N E WS L E T T ER
OF THE EUROPEAN MATHEMATICAL SOCIETY

FeatureTensor Product and Semi-Stability
Interviews
Peter Sarnak
Gigliola Staffilani
Obituary
Robert A. Minlos

Prague, venue of the EMS Council Meeting, 23–24 June 2018

S E
M
M
E S

EuropeanMathematicalSocietyJune 2018
Issue 108ISSN 1027-488X

S E
M
M
E S

New books published by the

Individual members of the EMS, member
societies or societies with a reciprocity agreement (such as the American, Australian andCanadian Mathematical Societies) are entitledto a discount of 20% on any book purchases, ifordered directly at the EMS Publishing House.

EuropeanMathematicalSocietyBogdan Nica (McGill University, Montreal, Canada)
A Brief Introduction to Spectral Graph Theory (EMS Textbooks in Mathematics)
ISBN 978-3-03719-188-0. 2018. 168 pages. Hardcover. 16.5 x 23.5 cm. 38.00 EuroSpectral graph theory starts by associating matrices to graphs – notably, the adjacency matrix and the Laplacian matrix. The
general theme is then, firstly, to compute or estimate the eigenvalues of such matrices, and secondly, to relate the eigenvalues
to structural properties of graphs. As it turns out, the spectral perspective is a powerful tool. Some of its loveliest applications
concern facts that are, in principle, purely graph theoretic or combinatorial.
This text is an introduction to spectral graph theory, but it could also be seen as an invitation to algebraic graph theory. The firsthalf is devoted to graphs, finite fields, and how they come together. This part provides an appealing motivation and context ofthe second, spectral, half. The text is enriched by many exercises and their solutions.
The target audience are students from the upper undergraduate level onwards. We assume only a familiarity with linear algebra
and basic group theory. Graph theory, finite fields, and character theory for abelian groups receive a concise overview and render
the text essentially self-contained.

Anne Thomas (The University of Sydney, Australia)
Geometric and Topological Aspects of Coxeter Groups and Buildings (Zürich Lectures in Advanced Mathematics)
ISBN 978-3-03719-189-7. 2018. 160 pages. Softcover. 17 x 24 cm. 34.00 Euro
Coxeter groups are groups generated by reflections, and they appear throughout mathematics. Tits developed the general theoryof Coxeter groups in order to develop the theory of buildings. Buildings have interrelated algebraic, combinatorial and geometricstructures, and are powerful tools for understanding the groups which act on them.
These notes focus on the geometry and topology of Coxeter groups and buildings, especially nonspherical cases. The emphasis
is on geometric intuition, and there are many examples and illustrations. Part I describes Coxeter groups and their geometricrealisations, particularly the Davis complex, and Part II gives a concise introduction to buildings.
This book will be suitable for mathematics graduate students and researchers in geometric group theory, as well as algebra andcombinatorics. The assumed background is basic group theory, including group actions, and basic algebraic topology, together
with some knowledge of Riemannian geometry.
Local Representation Theory and Simple Groups (EMS Series of Lectures in Mathematics)
Radha Kessar (City University of London, UK), Gunter Malle (Universität Kaiserslautern, Germany) and Donna Testerman(EPF Lausanne, Switzerland), Editors
ISBN 978-3-03719-185-9. 2018. 369 pages. Softcover. 17 x 24 cm. 44.00 EuroThe book contains extended versions of seven short lecture courses given during a semester programme on “Local Representation Theory and Simple Groups” held at the Centre Interfacultaire Bernoulli of the EPF Lausanne. These focussed on modular
representation theory of finite groups, modern Clifford theoretic methods, the representation theory of finite reductive groups,
as well as on various applications of character theory and representation theory, for example to base sizes and to random walks.
These lectures are intended to form a good starting point for graduate students and researchers who wish to familiarize themselves with the foundations of the topics covered here. Furthermore they give an introduction to current research directions,
including the state of some open problems in the field.

Lectures in Model Theory (Münster Lectures in Mathematics)
Franziska Jahnke (Universität Münster, Germany), Daniel Palacín (The Hebrew University of Jerusalem, Israel) and
Katrin Tent (Universität Münster, Germany), Editors
ISBN 978-3-03719-184-2. 2018. 222 pages. Softcover. 17 x 24 cm. 38.00 Euro
Model theory is a thriving branch of mathematical logic with strong connections to other fields of mathematics. Its versatility
has recently led to spectacular applications in areas ranging from diophantine geometry, algebraic number theory and grouptheory to combinatorics.
This volume presents lecture notes from a spring school in model theory which took place in Münster, Germany. The notes are
aimed at PhD students but should also be accessible to undergraduates with some basic knowledge in model theory. They contain the core of stability theory (Bays, Palacín), two chapters connecting generalized stability theory with group theory (Clausen
and Tent, Simon), as well as introductions to the model theory of valued fields (Hils, Jahnke) and motivic integration (Halupczok).

European Mathematical Society Publishing House
Seminar for Applied Mathematics, ETH-Zentrum SEW A21

Scheuchzerstrasse 70
CH-8092 Zürich, Switzerlandorders@ems-ph.org
www.ems-ph.org

Contents

Editorial Team
Gemma Huguet

Editor-in-Chief
Valentin Zagrebnov
Institut de Mathématiques deMarseille (UMR 7373) – CMI
Technopôle Château-Gombert
39, rue F. Joliot Curie
13453 Marseille Cedex 13,
Francee-mail: Valentin.Zagrebnov@univ-amu.fr

Copy Editor
Chris Nunn
119 St Michaels Road,
Aldershot, GU12 4JW, UK
e-mail: nunn2quick@gmail.com(Research Centres)

Departament de Matemàtiques
ETSEIB-UPC
Avda. Diagonal 647
08028 Barcelona, Spaine-mail: gemma.huguet@upc.edu

Vladimir R. Kostic(Social Media)

Department of Mathematics
and Informatics
University of Novi Sad
21000 Novi Sad, Serbia
Octavio Paniagua Taboada
FIZ Karlsruhe, Franklinstr. 1110587 Berlin, Germany

Jean-Paul Allouchee-mail: octavio@zentralblatt-math.org(Book Reviews)

IMJ-PRG, UPMC
4, Place Jussieu, Case 247
75252 Paris Cedex 05, France
e-mail: jean-paul.allouche@imj-prg.fr

Jean-Bernard Bru

Ulf Persson(Social Media)

Matematiska Vetenskaper
Chalmers tekniska högskola
S-412 96 Göteborg, Sweden
e-mail: ulfp@math.chalmers.se(Contacts with SMF)

Departamento de Matemáticas
Universidad del País Vasco
Apartado 644
48080 Bilbao, Spaine-mail: jb.bru@ikerbasque.org

Fernando Pestana da Costa(Societies)

Vladimir L. Popov(Features and Discussions)

Steklov Mathematical InstituteRussian Academy of Sciences
Gubkina 8119991 Moscow, Russia
e-mail: popovvl@mi.ras.ru

Depto de Ciências e Tecnologia, Secção de Matemática,
Universidade Aberta
Rua da Escola Politécnica, nº
141–147
1269-001 Lisboa, Portugal

Michael Th. Rassias

e-mail: fcosta@uab.pt

e-mail: michail.rassias@math.uzh.ch

Jean-Luc Dorier(Problem Corner)

Institute of Mathematics
University of Zurich
Winterthurerstrasse 1908057 Zürich
Volker R. Remmert(Math. Education)

FPSE – Université de Genève
Bd du pont d’Arve, 40
1211 Genève 4, Switzerlande-mail: Jean-Luc.Dorier@unige.ch(History of Mathematics)

IZWT, Wuppertal University
D-42119 Wuppertal, Germany
e-mail: remmert@uni-wuppertal.de

Vladimir Salnikov

Javier Fresán(Young Mathematicians’ Column)

Centre de Mathématiques
Laurent Schwartz
École Polytechnique
91128 Palaiseau, France

e-mail: javier.fresan@polytechnique.edu

Newsletter No. 108, June 2018e-mail: vladimir.slk@gmail.com(zbMATH Column)

Editors

EuropeanMathematicalSociety(Young Mathematicians’ Column)

La Rochelle University
LaSIE, Avenue Michel Crépeau
17042 La Rochelle Cedex 1,
Francee-mail: vladimir.salnikov@univ-lr.fr

Dierk Schleicher(Features and Discussions)

Research I
Jacobs University Bremen
Postfach 750 561
28725 Bremen, Germany
e-mail: dierk@jacobs-university.de

Scan the QR code to go to theNewsletter web page:
http://euro-math-soc.eu/newsletter

EMS Newsletter June 2018

EMS Agenda / EMS Scientific Events........................................... 2
Editorial – On the Road to MSC 2020 - A. Bannister et al............. 3
EMS Secretariat......................................................................... 4
New Editor Appointed................................................................. 5
Report from the Executive Committee Meeting - R. Elwes............. 5
Tensor Product and Semi-stability: Four Variations on a Theme M. Maculan ........................................................................... 8Interview with Peter Sarnak - M. Th. Rassias ............................... 16
Mathematics as a Positive Mental Place – An Interview with
Gigliola Staffilani - R. Natalini................................................... 19
Robert Adol’fovich Minlos (1931–2018) – His Work and Legacy C. Boldrighini et al. ................................................................. 22
Supporting Young Researcher Families in Switzerland M. Podkopaeva & O. Chekeres................................................ 28
The Floer Center of Geometry - A. Abbondandolo......................... 30
The Mathematical Society of Serbia - V. Mićić & Z. Kadelburg ..... 32
Is the System of Scientific Publications on the Eve of a Revolution?
And if so, Toward What? - M. Farge & F. Hélein........................ 35
Can Statistics Predict the Fields Medal Winners? - A. Bannister& Olaf Teschke....................................................................... 40
ICMI Column – The Mathematical Legacy of Jean-Pierre Kahane J. Barral, J. Peyrière & H. Queffélec......................................... 43
ERME Column - R. Chorlay & J. Cooper....................................... 47
Our Missing Teachers - J. Ewing ............................................... 49
News on the zbMATH Interface - O. Paniagua Taboada................ 54
Book Reviews............................................................................ 55
Personal Column........................................................................ 60
The views expressed in this Newsletter are those of theauthors and do not necessarily represent those of the
EMS or the Editorial Team.
ISSN 1027-488X© 2018 European Mathematical Society
Published by the
EMS Publishing House
ETH-Zentrum SEW A21
CH-8092 Zürich, Switzerland.
homepage: www.ems-ph.org
For advertisements and reprint permission requests
contact: newsletter@ems-ph.org

1

EMS Agenda

EMS Executive Committee
President
Prof. Pavel Exner(2015–2018)
Doppler InstituteCzech Technical UniversityBřehová 711519 Prague 1
Czech Republic
e-mail: ems@ujf.cas.cz

Vice-Presidents
Prof. Volker Mehrmann(2017–2018)
Institut für Mathematik
TU Berlin MA 4–5
Strasse des 17. Juni 136
10623 Berlin
Germany
e-mail: mehrmann@math.TU-Berlin.DE

Prof. Armen Sergeev(2017–2020)
Steklov Mathematical InstituteRussian Academy of Sciences
Gubkina str. 8
119991 Moscow
Russiae-mail: sergeev@mi.ras.ru

Secretary
Prof. Sjoerd Verduyn Lunel(2015–2018)
Department of Mathematics
Utrecht University
Budapestlaan 63584 CD Utrecht
The Netherlands
e-mail: s.m.verduynlunel@uu.nl

Treasurer
Prof. Mats Gyllenberg(2015–2018)
Department of Mathematics
and Statistics
University of Helsinki
P. O. Box 68
00014 University of Helsinki
Finlande-mail: mats.gyllenberg@helsinki.fi

Ordinary Members
Prof. Nicola Fusco(2017–2020)
Dip. di Matematica e Applicazioni
Complesso Universitario diMonte Sant’ Angelo
Via Cintia
80126 Napoli
ItalyProf. Stefan Jackowski(2017–2020)
Institute of Mathematics
University of Warsaw
Banacha 2
02-097 Warszawa
Polande-mail: sjack@mimuw.edu.pl

Prof. Vicente Muñoz(2017–2020)
Departamento de Geometria y
Topologia
Facultad de Matematicas
Universidad Complutense de
MadridPlaza de Ciencias 328040 MadridSpaine-mail: vicente.munoz@mat.ucm.es

Prof. Beatrice Pelloni(2017–2020)
School of Mathematical &Computer Sciences
Heriot-Watt University
Edinburgh EH14 4ASUKe-mail: b.pelloni@hw.ac.uk

Prof. Betül Tanbay(2017–2020)
Department of Mathematics
Bogazici University
Bebek 34342 Istanbul
Turkeye-mail: tanbay@boun.edu.tr

EMS Secretariat
Ms Elvira Hyvönen
Department of Mathematics
and Statistics
P. O. Box 68(Gustaf Hällströmin katu 2b)
00014 University of Helsinki
FinlandTel: (+358) 2941 51141
e-mail: ems-office@helsinki.fi
Web site: http://www.euro-math-soc.eu

EMS Agenda
2018
29 – 30 JulyIMU General Assembly, São Paulo, Brazil
9 – 11 November
EMS Executive Committee Meeting, Barcelona, SpainEMS Scientific Events
2018
2 – 6 July
The 12th International Vilnius Conference on Probability Theory
and Mathematical Statistics and2018 IMS Annual Meeting on Probability and Statistics
Vilnius, Lithuania
9 – 13 JulyEMS Summer School “4rd Barcelona Summer School on
Stochastic Analysis”
Barcelona, Spain
9 – 14 July
Activity Endorsed by the EMS:
Young African Scientists in Europe (YASE), Toulouse, France
9 – 14 July
4th EU/US Summer School on Automorphic Forms andRelated Topics
Budapest, Hungary
23 – 27 July
11th European Conference on Mathematical and Theoretical
Biology (ECMTB 2018), Lisbon, Portugal
1 – 9 August
ICM 2018
Rio Centro Convention Center, Rio de Janeiro, Brazil
19 – 26 August
Helsinki Summer School on Mathematical Ecology and
Evolution
Turku, Finland

EMS Publicity Officer
Dr. Richard H. Elwes
School of Mathematics
University of Leeds
Leeds, LS2 9JTUK
e-mail: R.H.Elwes@leeds.ac.uk

e-mail: n.fusco@unina.it

2

EMS Newsletter June 2018

Editorial

Editorial – On the Road to MSC 2020
Adam Bannister (FIZ Karlsruhe, Berlin, Germany), Fabian G. Müller (FIZ Karlsruhe, Berlin, Germany),
Mark-Christoph Müller (Heidelberg Institute for Theoretical Studies, Heidelberg, Germany) and
Olaf Teschke (FIZ Karlsruhe, Berlin, Germany)
Eighteen months ago, the beginning of the revision of theMathematical Subject Classification was announced.1
Since then, the mathematical community has alreadycontributed a number of suggestions on the public wiki
available at https://msc2020.org/. In this article, we will
give a brief overview of the current usage of the MSC,
analyse some data related to its effectiveness and precision, relate it to topic clusters generated by data mining
techniques and indicate some trends that have become
visible in the course of the revision.

Current usage of the Mathematical Subject
Classification (MSC)
While the raw scheme had already been introduced in theearly volumes of Mathematical Reviews, the current shape,
as a joint effort of both mathematical reviewing services,evolved in 1980. This came after an initiative of Bernd
Wegner to incorporate the system into Zentralblatt MATH
and to maintain regular collaborative revisions. Since
then, MSC has been primarily used by MathSciNet andzbMATH reviewers and editors to classify the mathematical research literature, as well as being adapted by classicaland digital libraries and journals. Several recent developments in the zbMATH database, such as author, journaland citation profiles or filter functions, have utilised thesubject information beyond its original raison d’être.
How reliable is the MSC?
MSCs are assigned to books and papers by authors,
reviewers and editors, with the final classification
approved by MathSciNet and zbMATH section editors.
Naturally, as a human enterprise, such assignments may
be subjective. Hence, it is natural to ask about the degreeof subjectivity that comes along with a classification performed by hand and whether it is possible to derive conclusions for the revision from the degree of vagueness. To
determine this, a comparison of MSC2010 assignments
has been made for 78,063 articles published between
2010 and 2016 in journals indexed cover-to-cover by both
MathSciNet and zbMATH. For this corpus, both services
coincided at the top level MSC for 62,951 documents,and even for 40,244 at the level of the overall MSC. Moreprecisely, the average F12 score for the coincidence of
MathSciNet and zbMATH classifications turned out tobe 0.83 at the top level, 0.72 at the second level and 0.58
at the third level of the first assigned MSC.3 The concord-

ance turned out to be significantly larger when permutations were taken into account; indeed, the largest differences by far occurred in the cross-subject MSC sections
like 00 (General), 97 (Education), 58 (Global analysis),
19 (K-Theory) and 37 (Dynamical systems). Interestingly,
three of them were introduced in the 1991 and 2000 MSC
revisions. Hence, while the MSC has overall becomeless tree-like, with more cross-references introduced in
the last revisions, it seems that a large proportion of thearticles still fit conveniently into the more classical hierarchical structure of the main subjects. Consequently, it
seems justified that there has been no introduction of a
new top-level MSC in 2010 and there also seems no needto do so in 2020.

The unreasonable effectiveness of the MSC
The relative reliability of the top-level MSCs can also
be derived from the cross-citation Figure 5 in Bannister and Teschke,4 which shows a strong concentration ofreferences to articles with the same MSC. In this sense,
the main subjects can also be seen as most natural clusters of the citation graph. Naturally, due to the interconnected nature of mathematics, this effect is less significant
for more granular MSC levels. However, the question
remains of whether there are automated ways to organise
mathematical literature into subjects. Apart from graphtheory approaches, the last decade has seen tremendous
progress in topic modelling by data mining and machinelearning techniques. An experiment performed by the
Heidelberg Institute of Theoretical Studies (HITS) created several clusters using the ToPMine tool.5 Human
evaluation showed that it performed reasonably well
for applied areas (producing, for example, a cluster containing Bayesian inference, posterior distribution and
the Gibbs sampler, roughly corresponding to 62F15) but
was quite limited for pure mathematics (e.g. it joined thenotions of pull back and container loading from category
theory and operations research and created the cluster“hyperplane arrangement, traffic jam, speed of light” ofhitherto unknown mathematical semantics). Some of theeffects may derive from the fact that publication numbers
are extremely unevenly distributed in mathematical areas
and automated methods tend to underperform for areas
with relatively small publication numbers, which are often,
however, very important within the mathematical corpus.
3

1

E. G. Dunne and K. Hulek, MSC2020 – announcement of theplan to revise the Mathematics Subject Classification. Eur.
Math. Soc. Newsl. 101, 55 (2016).
2 Weighted harmonic mean of the fractions of MSC codes in
one set that also occur in the respective other.

EMS Newsletter June 2018

MSC codes have three levels of increasing granularity, denoted by two digits, a letter and two more digits.
4 A. Bannister and O. Teschke, An Update on Time Lag in
Mathematical References, Preprint Relevance, and Subject
Specifics. pp. 41–43. Eur. Math. Soc. Newsl. 106, 41–43 (2017).
5 http://illimine.cs.uiuc.edu/software/topmine/.

3

Editorial
Developments toward MSC2020
Taking the mentioned limitations into account, quantitative methods such as those mentioned above can be used
to create suggestions for the MSC2020 revision. Phrases
that have occurred much more frequently since 2010
have often included developments in the applicationsof mathematics (which tend to be both more numerous
in publications and more fluid in topic denomination),
for instance “loop quantum gravity” and “PT symmetry”
in quantum theory, “scaling limits” arising both in stochastics and physics, “exponential stability” in control
theory, “quantum circuits” and “quantum games”, as well
as “sparse graphs”, “spatial graphs”, “circulant graphs”
and “phylogenetic trees” connected to the rise of network research, “copulae models” in statistics, “character
varieties” in algebraic geometry and topology and thecluster “Khovanov/Heegaard-Floer/HOMFLY homology” from topology, along with transcending techniques
like “matrix factorisation”. Copulae and character varieties have already been independently proposed in the
MSC2020 wiki, as well as many new developments notdetected by automated methods, such as “numerical algebraic geometry”, “higher categories”, “topological data
analysis” and “computer-assisted proofs”. On the other
hand, several recent concepts (like homotopy type theory) are still missing, so please engage in the joint effort
and contribute to the MSC2020 wiki at https://msc2020.
org/, which will remain open until August 2018!

Adam Bannister [adam.bannister@fizkarlsruhe.de] has a postgraduate diploma
in Geographic Information Systems andcurrently works on the Scalable Author Disambiguation for Bibliographic Databases
at zbMath in cooperation with Schloss Dagstuhl and Heidelberg Institute for Theoretical Studies.
FabianMüller[fabian.mueller@fizkarlsruhe.de] studied mathematics andcomputer science at Humboldt-Universität,
Berlin. After finishing his doctoral studies
in algebraic geometry in 2013, he started
working at zbMATH, where he is responsible for coordinating IT development efforts.
Dr. Mark-Christoph Müller [mark-christoph.mueller@h-its.org] is a research associate in natural language processing at the
Heidelberg Institute for Theoretical Studies
and currently works on the Scalable Author
Disambiguation for Bibliographic Da­tabases project, in cooperation with Schloss
Dagstuhl and FIZ Karlsruhe.
Olaf Teschke [olaf.teschke@fiz-karlsruhe.
de] was a member of the Editorial Boardof the EMS Newsletter from 2010–2017, responsible for the zbMATH Column.

EMS Secretariat
Dear EMS member,
We kindly ask you to check your personal data in the EMS membership database and update it if necessary:
http://euro-math-soc.eu/members.
If you have never logged in to the EMS membership database before just go to http://www.euro-math-soc.eu/
members/ and click the link “Create a new account”. Insert your EMS ID number and a password will be sent
to your email address that you submitted while registering for the EMS. After that you can login to the database(same address as above) with your EMS ID number and the password given in the email. The EMS ID number
can be found in the welcome letter sent to you by the EMS Office or on the address sticker of the EMS Newsletter
sent to you by the EMS Publishing House.
If you would like to use the extended services of our webpage, you need to register first and create a user accountat our webpage:
http://www.euro-math-soc.eu/user/register
The EMS website is not related to the members database.
Elvira Hyvönen, EMS Secretariat
E-Mail: ems-office@helsinki.fi, Phone: (+358) 2 941 51141

4

EMS Newsletter June 2018

EMS NewsNew Editor Appointed
Ulf Persson received his PhD
at Harvard in 1975 under David
Mumford. His dissertation wasentitled “Degenerations of Algebraic Surfaces”. Persson’s professional publications have been
almost exclusively in algebraic
geometry and especially on surfaces. He is inordinately proudof having introduced the notionof the ‘geography of surfaces’,
where the notion of ‘geography’ has caught on in othercontexts. Persson has been based in Sweden since 1979

but did many stints as a visitor to a variety of American
universities during the 1980s. In recent years his activities
have widened. He founded the Newsletter of the Swedish Mathematical Society during his presidentship andhas been its main editor for most of the time since then.
Ulf Persson has also been an editor of the EMS Newsletter (2010-2014). He is fond of conducting somewhat
idiosyncratic interviews with mathematicians, some ofthem appearing in this Newsletter but the more extreme
appearing in the Newsletter of the Swedish Mathematical Society. As is not unusual for people who are aging,he has picked up his youthful interest in philosophy andhas published a book and an article on Popper.

Report from the
Executive Committee Meeting in
Portoroz, 24–26 November 2017
Richard Elwes, EMS Publicity Officer
Last Autumn, the EMS’s Executive Committee (EC)plus guests gathered in the Grand Hotel Bernardin in
Portoroz on the stunning Slovenian Coast, on the kind
invitation of the Society of Mathematicians, Physicists, &Astronomers of Slovenia (DMFA) and the University of
Primorska. The choice of venue was significant: in July
2020 the same site will host the 8th European Congressof Mathematics (8ECM). The congress was therefore at
the forefront the committee’s mind throughout the meeting; in particular, it offered those who had not alreadydone so the opportunity to explore the area and facilities,
under the guidance of the congress’s local organisers.
On Friday afternoon, the committee was welcomedby Boštjan Kuzman, President of the DMFA’s National
Committee for Mathematics, who told us about his society’s history and activities, as well as some general history
of Slovene mathematics. The DMFA was founded in 1949
(when Slovenia was part of Socialist Federal Republic of
Yugoslavia). Its first honorary member was Josip Plemelj(1873–1967), the First Rector of University of Ljubljana,famous for his solution to the Riemann–Hilbert problem
concerning the existence of a differential equation witha given monodromy group. In 1992, after the break-up
of Yugoslavia, the DMFA became a member society of
the EMS. Today, the DMFA represents Slovenia in international settings, promotes research, organizes national
and international mathematics competitions (including

EMS Newsletter June 2018the 2006 International Mathematical Olympiad), publishes the journal Ars Mathematica Contemporanea
alongside other journals, textbooks, and problem compendia, organises conferences for researchers, seminars
for teachers, and assorted mathematical outreach events.
Tomaž Pisanski then introduced some other facets ofSlovene mathematical life, including the new Slovenian
Discrete and Applied Mathematics Society (SDAMS),
which was established in 2017 along with its journal The
Art of Discrete and Applied Mathematics whose firstedition was published in 2017.

Officers’ Reports and Membership
The meeting opened with some words of welcome fromthe chair, EMS President Pavel Exner, who related his
activities since the Spring meeting (many featuring in later items on the agenda). He closed with some reflections
on the political situation around the world, noting that
the EMS has a duty to raise its voice when mathematics
is under threat, and the unique organizational structureof the EMS makes our voice strong. At the same time,the EMS is not principally a political organization, andto maximise our impact, we must be selective in deciding
when to speak out.
The EMS Treasurer Mats Gyllenberg presented his
report on the society’s 2017 income and expenditure.
The financial situation is healthy; however less money

5

EMS Newswas spent on scientific projects than intended, something
to bear in mind when allocating future funds. The EMS
Secretary Sjoerd Verduyn Lunel then reported on preparations for the next EMS Council meeting (23–24 June2018 in Prague).
The committee was delighted to approve an application for Institutional Membership from the Mathematical
Institute, University of Oxford. Contrastingly, it regretted a notification of withdrawal from the French Statistical Society. Several other member societies remain
severely in arrears on their fees, and at the next Council
meeting it may sadly be necessary to propose that certain societies have their EMS membership terminated.
The committee was pleased to approve a list of 155 newindividual members.

Scientific Meetings
The committee heard an encouraging presentation from
Klavdija Kutnar, deputy chair of the local organisingcommittee of 8ECM, on plans for the 2020 congress.
The President thanked her and chair Tomaž Pisanski
for all the work their committee has already done, andwished them continuing success as the congress draws
nearer. The EC then began the important discussion of
the leadership and personnel of the congress’s Scientificand Prize Committees, based on candidates proposed bymember societies. The society is of course seeking top
mathematicians for these roles, but also individuals withbroad mathematical interests, who as part of their duties
will attend the ECM in person. Good progress was made,and the discussion will continue at the EC’s next meetingin Spring 2018.
The committee then discussed applications for support for scientific events in 2018, based on evaluations
from the Meetings committee. The EMS distinguishedspeakers for 2018 will be Benoit Perthame at the Joint
EMS-FMS-ESMTB Mathematical Weekend in Joensuu,
Finland (4–5 January 2018) and Andre Neves at the at
the 7th Iberoamerican Congress on Geometry in Valladolid, Spain (22–26 January 2018), which will also be
supported as an EMS Special Activity. The EC agreed tosupport the following eight Summer Schools:- Advanced Techniques in Mathematical Modellingof Tumour Growth, Centre de Recerca Matemàtica
(CRM), Barcelona, 3–6 April 20186

- New Results in Combinatorial & Discrete Geometry,
CRM, Barcelona, Spain, 7–11 May 2018
- EMS-EWM Summer School “Nonlocal interactions in
Partial Differential Equations and Geometry”, Institut
Mittag Leffler, Stockholm, 21–25 May 2018
- EMS-IAMP summer school in mathematical physics:
Universality in probability theory and statistical mechanics, Ischia, Italy, 11–15 June 2018
- Géométrie Algébrique en Liberté XXVI, Strasbourg,
France, 18–22 June 2018
- 16th School on Interactions between Dynamical Systems and Partial Differential Equations, CRM, Barcelona, 25–29 June 2018
- 4th Barcelona Summer School on Stochastic Analysis,
CRM, Barcelona, Spain, 9–13 July 2018
- “Building Bridges”, the 4th EU/US Summer School on
Automorphic Forms and Related Topics, Alfréd Rényi
Institute, Budapest, 9–14 July 2018
- Helsinki Summer School on Mathematical Ecology
and Evolution, Turku, Finland, 19–26 August 2018.
The committee additionally agreed to support the 12th
International Vilnius Conference on Probability Theory
and Mathematical Statistics, Lithuania 2–6 July 2018,as well as the Emil Artin International Conference,
Yerevan, Armenia, 27 May–2 June 2018. The President
recalled that the EMS-Bernoulli Society Joint Lecture2018 will be given at the European Conference on Mathematical and Theoretical Biology in Lisbon, 23–27 July
2018, by Samuel Kou.

Officers, Standing Committees, and Projects
The President reminded the committee that his term
and that of Vice-President Volker Mehrmann, along withthose of the Treasurer and Secretary, will end at the endof 2018. The next Council meeting in June 2018 will electa new President and Vice-President; while the current
Treasurer and Secretary declared that they would beavailable for re-election at the Council’s pleasure.
Several of the EMS’s standing committees also
required replenishment of their membership and leadership, and the Executive Committee was pleased tomake a number of appointments, including Stéphane
Cordier and Koby Rubenstein as Chair and Vice-Chair
respectively of the Applied Mathematics Committee,
Leif Abrahamsson and Sophie Dabo as Chair and ViceChair of the Committee for Developing Countries, Jiří
Rákosník and Dirk Werner as Chair and Vice-Chair of
the Ethics Committee, Zdenek Strakos as Vice-Chair ofthe Meetings Committee, and Silvia Benvenutti as ViceChair of the Committee for Raising Public Awareness ofMathematics.
The outgoing President of the Applied Mathematics Committee, José Antonio Carrillo, in attendance as aguest, delivered a short report on his committee’s activities including the exciting forthcoming Year of Mathematical Biology 2018. On behalf of the society, the President thanked him for his hard work over several years.
Mats Gyllenberg, liaison officer for the Committee for Developing Countries (CDC) delivered a short

EMS Newsletter June 2018

EMS News
report on the first round of applications for the EMSSimons for Africa programme. The EC also discussed the
problem of donations to the CDC decreasing over recent
years. This is due largely to EMS members increasingly
paying their dues through their local societies, rather
than through the EMS webpage, where donations to the
CDC are solicited.
The Executive Committee also discussed reports fromthe Committees on Education, Ethics, Meetings, European Solidarity, Publishing and Electronic Dissemination,
Raising Public Awareness, and Women in Mathematics.
Discussions followed on other projects the EMS
is involved with, including the online Encyclopediaof Mathematics (www.encyclopediaofmath.org), EUMATHS-IN (European Service Network of Mathematics
for Industry and Innovation), the Global Digital Mathematics Library, Zentralblatt MATH (www.zbmath.org).
The society’s own newsletter was also discussed, with
two new appointments made to the editorial team. The
EMS’s e-news, social media platforms, and other communications and publicity channels were also reviewed.

