demographically representative. Geographical of approaches to recent authors in mathematics journals, representation was dominated by Europe (54%) and societies and mathematics departments worldwide.1 North America (25%). Other respondents selected Oce- ania (11%), Asia (6%), South America (4%) and Africa 1 Authors’ email addresses were extracted from issues of the following journals in the years 2014–16: Acta Appl. Math., (0.5%) as locations. Acta Inf., Acta Math. Sin. (Engl. Ser.), Adv. Comput. Math., BIT, Calc. Var. Partial Differential Equations, Comput. Math. 2.2 Appetite for change Organ. Theory, Comput. Math. Model., Funct. Anal. Appl., On a five point scale from 1 being “the status-quo is com- Graphs Combin., Invent. Math., J. Algebraic Combin., J. En- pletely acceptable” and 5 being “almost all [journals] grg. Math., J. Math. Sci. (N.Y.), J. Theoret. Probab., Manu- need serious work”, 78% of respondents selected 3, 4 scripta Math., Monatsh. Math., Numer. Math., Potential Anal., Probab. Theory Related Fields, Statist. Papers, Theoret. and or 5. Amongst respondents, there is a strong desire for Math. Phys. Mathematics departments were chosen with no particular plan from universities in China, Czechia, Israel, Ja- 2 Thanks to Ben Rohrlach for additional exploratory analysis pan, Sweden, Turkey, Azerbaijan, Iran and South Africa. and help with R.

46 EMS Newsletter March 2017 Discussion change. Free text answers describing the major perceived problems revealed serious concerns that suggest system- ic issues: almost 200 journals from 57 publishers were mentioned by name as needing serious improvement. These ranged from journals at large commercial publishers and university presses to small Open Access journals that do not charge an Article Processing Charge (APC), over the whole spectrum of prestige. Table 1 gives a clas- sification of the stated issues into main categories (from the 466 respondents who named a journal). Of particular concern is the number of respondents who had concerns with the quality of peer review. For example, 126 journals or publishers were named as being unsatisfactory in the time taken for refereeing or the time taken from accept- Figure 1: Number of suggestions per publisher, by category and port- ance to publication. folio size (small horizontal jitter added for clarity).

Table 1: Distribution of free-form comments by area where improvement is needed.

Issue N% peer review quality 139 30 efficiency 115 24 price 101 21 other quality 83 17 access 72 15 ethics 35 7 governance 27 6 unclear 15 3 Figure 2: Stacked diverging bar chart of Likert scale responses.

On this question, those who had acted as editors did not lowed by the reputation of editors and historical repu- differ substantially from those who had not. To protect tation, and selectivity (median 4), then Journal Impact anonymity, the survey did not ask which journals editors Factor (JIF), Open Access status and external rankings worked for but with over 330 editors this sample must (median 3). The publisher had the lowest median ranking include many associated with traditionally run journals. (2), with a mode of 1. Figure 1 plots the suggestions for each publisher in When we asked for the respondents’ assessment of the each category by the size of the mathematics journal importance of these factors in the community’s view, a portfolio (or rather, by log10 of the number of mathemat- striking pattern emerged, as shown in Figure 3. For factors ics journals to account for the two orders of magnitude that might be considered as traditional markers of prestige range: 1–202 journals). Any publisher with at least five (publisher, external rankings, JIF), respondents believe journals suggested is labelled. One would expect pub- they matter more to the community than they do to them- lishers with larger mathematics portfolios to garner selves. That is, respondents tend to believe themselves less more criticism but there is essentially little trend among influenced by such“external” factors than the community. publishers excluding Elsevier and Springer. Even though For other “traditional” markers (editors’ reputation, his- Elsevier publishes less than half as many mathematics torical reputation, degree of selectivity), this was less pro- journals as Springer, its journals get more suggestions for nounced but the tendency is in the same direction. improvements in all categories but one.

2.3 Which attributes of journals contribute to journal prestige? A diversity of studies continue to show that journal reputation or prestige is an important factor for authors in selecting a journal. In two sets of questions, we asked respondents how important they thought specific aspects were for journal reputation and how important they thought those same aspects were for the community’s view of reputation. Results are summarised in Figure 2. The most important factor for respondents was the quality of peer review (median rank 5). This was fol- Figure 3: Respondents’ beliefs about community opinion on issues.

EMS Newsletter March 2017 47 Discussion

When asked about Open Access (OA), respondents implied strongly that it was more important to them than the community. Combined together, this shows that our respondents believe their colleagues to be more influenced by traditional markers and less interested in OA than they are. These differences matter. Change is risky. If mathematicians are pessimistic about their colleagues’ desire for change then working for change is much less appealing. It is one thing for the status quo to be supported by peer pressure but it appears it may be supported by the perception of peer pressure. Finally, the difference between personal and community views on the importance of the peer review process was both striking and disturbing. By a strong mar- gin, most respondents view the quality of peer review as more important to themselves than they believe it is to the community. If this is true beyond our sample, it Figure 4: Importance of journal aspects: editors and non-editors. is concerning because it suggests that individuals do not see the community as a whole as driven by high standards. While this is potentially a result of sample bias, further investigation of this finding should be carried out.

2.4 Changing practice If there is change, what should it look like? When asked to rate the importance of elements of journal publishing, high ethical standards and timely and thorough peer review were rated the most important (median 5). All other factors (Open Access, low cost of publication, non- profit status, transparent costings, community control Figure 5: Support for new practices. and use of modern internet technologies) had a median ranking of 4. The most frequent ranking (mode) was 5 for all of these questions, apart from low cost. Perhaps more should be paid by library consortia and a quarter saying informatively, there is greater distribution in responses they were “OK if they are sufficiently low”. Respondents for those lower ranked priorities. In terms of the specif- were, however, united on one issue. Only 2% believed ics of change, editors are less keen on Open Access than that they were “not a problem, and competition in the non-editors. This may be related to their having a sub- journal market will take care of them”. stantially stronger view that author payments for publication are unacceptable (see Section 2.5). 3 Discussion In terms of new practices, almost a quarter of Overall, we interpret these results as showing that respondents supported open peer review as a default respondents are strongly in favour of change in the (with opt-out) and half supported post publication publishing system but pessimistic about the support the review with moderated comments and commenter iden- efforts for such change would get from their colleagues. tities revealed. Nearly half supported the publication of There is strong support for high(er) ethical standards anonymous referee reports, suitably presented, to help and high quality peer review, and substantial support for readers. Free-form responses were also allowed and, rather radical changes to the way journals operate. These of the 53 constructive suggestions made, 11 mentioned issues are also the subject of serious concerns raised in double-blind refereeing. Editors were clearly less favour- free-text answers. Editors and publishers should take able towards open review (26% vs. 38%) and community note of these concerns, alongside the demand for greater election of editors (31% vs. 43%) than non-editors. Inter- transparency in editor selection and editorial processes. estingly, editors were slightly more supportive of banning On several of these issues, editors’ views diverge from monetary payments to editors (45% vs. 41%) and of edi- that of the community and this should be a subject of tor term limits (31% vs. 29%). some concern. However, there is substantial agreement between editors and non-editors on many issues. 2.5 Funding of increased access When asked what should happen if efforts by edi- Because mathematics is a discipline with relatively little tors to reform a journal are blocked by the publisher, funding and therefore has limited discretionary resourc- over half of respondents favoured resigning to join a es, it is commonly believed that there is a strong aversion better journal (29%) or to create a new one (32%). to author publication charges (APCs). However, opin- Only a very small proportion (4.5%) favoured settling ions on APCs were split, with (roughly) a quarter believ- for the status quo. For this set of respondents at least, ing them unacceptable in principle, a quarter saying they the appetite for change is there and community support

48 EMS Newsletter March 2017 Discussion for bold moves by editors on behalf of the community [3] S. Dallmeier-Tiessen, B. Goerner, R. Darby, J. Hyppoelae, P. Igo- is strong. Kemenes, D. Kahn, and S. Lambert. Open access publishing – mod- To our knowledge, no previous study has sought to els and attributes. 27:93–103, 2010. compare the views of individuals with their views of the [4] D.J. Solomon. A survey of authors publishing in four megajournals. community. Although it may reflect a sampling bias, it is PeerJ, 2:e365, 2014. [5] Taylor & Francis. Open Access Survey: Exploring the view of Tay- striking that respondents to this survey show a strong ten- lor and Francis and Routledge authors. http://www.tandf.co.uk/ dency to claim views that are more aligned with change journals/explore/Open-Access-Survey-March2013.pdf. than those they believe the community hold, particularly [6] Carol Tenopir, Elizabeth Dalton, Lisa Christian, Misty K. Jones, on Open Access and traditional measures of prestige and Mark McCabe, MacKenzie Smith, and Allison Fish. Imagining a quality. This is in sharp contrast to their views on peer gold open access future: Attitudes, behaviors, and funding scenar- review, where there appears to be pronounced scepticism ios among authors of academic scholarship college and research on the importance other members of the community libraries. College & Research Libraries, September 2017. http:// place on the quality of peer review. www.infodocket.com/2016/10/06/research-article-imagining-a- gold-open-access-future-attitudes-behaviors-and-funding-scenari- 3.1 How is Europe different? os-among-authors-of-academic-scholarship-preprint/. We recalculated some of the results for the subset of data in which the respondent indicated they work in Europe. Cameron Neylon [Cameron.Neylon@cur- One difference observed is that European respond- tin.edu.au] is a professor of research com- ents were somewhat keener on Open Access than non- munications at Curtin University. Europeans (the distribution of European answers sto- chastically dominated the non-European). In terms of demographics, there were more PhD students and fewer editors in the European respondent set than the non- European but the differences were not very large. How- David M. Roberts [david.roberts@adelaide. ever, we have not delved into this issue rigorously and edu.au] is a visiting mathematician at the leave it to our European colleagues to analyse our pub- University of Adelaide. licly available data.

References [1] B.-C. Björk and D.J. Solomon. How research funders can finance APCs in full OA and hybrid journals. Journal of the American Soci- Mark C. Wilson [[email protected]] ety for Information Science and Technology, 27:93–103, 2014. is a senior lecturer at the Department of [2] Bo-Christer Björk and David Solomon. Developing an effective Computer Science, University of Auckland. market for open access article processing charges. URL: https:// wellcome.ac.uk/sites/default/files/developing-effectivemarket-for- open-access-article-processing-charges-mar14.pdf (as of: 9 Novem- ber 2016), 2014.