Funding, Political, and Scientific Organisations
The President gave an update of recent European
Research Council (ERC) developments, with 2017 being
its tenth anniversary year. The committee welcomed thenews that former EMS President Jean-Pierre Bourguignon’s term as President of the ERC has been extended.
The committee discussed recent developments
regarding Horizon 2020 (and its successor framework),
noting the importance of mathematicians registering as
possible evaluators of Marie Curie proposals (a separate
database from that of ERC evaluators).
Following the recent large increase in the membership fee for the Initiative for Science in Europe, after
extensive discussions the committee agreed to terminate
the EMS’s membership of that body. The committee
will investigate other pathways for political lobbying inEurope, given the importance of that task.
The President reported on the latest developments
at the International Mathematical Union (IMU), in particular that Carlos Kenig has been proposed as the next
IMU President for the period 2019–2022, and that Helge
Holden intends to stay for a second term as Secretary. At
the 2018 ICM in Rio de Janeiro, the general assembly will
decide on the site for ICM 2022. The remaining two bids
from Paris and Saint Petersburg, both being European,
have each been endorsed by the EMS. Volker Mehrmann(member of the board of the International Council for
Industrial and Applied Mathematics (ICIAM)) reported
on the latest developments there. The President Elect is
Ya-xiang Yuan, the next ICIAM conference will be in2019 in Valencia (Spain), with ICIAM 2023 to be hosted
in Tokyo. The EMS’s relationship with various prize committees and research centres was also discussed, including proposing several nominations.
Future Society Meetings and Closing Remarks
The Executive Committee’s next official meeting was
set for 23–25 Match 2018 in Rome, before which it was

EMS Newsletter June 2018agreed to meet for an informal retreat in Joensuu (Finland), 6–7 January 2018 to discuss the society’s longterm strategic goals. The annual meeting of Presidents
of EMS member societies will take place in Maynooth,Ireland, 14–15 April 2018 on the generous invitationof the Irish Mathematical Society. (In 2019 the corresponding meeting will take place at CIRM in Marseille,
France.)The meeting closed with the committee’s warm
thanks to the Society of Mathematicians, Physicists andAstronomers of Slovenia and to the University of Primorska, in particular to Klavdija Kutnar, Tomaž Pisanski,
and Boštjan Kuzman, both for their impeccable organisation of this meeting and also for their ongoing efforts
for the 2020 European Congress, to which the wholesociety is greatly looking forward. ed at the next Council.
The committee was pleased to approve a list of 154 newindividual members.

Call for the
Ferran Sunyer i
Balaguer Prize 2019
The prize will be awarded for a mathematical monograph of an expository
nature presenting the latest developments in an active area of research inmathematics. The monograph must be
original, unpublished and not subject
to any previous copyright agreement.
The prize consists of 15,000 Euros
and the winning monograph will be
published in Springer Basel’s Birkhäuser series “Progress in Mathematics”.
DEADLINE FOR SUBMISSION:
30 November 2018, at 1:00 pm
http://ffsb.iec.cat

Award of the 2018 Prize
The 2018 Prize was awarded toMichael Ruzhansky (Imperial College
London) and Durvudkhan Suragan(Nazarbayev University, Kazakhstan)for the monograph Hardy inequalities
on homogeneous groups (100 years ofHardy inequalities).

7

Feature

Tensor ProductProduct andand Semi-stability:
Semi-stability:
Four Variations
Variations on
onaaTheme
Theme
Marco
Maculan(Université(UniversitéPierre
PierreMarie
Curie,
Paris,
France)Marco Maculan
et et
Marie
Curie,
Paris,
France)Chevalley proved that in characteristic 0 the tensor productof semi-simple representations is semi-simple. This result has
analogues in rather diverse contexts: three of them are presented here in independent sections, focusing on the differences of the frameworks and the similarities of the proofs.
Algebraic groups will play a crucial role, sometimes in unexpected ways.
1

Representations

Let G be a group and k a field. In this note, a representationof G is a finite-dimensional k-vector space V together with agroup homomorphism ρ : G → GL(V).
Semi-simple representations
A representation is said to be:• irreducible if there are exactly two sub-vector spaces of V
stable under the action of G: the zero subspace 0 and thewhole vector space V. In particular the zero representation
is not considered to be irreducible.
• semi-simple if it can be decomposed into irreducible ones:
there are irreducible sub-representations V1 , . . . , Vn of V
such that V = V1 ⊕ · · · ⊕ Vn . This is equivalent to sayingthat for every G-stable subspace W of V there is a G-stablesupplement W  .
Theorem 1 (Chevalley [8], p. 88). Suppose char(k) = 0. The
tensor product of semi-simple representations V1 , V2 of G issemi-simple.
The proof of Chevalley’s theorem is a beautiful application of the theory of linear algebraic groups (that is, groups of
matrices defined by polynomial equations), even though thegroup G may not at all be of this form.
Indeed, in order to prove theorem 1, one may suppose thatthe field k is algebraically closed and look at GL(V1 ), GL(V2 )as algebraic groups.
Then one can suppose G to be itself a linear algebraic
group. For, it suffices to take the Zariski-closure Ḡ of the image of G in GL(V1 ) × GL(V2 ), namely the set of points
x ∈ GL(V1 ) × GL(V2 )such that f (x) = 0 for all polynomial functions f vanishingidentically on the image of G. The semi-simplicity of the representation V1 ⊗k V2 is equivalent for G and Ḡ, because it isa condition that can be expressed as the vanishing of somepolynomials.
Now the theory of linear algebraic groups applies: there
is an algebraic subgroup radu (G) of G called the unipotent
radical which is connected, unipotent (meaning that all the
eigenvalues of its elements are 1), normal and contains any

8other subgroup of G with these three properties. The unipotent
radical controls the semi-simplicity of the representations ofG:
Theorem 2 (Weyl [33]). Suppose char(k) = 0. An algebraic1
representation of a linear algebraic group G is semi-simple if
and only if every element of the unipotent radical acts as theidentity on V.
Applying the preceding fact, the proof of Chevalley’s theorem is easily achieved: since the representations V1 and V2
are supposed to be semi-simple, an element g of the unipotent
radical of G acts as the identity on V1 and V2 . Therefore goperates trivially on V1 ⊗k V2 too.
Reductive groups
Weyl’s theorem is usually formulated as follows: an algebraic representation of a reductive group, i.e. a linear algebraic group whose unipotent radical is trivial, is semi-simple.
Examples of reductive groups are GLn , SLn , SOn , Sp2n . In
general, a reductive group is the extension of a semi-simple
group by a product of copies of the multiplicative group k∗ ,the algebraic equivalent of C∗ . An example of a non-reductive
group is the additive group k 2 , which corresponds to the complex Lie group C.
Over the complex numbers, a reductive group G can beseen also as a complex Lie group. Therefore two topologies
cohabit in it: the one coming from the topology of C (the“usual topology”), and the Zariski one (which is coarser).
This is true for every algebraic group, but there is a topological property which is specific to reductive groups: they
contain subgroups which are dense for the Zariski topology
and compact for the usual one.
For instance, a Zariski-dense compact subgroup of GLn (C)is the group U(n) of unitary matrices; on the other hand, theunique compact subgroup of C is the trivial one (which is not
Zariski-dense).
For a semi-simple group G one uses É. Cartan’s classification of semi-simple Lie algebras to construct a Zariskidense compact subgroup K: indeed the Killing form on Lie G
is non-degenerate, and K corresponds to a maximal real Lie
sub-algebra on which the form is negative definite.
1 This means that the map G → GL(V) is polynomial in the coefficientsof the matrices of G.
2 The additive group k can be seen as a subgroup of GL (k) through the2embedding


1 a
a −→
.
0 1

EMS Newsletter June 2018

FeatureWeyl’s Unitarian Trick
Let us go back to the proof of theorem 2, in the second reformulation I gave above. Quite often in algebraic geometry thecharacteristic 0 hypothesis is the shade of transcendental techniques, that is, analysis on real or complex numbers. Theorem
2 is an example of this phenomenon.
Indeed, in order to prove it one reduces to the case where kis C: this is possible because the polynomial equations defining G and its representation G → GL(V) involve only finitely
many terms that are transcendental over Q.
Given a G-stable subspace W of V, finding a G-stable supplement amounts to construct a G-equivariant linear projection V → W.
Here comes a remarkable argument, refined gradually byHilbert, Hurwitz and Weyl, who at last named it Unitarian
Trick. Pick one linear projection π : V → W and average it ona Zariski-dense compact subgroup K of G:π̃(v) :=gπ(g−1 v) dµ(g), (v ∈ V),
K

where µ is the Haar measure of total mass 1 on K. The function π̃ is equivariant for the action of K, but by Zariski-densityof K, the projection π̃ is also G-equivariant.
A motto summarizing the previous proof might be: the subgroup K is small for the usual topology (which permits to integrate), but big enough to control the representations of G
(which depends only on the Zariski topology).

Characteristic p > 0
Over a field k of characteristic p > 0 (assumed algebraicallyclosed, for simplicity), both theorems 1 and 2 are false.
On the positive side, every representation V of an algebraic
group G is semi-simple in the following cases:• G is a finite group whose order is prime to p: given a Gstable subspace W ⊂ V and a projection π : V → W, thefunction1 
gϕ(g−1 v),
π̃(v) :=
#G g∈Gis G-equivariant;• G is the multiplicative group k∗ : in this case V decomposes
as

V=
Va ,
a∈Z

where Va is the subspace where t ∈ k∗ acts by ta .
Nagata [11, Théorème IV.3.3.6] shows that there are no otherpossibilities: given G an algebraic group such that every representation is semi-simple, the identity component G0 is isomorphic to Grm and G/G0 has order prime to p.
Concerning theorem 1 one has the following:
Example 3. Let k be a field characteristic p > 0 and V = k2 .
Consider the k-vector space V(d) of homogeneous polynomials of degree d on V: it is of dimension d + 1, a basis being
given by the monomials
xd , xd−1 y, . . . , xyd−1 , yd .

The group G = SL2 (k) acts on V(d) through the contragradient representation: explicitly, for a polynomial f (x, y) and an
invertible matrix


α β
g=
,
γ δ

EMS Newsletter June 2018with α, β, γ, δ ∈ k such that αδ − βγ = 1,
(g f )(x, y) = f g−1 (x, y) = f (δx − βy, −γx + αy).

The representation V(d) is irreducible for d = 1, . . . , p−1. For
d = p the map ϕ : V(1) → V(p), associating to a linear formf its p-th power f p , is linear, injective and G-equivariant. The
image of ϕ is the kernel of the map ψ : V(p) → V(p − 2) given
by
1 ∂f
f −→
.
y ∂x
The exact sequence of representationsϕψ

0 −→ V(1) −→ V(p) −→ V(p − 2) −→ 0,

obtained in this way does not split if the field k has at least 3
elements. In particular V(p) is not semi-simple. If d1 , . . . , dn
are integers ranging between 1 and p − 1 such that their sum
is p, the multiplication map f1 ⊗ · · · ⊗ fn → f1 · · · fn induces
a surjection W := V(d1 ) ⊗ · · · ⊗ V(dn ) → V(p). It follows that
W is not semi-simple.
However Serre shows that these problems do not occur assoon as the dimension is small enough:Theorem 4 (Serre [29]). Let W1 , . . . , Wm be semi-simple representations of a group G on a field of characteristic p > 0.
Ifm

dim Wi < p + m,i=1then W1 ⊗ · · · ⊗ Wm is semi-simple.

According to Example 3 the condition in the theorem above
is sharp. Various generalizations of the result of Serre can befound in [30, 10, 2].
2Vector bundles on Riemann surfaces

Representations of the fundamental group
Let X be a compact Riemann surface of genus g and X̃
its universal covering. The fundamental group π1 (X, x)(with respect to a base point x) is generated by 2g loops
a1 , . . . , ag , b1 , . . . , bg satisfying the relation−1−1 −1
a1 b1 a−1
1 b1 · · · ag bg ag bg = 1.

To a representation ρ : π1 (X, x) → GLr (C) of the fundamental group, one associates the vector bundle E(ρ) of rank r
on X obtained as the quotient of X̃ × Cr through the actiong( x̃, v) = (g x̃, ρ(g)v).

Theorem 5 (Narasimhan–Seshadri [25]). Suppose ρpreserves a hermitian norm on Cr . Then:
1. E(ρ) is a direct sum of simple bundles, and E(ρ) is simple
if and only ρ is irreducible;
2. if E(ρ) is simple, then it does not admit non-zero global
holomorphic sections;3. deg E(ρ) = 0 and E(ρ) is semi-stable: deg F ≤ 0 for every
sub-vector bundle F of E(ρ).
Moreover, every stable vector bundle of degree 0 arises froma unitary representation of the fundamental group of X.
In the previous statement, a vector bundle E is said to be:• simple if the only endomorphisms of E are homotheties;

9Feature• semi-stable (resp. stable) if for every sub-vector bundle Fof E different from 0 and E,
deg F deg E
≤
,
rk F
rk E

(resp. <).

E
The number µ(E) := deg
rk E is called the slope of E.
Remark that a simple bundle E cannot be written as the
direct sum of two proper sub-bundles.
Statements (1) and (2) are consequences of the following isomorphism, for unitary representations ρi : π1 (X, x) →GL(Vi ) and i = 1, 2,∼
Homπ1 (X,x) (V1 , V2 ) −→ Hom E(ρ1 ), E(ρ2 ) .

Statement (1) is obtained thanks to Weyl’s Unitarian Trick, bydecomposing the representation ρ into its irreducible components. In order to get a taste of how properties of the representation ρ transfer to those of the vector bundle E(ρ), let me
sketch the proof of (3).

The line bundle L := r E(ρ) is associated to the determinant representation det ρ : π1 (X, x) → C× of ρ, which is unitary. If the degree of L were positive, by the Riemann-Rochtheorem, some positive enough multiple L⊗d of L would admit
non-zero global sections: this would contradict (2), as L⊗d is
indecomposable. One concludes by applying the same reasoning to the determinant of the representation ρ∗ contragradientto ρ.
Take F to be a sub-vector bundle of E(ρ) and, arguing bycontradiction, assume that the degree of F is > 0. One can further suppose that F is a line bundle: if s denotes the rank of F,the s-th exterior power s F is a sub-line bundle of the vecs
E(ρ), which is associated to the representation
tor bundlesρ.
As before, by the Riemann–Roch theorem, there is a positive integer d ≥ 1 such that L⊗d has non-zero sections. Decompose the unitary representation ρ⊗d into irreducible ones
ρ1 , . . . , ρn . This corresponds to writing E(ρ)⊗d as the sum of
the simple vector bundles E(ρi ). The projection of L ontoE(ρi ) has to be zero for all i = 1, . . . , n: otherwise E(ρi ) would
have non-zero sections, which is impossible according to (2).
This implies that F is the trivial bundle, contradicting the hypothesis of having positive degree.
The results of Narasimhan–Seshadri have been for quite along time the only tool to prove the following result:
Theorem 6. Let E, F be semi-stable vector bundles on X.
Then E ⊗ F is semi-stable.
The key situation is when E and F are stable of degree
0: if this is the case, thanks to the theorem of NarasimhanSeshadri, E and F are associated to unitary representations
ρE , ρF of the fundamental group. The tensor product representation ρE ⊗ ρF is unitary, thus E ⊗ F is semi-stable.
Characteristic p > 0
On a field of positive characteristic, compact Riemann surfaces are replaced by smooth projective curves.
To fix ideas, let f ∈ F p [x0 , x1 , x2 ] be an irreducible homogeneous polynomial and consider the locus in P2 (F̄ p ) where it
vanishes:
X = [x0 : x1 : x2 ] ∈ P2 (F̄ p ) : f (x0 , x1 , x2 ) = 0 .

10Suppose that X does not have singular points: this means that∂fdoes notfor every point x ∈ X, some partial derivative ∂xivanish at x.
The genus g of X can be computed as for Riemann surfaces:
if f is of degree d, then(d − 1)(d − 2).
2
However X carries something that a Riemann surface does
not, the Frobenius endomorphism. It is the map sending apoint [x0 : x1 : x2 ] ∈ X to [x0p : x1p : x2p ]. Note that thelatter point still belongs to X becauseg=f (x0p , x1p , x2p ) = f (x0 , x1 , x2 ) p = 0,(one uses here that f has coefficients in F p ).
There is no obvious way to port unitary representations of
the fundamental group into this context. Rather, one takes thepoint of view of semi-stable vector bundles, whose definition
can be translated word by word. Yet the statement analogous
to theorem 6 is false and, as one can guess, examples comefrom symmetric powers:
Example 7. Let E be a vector bundle on X and consider its
i-th symmetric power Symi E.
The naive idea of embedding E into Sym p E by raising
sections to the p-th power does not work this time: the map
E → Sym p E, (x, s) → (x, s p ), where x is a point of X and s isa section of E over x, is not a bundle map. Nonetheless it induces an inclusion of the pull-back Fr∗X E of E (the so-called
Frobenius twist) as a sub-vector bundle of Sym p E.
If the genus of X is at least 2, there are plenty of semistable vector bundles on X such that their Frobenius twist is
not semi-stable3 : for such a vector bundle E, the above discussion shows that Sym p E is not semi-stable.
There is also an analogue of Serre’s theorem for semistable vector bundles (Balaji–Parameswaran [3], Ilangovan–
Mehta–Parameswaran [18]):
Theorem 8. Let E1 , . . . , En be semi-stable vector bundles on
X. If
n

rk Ei < p + n,i=1then E1 ⊗ · · · ⊗ En is semi-stable.

Back to characteristic 0
In view of the preceding example, an algebraic proof of theorem 6 has to break down in positive characteristic. Let me
detail the one discovered by Ramanan–Ramanathan [26]: itis particularly interesting from our point of view because it
makes intervene algebraic groups (where a priori there arenone).
Let V1 and V2 be semi-stable vector bundles on a compact
Riemann surface X, and let W be a sub-vector bundle of the
3 For instance, consider the push-forward V := (Fr ) O of the trivial
X ∗ X
bundle OX . A simple computation showsdeg V = (p − 1)(g − 1) > 0.

By the Harder–Narasimhan filtration, V contains a semi-stable vector bundle
E of positive degree. The kernel K of the induced map Fr∗X E → OX has
degree ≥ deg Fr∗X E. Having smaller rank, the slope of K is bigger than theslope of Fr∗X E.

EMS Newsletter June 2018

Feature
is easy to obtain when W is “special” (for instance if W is of
the form W1 ⊗ W2 for sub-vector bundles Wi ⊂ Vi ) or if it is“generic” (meaning that its intersection with some filtration
has small dimension).
This heuristics is made precise by Ramanan–Ramanathan:replacing the locutions “special” and “generic” respectively
by unstable and semi-stable in the sense of Geometric Invariant Theory (GIT) permits to derive (1) in both cases. The algebraic group that enters into play is GLr1 ,K × GLr2 ,K where
K is the field of meromorphic functions on X and ri the rankof Vi .
What goes wrong in characteristic p is the characterizationof semi-stability they use (the Hilbert–Mumford criterion [17,24], in the version of Kempf [20] and Rousseau [27]) thatneeds the field K to be perfect.
If X is a compact Riemann surface, then K is perfect because of characteristic 0. If X is a smooth projective curve
over a field k of characteristic p, K is a finite extension of the
field k(t) of rational functions in one variable, thus imperfect.
3

Weakly admissible isocrystals

Counting points with eigenvalues
Let f (x1 , . . . , xn ) be a polynomial with integral coefficients.
The set of its solutions modulo p,V(F p ) := x = (x1 , . . . , xn ) ∈ Fnp : f (x) = 0 ,is finite. What can we say about its cardinality?
There is a fruitful way to look at solutions modulo p. Consider the affine variety defined by f ,V := x ∈ F̄np : f (x) = 0 .

As an element a ∈ F̄ p belongs to F p if and only if a p = a,a point (x1 , . . . , xn ) ∈ V lies in V(F p ) if and only if xip = xi
for all i. In other words, V(F p ) is the set of fixed points of the
Frobenius endomorphism of V (i.e. the map raising to the p-th
power the coordinates of the points of X).
A brilliant idea of Weil was to relate this point of view withthe following form of the “Lefschetz fixed point theorem”,
known as the Lefschetz–Hopf theorem:
Theorem 9. Let X be a compact manifold and F : X → X acontinuous map. If the set Fix(F) of fixed points of F is finite,
then




i(x, F) =(−1)i Tr F∗ | Hi (X, Q) ,x∈Fix(F)i≥0where i(x, F) is the index of F at a fixed point x and Hi (X, Q)
the i-th rational homology group.

Weil conjectured that if there were a cohomology theory
that behaved well enough (called in this day and age “Weilcohomology theory”) one could compute the number of rational points by means of an analogue of the Lefschetz fixed
point theorem.
Constructing such a cohomology theory has been one ofthe driving forces behind the work of Grothendieck and his
school. The results in SGA 4 and 5 say that there is actually
one for each prime   p: the étale -adic cohomology, withcoefficients in the field Q of -adic numbers. Grothendieck inparticular was able to prove an analogue of Lefschetz’s fixed
point theorem:

EMS Newsletter June 2018

Theorem 10 (Grothendieck’s trace formula [16]). Let X be a
projective4 variety over F p . Then,#X(F p ) =(−1)i Tr FrX | Hiét (X, Q ) ,i≥0

where X(F p ) is the set of points of X having coordinates in F p .
Let us go back to counting the points of V(F p ). The preceding theorem suggests to pass to the projective closure of
V: consider the homogeneous polynomial associated to f ,f˜(x0 , . . . , xn ) := xd f (x1 /x0 , . . . , xn /x0 ),0where d is the degree of f , and
X := x = [x0 : · · · : xn ] ∈ Pn (F̄ p ) : f˜(x) = 0 .

With this at hand, the philosophy can be restated as follows:in order to compute the number of rational points of X (i.e.
those with coordinates in F p ), one has to estimate the sizeof the eigenvalues of the Frobenius acting on the -adic étale
cohomology.
If X is non-singular, the following facts are consequencesof Deligne’s proof of the last of Weil’s conjectures (called
Riemann hypothesis):
1. the set of eigenvalues of FrX on Hiét (X, Q ) does not dependon ;
2. an eigenvalue of FrX on Hiét (X, Q ) is an algebraic integer,
which isnot divisible by  and whose complex absolutevalue is pi (with respect to any complex embedding).

Divisibility properties
A question left aside is the order of divisibility by p of theeigenvalues of the Frobenius, or in other words, their p-adic
valuation. A first result of this kind is the following:
Theorem 11 (Chevalley-Warning [28]). Suppose deg f < n.
Then #V(F p ) is divisible by p.

In order to study such a question it seems appropriate to ask
the cohomology theory to produce Q p -vector spaces. Unfortunately p-adic étale cohomology does not work as one would
like and one has to consider crystalline cohomology.
From now on suppose X non-singular and that f is not divisible by p. The polynomial f we started with, as well as its
homogeneization f˜, have integral coefficients. Consider the
projective variety, defined over Q,
Y := x = [x0 : · · · : xn ] ∈ Pn (Q̄) : f˜(x) = 0 .
As a consequence of Berthelot’s results on crystalline cohomology, the algebraic de Rham cohomology groups of Y, or
better said their extension to Q p ,HqdR (X/Q p ) := HqdR (Y/Q) ⊗ Q p ,come equipped with a Frobenius operator FrX .
A classical conjecture of Katz proved by Mazur relates thep-adic absolute values of the eigenvalues of FrX to the Hodge
numbers of Y,
hi,q−i := dimQ Hq−i (Y, ΩiY ),where ΩiY denotes the bundle of differential i-forms on Y. Order the eigenvalues αi of FrX so that ord p (αi ) ≤ ord p (αi+1 ),and set βi = h0,q + h1,q−1 + · · · + hi,q−i .

4 The results holds more generally for any algebraic variety assuming thatthe cohomology is taken with compact supports.

11

Feature
Theorem 12 (Katz’s conjecture [19], Mazur [22, 23]). With
the notations introduced above,
ord p (α1 ) + · · · + ord p (αt )≥ 0 · h0,q + 1 · h1,q−1 + · · · + ihi,q−i + (i + 1)(t − βi ),(2)

0 1 2 3 4 5 6

0 1 2 3 4 5 6

0 1 2 3 4 5 6

rk p (A) = 3

rk p (A) = 2

rk p (A) = 1

where βi < t < βi+1 .

There is a more geometric way to state theorem 12. Consider the Newton polygon5 of the characteristic polynomial
det(id − FrX ·t) and the polygon associated to the Hodge filtration of HqdR (X/Q p ):F i HqdR (X/Q p ) :=

q

j=i

Hq− j (Y, ΩYj ) ⊗ Q p .

slopeon theinterval

Hodge polygonord(αi )i−1

j=1 m j ,i

j=1 m j

i


i−1

j=0 h

j,q− j

,

i

j=0 h

j,q− jwhere mi is the multiplicity of the eigenvalue αi . Theorem 12
becomes “the Newton polygon lies above the Hodge polygon”.
Example 13. Suppose X is a smooth projective curve of
genus g. Let me collect here some information on the Newton
polygon of X:• By Serre duality, h0,1 = h1,0 = g.
• Poincaré duality implies that the Newton polygon ends at(2g, g), as the Hodge polygon;• Let A = Jac(X) be the jacobian variety of X. For a positive
integer n denote by A[n] the subgroup of n-torsion pointsof A(F̄ p ). There is an integer 0 ≤ rk p (A) ≤ g, the p-rank ofA, such thatnrk p (A) if n is a power of p,#A[n] = n2g
if n is prime to p.

The previous information describe completely the Newton
polygon of X for g = 1, 2. However, starting from g = 3
the situation is more involved (see Figure 1).

Filtered isocrystals
A strengthening of Mazur’s theorem, conjectured by Fontaine[15] and proved by Faltings [12], states that the inequality (2)holds for every sub-vector space W ⊂ HqdR (X/Q p ) stable under the action of the Frobenius endomorphism:
i dimQ p (W i /W i+1 ) ≤ ord p det(FrX | W),iwhere W i := F i HqdR (X/Q p ) ∩ W. Moreover, the fact that both
polygons have the same end-points says that the previous inequality is an identity for W = HqdR (X/Q p ).
A filtered isocrystal over Q p is a linear algebraic datum
miming the above situation: it is a triple (V, ϕ, F • V) made of
5 Given a polynomial

f (t) = a0 + a1 t + · · · + ad td with coefficients in
Q p , its Newton polygon is the convex hull of the points (i, ord p (ai )), i.e. thesmallest convex function on the interval [0, d] such that the points (i, ord p (ai ))lie above (or on) its graph.

12

0 1 2 3 4 5 6

0 1 2 3 4 5 6

rk p (A) = 0
A not super-singularrk p (A) = 0
A super-singular

0 1 2 3 4 5 6

Figure 1: Newton polygons for a curve of genus 3

Concretely, these polygons are piecewise linear functions
starting at (0, 0) and with slopes
Newton polygon

Newton polygon
Hodge polygon

a finite-dimensional Q p -vector space, a linear map ϕ : V → V
and a decreasing filtration F • V = (F i V)i∈Z . It is said to be
weakly admissible if, for every linear subspace W ⊂ V stable
under ϕ,i dimQ p (W i /W i+1 ) ≤ ord p det(ϕ | W),iiwhere W := F i V ∩ W, with equality for W = V.6
Weakly admissible filtered isocrystals are meant to be oneof the possible p-adic analogues of Hodge structures. Becauseof the Künneth formula for cohomology, it is natural to ask
whether the tensor product of weakly admissible isocrystals is
weakly admissible. A partial affirmative answer was given by
Lafaille [21], while the general result was proven by Faltings[13, 14] and Totaro [32].
The role of semi-stable vector bundles
In order to explain the approaches of Faltings and Totaro, letme consider a slightly different situation.
Let V be a finite dimensional Q p -vector space, K a finite
extension of Q p and F • V = (F i V) a filtration of the K-vector
space V ⊗Q p K by K-vector spaces. For a Q p -linear subspace
W ⊂ V definedeg W =i dimQ p (W i /W i+1 ),
(3)
i

i

i

where W := F V ∩ (W ⊗Q p K).
The filtered vector space (V, F • V) is said to be semi-stableif, for all non-zero Q p -linear subspace W ⊂ V,
deg V
deg W
≤ µ(V) :=.
µ(W) :=
dim W
dim V

Example 14. Suppose V = Q2p . A point x ∈ P1 (Q̄ p ) corresponds to a vector line L ⊂ V ⊗Q p K for a suitable finite
extension K of Q p . The filtration
F 2 V := 0⊂

F 1 V := L⊂

F 0 V := V ⊗Q p K

satisfies µ(V) = 32 and µ(W) = 2 − dimK L ∩ (W ⊗Q p K) for
every Q p -vector line W ⊂ V. In particular (V, F • V) is semistable if and only if the line L is not defined over Q p .
The set of semi-stable filtrations on V is Drinfeld’s upper half-plane Ω1Q p := P1 (Q̄ p )  P1 (Q p ). It owes its name to
Poincaré’s upper-half plane h = {z ∈ C : Im z > 0} which canbe seen as the “upper-half” of P1 (C)  P1 (R).
6 Actually this definition is too restrictive, as one has to let the filtration
F • V to be defined only on a finite extension of Q p .

EMS Newsletter June 2018

Feature
To have a picture in mind, Drinfeld’s upper half-plane (orbetter the Berkovich analytic space attached to it) retracts onto
the Bruhat-Tits building of SL2 (Q p ); see Figure 2.

Diophantine approximation
Before diving into the analogy with vector bundles, let me explain how these concepts arise in Diophantine approximation.
Theorem 15 (Thue [31]). Let α ∈ R be an algebraic numberof degree d ≥ 2. Given ε > 0 there are only finitely many
rational numbers p/q ∈ Q such thatα − p  < 1 ,q  q d2 +1+ε
where (p, q) = 1 and q > 0.

Figure 2: The Bruhat-Tits building of SL2 (Q p ) for p = 2

Suppose that K is a Galois extension of Q p and consider
the filtrations F1• V, . . . , Fn• V conjugated to F • V under the action of the Galois group Gal(K/Q p ). Faltings’ idea is to associate to these filtrations a vector bundle E(V) on a compact
Riemann surface Y such that E(V) is semi-stable (as a vector bundle) if and only if the filtered vector space (V, F • V)
is. Since this construction is compatible with tensor product,
one draws the result thanks to the theorem of Narasimhan–
Seshadri.
The proof of Totaro is again based on the tensor product ofsemi-stable bundles, but instead of relying on the “analytic”
result of Narasimhan-Seshadri, it follows the “algebraic” approach of Ramanan–Ramanathan.
Totaro characterizes semi-stability of filtrations in terms ofGeometric Invariant Theory. It is just a matter of unwinding the definitions: the reader familiar with GIT recognizes
the quantity appearing in (3) as Mumford’s µ-coefficient in
the case of filtrations [24, §4.4]. Then he uses Ramanan–
Ramanathan’s dichotomy to derive the wanted inequality.
4

Hermitian vector bundles on arithmetic curves

Euclidean lattices
An Euclidean lattice of rank n is a finitely generated subgroupΓ of Rn such that Rn /Γ is compact. On Rn /Γ there is a uniquemeasure µΓ such that, for all continuous functions f : Rn → R
with compact support,f˜(ξ) dµΓ (ξ),
f (x) dλ(x) =
Rn

Rn /Γ
n


where λ is Lebesgue measure on R and f˜(ξ) = Γ+x=ξ f (x).
The volume of Rn /Γ with respect to this measure is called
the covolume of Γ. If γ1 , . . . , γn is a basis of Γ, the covolume
covol(Γ) can be computed as | det(γ1 , . . . , γn )|.
There is a more intrinsic definition of a Euclidean lattice: it
is a couple Ē = (E,  · E ) made of a free abelian group E of
finite rank together with a hermitian norm  · E on E ⊗Z R.
Following conventions that have become usual in Arakelov
geometry, define:the rank of Ē
rk(Ē) := dimR E ⊗Z R,
the degree of Ē
the slope of Ē Ē) = − log covol(E),
deg(
 Ē.
µ̂(Ē) = (rk Ē)−1 deg

EMS Newsletter June 2018

The exponent d2 + 1 has been successively sharpened by√ √Siegel, Dyson and Roth respectively to 2 d, 2d and 2, thelatter being optimal because of Dirichlet’s theorem on approximation of real numbers. However all these proofs follow thescheme of Thue’s argument, which can be roughly split infour steps:
1. Construct a polynomial f in n variables with small integral
coefficients vanishing “as much as possible” on the point(α, . . . , α).
2. Prove that the polynomial f does not vanish too much on
n-tuples x = (x1 , . . . , xn ) made of rational approximations.
Or better said some derivative g of f does not vanish at x.
3. Bound from above |g(x)| in terms of the order of vanishingof g at (α, . . . , α) by looking at its Taylor expansion.
4. Bound from below |g(x)| by using that a positive integer is≥ 1.
When there are too many good approximations of α to exist,
bounds in 3 and 4 are in contradiction.
In his Bourbaki report on the work of Masser and Wüstholzon periods and isogenies of abelian varieties (having amongits consequences Mordell’s Conjecture, by the time already atheorem of Faltings), Bost reinterprets in terms of Euclidean
lattices the preceding steps.
The point is to use a simple fact on the behaviour of slopes
with respect to linear maps:
Lemma 16 (Slopes inequality). Let Ē, F̄ be Euclidean lattices and ϕ : E → F an injective map of abelian groups. Then,µ̂(Ē) ≤ µ̂max (F̄) + log ϕsup ,where µ̂max (F̄) is the maximum of the slopes of sub-lattices ofF̄ and ϕsup is the operator norm of ϕ.