European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum SEW A21, CH-8092 Zürich, Switzerland [email protected] / www.ems-ph.org

Nicolas Raymond (Université de Rennes, France) Bound States of the Magnetic Schrödinger Operator (EMS Tracts in Mathematics Vol. 27) ISBN 978-3-03719-169-9. 2016. 394 pages. Hardcover. 17 x 24 cm. 64.00 Euro This book is a synthesis of recent advances in the spectral theory of the magnetic Schrödinger operator. It can be considered a catalog of concrete examples of magnetic spectral asymptotics. Since the presentation involves many notions of spectral theory and semiclassical analysis, it begins with a concise account of concepts and methods used in the book and is illustrated by many elementary examples. Assuming various points of view (power series expansions, Feshbach–Grushin reductions, WKB constructions, coherent states decompositions, normal forms) a theory of Magnetic Harmonic Approximation is then established which allows, in particular, accurate descriptions of the magnetic eigenvalues and eigenfunctions. Some parts of this theory, such as those related to spectral reductions or waveguides, are still accessible to advanced students while others (e.g., the discussion of the Birkhoff normal form and its spectral consequences, or the results related to boundary magnetic wells in dimension three) are intended for seasoned researchers.

EMS Newsletter March 2017 49 Research Centres M&MoCS – International Research Center on Mathematics and Mechanics of Complex Systems

Francesco dell’Isola (University of Rome “La Sapienza”, Italy and International Research Center on Mathematics and Mechanics of Complex Systems, L’Aquila, Italy), Luca Placidi (International Telematic University Uninettuno, Italy and International Research Center on Mathematics and Mechanics of Complex Systems, L’Aquila, Italy) and Emilio Barchiesi (University of Rome “La Sapienza”, Italy and International Research Center on Mathematics and Mechanics of Complex Systems, L’Aquila, Italy)

The Centre engineering. Lectures were complemented by discus- The International Research Center for Mathematics & sion sessions to foster lively interactions amongst par- Mechanics of Complex Systems (M&MoCS) is a research ticipants. Moreover, the centre, in collaboration with the centre of the University of L’Aquila. The centre was CNRS International Associate Laboratory Coss&Vita, established in 2010 by the Dipartimento di Ingegneria the Paris Federation of Mechanics Labs and the GDR delle Strutture, delle Acque e del Terreno (DISAT), the MeGe (French Research Network), has organised, since Dipartimento di Matematica Pura e Applicata (DMPA) its foundation, many workshops in Arpino, Alghero of the University of L’Aquila and the Dipartimento di and Catania. Since 2014, the centre has organised two Strutture of Roma Tre University, with the financial and EUROMECH colloquia, and a third one “Generalized logistic aid of Fondazione Tullio Levi-Civita. The current and microstructured continua: new ideas in modeling director of the centre is Francesco dell’Isola, who is a full and/or applications to structures with (nearly-)inex- professor of mechanics of solids at “La Sapienza” Uni- tensible fibers” is planned for 3-8 April 2017 in Arpino versity of Rome. The previous director of the centre was (Italy). In the framework of a joint effort with Warsaw Angelo Luongo, who is a full professor of mechanics of University of Technology and among the many seminars solids at the University of L’Aquila. organised at the centre, an introductory course in analyt- The mission of the centre is the development and ical continuum mechanics and computational mechanics dissemination of scientific knowledge. In order to pur- is offered every year. Many other events are scheduled sue these goals, the centre: (a) conducts and coordi- throughout the year. nates research activities; (b) promotes initiatives for the To find out more information, please visit the website enhancement of scientific liaison between researchers http://memocs.univaq.it. in mathematical fields and researchers in solid and fluid mechanics, operating both in Italy and abroad; (c) pro- Research and facilities motes, supports and organises highly qualified educa- Research activities carried out at the centre are directed tional activities, such as training, Master’s and doctorate toward the formulation of computationally tractable courses; (d) encourages the promotion of mathematics mathematical models to predict phenomena occurring and mechanics of complex systems through publications, in complex systems and address their numerical solu- conferences, seminars and exhibitions; and (e) carries tion. Experimental research is also being carried out at out consultancy and research activities for organisations the centre facilities. The research mainly concerns: vari- and institutions. ational and optimisation methods, gamma convergence, homogenisation techniques for periodic media, mechan- Workshops and summer schools ics of fluids and solids, vibration control by means of From 2011 to 2013, the centre offered the Sperlonga piezoelectric actuators, composite materials, landmine Summer Schools on Mechanics and Engineering Scienc- detection, biomechanics of growing tissues, fluid dynam- es: courses and seminars, organised with an interdiscipli- ics and transport phenomena, kinetic theory, vibrations nary attitude, aimed at introducing young scientists to and waves in continuous and multi-phase media, plastic- present-day developments in mechanics at the interface ity, damage mechanics, continuum mechanics, stability with mathematics, physics, materials science, biology and and control of structures, identification of materials and

50 EMS Newsletter March 2017 Research Centres mechanical systems, dynamical systems and bifurcation remote driving. It could potentially be employed in civil theory, fluid dynamics models for the analysis of traffic defence or for the coast guard, etc. flows and the social sciences, and numerical differential modelling of the mechanical and electromagnetic Vibrations Laboratory response of biological materials and nano-structures. The Vibrations Laboratory conducts research in the In the following subsections, we briefly summarise fields of smart structures, vibration control, noise gener- the main activities of the experimental laboratories. ation and transmission, and structural integrity monitor- ing. In particular, research focuses on: (a) non-destruc- Laboratory of Materials and Structures Testing tive fresco integrity diagnosis with Doppler vibrometer The activities of the Laboratory of Materials and scanning lasers; (b) development of acoustic-vibrators Structures Testing concern experiments on materials for the location of historically significant fresco and and structures for the purpose of consultancy, applied plaster defects; (c) development of lighter and cheap- research and teaching. The aim of the laboratory is to er flexible robotic systems; (d) development of smart provide the construction industry with a diagnosis of structures, equipped with piezoelectric sensors such as the state of degradation of civil works, thus providing an thin plates or panels for noise and vibration control; (e) assessment of the residual life of structures. finite element modelling for the design of structures in the civil and industrial sectors, such as wood laminated Laboratory of Structures and Smart Materials orthotropic structures; and (f) energy analysis of build- The Laboratory of Structures and Smart Materials is ings. actively engaged in the study and prototyping of smart structures. The research group addresses the issue of Humanitarian Demining Laboratory (HDL) mechanical structure vibration damping using piezoelec- The main aim of the Humanitarian Demining Labo- tric transducers coupled with electronic systems and the ratory (HDL) is to develop new anti-personnel mine research is directed toward linear and nonlinear control detection devices for humanitarian demining. Experi- of structural dynamics. Applications considered include mental activity is being carried out on a promising the design of soundproofing systems, wing and blade original active thermal technology based on localised flutter control, the identification of structural damage heating pulses and temperature sensing. With the aim and the design of smart systems able to self-monitor the of developing a multi-sensor platform employing data evolution of their constitutive parameters. Uncertainty fusion and collaborating robotic agents, vibrometric/ modelling in inhomogeneous structures with unknown acoustic and GPR techniques are also employed. For inhomogeneity and stimulated by piezoelectric actua- experimental purposes, a computer-controlled cart that tors and the analysis of metals subject to the action of can move over a sand box while holding a heater and external loads and induced structural change (such as other instrumentation has been realised. In the sand anisotropy and strength of material) are also investi- box, two accurate low-metal-content mine surrogates gated. of different materials are hidden, together with another object. An outdoor “minefield” has also recently been Naval Structures and Onboard Instrumentation realised. Laboratory The Naval Structures and Onboard Instrumentation Functional Multiscale Metamaterials and Smart Laboratory (officially known as Laboratorio Strutture Systems Lab Navali e Strumentazioni di Bordo) conducts research The activities carried out in the Functional Multiscale on prototypes of ship structures (surface and under- Metamaterials and Smart Systems Lab range from water vehicles). The research activity is focused on the numerical modelling to experiments on micro- and areas of control and vibration damping in the field of nano-structural and functional materials. The materials marine structures and stability of ship structures subject have many applications, from biomaterials to energy to the action of fluid waves and impact. The laboratory harvesting. The laboratory has equipment for electron, is directly involved in the development and realisation ionic and atomic force microscopy (including advanced of the project SEALAB. The SEALAB project includes technologies for spectroscopy and nanoindentation) for the construction of an experimental surface marine the structural characterisation of micro- and nano- inno- mobile station, functioning as a test bed for marine vative materials. technologies. It is intended to develop, validate, refine and eventually patent new design solutions, devices and MEMOCS Journal innovative systems in the field of marine engineering. At The International Research Center for the Mathemat- the same time, SEALAB aims to bring together, in a sin- ics and Mechanics of Complex Systems has founded gle project, the efforts and skills of university and indus- the homonymous journal Mathematics and Mechanics try research. Although SEALAB was born with the goal of Complex Systems, abbreviated to MEMOCS, for the of providing an experimental platform to develop new benefit of the community of researchers in mechanics solutions, the vehicle being developed can be used in an and mathematics. MEMOCS is peer-reviewed, indexed HSU (high-speed unmanned) version as a coastal patrol in all major databases and free to both authors and read- in autonomous and/or high speed (close to 200 km/h) ers. It publishes articles from diverse scientific fields with

EMS Newsletter March 2017 51 Research Centres a specific emphasis on mechanics. Articles must rely on Francesco dell’Isola received cum laude his the application or development of rigorous mathemati- degree in physics at the University of Naples cal methods. Indeed, the journal is intended to foster a “Federico II” in 1986. At the same universi- multidisciplinary approach to knowledge firmly based ty, in 1992, he received a PhD in mathemati- on mathematical foundations. It will serve as a forum cal physics with a thesis on the rational ther- where scientists from different disciplines meet to share modynamics of nonmaterial bidimensional a common, rational vision of science and technology. continua. Since 2006, he has been a full pro- It is intended to support and divulge research whose fessor of solid and structural mechanics at “La Sapienza” primary goal is to develop mathematical methods and University of Rome. Professor dell’Isola has authored tools for the study of complexity. The journal also fos- more than 150 papers in international journals. ters and publishes original research in related areas of mathematics of proven applicability, such as variational Luca Placidi graduated cum laude in phys- methods, numerical methods and optimisation tech- ics at the University of Naples “Federico II” niques. Besides their intrinsic interest, such treatments in 2001 and in engineering at the Virginia can become heuristic and epistemological tools for fur- Polytechnic Institute in 2002. He received ther investigations and provide methods for deriving a PhD in 2004 from the Technical Univer- predictions from postulated theories. Papers focusing on sity of Darmstadt and a second one in 2006 and clarifying aspects of the history of mathematics and from “La Sapienza” University of Rome. science are also welcome. All methodologies and points He has authored five books and more than 40 papers in of view, if rigorously applied, are considered. journals. Since 2011, he has been an assistant professor at To find out more information, please visit the website the International Telematic University Uninettuno. http://memocs.org. Emilio Barchiesi completed cum laude his M&MoCS MSc in mathematical engineering at the DICEAA University of L’Aquila (Italy) in 2016, de- Università degli Studi dell’Aquila fending a thesis on the numerical identifica- Via Giovanni Gronchi 18 tion of mathematical models for the descrip- 67100 L’Aquila, Italy tion of engineering fabrics. He is currently Tel. 06.90.28.67.84 Fax 0773.1871016 pursuing his PhD studies in theoretical and Website: http://memocs.univaq.it applied mechanics at “La Sapienza” University of Rome. Email: [email protected] His main research interests lie in homogenisation theory, computational mechanics, higher gradient continua and variational methods.