Thue’s four steps argument is translated as follows. Step 2,
which is the one with geometric content, corresponds to theinjectivity7 of the map ϕ. Step 3 corresponds to the upperbound of ϕsup and step 4 to bounding the slopes of Ē and F̄.
A step that seems gone missing is the first one, which isindeed a little different. Instead of picking a particular polynomial, the slopes inequality allows to consider the whole
space of polynomials with the wanted vanishing property. In
the concrete case of Masser and Wüstholz this permitted Bost
to replace fine considerations on theta functions by geometricarguments of Moret–Bailly.
Semi-stable lattices
A Euclidean lattice Ē is said to be semi-stable if µ̂(F̄) ≤ µ̂(Ē)
for all non-zero sub-lattices F ⊂ E with the induced Euclidean norm.
7 In this situation ϕ is the evaluation map of a polynomial at α (or better,some truncated Taylor expansion of f around α).

13Feature
Question (Bost). Is the tensor product8 of semi-stable lattices
semi-stable?
Results towards a positive answer to this question havebeen proved by André [1], Bost, de Shalit–Parzanovski, Chen[7] and Bost–Künneman [6]. The best available result is thefollowing:
Theorem 17 (Bost-Chen [5]). Let Ē, F̄ be semi-stable Euclidean lattices. Then,1µ̂max (Ē ⊗ F̄) ≤ µ̂(Ē) + µ̂(F̄) + min (rk E), (rk F) ,2

where, for an integer n ≥ 2, (n) = ni=2 1i ≤ log n.
Furthermore, if rk E · rk F ≤ 9, then Ē ⊗ F̄ is semi-stable.
Analogy with the projective line?
The field of rational numbers ressembles somehow to the fieldof rationals functions in one variable. This analogy works bymaking correspond points of P1 to equivalence classes of absolute values on Q. Stressing this point of view, the avatar
over Q of vector bundles are Euclidean lattices.
Nonetheless the theory of vector bundles on P1 is rather
poor: a vector bundle on P1 is a direct sum of line bundles9 ,and semi-stable vector bundles are all of the form O(d)⊕n .
This is not at all the situation for Euclidean lattices: theset of isomorphism classes of Euclidean lattices of rank n is
the double quotient GLn (Z)\ GLn (R)/On (R), where On (R) is
the group of orthogonal matrices of size n. Moreover, if one
associates to a point τ in the upper half plane the lattice generated by 1 and τ, then the region of semi-stable lattices ofrank two is the closed exterior of the Ford circles in the strip
0 < Im τ ≤ 1 [1].

inheriting the standard scalar product of R3 . Consider the lattice E = H ∩ Z3 . Identifying H with the standard Euclidean
plane R2 , E corresponds to the vertices of a planar tessellation
by equilateral triangles.
The group of isometries of E is the group of permutations
on 3 elements, which acts irreducibly on H.
Other examples of irreducible Euclidean lattices comefrom reductive groups! Indeed, over an algebraically closed
field, reductive groups are classified by a combinatorial
datum called the root system. Simple Lie algebras correspondto irreducible root systems, which give rise to irreducible
Euclidean lattices: the example above is the root system of
SL3 .
Theorem 19 (Bost).
semi-stable.

1. An irreducible Euclidean lattice is

2. The tensor product of irreducible Euclidean lattices isirreducible.
The first statement follows from the existence of theHarder–Narasimhan filtration: there is a sub-lattice Ēmax
realizing the biggest slope and which is maximal for thisproperty. By maximality, it is unique hence stable under theaction of Aut(Ē). Therefore, if Ē is irreducible, Ē = Ēmax .
The second is a straightforward consequence of the following classical result:
Theorem 20. Let k be a field of characteristic 0 and Vi anirreducible representation of a group Gi over k (i = 1, 2).
Then V1 ⊗k V2 is an irreducible representation of G1 × G2 .
Theorem 20 follows from the formula
EndG1 ×G2 (V1 ⊗k V2 ) = EndG1 (V1 ) ⊗ EndG2 (V2 )Figure 3: The exterior of Ford circles

Irreducible Euclidean lattices
Given a Euclidean lattice Ē, denote by Aut(Ē) the group ofelements g ∈ GL(E) respecting the Euclidean norm: it is a
finite group. A Euclidean lattice Ē is said to be irreducible if
the representation Aut(Ē) → GL(E ⊗Z Q) is.
Example 18. Consider the real plane
H = (x1 , x2 , x3 ) ∈ R3 : x1 + x2 + x3 = 0 ,8 The Euclidean norm on the tensor product is defined as follows: given

Euclidean spaces V, W with scalar products ·, ·V and ·, ·W , the tensor product V ⊗R W is given the structure of Euclidean space by defining
v ⊗ w, v ⊗ w V⊗W = v, v V · w, w W .

9 In the modern literature, this theorem is sometimes referred to as theGrothendieck–Birkhoff decomposition. However, it is easily deduced fromthe Elementary Divisors theorem for the ring k[t, t−1 ], due to Dedekind–Weber [9], or its complex analytic analogue discovered by Birkhoff [4].

14

(4)

and Schur’s Lemma: if k is algebraically closed and V is anirreducible representation of a group G, then EndG (V) = k.
Formula (4) is a special case of Jacobson’s density theorem.
Another proof, similar to that of Chevalley’s theorem, goesas follows. One may assume that G1 and G2 are reductive
groups and the ground field is C. For i = 1, 2 let Ki be a
Zariski-dense compact subgroup of Gi . Write a G-equivariant
endomorphism ϕ of V1 ⊗k V2 as
ϕ=

N

λ=1

α1λ ⊗ α2λ ,with αiλ ∈ End(Vi ). By G-equivariance of ϕ and Fubini’s theorem:ϕ(x1 ⊗ x2 ) =gϕ(g−1 (x1 ⊗ x2 )) dµ1  dµ2 (g)K1 ×K2=Nλ=1α̃1λ ⊗ α̃2λ (x1 ⊗ x2 ),where µi is the Haar measure of total mass 1 on Ki andα̃iλ (x) :=gi αiλ (g−1i xi )dµi (gi ),Ki

is Gi -equivariant.

EMS Newsletter June 2018

Feature
where µi is the Haar measure of total mass 1 on Ki andα̃iλ (x) :=gi αiλ (g−1i xi )dµi (gi ),Ki

is Gi -equivariant.

Ramanan–Ramanathan method, again
Chen translated the argument of Ramanan–Ramanathan, as
elaborated by Totaro, in the context of Arakelov geometry inorder to give the following bound:nµ̂max (Ē1 ⊗ · · · ⊗ Ēn ) ≤(µ̂(Ēi ) + log rk Ei ).
i=1

The error term is Chen’s inequality comes for a very simple
reason: the operator norm of the projection V ⊗ dim V → det V
is dim V!.
References[1] Y. André. On nef and semistable Hermitian lattices, and their behaviour under tensor product. Tohoku Math. J. (2), 63(4):629–649,
2011.
[2] V. Balaji, P. Deligne, and A. J. Parameswaran. On complete reducibility in characteristic p. L’épijournal de Géométrie Algébrique, Volume
1, 2017.
[3] V. Balaji and A. J. Parameswaran. Tensor product theorem for
Hitchin pairs – an algebraic approach. Ann. Inst. Fourier (Grenoble),61(6):2361–2403 (2012), 2011.
[4] G. D. Birkhoff. A theorem on matrices of analytic functions. Math.
Ann., 74(1):122–133, 1913.
[5] J.-B. Bost and H. Chen. Concerning the semistability of tensor products in Arakelov geometry. J. Math. Pures Appl. (9), 99(4):436–488,
2013.
[6] J.-B. Bost and K. Künnemann. Hermitian vector bundles and extension groups on arithmetic schemes. I. Geometry of numbers. Adv.
Math., 223(3):987–1106, 2010.

[17] D. Hilbert. Über die vollen Invariantensysteme. Math. Ann.,
42(3):313–373, 1893.
[18] S. Ilangovan, V. B. Mehta, and A. J. Parameswaran. Semistability andsemisimplicity in representations of low height in positive characteristic. In A tribute to C. S. Seshadri (Chennai, 2002), Trends Math.,pages 271–282. Birkhäuser, Basel, 2003.
[19] N. M. Katz. On a theorem of Ax. Amer. J. Math., 93:485–499, 1971.
[20] G. R. Kempf. Instability in invariant theory. Ann. of Math. (2),
108(2):299–316, 1978.
[21] G. Laffaille. Groupes p-divisibles et modules filtrés: le cas peu ramifié.
Bull. Soc. Math. France, 108(2):187–206, 1980.
[22] B. Mazur. Frobenius and the Hodge filtration. Bull. Amer. Math. Soc.,
78:653–667, 1972.
[23] B. Mazur. Frobenius and the Hodge filtration (estimates). Ann. of
Math. (2), 98:58–95, 1973.
[24] D. Mumford. Geometric invariant theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Band 34. Springer-Verlag,
Berlin-New York, 1965.
[25] M. S. Narasimhan and C. S. Seshadri. Stable and unitary vector bundles on a compact Riemann surface. Ann. of Math. (2), 82:540–567,
1965.
[26] S. Ramanan and A. Ramanathan. Some remarks on the instability flag.
Tohoku Math. J. (2), 36(2):269–291, 1984.
[27] G. Rousseau. Instabilité dans les espaces vectoriels. In Algebraic
surfaces (Orsay, 1976–78), volume 868 of Lecture Notes in Math.,pages 263–276. Springer, Berlin-New York, 1981.
[28] J.-P. Serre. Cours d’arithmétique, volume 2 of Collection SUP: “LeMathématicien”. Presses Universitaires de France, Paris, 1970.
[29] J.-P. Serre. Sur la semi-simplicité des produits tensoriels de représentations de groupes. Invent. Math., 116(1-3):513–530, 1994.
[30] J.-P. Serre. 1998 Moursund Lectures at the University of Oregon.
arXiv:math/0305257 May 2003.
[31] A. Thue. Über Annäherungswerte algebraischer Zahlen. J. Reine
Angew. Math., 135:284–305, 1909.
[32] B. Totaro. Tensor products in p-adic Hodge theory. Duke Math. J.,
83(1):79–104, 1996.
[33] H. Weyl. Theorie der Darstellung kontinuierlicher halb-einfacher
Gruppen durch lineare Transformationen. I. Math. Z., 23(1):271–309,
1925.

[7] H. Chen. Maximal slope of tensor product of Hermitian vector bundles. J. Algebraic Geom., 18(3):575–603, 2009.
[8] C. Chevalley. Théorie des groupes de Lie. Tome III. Théorèmes
généraux sur les algèbres de Lie. Actualités Sci. Ind. no. 1226. Hermann & Cie, Paris, 1955.
[9] R. Dedekind and H. Weber. Theorie der algebraischen Functionen
einer Veränderlichen. J. Reine Angew. Math., 92:181–290, 1882.
[10] P. Deligne. Semi-simplicité de produits tensoriels en caractéristiquep. Invent. Math., 197(3):587–611, 2014.
[11] M. Demazure and P. Gabriel. Groupes algébriques. Tome I: Géométrie
algébrique, généralités, groupes commutatifs. Masson & Cie, Éditeur,
Paris; North-Holland Publishing Co., Amsterdam, 1970. Avec un appendice ıt Corps de classes local par Michiel Hazewinkel.

Marco Maculan [marco.maculan@imjprg.fr] received his PhD in Paris in 2012.
He is now a Maître de Conférences atUniversité Pierre et Marie Curie. His fieldof interest is arithmetic geometry, especiallyin connection with p-adic analysis andalgebraic groups.

[12] G. Faltings. Crystalline cohomology and p-adic Galois-representations. In Algebraic analysis, geometry, and number theory (Baltimore,
MD, 1988), pages 25–80. Johns Hopkins Univ. Press, Baltimore, MD,
1989.
[13] G. Faltings. Mumford-Stabilität in der algebraischen Geometrie. In
Proceedings of the International Congress of Mathematicians, Vol. 1,2 (Zürich, 1994), pages 648–655. Birkhäuser, Basel, 1995.
[14] G. Faltings and G. Wüstholz. Diophantine approximations on projective spaces. Invent. Math., 116(1-3):109–138, 1994.
[15] J.-M. Fontaine. Modules galoisiens, modules filtrés et anneaux deBarsotti-Tate. In Journées de Géométrie Algébrique de Rennes.
(Rennes, 1978), Vol. III, volume 65 of Astérisque, pages 3–80. Soc.
Math. France, Paris, 1979.
[16] J. Giraud, A. Grothendieck, S. L. Kleiman, M. Raynaud, and J. Tate.
Dix exposés sur la cohomologie des schémas, volume 3 of Advanced
Studies in Pure Mathematics. North-Holland Publishing Co., Amsterdam; Masson & Cie, Editeur, Paris, 1968.

EMS Newsletter June 201815

Interview

Interview with Peter Sarnak
Michael Th. Rassias (University of Zürich, Switzerland)Peter Sarnak is the Eugene Higgins Professor of Mathematics at
Princeton University, as
well as a Professor at the
School of Mathematicsof the Institute for Advanced Study, Princeton.
Born in 1953 in Johannesburg, South Africa,
he studied at the University of Witwatersrand,
obtaining a BSc in 1974
and a postgraduate BSc(Hons) in 1975.
Peter Sarnak.
He obtained his PhDfrom Stanford University in 1980 under the direction of Paul Cohen.
Sarnak is known for his groundbreaking contributions
to number theory and problems of mathematical analysis
motivated by number theory. The techniques he employs
in his research, as well as his interests, are surprisingly
wide-ranging, from zeta functions and automorphic forms
to mathematical physics and quantum computation.
Throughout his career, he has received several prestigious awards, including the Pólya Prize (1998), the Ostrowski Prize (2001), the Levi L. Conant Prize (2003), theCole Prize (2005) and the Wolf Prize (2014), and has been
awarded with honorary doctorates from several universities. He has also been elected as a member of several important academies and societies, including as a Member of
the National Academy of Sciences (USA) and a Fellow of
the Royal Society (UK). Peter Sarnak has also supervised
more than 50 PhD students!
M. Th. Rassias: If I remember correctly, as a high school
student you were very much involved with chess competitions and you did not get really interested in mathematics until your undergraduate years. Is this related
to how mathematics was taught to you in your early lifein South Africa or was it just a matter of taste at the
time? Provided with other inputs, might you have been
participating in mathematics problem-solving competitions during your high school years instead?
P. Sarnak: The mathematics that I was exposed to in highschool in South Africa in the late 1960s was mostly routine and, while it came easily to me, I wasn’t aware of anyof the challenges that mathematics had to offer. On theother hand, chess was a challenge and one that was decided with an immediate outcome. I was drawn to it froman early age and probably hit my peak aged 16. My father
was very supportive of my involvement in chess competitions until I declared my intention of going abroad af-

16

Celebration at the conference in honor of Peter Sarnak’s 61st birthday
at Princeton.

ter school and trying to make it as a professional chessplayer. He insisted that I go to university first to study
and, indeed, once I got there (at the University of Witwatersrand in Johannesburg) and was introduced to realmathematics and especially abstraction, I was quickly
drawn to its beauty and challenge. I carried on playing
chess competitively until my early 20s but, after that anduntil today, mathematics has been my passion.
Were you exposed to any other scientific disciplines before choosing mathematics? (If yes, was that beneficial?)
As I mentioned, the subjects that came easily to me inhigh school were mathematics and science (the lattermeaning basic physics and chemistry). Being a professional mathematician was not on my radar (or of thosearound me). So, I was planning to major in physics. However, in my first year, I had a miserable time in the laboratory and a good friend of mine Eddie Price, who was
one of the top chess players in South Africa at the timeand also a lecturer in the physics department of the university, told me that if it is the theoretical side of physics that I enjoy then I would be better off doing appliedmathematics. So, I majored in mathematics and appliedmathematics and, while it is “pure” mathematics that has
driven my main interests and research, my early introduction to applied mathematics played a key role in me
having broad mathematical interests and an appreciationof its applications.
Do you believe that the strategic way of thinking you
cultivated as a professional chess player helped you
when approaching mathematical problems and, if so,in what way?
The deductive analytic and positional reasoning, and therich chess theory that human chess players employ (I say

EMS Newsletter June 2018

Interview
ever, when one gets down to understanding as to how anargument works, you find that the fundamental ideas andtools are much more limited than what might be apparent at first. In Hilbert’s time, say, one person, like himself
or his student Weyl, could have a good understanding oflarge portions of central mathematics. While today thismight seem impossible, our mathematical universe is still
small and the cornerstones from which major developments and changes are taking place at any given period
are quite limited.

Peter Sarnak with Henryk Iwaniec.

human since, if I understand correctly, the recent computer programs such as the machine learning “alpha go”,
which are stronger than any program, apparently usevery little “rich human chess theory”), have a lot of similarities to parts of mathematics. It is certainly very goodtraining for the mind and for mathematics, in a similar
way to mathematics competitions, but like the latter it isneither necessary nor sufficient for being a good mathematician.
Was there a specific paper, book, lecture or even theorem you came across that won you over to mathematics? What was the spark?
What won me over as a first year undergraduate studying mathematics was abstraction and specifically thatconceptual thinking can make the solution of a problem
and understanding of a theory completely transparent. I
remember the first course in abstract linear algebra as aspark, and also a topology course that drew me to wantto learn and understand much more.
One may state that mathematics has witnessed a great
expansion over the last, say, 100 years, with many different areas emerging and various methods discovered,bridging seemingly different fields. You once said that:“Mathematics is really very small, not big. There aren’tthat many great ideas and people use the same idea over
and over again in different contexts.” As a mathematician who has worked in various areas of mathematics
and who has used a variety of techniques in order totackle difficult problems, do you believe that the more
one matures in mathematics, the greater the unity one
sees in it?
I continue to hold the opinion in the quote that you
mention, though now, some years later, with the explosion of papers on the arXiv, it seems harder to hold thatview. Mathematical research is looking more like othersciences, with many papers having multiple authors andeven some research being done in what looks like mathematics laboratories. Some of this is natural, thanks tovarious tools becoming very specialised and complicatedand so it is not surprising that technical projects draw
contributions from people with different expertise. How-EMS Newsletter June 2018

Is there a mathematician who influenced you the most,either through collaborations or interactions or even by
studying their work?
Firstly, Paul Cohen, who was my thesis advisor, had a major influence on my mathematical taste, knowledge, insight and intuition. His view of the unity of mathematics(and that one really need not stick to a small sub-field)
made a big impression on me. Together, Paul and I studied a good portion of Selberg’s works, and the core of
my own work is very much shaped and influenced by
Selberg’s ideas.
I remember you saying to us in a work group seminar
at Fine Hall, Princeton, around three years ago, thatyou believed that the problem of factoring integers into
primes must not be as hard as we think and that at somepoint someone will find an easy way to factor in polynomial time. Why is that?
There is no theoretical evidence that factoring is difficult and if an efficient algorithm to factor were found, no
problem that is expected to be difficult (e.g. an NP complete problem) would follow. At this point in time and as
far as is known in the public domain, there is no known
efficient factoring algorithm, and the evidence that is offered to it being “hard” is that smart people have triedand failed. If our attitude in mathematics is that smart
people have failed to solve a problem therefore it cannot
be solved, we would be out of business as far as attacking
the central unsolved problems.
Us mathematicians working in the trenches musthave some beliefs as to what is true and what can beproved in order to proceed in our efforts, and my belief
as far as factoring goes is that it can be done efficiently.
In fact, one can take Shor’s quantum factoring algorithm
as evidence that factoring can be done efficiently with aclassical computer.
Speaking about primes, you have formulated a very
intriguing conjecture related to the Möbius function,
which has captured the interest of many mathematicians. Would you like to describe it for the readers who
do not belong to the world of number theory?
The parity of the number of prime factors of a number
n is an elusive quantity that carries a lot of information(even the complexity of computing this parity appears tobe as difficult as factoring n).
If n is square-free, the Möbius function m(n) is(–1)^(parity), while m(n) is zero for numbers that are notsquare-free. As a function of n, m(n) is apparently very

17

Interview
random. For example, the partial sums of m(n) for n up to
N are of order roughly sqrt(N) if and only if the Riemannhypothesis is true. More generally, the sum of m(n).f(n)for n up to N is expected to have some cancellation for abounded function f(n) if the latter is of “low complexity”;
this heuristic is known as the Möbius randomness principle and goes back at least to I. M. Vinogradov. My “Möbius disjointness conjecture” that you mention makes
this heuristic precise by realising f(n) as an observablesequence in a (topological) dynamical system.
If the system is deterministic (i.e. has zero entropy)
then there should be cancellation. The primary tool for
estimating such sums when f(n) is not a multiplicative
function of n (f(n) is multiplicative if f(mn) = f(m).f(m)when m and n are relatively prime) is Vinogradov’s bilinear method. For f(n) an observable in a dynamical system, Vinogradov’s bi-linear sum is a Birkhoff sum for thejoining of the system with itself.
This allows for dynamical ideas to be brought into
the study of these sums and, as a consequence, there has
been a lot of progress proving this Möbius disjointness
conjecture for many deterministic systems. For logarithmically averaged versions of the conjecture, there is evenmore progress, thanks to works of Matomaki/Radziwill,
Tao and Frantzikinakis and Host. In the latter, notions
from Furstenberg’s non-conventional ergodic averages
and his dynamical disjointness play a central role.
It is really impressive that in your work you are able touse techniques from seemingly independent areas in order to solve a problem you are working on. In that manner, you have studied and discovered some fascinatinginterconnections. One example is the association of the
problem of sums of squares in number theory with objects from quantum computation, such as the so-called
Golden Gates. Would you like to discuss this topic?
Yes, I would be happy to elaborate on this topic. It turns
out that for the construction of universal quantum gates,
one is faced with the problem of providing an optimal
set of topological generators for the groups G = SU(2)and SU(4) (the first for single qubits and the second
for two qubits). By optimal, we mean that the words oflength m in the generators cover G (with its bi-invariant
metric) optimally for all large m and that there is an efficient algorithm to find the best approximation of length
m for any g in G. Standard quantum computation textbooks give specific generators for SU(2), such as “Clifford matrices plus T”. It turns out that these generate an
S-arithmetic unitary group defined over a real quadratic
number field and it is precisely this that makes themgood generators. In recent work, Parzanchevski and I
show how, for each of the (finite) symmetry groups of
the Platonic solids, one can add an involution yielding anessentially optimal generating set for SU(2). Interestingly, the proof of the optimal (almost) covering property
makes use of the Ramanujan conjectures established by
Deligne. An heuristic algorithm developed by Ross andSelinger to navigate with the Clifford and T matrices
can be adapted for all these “arithmetic Golden Gates”,
as I like to call them, and leads to efficient navigation.

18

Peter Sarnak with young mathematicians.

While the mathematics of S-arithmetic unitary groups
is available and provides such Golden Gates for SU(4)(Parzanchevski), efficient navigation remains open inthis case.
How do you see the future of applications of number
theory to other fields of science?
In Hardy’s “Mathematicians Apology”, he points to thetheory of numbers as the epitome of “pure mathematics”
and being very far from applications.
The modern digital and computer world has provenhim to be quite mistaken. The applied area that is often
pointed to as far as applications of number theory go iscryptography (for example, RSA and factoring). However, it runs much further than that. In fact, any fundamental problem that is discrete in nature (e.g. quantum
mechanics) will, when studied down to its finest features,
become one of number theory. There are many examples,such as the quantum gates above, where number theory
problems that emerge are also ones that have been identified by number theorists as fundamental. I like to thinkthat the reason for this is that in number theory (and inmathematics more generally), mathematicians are looking for the deeper truths about whole numbers and,while we are not motivated by applications, these comenaturally in cases where our insights are fundamental
features of the objects that we study. In any case, whenthings work out this way, it is particularly pleasing.
My colleague Michail Aizenman (a mathematicalphysicist) once commented in a lecture on random matrix theory and zeros of the zeta function that “number
theory is the final frontier of science”. He doesn’t have toconvince me of that!
Mentoring young researchers can be an important aspect of the life of a mathematician. Hilbert, for example, supervised 69 PhD students throughout his life. It issaid that he enjoyed interacting with students and used
to go for long walks with them to discuss mathematics.
You have already supervised more than 50 PhD students
and I have personally witnessed the very close relations
you have with them. Would you like to talk about this
aspect of your mathematical life?

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Interview
Yes, I have guided quite a number of PhD students over
the years and I am very fortunate to have had this opportunity. For me, teaching, communicating and mentoring
are an integral part of doing mathematics. Very often, I
learn as much from these exchanges as do those being
mentored. Over time, this reciprocal activity of guiding
many PhD students has allowed me to learn and appreciate a much wider landscape of mathematics and it
has opened doors to finding unexpected connections between disparate areas.
Directly and indirectly, the students I have mentored
have played a big role in what I have managed to do. My
role as a senior mentor is mostly that of being a coach:one provides encouragement and makes sure that theperson being mentored is working on interesting problems and that they are aware of the basic tools that areavailable and what is known.
Note: The copyright of the pictures featured in this interview is held by C. J. Mozzochi, Princeton, NJ. We thankhim very much for giving us the permission to publish
them in this interview.
Michael Th. Rassias is on the Editorial Board of the EMS
Newsletter.

Mathematics as a Positive
Mental Place – An Interview with
Gigliola Staffilani
Roberto Natalini (Consiglio Nazionale delle Ricerche, Rome, Italy, Chair of the Raising Public AwarenessCommittee of the EMS)
Gigliola Staffilani is an Italian mathematician working
in the USA as the Abby Rockefeller Mauze Professor of
Mathematics at the Massachusetts Institute of Technology.
Her research concerns harmonic analysis and partial differential equations. In 2014, she was elected to the American Academy of Arts and Sciences.
Roberto: Gigliola, could you describe your background,your early education and your life as a child in Italy?
Gigliola: I was born in a small town in Abruzzo. My parents were farmers and I lived with my family and that of
my dad’s brother. I really enjoyed playing outdoors with
my dear friend Lina, who lived next door.
I was very good in school, in particular math. I was
also very competitive and I was very unhappy if I did not
get the highest marks. My brother is 10 years older thanme and he was the first in the family to go to high school
and then university. There were no books at home but hesubscribed to “Le Scienze” and so, from very early on, I
started reading about amazing discoveries in science. I
couldn’t understand much of what I was reading but I
loved the short biographies of the scientists. It was during this time that I learned about Princeton, Stanford,
Harvard, MIT… I loved my childhood but unfortunately,when I was 9, my father, who was 43, got sick with advanced colon cancer and died in less than a year. I wasdevastated and I lost my carefree spirit. In order not tothink about my loss during my spare time I decided tostart solving math problems from my school book and I
continued doing it well into high school (and somehow
this worked).

EMS Newsletter June 2018

What did you think about mathematics when you werea child?
I loved the fact that mathematics was completely logic –no surprises there. I liked the fact that I could control it
and that a proof was not subjective or emotional. I had
enough negative emotions around me and I just neededa mental place where no emotions were taking over everything.

Gigliola Staffilani – image from the video “Truth Values: One Girl’s
Romp Through M.I.T.’s Male Math Maze”, as part of the institute’s
150th anniversary celebration, December 2012.

Could you describe the beginning of your career inmathematics in the United States? Was it hard to startas an emigrant in a new and unknown country?
I think that the first and probably biggest “culture shock”
came when I moved from my little town to Bologna. The
move from Bologna to the University of Chicago wasin a way simpler, in spite of the fact that I encountered

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Interview
an endless lists of obstacles. I didn’t have to prove anything to anybody in Chicago and I was completely naive
about American culture. The first obstacle I encounteredonce I arrived on campus was that I couldn’t register as
a student because I had not taken the TOEFL exam. In
fact, I didn’t know any English. As a consequence, I only
had a “prospective student visa” – not enough to register. Since I couldn’t register, I couldn’t live as a student
in the International House, so I found myself basicallyhomeless in one of the most dangerous campuses in the
US. I decided to look for a room to rent as I had saved
some money working during the previous two Summers.
I found one unfurnished and I took it; at least I had a roof
over my head. Fortunately, though, the chair of the mathdepartment allowed me to register as a student in spiteof the visa; he was just hoping that I would give up after
a couple of weeks and solve the problem. But I stayed
for a whole month. Then, a second obstacle appeared. I
didn’t receive the first cheque of my fellowship. I learned
later that this was again a consequence of the fact that officially I was a “prospective student”. At this point, I was
ready to give up and, with enormous disappointment, I
decided to use the public phone in the math department
to make a reservation to fly back to Italy. While at thephone, my registration advisor, Professor Sally, walked
by. He noticed that I was upset and he signalled to me
to end the phone call and talked to me. With my broken
English, I explained the situation and with total ease hewalked me to his office and gave me a personal cheque ofabout $1,500, which was the equivalent of my first monthof fellowship. Very often, I think about this event: a completely random and lucky coincidence that may not have
happened at all. If so, my life would have been completely different!

Could you explain, for an educated but not specialist audience, the core of these works about dispersive equations
you performed in the first part of your academic career?
When I started talking to my advisor at the University of
Chicago, Carlos Kenig, he explained to me that I could
work in one of two areas that he was an expert in: elliptic
equations, on which incredible progress had been madein the preceding years and where the problems left open
were really hard, or dispersive equations, on which he had
started working more recently and where many problems
were completely open. He added that he didn’t reallyknow if this direction would become central in analysis. I
decided to take the second option and I am glad I did because, indeed, thanks also to the work in this area by Jean
Bourgain and Terence Tao, dispersive equations becamevery important. The main questions that I addressed withcollaborators were on existence, uniqueness and stability(well-posedness) of rough solutions to dispersive equations, such as the Schrödinger of the KdV equations. We
were interested in rough solutions because one would
like to assume that only the mass (L2 norm) or the energy(related to the H1 norm) are bounded for these solutions.
As a first step, one would prove well-posedness in a smallinterval of time but the next and harder step is to understand what happens when time evolves arbitrarily far. To
answer this question, with my collaborators Colliander,
Keel, Takaoka and T. Tao, we invented the concept of “almost conservation laws”, which was then developed inmany different contexts by us and other researchers.

You are now a worldwide recognised expert in harmonic
analysis and dispersive partial differential equations.
How did it happen that you started to work in this area?
Why do you like it?
I started working on harmonic analysis as a student in
Bologna when I was writing my thesis on certain Green’s
functions. I like analysis; for me, it is way more flexible
than algebra. Harmonic analysis, in particular, allows you
to reduce many problems to understanding a variety ofinteractions between simple functions and then reassemble them in a clever way to deduce properties for generalfunctions. I think it is a very powerful and flexible tool.

Could you mention the most important of your mathematical results and why it is important to you?
For me, the proof of global well-posedness for the energy
critical nonlinear Schrödinger equation in 3D is my mostimportant result. I think it is important because we had
to find a missing ingredient, now known as the interaction Morawetz inequality, which is actually a fundamental identity that had not been discovered till then.

Gigliola Staffilani during a workshop at the Institut des Hautes Études
Scientifiques, Paris, July 2016.

20

What are the main contributions you have made in yourfield – the main original ideas?
I would say that the idea of the “almost conservationlaws” is what I like the most.

You collaborate with some other well-known mathematicians, such as James Colliander, Markus Keel, Hideo
Takaoka and Terence Tao, and I read that you are
known as the “I-team”. Could you explain the meaning
and the origin of this name?
We are called the I-team because, in one of the original
papers on “almost conservation laws”, we used a multiplier operator that for no special reason we called “I”. I
guess we had run out of other good letters by that point.
You were appointed as a professor of pure mathematics
at MIT. I believe you are one of the few women to get
this kind of position. What is your feeling about that? Is
the situation changing?
When I arrived at MIT, there was only another woman
in applied mathematics; I was the only one in pure math-

EMS Newsletter June 2018

Interview
ematics. Today we are a total of five. So, a little improvement but not much if one considers that there are a totalof 53 professors in our department.
How is the environment in your department and how is
it important to you for your work?
I love my department; it is very “democratic” and people
listen. Of course, there are discussions but they are constructive. There are no groups fighting against each otherand everybody is invested in having the best set-up forstudents, postdocs and professors that we can possibly
have. This, for me, is absolutely fundamental. I need tofeel happy when I go to my office; otherwise, I would bea terrible researcher, teacher and mentor.
What about more recent problems you have considered?
What is the core of your activity nowadays?
Recently, I have been introducing a little more probability into my work. Often, when working with rough data,
one can prove that there are special counterexamples forwell-posedness. But, if one is a little less greedy, one may
be happy to claim that for “almost all initial data”, wellposedness is available. Of course, one has to make senseof the “almost all” but this is what probability is for. I
have also been working on the integrability structure fora certain hierarchy of so-called dispersive equations thatmodel Bose-Einstein condensation in the framework ofGross-Pitaevskii theory.
You have been awarded many honours and prizes.
Which one is the most important to you?
I would say being elected to the American Academy ofArts and Sciences has been really great. It is such an historical organisation that I feel like I am part of history
itself. Also, as a member, I get to discuss possible directions in education that may one day affect many people,
so it is a bit like “giving back” to society.
How much in your work is intuition and how much isjust hard work?
I think, in my case, it is 50%-50%. I believe that intuition
comes when you have cleared up your brain to receive it.
To clear up your brain, you need to work hard to eliminate all those attempts that do not lead anywhere.
How do you organise your work? Do you follow a routineor does it vary a lot according to external conditions?
Recently, I have been working a lot with senior collaborators and postdocs. It is difficult to juggle everything so I
try to set aside certain times with certain groups that are
essentially fixed every week. So, I would say that I follow
a routine.
According to you, what is the situation of women inmathematics around the world? Is there any difference
between Europe and the United States?
I can compare maybe Italy and the US. I think in bothcountries there are too few women at the level of full
professor. But, in Italy, I do not think that people believethat the reason is that women do not have the same tal-

EMS Newsletter June 2018

Gigliola Staffilani and her family, from the interview on the Italian site
MaddMaths!, December 2010.

ent as men. Unfortunately, in the US, people still thinkthat women are not good at math in general and that not
being good at math for a girl is totally acceptable. Thissocial belief is really difficult to change I am afraid.
You are committed to reducing the gap between womenand men in mathematics. What are your actions in thisdirection?
I strongly believe in diversity, in all its shades: gender,race and family background. I believe that when there
are no role models, it is very difficult to imagine yourself
in a certain position, so I am a strong supporter of having role models as mentors. At MIT, I organise a lunch
seminar, where I invite senior women mathematicians,
working in academia or industry, to come and recountto the women in the department (from undergraduates
onward) how they arrived to the place they are now. In
doing so, they also explain, in general terms, the mathematics they use in their research or their job.
What do you do outside math? Do you have hobbies?
I really do not have much time for myself but when I
do I like to go hiking, take care of my small city garden
and, most of all, spend time chatting with my kids and myhusband.
How is your relation with Italy now? Are you still intouch with your country?
I love to visit Italy, either for work or personal reasons. I
am in touch with a few mathematicians there and I havelectured in a few Summer Schools as well. In July, in fact,
I will be in Rome for a week!

Roberto Natalini has been the Director of
the Istituto per le Applicazioni del Calcolo“Mauro Picone” of the National ResearchCouncil of Italy since 2014. His research interests include fluid dynamics, road traffic,
semiconductors, chemical damage of monuments and biomathematics. He is Chair of
the Raising Awareness Committee of the European Mathematical Society.

21

Obituary

Robert Adol’fovich Minlos(1931–2018) – His Work and Legacy
Carlo Boldrighini Istituto Nazionale di Alta Matematica (INdAM), Roma, Italy), Vadim A.Malyshev, Lomonosov
Moscow State University, Russia), Alessandro Pellegrinotti (Università Roma Tre, Italy), Suren K. Poghosyan(Institute of Mathematics of the NAS RA, Yerevan, Armenia), Yakov G. Sinai (Princeton University, USA),
Valentin A. Zagrebnov (Institut de Mathematiques de Marseille, France), Elena A. Zhizhina (Institute forInformation Transmission Problems, Moscow, Russia)
On 9 January 2018, therenowned mathematician Professor Robert Adol’fovich Minlos passed away at theage of 86. An eminent
researcher and outstanding teacher, he
was a world-renowned
specialist in the areaof functional analysis,
probability theory andcontemporary
mathematical physics.
R.A. Minlos wasborn on 28 February
Robert Adol’fovich Minlos.
1931 into a family asso(Courtesy of A. Kassian)ciated to the humanities. His father Adol’fDavidovich Miller was known as a lecturer and author
of English dictionaries and manuals. His mother Nora
Romanovna (Robertovna) Minlos was an historian-ethnographer. Her brother Bruno Robertovich Minlos, witha PhD in historical sciences, was a specialist in the history
of Spain. This is perhaps why Robert Adol’fovich loved
poetry, wrote verses himself, was a fervent theatre-goerfrom his school years and was seriously occupied withpainting from the age of 40.
Nothing foretold a mathematical future but when
he was 15, the young Robert accidentally saw a poster
about the Moscow Mathematical Olympiad for schoolchildren. He participated in it, obtained the second prize
and, inspired by that, began to attend the school club led
by E. B. Dynkin. In 1949, Robert joined the Faculty of
Mechanics and Mathematics of the Moscow State University. He continued to participate in Dynkin’s seminar,
which, together with A. S. Kronrod’s seminar, had a great
influence on him as an undergraduate student.
R.A. Minlos prepared his first scientific paper (equivalent to a Master’s degree thesis) in 1950 while participating in the Moscow State University seminar on thetheory of functions of a real variable (under the leadership of A. S. Kronrod). However, the real scientificinterests of the young mathematics student began toform after he became acquainted to I. M. Gelfand. Their
joint publication “Solution of the equations of quantum
fields” (Doklady Akad. Nauk SSSR, n.s., 97, 209–212,221954) became Minlos’ diploma thesis in mathematics. It
was devoted to the functional or, in mathematical physics
language, the path integral, which has a direct relation toquantum physics.
As Minlos himself admitted: “My further life in mathematics was predetermined by that work because I was
subsequently mainly occupied with mathematical physics.” There were, nevertheless, more works on random
processes, on measure theory and on functional analysis.
Very soon, one of his papers, “Extension of a generalised random process to a completely additive measure” (Doklady Akad. Nauk SSSR, 119, 439–442, 1958),
brought Minlos to worldwide fame. It became the basisof his Candidate (equivalent to PhD) Dissertation “Generalised random processes and their extension to a measure’’, which was published in Trudy MMO, 8, 497–518,
1959. This result, which is important for the theory of random processes, as well as for functional analysis, is now
known as the Minlos theorem on the extension of cylindrical measures to Radon measures on the continuous
dual of a nuclear space, i.e. the continuation of a process
to a measure on spaces adjoint to nuclear spaces.
The connection of Minlos to mathematical physics atthat time manifested in the publication (jointly with I. M.
Gelfand and Z.Ya. Shapiro) of the monograph Representations of the rotation and Lorentz groups and their applications (1958), which was later translated from the Russianby Pergamon, London, in 1964. Note that the monographappeared in 1958,
just at the time
when the need forphysicists to understand representation theory wasstrongly motivatedby the discovery ofelementary particle symmetries, aswell as the role oftheir spins and symmetries related torelativistic Lorentz
transformations.
From 1956 toSeminar of F. A. Berezin and R. A. Minlos at
1992, R.A. Minlos the Faculty of Mechanics and Mathematics –was employed by MSU, 1959.
the Department of (Courtesy of N. D. Vvedenskaya)

EMS Newsletter June 2018

Obituary
the Theory of Functions and Functional Analysis of theFaculty of Mechanics and Mathematics at Moscow State
University (MSU). In that period, there was a need toorganise a joint seminar with F.A. Berezin, primarily to
discuss the mathematical problems of quantum mechanics and quantum field theory.
A real advance of activity in the field of mathematical physics at the Faculty of Mechanics and Mathematics of MSU was achieved with the organisation in 1962
by R.A. Minlos and R. L. Dobrushin of a seminar on
statistical physics. It soon became widely known in theSoviet Union and abroad as the Dobrushin–Malyshev–
Minlos–Sinai seminar. The quantum aspects of statisticalmechanics at the seminar were primarily associated to
the name of R.A. Minlos. The seminar lasted until 1994
and had a huge impact on the world of modern mathematical physics. Almost all the celebrated specialists inthe field visited Moscow during the lifespan of the seminar. Besides traditional scientific contacts with Socialist
European countries, a fruitful collaboration was alsoestablished with colleagues from other countries. However, the most intensive contacts were within the country,
involving almost all the republics. For example, many of
Minlos’ PhD students came from Uzbekistan.
The beginning of the 1960s was extremely fruitfulfor Robert Adol’fovich. Initially, there were new results
obtained jointly with L. D. Faddeev on the quantum
mechanical description of three particles (1961). Thiswas followed by two articles devoted to the study ofthe thermodynamic limit in classical statistical physics(1967). There, R.A. Minlos suggested the first rigorous
mathematical definition of the limiting Gibbs distributions for an infinite system of interacting classical particles and also analysed the properties of such distributions (Funct. Anal. Appl., 1, 140–150 and 206–217, 1967).
His approach was very close to the classical Kolmogorov construction of random processes (fields). This result
anticipated the origin of the Markovian understandingof Gibbs random fields in the sense of Dobrushin–Lanford–Ruelle (1968).
The result (together with Ya. G. Sinai) of the appearance of phase separation in lattice systems at low temperatures (Math. USSR-Sb., 2, 335–395, 1967; Trudy
MMO, 17, 213–242, 1967, and 19, 113–178, 1968) was offundamental importance for the mathematical theoryof phase transitions. It formed the basis of Minlos’ doctoral dissertation, which he submitted for habilitationin 1968. In another joint work with Ya. G. Sinai (Theor.
Math. Phys., 2, 167–176, 1970), the foundation was laid
for a new approach to the study of spectral propertiesof many-particle systems. In combination with cluster
expansions, this approach drove significant progress inthe description of properties of such infinite systems,
including the spectrum of elementary particles of quantum fields and the mathematical description of the quasiparticle picture in statistical physics.
The new powerful method of cluster expansions
was, from the very beginning, a central one in the list ofinterests of Robert Adol’fovich. The results of a large
series of papers in this topic by R.A. Minlos, V.A. Maly-EMS Newsletter June 2018

shev and their students have been summarised in two
monographs: Gibbs Random Fields: Cluster Expansions(Springer 1991, translation of the 1985 Russian edition)and Linear Infinite-particle Operators (Amer. Math. Soc.
1995, translation of the 1994 Russian edition). As wasoutlined in the book Gibbs Random Fields, the methodof cluster expansions provides, besides a constructionof the limiting Gibbs measure, a cluster representationof the projections of the limiting Gibbs measure onto
bounded regions.
A famous peculiarity of the Dobrushin–Malyshev–
Minlos–Sinai seminar was not only its duration of aboutfour hours, which was amazing for foreign guests, or theassertive directness in communicating with lecturers but
also the opportunity to obtain from the discussions someinteresting problems to be solved. In essence, the seminar was functioning as a machine, generating questions
and a possible way to convert them into answers. Robert Adol’fovich was always one of the sources of interesting questions and open problems. The list of projects
thus originated includes, for example, cluster expansions
and their applications to the problem of uniqueness/nonuniqueness of Gibbs states, the quantum three-particle
problem, the Trotter product formula for Gibbs semigroups, the study of infinite-particle operators spectra,
the analysis of the quasi-particle picture in statisticalphysics and many others.
The Dobrushin–Malyshev–Minlos–Sinai seminar
had a tradition of studying and discussing new important publications on mathematical results and problems
in statistical physics. The Trotter product formula problem for Gibbs semigroups was originated after discussing the Pavel Bleher report about new techniques, reflection positivity and infrared bound estimates (launched
by Fröhlich–Simon–Spencer (1976–1978) to prove theexistence of phase transitions). In the case of quantum
systems, this technique involves the Trotter productformula approximation of the Gibbs density matrix.
In this context, Robert Adol’fovich posed a question
about the topology of convergence of the Trotter product formula because, to obtain the infrared bound, one
has to interchange the trace and the limit of the Trotter approximants. In fact, this operation is not harmfulfor quantum spin systems since the underlying Hilbert
spaces are finite-dimensional but it does produce a
problem, for example, in the case of unbounded spins.
A typical example is the problem of the proof of infrared bounds for the case of structural phase transitions in
one-site double-well anharmonic quantum crystals withharmonic interaction between sites (Fröhlich, 1976).
Then, the interchange is possible only when the Trotter product formula converges in trace-norm topology.
For a particular case of anharmonic quantum crystals,
the convergence of the Trotter product formula in the
trace-norm topology was proved via the Feynman-Kac
representation for Schrödinger (Gibbs) semigroups. The
abstract result, which also includes a generalisation to
trace-norm Trotter–Kato product formula convergence,is due to H. Neidhardt and V.A. Zagrebnov (1990). So,
the answer to the question posed by Robert Adol’fovich

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Obituary
was solved affirmatively in favour of trace-norm topology for the case of Gibbs semigroups.
At the end of the 1990s, Robert Adol’fovich returnedto the question of quantum phase transitions, in the areaof anharmonic quantum crystals (which was alreadywell-known to him) but from the opposite direction. It is
known that in contrast to classical systems, phase transitions in their quantum analogues may disappear due tointrinsic quantum fluctuations, which may lead to important tunnelling in the double-well potential. A particular
manifestation of that is the elimination by these fluctuations of the order parameter even at zero temperature,
whereas it is non-zero in the classical limit when the
Planck constant  = 0. A typical example is the above
structural phase transition in one-site double-well potential anharmonic quantum crystals with harmonic interaction between sites for particles of mass m in each site.
Moreover, experimental data for crystals close to thismodel manifest a so-called “isotopic effect”: the orderparameter of the structural phase transition for samples
with light masses m < mc disappears, a fact which warms
up interest in the mathematical aspect of this phenomenon.

R. A. Minlos with co-authors N. Angelescu and V. A. Zagrebnov on avisit to Dublin Institute for Advanced Studies, 2000.

During his visits to Dublin, Leuven and Marseilles, Robert Adol’fovich, in collaboration with E.A. Pechersky,
A. Verbeure and V.A. Zagrebnov, addressed the proof ofthe existence of a critical mass mc such that below thisthreshold the quantum state of the system is in a certain
sense trivial, or at least the order parameter is trivial.
In two papers, with A. Verbeure and V.A. Zagrebnov(2000) and then with E.A. Pechersky and V.A. Zagrebnov (2002), R.A. Minlos proposed using cluster expansion techniques for the small parameter ξ = √⎯m / . Then,the classical limit corresponds to ξ → ∞ and the quantum
regime, with zero order parameter for any temperature,⎯corresponds to ξ < √⎯mc / . Since the structural phasetransition in the model manifests as the displacement
order parameter, these papers consider projection of thefull quantum state on the commutative coordinate *-subalgebra Aq of bounded functions of displacements on thelattice. Then, the Feynman–Kac–Nelson formula for the
Gibbs semigroup kernel allows one to show that this pro-24

jection reduces to a classical ensemble of weakly interacting (for m < mc) Ornstein–Uhlenbeck trajectories.
Using the cluster expansion technique, the exponential
mixing of the limit state with respect to the lattice grouptranslations was proven for all temperatures, including
zero, if m < mc (2000). To check that in this domain oflight masses (high quantumness) the order parameteris zero for all temperatures, including the ground state,
R.A. Minlos and his coauthors, in the 2002 paper, usedthe external sources h conjugated to the local displacements instead of localising the trajectories boundary
conditions, as in the 2000 paper. This allows the analysis,
for any temperature θ, of the free-energy density function f (θ, h) for free or periodic boundary conditions. It is
proved in the 2002 paper that there exists a radius h0(m)such that h |→ f (θ, h) is holomorphic in the disc { h  C}:| h | < h0(m)} for any θ ≥ 0 if m < mc. Moreover, the Gibbs
expectations: h |→ 〈 A 〉(θ, h), are holomorphic in the same
disc for any bounded operator A of a quasi-local *-algebra A of observables. The analyticity, in particular, yields
that the displacement order parameter is equal to zerofor h = 0 and for any temperature θ ≥ 0 if m < mc.
This was also a period when, during his visits to KU
Leuven and CPT Marseilles, Robert Adol’fovich got into
an argument with A. Verbeure about the mathematical sense of the notion of quasi-particles in many-bodyproblems and of the corpuscular structure of infinite system excitations. In the framework of quantum statisticalmechanics, an attractive way to promote this notion was
based on the non-commutative central limit theorem
for collective excitations. This concept yields a plausible(for physics) picture of boson quasi-particles excitations(phonons, magnons, plasmons, etc.) in the corresponding
Fock spaces (A. Verbeure et al. (1995)).
On the other hand, in their book Linear infinite-particle operators, V.A. Malyshev and R.A. Minlos proposed
the description of a quasi-particle picture based on theconstruction by cluster expansions of the lower branchesof the spectrum of infinite many-body systems with goodclustering. This idea goes back to the paper by R.A. Minlos and Ya. G. Sinai “Investigation of the spectra of somestochastic operators arising in the lattice gas models”(1970). There, a new approach to studying the spectral
properties of the transfer matrix in general lattice models at high temperatures was developed. For translationinvariant systems, the lowest branch of the spectrum
enumerated by momentum corresponds to one-quasiparticle excitations above the ground state. Then, in thesimplest case, the energy of these excited states is completely defined by the momentum. This is called a dispersion law for quasi-particles, which is also well-known
for boson quasi-particles. If the system possesses a goodclustering, one can construct separated translation-invariant two-, three- and more (interacting) quasi-particles
excited states, which are combinations of branches withbands of continuum spectra. Robert Adol’fovich called
this property of excitations “The ‘corpuscular’ structureof the spectra of operators describing large systems” (thetitle of his paper in Mathematical Physics 2002, Imperial
Coll. Press, 2000).

EMS Newsletter June 2018

Obituary
The technique developed by V.A. Malyshev and R.A.
Minlos allowed the study of the corpuscular structure ofgenerators of stochastic dynamics. Their approach was
also applied to generators of stochastic systems: Glauber
dynamics, the stochastic models of planar rotators, thestochastic Ising model with random interaction and otherlattice stochastic models with compact and non-compact
spin spaces, as well as stochastic dynamics of a continuous gas and other stochastic particle systems in the continuum. Using this technique, one can find spectral gaps
and construct lowest one-particle invariant subspaces of
the generator that determine the rate of convergence tothe equilibrium Gibbs state. Moreover, it also allowed
the study, in detail, of the spectrum branches of infiniteparticle operators on the leading invariant subspaces
and, in particular, the construction of two-particle boundstates of the cluster operators. These results led to theunderstanding that a wide class of linear infinite-particle
operators of systems in a regular regime have a corpuscular structure.
Visiting Leuven and Marseilles, Robert Adol’fovich
proposed elucidating the concept of corpuscular structure of spectral branches for several particular models
on the lowest level of one-particle elementary excitations. This programme was performed in papers with
N. Angelescu and V.A. Zagrebnov (2000, 2005) and then
all together with J. Ruiz (2008), for lattice models andfor polaron-type problems. More activity and resultsin this direction were due to intensive collaboration ofRobert Adol’fovich with the group of H. Spohn, where
he studied spectral properties of Hamiltonians for quantum physical systems, in particular for Nelson’s model ofa quantum particle coupled to a massless scalar field.
Another long-term and fruitful collaboration of R.A.
Minlos was with the Bielefeld group, essentially with
Yu. G. Kondratiev and his co-authors and pupils. Firstly,
they generalised the original method (adapted for lattices) to functional spaces to control general particle configurations. This allowed the extension of their analysis
from lattice to continuous systems. In the paper “Oneparticle subspace of the Glauber dynamics generator
for continuous particle systems” (2004), they studied, in
detail, the spectrum of the generator of Glauber dynamics for continuous gas with repulsive pair potential. To
this end, the invariant subspaces corresponding to the
corpuscular structure were constructed.
The ideas and technical tools elaborated in this paper
were used in a number of other projects (Yu. Kondratiev,
E. Zhizhina, S. Pirogov and O. Kutoviy) on equilibriumand non-equilibrium continuous stochastic particle systems. This, in particular, concerns a delicate continuous
models problem of thermodynamic limit. One unexpected application of this technique concerns image restoration processing. Robert Adolfovich, in collaboration with
X. Descombes and E. Zhizhina, actively participated in
the INRIA project on the mathematical justification of a
new stochastic algorithm for object detection problems.
The result was summarised in the article “Object extraction using stochastic birth-and-death dynamics in continuum” (2009).

EMS Newsletter June 2018

In addition to the Dobrushin–Malyshev–Minlos–
Sinai seminar in the 1970s, there was also a regulartutorial seminar, which was led by Robert Adol’fovich
once a week. This was a very good opportunity to learn
elements of topological vector spaces, in particular the
Minlos theorem about the extension of a generalised
random process to a measure on spaces adjoint to nuclear spaces. The seminar also covered elements of mathematical statistical physics in the spirit of the famous“Lectures on statistical physics’’ in Uspekhi Math.
Nauk (1968). These lectures of Robert Adol’fovich very
quickly became a textbook for many students and scientists interested in mathematical statistical physics. In
these lectures, Robert Adol’fovich systematically used
the notion of configuration space, which appeared in his
earlier work, where he gave the mathematical definition of the limiting Gibbs measure as a measure on thespace of locally finite configurations in Rd. This concept
is technically very useful and is close to modern random
point process theory.
R.A. Minlos, Ya. G. Sinai and R. L. Dobrushin were
often invited by the Yerevan State University and theInstitute of Mathematics of the Armenian Academy of
Sciences to give lecture courses on statistical mechanics.
They all had PhD students working at the Institute ofMathematics in Yerevan. This was the main motivation
for the Institute of Mathematics to organise regular conferences (every 2–3 years) in Armenia under the name“Probabilistic methods in modern statistical physics”.
The first one was held in 1982 and the last one in 1988,
three years before the collapse of the Soviet Union.

R. A. Minlos with participants of the conference “Probabilistic methods in modern statistical physics”, Yerevan, Lake Sevan, 2006.

The conferences restarted in 1995 at the international
level. Robert Adol’fovich participated (as a rule, with his
students) in all of them, including the conference in Lake
Sevan in 2006. He always supported the conferencesin Armenia by being a permanent member of the programme committee and one of the main speakers, formulating new problems and generating interesting ideas,
questions and discussions. Unfortunately, he was not able
to participate at the conferences after 2006.
In the early 1990s, Robert Adol’fovich began his
collaboration with Italian institutions and mathematicians. He was a guest of the Department of Mathematics at the University of Rome “La Sapienza” many times

25Obituary

R. A. Minlos with participants of the conference: S. Lakaev, V. Zagrebnov, H. Suqiasian and B. Nahapetian, Yerevan, Lake Sevan, 2006.

and he also visited other institutions in Trieste, Naples,
L’Aquila and Camerino. During his stay at “La Sapienza”, he read a course on the mathematical foundationsof statistical mechanics, which was published as a bookby the American Mathematical Society in 2000 under
the title Introduction to mathematical statistical physics’.
In Rome, he began a collaboration with C. Boldrighini
and A. Pellegrinotti on models of random walks (RW)in interaction with a random environment fluctuating intime (“dynamic environment”).
At that time, several important results on random
walks in a fixed environment had already been obtained,by Solomon, Kesten, Sinai and others, but very little was
known for dynamic environments. Following the usual
terminology, the behaviour of RW for a fixed choice ofthe history of the environment is called “quenched” andits distribution induced by the probability measure of theenvironment is called “annealed”. A first result had beenobtained by C. Boldrighini, I.A. Ignatyuk, V.A. Malyshev and A. Pellegrinotti on the annealed model of adiscrete-time random walk on a d-dimensional latticein mutual interaction with a dynamic random environment. Robert Adol’fovich proposed applying the resultsthat he had obtained, together with V.A. Malyshev andtheir students, on the spectral analysis of the transfer
matrix for perturbed homogeneous random fields. The
approach proved to be very fruitful and in the following years 1993–1996, several results (C. Boldrighini et al.
(1994)) were obtained on the annealed RW, on the convergence to a limiting measure for the field “as seen fromthe particle”, on the decay of the space-time correlation
for the random field in interaction with the RW and on
the RW of two particles in mutual interaction with a random environment.
It was then possible, with the help of some tools ofcomplex analysis of which Robert Adol’fovich had a deep
knowledge, to deal with the quenched model of the RW.
After the first results of a perturbative approach (C. Boldrighini et al. (1997)), a complete non-perturbative
answer could be obtained for the case when the components of the dynamic environment ξ = {ξ (x, t) : (x, t)  Zd
 Z} are independent, identically distributed randomvariables (C. Boldrighini et al. (2004)). Unlike the casewith fixed environment, the quenched RW in dynamic
environment behaves almost surely as the annealed RW
in all dimensions d ≥ 1. In low dimension d = 1, 2, the random correction to the leading term of the RW asymp-

26

totics is of an “anomalous” large size. A quenched local
limit theorem was also obtained, with an explicit dependence on the field as seen from the particle. The results
were then extended to models of directed polymers indimension d > 2 below the stochastic threshold. Results
for a quenched model of RW in a dynamic environment
with Markov evolution were also obtained in dimensiond > 2 by cluster expansion methods (C. Boldrighini et al.
(2000)). Further results on models of RW in a dynamic
environment were obtained by Zeitouni, Rassoul-Agha,
Liverani, Dolgopyat and others.
Later on (in collaboration with F. R. Nardi), it waspossible to derive Ornstein–Zernike asymptotics for thecorrelations of a Markov field in interaction with a RW
(C. Boldrighini et al. (2008)) and also for a general “twoparticle” lattice operator (C. Boldrighini et al. (2011)).

R. A. Minlos with A. Pellegrinotti and C. Boldrighini, Yerevan, 2006.

In the last few years, the interest of Robert Adol’fovich
in the study of random walks in a dynamic random environment did not fade and several difficult problems weresolved. They concern extensions to continuous space(C. Boldrighini et al. (2009)), to continuous time and tothe case of long-range space correlations for the environment (in collaboration with E.A. Zhizhina).
Robert Adol’fovich was a wonderful teacher and apatient and wise mentor. Directness, accessibility andenthusiasm attracted numerous students and followers
to him. Many of his later PhD students made their firstacquaintance with special branches of mathematics andmathematical physics due to the tutorial seminar at theFaculty of Mechanics and Mathematics at MSU. There,
they benefited from direct generous contact with the
Master. This student seminar was combined with lecturesand scientific seminars guided by Robert Adol’fovich,
together with F.A. Berezin and then with V.A. Malyshev.
The lecture notes gave rise to many nice and populartutorial books, for example Introduction to mathematical statistical physics, published by R.A. Minlos in Univ.
Lect. Series, Vol. 19, AMS 2000. Many of Minlos’ former students successfully continue research in different
branches of mathematics and mathematical physics, forexample: S. K. Poghosyan and E.A. Zhizhina – spectral
theory of infinite systems and mathematical problems ofstatistical mechanics; S. Lakaev – operator spectral theory and mathematical quantum mechanics; A. Mogilner

EMS Newsletter June 2018

Obituary– mathematical biology; E. Lakshtanov – infinite particle systems; and D.A. Yarotsky – random processes andspectral theory of infinite systems.
Besides the students and the tutorial seminar, Robert
Adol’fovich was in contact with followers and co-authors
assisting the crowded Dobrushin–Malyshev–Minlos–
Sinai research seminar. There, Minlos initiated a numberof projects, often related to discussions during the seminar. Always attentive and gentle, Robert Adol’fovichshared his enthusiasm to encourage followers in solving
the problems.
In this way, Minlos launched the project “Cluster
expansions” with V.A. Malyshev. In fact, this happenedby accident when they were both in a lift while attending a seminar in the tall main MSU building. Less unusual were the origins of the projects “On the spectral
analysis of stochastic dynamics” with E.A. Zhizhina
and “Gibbs semigroups” with V.A. Zagrebnov, which infact started from questions formulated during and after
the seminar. The origins of many of them were due toactive contacts made by Robert Adol’fovich travelling toother research centres. This is, for example, the case forthe project “Application of the spectral analysis of thestochastic operator to random walks in dynamic random
environments” with C. Boldrighini and A. Pellegrinotti
and “Spectral properties of multi-particle models” with
H. Spohn, as well as “Infinite dimensional analysis” and“Stochastic evolutions in continuum” with Yu. G. Kondratiev.
To his students and collaborators, Robert Adol’fovich
was a Master, who, like a brilliant sculptor, could createa mathematical masterpiece from a shapeless block by
cutting off the excess. Sometimes, it brought not just afeeling of amazement but a sense of miracle when, asa result of some incredible expansions, evaluations, virtuoso combinations with various spaces and other technical refinements, complex infinite-dimensional and infinite-particle systems took an elegant, precise and easily
understandable form.
In this connection, problems related to the theoryof operators and to quantum physics should be especially noted. This theme began in his joint paper with
I. M. Gelfand and, since then, it has permanently been
the focus of Minlos’ attention. In 2010, together with his
old co-author and friend V.A. Malyshev, he turned to afundamental question in quantum chemistry: what is theinteraction between atoms? (Theoretical and Mathematical Physics, 162, 317–331, 2010.) However, his favourite
subject since the 1960s and until recently has been thequantum three-body problem and point interaction. A
long paper (“A system of three quantum particles withpoint-like interactions”, Russian Math. Surveys, 69, 539–
564, 2014) was published by R.A. Minlos on this topic.
A recent paper by Robert Adol’fovich was dedicated to another of his favourite subjects: the random
walk in a random environment (“Random walk in
dynamic random environment with long-range spacecorrelations”, Mosc. Math. J., 16:4 (2016), 621–640, with
C. Boldrighini and A. Pellegrinotti). His very last manuscript (with C. Boldrighini, A. Pellegrinotti and E.A.

EMS Newsletter June 2018

Zhizhina) was also on this subject: “Regular and singular continuous time random walk in dynamic random
environment”.
Robert Adol’fovich selflessly served science and, ineveryday life, was a generous and friendly person. He
gladly shared his enthusiasm and energy with his students
and colleagues. In addition to the accuracy of reasoning
and complicated techniques involved, there is always abeautiful idea and harmony in his works. It is interesting
to mention his response to the question of Natasha Kondratyeva: “What three mathematical formulas are themost beautiful to you?” He gave the answer: “The Gibbs
formula, the Feynman–Kac formula and the Stirling formula.” And those are the formulae that were widely usedby Robert Adol’fovich in his works.

R. A. Minlos, Moscow 2016 (Courtesy of E. Gourko).

Robert Adol’fovich was notable for his figurative Russian language and good wit, often with subtle mathematical humour. In the 1980s, in a conversation with Roland
Dobrushin at the Fourth Vilnius Conferences on Probability Theory and Mathematical Statistics (1985), he
expressed his doubt that “the life of a Soviet citizen iscomplete with respect to the norm of the anti-alcohol
campaign”. A campaign was ongoing at that time in the
country under the slogan “Sobriety is the norm of ourlife!” and was visible everywhere on white-red streamers.
Since then, this allusion to the completeness of life andnormed spaces has entered into the folklore of the mathematical community.
Always surrounded by relatives and intimates, andalso by loving pupils, colleagues and friends, Robert
Adol’fovich Minlos lived a complete life. In each of those
who knew Robert Adol’fovich, he left a bright drop of
memory of himself.

27

Young Mathematicians’ Column (YMCo)

Supporting Young Researcher
Families in Switzerland
Maria Podkopaeva and Olga Chekeres (both University of Geneva, Switzerland)
The life of a PhD student or postdoc in mathematics (orany other subject) can be challenging in many ways, andmore so if you have a family, especially if you have young
children. In Switzerland, there exist several institutions
that offer help in such situations. Some measures, specifically tuned to young researchers in mathematics andtheoretical physics, are proposed by NCCR SwissMAP.
The Swiss National Centre of Competence in
Research (NCCR) SwissMAP1 is a programme supporting mathematics and theoretical physics in Switzerland,
created in the context of the NCCR projects of the Swiss
National Science Foundation. It brings together around40 professors from Swiss universities and CERN andtheir groups (altogether around 200 people). In addition to its research curriculum, the goal is to attract moretalented young people to mathematics and theoreticalphysics in Switzerland. This is done via a full cycle of educational activities: from a wide variety of outreach projects for schools, some of which have been described in aprevious issue of the EMS Newsletter,2 through masterclasses (year-long in-depth Master’s level programmes
for internationally selected students), to opportunities and training for PhD students and postdocs. NCCR
SwissMAP also has several programmes for supporting
young researchers with families.