European Mathematical Society Publishing House e boo published by the Seminar for Applied Mathematics ETH-Zentrum SEW A21, CH-8092 Zürich, Switzerland [email protected] / www.ems-ph.org

Dynamics Done with Your Bare Hands. Lecture notes by Diana Davis, Bryce Weaver, Roland K. W. Roeder, Pablo Lessa (EMS Series of Lectures in Mathematics) Françoise Dal’Bo (Université de Rennes I, France), François Ledrappier (University of Notre Dame, USA) and Amie Wilkinson (University of Chicago, USA), Editors ISBN 978-3-03719-168-2. 2016. 214 pages. Hardcover. 17 x 24 cm. 36.00 Euro This book arose from 4 lectures given at the Undergraduate Summer School of the Thematic Program Dynamics and Boundaries held at the University of Notre Dame. It is intended to introduce (under)graduate students to the field of dynamical systems by emphasizing elementary examples, exercises and bare hands constructions. The lecture of Diana Davis is devoted to billiard flows on polygons, a simple-sounding class of continuous time dynamical system for which many problems remain open. Bryce Weaver focuses on the dynamics of a 2x2 matrix acting on the flat torus. This example introduced by Vladimir Arnold illustrates the wide class of uniformly hyperbolic dynamical systems, including the geodesic flow for negatively curved, compact manifolds. Roland Roeder considers a dynamical system on the complex plane governed by a quadratic map with a complex parameter. These maps exhibit complicated dynamics related to the Mandelbrot set defined as the set of parameters for which the orbit remains bounded. Pablo Lessa deals with a type of non-deterministic dynamical system: a simple walk on an infinite graph, obtained by starting at a vertex and choosing a random neighbor at each step. The central question concerns the recurrence property. When the graph is a Cayley graph of a group, the behavior of the walk is deeply related to algebraic properties of the group.

52 EMS Newsletter March 2017 Societies Join the London Mathematical Society (LMS)

Elizabeth Fisher (London Mathematical Society, London, UK)

About the LMS - Thirteen international peer-reviewed journals, seven of As a UK-wide learned society for mathematics, the Lon- which are in collaboration with other learned societies don Mathematical Society (LMS) advances, disseminates and institutions, and two book series (lecture notes and and promotes mathematical knowledge, both nationally student texts): https://www.lms.ac.uk/publications. and internationally. Its activities include: - Scientific lectures and meetings for research mathematicians, including meetings at the ECM and the ICM: - Grants to support conferences and collaborative re- https://www.lms.ac.uk/events/society-meetings. search between UK based and non-UK based math- - Representation of mathematics research and educa- ematicians, e.g. Research in Pairs (https://www.lms. tion to Government and other national policymakers ac.uk/grants/research-pairs-scheme-4). and sponsors: https://www.lms.ac.uk/policy/policy-con- - Training opportunities aimed at Young Mathemati- sultations. cians and Early Career Researchers, e.g. LMS-CMI - Participation in international mathematical initiatives Research Schools: https://www.lms.ac.uk/events/lms- and promoting the discipline more widely, e.g. sup- cmi-research-schools. porting mathematics in Africa through Mentoring Af- - Prestigious prizes to recognise achievements in math- rican Research in Mathematics (MARM) and grants ematics research, with nine Whitehead Prize winners to conferences at the African Mathematics Millennium going on to win EMS Prizes: https://www.lms.ac.uk/ Science Initiative (AMMSI): https://www.lms.ac.uk/ prizes. grants/international-grants#Africa.

LMS Members at the Annual General Meeting, London, November Simon Donaldson gives a talk at the South West & South Wales 2016. Regional Meeting, Bath, December 2016.

Participants at the LMS Women in Mathematics Day, Edinburgh, Speakers at the Research School, Belfast, September 2016. April 2016.

EMS Newsletter March 2017 53 Societies

Further information about recent LMS activities can also The LMS is pleased to have reciprocal agreements with be found in the Society’s 2015–16 Annual Review: www. 20 mathematical societies. Any non-UK based EMS lms.ac.uk/sites/lms.ac.uk/files/files/files/reports/LMS%20 members thinking of joining the LMS are advised to Annual%20review%202016%20web.pdf. check the list to see if their society is included when applying: https://www.lms.ac.uk/membership/member The LMS and the European Mathematical Society ship-categories#Reciprocity. The LMS has been a member society of the EMS since 1990. The collaboration between the LMS and the EMS How to join has included: The LMS welcomes applications via its online form: https://www.lms.ac.uk/membership/online-application, - Hosting the EMS Ethics Committee in 2013 and both and applicants are formally elected to membership at the EMS Executive Committee and the EMS Applied one of the society meetings of the LMS. Mathematics Committee in 2014. Do visit the website for full details on the applica- - Travel Grants to European Congresses of Mathemat- tion process (https://www.lms.ac.uk/membership/how- ics. join) and the current fees (https://www.lms.ac.uk/sites/ - Participation of LMS delegates at the EMS Council lms.ac.uk/files/Membership/Subscription%20Rates%20 Meetings. and%20Notes%202016-17%20updated.pdf).

This partnership was recently celebrated at a Joint Meet- Benefits of LMS membership include: ing held at the University of Birmingham in September - Membership of a vibrant, national and international 2015 to mark the societies’ respective anniversaries: 25 mathematics community. years for the EMS and 150 years for the LMS. - Networking opportunities.

LMS Immediate Past President Terry Lyons FRS (Oxford), EMS- LMS Meeting Organiser Chris Parker (Birmingham) and EMS Presi- dent Pavel Exner (Academy of Sciences of the Czech Republic) at the LMS President Simon Tavaré (Cambridge) welcomes new LMS mem- Joint LMS-EMS Anniversary Meeting, Birmingham, September 2015. bers at the 7ECM in Berlin, July 2016

Membership of the LMS The LMS has a membership of 2,800 members that make - Opportunities to influence national policy. up a vibrant international mathematical community, with - Full voting rights in society elections – your chance to 20% of the LMS membership based outside the UK, shape the future of the LMS. including 234 members based in other European coun- - A complimentary monthly newsletter – available in tries. print and online. Membership of the LMS falls into three categories: - Regular members-only LMS e-Updates. - Opportunities to attend events hosted by the society. - Ordinary membership: for academic staff, mathemati- - Free online subscriptions to the Bulletin of the Lon- cians in other occupations and all those with a deep don Mathematical Society, the Journal of the London commitment to mathematics and an interest in math- Mathematical Society, the Proceedings of the London ematical research. Mathematical Society and Nonlinearity (published - Associate membership: for undergraduates, postgradu- jointly with the Institute of Physics). ates and early career mathematicians who are within - Members discount on other selected LMS publica- three years of completing their PhD. tions: 25% discount on the LMS Lecture Note Series - Reciprocity membership: for those not normally resi- and 25% discount on LMS Student Texts. dent in the UK and also members of some overseas - Use of the Verblunsky Members’ Room at De Morgan mathematical societies. House, Russell Square, London.

54 EMS Newsletter March 2017 Societies

- Use of University College London Library, where the society’s library is housed. - The opportunity to sign the LMS Members’ Book, which dates back to 1865 when the society was founded and which contains signatures of members throughout the years, including Augustus De Morgan, Henri Poin- caré, G. H. Hardy and Mary Cartwright.

Contact the LMS We would be pleased to hear from you. For queries about membership, do email us at [email protected] and for regular news from the LMS, follow us on Twitter @ LondMathSoc.

And what about Brexit? EMS President Pavel Exner and LMS Vice-President John Greenlees Following the Brexit vote, the LMS would like to express at the UK Parliamentary Links Day with Science, June 2016. its support and solidarity with the 31% of the UK mathematical academic community from other EU states. We is a member, is also available at https://www.lms.ac.uk/ would like to reassure our EU friends and colleagues, news-entry/05072016-1201/council-mathematical-scienc- wherever they are based, that we are keen to maintain es-eu-referendum-statement. our close mathematical and professional relationships with the rest of Europe, however the Brexit process pro- ceeds. The EU referendum statement from the Council Elizabeth Fisher for the Mathematical Sciences (CMS), of which the LMS LMS Membership & Activities Officer

Aard innin monoraphs published by the European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum SEW A21, CH-8092 Zürich, Switzerland [email protected] / www.ems-ph.org

Augusto C. Ponce (Université catholique de Louvain, Belgium) N

Elliptic PDEs, Measures and Capacities (EMS Tracts in Mathematics Vol. 23) M

ISBN 978-3-03719-140-8. 2016. 463 pages. Softcover. 17 x 24 cm. 58.00 Euro Partial differential equations (PDEs) and geometric measure theory (GMT) are branches of analysis whose connections are usually not emphasized in introductory graduate courses. Yet, one cannot dissociate the notions of mass or electric charge, naturally described in terms of measures, from the physical potential they generate. Having such a principle in mind, this book illustrates the beautiful interplay between tools from PDEs and GMT in a simple and elegant way by investigating properties like existence and regularity of solutions of linear and nonlinear elliptic PDEs. This book invites the reader to a trip through modern techniques in the frontier of elliptic PDEs and GMT, and is addressed to graduate students and researchers having some deep interest in analysis. Most of the chapters can be read independently, and only basic knowledge of measure theory, functional analysis and Sobolev spaces is required.