Dual careers
Everybody knows that the lifestyle of a postdoc (or astudent involved in an exchange programme) is not very
well suited for someone with a family. The positions areshort-term and spouses have their own careers to follow.
Obviously, dual career measures are more effective at ahigher level but it is not very often that both partners
work within the relatively narrow field on the border ofmathematics and theoretical physics. When it does happen (so far, in four years of SwissMAP’s existence, we’ve
had one case, with a Master’s student in mathematics
married to an exchange PhD student in physics, and one
more case is being discussed right now), we are happy tohelp by providing dual career scholarships or supporting
other arrangements.
Young children
Being a PhD student or a postdoc with a baby can be
tough. If you plan to pursue a career in maths, you haveto work on your CV, publish papers and participate inconferences, schools and workshops, often abroad. You12

http://nccr-swissmap.ch/.
EMS Newsletter No. 100, June 2016.

28

are probably also expected to teach, since part of yoursalary usually comes from being hired in your department as a teaching assistant. All this can be a major challenge if you are a young mother: first, you are out of thegame for the several months of your maternity leave andthen, even if you have a place at kindergarten, you stillneed (and want!) to spend lots of time with your child.
During their first year, there are all-night feedings, teething, upset stomachs, etc., and later on, you need to keep
up with an energetic toddler. This does not leave muchspace for research.
To help, we have introduced several schemes of support. In the mobility aspect, if you want to attend a conference or a school, SwissMAP proposes covering theextra costs for travelling with your baby or temporary
childcare while you are away. The same is offered toexternal researchers who visit events organised by SwissMAP.
Last year, we launched the “buy out of teaching”
grants.3 Once again, when you are an assistant at university, in addition to your research you have teaching3

http://nccr-swissmap.ch/articles/family-support-grants.

EMS Newsletter June 2018

Young Mathematicians’ Column (YMCo)
duties, which often take a lot of time: the TA sessions
themselves, plus the preparation, plus correcting homework… With SwissMAP family support grants, young
mothers can be liberated from their teaching (or othernon-research) duties so that they do not have to sacrifice
their research time, which is already decreased becauseof their new function as a mother. After all, research isthe priority of a scientist and, unlike teaching, it is hardto catch up if you take a long break.
The first person to receive such a grant was Olga
Chekeres, a PhD student in Geneva, whose son wasborn in February 2017. Here is what she thinks about themother/PhD student duality.
You are a mother of two, one of whom is still a baby. How
does this affect your work as a PhD student? What are themain challenges?
As a mother of two children, with one still a baby, I experience great difficulties managing time, like any working
mom I guess. Every day, I have to bring the baby to the
daycare in the morning and then not to be late to pickhim up in the evening; my effective working hours are
reduced. Also, when I am at home, normally there is no
way I can do anything for my research – do some reading
or write down something – and this is a lot of time gonefrom my PhD.
Another problem is psychological – to manage making plans. It seems simple, but it took me some time torealise that I don’t belong to myself. I have to fix it in
my mind that when I leave the office I allow myself not
to work, because this time belongs to my kids. And if it
happens that the baby goes to bed early and easily andgives me time to enjoy doing something for my project, I
would rather take this as a happy bonus, instead of making plans to get him to sleep at 21:00 and then being upset
if this doesn’t work.
And, of course, finances. The nursery consumes a significant part of the income.
Were SwissMAP programmes for supporting families
useful for you? In what way?
SwissMAP programmes were really useful. One of themliberated me from giving classes, compensating the teaching part of my salary. It really helps to focus more on myresearch, since my working time is reduced.
Another programme reimburses the expenses for a
daycare centre or a baby-sitter if I attend a conference,
which is really a great thing, allowing me to stay updated
and to participate in academic life.

to bring the baby to a doctor and then return back towork, etc., etc. And even on a daily basis, just the trajectory home – nursery – office would consume much moreworking time, which is already reduced.
Our aim is to be as flexible as possible when it comesto supporting young researchers. The schemes describedabove are already formalised and functioning but wealways keep our eyes open for new ideas and try to finda way to help in any special situations (thankfully, theSwiss Science Foundation is usually quite forthcomingwith respect to such a personalised approach).
Maria Podkopaeva is a research associate
at the University of Geneva working as ascience officer for NCCR SwissMAP. She
obtained her PhD at the University of Geneva in 2012.

Olga Chekeres is a PhD student at the
University of Geneva under the supervision of Anton Alekseev. Her research field
is mathematical physics.

Do you see other aspects of reconciling family life andyour career, where SwissMAP or your university could
help?
Actually, any financial support is always useful.
Organising a parking space for working moms might
be helpful. For me, this problem was solved with the helpof a friend, otherwise renting a place or paying per houris extremely expensive. Without a car, there is no mobility to come quickly to the nursery if anything happens,

EMS Newsletter June 201829

Research Centres

The Floer Center of Geometry
Alberto Abbondandolo (Ruhr University, Bochum, Germany)

The Floer Center of Geometry has been active at the Ruhr
University of Bochum since 2011. It is named after Andreas Floer, who was first a student and later a professor at
the Faculty of Mathematics in Bochum and whose ideas
profoundly influenced the fields of symplectic geometry
and low dimensional topology.

Andreas Floer
Andreas Floer was born in 1956 in the German town ofDuisburg, where the river Ruhr meets the Rhine, in the
middle of a densely populated mining and industrial area.
At the time of his birth, there was not a single university
in the whole Ruhr area but that situation was soon going
to change. Local and national governments had understood that coal and heavy industry were not forever andhad started to diversify. One of the first effects of thisdiversification was the foundation of the Ruhr Universityof Bochum in 1962, the “oldest of the new universities”,
that is, the first of the new public universities that werebuilt in Germany after World War II. Classes in Bochum
started in 1965 and, a decade later, Floer enrolled as a
student in mathematics. He specialised in algebraic topology and dynamical systems, two fields that were investigated in Bochum under the chairs of Heiner Zieschangand Eduard Zehnder. He received his diploma in 1982
under the supervision of Ralph Stöcker.
He then spent a year and a half in Berkeley, where hestarted his PhD studies working with Clifford Taubes andAlan Weinstein. He then returned to Bochum, where hereceived his doctoral degree in 1984 under the supervision of Eduard Zehnder. He later had research positions
at SUNY in Stony Brook and at the Courant Instituteof New York University, before becoming an assistantprofessor at Berkeley in 1988. In the Autumn of 1990,
he returned to Bochum as a full professor. He tragicallytook his life on 15 May 1991.
Floer’s work had a deep impact in symplectic geometry, gauge theory and low dimensional topology. Mostof the recent progress in these fields would be simply
unthinkable without the seminal ideas that shape what
is now known as Floer homology. Floer homology is nota single theory but rather a new approach to the study ofcritical points of certain geometric functionals and a wayof producing algebraic invariants out of them.
The first version of Floer’s theory was motivated by
the Arnold conjecture on the number of fixed points ofHamiltonian diffeomorphisms. In order to attack thisconjecture, one has to prove lower bounds on the number of critical points of a functional (the action functional from Hamiltonian mechanics), which presents severe
analytical difficulties. It is a functional defined on thespace of closed loops on a symplectic manifold and its
second differential at critical points is strongly indefinite,30

meaning that maximal subspaces on which this bilinear
form is positive or negative definite are both infinite
dimensional. Floer observed that a suitable gradientequation for this functional is given by an elliptic partial
differential equation, which is a lower order perturbationof the nonlinear Cauchy-Riemann equation that Gromov had introduced and studied in 1985. Starting fromthis observation, Floer developed a Morse theory for theaction functional based on an algebraic counting of thespaces of finite energy solutions of this elliptic equation,and eventually proved the Arnold conjecture for a classof closed symplectic manifolds. Together with Helmut
Hofer, he also realised that his Floer homology could
also be used to produce new powerful invariants fordomains inside symplectic manifolds.
Floer soon realised that his ideas could also be applied
to the study of the Chern–Simons functional, a functional on the space of SU(2) and SO(3)-connections overa three-manifold whose critical points are flat connections. This led him to the definition of the instanton Floer
homology of a three-manifold, a theory which plays amajor role in the study of four-manifolds with boundary.
A quarter of a century after Floer’s work, his ideas
permeate symplectic geometry and low dimensional
topology. Lagrangian intersection Floer homology, contact homology, embedded contact homology, Heegaard
Floer homology and Seiberg-Witten Floer homology are
just a few of the theories that grew out of the work of thisextremely talented mathematician.

EMS Newsletter June 2018

Research Centres
The Floer Center
The Floer Center of Geometry was opened in Bochum
in 2011, 20 years after Andreas Floer’s tragic death. It isbased at the Faculty of Mathematics of the Universityof Bochum and brings together the research activity ofprofessors, postdocs and PhD students working in algebra, algebraic geometry, complex geometry, differential
geometry, dynamical systems, symplectic geometry andtopology.
The opening ceremony took place in December 2011,
with colloquium talks by Helmut Hofer, Stefan Nemirovski and Andrew Ranicki, who were introduced by
Gerd Laures (initiator of the foundation of the Floer
Center of Geometry and its first director until 2013).
One of the main events of the Floer Center of Geometry is the annual Floer Lectures, which, over the last few
years, have been given by Kai Cieliebak, Alex Ritter,
Claude Viterbo, Frol Zapolski, Yakov Eliashberg, Hansjörg Geiges, Stefan Nemirovski, Ralph Cohen, Thomas
Kragh, Michael Weiss and Thomas Willwacher.
In February 2017, the Floer Center of Geometry
hosted a very special edition of “Geometric DynamicDays”, an annual event involving several German universities. The speakers of this edition, Helmut Hoferand Eduard Zehnder, have been close collaborators andfriends of Andreas Floer since his undergraduate years.
Their lectures on “Pseudoholomorphic Curves in Hamiltonian Dynamics and Symplectic Geometry” and “Thebeginnings of symplectic topology at the RUB” can bewatched online athttp://www.ruhr-uni-bochum.de/ffm/Lehrstuehle/
Lehrstuhl-VII/gdd17.html.

Since 2017, the Floer Center of Geometry offers Floer
Postdoctoral Research Fellowships: three-year postdoc
positions for young mathematicians working in the core
fields of the centre. The first postdoctoral fellow Lara
Simone Suarez, an expert in Lagrangian Floer homology
and Lagrangian cobordism, was hired in October 2017.
The Floer Center of Geometry is funded by the University of Bochum through the Faculty of Mathematics
and by diverse grants of the Deutsche Forschungsgemeinschaft (DFG). It serves also as a platform to coordinate the activities of the Collaborative Research Centre on “Symplectic Structures in Geometry, Algebra andDynamics”, a collaboration scheme funded by the DFG
and based at the Universities of Bochum and Cologne.

References[1] Floer Center of Geometry’s web site. http://www.floer.rub.de/index.html.en.
[2] S. K. Donaldson. On the work of Andreas Floer. Jahresber. Deutsch.
Math.-Verein., 95(3):103–120, 1993.
[3] Helmut Hofer, Cliord H. Taubes, Alan Weinstein, and Eduard
Zehnder, editors. The Floer Memorial Volume, Progress in Mathematics Vol. 133. Birkhäuser Verlag, Basel, 1995.

Alberto Abbondandolo [alberto.abbondandolo@rub.de] is a professor of analysis at the Ruhr University of Bochum andthe Director of the Floer Center of Geometry. He works in symplectic geometry, dynamical systems and global analysis.

The next edition of the Floer Lectures will take place on
21–22 June 2018, with four talks by Victor Guillemin andPaul Seidel.

New journal published by the

ISSN print 2520-2316
ISSN online 2520-2324
2018. Vol. 1. 4 issues
Approx. 400 pages.
17.0 x 24.0 cm
Price of subscription:
198 ¤ online only /238 ¤ print+online

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

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 ofPennsylvania, 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), Andrea Montanari (Stanford
University, USA), Jelani Nelson (Harvard University, Cambridge, USA), Philippe Rigollet (MIT,
Cambridge, USA), Rachel Ward (University of Texas, Austin, USA)
Aims and Scope
Mathematical Statistics and Learning will be devoted to the publication of original and highquality 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.

EMS Newsletter June 201831

Societies

The Mathematical Society of Serbia
Vladimir Mićić and Zoran Kadelburg (University of Belgrade, Serbia)
This year, the Mathematical Society of Serbia (MSS) (or Društvo
Matematičara Srbije in Serbian)celebrated its 70th anniversary. But
is this a long or short period?

Preliminaries
Liberation and restoration of the State of Serbia (in thefirst half of the 19th century) was followed by a serious
movement toward educational and scientific development in the Principality as well as the Kingdom of Serbia. A number of Serbian mathematicians had PhDs
from European scientific centres (e.g. Paris, Wien andBudapest) and provided an essential contribution to thisdevelopment, which included the advancement of mathematical sciences at university. In the final decades of the
19th century and the first four decades of the 20th century, one can see the presence of Serbian mathematicians(e.g. Mihailo Petrović, Nikola Saltikov, Anton Bilimović,
Radivoje Kašanin, Tadija Pejović and Jovan Karamata)
as authors of papers in scientific mathematical journals,
as well as participants of the ICM and the ICME, with
Mihailo Petrović present at the founding meeting of
the IMU in 1920. In 1926, a “Mathematical Club“ at the
University of Belgrade was founded, directed by Anton
Bilimović; it dealt with the scientific achievements anddiscussions of members, including their applications for
publication in journals. In 1937, this club had turned into
the “Yugoslav Mathematical Society“, gathering together approximately 100 members with Tadija Pejović as itspresident. An additional list of mathematical publications,
intended for secondary school students (“Matematičkilist”) and researchers in mathematics (“Matematički
vesnik”), as well as activities of the Union of Mathematics Students, complete this sketch of mathematical life in
Serbia during the period before the Second World War.
Foundation of the MSS – the first years
The foundational meeting of the society took place on 4
January 1948 in Beograd; Tadija Pejović was elected as its
first president. There was a basic decision to realise threeparallel activities: scientific work in mathematics, mathematics education, and popularisation of mathematics. As
physicists were included, the same
scheme was also projected for physics (till 1981). At the very beginning,
the scientific journal Vesnik Društva
matematičara i fizičara Srbije wasestablished and Jovan Karamata wasnamed the editor-in-chief; its publication started in 1949. As Matematički
Vesnik, it has been a permanent presence in the scientific life of Serbia
Tadija Pejović
since 1964. The MSS initiated the32

foundation of the Union of Yugoslav Societies of Mathematicians andPhysicists (UYSMP) and organised
the first Congress of Mathematicians
and Physicists of Yugoslavia. These
initiatives were realised in 1949 and,later on, UYSMP represented all thesocieties on international boards andorganisations, including the IMU, theICMI and the BMU (Balkan MathJovan Karamata
ematical Union). Publishing the necessary journals and other publications (intended for the whole country) was initiated andpromptly realised. Nastava matematike i fizike, covering
problems of education at all levels, has been publishedin Beograd since 1952 (it is now modified and entitled
Nastava matematike).

The years that followed
The original scheme of MSS activities, enriched andmodernised, has been preserved in subsequent years.
Scientific section
In the scientific domain, the MSS has been permanently
present as a responsible standard-bearer in a wide rangeof activities (including international connections and collaboration), such as: publication of the scientific journal;
regularly organised congresses of mathematicians; symposia and seminars (national and international) considering actual problems in several mathematical disciplines;
organisation of a number of scientific sessions; and coordination of collaborations between various participantsin the scientific life of the country (mathematical departments, institutes and other partners).
The scientific journal Matematički Vesnik mentioned
in the introductory section has been in existence for 69
years (with 210 issues so far) and has been an importantstage for the publication of original research papers, thepresentation of new results and achievements in mathematical sciences and the provision (till 1980) of information about the activities of the MSS. The editorial
board has been (and is) of international character, as is
the majority of invited referees. The list of authors hasvaried from predominantly domestic to predominantly
foreign authors; at present, it is somewhat balanced.
The complete list of editors-in-chief is Jovan Karamata,Dragoljub Marković, Zlatko Mamuzić, Dušan Adnadjević, Zoran Kadelburg, Mila Mršević, Ljubiša Kočinac and
Neda Bokan. The journal is available in electronic formand can be found at www.vesnik.math.rs.
Scientific meetings, organised by the MSS itself or bysome other institution with an essential MSS contribution, can be classified into (a) national congresses (13so far with the 14th in preparation), Yugoslav as well

EMS Newsletter June 2018

Societies
as Serbian, with significant international participation,and (b) international seminars and symposia. The latter
have covered: Differential and Partial Differential Equations; Numerical Solution of Differential Equations;
Coordination of Mathematics and Physics Instruction;
the 5th Balkan Congress of Mathematicians; Topology
and its Applications (five symposia); Complex Analysis
and Applications (three symposia); and Mathematical
Analysis and its Applications (two symposia with thethird in preparation). One should take into account thatthe majority of these meetings were organised during
the time of the Cold War (before 1990), a period whencontact, even between scientists, was controlled and, ina way, extremely reduced. As Yugoslavia (and therefore
Serbia) was a non-alignment country, we were able toinvite colleagues from both sides of the “Iron Curtain”
and to initiate creative official and private meetings andfurther collaboration. In the shortened list of foreign
participants, one can recognise the names of numerous
significant mathematicians of this period: L. Ahlfors,
L. Aizenberg, P. S. Aleksandrov, T. Ando, M. Antonovskij, V. G. Boltjanskij, K. Borsuk, G. Choquet, R. Courant, A. Csaszar, W. Hengartner, G. Henkin, E. Hille,
Y. Komatzu, I. Korevaar, W. Rudin, G. Sansone, W. Sierpinski, M. H. Stone, M. Vuorinen and V.A. Zorič. Since
the 10th congress (Beograd 2001), the scientific committees of the congresses have awarded special prizes foryoung researchers (to Danko Jocić, Vladimir Dragović,Božidar Jovanović and Dragana Cvetković-Ilić).
Another in the range of scientific activities led by the
MSS was a serious contribution to originating a number of “schools”, characterising topics of mathematics
achievement in Serbia, such as topology and set theory,complex analysis and distribution theory.
Particular thanks go to certain members of the MSS:
Vojin Dajović, Djuro Kurepa, Konstantin Orlov, Zlatko
Mamuzić, Bogoljub Stanković and Milica Ilić-Dajović,due to whose enthusiasm and essential contribution themeetings and activities mentioned
have been successfully realised.

Vojin Dajović

Djuro Kurepa

Teaching section
Systematic efforts toward improvement of teaching and learning inmathematics have been present in
MSS activities from the very beginning. In the early 1960s, thanks to theinitiative and personal devotion ofVojin Dajović, an official documentof the state authorities was accepted
containing explicit recommendations for “paying special care to themathematics education and development of the mathematical culture”.
This was a strong impulse for generalaffirmation of mathematics. It has
been followed by further confirmation of programmes with a sciencemathematical specialisation in highschools, including the foundation

EMS Newsletter June 2018

in Beograd of a specialised Mathematical High School
and an increase of interest in studying mathematics. The
first steps introducing numerical methods and computer
programming into the curricula of secondary schools anduniversities were made during these years, too.
Nastava matematike, the MSS Serbian language journal (and its bulletin) continued to be published, with theaim of including theoretical problems of learning andteaching practices in mathematics, the exchange of goodpractices, information about tendencies in mathematics educational practices around the world, etc. The listof editors of the journal over 65 years of mathematical life in Serbia is: Ivan Bandić, Djuro Kurepa, Milica
Ilić-Dajović, Miroslav Živković, Milosav Marjanović,
Vladimir Mićić and Zoran Kadelburg. The MSS scientific journal The Teaching of Mathematics was founded
in 1998 for research papers in mathematics teaching and,over the past 20 years, has acquired wide support in Serbia and abroad. The international character of both theeditorial board and the majority of invited referees canbe taken as an additional guarantee of permanency. The
complete list of editors is relatively short: Milosav Marjanović, Vladimir Mićić and Zoran Kadelburg. The active
site of the journal is www.teaching.math.rs.
In the early 1960s, traditional “state seminars for professional improvement of primary and secondary schoolmathematics teaching” were established (from the late1980s also including informatics). Their programmesconsist of a few plenary lectures, several special working sections and a number of master classes. In recent
years, since the process of systematic licensing of teachers within the educational system has started, the character of these seminars has been, in some way, formalised
and their role in the professional progress of mathematics and informatics teachers has been significantly
improved. They are supplemented by regional seminarsof similar character. The permanent increase in the number of master classes in informatics is evident.

Popularisation section
Ever since the founding of the MSS, systematic activities
have been present for the identification, fostering anddevelopment of students, at all levels, who are gifted andinterested in mathematics. A range of “out of the classroom” activities have been realised. Following the practiceof mathematical societies and other institutions in many
countries, the necessary literature was published and asystem of competitions was organised. For secondary
school students, the journal Tangenta has been publishedsince 1995. The same function for primary school pupils
was assigned to Matematički list, published in Beograd
since 1967. These two journals have been accompanied by
55 issues of the series Materijali za mlade matematičare,which is intended for young mathematicians. At least
twice a year, MSS organises a “Summer (Winter) schoolfor young mathematicians”, a kind of master class forinterested students. They have been organised by MSS
branches from mathematical centres of Serbia, the majority of them by the Valjevo branch and managed by Vojislav
Andrić, the current president of the MSS.

33

Societies
From the first unofficial competition for secondary
school students (Beograd, 1958) to participation at the
7th IMO (Poland, 1963), the route was direct and prompt.
This was similar for higher primary school pupils, aged
between 9 and 15, who started with a Serbian competition in 1967. Their first participation in an international
event took place in 1997 at the Junior Balkan Mathematical Olympiad, held in Beograd and organised by
the MSS. Young mathematicians of Serbia, through the
MSS, are regular participants of the famous international
competition “Kangaroo without frontiers” (the numberof participants has increased to 36,000 in 2018).
Thanks to its specific geopolitical position, the MSS,
as the host of the 9th IMO (1967), was able to invite several countries from Western Europe to participate; withthe presence of young mathematicians from the United
Kingdom, France, Italy and Sweden, the IMO seized theopportunity for a mathematical competition with representation from more than just Eastern European countries. The next contribution of the MSS to the broadeningof the IMO movement was realised in 1977, when the
MSS, as host of the 19th IMO, invited mathematiciansfrom some non-European countries; with the presenceof young mathematicians from Algeria, Brazil and Cuba,
the IMO turned into a worldwide event. These organisational steps were accompanied by some very good resultsfor Serbian students at the IMOs, including 15 gold med-als, with Teodor von Burg achieving four of them, which
resulted in him holding first position in the Hall of Fameof the IMO for several years.
The Mathematical Society of Serbia will continue allof its activities over the forthcoming years. In particular,
in 2018, the 14th Serbian Mathematical Congress will beorganised, the 35th Balkan Mathematical Olympiad will
be held in Beograd and there will be an event marking
the 150th anniversary of our great mathematician Mihailo Petrović.
Vladimir Mićić (1936) [vladimic@mts.rs] isa professor of advanced engineering mathematics and didactics of mathematics at the
University of Belgrade. He has been activewith the MSS for 55 years. He is an ex-president of the society and is its first Honoured
Member.
Zoran Kadelburg (1950) [kadelbur@matf.
bg.ac.rs] is a professor of mathematical analysis and functional analysis and is an ExDean of the Faculty of Mathematics at the
University of Belgrade. He has been involvedin the complete range of MSS activities for
50 years and is an ex-president of the society.

MATRIX - Call for Programs
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Program proposals are due by Friday 9th November 2018. Applications can be submitted athttps://www.matrix-inst.org.au/call-for-programs/
Further enquiries: office@matrix-inst.org.au

34

EMS Newsletter June 2018

Raising Public AwarenessIs the System of Scientific Publications
on the Eve of a Revolution? And if so,
Toward What?
Marie Farge (École Normale Supérieure, Paris) and Frédéric Hélein (Université Paris Diderot - Paris 7, France)

Today, researchers benefit from an extremely powerful electronic distribution system for scientific articles,
which allows any researcher (whose institution has paid
the subscription fees) to access, via the internet, most of
the articles they need, instantaneously and regardless of
one’s location. Electronic publishing has indeed enabledlibraries to subscribe electronically to a very large number of journals, through so-called big deals, at rates thatwere initially affordable, especially in cases where theselibraries partnered in consortia to negotiate with publishers.
However, behind this simplicity for the researcher
lies a digital infrastructure much more complex than thatof traditional libraries acquiring and preserving printed
articles on paper and, behind the apparent impression of
gratuity that the researcher may have, there are exorbitant and constantly increasing costs and contracts thathave become opaque. The creation of journal bundles
has resulted in pricing that no longer has any relation
to the real cost of production for the publisher, and thedematerialisation of articles has allowed commercial
publishers to multiply tolls according to usage: pay toread the papers to which your library has subscribed thisyear, pay to read the archives of journals to which it subscribed in previous years, pay to publish in Open Access,etc. At the same time, electronics has allowed large publishers to make enormous reductions in the costs of producing an electronic paper (and any researcher knowsthat the raw material and its evaluation are provided
free of charge).
In economic terms, this development has mainly benefited big publishers, whose profit margins far exceed
those of pharmaceutical companies or banks, but it isincreasingly weighing on public finances. Indirectly, it
is also the small publishing houses, private or academic,
that bear the costs because it is by cancelling subscriptions with these small publishers that libraries manage
to balance their budgets. This maintains a vicious circle,
which results in a growing concentration of scientificjournals in the hands of the largest publishers. These
economic mechanisms have had an impact on the quality of scientific publications, as major publishers have acommercial interest in encouraging the proliferation ofjournals and articles. A visible effect of this is that many
scientific communities are saturated by the number of
papers and the quality of peer review has fallen. Anothereffect is the multiplication of predatory journals (there
are nearly 10,000 today, all disciplines combined).

EMS Newsletter June 2018

The Open Access project
Since the early 2000s, and particularly since an initiative [1] launched in Budapest in 2002, researchers havedreamed of building a model for distributing articles
free of charge for all (thanks to the internet) and thus
freeing themselves from the big publishers. Although at
first glance this dream seemed within reach, it is far frombeing realised as, even today, more than 80% of articles
are not published in Open Access. The first reason is thatany system allowing rapid dissemination (e.g., the arXiv)
has a cost, even if it is small in comparison to the one
charged by publishers.
Publishers, who initially feared the Open Accessmovement, have managed to turn the Open Access project to their advantage by proposing a model in which
the author pays publication fees, often called “APC” (forArticle Processing Charges), so that their article is immediately Open Access. However, it would be more correct
to call APC a licence fee (which we will do in the following) because the amount usually has no relation to thereal cost of the service.
There is therefore still a long way to go to build anOpen Access publishing model, in which researchers andtheir institutions regain full possession and free use of thedata they themselves have produced, and in which thecosts associated with their publication and dissemination
are charged at the right price. Indeed, the transition is tremendously complex to set up between, on the one hand,
extremely well organised multinationals and, on the other hand, scientists and institutions, divided according totheir disciplines, their institutions and their countries andwhere dialogue is not necessarily well organised nor does
information circulate very well.
Moreover, a fundamental obstacle is that scientificjournals combine several essential functions, inheritedfrom the age of printing, namely:- Evaluation process. This task is mainly carried out by
researchers, as members of editorial boards or as referees. It should, however, be noted that this task involves
secretarial work, which is usually carried out by a secretary, but also sometimes by researchers, who may beassisted by software. Secretariat funding is provided by
the publisher or a public institution, in varying proportions depending on the journals.
- Label provided by the reputation of the journal.
- Referencing and notification of an article. Several actors contribute to this function: the journal itself, data­

35

Raising Public Awarenessbases, such as Web of Science and Scopus for mostdisciplines and MathSciNet and Zentralblatt for mathematics, and libraries, as well as institutional reference
systems that are developing.
- Diffusion and promotion.
- Archiving.
The latter two functions can be performed by journals
or by archive platforms, such as arXiv, but also, as far asarchiving is concerned, by public institutions and nonprofit organisations, such as CLOCKSS [2] and PKPLOCKSS [3], not forgetting libraries for printed versions.
It can be seen that almost all of these functions areservices that can be the subject of a call for competition
on a market, or be taken over by public institutions or
non-profit organisations. All of them are services except… the label of a journal, which is attached to recognitionby the community and which is thus unique. This is thereason why the market for scientific journals is essentiallywithout competition, and this is the main explanation forprices unrelated to the cost of services. As long as journals combine these functions, it will always be difficult tobring subscription prices down to reasonable levels.

Where do we stand?
After two rather calm decades, it seems that we havereached an unstable situation, and are probably on theeve of great upheaval. Indeed, tough negotiations withpublishers are multiplying (as we will see later), institutions are unsubscribing from big deals (see a list of
institutions that have cancelled their subscription to big
deals in [4]) and the Sci-Hub pirate platform is enjoying
worldwide success, with hundreds of thousands of illegal downloads of articles per day. In Europe, the pressure is increasing because the European Union has set,
under Horizon 2020, the objective of free disseminationof European scientific production. Finally, the development of alternative solutions is accelerating, as we shallsee next.
An overview of countless innovations
Open Archive platforms like arXiv [6], which was created in 1991 by Paul Ginsparg, were long confined tocertain science disciplines, such as high-energy physics,
1

Open archive platforms were developed in the last century
by researchers to share their preprints and reprints (note thatthose years publishers where providing reprints to authors
for free and asked them to send them to colleagues for advertising). The SPIRES High Energy Physics database, developed at SLAC-Stanford University in the ’70s, was the firstmade accessible via the Web in 1991 and replaced by Inspire[10] in 2012. It was followed by the database ADS [5] developed for astrophysics in 1988 and transferred to the Web in
1994. In 1991 arXiv [6] was created by Paul Ginsparg at Los
Alamos National Laboratory for physics. In 1994 Michael
Jensen founded SSRN [7] for social sciences and humanities(in 2013 it was the largest open repository in the world andin 2016 it was sold to Elsevier). In 1997 PubMed [8] was designed for medicine and RePEc [9] for economy.

36

computer science, mathematics and economy.1 Today,
the situation is changing rapidly as new platforms arebeing created, such as bioRxiv [12] in biology in 2013,
ChemRxiv [13] in chemistry in 2016, and EarthArXiv[14] and ESSOAr [15] in geosciences in 2018. In addition,
recent agreements between institutions to accelerate
the development of these archives and to federate them,
together with initiatives such as ASAPbio [16], are aimed
at encouraging researchers in biology and medicine todeposit their preprints on public archives. These initiatives have an international dimension.
This rise in the power of Open Archives can be largely, but not only, explained as a reaction to abusive tariffs
by major publishers or to licence fees for publishing inOpen Access. It also appears that this flexible and commercially unfettered method of dissemination is incomparably faster than the journal system and better protects researchers from plagiarism.
Open Access journals without licence fee, often referred
to as Diamond Open Access journals [17], continue togrow. The existence of such journals is most often basedon projects, such as Episciences [18] or Cedram [19](whose activities are expanding with the creation of theCentre Mersenne [20]) for mathematics. Similar projects
exist in all disciplines (especially in the humanities, which
is at the forefront of this movement), such as: the Public
Knowledge Project [21], which develops Open Journal
Systems, an open source software to publish journals;
organisations such as Knowledge Unlatched [22] andOpen Library of Humanities [23], which propose crowdfunding for the publication of Open Access books andjournals; OpenEdition [24], which provides platforms
for books, journals and blogs; and the OA Cooperative
Project [25]. In biology, the mega journal PLOS One[26], supported by a non-profit organisation, is an Open
Access journal that was free of publication fees in its early days (but unfortunately now charges publication feesof the order of $1,500 per article). In Latin America, theSciELO [27] platform, founded by Brazil, includes 1285
journals, most of which are Open Access. These projects
are supported by various foundations and organisations,such as LingOA [28], MathOA [29], the Foundation
Compositio Mathematica [30], SPARC [31] and SCOSS[32].
These Open Access journals can either be a newjournal, or an already existing title whose editorial
board resigns or leaves a commercial publisher so thatthe journal can continue in an open framework. This iscalled emancipation of a journal [33]. The most notable
example of emancipation of a mathematical journal was
Topology, which became in 2006 the Journal of Topology, but there have been other journals before and after,
including Acta Mathematica (since 2017). A list of emancipated Open Access journals can be found on the Journal Declaration of Independance [34]. A list of Open
Access mathematical journals without publication fees
can be found on the Cimpa website [35]. (See also, forexample, the Directory of Open Access Journals [36] forall fields.)