Yves Cornulier (Université Paris-Sud, Orsay, France) and Pierre de la Harpe (Université de Genève, Switzerland) N

Metric Geometry of Locally Compact Groups (EMS Tracts in Mathematics Vol. 25) M ISBN 978-3-03719-166-8. 2013. 243 pages. Softcover. 17 x 24 cm. 62.00 Euro The main aim of this book is the study of locally compact groups from a geometric perspective, with an emphasis on appropriate metrics that can be defined on them. The approach has been successful for finitely generated groups, and can favourably be extended to locally compact groups. Parts of the book address the coarse geometry of metric spaces, where ‘coarse’ refers to that part of geometry concerning properties that can be formulated in terms of large distances only. This point of view is instrumental in studying locally compact groups. Basic results in the subject are exposed with complete proofs, others are stated with appropriate references. Most importantly, the development of the theory is illustrated by numerous examples, including matrix groups with entries in the the field of real or complex numbers, or other locally compact fields such as p-adic fields, isometry groups of various metric spaces, and, last but not least, discrete group themselves. The book is aimed at graduate students and advanced undergraduate students, as well as mathematicians who wish some introduction to coarse geometry and locally compact groups.

EMS Newsletter March 2017 55 Mathematics Education ICMI Column

Jean-Luc Dorier (University of Geneva, Switzerland)

Call for nominations for the 2017 ICMI Felix Klein 2020 (see http://www.icme14.org/ and watch out for and Hans Freudenthal Awards upcoming announcements and further information). Since 2003, the International Commission on Mathe- The ICMI is calling for intentions to bid for ICME 15 matical Instruction (ICMI) has awarded biannually two (to be held in July 2024). Mathematics education and/or awards to recognise outstanding accomplishments in mathematics associations of potential host countries are mathematics education research: the Felix Klein Medal invited to consider hosting the congress. and the Hans Freudenthal Medal (http://www.mathunion. org/icmi/activities/awards/introduction/). Important dates: The Klein and Freudenthal Awards Committee con- - Preliminary declaration of intent to present a bid to sists of a Chair (Professor Anna Sfard), who is nominat- act as host for ICME 15 should be received by the Sec- ed by the President of the ICMI, and five other members retary-General of the ICMI (abraham.arcavi@weiz- who remain anonymous until their terms have come to mann.ac.il) by 1 December 2017. an end. The committee is currently entering the 2017 - Firm bids should reach the Secretary-General by 1 No- cycle of selecting awardees and welcomes nominations vember 2018 (in 12 hard copies). for the two awards from individuals or groups of individuals in the mathematics education community. For further details please see http://www.mathunion.org/ Practical information can be seen at http://www.mat- icmi/conferences/icme-international-congress-on-math- hunion.org/icmi/activities/awards/call-for-awards-2017/. ematical-education/icme-15-2024/. All nominations must be sent no later than 15 April 2017. Subscribing to the ICMI Newsletter Most of the previous material was first published in the The USA–Finland Workshop ICMI Newsletter. A few months ago, the U.S. National Commission on There are two ways of subscribing to ICMI News: Mathematics Instruction in collaboration with the Uni- versity of Helsinki held a Workshop on Supporting Math- 1. Visit www.mathunion.org/index.php using a Web ematics Teachers and Teaching in the United States and browser and go to the “Subscribe” button to sub- Finland. The bilateral meeting of U.S. and Finnish math- scribe to ICMI News online. ematics educators was held on 1–2 August at the Univer- 2. Send an email to [email protected] sity of Helsinki in Helsinki, Finland, and was attended with the Subject-line: “Subject: subscribe”. by 30 experts from both nations and approximately 70 international experts virtually. The workshop was spon- All previous issues can be seen at http://www.mathunion. sored by Åbo Akademi University, Högskolestiftelsen i org/pipermail/icmi-news. Österbotten, the National Science Foundation and Sven- sk-Österbottniska samfundet. Videos of the workshop sessions, presentations (in PDF format for download) and background readings are NOW available at http://sites.nationalacademies.org/ PGA/biso/ICMI/PGA_173314.

The 69th CIEAEM Conference The 69th Conference of the Commission Internationale pour l’Étude et l’Amélioration de l’Enseignement des Mathématiques (International Commission for the Study and Improvement of Mathematics Teaching) will take place on 15–19 July 2017 at the Freie Universität Berlin, Germany. The theme of the event is “Mathematisation: social process & didactic principle”. The Call for Papers was distributed in December 2016. For more details, see http://www.cieaem.org/.

ICME 14 and ICME 15 The next International Congress on Mathematical Edu- cation ICME 14 will be held in Shanghai on 12–19 July

56 EMS Newsletter March 2017 Book Reviews Book Reviews

growth rate of the values of the potential along periodic Frédéric Paulin, Mark Pollicott and orbits of increasing period (the possibility of non-com- Barbara Schapira pact manifolds requires a restriction to periodic orbits that meet a given compact set, though it is shown that the Equilibrium States in Negative resulting rate does not depend on the choice of set). The Curvature topological pressure (in other contexts also called the variational pressure) is defined as the supremum of the Astérisque, 2015 measure-theoretic entropy plus the averaged potential 289 p. with respect to the measure, where the supremum is tak- ISBN 978-2-85629-818-3 en over all invariant Borel probability measures. A measure realising the supremum is called an equilibrium state Reviewer: Katrin Gelfert for the potential. It is shown that all these quantities are well defined and, moreover, that all three of them agree The Newsletter thanks zbMATH and Karin Gelfert for the (which extends [A. Manning, Ann. Math. (2) 110, 567–573 permission to republish this review, originally appeared as (1979; Zbl 0426.58016)], [D. Ruelle, Bol. Soc. Bras. Mat. Zbl 1347.37001. 12, No. 1, 95–99 (1981; Zbl 0599.58038)] and [W. Parry, Lect. Notes Math. 1342, 617–625 (1988; Zbl 0667.58056)] This book studies Gibbs measures (or states) for the geo- in the case when M is compact). See Chapters 3, 4 and 6. desic flows of negatively curved Riemannian manifolds. The book develops a Patterson–Sullivan theory for Its framework allows, in particular, the removal of the Gibbs measures (Chapter 3). Here, the Gibbs measure compactness assumption on the manifold. The authors mF of F is defined throughMohsen ’s extension [Ann. consider a Hölder continuous function (also called a Sci. Éc. Norm. Supér. (4) 40, No. 2, 191–207 (2007; potential) F defined on the unit tangent bundle T 1M Zbl 1128.58008)] of the Patterson–Sullivan construc- and prove many results on the existence, uniqueness and tion [S. Patterson, Acta Math. 136, 241–273 (1976; Zbl finiteness of the Gibbs measure of F. This very compre- 0336.30005) and D. Sullivan, Publ. Math., Inst. Hautes hensive monograph not only includes detailed proofs Étud. Sci. 50, 171–202 (1979; 0439.30034)]. In the case and background material but also highlights possible when M is compact, they recover, up to some scalar mul- further developments and open problems. tiple, for F = 0, (the Patterson-Sullivan construction of) Gibbs measures have a thermodynamic origin and the Bowen–Margulis measure and, for F being the (un) were introduced for hyperbolic dynamical systems stable Jacobian (that is, the pointwise exponential growth by Ya. G. Sinai [Usp. Mat. Nauk 27, No. 4(166), 21–64 rate of the Jacobian of the geodesic flow restricted to the (1972; Zbl 0246.28008)], R. Bowen [Equilibrium states strong (un)stable manifold), the Liouville measure. One and the ergodic theory of Anosov diffeomorphisms. 2nd fundamental result of the book is the Variational Prin- revised ed. Berlin: Springer (2008; Zbl 1172.37001)] and ciple: in the case when the Gibbs measure of F is finite D. Ruelle [Thermodynamic formalism. The mathemati- then this (properly normalised) measure is the unique cal structures of equilibrium statistical mechanics. 2nd equilibrium state for F; otherwise, there exists no equi- edition. Cambridge: Cambridge University Press. (2004; librium state.

Zbl 1062.82001)], revealing their intimate relation with Under the hypotheses that the Gibbs measure mF is symbolic dynamical systems over finite alphabets. In finite and mixing, Dirac masses on an orbit weighted by the case of the non-wandering set of the geodesic flow the potential are asymptotically equidistributed toward (and, in particular, when M is compact), this reduction the product of Patterson densities on the sphere at infin- ~ to a symbolic system allows a detailed analysis of Gibbs ity  M, which in turn provides counting asymptotics of (equilibrium) measures and the (weighted) distribution orbit points. This type of result translates to results on of periodic orbits. No such coding theory that does not the equidistribution of periodic orbits and on counting lose geometric information is known in the non-compact of periodic orbits (which, in the compact case, are due to case. The authors circumvent this difficulty by geometri- R. Bowen [Am. J. Math. 94, 413–423 (1972; 0249.53033)]); cally constructing and studying Gibbs measures in this see Chapter 9. The authors show the equivalence of the general case. following properties: The guiding objects are three numerical invariants associated to the weighted dynamics. The critical expo- - The Poincaré series of (G, F) diverges at the critical ex- nent is defined by means of the fundamental group G of ponent. ~ M acting by isometries on the universal coverM of M - The conical limit set of G has positive Patterson meas- and measures the exponential growth rate of the orbit ure. 1 ~ ~ ~ points of the group weighted by the (lift to T M of the) - The action of G on M × M is ergodic and conserva- potential F. The Gurevich pressure is the exponential tive.

EMS Newsletter March 2017 57 Book Reviews

- The geodesic flow is ergodic and conservative with re- Math. Fr., Nouv. Sér. 95, 96 p. (2003; Zbl 1056.37034)] and spect to the Gibbs measure of F (Chapter 5). builds essentially on work by the last author [Propriétés ergodiques du feuilletage horosphérique d’une variété à Geometric criteria for the finiteness and the mixing prop- courbure négative. Orléans: Univ. Orléans (PhD Thesis) erty of mF are also discussed (Chapter 8). One particular (2003)]. and very natural focus is on the case when F is the (un) stable Jacobian (Chapter 7). Katrin Gelfert ([email protected]) is a pro- Further investigations concern the ergodic theory fessor at the Department of Mathematics of the strong unstable foliation of T 1M (Chapter 10). at the Federal University of Rio de Janeiro,

Again, under the hypotheses that mF is finite and mix- Brazil, where she has been since 2010. She ing, there exists an (up to scalar multiple) unique family received her PhD in mathematics in 2001 of transverse measures for the strong unstable foliation (TU Dresden, Germany). Her research in- that is quasi-invariant for the holonomy map (taking into terests include dynamical systems theory account the potential F), which enables unique ergodic- and ergodic theory. ity results. This extends work by T. Roblin [Mém. Soc.

Martin Schlichenmaier it to multi-point cases. This chapter also defines almost grading for the type of algebras introduced earlier. Krichever-Novikov Type Chapter 4 deals with some technical but useful details Algebras. of almost grading, like the proof of existence of an almost Theory and Applications grading. For this, the author uses Riemann–Roch type arguments to show the existence of certain basis ele- De Gruyter, 2014 ments that give the almost-graded structure. In Chapter xv, 360 p. 5, there are explicit expressions for the homogeneous ISBN 978-3-11-027964-1 basis elements. The following chapters deal with representations Reviewer: Alice Fialowski of algebras. Firstly, in Chapter 6, central extensions of Krichever–Novikov type algebras are studied: their rela- The Newsletter thanks zbMATH and Alice Fialowski tion to Lie algebra cohomology and their construction for the permission to republish this review, originally for geometrically induced Lie algebras. Central exten- appeared as Zbl 1347.17001. sions appear naturally in quantisations of classical field theories. Chapter 7 studies fermionic Fock space repre- The goal of this book is to acquaint the reader with sentations or, equivalently, semi-infinite wedge represen- Krichever–Novikov type infinite dimensional Lie alge- tations of these algebras. One can also obtain a represen- bras, to study their properties and to highlight the most tation of a certain central extension of the full algebra important applications. of differential operators with the help of a regularisa- The book is excellent for studying the topic. It has 14 tion procedure. As a technical tool, infinite dimensional chapters, an extended bibliography and an index. The matrix algebras are used. only prerequisites are the theories of Lie algebras and In Chapter 8, the author shows that the semi-infinite Riemann surfaces. To make the presentation self-con- wedge form also comes with the representation of an tained, Chapter 1 recalls the basics of Lie algebras, super- algebra that has a Clifford algebra-like structure. The algebras, Virasoro and current algebras, and Riemann corresponding field theory system is called the b–c sys- surfaces of genus g ≥ 0. In Chapter 2, algebraic structures tem. In a mathematical context, the operators b and c of specific meromorphic objects on a compact Riemann are defined via anticommutators. In this chapter, arbi- surface are studied. Such algebras are called Krichever– trary representation spaces of these field operators are Novikov type if they are holomorphic outside a fixed set considered and the energy momentum tensor is defined. of points. They form an important class of infinite dimen- Its “modes” define a representation of an almost-graded sional Lie algebras, which are far from being understood central extension of a vector field algebra. The 2-point in general. Krichever–Novikov type algebras nicely com- and the multi-point situations are discussed. bine geometric and algebraic properties. Chapter 9 contains the detailed study of higher genus These algebras are, in general, not graded, which is current algebras and their central extensions. The almost- often useful for dealing with infinite dimensional algebras. graded central extensions are classified. These algebras Instead, as Krichever and Novikov observed, a weaker correspond to gauge symmetries. As examples of repre- concept ‘almost grading’ can be introduced, which makes sentations, the Verma modules and fermionic Fock space the necessary constructions possible. The definition is representations are introduced. Chapter 10 presents the given in Chapter 3. The original almost-graded structure Sugawara construction for arbitrary genus in the multi- is given for the 2-point situation and the author extends point situation. This construction relates gauge symmetry

58 EMS Newsletter March 2017 Book Reviews to conformal symmetry, which is realised by vector field tion, the vanishing cohomology space does not imply that algebras and their central extensions. the algebra is rigid in the geometric sense. The last four chapters cover the main applications Chapter 13 introduces Lax operator algebras, which of the theory. Chapter 11 presents the author’s and O. are a new class of global current type algebra. They are Sheinman’s global operator approach to Wess–Zumino– related to integrable systems. In these algebras, addition- Novikov–Witten models and the Knizhnik–Zamolod- al singularities are allowed. It is possible for such Lax chikov connection. Their approach uses global objects, operator algebras to introduce almost grading and classi- which are Krichever–Novikov algebras and their repre- fy associated almost-graded central extensions. The con- sentations. A crucial point is that a certain subspace of struction works so far for gl(n), sl(n), so(n), sp(2n) and the Krichever–Novikov algebra of vector fields can be G2. In the last chapter, there are further developments identified with tangent directions on the moduli space of and related subjects, mainly noted through references. the geometric data. One can define conformal blocks and The book should convince the reader that, beside also, with the global Sugawara construction, the higher Krichever–Novikov type algebras being mathematically genus multi-point Knizhnik–Zamolodchikov connection. very interesting infinite dimensional geometric examples, It should be mentioned that this global construction (so they are important in conformal field theory, integrable far) only works over an open dense subset of the moduli systems, deformations and many other topics. space. The approach of other authors provides a valid theory for the compactified moduli space. Alice Fialowski ([email protected]) is a Another application of Krichever–Novikov type professor at the Institute of Mathematics of algebras is related to deformations. This is the material of the University of Pécs and Eötvös Loránd Chapter 12 and the results have been investigated jointly University Budapest, Hungary. She received with A. Fialowski. It turns out that Witt/Virasoro alge- her candidate’s degree at Moscow State bra deforms into elliptic vector field Krichever–Novikov University in 1983, under the supervision algebras. These families are locally nontrivial, in spite of A. A. Kirillov. She became a professor in of the fact that Witt/Virasoro algebra is formally rigid, the USA in 1994. Her research interests are because the second cohomology group with values in the Lie theory, cohomology and deformation adjoint module is trivial. The same is true of current alge- theory, with applications in mathematical bras. The point is that, in the infinite dimensional situa- physics.

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ISSN print 1661-7207 Editor-in-Chief: ISSN online 1661-7215 Rostislav Grigorchuk (Texas A&M University, College Station, USA and Steklov Institute of 2017. Vol. 11. 4 issues Mathematics, Moscow, Russia) Approx. 1500 pages Aims and Scope 17.0 x 24.0 cm Groups, Geometry, and Dynamics is devoted to publication of research articles that focus on Price of subscription: groups or group actions as well as articles in other areas of mathematics in which groups or 368¤ online only group actions are used as a main tool. The journal covers all topics of modern group theory 428 ¤ print+online with preference given to geometric, asymptotic and combinatorial group theory, dynamics of group actions, probabilistic and analytical methods, interaction with ergodic theory and operator algebras, and other related fields.

ISSN print 1661-6952 Editor-in-Chief: ISSN online 1661-6960 Alain Connes (Collège de France, Paris, France) 2017. Vol. 11. 4 issues Aims and Scope Approx. 1500 pages The Journal of Noncommutative Geometry (JNCG) covers the noncommutative world in all 17.0 x 24.0 cm its aspects. It is devoted to publication of research articles which represent major advances in Price of subscription: the area of noncommutative geometry and its applications to other fields of mathematics and 338¤ online only theoretical physics. Topics covered include in particular: Hochschild and cyclic cohomology; 398 ¤ print+online K-theory and index theory; measure theory and topology of noncommutative spaces, operator algebras; spectral geometry of noncommutative spaces; noncommutative algebraic geometry; Hopf algebras and quantum groups; foliations, groupoids, stacks, gerbes.

EMS Newsletter March 2017 59 Problem Corner

174. Prove, disprove or conjecture: SolvedSolved and Un- 1. There are infinitely many primes with at least one 7 in their decimal expansion. 2. There are infinitely many primes where 7 occurs at least solvedand Unsolved Problems 2017 times in their decimal expansion. 3. There are infinitely many primes where at most one-quarter MichaelProblems Th. Rassias (Institute of Mathematics, University of the digits in their decimal expansion are 7s. of Zürich, Switzerland) 4. There are infinitely many primes where at most half the Michael Th. Rassias (University of Zürich, Switzerland) digits in their decimal expansion are 7s. 5. There are infinitely many primes where 7 does not occur in God created the natural numbers. The rest is the work of man. their decimal expansion. Leopold Kronecker (1823–1891) Note. Let p be a prime. Then, the decimal expansion of 1/p is The column Solved and Unsolved Problems will continue present- often called the “decimal expansion of p”. ing six proposed problems and two open problems, as has been done (Steven J. Miller, Department of Mathematics and Statistics, over recent years. The set of proposed and open problems in each is- Williams College, Williamstown, MA, USA) sue will be devoted to a specific field of mathematics. In every issue featuring this column, solutions will be presented to the proposed problems from the previous issue along with the names of solvers. 175 Possible progress toward the solution of any of the open problems . Show that there is an infinite sequence of primes p1 < p2 < p < such that p is formed by appending a number in front proposed in this column will also be featured. The goal of the Solved 3 ··· 2 and Unsolved Problems column is to provide a series of intriguing of p1, p3 is formed by appending a number in front of p2 and so proposed problems and open problems ranging over several areas on. For example, we could have p1 = 3, p2 = 13, p3 = 313, of mathematics. Effort will also be made to present problems of an p4 = 3313, p5 = 13313, .... Of course, you might have to add interdisciplinary flavour. more than one digit at a time. Find a bound on how many digits The column in this issue is devoted to number theory. As is well you need to add to ensure it can be done. known, number theory is one of the oldest and most vibrant areas of (Steven J. Miller, Department of Mathematics and Statistics, pure mathematics. Over the last few decades, it has also found im- Williams College, Williamstown, MA, USA) portant applicability in various scientific domains such as cryptog- raphy, coding theory, theoretical computer science and even nuclear physics and quantum information theory. 176. Consider all pairs of integers x, y with the property that xy 1 is divisible by the prime number 2017. If three such integral− pairs lie on a straight line on the xy plane, show that both the − I Six new problems – solutions solicited vertical distance and the horizontal distance of at least two of such three integral pairs are divisible by 2017. Solutions will appear in a subsequent issue. (W. S. Cheung, Department of Mathematics, The University of Hong Kong, Pokfulam, Hong Kong) 171. Prove that every integer can be written in infinitely many ways in the form