EMS Newsletter June 2018

Raising Public Awareness
Lastly, there are some very interesting innovations interms of evaluation, including Open Peer Review [17, 37].
The principle is to organise the evaluation of an article
in an open way, i.e. by making public the reports of thereferees, the authors’ responses to the referee and eventhe contributions of other researchers. Such a system maygive rise to fears, but many variants exist, and the result
will depend on the quality of the editorial board thatoversees this process and on the adjustment of its details(in particular, one can choose to keep references anonymous and to make public only the positive reports, inwhich case one should really speak of recommendations).
Among the first experiments on Open Peer Review
are Copernicus in 2001 [38] and F1000Research [39] (anon-profit organisation currently with low publicationfees) and SciPost [40] (without publication fees); they
seem to lead to a significant improvement in the qualityof reviews. A similar experience, perhaps even more innovative, is “Peer Community in…” [41], a recommendation
platform that is not a journal, offering positive evaluations of preprints or articles. These projects take up an
idea already proposed in the Self Journal of Science project [42], which has unfortunately had difficulties getting
off the ground. The interest in these latter projects is todecouple evaluation (and its associated label) from therest of the services.
We could also mention the European Digital Mathematical Library [43], and the platform Dissemin [44]which detects papers behind pay-walls and invites theirauthors to upload them in one click to an open repository, in order to boost open access while respecting legal
constraints [44]. Another interesting innovation is the
DOAI (Digital Open Access Identifier) which is an alternate DOI (Digital Object Identifier) resolver that points
to a free version of the requested article, when available,
instead of its version under pay-wall [45].
The previous list of innovations, institutions and tools
is by no means exhaustive since this would be impossible, as new innovations are emerging every month. The
term bibliodiversity has been proposed to refer to thisproliferation and led to the “The Jussieu Call” [46], an
initiative aimed at supporting these alternative solutions,which has been signed by more than 100 institutions
from 13 countries.

The temptation to contract with major publishers for Open Access publishing(This section reproduces a large portion of the article by
the authors, “Transition vers l’Accès Libre: le piège desaccords globaux avec les éditeurs”, which was published in
the French newspaper Mediapart on 14 April 2018 [47].)
A worrying policy in some countries is to conclude
national agreements with large multinational publishing companies to pre-pay licence fees for Open Accesspublishing. As it happens, the countries at the forefrontof this movement are essentially those in which these
global publishing giants are mostly established, namely
the United Kingdom, the Netherlands and Germany. We
are therefore entitled to ask ourselves whether this is a

EMS Newsletter June 2018

coincidence and we propose here hypotheses to interpret
this “coincidence”.

Let us put ourselves in the shoes of a multi­
national scientific publishing company
To understand this situation, it is useful to adopt the
point of view of the major commercial publishing groups
and ask what would be the most effective strategy forthese groups to adapt to these changes whilst maintaining or increasing their profits:- Firstly, preserve current income, without rushing but
simply by taking advantage of the inertia of the systemthat exists today, with more than 80% of articles published in journals with subscription, whilst multiplying
the tolls to access articles on the electronic platformsof the publishers.
- Then, invest for the future by developing, just like
Google, new “services” based on algorithms to mine thelarge amount of data accumulated. This is the direction
taken by Elsevier, the largest multinational publishingcompany, which buys back several start-ups that havedeveloped such services each year. The amount of dataprocessed can come both from the vast scientific corpus contained in papers and from the data concerning
the researchers themselves (as authors or experts). In
this “market”, Elsevier competes with Clarivate (formerly Thomson–Reuters).
- Finally, build an Open Access model that is stable andthat guarantees a firm’s profits to be as large as thosethey currently make with subscriptions. We will see
how delicate this operation is.
Indeed, for the major publishing groups, Open Accesshas remained, since 2000, both a source of additional
income and a source of concern.
- It is a source of additional income thanks to the payment of licence fees by researchers (or their institutions) for each article published in Open Access(noting, in passing, that these fees may correspond toarticles that are either published in journals that are
totally Open Access or hybrid journals, i.e. for whichlibraries already pay for subscriptions!).
- It is a source of concern because of the risk that researchers and institutions might emancipate themselves from the current system of journals by building their own Open Access system (for example, thisconcern can be seen directly through Elsevier’s shareprice, which fell at the time of the Budapest Initiative in2001 and then on other similar occasions). To avoid thisrisk, these multinationals must build a stable Open Access publishing model that allows them to collect fees
as cost-effectively as the current subscription model
but, at the same time, does not provoke a hostile anddestabilising reaction from researchers and their institutions.
With the current subscription system, a publisher
receives, on average, more than 5000 euros per article

37

Raising Public Awarenesspublished, this revenue being even higher in the case of
Elsevier. Therefore, to maintain comparable profit margins with the Open Access publishing system, such a
publisher would have to charge royalties at an average
price of the same order of magnitude. On a laboratory
scale, such rates would result in an extremely high bill(and in mathematics, even charging 2000 or 2500 euros,
the fees for all items in some laboratories, would absorb
the entire budget!). Such a solution is therefore unrealistic. Moreover, even if the budgetary limits of the laboratories were disregarded, this would have a devastating
effect on the publisher’s image because it would reveal
to researchers, in a concrete way, the scandalous levelof fees charged, of which, in the majority of cases, they
are unaware. Moreover, the fees give those who control
the credits excessive power, which is likely to degrade
relations between colleagues. Finally, the development ofsuch a model risks, even before it has come to an end,
provoking an acute awareness and reaction from thescientific community, which would compromise the commercial publisher’s projects. Indeed, there are now more
and more initiatives from the scientific community that
could well be the premise for such a reaction.
The best solution for these publishers therefore
remains to obtain payment of fees from their traditional interlocutors, libraries, which already have large
budgets capable of supporting expensive subscriptions,and national agencies or institutions. Thus, the fees
will not weigh directly on laboratories’ budgets, willremain invisible to researchers and, even better, will“smoothly”replace the astronomical sums that libraries
are used to paying. Publishers will thus be able to continue their “business as usual” in complete security.
The first country where such a model was tested was
the United Kingdom. The British have been engaged
since 2013 in a transition programme toward Open
Access, combining the use of Open Archives and thepayment of fees to publishers. This required the establishing of a complex protocol for the institutions and thecreation of a special fund to finance the additional costs
involved. Despite a political will to control overall costs(subscriptions and fees), it is clear that no expenditure
could be contained. Worse still, the system of reimbursement of fees to universities by the special fund has created a bureaucracy whose cost has added to the bill. Thereis therefore quite a bit of discontent with this system in
the United Kingdom. This experience has encouraged
the major publishing groups and countries tempted bythis direction to move towards global agreements on acountry scale, the bill of which would be paid by libraries.
This is the path taken by the Netherlands, by concluding a first agreement at the end of 2014 with Springer that
integrates a subscription to a bundle of journals with theright for Dutch researchers to publish Open Access at no
additional cost (these are therefore included in the subscription invoice). Similar agreements have since beenconcluded with other publishers and in other countries:in Austria and Germany at the end of 2015, in Sweden in
2016, in Finland in 2018, etc. The type of contract varies
but there is a shift toward contracts in which the propor-

38

tion of fees for Open Access publications is becoming
increasingly important. Thus, from the end of 2016, Germany embarked on an even more radical path: insteadof wanting to conclude mixed agreements, concerning
both subscriptions to read and fees to publish, Germany
believes that it is no longer a question of paying to readbut just paying to publish.
In any event, it is striking to observe that this movement toward global contracts including Open Accessfees (which, as we have seen, is certainly the most satisfactory solution for multinational publishing companies) is developing mainly from the United Kingdom,
Germany and the Netherlands and in the geographicalarea around these countries. But it is also striking to note
that, with the exception of the American Chemical Society, the main publishing companies (Elsevier, Springer
Nature, Wiley and Taylor & Francis) are precisely located in these three countries. It should also be noted that
small scientific, commercial or academic publishers are
not involved in these agreements and risk, once again,
paying the price for these developments.
There may be several explanations for this coincidence: the result of lobbying by these publishing firms
cannot of course be excluded, but the most plausible
explanation is the conflict of interest situation in whichthese three countries naturally find themselves. Indeed,public institutions in these countries must certainly try
to reduce, if not contain, the bill they pay to publishers
but, at the same time, we can assume that it is difficult
for them to make choices that would harm the multinationals based in these countries, not so much for fear of
weakening them as for fear that these companies would
threaten to relocate their activities to other countries.
In Germany, the Max Planck Society (Max Planck
Gesellschaft), the main German research institution,
which has played an active role in promoting Open
Access since 2003, is also in an ambiguous situation.
Stefan von Holtzbrinck, who owns more than half ofSpringer­Nature’s capital, is a member of its Board of
Directors. Stefan von Holtzbrinck is also the president
and co-founder of the Max Planck Förderstiftung, a foundation that financially supports the Max Planck Society.
With regard to Germany, it is important to distinguish
two trends that are developing in independent directions:- On the one hand, the German institutions have joined
forces within the DEAL consortium [48] to negotiate
hard with Elsevier, Springer Nature and Wiley, in order to obtain significant price reductions and to obtaintransparent contracts (since, until now, the details ofthese contracts have remained confidential, an opacity that benefits publishers). Having failed to reach an
agreement with Elsevier at the end of 2016, the consortium decided that, as of January 2017, all contracts with
Elsevier that were due to expire would be terminated,
resulting in a standoff with this publisher. Of course,
we can only applaud this exemplary determination.
- On the other hand, as we have seen, the Germans
demand contracts that guarantee them free access tojournals but they agree to pay to publish in Open Ac-

EMS Newsletter June 2018

Raising Public Awarenesscess. This radical position is inspired by a strategy developed by the Max Planck Society, which is the subjectof a promotional campaign inviting institutions aroundthe world to rapidly switch to an Open Access publication system, even if this means paying licence feesin advance. The Max Planck Society thus proposes toinstitutions around the world to commit to this pathby signing the Initiative OA2020 for Open Access 2020[49] – not to be confused with the H2020 programmeof the European Union for financing research! Notethat this initiative also calls for transparency of costs– currently refused by publishers – and for a certain
diversity of models.

tries by signing agreements with major publishing groups
that would include the advance payment of licence feesfor Open Access publishing. We must avoid this scenario,
which would further strengthen the hold of these large
groups.
In addition, the transition to Open Access must not
replace commercial barriers to read with barriers to publish. Research results and, more generally, data producedby public institutions must stop being privatised, which
does not exclude using private providers to disseminate
and make them visible.

References[1]

For the moment, no final agreement with the publishers
has been concluded. It seems that the reason for this is,
not so much the reverse model (paying to publish insteadof paying to read), but the price to pay and the complexity of the deals. As long as publishers are not asked tosignificantly reduce their turnover, they have every reason to be satisfied to the extent that they are offered astable solution. Indeed, Springer and Wiley affirm theirwill to bring these negotiations to a successful conclusion
and, even if relations with them are much more tense,
Elsevier clearly indicated in a note [50], made public in
September 2017, that they do not disagree with the principle but on the price and details of the implementationof the changeover.
On the other hand, the selling point [51] underlying
the OA2020 initiative, promising a significant reduction in
tariffs, unfortunately seems simplistic and it is hard to see
what serious element could support it once one market
without competition is replaced by another market without competition. On the contrary, the risk seems greatthat the result will be the creation of a new Open Accesspublishing model in which large companies will be able tocontinue to dictate their financial terms (see [52]).

As a conclusion
As we have seen, Germany has been engaged since January 2017 in a tough negotiation with Elsevier. Likewise,
the French institutions, grouped within the Couperin
consortium, have recently followed a similar approachwith Springer (but for a big deal subscription contractthat does not include Open Access). Failing to obtaina significant reduction in Springer’s rates, these institutions terminated their contracts in January 2018 … until
this publisher become reasonable. Finally, many institutions [53] around the world have simply cancelled theirsubscriptions to big deals. All these steps may signal achange in the power balance. It is interesting to note thatin Germany, after cutting access to institutions that had
terminated their contracts for a few weeks, Elsevier finally reinstated them in February 2017. Similarly, in France,
Springer has not cut off access to French institutions
since January 2018.
On the other hand, there is a risk that in many countries, decision-makers and library managers, eager tomake a transition to Open Access, will give in to temptation to follow the examples of northern European coun-

EMS Newsletter June 2018

[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
[30]
[31]
[32]
[33]
[34]
[35]
[36]
[37]
[38]
[39]
[40]
[41]
[42]
[43]
[44]
[45]

Budapest Open Access Initiative: http://www.budapestopen
accessinitiative.org/read.
CLOCKSS: https://www.clockss.org/.
PKP-LOCKSS: https://pkp.sfu.ca/pkp-lockss/.
SPARC Big Deal Cancellation Tracking: https://sparcopen.org/
our-work/big-deal-cancellation-tracking/.
Astrophysics Data System: http://adsabs.harvard.edu/.
ArXiv: https://arxiv.org/.
SSRN: http://www.ssrn.com/en/.
PubMed: https://www.ncbi.nlm.nih.gov/pubmed/.
Research Papers in Economics (RePEc): http://repec.org/.
Inspire: http://inspirehep.net/?ln=en.
PubMed: https://www.ncbi.nlm.nih.gov/pubmed/.
bioRxiv: https://www.biorxiv.org/.
ChemRxiv: https://chemrxiv.org/.
EarthArXiv: https://eartharxiv.org/.
ESSOAr: https://www.essoar.org/.
ASAPbio: http://asapbio.org/about-2.
M. Farge, Scholarly publishing and peer-reviewing in open access,https://zenodo.org/record/1227093.
Episciences: https://www.episciences.org/.
Cedram: http://www.cedram.org/.
Centre Mersenne: https://www.centre-mersenne.org.
Public Knowledge Project: https://pkp.sfu.ca/.
Knowledge Unlatched: http://www.knowledgeunlatched.org/.
Open Library of Humanities: https://www.openlibhums.org/.
OpenEdition: https://www.openedition.org/.
OA Cooperative Project: http://oa-cooperative.org/about.html.
PLOS ONE: http://journals.plos.org/plosone/.
SciELO: http://www.scielo.org/php/index.php?lang=en.
LingOA: https://www.lingoa.eu/.
MathOA: http://www.mathoa.org/.
Foundation Compositio Mathematica: https://compositio.nl/
stichting2.html.
SPARC: https://sparcopen.org/.
SCOSS: http://scoss.org/.
I. Daubechies, IMU blog: https://blog.wias-berlin.de/imu-journals/.
Journal Declarations of Independence: http://oad.simmons.edu/
oadwiki/Journal_declarations_of_independence.
CIMPA: https://www.cimpa.info/en/node/62.
DOAJ (Directory of Open Access Journals): https://doaj.org/.
J. Bordier, OpenEdition experiments open peer review protocols
for scientific articles, https://oep.hypotheses.org/1661.
Copernicus: https://publications.copernicus.org/.
F1000Research: https://f1000research.com/.
SciPost: https://scipost.org/about.
Peer Community in…: https://peercommunityin.org/.
Self Journal of Science: http://www.sjscience.org/.
EuDML: https://eudml.org/.
Dissemin: https://dissem.in/.
Digital Open Access Identifier: http://doai.io/.

39

Raising Public Awareness[46] Jussieu Call: http://jussieucall.org/index.html.
[47] https://blogs.mediapart.fr/edition/au-coeur-de-la-recherche/
article/130418/transition-vers-l-acces-libre-le-piege-des-accordsglobaux-avec-les-editeur.
[48] DEAL: https://www.projekt-deal.de/about-deal/.
[49] OA2020: https://oa2020.org/.
[50] Elsevier: https://www.elsevier.com/connect/working-towards-atransition-to-open-access.
[51] R. Schimmer, K. K. Geschuhn, A. Vogler, Disrupting the subscription journals’ business model for the necessary large-scale transformation to open access, http://pubman.mpdl.mpg.de/pubman/
faces/viewItemOverviewPage.jsp?itemId=escidoc:2148961.
[52] https://poynder.blogspot.fr/2018/03/the-open-access-big-dealback-to-future.html.
[53] SPARC: Big Deal Cancellation Tracking, https://sparcopen.org/
our-work/big-deal-cancellation-tracking/.

Marie Farge [marie.farge@ens.fr] is a research director at CNRS-INSMI and has
worked at LMD-ENS Paris since 1981. She
has a Master of Science from Stanford University, a PhD in physics from Université
Paris VII and a PhD in mathematics from
Université Paris VI. She studies turbulent
flows of incompressible fluids and hot plasmas by direct

numerical simulation and their applications in geophysics,
nuclear fusion and aerodynamics of insect flight. In 1988,
she pioneered the use of wavelets in fluid mechanics andparticipated in the creation of the journal ACHA (Applied
and Computational Harmonic Analysis) in 1992. She has
been member of the Ethics Committee of CNRS, memberof RISE (Research Innovation and Science Experts) of
the European Commission, and is a member of the Council of the Academia Europaea.
Frédéric Hélein [helein@math.univ-parisdiderot.fr] is a professor at the University
Paris Diderot. He first worked on the analysis of harmonic maps between Riemannian
manifolds, the asymptotic analysis of theGinzburg–Landau equation and the studyof some completely integrable systems in
differential geometry. His recent work concerns mathematical physics, especially the study of the Hamiltonian
structure of field theories and gauge theories within a differential geometric viewpoint. He is the scientific chair of
the French Réseau National des Bibliothèques de Mathématiques and participates in negotiations with publishers
and in various discussion groups working on scientificpublications policy.

Can Statistics Predict the Fields Medal
Winners?
Adam Bannister and Olaf Teschke (FIZ Karlsruhe, Berlin)
With the upcoming ICM in Rio de Janeiro, the seasonal
speculation about who will receive the 2018 Fields Medals at the opening ceremony is once more in full swing.
With the big data industry measuring us in all possible ways, a natural question might be whether statistical approaches could possibly predict the committee’s
choice of the Fields Medallists. We undertake someexperiments here to see which predictions are provided
by standard approaches based on data from zbMATH
and linked data sources.

“The limits of statistics are illustrated by the numbers
contained in the penultimate line of [BD14, Table 1].
(It is not commented on in the paper.) While most of
the Fields Medallists and contenders are happily alive,
Figure 3 shows a disturbing pattern about those whohave passed away […Namely, an average age of deathof 74.0 for Fields Medallists compared to only 66.3 forcontenders…] Thus, if you got a Fields Medal, you can
expect to enjoy your extra US$120,000 per year foralmost eight more years.”Fields Medal is small data
One obvious obstacle, however, is that the set of Fields
Medallists is small by its very nature and may easily defy
statistics with all kinds of outliers. As a nice example, one
could recommend reading Borjas-Doran’s study on a statistical decline of Fields Medallists’ productivity [BD14]
and Kollár’s amusing review [K15]. Without further discussing the fundamental problem of measuring a mathematician’s productivity by publication and citation numbers – the fallacies of this approach have been frequently
discussed in the newsletter, for example in [BT17] – wejust note here the very last observation in [K15]:Firstly, we may take this as an illustration of how seemingly exact science is often perturbed by possibly unreliable data. It was, for us, impossible to reproduce theaverage age of death of 74 from [BK14] ([K15] does
not comment on this figure). Submitted in 2014 before
the death of Grothendieck, the nine Fields Medallists
deceased at that time reached an average age of 78.5. (Acloser look at the appendix of [BK14] reveals that the1936 medallists Ahlfors and Douglas seem to have been
excluded from the study but that has almost no effecton this average). Secondly, as we are all sadly aware, thisfigure has been significantly affected since then by the40

EMS Newsletter June 2018

Raising Public Awarenesspassing of several medallists, reducing the average age
at death by more than four years and wiping out a largepart of this statistical effect.

Which data can reasonably be taken into
account?
There is basically no formal limitation to being a Fields
Medal candidate except for the famous age rule. However, even this simple condition requires some work –for most of the (approximately) one million authors in
zbMATH, the age is simply unknown. Reasonable estimates are often possible based on publication history
but in many cases may lead to gross errors: while quite
a few mathematicians have already published relevant
research in their teens, others may be over 30 at theirfirst zbMATH entry. This happens frequently in borderareas when most publications are outside the scope ofthe database or for mathematicians suffering from political suppression, e.g. from the Nazi regime, Stalinist terror
or during the Cultural Revolution in China.
Needless to say, birth dates derived from publication history are therefore not suitable to check therather clear-cut Fields Medal age limit. Fortunately,
the zbMATH author database is linked to many othercollections, some of which – like Wikipedia, MacTutor,
GND and MathNet.Ru – provide birth date information.
Additionally, the communities pursuing these services
usually do a reliable job in relevance decisions. Overall,
zbMATH contains links to data resources providing sufficient age information for almost 13,000 authors – only
a fraction of the overall million but covering all Fields
Medallists and likely candidates. Only 252 of them are at
most 40 years old.
The harder task is to identify relevant features for thestatistical model. Modern databases like zbMATH offer
various facets that can be taken into account – not justquantitative publication and citation information but also
granular subject information, or co-author and reference
networks. Unfortunately, many of these quantities can bedual-edged: a high publication number may be obtainedby people like Yau, Bourgain or Tao, or by more notorious
representatives of the class of prolific writers.1 High citation numbers may be related to a lasting impact of results
but they are also much affected by subject and community
custom. Even more importantly, they come with a massive
time delay [BT17]. Subject information is certainly valuable (statistically, the Fields Medals are far from evenly distributed within MSC subjects) but will usually not reflect
breakthroughs that may define new areas in the future.
Publication sources are certainly meaningful – prize winners will almost inevitably have a distinctive record in the
Annals, Inventiones, etc. – but are significantly limited bypublication delay, with many relevant results appearing
only after several years (the committee of course being
aware of them). Close collaboration or citation distance1

Currently, Yau, the Fields Medallist with the most publications in the zbMATH database (authoring on average apaper every two weeks over the past few years), ranks onlyaround 50th place in this list.

EMS Newsletter June 2018

to former prize winners may indicate that you are actively
involved in pursuing cutting-edge research but could also
be an indication of a supporting role rather than a unique
individual effort qualifying for the medal.
Less ambiguous features would be existing prizes likethe EMS Prizes (which are also connected to an age limit
and have a distinct overlap with later Fields Medallists
from Europe) but many prizes have a shorter history
than the Fields Medal, as well as regional restrictions,
thereby further complicating the involved statistics. Fortunately, a substantial list of prize winners is available
for analysis via the zbMATH connection with Wikidata;
others (like the EMS prizes) have been added manually.
The same holds for the information on being an invited
speaker to an ICM, which may reasonably be treated likewinning a global prize.
Finally, we emphasise that no data generated by user
searches were taken into account due to our strict dataprotection policy [HT14]. As outlined there, one could
expect rather distinctive results, especially if IP information were analysed (which is ruled out). A rough approximation might be obtained by taking Google search data
into account (although this would most likely not reflect
the committee’s procedure well). Currently, this would
see Simon Brendle, Hugo Duminil-Copin, Alessio Figalli, Ciprian Manolescu, Fernando Codá Marques, Sophie
Morel, Peter Scholze, Maryna Viazovska and Geordie
Williamson as the most likely candidates (in alphabeticorder, with Peter Scholze leading).2 A closer look at thetrends indicates that most of the queries are correlated toprize announcements, hence one might expect that this iscovered by the above features.

Methodology
Educated humans will usually overcome most of theseobstacles, e.g. a closer look will easily distinguish deep
results from superficial mass publications with bulk references. Automatic recognition is, however, still limited
in addressing such questions. Approaches like neuralnetworks have made tremendous progress over the past
years but still encounter problems, for example in distinguishing art from pornography (a somehow related question), despite the fact that technology in image processing has become more advanced and much more data are
available. Some tools to recognise “maths pornography”
might help editors, reviewers and readers but there has
not been much activity toward this yet. Moreover, bigdata approaches would ideally require billions of samples as training data, far more than the currently available mathematical publications (although several groupsof authors, in an often undervalued effort, are very
active in enlarging the available datasets). The problemof scarce data applies not just to bibliometrics but evenmore to the other features mentioned, so there currently
seems no hope of applying neural network technology to
the Fields Medal prediction.
2

This kind of crowd-sourced projection also agrees well withcertain internet polls, e.g. https://poll.pollcode.com/44839318_
result?v.

41

Raising Public Awareness
Instead, we just put the available data into a support
vector machine model. We defined data slices for the
information available at the time of the congress. First,
we trained a model based on previous years’ winners. We
then used this model to analyse our candidates for thisyear. We repeated this procedure 20 times and averaged
the results to remove any outliers created by an imbalance in the splitting of testing and training data. Thisaveraging of the results is due purely to the small sample
of data available to train a model; in some runs, we could
be unlucky enough to get no positive examples in our
training set.
As we have used prizes first awarded in the 1990s, we
also had to limit our Fields years to 1994 or later. We took
into consideration the EMS Prizes, the Bôcher Memorial
Prize, the Coxeter-James Prize, the Fermat Prize, the SASTRA Ramanujan Prize, the Oswald Veblen Prize, the Clay
Research Award, the Wolf Prize and the Salem Prize.
As a by-product, we obtained measures for the significance of the different features.

In contrast, other prizes and ICM invitations are themost predictive sole features, which produce distinctive
projections and were more than 97% successful for past
test sets. By adding further features like collaborationand citation distance to former Fields Medallists, prize
winners and invited speakers, the success rate for test sets
can be improved further (as is natural when dimensions
are added) up to greater than 99.3% but with decreasing sharpness of prediction. The differences also indicate
a possible bias toward more collaborative communities,
after adding the distance features, and a bias against
recent, yet unpublished achievements. Table 1 shows thefigures for the leading contenders in the respective models.3
From this, one might reasonably predict that Peter
Scholze is a strong favourite to win a Fields Medal but theothers remain highly competitive, with different models
producing very different outcomes. Geordie Williamson
and Bo’az Klartag seem to have the most consistent statistical chances from the field.

Results
Not unexpectedly, sole bibliometric features turn out
to be almost non-predictive. In a model that takes justcitation figures, journals or MSC subjects into account(the latter two features should be at least included toadjust citations numbers [BT17]), one can generate ahigh-dimensional (due to the variety of journals medallists have published in) linear model that is adjusted tothe past but generates only individual winning probabilities of about 1% and less in the projection (with Jeremy Blanc, Anton Koroshkin, Luis Pedro Montejano
and Evgeny Sevostyanov leading by slim digits a basically even field). Hence, citation-based hiring will most
likely lead to missing a future Fields Medallist (actually,
it will perform at most marginally better than randomly
picking a mathematician younger than 40 years with a
Wikipedia entry).

Does the committee’s composition matter?
Of course, the decision is solely made by the committee
members, whom we can expect to weight mathematical achievement over superficial facets. Since assuming
responsibility for the committee, the IMU has put much
effort into creating a balanced composition of prize committee with respect to aspects like geography or researcharea, and the difficulty of obtaining significant projections may serve as a good illustration. Of course, thecomposition of the 2018 committee cannot be used for
projections since it is revealed only at the ICM (except
for IMU president Shigefumi Mori, who is an ex-officio
3

Important caveat: Since we didn’t use 4-years age slices of in
the model to avoid more sparsity effects, the resulting probability reflects the chance of winning a Fields Medal in the
future, not necessarily at the next Congress.

Table 1. Projected winners in different models

Peter Scholze
Geordie Williamson
Bo’az Klartag
Simon Brendle
Hugo Duminil-CopinPeter Pal Varju
Sophie Morel
Alessio Figalli
Ciprian Manolescu
Maryna Viazovska
Fernando Coda Marques

42

Prize

Prize +
Invitation

Prize +
Invitation +Coauthor

Prize +
Invitation +Coauthor +Citation

All features

64%
56%
50%
49%
5%
5%
6%
5%
5%
1%
< 1%

81%
82%
38%
30%
6%
6%
6%
6%
6%
1%
2%

91%
25%
15%
2%
14%
11%
9%
9%
9%
9%
1%

34%
10%
14%
< 1%
11%
2%
< 1%
4%
3%
1%
< 1%

86%
2%
16%
1%
20%
16%
6%
2%
< 1%
< 1%
3%

EMS Newsletter June 2018

Mathematics Education
member). However, one may ask whether the knowledgeof the composition of past committees4 would have significantly influenced the projections. The figures show
only modest changes when adding the committee information, hence a significant “committee bias” cannot be
confirmed via this statistical approach.

Conclusions
There are several facets of public information available that may serve as features for statistical predictions
about Fields Medal winners but many come with certain
disadvantages. Taking different reasonable models into
account, the formal statistical approach may provide
some educated guesses with reasonable probabilities but
a rather high uncertainty remains, certainly sufficient tokeep the tension about the disclosure of the winners at
the ICM.
Perhaps the most important caveat is, however, thatthe statistical method will only succeed in carrying forward past trends to the future. As is well known, this is
one of its major drawbacks, which may preserve or evenworsen existing discriminations [O16]. Due to these4This information is available at the IMU site; the authors liketo thank the MathOverlow community for clarifying a question related to the 1962 committee.

effects, we didn’t include available data features like gender or country of origin into the model because this would
almost certainly generate further intrinsic bias. Since thecomposition of the Fields Medallists has grown significantly more diverse over the past few years (reflecting
the development of the mathematical community), statistical predictions will most likely have a conservative
bias compared to the actual decisions and the committee
will likely succeed in proving statistical guesses at least
partially wrong.

References[BT17] A. Bannister, O. Teschke, An update on time lag in mathematical references, preprint relevance, and subject specifics. Eur.
Math. Soc. Newsl. 106, 37–39 (2017; Zbl 06853068).
[BK14] G. J. Borjas, K. B. Doran, Prizes and Productivity: How Winning
the Fields Medal Affects Scientific Output, J. Human Res. 50,
No. 3, 728–758 (2015).
[K15] J. Kollár, Is there a curse of the Fields Medal? Notices Am.
Math. Soc. 62, No. 1, 21–25 (2015; Zbl 1338.01086).
[O16] C. O’Neil, Weapons of math destruction. New York, NY: Crown
Random House (2016; Zbl 06801031).
[HT14] J. Holzkämper, O. Teschke, Guarding your searches: data protection at zbMATH. Eur. Math. Soc. Newsl. 92, 54–55 (2014; Zbl
1302.68110).

See page 4 for a photo and biography of the authors.

ICMI Column – The Mathematical
ICMI Column – The Mathematical
Legacyof Jean-Pierre
Jean-PierreKahane
KahaneLegacy ofJulien Barral (Paris, France), Jacques Peyrière (Université Paris-Sud, Orsay, France)
JulienBarral(Université
Paris 13,
France), Jacques(Université
Paris-Sud, Orsay, France)
and
Hervé
Queffélec
(Université
desVilletaneuse,
Sciences et Technologies
de Lille,Peyrière
Villeneuved’Ascq, France)and Hervé Queffélec (Université des Sciences et Technologies de Lille, Villeneuve d’Ascq, France)
Jean-Pierre Kahane (1926–2017), an alumnus of École Normale Supérieure, was appointed as a professor at Montpellier
University just after his PhD thesis (prepared at CNRS) andat the Faculté des Sciences d’Orsay in 1962, which was then apart of the Sorbonne and later linked to Université Paris-Sud.
He has been the recipient of numerous awards (the enumeration of which would be tedious) and he was a member of the
Académie des Sciences de Paris.
His particularly intense and fruitful scientific activity did
not prevent him from assuming various positions to the benefit of the community: for instance, he served as the President of Université Paris-Sud, the President of the MIDIST (aGovernment agency) and the President of the CIEM. Again,
it would be tedious to list all the responsibilities he has assumed.
He was fully concerned with the teaching of mathematics.
Indeed, he thought of it as intimately linked with research. We
refer to the preceding article by Michèle Artigue for an account on this subject. He also had a special interest in the history of mathematics. In particular, he revisited some of Plato’s
writings.