12 32 52 (2k + 1)2 ± ± ± ±···± II Two new open problems (on ζ-functions) by Preda for some choices of signs + and . Mihailescu,˘ Mathematisches Institut, Göttingen, − Germany (Dorin Andrica, Babes¸sBolyai University, Cluj-Napoca, Romania) Let K be a number field, let I(K) denote the set of integral ideals of K, including the trivial ideal 1 = (K), let P(K) I(K) denote the 172. Show that, for every integer n 1 and every real number principal ideals and let (K) be theO ideal class group⊂ of K. Denote by ≥ a 1, one has N = N the absolute normC N and let d = [K : Q]. The Dedekind ≥ K K/Q n a ζ-function of K is 1 1 a 1 1 1 1 k < 1 + . ζ (s) = . (1) 2n ≤ na+1 − a + 1 2n 2n K N a s k=1 a I K ∈ | | If K = Q then (László Tóth, University of Pécs, Hungary) 1 ζ (s) = ζ(s) = K ns n 1 173 ≥ . Let cn(k) denote the Ramanujan sum, defined as the sum of is the Riemann ζ-function. More precisely, the Dirichlet series above kth powers of the primitive nth roots of unity. Show that, for any define the respective ζ-functions on the half plane H1 = s C : integers n, k, a with n 1, { ∈ ≥ (s) > 1 , on which the series are absolutely convergent. They have aℜ pole at }s = 1 and it is proved by means of Mellin transforms that c (k)an/d 0 (mod n). d they have an analytic continuation to C, with no other singularity d n ≡ | except for s = 1. The properties of the ζK (s) have been investigated (László Tóth, University of Pécs, Hungary) by a series of classical mathematicians, including Dirichlet, Weier- straß and Hecke. We refer to Lang’s Algebraic Number Theory [La],

60 EMS Newsletter March 2017 Problem Corner

* Chapters V–VIII, for a review of the classical results on ζK (s). Let 177 . We keep the notation introduced above, in particular the K (K) be a class. notation in (2) and (3). ∈C a1 a2 b1 b2 (i) Prove that γK = c1R ∆ + c2S (E) ∆ , with constants The counting function J(K, t) for t R>1 plays an essential role · ∈ c , c > 0 and powers a , a , b , b Q, which do not de- in this context. It counts the number of ideals a I(K) K that have 1 2 1 2 1 2 ∈ ∈ ∩ pend on the extension degree d. norm less than t. This is done by choosing some fixed ideal b K 1 − (ii) Prove that there is an additional constant 0 < C < 1 such and counting the number of ideals (α) P(K) b that have∈ norm b ∈ ∩ that bounded by N(α) < t N( ) . It is an elementary fact (proved in [La]) 1 1/d | | | | J(K, t) ρ t > Cγ t − that these ideals are in one-to-one correspondence to the ideals of K | − K | K with norm less than t. Let E = ×(K) be the global units; certainly, for all t > ∆. for α (K), the principal idealO (α) P(K) is generated by any We continue our investigation of the counting function J for arbi- ∈O ∈ element of the orbit αE of α under the action of the units by multi- trary number fields K with a problem on the geometry of numbers. plication. We are thus reduced to the problem of counting orbits of For a given class K, one can consider the lattices La spanned by numbers α b under the action of the units. Here enters the geom- some ideal a K as a Z-module in Minkowski space. We are ∈ ∈ etry of numbers. For details of the classical estimates, we refer the interested in determining how close such a lattice can come to or- reader to any detailed deduction of the classical results in any book thonormal lattices, if we allow a to take all the ideals in K as its on algebraic number theory that also treats analytical results – the value. The following definition will introduce quantitative mea- account of Lang is a possible example. sures for the “distance” of a lattice to an orthonormal one. Let n n n Λ R be a full lattice, let (vi) R be a spanning set of Briefly, the numbers of the field K have two representations in ⊂ i=1 ⊂ generators and let Av be the matrix with these vectors as columns. r+1 R , with r = r1 + r2 1 the Dirichlet rank of the units. The first n Let the Euclidean norm of a matrix B = (bi, j)i, j=1 be the norm − r+1 δi r+1 r1 representation is µ : K× R via x ( σi(x) )i=1 , with (σi)i=1 → �→ | | r+1 B = b2 and let BT denote the transpose. Then we define an enumeration of the real embeddings of K and (σ j) an enu- i, j i, j j=r1+1 || || meration of representatives of pairs of complex conjugate embed- the orthonormality defect of this base by �� dings; the exponents are δ = 1 for real embeddings and δ = 2 i j T for complex embeddings. The map µ is continued by an additive one ωv(Λ) = inf A A λI . λ R+ || · − || r+1 r+1 ∈ λ : R R , defined by λ(µ(x))k = log( µ(x)k ) for k = 1, 2,...,r → 1/d | | Λ = and λ(µ(x)) + = N(x) . The fundamental classical result deduced The orthonormality defect of the lattice is defined by ω( ): r 1 | | by investigating J(K, t) under these maps is infv ωv(Λ), the infimum being over all bases of Λ. 1 Now, let a class K (K) be fixed and b K− I(K) be any in- 1 1/d ⊂O b ∈ ∩ r+1 J(K, t) = ρK t + O(t − ). (2) tegral ideal. The image of under the map µ is a lattice Lb R . 1/d ⊂ Let sb = N(b) and normalise the lattice to Lb′ = Lb/sb, a lattice | | b The constant ρK is completely determined in terms of the data of of volume one. The orthonormality defect of is naturally given the field, which are ∆, R, w – the discriminant, the regulator and the by ω(b) = ω(Lb′ ). For our counting function, the choice of b is number of roots of unity of the field respectively. It is independent arbitrary. We may multiply b by field elements (not necessarily in- of K and its value is, with these notations, tegral) and obtain ideals of the same class. This leads to defining the orthonormality defect of the class K by 2r+1πr2 R ρK = . ω(K) = inf ω(b). (4) w √∆ b K 1 I(K) ∈ − ∩ 1 The order of magnitude of the error term is determined by a crude The defect of the class K is thus defined by means of ideals in K− . argument involving the fact that the fundamental domain D(1) The second problem concerns orthonormality defects of classes. ⊂ Rr+1 used for estimating J(K, t) is Lipschitz-parametrisable. One can rephrase the formula above by stating that there certainly exists some constant γK (K), depending only on K and possibly also on the class K, such that 178* 1 1/d . J(K, t) ρ t γ (K) t − , for t > ∆. | − K |≤ K · (i) Find an optimal estimate for the orthonormality defect ω(K) of a class K (K). It is important to choose a lower bound for t in order to obtain an (ii) Prove or disprove∈C that the radii verify an ultrametric inequal- accurate order of magnitude but the bound ∆ chosen in our defini- ity tion is not stringent. One may expect, for reasons discussed in the ω(K K′) max(ω(K),ω(K′)) . Remarks below, that these constants are quite small. However, the · ≤ present methods of estimates, which have only recently been worked Remarks: The Riemann1 zeta function ζ(z) has a Laurent expan- out by van Order and Murty [MO] to the effect of obtaining explicit sion in a neighbourhood of its simple pole at z = 1: bounds on γK (K), yield excessively large values for the bound. We shall make the definition of our constant uniform to make it indepen- 1 ∞ ( 1)n ζ(z) = + − γ (z 1)n, (5) dent of the class and then state our first problem, which is a conjec- z 1 n! n − − n=0 ture. We define the constant γK by � where γn are the Stieltjes constants 1 1/d γ := inf γ R : J(K, t) ρ t γ t − . (3) K >0 K m t> √∆,K (K) ∈ | − |≤ · n n+1 ∈C ln k ln m � � γn = lim , n = 0, 1,... (6) n k − m + 1 →∞  k=1  We define the surface of the units as follows: for a fundamental sys- �   tem of units ui E(K), we let S (�u) be the surface of the fundamental   ∈ Clearly, γ0 is the Euler-Mascheroni constant and note that all parallelepiped of the lattice spanned by the vectors wi := λ(µ(ui)). the terms of the sequence (γn)n 0 are Euler-Mascheroni type con- The surface S (E(K)) = inf�u S (�u), the infimum over all the funda- ≥ stants. Here are the first decimals of γn for n = 0, 1, 2, 3, 4, 5. mental systems of units of K.

EMS Newsletter March 2017 61 Problem Corner

a b m+2 a m+1 b It follows that 3x 5 = 2 5 and 3x + 5 = 2 − 5 − , where a γ0 = 0.5772156649 ...,γ1 = 0.0728158454 ..., − · · − and b are non-negative integers, hence γ = 0.0096903631 ...,γ = 0.0020538344 ..., 2 3 m+2 a m+1 b a b 2 − 5 − 2 5 = 10, (8) γ4 = 0.0023253700 ...,γ5 = 0.0007933238 .... · − · that is, m+1 a m b a 1 b 1 An elementary proof of the expansion (1) can be obtained by the 2 − 5 − 2 − 5 − = 1. (9) · − · Euler–Maclaurin summation formula. In the paper [AT], formula We consider the following cases for equation (9). (1) and some asymptotic evaluations were obtained by using the m m+1 b b 1 Laplace transform. The behaviour of these constants suggests that Case 1: a = 1. We obtain 2 5 − 5 − = 1. If b = 1 then it the error term in (2) might be small, despite our present incapacity follows that 5m = 2, which is not· possible− for m 1. ≥ of ﬁnding appropriate estimates – hence the relevance of these two m m 1 If b = m, we obtain 2 5 − = 1, which is not possible because research problems. m 1 m − 5 − > 2 for m 1. ≥ m b m b 1 References Case 2: a = m+1. It follows that 5 − 2 5 − = 1. If b = 1, we get 5m 1 2m = 1, which is not possible because− · 5m 1 > 2m+1 > 2m + 1 [AT] D. Andrica and L. Tóth, Some remarks on Stieltjes con- − − when−m 1. stants of the zeta function, Stud. Cerc. Mat. 43 (1991) 3–9; ≥ MR 93c:11066. If b = m, we obtain 2m+2 5m = 0, which is not possible. [Br] W. E. Briggs, Some constants associated with the Riemann · In conclusion, the only solutions are m = 1, n = 1 and m = zeta-function, Michigan Math. J. 3 (1955–56) 117–121; 2, n = 1. MR 17,955c. [La] S. Lang: Algebraic Number Theory, Spinger GTM 110, (1986). Also solved by Panagiotis T. Krasopoulos (Athens, Greece), Hans [MO] M. Ram Murty and J. Van Order: Counting integral ideals in J. Munkholm, Ellen S. Munkholm (University of Southern Denmark, a number ﬁeld. Odense, Denmark), F. Plastria (BUTO-Vrije Universiteit Brussel), José Hernández Santiago (Morelia, Michoacan, Mexico)

164. Prove that every power of 2015 can be written in the form x2+y2 x y , with x and y positive integers. − III Solutions (Dorin Andrica, Babe¸s-Bolyai University, Cluj-Napoca, Romania)

163. Find all positive integers m and n such that the integer Solution by the proposer. We have 2015 = 5 13 31. Because 31 · · am,n = 2 ...2 5 ...5 is congruent to 3 modulo 4, it follows that 31 divides both x and y, n n m time n time etc. We get x = 31 x1, y = 31 y1 and replace in the equation to obtain x2 + y2 = 5n 13n(x y ). But 5 13 = 65 = 82 + 12, hence is a perfect square. 1 1 1 1 5n 13n = (82 + 12·)n = a2−+ b2, where· we can assume that a > b . (Dorin Andrica, Babe¸s-Bolyai University, The· equation is equivalent to (x + y )2 + (x y )2 2 5n 13n(x 1 1 1 − 1 − · · 1 − Cluj-Napoca, Romania) y ) + (5n 13n)2 = (5n 13n)2, that is, 1 · · 2 n n 2 n n 2 (x1 + y1) + (5 13 x1 + y1) = (5 13 ) . 2 · − · Solution by the proposer. We have a1,1 = 25 = 5 and a2,1 = 225 = The last equation is Pythagorean and we select solutions as 2 15 . In the ﬁrst step, we will show that if am,n is a perfect square then n n 2 2 2 2 n n = 5 13 = a + b , x1 + y1 = a b , 5 13 x1 + y1 = 2ab , n 1. We can write · − · − m+n 1 n n 1 where a and b are positive integers such that am,n = 2(10 − + + 10 ) + 5(10 − + + 1) ··· ··· n n 2 2 n 2 2 10m 1 10n 1 5 13 = (8 + 1 ) = a + b and a > b . = 2 10n − + 5 − . · · · 9 · 9 It follows that

2 2 2 Therefore, the relation am,n = x is equivalent to x = a ab = a(a b), y = ab b = b(a b) . 1 − − 1 − − 2 10m+n + 3 10n 5 = (3x)2. (7) Finally, it follows that the equation is solvable and has solution · · − = n n If n 2, it follows that 3x is divisible by 5, hence x = 5x for some (x, y) (31 a(a b), 31 b(a b)) . ≥ 1 − − positive integer x1. Replacing in equation (7), we get the equation For example, for n = 1, we have a = 8, b = 1, hence we get the = m+n m+n 1 n n 1 2 solution to the reduced equation modulo 31, (x1, y1) (8(8 1), 2 2 5 − + 3 2 5 − 1 = 5(3x1) , 1(8 1)) = (56, 7). Finally, it follows that the equation is solvable− · · · · − − and has solution which is not possible. Now, we will prove that for m 3 the integer a = 2 ...2 5 is (x, y) = (31 56, 31 7) = (1736, 217) . ≥ m,1 · · m time not a perfect square. For n = 1, equation (7) is equivalent to 2 10m+1 + 25 = (3x)2, Also solved by Mihály Bencze (Brasov, Romania), Panagiotis T. Kra- · sopoulos (Athens, Greece), Hans J. Munkholm, Ellen S. Munkholm that is, (University of Southern Denmark, Odense, Denmark), F. Plastria 2 10m+1 = (3x 5)(3x + 5). (BUTO-Vrije Universiteit Brussel) · −

62 EMS Newsletter March 2017 Problem Corner

165. Find the smallest positive integer k such that, for any n k, Solution by the proposer. Define A := x [0, 100] : f (x) x . ≥ { ∈ ≥ } every degree n polynomial f (x) over Z with leading coefficient 1 Since 0 A, A φ, let a := sup A. Clearly, a < 100. For any ε>0, ∈ must be irreducible over Z if f (x) = 1 has not less than n + 1 there exists x A such that a ε0 is arbitrary, we have a f (a). ≤ Suppose f (a) a = δ>0. Then, for any x (a, a + δ) [0, 100], Solution by the proposer. Suppose f (x) is a degree n polynomial with − ∈ ∩ n x does not belong to A and, by the monotonicity of f , we have leading coefficient 1 such that f (x) = 1 has at least 2 + 1 distinct | | = integral roots. Assume that f (x) is reducible, say, f (x) g(x)h(x), f x f a = a + δ> f a + δ f x , n ( ) ( ) ( ) ( ) with deg g deg h. Clearly we have deg g 2 . ≥ ≥ ≤ ≤ n Suppose f (xi) = 1 for i = 1,...,m with m + 1, where xi Z | | ≥ 2 ∈ which is absurd. Thus f (a) = a. are distinct. We have g(xi) = 1 for all i = 1,...,m. Without loss ± of generality, assume that g(xi) = 1 for 1 i ℓ, g(x j) = 1 for ℓ + 1 j m, and ℓ m . ≤ ≤ − ≤ ≤ ≥ 2 Also solved by A. M. Encinas (Universitat Politècnica de Catalunya, Then, Spain), Laurent Moret-Bailly (IRMAR, Université de Rennes 1, g(x) 1 = (x x )(x x ) (x x )P(x) − − 1 − 2 ··· − ℓ France), F. Plastria (BUTO-Vrije Universiteit Brussel, Belgium). for some polynomial P(x). Observe that n n ℓ deg g < + 1 m . ≤ ≤ 2 2 ≤ 167. Show that, for any a, b > 0, we have Since 2 2 g(x j) = 1 ℓ + 1 j m , 1 min a, b b a 1 max a, b − ∀ ≤ ≤ 1 { } − ln b + ln a { } 1 . we have 2 − max a, b ≤ a − ≤ 2 min a, b − { } { } (10) (x x )(x x ) (x x )P(x ) = 2 ℓ + 1 j m , j − 1 j − 2 ··· j − ℓ j − ∀ ≤ ≤ (Silvestru Sever Dragomir, Victoria University, and so Melbourne City, Australia)

(x x )(x x ) (x x ) 2 ℓ + 1 j m . ( ) j − 1 j − 2 ··· j − ℓ | ∀ ≤ ≤ ∗ If ℓ 4, (x x )(x x ) (x x ) is a product of 4 or more Solution by the proposer. Integrating by parts, we have ≥ j − 1 j − 2 ··· j − ℓ distinct non-zero integers and so its absolute value is 4 and cannot ≥ b b t b a divide 2. Hence ℓ 3. − dt = − ln b + ln a (11) ≤ t2 a − If ℓ = 3, (*) reduces to a for any a, b > 0. (x x )(x x )(x x ) 2 . j − 1 j − 2 j − 3 | If b > a then Observe that there can be at most one a Z satisfying ∈ 1 (b a)2 b b t 1 (b a)2 (a x1)(a x2)(a x3) 2 . − − dt − . (12) − − − | 2 a2 ≥ t2 ≥ 2 b2 a Thus, we must have m = 3 or 4. If ℓ 2, since ℓ m , we also have m 4. If a > b then ≤ ≥ 2 ≤ n b a a Since m 2 + 1, we have n 7. b t b t t b ≥ ≤ − dt = − dt = − dt This shows that, for any n > 7, if f (x) = 1 has not less than t2 − t2 t2 a b b n + 1 distinct integral roots then f (x) is| irreducible.| 2 and Finally, observe that k = 7. In fact, for n = 7, the function f (x) 1 (b a)2 a t b 1 (b a)2 deﬁned by − − dt − . (13) 2 b2 ≥ t2 ≥ 2 a2 h(x) = 1 + x(x 3)(x 2)(x 1) b − − − Therefore, by (12) and (13), we have for any a, b > 0 that g(x) = 1 + x(x 3)(x 1) − − f (x) = g(x)h(x) b b t 1 (b a)2 1 min a, b 2 − dt − = { } 1 is reducible, whereas f (x) = 1 when x = 0, 1, 2, 3. So k cannot be t2 ≥ 2 max2 a, b 2 max a, b − a made smaller. | | { } { } and

Also solved by Mihály Bencze (Brasov, Romania), F. Plastria b b t 1 (b a)2 1 max a, b 2 dt = . (BUTO-Vrije Universiteit Brussel) −2 2− { } 1 a t ≤ 2 min a, b 2 min a, b − { } { } 166. Let f : R R be monotonically increasing ( f not neces- By the representation (11), we then get the desired result (10). → sarily continuous). If f (0) > 0 and f (100) < 100, show that there exists x R such that f (x) = x. ∈ (Wing-Sum Cheung, The University of Hong Kong, Also solved by Panagiotis T. Krasopoulos (Athens, Greece), John Pokfulam, Hong Kong) N. Lillington (Wareham, UK), F. Plastria (BUTO-Vrije Universiteit Brussel)