EMS Newsletter June 2018

Jean-Pierre Kahane acted as the advisor for numerous
PhD theses. He was a patient and attentive advisor; he fully
respected the personality of his students. He was a talentedorganiser and coordinator. He is one of the founders of themathematics department at Orsay. He created and energiseda seminar and a team. Due to his international fame, he wasable to attract the most world-wise renowned visitors to Orsay.
His first studies deal with the connections between themean periodic functions of Laurent Schwartz, entire functions
and quasi-analyticity. Of course, this echoes the preoccupations of Szolem Mandelbrojt, his PhD advisor. From him, heinherited a taste for Dirichlet series, on which he kept an eye
throughout his career.
Soon, he won international renown, as shown by thelist of his early collaborators. Nevertheless, in the 1960s,
Jean-Pierre Kahane was (in France) a somewhat singularmathematician, or at least he felt so. The preface of his
book with Raphael Salem, “Ensembles parfaits et séries
trigonométriques”, is the best description of his view of mathematics. Here is an excerpt:43Mathematics Education
Il y a quelques dizaines d’années, ce livre aurait pu se passerde cette préface, qui est écrite en guise d’apologie. Aujourd’hui,
venant à un moment où la plupart des mathématiciens – et les
meilleurs – s’intéressent surtout aux questions de structure, ilpeut paraître suranné et ressembler en quelque sorte à un herbier. Les auteurs se doivent donc d’expliquer que leur proposn’est en aucune façon réactionnaire. Ils savent la beauté desgrandes théories modernes, et que leur puissance est irremplaçable, car sans elles on serait souvent condamné (comme l’a dit
Lebesgue) à renoncer à la solution de bien des problèmes à énoncés simples posés depuis fort longtemps. Mais ils pensent que,sans ignorer l’architecture qui domine les êtres mathématiques, il
est permis de s’intéresser à ces êtres eux-mêmes qui, pour isolés
qu’ils puissent paraître, cachent souvent en eux des propriétés,qui, considérées avec attention, posent des problèmes passionnants . . .

The objects that this text refers to are generically called thinsets (Kronecker, Sidon, Helson sets, uniqueness sets, etc.), thestudy of which has been very intensive in his team for morethan a decade.
This book was published in 1962, with a second issue in
1994. It is worth noting that such a preface would nowadays
be unnecessary; indeed, this view of mathematics, in particular due to Jean-Pierre Kahane, is widespread.
He left a huge volume of work, of which the main topics
are:• Harmonic analysis and Fourier series, in connection withfunctional analysis and number theory.
• Probabilistic methods in analysis: fine structure of Brownian motion and construction of fractal sets.
• Harmonic analysis methods in analytic number theory.
• Mandelbrot martingales and multiplicative chaos.
Rather than try and describe all of his contributions in thislimited space, it is preferable to put the stress on his works
that have had the greatest influence. Most of his works are easily accessible via the book “Selected works”(Kendrick press,2009).
The Kahane–Katznelson–de Leeuw theorem
If the sequence (cn )n∈Z is square summable, there exists a
continuous 2π-periodic function such that, for all n ∈ Z,|f (n)| ≥ |cn |. This brings a definitive conclusion to lots of previous works. The proof mixes probability and combinatorics,
a cocktail that he was fond of. Probabilities (the Khintchineinequalities) give | 
f (n)| ≥ |cn | and f ∈ p<∞ L p . The difficulty
is to get f ∈ L∞ . This is achieved via the use of combinatorics.
This result teased several followers: Kisliakov showedthat this result still holds for the disc and bidisc algebras. The
problem seems to still be open in higher dimension; later on,
in 1997, Françoise Piquard proved a similar result for infinite
matrices.
The Khintchine–Kahane inequalities
This is a highly non-straightforward extension of Khintchine
inequalities to vector random variables. The idea is expressed“à la Kahane” in his book “Some random series of functions”:
If the probability that a sum of the type Rademacher be large issmall, then the probability it be very large is very small.

More precisely, if M is the supremum of the norms of partial
sums of a Rademacher series rn xn (where the xn lie in an

44

arbitrary Banach space) then, for t > 0 large enough, one hasP(M > 2t) ≤ P(M > t)2 . It results that the sum S of thisseries belongs to the Orlicz space Lψ1 , where ψ1 (x) = e x − 1.
This seems not to be as good as in the scalar case, where the2correct Orlicz function is ψ2 (x) = e x − 1. But, as Kwapien
showed, this result is self-improving so as to give the optimal
Orlicz function ψ2 .
Slow points of Brownian motion B(t)This is a long story. One dimensional Brownian motion was
known to have, with probability 1, continuous trajectories, almost uniformly Hölder 1/2 (not 1/2, of course, because of theiterated logarithm law). It was also known (Paley–Wiener–
Zygmund) that it is a.s. everywhere non-differentiable andeven everywhere non-Hölder α if α > 1/2. Dvoretzky, asked
by Kahane about the case α = 1/2, proved in 1963 the following result. Set|B(t + h) − B(t)|L(t) := lim sup·|h|1/2h→0Then, a.s. for all t, L(t) > 0. In 1974, Orey and Taylor provedthat L(t) = ∞, and a little more, can happen (such points are
so-called “fast points”). Shortly afterwards, by using an astonishingly inventive method, Kahane proved that slow points
exist, i.e. points at which the Hölder exponent is 1/2. The
proof is detailed in the second issue of his book “Some random series of functions”.
Solution of the Bateman–Diamond conjecture
J.-P. Kahane has always been fascinated by the issue, in thejob of a mathematician, of the direction in which one shouldlook for an answer? No, and you look for a counterexample.
Yes, and you look for a general proof . . . In this respect, heoften quoted Carleson’s theorem (the solution of the Lusin
conjecture) on almost everywhere convergence of Fourier series of square-summable functions on the circle. According tohim, Carleson rather believed the answer to be negative. But
each attempt in this sense had run into a stone wall. Here was
Kahane’s comment in 2011:
Finally, the walls he ran into nearly built a POSITIVE answer tothe Lusin conjecture!

Quite in this spirit, he obtained, in 1997, when over 70, a positive answer to a conjecture of Bateman and Diamond about
generalised prime numbers of Beurling. Here are some details. You start from a discrete and multiplicatively independent subset P of the half-line (1, +∞) (the generalised prime
numbers) and then consider the multiplicative semi-group N
generated by P (the generalised integers), as well as their respective counting functions
P(x) = card P ∩ [1, x]

andN(x) = card N ∩ [1, x].

Beurling showed that by setting N(x) = Dx + xε(x), with Da positive constant, the assumption ε(x) = O (log x)−α forsome α > 3/2 is enough to imply the Prime Numbers Theorem (PNT):
x.
P(x) ∼
log x
In this context, the less demanding condition
 ∞

 dx
<∞
(1)
ε(x) log x 2x1

EMS Newsletter June 2018

Mathematics Education
naturally emerges. Bateman and Diamond asked whether
Condition (1) implies the PNT. After failing to build a counterexample and having analysed the reasons of that failure,
Kahane was led, in Carleson’s style, to a positive answer tothe conjecture. The (quite elegant) method uses Fourier analysis and, notably, the non-vanishing of ζ, the zeta function
naturally associated with N, on the line s = 1; this is by interpreting the hypothesis as the membership of t → ζ(1+it) in
the Sobolev space H 1 and making use of the local propertiesof functions in that space, notably the propertyf ∈ H 1 ⇒ | f (t0 + h) − f (t0 )| = o(|h|1/2 ).
(Surprisingly enough, this is somewhat reminiscent of slow
points of Brownian motion.) Jean-Pierre Kahane was particularly proud of this achievement.
The Mandelbrot martingales
Let c ≥ 2 be an integer and W a non-negative random variableof expectation 1. We consider a family W j1 , j2 ,··· , jn , indexed by
the finite sequences of numbers between 1 and c, of independent random variables equidistributed with W. Each sequence j1 , j2 , · · · , jn determines a c-adic subinterval I j1 , j2 ,··· , jnof [0, 1]. Let µn be the measure whose density with respect
to the Lebesgue measure on each interval I j1 , j2 ,··· , jn is the constant W j1 W j1 , j2 · · · W j1 , j2 ,··· , jn . Its total mass is the random variable
Yn = c−n(2)
W j1 W j1 , j2 · · · W j1 , j2 ,··· , jn .
1≤ j1 j2 ··· jn ≤cThis is a non-negative martingale so it converges a.s. to a random variable Y. Of course, the expectation of Y does not exceed 1. It may happen that Y = 0 with probability 1. When the
expectation of Y equals 1, the sequence of measures µn has,with probability 1, a non-trivial weak limit µ. This construction was devised by B. Mandelbrot to statistically describe
turbulent fluids – in this case, µ is interpreted as energy dissipation – or, more generally, to account for intermittency. So, it
was crucial to determine whether the expectation of Y differsfrom 1 and Mandelbrot constructed a few conjectures. These
have been answered positively by Kahane. Here are the results.
– For p > 1, the martingale (2) converges in L p if and only if
E W p < c p−1 (1974).
– The martingale converges in L1 if and only if E W log W <log c (1976).
– When this martingale does not converge in L1 , it completely degenerates, i.e. E Y = 0.
The 1974 result was already difficult but the one in 1976 is atrue feat of strength.
These works had longlasting influence. Other proofs havebeen devised and lots of generalisations have been studied.
Multiplicative chaos
In 1974, B. Mandelbrot introduced the previous martingales
as a toy model for energy dissipation in fully developed turbulence. The more realistic model he had considered consistedof a basis-free lognormal multiplicative process, which was
rigorously founded by Kahane in “Sur le chaos multiplicatif”
in 1985, where Gaussian multiplicative chaos theory is elaborated. Then, Kahane unified both models in the more abstract

EMS Newsletter June 2018

T -martingale setting, of which there is a short presentation
here.
One considers (Ω, A, P) a probability space, (T, d) a locally compact metric space andQ = (Qn : T × Ω → R+ )n≥1

a sequence of measurable maps such that for all t ∈ T ,(Qn (t, ·))n≥1 is a martingale of expectation 1.
Given σ any Radon measure on T , the sequence (Qn σ)n≥1
vaguely converges almost surely to a Radon measure Qσ and,as distributions E(Qn σ) and σ satisfy
E(Qσ) ≤ σ

(by Fatou’s Lemma),

if Qσ  0 with positive probability, one says that Qσ is nondegenerate. Basic questions are then:
1. When is Qσ non-degenerate and when does the equality
E(Qσ) = σ hold? In the case of degeneracy, how can Qσnbe renormalised to get a non-trivial limit, at least in law?
2. Suppose Qσ is non-degenerate.
• Which moments of the total mass of Qσ restricted tocompact sets are finite?
• What is the Hausdorff dimension of Qσ?
• What is the multifractal nature of Qσ, i.e. what are theHausdorff dimensions of the sets


 





log Qσ B(t, r)E(α) = 
t
∈
T
:
lim
inf
=
α
?



+


r→0
log(r)


Gaussian mutiplicative chaos as defined by Kahane corresponds to
n



1
Pk (t), with Pn (t) = exp Xn (t) − EXn2 (t) ,Qn (t) =2
k=1

where Xn (t) are independent Gaussian centered random functions on T . The distribution of Pn depends only on the correlation functionpn (t, s) = E Xn (t)Xn (s)and the distribution of Qσ only depends on∞q(t, s) =pn (t, s)n=1

and not on the order of summation if the pn are non-negative.
This non-trivial achievement is based on fundamental inequalities that Kahane obtained in this context. Let (Xi )1≤i≤nand (Yi )1≤i≤n be two centered Gaussian vectors such that
∀ i, j,E(Xi X j ) ≤ E(Yi Y j ).

Then, for all non-negative weights (pi )1≤i≤n and all convex(resp. concave) functions F : R+ → R, with almost polynomial growth at infinity,




n
n
 
 
1
1
2 
2 


E F
pi eXi − 2 E(Xi )  ≤ (resp. ≥)E F
pi eYi − 2 E(Yi )  .
i=1

i=1

A case of particular interest is when the Gaussian field∞
n=1 Xn (t) (seen as a distribution) is log-correlated, i.e.
1q(t, s) = u log+
+ O(1) (u > 0).
(3)d(t, s)Then, if1

sup N diam(B), B < ∞,2
B
where N(δ, B) stands for the covering number of the closed
ball B by closed balls of radii δ, Kahane shows that:45

Mathematics Education(1) If dim T < u/2 then for all σ one has Qσ = 0.
(2) If dim T > u/2 then the infimum of the dimensions ofrandom Borel sets on which the non-vanishing measures
Qσ are concentrated is dim T − u/2.
Moreover, when (T, d) is the Euclidean space Rd and σ =Lebesgue then dim T < u/2 can be replaced by dim T ≤ u/2
in (1). These results are partly based on Peyrière’s approachto the dimensions of limits of Mandelbrot martingales andKahane’s observation of a deep principle of composition for
multiplicative chaos.
Questions of non-degeneracy, moments, dimension and multifractal analysis were then solved for various classes of models of T -martingales (works by Kahane, Holley and Waymire,
Molchan, Falconer, Fan, Barral and Mandelbrot, Bacry andMuzy, Rhodes, Sohier and Vargas, and Barral and Jin), including log-correlated Gaussian multiplicative chaos.
The answer to the renormalisation question came recently
for Mandelbrot martingales (in works by Aidekon and Shi,
Webb, Madaule, and Barral, Rhodes and Vargas) and for somelog-correlated Gaussian multiplicative chaos on Rd (in works
by Duplantier, Rhodes, Sheffield and Vargas, and Madaule,
Rhodes and Vargas). Both situations led to similarly beautiful
results. In the latter case, if we take into account the dependence of Qn in the parameter u, if Xn ∼ N(0, 1) and σ isthe Lebesgue measure√then, in the critical case u = 2d, thenormalised sequence n(Qu,n σ) weakly converges in probability to a positive continuous randommeasureµ of dimen√
√
√
3 u
√

( √u − d)2 nsion 0, while if u > 2d then n 2 2d e 2(Qu,n σ) weakly
converges in law to a purely atomic stable random measure,
whose atoms are placed according to a Poisson point processwith intensity given by µ. Note that complex T -martingales
and their renormalisation have also been investigated.
Gaussian multiplicative chaos theory was further developed over the last decade with two related motivations:constructing a theory associating random measures to nonnegative kernels q(t, s) of positive type but not necessarily of
σ-positive type (works by Robert and Vargas, Duplantier andSheffield, Rhodes and Vargas, Shamov and Berestycki); anddeveloping a rigorous probabilistic theory of quantum field
theory (works by Duplantier, Miller and Sheffield, David, Kupianen, Rhodes and Vargas, and Aru, Hung and Sun). It turns
out that, in certain cases, not only can one characterise thefiniteness of moments but one can determine the law of thetotal mass! (See recent works by Kupianen, Rhodes and Vargas, and Rémy.)Random coverings, multiplicative chaos and subordinators
Multiplicative chaos is related to some random covering problems, which were among the favourites of Kahane. Alongwith his students Billard and El Hélou, he was among themain protagonists (with Erdös, Hawkes, Mandelbrot andShepp) in the study and solution of a random covering problem on the circle raised by Dvoretzky in 1956, and of its extension to tori. Kahane also worked on the closely related andmore tractable Poisson covering problem in Euclidean spaces
introduced by Mandelbrot (1972). Let us discuss the lattermodel in dimension 1.
Denote the Lebesgue measure on R by λ and fix a positive Borel measure µ on (0, ∞). Consider a Poisson point pro-46

cess of intensity λ ⊗ µ, which is a random discrete subset of
R × (0, ∞) such that the number of its point in a given Borel
set B is a Poisson r.v. of parameter λ ⊗ µ(B) and the numberof points associated with disjoint Borel sets are independent.
Denote the points of this process by (xi , yi ) and associate tothem the open intervals (xi , xi + yi ). Let G = i (xi , xi + yi ).
The problem is to determine whether G = R a.s. or not. The
answer was given by Shepp (1972), simultaneously with theanswer to Dvoretzky’s problem:
G = R a.s. or G  Ra.s. according towherek(t) = exp

 ∞
t

 ∞
0

e−t k(t) dt = ∞ or not,µ(y, ∞)dy.

Kahane revisits this result in the late 1980s. He gives a new
proof based on a stopping time idea due to Janson and alsouses a potential theory approach to get the following finer
result. If K is a compact subset of R then K ⊂ G a.s.
 or K  n≥1 i: yi <1/n (xi , xi + yi ) a.s. according to whether
Capk (K) = 0 or not. Multiplicative chaos comes into playwhen Capk (K) > 0, i.e. there exists a Borel probability measure σ on K such thatk(t − s) dσ(t)dσ(s) < ∞.
(4)
K2
Indeed, for n ≥ 1, setting Gn = i: yi ≥1/n (xi , xi + yi ) andQn (t) =1 − 1Gn (t),1 − P(t ∈ Gn )(4) is equivalent to L2 -convergence of the K-martingale(Qn σ)(K).
Now, consider G+ = i: xi >0 (xi , xi +yi ). In the same period,
Kahane shows that when
 1
 1

exp
µ(y, ∞)dy dx < ∞,0xthere exists some Radon measure σ supported on R+ suchthat Qσ is supported on R+ \ G+ , the set of uncovered points,
and Qσ is the image of the Lebesgue measure by some Lévy
subordinator. This implies a result by Fitzsimmons, Firstedt
and Shepp (partially proved by Mandelbrot), claiming that,in this case, R+ \ G+ is the closure of the range of this subordinator but provides a new interpretation in terms of multiplicative chaos. In particular, an interesting consequence of
this approach is that local times of Brownian motion are theindefinite integrals of limits of R-martingales. It should be
noted that the topic of random coverings is currently particularly active.
J.-P. Kahane’s influence turned out to be amazing. Firstly,
he was always ready to discuss and talk about mathematics. We remember J. P. Serre asking him: “We were wondering if we could get from you a talk on Brownian motion
next week?” He instantly replied: “I agree." And he delivered
several talks at the Bourbaki seminar, notably on Beurling’s
works, to whom he first asked for agreement. Facing a lackof reply from the latter, he sent a message: “In three days,
I present your work at the Bourbaki seminar,” no longer asking for agreement. He then received agreement with an abrupt

EMS Newsletter June 2018

Mathematics Education“OK”. Much more recently (in 2014), J. P. Kahane delivered
a talk on Brownian motion, white noise and Langevin’s equation at the “BNF” (Bibliothèque Nationale de France), in frontof a large audience comprised of many teenagers. His aptitude
for “doing mathematics" and writing equations, while remaining sufficiently general to avoid the young listeners getting
lost, was impressive.
J.-P. Kahane wrote four advanced books:• “Ensembles parfaits et Séries Trigonométriques” with
R. Salem (1963); second edition in 1994. This book has notaged. In particular, it provides a clear account on capacities
and Hausdorff dimension.
• “Séries de Fourier absolument convergentes” (1970). This
book contains magnificent applications of Banach algebras(Shilov idempotents theorem) to the zero sets of analytic
functions in the Wiener class.
• “Some random series of functions” (1968, reissued in1985). This book is still an obligatory reference.
• “Séries de Fourier et Ondelettes” with P. G. Lemarié (1998,
reissued in 2016).
This last work shows the taste and ability of J.-P. Kahane forhistory, focusing on the development of ideas rather than on
persons. Before the wavelet part, written by P.-G. Lemarié,
he writes about the contributions of the great precursors:
Euler, Fourier, Dirichlet, Weierstrass, Cantor, Riemann andLebesgue. His analysis of their ideas and achievements is very
exciting.
Jean-Pierre Kahane will remain an example thanks to his
enthusiasm, his quest for invention and his preoccupation to

explain and transmit his knowledge and experience. His influence will continue for a long time.
Julien Barral [barral@math.univ-paris13.fr]
is a Professor of Mathematics at Paris 13
University. He is Chairman of Laboratoired’Analyse, Géométrie et applications (CNRS
UMR 7539). He graduated from Paris-Sud
University, where he got his PhD under
Jacques Peyrière supervision. He works in probability, ergodic theory and fractal geometry, with a special interest to
multifractals.
Jacques Peyrière [jacques.peyriere@math.
u-psud.fr] is an Emeritus Professor of Mathematics at Paris Sud University (Orsay). His research interests are Harmonic Analysis, Probability Theory, and Fractals. He is a former
student of Jean-Pierre Kahane.
Hervé
Queffélec
[herve.queffelec@
univ-lille1.fr] is now an Emeritus Professor at the University of Lille, after having
been a “maître de conferences” at the University of Orsay till 1992 and a Professor at
the University of Lille 1 from 1992 to 2011.
His research concerns are harmonic analysis in connection
with analytic number theory and Dirichlet series, as well as
Banach spaces of analytic functions and their operators. He
is a former student of Jean-Pierre Kahane.

ERME Column
Renaud Chorlay (Sorbonne Université, Paris, France) and Jason Cooper (Weizmann Institute of Science,
Rehovot, Israel)

ERME Thematic Working Groups
The European Society for Research in Mathematics Education (ERME) holds a biennial conference(CERME), where research is presented and discussed in
Thematic Working Groups (TWGs). We continue here
the initiative (which began in the September 2017 issue)of introducing the working groups, focusing on ways inwhich European research in the field of mathematics
education may be interesting or relevant for researchmathematicians. Our aim is to extend the ERME community with new participants, who may benefit fromhearing about research methods and findings and who
may contribute to future CERMEs.

EMS Newsletter June 2018

Introducing CERME’s Thematic Working Group
12 – History in Mathematics Education
Group leader: Renaud Chorlay
Even though the inception of this TWG is recent(CERME6, 2009), it has deep institutional roots within
the mathematics education research community. Indeed,the History and Pedagogy of Mathematics study group(HPM [1]) was founded at the 1972 ICME conference;
it has been organising satellite conferences of the ICME
meetings since 1984 and has several active regional
branches (e.g. HPM-Americas and European Summer
Universities). At CERME10 (2017), 16 papers and two

47

Mathematics Education
posters were presented in TWG12, covering a large
range of European countries (from Ireland to Russia)
and beyond (Brazil, Mexico and the U.S.). A survey has
shown that this TWG has attracted newcomers to the
CERME community from the HPM community, sincenine participants were CERME first-timers yet only two
had never attended an HPM-related event. The strengthof the historical and HPM communities varies greatly
among countries and these meetings play a crucial rolefor researchers working in relative isolation.
The work carried out in TWG12 lies at the intersection of two different fields of academic research: mathematics education and history of mathematics, a specificsituation which calls for versatility and methodologicalvigilance [2]. The historical approach enables researchers
to consider mathematics not only as a collection of facts
and methods but as a multifaceted human endeavour. At
the interface between the two fields, TWG12 meetings
allow the dissemination of recent results and renewed
perspectives from historical research, including: empirical and theoretical investigations into the variety of science-makers, the nature of the work collectives and theepistemological cultures shaping the engagement withmathematics; studies on the forms and the meaning ofalgorithmic and diagrammatic thinking; studies on therole of examples, numerical tables and problems, either
in themselves or as organised collections. The underlying
rationale is not that of a parallelism between historical“development” and cognitive development of the learner but that – on a par with mathematical knowledge –historical knowledge provides relevant tools and insights
for all facets of didactical research. This was reflected in
the papers presented at CERME10, with an emphasis
on argumentation in numerical and algebraic contexts.
It should be noted that there was little intersection withwhat was covered in TWG8 (Affects and mathematicalthinking) and TWG10 (Diversity and mathematics education), in spite of the fact that it is not uncommon foroutsiders to the HPM research community – including
policymakers and curriculum-designers – to ascribe such
pedagogical goals to the historical perspective in teaching.
For a few years now, research perspectives on thedesign of tasks using historical documents – either in theclassroom or in teacher-training – have evolved significantly. Even though there is still room for new insights
into the “why” question and for accounts of small-scale
design-processes, the emphasis has shifted to place
greater attention on the “how” question [3], on the theoretical analysis of the nature of the expected educational
effects and on the importance of empirical studies into
actual effects. As far as higher education is concerned,
the latest developments of the TRIUMPHS [4] project
were discussed at CERME10. This five-year project,funded by the National Science Foundation in the U.S.,
will create 25 full-length Primary Source Projects (PSPs)and 30 one-day “mini-PSPs”, allowing students to study“from the masters” (such as Euler, Cauchy and Cantor).
The project includes an extensive “research with evaluation” study, which will enable both formative and sum-48

mative evaluation of the project activities. By the end of
the project, it is expected that some 50 instructors andover 1000 students will have participated.
Two other promising research topics were discussed
at the conference. One covers the “how” question,
prompting us to carry out empirical investigations into
the actual practices of “ordinary” teachers attempting
to integrate historical elements in the classroom. The
other highlights the potential of studies combining history of mathematics education and history of didacticaltheories for the networking of theories in mathematics
education.

References[1] http://www.clab.edc.uoc.gr/HPM/.
[2] Chorlay, R, de Hosson, C. (2016). History of Science, Epistemology
and Mathematics Education Research. In B. Hodgson, A. Kuzniak,
J.-B. Lagrange (Eds.) The Didactics of Mathematics: Approaches
and Issues (pp. 155–189). Springer International Publishing, Switzerland.
[3] Jankvist, U. (2009) A categorization of the ‘whys’ and ‘hows’ ofusing history in mathematics education. Educational Studies inMathematics 71(3):235–263.
[4] Transforming Instruction in Undergraduate Mathematics via Primary Historical Sources, http://webpages.ursinus.edu/nscoville/
TRIUMPHS.html.

Renaud Chorlay is a senior lecturer in history and didactics of mathematics at Sorbonne Université (Paris) and a researcher
at the Laboratoire de Didactique André
Revuz. His research interests in history cover the modern period (emergence of qualitative approaches in analysis, and structuralism beyond algebra). His didactical work covers theuse of historical documents in the classroom and teacher
training, and the transition from calculus to analysis.
Jason Cooper is a research fellow at the
University of Haifa’s Faculty of Education.
He is also a researcher at the Weizmann
Institute’s Department of Science Teaching.
His research concerns various aspects ofteacher knowledge, including roles of advanced mathematical knowledge in teaching and contributions of research mathematicians to theprofessional development of teachers. He has been amember of the ERME board since 2015.

EMS Newsletter June 2018

Mathematics Education

Our Missing Teachers
John Ewing (Math for America, New York, USA)It was another amiable conference, scheduled over twodays in a hotel that adjoined the campus of a large university. The topic was mathematics education, from elementary through to high school. The roughly 30 speakers
were almost all people I knew well, some for many years.
They were distinguished mathematics or education faculty from universities across the United States, with a few
from Europe and Asia. Their talks and panels covered
many topics from curriculum to pedagogy, from elementary school to high school and from policy to practice. I
had been looking forward to the event.
As I settled into my seat near the back of the room, I
looked at the list of participants. One aspect of the conference was strikingly familiar – the speakers included
not a single practising classroom teacher. Only one wason a panel and that was because he was connected to auniversity teacher training programme.
Something similar happens at most conferences andworkshops on K-12 mathematics education held in the
United States (and many, although not all, other countries). The speakers are university faculty, education
reformers, superintendents of districts, CEOs of corporations, even politicians – almost anyone other than practising teachers, the people who carry out the day-to-day
work of K-12 education. When a handful of teachers areincluded, it’s because they have some other role.
I have spent most of my life in or associated to universities. For most of that time, I never saw anything
unusual about the missing classroom teachers. I worried
regularly that research mathematicians were often missing from education events and projects. Reforming K-12
education should be a partnership between mathematics
education and mathematics faculty, I insisted. Education
policy disconnected from content loses its way, giving rise
to “school mathematics” rather than “mathematics.” But
I never thought that failing to involve classroom teachers
might be a problem.

The problem
The missing classroom teachers are a problem, of course,
and for the same reason that missing mathematiciansare. Talking about education, either policy or practice,without actual practitioners is just as nutty as talking
about education without experts in content. It’s easy tolose perspective, to misunderstand the consequences ofactions and to misjudge the difficulty of success. Whyleave out people who can provide such perspective?
But the absence of classroom teachers is a symptomof a more serious problem: we omit teachers because
we do not think of teachers as professionals – masters
of a discipline with special expertise and craft. We don’t
think of them in the same way we think of, say, medical
doctors, engineers, or university mathematicians. Indeed,

EMS Newsletter June 2018

we don’t think of teaching as a profession in the way
we think about other professions. The public often seesclassroom teachers as workers who follow instructions
provided by someone else – the real experts. Teachers arelike education contractors, not education architects. And
because we don’t think of them as professionals, we don’t
treat them that way.
At this point, some readers will be shaking their heads.
“Not all classroom teachers are professionals,” they protest. Of course, they are right, some are not. There area great variety of classroom teachers. In many countries, including the United States, most teachers in lower
grades are generalists, with little specialised training inmathematics. While they may be professional teachers,
they are hardly professional mathematics teachers (orprofessional X teachers for any value of X). How canthey contribute to policy discussions about mathematics education? Teachers in the upper grades vary as well.
Some have only modest backgrounds in mathematics
and often teach routine courses in routine ways. Somehave lost their edge over the years and become dull.
Some may be knowledgeable in their subject but awful
in their craft, unable to unpack ideas that are familiar tothem. Some may be truly dreadful. Yes, yes, yes.
But some mathematics teachers are professionals. In
the lower grades, they know their subject in surprisingly
deep ways and are, in every respect, teachers of mathematics, inspiring young students. In upper grades, they are
not only teachers of mathematics but mathematicians as
well, with both insight and devotion to their subject. They
are experts in their craft. They inspire their students.
They guide and mentor their colleagues. They continue
to learn, both mathematics and pedagogy, throughout
their careers. These teachers have all these characteristics and they are professionals in every sense of the word.
They deserve to be treated accordingly. When we treat
them otherwise, we send a message that their professionalism isn’t valued.
If we don’t value our most accomplished teachers,
they will not stay in teaching. If we don’t treat them as
professionals, the profession itself becomes unattractive
and we will lose future accomplished teachers as well. If
we want more high-quality teachers, we had better value
the ones we have.

A secondary problem
This situation illustrates a second problem in education.
When we look at teachers, we almost always focus on
the weakest – the ones who are deficient in some way,
who need repair, who represent some failure. It is hardto think of teachers as professionals who can contributeto education more broadly because we are focused onthat awful algebra teacher who barely knew mathemat-

49

Mathematics Education
ics and ruined mathematics for our son or daughter. We
may remember a few great teachers in our own lives butwhen we talk about education, we think about the teachers who are deficient and need repair.
This is an unconscious theme in modern K-12 education. Reform has become a simple formula: find what’s
broken and fix it. Find the weakest teachers, the poorest
schools, the most troubled students. Expose them, repair
them, get rid of them and education will get better. Thisis educational reform today.
But it’s a remarkably short-sighted view, strangely
peculiar to K-12 education. In the business world, one
doesn’t only focus on what doesn’t work well. To build athriving business, you fix things to be sure, but you alsofind things that already work well. You build great businesses on those things, using them as the cornerstones.
Similarly, a university president would never arrive on
campus and immediately focus on faculty who are weakest. Great universities get built using the most prestigious
and accomplished. Why do we try to improve education
by focusing only on the rubble of failure? In life, excellence is built on excellence. Education is no different.

tion with colleagues from a wide range of schools. Fellows participate actively and enthusiastically in ways thatsuit their individual professional needs. They also receive
an annual stipend (currently US $15,000), which is meant
to recognise their achievement and to compensate for
their substantial commitment. Teachers can also apply torenew their fellowship.
New York City’s school system is large: 1.2 million
students in 1800 schools taught by about 75,000 teachers. MfA now has over 1000 teachers – about 10% of themaths and science teachers in the city. Is that the rightpercentage? Perhaps. It seems about right in New York
City. The larger the number, the greater the effect, ofcourse. On the other hand, it is essential to have accomplished teachers to make this programme work. In New
York, 10% seems to strike the right balance.
While the basic idea is simple, the details are more
complicated. They are important! It would be easy toallow a programme like this to drift into something
resembling traditional education reform, for example,
by selecting teachers who were not accomplished or by
converting the workshops into “teacher repair”. The lureof traditional reform is powerful and one must safeguard
against it. Here are some details:

Math for America
Math for America (MƒA) sets out to address these problems. The idea is embarrassingly simple – find genuinely
accomplished teachers, give them opportunities modelled on professional life in universities, and trust them totake advantage of the opportunities. In short, find teachers who deserve to be treated as professionals and treat
them that way.
(“Math for America” is a misnomer in two ways. The
programme includes science teachers as well as mathematics, in roughly equal numbers. Also, it is only forteachers in New York City rather than all America.)MƒA offers a 4-year master teacher fellowship forexperienced mathematics and science teachers who are
currently in the classroom (and who continue to teach).
Teachers apply, are selected, and join a community ofmaths and science teachers spanning all the grades,
although predominantly in grades 6-12. The community
tries to model scholarly life in a vibrant university – workshops, seminars and lectures, covering maths or science,
policy or pedagogy and abstract or practical topics. It alsooffers the opportunity to exchange ideas and informa-

Selection
The selection process is important because MƒA’s successdepends upon finding accomplished teachers. Selection
begins with a lengthy application consisting of information about a teacher’s education (including transcripts)
and career. Applicants submit a short essay and a lesson
study. They have three people write detailed letters ofrecommendation. They take a standard (undemanding)test of content that is specific to their specific discipline.
One element is notably and deliberately missing fromthe selection process – test scores from a teacher’s students. The application is structured in a way that helps
teachers decide whether the fellowship is right for themand encourages them to stop if it’s not. We don’t encourage applications simply so we can brag about our low
acceptance rates.
Applicants who make it through the first part of theprocess are invited for an interview. At that interview,
they are first assigned to work in small groups, creating
and presenting a piece of mathematics or science they

Typical evening at MƒA.