EMS Newsletter March 2017 63 Problem Corner

168. Let f : I C be an n-time diﬀerentiable function on the and → interior I˚ of the interval I, and f (n), with n 1, be locally abso- ≥ b lutely continuous on I˚. Show that, for each distinct x, a, b I˚ and f (n+1) (t)(t x)n dt ∈ − for any λ R 0, 1 , we have the representation x ∈ \{ } 1 = (b x) f (n+1) ((1 s) x + sb) ((1 s) x + sb x)n ds f (x) = (1 λ) f (a) + λ f (b) − − − − − 0 n 1 1 n+1 (n+1) n + (1 λ) f (k) (a)(x a)k + ( 1)k λ f (k) (b)(b x)k = (b x) f ((1 s) x + sb) s ds. k! − − − − − 0 − k=1 + S n,λ (x, a, b) , (14) The identities (16) and (17) can then be written as

where the remainder S n,λ (x, a, b) is given by n 1 f (x) = f (k) (a)(x a)k k! S (x, a, b) k= − n,λ 0 1 1 1 1 + + := (1 λ)(x a)n+1 f (n+1) (1 s)a + sx (1 s)n ds + (x a)n 1 f (n 1) ((1 s) a + sx)(1 s)n ds (18) n! − − − − n! − 0 − − 0 1 + ( 1)n+1 λ (b x)n+1 f (n+1) (1 s)x + sb snds . (15) and − − 0 − n ( 1)k f (x) = − f (k) (b)(b x)k (Silvestru Sever Dragomir, Victoria University, k! k=0 − Melbourne City, Australia) (b x)n+1 1 + ( 1)n+1 − f (n+1) ((1 s) x + sb) snds. (19) − n! 0 − Solution by the proposer. Using Taylor’s representation with the integral remainder, we can write the following two identities: Now, if we multiply (18) by (1 λ) and (19) by λ and add the resulting equalities, a simple calculation− yields the desired identity (14) n 1 1 x f (x) = f (k) (a)(x a)k + f (n+1) (t)(x t)n dt (16) with the reminder from (15). k! n! k= − a − 0 and Also solved by Mihály Bencze (Brasov, Romania), Panagiotis T. Kra- n sopoulos (Athens, Greece), John N. Lillington (Wareham, UK) ( 1)k ( 1)n+1 b f (x) = − f (k) (b)(b x)k + − f (n+1) (t)(t x)n dt k! n! k= − x − 0 Remark 1. Note that Problems 155 and 159 were also solved by John (17) N. Lillington (Poundbury, Dorchester, UK) for any x, a, b I˚. ∈ For any integrable function h on an interval and any distinct Remark 2. K. P. Hart noted that the answer to problem 157 can numbers c, d in that interval, we have, by the change of variable be found in the article by Freudenthal and Hurewicz from 1936, t = (1 s) c + sd, s [0, 1] , that https://eudml.org/doc/212824. − ∈ d 1 h (t) dt = (d c) h ((1 s) c + sd) ds. − − c 0 Therefore, We wait to receive your solutions to the proposed problems and x ideas on the open problems. Send your solutions both by ordinary f (n+1) (t)(x t)n dt − mail to Michael Th. Rassias, Institute of Mathematics, University of a 1 Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland, and = (x a) f (n+1) ((1 s) a + sx)(x (1 s) a sx)n ds by email to [email protected]. − − − − − 0 We also solicit your new problems with their solutions for the next 1 = (x a)n+1 f (n+1) ((1 s) a + sx)(1 s)n ds “Solved and Unsolved Problems” column, which will be devoted to − 0 − − Discrete Mathematics .

64 EMS Newsletter March 2017 FROM FRENET TO CARTAN: THE METHOD OF MOVING FRAMES Jeanne N. Clelland, University of Colorado An introduction to the method of moving frames as developed by Cartan, at a level suitable for beginning graduate students familiar with the geometry of curves and surfaces in Euclidean space. The main focus is on the use of this method to compute local geometric invariants for curves and surfaces in various 3-dimensional homogeneous spaces, including Euclidean, Minkowski, equi-affine, and projective spaces. A special feature of this book is the inclusion of detailed guidance regarding the use of the computer algebra system Maple™ to perform many of the computations involved in the exercises. Graduate Studies in Mathematics, Vol. 178 Apr 2017 414pp 9781470429522 Hardback €85.00

GAME THEORY, ALIVE Anna R. Karlin, University of Washington & Yuval Peres, Microsoft Research Presents a rigorous introduction to the mathematics of game theory without losing sight of the joy of the subject. This is done by focusing on theoretical highlights and by presenting exciting connections of game theory to other fields such as computer science, economics, social choice, biology, and learning theory. Both classical topics, such as zero-sum games, and modern topics, such as sponsored search auctions, are covered. Along the way, beautiful mathematical tools used in game theory are introduced, including convexity, fixed-point theorems, and probabilistic arguments. Apr 2017 396pp 9781470419820 Hardback €87.00

IT’S ABOUT TIME Elementary Mathematical Aspects of Relativity Roger Cooke, University of Vermont This book has three main goals: to explore a selection of topics from the early period of the theory of relativity, focusing on particular aspects that are interesting or unusual; to provide an exposition of the differential geometry needed to understand these topics; and to reflect on the historical development of the subject and its significance for our understanding of what reality is. The book also takes note of historical prefigurations of relativity, such as Euler’s 1744 result that a particle moving on a surface and subject to no tangential acceleration will move along a geodesic, and the work of Lorentz and Poincaré on space-time coordinate transformations between two observers in motion at constant relative velocity. Apr 2017 410pp Hardback 9781470434830 €87.00

PUSHING LIMITS From West Point to Berkeley & Beyond Ted Hill, Georgia Tech Challenges the myth that mathematicians lead dull and ascetic lives. It recounts the unique odyssey of noted mathematician Ted Hill, who overcame military hurdles at West Point, Army Ranger School and the Vietnam War, and survived many civilian escapades. From ultra-conservative West Point in the ‘60s to ultra-radical Berkeley in the ‘70s, and ultimately to genteel Georgia Tech in the ‘80s, this is the tale of an academic career as noteworthy for its offbeat adventures as for its teaching and research accomplishments. It brings to life the struggles and risks underlying mathematical research, the unparalleled thrill of making scientific breakthroughs, and the joy of sharing those discoveries around the world. Hill’s book is packed with energy, humor, and suspense, both physical and intellectual. Apr 2017 334pp Hardback 9781470435844 €53.00

Free delivery worldwide at eurospanbookstore.com/ams AMS is distributed by Eurospan|group CUSTOMER SERVICES: FURTHER INFORMATION: Tel: +44 (0)1767 604972 Tel: +44 (0)20 7240 0856 Fax: +44 (0)1767 601640 Fax: +44 (0)20 7379 0609 Email: [email protected] Email: [email protected] New books published by the Individual members of the EMS, member S societies or societies with a reciprocity agree- E European ment (such as the American, Australian and M M Mathematical Canadian Mathematical Societies) are entitled to a discount of 20% on any book purchases, if E S Society ordered directly at the EMS Publishing House.

Walter Schachermayer (Universität Wien, Austria) Asymptotic Theory of Transaction Costs (Zürich Lectures in Advanced Mathematics) ISBN 978-3-03719-173-6. 2017. 160 pages. Hardcover. 17 x 24 cm. 34.00 Euro A classical topic in Mathematical Finance is the theory of portfolio optimization. Robert Merton’s work from the early seventies had enormous impact on academic research as well as on the paradigms guiding practitioners. One of the ramifications of this topic is the analysis of (small) proportional transaction costs, such as a Tobin tax. The lecture notes present some striking recent results of the asymptotic dependence of the relevant quantities when transaction costs tend to zero. An appealing feature of the consideration of transaction costs is that it allows for the first time to reconcile the no arbitrage paradigm with the use of non-semimartingale models, such as fractional Brownian motion. This leads to the culminating theorem of the present lectures which roughly reads as follows: for a fractional Brownian motion stock price model we always find a shadow price process for given transaction costs. This process is a semimartingale and can therefore be dealt with using the usual machinery of mathematical finance.

Hans Triebel (University of Jena, Germany) PDE Models for Chemotaxis and Hydrodynamics in Supercritical Function Spaces (EMS Series of Lectures in Mathematics) ISBN 978-3-03719-171-2. 2017. 138 pages. Hardcover. 17 x 24 cm. 32.00 Euro This book deals with PDE models for chemotaxis (the movement of biological cells or organisms in response of chemical gradients) and hydrodynamics (viscous, homogeneous, and incompressible fluid filling the entire space). The underlying Keller–Segel equations (chemotaxis), Navier–Stokes equations (hydrodynamics), and their numerous modifications and combinations are treated in the context of inhomogeneous spaces of Besov–Sobolev type paying special attention to mapping properties of related nonlinearities. Further models are considered, including (deterministic) Fokker–Planck equations and chemotaxis Navier–Stokes equations. These notes are addressed to graduate students and mathematicians having a working knowledge of basic elements of the theory of function spaces, especially of Besov-Sobolev type and interested in mathematical biology and physics.

O G R N A P O

Vincent Guedj and Ahmed Zeriahi (both Université Paul Sabatier, Toulouse, France) H M

A Degenerate Complex Monge–Ampère Equations (EMS Tracts in Mathematics, Vol. 26) D W A R ISBN 978-3-03719-167-5. 2017. 496 pages. Hardcover. 17 x 24 cm. 88.00 Euro Complex Monge–Ampère equations have been one of the most powerful tools in Kähler geometry since Aubin and Yau’s classical works, culminating in Yau’s solution to the Calabi conjecture. A notable application is the construction of Kähler-Einstein metrics on some compact Kähler manifolds. In recent years degenerate complex Monge–Ampère equations have been intensively studied, requiring more advanced tools. The main goal of this book is to give a self-contained presentation of the recent developments of pluripotential theory on compact Kähler manifolds and its application to Kähler–Einstein metrics on mildly singular varieties. After reviewing basic properties of plurisubharmonic functions, Bedford–Taylor’s local theory of complex Monge–Ampère measures is developed. In order to solve degenerate complex Monge–Ampère equations on compact Kähler manifolds, fine properties of quasi-plurisubharmonic functions are explored, classes of finite energies defined and various maximum principles established. After proving Yau’s celebrated theorem as well as its recent generalizations, the results are then used to solve the (singular) Calabi conjecture and to construct (singular) Kähler–Einstein metrics on some varieties with mild singularities. The book is accessible to advanced students and researchers of complex analysis and differential geometry.

Absolute Arithmetic and F1-Geometry Koen Thas (University of Gent, Belgium), Editor ISBN 978-3-03719-157-6. 2016. 404 pages. Hardcover. 17 x 24 cm. 68.00 Euro It has been known for some time that geometries over finite fields, their automorphism groups and certain counting formulae involving these geometries have interesting guises when one lets the size of the field go to 1. On the other hand, the nonexistent field with one element, F1, presents itself as a ghost candidate for an absolute basis in Algebraic Geometry to perform the Deninger–Manin program, which aims at solving the classical Riemann Hypothesis. This book, which is the first of its kind in the F1-world, covers several areas in F1-theory, and is divided into four main parts – Combi- natorial Theory, Homological Algebra, Algebraic Geometry and Absolute Arithmetic. Topics treated include the combinatorial theory and geometry behind F1, categorical foundations, the blend of different scheme theories over F1 which are presently available, motives and zeta functions, the Habiro topology, Witt vectors and total positivity, moduli operads, and at the end, even some arithmetic. Each chapter is carefully written by experts, and besides elaborating on known results, brand new results, open problems and conjectures are also met along the way. The diversity of the contents, together with the mystery surrounding the field with one element, should attract any mathematician, regardless of speciality.

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