An MƒA workshop on puzzles.

50

EMS Newsletter June 2018

Mathematics Education
can choose themselves (because it’s interesting and not
because they teach it). Afterwards, they converse about
the presentations, asking each other questions, answering them and offering observations. A three-person team(a mathematician, an educator and an MƒA representative) observe all this. The team then interviews each candidate separately, following up on what’s been observed.
The process allows us to gauge with remarkable accuracy a teacher’s mathematical or scientific knowledge,
their ability to communicate that knowledge and theirapproach to collaborative work. At the end, we knoweach candidate’s strengths and weaknesses.
Final acceptance is carried out by MƒA staff, using
the complete dossier for each applicant and a set of carefully developed rubrics, along with sound judgment.

Scholarship
During most evenings, the MƒA teachers swarm throughout our New York City offices, which include a lounge, asmall library and a number of seminar rooms and breakout areas. This is meant to approximate the facilities of ahigh-quality maths or science department.
Some might describe what MƒA teachers do during
these evenings as “professional development”. I avoid
that term. Traditional professional development is often
dull and dreary, disconnected from a teacher’s discipline,
and aimed at fixing deficiencies or providing tips on how
to improve test scores. MƒA’s workshops, seminars andmini-courses are meant to be intellectually engaging.
Some are directly connected to instruction, but many
are about maths or science – a recent research result oran interesting topic – or about education challenges andpolicies. A few are single sessions but most are given insequences of three or more sessions. In the 2017-18 academic year, MƒA offered over 400 “courses” like this,taking place in nearly 1000 sessions.
Here is the most important part: two-thirds of thesecourses are created and led by the teachers themselves.
MƒA provides the infrastructure. We put together thecatalogue, determine the schedules and provide thefacilities, but the teachers themselves drive most of thisforward – in the same way that faculty and students drive
seminars and colloquia in a healthy university.
Of course, some workshops are run by people fromthe outside, including mathematicians, scientists and educators. But even these workshops are inspired by teacher
interest and aimed at intriguing, rather than fixing, teachers.
For many teachers, especially those in small schools
with few colleagues in their field, MƒA’s scholarly community serves as their essential connection to their discipline. It makes them feel like mathematicians and scientists. It makes them feel professional.
Interaction
Teaching is a lonely job. People are sometimes surprised
when I say this. They picture teachers surrounded bydozens of students and they wonder how anyone canbe lonely with all those students. But teachers interact
professionally with teachers and the modern structure of

EMS Newsletter June 2018

Old fashioned string art.

schools makes these professional interactions more andmore difficult.
One of the benefits of the MƒA community is theopportunity to interact with other teachers in new ways:
mathematics teachers talk to science teachers; middle school and elementary teachers talk to high school
teachers; and teachers at exclusive schools (in NYC they
are called “exam schools”) talk to those enrolling mainly
high-needs students. They all come together. This kindof professional interaction builds connections, which
creates networks that teachers can draw on for years tocome. It is a kind of interaction that seldom happens in
the everyday lives of teachers. At MƒA, it happens naturally.
We also foster such interaction. Part of our programme (about 15%) consists of teachers who are in thefirst few years of their teaching careers. We pair themwith master teachers who serve as mentors, not merely in name but in fact. Both junior and senior teachers
profit from these relationships. Many of our workshops
are offered in a special format – Professional Learning
Teams (PLTs) – that bring together 15–20 teachers overthe course of a semester or year. PLTs are always coled by a pair of teachers and adhere to a special format
designed to involve all participants in the ongoing discussions. We even run workshops on how to run workshops,
inside and outside of MƒA. We encourage MƒA teachers
to start their own PLTs in their schools.
University mathematicians are often unaware thatsuch routine interaction with their colleagues plays anessential role in their professional lives. It is an unremarkable part of life in a good university. For classroom
teachers, these interactions are a new experience thatchange the way that teaching feels.

Trust
None of this works without one final ingredient: trust.
MƒA fellows are required to participate in a modest
number of workshops, roughly one session per month.
(Most participate in far more!) We don’t require themto learn any particular content. We don’t ask them toacquire any particular skills. We are not fixing any particular deficiency. We trust them to decide for themselves
how they want to grow professionally.

51

Mathematics Education
Sample Mathematics Workshops
Spring 2018
Applying Ideas from Modern Algebra to Secondary
Teaching (MINI)
Combinatorial Game Theory (SSW)
Delving Deeper into Fraction Subconstructs andProcesses (PLT)
Dynamic Lesson Planning Using Geometer’s
Sketchpad (SSW)
Exploring Rational Tangles (SSW)
Fostering a Growth Mindset in Mathematics (PLT)
How to explain hard “Why” questions in Algebra
and Geometry using Calculus (MINI)Introduction to Category Theory (MINI)
Investigating Calculus Teaching and Learning (PLT)
Islamic Art and Geometry (MINI)The Mathematics of Gerrymandering (MINI)
Multi-criteria Decision Analysis for High School(SSW)
Made You Look – Statistics through Data
Visualizations (TT)Using Mathematical “Magic” to Engage Students in
Mathematics (MINI)
Vertical Alignment in High School Math (IG)For many of our teachers, this is their first experience
with this kind of trust. Teachers’ careers often progressfrom one mandated professional development experience to the next, many of them useless and some of themdreadful. They hear public figures proclaim how muchthey “respect” teachers, without extending even a modicum of trust in this sense. They are controlled, regimented and evaluated by a system designed with the least able
teachers in mind. MƒA tries to change that, at least in one
part of their professional lives. It is an important change.
One aspect of MƒA is not as essential as the previous four but still deserves a mention. Master teachers canrenew their fellowship after four years.
Renewals are not automatic. They require both anapplication and an interview (each different in naturefrom the initial ones). We expect more from masterteachers when they renew. They are meant to be leaders,
contributing more to MƒA, their colleagues and (possibly) their own schools. We make this a requirement but
not an overbearing one. The renewal itself depends on
our assessment of a teacher’s ability to take on this role.
Not all master teachers choose to renew and not all who
apply are accepted.
If we did not permit renewals, we could offer fellowships to more teachers. But we would lose part of whatmakes MƒA thrive. The senior master teachers who stayon for two or more fellowships play a special role in ourcommunity. They mentor those who are new to MƒA:They take the lead in proposing workshops and running
them. They model what an active professional life looks
like; and they often carry their MƒA experience outside
to their schools and to the rest of public education.
While renewals may not be essential, they make MƒAmore effective.

Similarly, we invite teachers to submit proposals forworkshops and courses based on their own ideas and notours. We vet these proposals, of course, but we trust theteachers to come up with good ideas. And they do!
Trust is a crucial ingredient in changing the way wethink about teachers. It is often confused with education
laissez faire – the proposition that teachers should do
whatever strikes their fancy in the classroom. But laissez
faire is impractical in most settings (including, it should
be added, in universities!). Professional trust is different
and more subtle. It means trusting teachers to control
their own professional lives, deciding what’s most important to them and how they want to develop their own
careers. Extending that trust is important.

Advocating
MƒA has evolved over time. The original programme wasconceived by Jim Simons and a group of financial mathematicians in 2004. The original implementation created afellowship that enticed mathematically talented individuals to become teachers – a year of training, four years ofcommitment, substantial stipends, and a community. The
programme was supported by an annual poker benefit
together with the Simons Foundation.

A 3D printer in the MƒA lounge.

A Professional Learning Team (PLT).

52

EMS Newsletter June 2018

Mathematics Educationteachers. There is no “treatment” that can be withheld
from a control group to see whether the dosage is correct. There is no simple statistic that measures what wewant to achieve. Some things are hard to measure withnumbers. Professionalism is one of them.
In two respects, though, MƒA is already successful:

MfA teachers in a breakout.

Gradually, MfA discovered that many highly accomplished mathematics teachers were already in classrooms. Many of them were leaving and keeping themseemed more efficient than creating brand new teachers.
The master teacher programme began to grow in 2012.
A similar programme was created for teachers early intheir careers. Science was added to broaden the appeal.
The scope was extended to include maths and scienceelementary teachers. Now, after 14 years of evolution,
MƒA has slightly more than 1000 teachers in its fellowships – about 10% of the maths and science teachers in
New York City. About half are in maths and half are inscience. Most are master teachers and some are early
career. Watching them work together each evening is
inspirational.
MƒA in New York City could serve as a model forother programmes elsewhere. We have tried to persuade
others to create similar programmes in other locations
with limited success. A few arose in other cities in the
U.S., with the largest in Los Angeles (nearly 100 teachers). A large, publicly-funded programme began several
years ago in the rest of New York State (which has roughly the same population as New York City). It currently
has over 900 teachers at nine sites around the state. We
are working with other states to help them create similar
publicly-funded programmes, not identical but similar toMƒA.
Advocating is a tough job, however. People sympathise with the goal but the approach can be jarring. That
we focus on excellent teachers seems counter-intuitive to
many people, especially education reformers. Why waste
resources on teachers who are already accomplished?
Why not concentrate on teachers who need help? And
many people find it hard to trust teachers in this way.
They can only remember that dreadful teacher they (ortheir child) recently encountered at school. Surely thatteacher doesn’t deserve a stipend or our trust!
If we really want to improve the quality of mathematics and science teachers, however, we need to find a way
to overcome these obstacles.

- In the U.S. today, experienced teachers leave teachingfor other careers at an alarming rate (about 8% eachyear). MƒA teachers leave at a far lower rate (3%). We
want experienced, accomplished teachers to stay inteaching even if it is only for four additional years.
- Throughout the world, teachers complain frequently
about shallow and useless professional development.
Teacher-led professional development that treatsteachers as professionals is a welcome change. MƒAteachers thrive in such an environment. Even the mostaccomplished teacher wants to grow professionallyand in MƒA they do.
But the ultimate goal is to change perceptions – to convince the public and teachers themselves that teaching
is not merely standing in front of a classroom and that
it is a profession requiring mastery of content and craft,which takes place over many years and is motivated bycuriosity, ambition and pride. Teaching is not preparing
students for tests. It is not following instructions. It is
not reciting facts or procedures. It is a profession and weshould welcome its accomplished professionals into themathematics community … and treat them like the professionals they are.
Perceptions only change over time, however. Achieving this goal will require persistence and patience – qualities that are frequently missing from educational reform
today. Fortunately, MƒA has both.
(For more details and background, see the Math forAmerica website at www.mathforamerica.org.)
All photo credits: Michael Lisnet, MƒA photographer.

John Ewing is the President of Math forAmerica, an organisation associated to andsupported by the Simons Foundation in
New York City. He was formerly the Executive Director of the American MathematicalSociety and, before that, he was on the faculty of Indiana University. Since joining MƒA
in 2009, he has gained both humility about the difficultyof mathematics education and certainty that improving it
requires us all to work together – mathematicians, maths
educators and classroom teachers – as equal partners.

Coda
Does MfA work? It’s too early to tell for sure. Unlike
traditional education reform, MƒA is not about fixing

EMS Newsletter June 2018

53

zbMATHNews on the zbMATH Interface
Octavio Paniagua Taboada (FIZ Karlsruhe, Berlin, Germany)
Our staff of developers and editors have updated andimproved several important features in zbMATH. All
these updates aim to provide our users a rich and modern
database interconnected with other worldwide databases
and services (e.g. digital libraries, Wikidata, ORCID andlinks to discussions in MathOverflow).

Reviewer service and compensation
During the last few months, we have updated several
components of the reviewer service and submission tool.
It is now possible to check the account balance and order
Springer books, as well as make donations to the EMS
book programme for developing countries via the interface. From January 2018, the financial compensation per
review will be increased to 3.00 EUR from the traditionalamount of 2.56 EUR. When this amount is used to order
Springer books, a discount of 50% is applied.
zbMATH Remote
zbMATH Remote (https://zbmath.org/remote_access/)
provides access to zbMATH from your mobile device
using your institution’s subscription, even when you are
not logged into the institution’s network. This access isvalid for a period of three months and can be renewed
whenever logged into your institution’s network. Whenconnected to a licensing network, you can activate remote
access to zbMATH by clicking on the ‘’Remote’’ link
located at the top right menu of the zbMATH website.

Here is an example of such a citation.

MSC2020zbMATH and Mathematical Reviews have worked
together to revise and improve the Mathematics Subject Classification (MSC) schema, which is used by thesereviewing services and publishers to categorise items inmathematics literature. Comments and suggestions canbe submitted through the website http://msc2020.org/.
zbMATH Atom FeedzbMATH now offers an additional way of keeping
researchers up-to-date with mathematical developmentsin their areas of interest via an atom feed. Every two
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This feed is an enhanced version of the classic web
feed format RSS and is supported by all current news
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If you are interested in a specific author, you canaccess this news feed through the author ID (displayed
at the top of each author profile, e.g. https://zbmath.org/
atom/ai/leibniz.gottfried-wilhelm). You can also accessthe atom feed via MSC code, which can be defined ascomplex and specific as you wish (e.g. https://zbmath.org/
atom/cc/81,17B,57R56 for quantum theory, Lie algebrasand Lie superalgebras, and topological quantum field
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zbMATH interface features
Search results can now be sorted using customised criteria such as publication year, document or review citation,and volume number.
Additional filters have been added. Formula searchhas been extended by the integration of arXiv full-texts,
making more than 160 million mathematical formulas
retrievable. A new interface available at MathOverflow
allows users to insert a citation into any question, answer
or comment, and enables linking back from zbMATH toa discussion there.

54

EMS Newsletter June 2018

Book Reviews
Tutorial videos
A series of videos have been produced in order to help
users exploit all of zbMATH’s potential. There are examples with complex queries, tricks and hints. New videos
will be added to cater for newly developed features in
zbMATH. You can watch these videos at the very bottomof the “About’’ page https://zbmath.org/about#id_5.
Octavio Paniagua Taboada is an editor
for zbMATH at the Berlin office of FIZ
Karlsruhe. He received a bachelor’s degree
and a Master’s in mathematics from the
National Autonomous University of Mexico (UNAM) and a PhD in mathematics
from the Université Paris-Sud XI, Orsay,
France. He has been an editor of zbMATH since January
2014 and of the EMS Newsletter since 2018.

Book Reviews
Jacques Sauloy
Differential Galois Theory
through Riemann-Hilbert
Correspondence: An Elementary
Introduction
AMS, 2016
275 p.
ISBN 978-1-4704-3095-5
Reviewer: Julien Roques
Linear differential equations are at the crossroads ofseveral areas of mathematics. This book starts withan exploration of analytic theory of (regular singular)
linear differential equations and concludes with aninvitation to the associated differential Galois theory.
It should be stressed that Sauloy’s intention is not to
expose differential Galois theory in its full generality,
for linear differential equations with coefficients in anarbitrary differential field K, but rather to focus on thecase K = C(z), taking advantage of the analytic tools at
our disposal in this context and making apparent somebeautiful interplay between analysis and algebra. The
intention of Sauloy is clearly to explain the meaning of
the adage “differential Galois theory is what algebra
can say about dynamics” and to convince the reader
that modern differential Galois theory has its roots inanalytic theory of differential equations (and, more precisely, in Riemann’s monodromy representations). And
it is a success.

EMS Newsletter June 2018

Here follows a summary of the contents of the book.
Part I is an introduction to complex analysis and aprelude to the analytic study of linear differential equations undertaken in Parts II and III.
Parts II and III are mainly concerned with analytic
theory of regular singular differential equations on
P1(C), say, of the forman (z) y(n) (z) + … + a1 (z) y’ (z) + a0 (z) y (z) = 0, (1)with a0(z), … ,an(z) ∈ C(z) and an(z) ≠ 0, and, more specifically, with their monodromy representation. Before
describing in more detail what can be found in Parts II
and III, here are a few words about the concept of monodromy, which is the hero of the book. The starting point is
the so-called Cauchy theorem. Denoting by  the set of
singularities in P1(C) of (1) and by U its complement in
P1(C), Cauchy’s theorem reads: for any z0 ∈ U$, the complex vector space Vz0 of germs of analytic functions at z0
that are solutions of (1 has dimension n. These solutions
possess a marvellous property: any f ∈ Vz0 can be continued analytically along any loop g in U based at z0 and thegerm of analytic function f g resulting from this process
is again a solution of (1), i.e. f g ∈ Vz0. In this way, we get
a C-linear action of the fundamental group p1(U, z0) on
Vz0, given, for all g ∈ p1(U, z0) and f ∈ Vz0, by [g] f := f g.
In other words, we get a finite-dimensional linear representationr : p1 (U, z0) → GL (Vz0).
This is the monodromy representation attached to (1).
The following two questions are natural:55

Book Reviews- Is sole knowledge of the monodromy representation
sufficient to reconstruct the differential equation we
started with?
- Is any finite-dimensional linear representation of thefundamental group p1 (U, z0) the monodromy representation of some differential equation?
In full generality, the answer to the first question is negative but becomes positive if we restrict our attention toa special class of differential equations, namely, to thosehaving only regular singularities. The answer to the second question is positive even if we restrict our attention
to differential equations having only regular singularities. These facts are usually formalised as a certain equivalence of categories, known as Riemann–Hilbert correspondance.
We are now in a position to describe the content ofParts II and III. Part II starts with Cauchy’s theorem,
continues with the proof of the analytic continuation
property of the solutions mentioned above and introduces the monodromy representation. Part III starts with thenotion of regular singularity and with a detailed study ofthis type of singularity. A local version of Riemann–Hilbert correspondance is then stated and proved. The statement and a sketch of the proof of (global) Riemann–Hilbert correspondance (over P1(C)) are then given. It is
worth noting that Part III also contains the determination of the monodromy representation of hypergeometric differential equations (following Riemann’s original
method).
With Part IV, the reader enters the world of differential Galois theory. The first few pages of this chapter
contain the definition of local differential Galois groups
and a proof of the fact that they are complex linear alge-

Walter van Assche
Orthogonal Polynomials andPainlevé Equations
Cambridge University Press, 2018
xii, 179 p.
ISBN 978-1-108-44194-0
Reviewer: Francisco Marcellán
The Newsletter thanks zbMATH and Francisco Marcellán for the permission to republish this review, originallyappeared as Zbl 06811550.
At the end of the 19th century some relevant mathematicians like Poincaré, Fuchs, Picard, Painlevé, among others,
were interested to find those nonlinear ordinary differential equations (ODE) such that their general solutions56

braic groups. The transition from Parts II and III to Part
IV is carried out in a very natural and smooth way via theso-called Schlesinger density theorem: the monodromy
is Galoisian and gives rise to Zariski-dense subgroups ofthe above mentioned local differential Galois groups in
the regular singular case.
The rest of Part IV is essentially concerned with theregular singular local universal Galois group; roughly
speaking, this universal Galois group is the algebraic hullof the local topological fundamental group and it can beused to formulate an “algebraic Riemann–Hilbert correspondance”.
The book ends with a selection of further developments and readings.
This is an accessible book, well-suited for students.
It is mainly self-contained. Some “advanced concepts”
are used, such as sheaves, categories and linear algebraic
groups but these concepts are introduced progressively
throughout the book, when needed, without any attemptof systematic presentation or maximal generality. This isa nice and efficient choice. Several exercises can also befound throughout the book.
The book is warmly recommended for those looking
for an accessible text about the analytic theory of regular
singular differential equations and the associated differential Galois theory.
Julien Roques is a maître de conférences at
the Institut Fourier of the Université Grenoble Alpes. He works on algebra, number
theory and functional equations, and on theirinteractions and applications.

are free from movable branch points (Painlevé property). The locations of possible branch points and criticalessential singularities of solutions can be independent
on the initial values. In the case of first order differential equations the Painlevé property only gives linear
differential equations, the Riccati differential equationand the equation associated with the Weierstrass elliptic
function. For second order differential equations, Picard raised the description of those nonlinear differential
equations yʺ (x) = R (x, y, yʹ), where R is a rational function, such that the Painlevé property holds. Paul Painlevé
found that, up to some simple transformations, there are
50 canonical forms for such ODEs. 44 of them can be
reduced to linear ODEs, Riccati equation or Weierstrasscase. The remaining six equations of such a list are called
Painlevé ODEs and their solutions are known as Painlevé transcendents. For them, the only movable singularities are poles (no essential singularities). A remarkable
overview on these topics is presented in [P.A. Clarkson,
Lect. Notes Math. 1883, 331–411 (2006; Zbl 1100.33006)].

EMS Newsletter June 2018

Book Reviews
On the other hand, discrete Painlevé equations are
nonlinear recurrence relations for which the continuous limit is one of the Painlevé ODE. A classification of
discrete Painlevé equations based on rational surfaces
associated with affine root systems was suggested by H.
Sakai [Commun. Math. Phys. 220, No. 1, 165–229 (2001;
Zbl 1010.34083)].
The book under review is focused on the relationshipbetween Painlevé equations and orthogonal polynomials. The basic fact is that the coefficients of the three termrecurrence relations (TTRR) the sequences of polynomials with respect to measures supported on the real line
together with some differential properties of the measure (Pearson equation) yield discrete Painlevé equations.
Moreover, if the measure depends on some time parameter, then you can deduce Painlevé ODE for the coefficients of the recurrence relation. This fact is connectedwith some integrable systems (Toda, Langmuir, among
others).
Two blocks constitute the skeleton of this book. In
the first one (five chapters) the author deals with the
Painlevé equations associated with the coefficients of
the TTRR that some families of orthogonal polynomialssatisfy. In the second one (two chapters), the attention is
focussed on rational solutions of Painlevé ODE whichappear for some choices of the parameters involved in
ODEs. Furthermore, the asymptotic behavior of orthogonal polynomials near critical points is presented byusing the Riemann-Hilbert approach.
Chapter 1 provides a basic background about orthogonal polynomials with a special emphasis on those associated with the so called semiclassical weights. Next,
the description of the continuous and discrete Painlevé
equations is presented.
In Chapter 2, some examples of Freud weights arestudied. First, in the case w(x) = exp (–x4 + tx2) a d – PI
equation appears for the coefficients of the TTRR. The
unicity of the positivity solution of such an equation withan initial condition x0 = 0 is analyzed and, consequently,
the value of x1 is determined. On the other hand, a differential-difference equation for the coefficients of the
TTRR is deduced and, by using the above equations, aPainlevé IV equation for such coefficients is obtained.
Chapter 3 deals with orthogonal polynomials associated with the measure w(z) = exp ((z + 1/z) t/2) supported on the unit circle. Now, the parameters of therecurrence relation that the corresponding orthogonal
polynomials satisfy (the so called Verblunsky coefficients) are related to a d – PII equation. They also satisfy
a Painlevé V equation, that can be reduced to a Painlevé
III. Examples of discrete orthogonal polynomials (generalized Charlier) are also analyzed and the corresponding
Painlevé equations for some choices of the parameters
are obtained. The unicity of solutions of d – PII and its
positivity according to some initial conditions is proved.
By using the formulation of the Riemann–Hilbert
problem for orthogonal polynomials a new proof of aclassical result about ladder operators for exponential
weights (see [Y. Chen and M. E. H. Ismail, J. Phys. A,

EMS Newsletter June 2018

Math. Gen. 30, No. 22, 7817–7829 (1997; Zbl 0927.33011)])is given. This approach is very useful in order to analyze
Painlevé equations for different families of semiclassical weights in continuous and discrete cases as done in
Chapters 4 and 5.
In Chapter 6 the author focuses the attention on
those Painlevé equations such that, for some choices ofthe parameters, either rational or special function solutions appear. For the Painlevé II and III they are given
in terms of the logarithmic derivative of two consecutive
Yablonskii- Vorobiev polynomials and Umemura polynomials, respectively. In other cases they can expressedas Wronskians of classical orthogonal polynomials. The
Painlevé equations II–VI also have solutions which canbe expressed in terms of classical special functions (Airy,
Bessel, parabolic cylinder, Kummer and Gauss functions,
respectively).
Finally, in Chapter 7 the connection between the density of zeros of orthogonal polynomials associated toweight functions with singularities, the construction of a
local parametrix at certain critical points in the asymptotic behavior of orthogonal polynomials and randommatrices is stated for Painlevé I, II, IV and V.
The presentation of this book is very friendly for ageneral audience interested in the theory of orthogonal
polynomials, nonlinear ODE and integrable systems. The
contents of the chapters are very pleasant taking into
account the main results and their proofs are given witha smart distribution. Many of the results described in thefirst block are based on the contributions by the author
and co-workers.
The book is based on lecture notes of courses andseminars in several higher education institutions. A listof 16 exercises with their solutions allows a dynamical
approach to the techniques described therein. Finally,an updated list of 156 references invites the reader toadvance in the learning of many questions contained inthis nice book.
Francisco Marcellán is professor of Applied Mathematics at Universidad Carlos
III de Madrid, Spain, as well as a senior researcher at Instituto de Ciencias Matemáticas (ICMAT) CSIC-UAM-UCM-UC3M.
His research interests are in orthogonal
polynomials, special functions, approximation theory, matrix analysis and their applications tointegrable systems and signal theory. At present he is the
President of the Royal Spanish Mathematical Society.

57

Book Reviews
Dan Burghelea
New Topological Invariants for
Real- and Angle-valued Maps.
An Alternative to Morse–Novikov
TheoryWorld Scientific, 2018
xvi, 242 p.
ISBN 978-981-4618-24-3

Reviewer: Dorin Andrica
The Newsletter thanks zbMATH and Dorin Andrica
for the permission to republish this review, originallyappeared as Zbl 06793944.
In the early 1920s, M. Morse discovered that the numberof critical points of a smooth real-valued function on amanifold is closely related to the topology of the manifold.
This became a starting point of the Morse Theory which is
now one of the basic and active parts of differential topology. We refer for instance to the book of L. I. Nicolaescu[An invitation to Morse theory. 2nd ed. Berlin: Springer(2011; Zbl 1238.57001)]. Circle-valued Morse theory is
originated from a problem in hydrodynamics studied by
S. P. Novikov in the early 1980s. It is a constantly growing
field of contemporary mathematics, also known as Novikov Theory, with applications and connections to manygeometrical problems such as Arnold’s conjecture in thetheory of Lagrangian intersections, fibrations of manifolds over the circle, dynamical zeta functions, and thetheory of knots and links in the three-dimensional sphere(see the monograph of A. Pajitnov [Circle-valued Morsetheory. Berlin: De Gruyter (2006; Zbl 1118.58007)]).
The present book presents a very interesting alternative to the above mentioned Morse–Novikov Theory,
symbolically called AMN Theory. The theory presented
by the author is of great interest in topology and dynamics, it has provided inspiration and has applications outside of mathematics, especially in data analysis and shaperecognitions in physics and computer science. In thisbook a nice space is a locally compact ANR (Absolute
Neighborhood Retract). Remark that finite dimensional
simplicial complexes and finite dimensional topologicalmanifolds are nice spaces but the class is considerablylarger. The author considers tame maps to be proper continuous maps f : X → R or f : → S1, defined on a nicespace X, which satisfies the following properties: (i) eachfiber of f is a neighborhood deformation retract; (ii) awayfrom a discrete set Σ ⊂ R or Σ ⊂ S1 the restriction of f
to X  f –1(Σ) is a fibration. In particular for t ∉ f –1(Σ)
there exists a neighborhood U of t such that for any tʹ ∈U the inclusion f –1(tʹ) ⊂ f –1(U) is a homotopy equivalence. Note that all proper simplicial maps and propersmooth generic maps defined on a smooth manifold, inparticular proper real or angle valued Morse maps, aretame. At least for spaces homeomorphic to simplicial
complexes the set of tame maps is residual in the space58

of all continuous maps and weakly homotopy equivalent to the space of all continuous maps (equipped withthe compact-open topology). Some refinements of thisnotion are also considered in the book. The main ideaof AMN Theory is to consider instead of critical points,
critical values for real-valued or circle-valued functions.
Instead of critical points and instantons between critical points, the AMN Theory introduces and calculates
barcodes. The closed trajectories are replaced by Jordan
cells or Jordan blocks whose direct sums calculate thehomological monodromies. It is remarkable that most of
the fundamental algebraic topology invariants which canbe recovered in the Morse–Novikov Theory from critical points, instantons, and closed trajectories, can equallywell recovered in the AMN Theory from barcodes andJordan blocks. The derived invariants are finite, computable by implementable algorithms in case the underlying space of the map has a triangulation and the map is
simplicial, and are, in some sense, the analogues of the setof trajectories between rest points and of closed trajectories of a generic vector field (which admits a Lyapunov
closed one form) on a smooth manifold. The book presents remarkable properties such as stability and Poincaré-duality of these invariants, and it relates them to theglobal algebraic topology of the space.
To give a general presentation of the results contained in the book, we give an overview on the content.
The material is organized into nine chapters including
Chapter 1 as the author description.
Preparatory Material: It reviews the linear algebra of
matrices and of linear relations, as well as few concepts andresults in topology: matrices, Fredholm maps and Fredholm
cross-ratio, an algorithm to calculate R(A,B)reg, ARNs,
tameness, regular and critical values, compact Hilbert
cube manifolds, infinite cyclic covers, simplicial complexes, cell complexes and incidence matrices, configurations,algebraic topology of a pair (X, x ∈ H 1(X; Z)).
Graph Representations: generalities on graph representations, the idecomposable representations (two basic
constructions, the $k[t –1},t]-module associated to a G2mrepresentation, the matrix M(r) and the representationru), calculation of indecomposables (an algorithm) (elementary transformations, algorithm for deriving barcodesfrom M(r), implementation of T1(i), T2(i), T3(i), T4(i)).
Barcodes and Jordan Blocks via Graph Representations: the graph representation associated to a map,
barcodes and Jordan blocks of a tame map (the configurations drf, the AM and AN spaces, the relevant exact
sequences), barcodes, Jordan cells, and homology, barcodes and Borel-Moore homology, calculations of barcodes and Jordan cells.
Configurations drf and d̂rf (Alternative Approach):
general considerations, the case of real-valued maps, thecase of angle-valued maps.
Configurations grf : general considerations, the case ofreal-valued functions, the case of angle-valued functions.
Monodromy and Jordan Cells: general considerations, geometric r-monodromy via linear relations, analgorithm for the calculation of Jordan cells of a simplicial angle-valued map.

EMS Newsletter June 2018

Book Reviews
Applications: relations with the classical Morse andMorse–Novikov theories (the Morse complex, the Novikov complex, chain complexes of vector spaces, the AM and
AN complexes for a Morse map), a few computational
applications (Novikov–Betti numbers in relation with Jordan cells, Alexander polynomial of a knot and generalizations).
Comments: relation to persistence theory, a measuretheoretic aspect of the configurations of drf, grf, an invitation.
The book ends with a rich bibliogrphy containing 56
suggestive references for the subject, nine of them representing the author’s contributions. A List of Figures and an
useful Index are also included.
The book under review is a very nice and valuable text
on the new AMN Theory. It is written in a clear mannerand it can help anyone who wants to learn this new theory
as well as its recent developments and applications.

Nikos I. Kavallaris, Takashi Suzuki
Non-Local Partial Differential
Equations for Engineering and
Biology. Mathematical Modelingand Analysis.
Springer, 2018
xix, 300 p.
ISBN 978-3-319-67942-6
Reviewer: Teodora-Liliana Rădulescu
The Newsletter thanks zbMATH and Teodora-Liliana
Rădulescu for the permission to republish this review,
originally appeared as Zbl 06817103.
This book presents some new results concerning nonlocal models arising in mathematical analysis. Basic concepts in mechanics, thermodynamics, game theory, andtheoretical biology are examined in detail. It starts witha review and summary of the basic ideas of mathematical modeling frequently employed in sciences and engineering. The first part of this monograph is devoted tothe investigation of some non-local models linked withapplications from engineering.
Chapter 1 focuses on the study of non-local models
associated with electrostatic micro-electro-mechanicalsystems (MEMS) control. The authors describe the twomain physical problems which build up the operation of
an idealized MEMS device: the elastic and the electric
problem. Next, the authors are concerned with the structure of the set of radially symmetric steady-state solutions,which are investigated along with their stability. Then, they
study the circumstances under which finite-time quenching occurs.
Chapter 2 discusses some non-local models describing
Ohmic heat production in various industrial processes. In
the first part of the chapter, the process of food sterilization through Ohmic heating is considered on the basis of a

EMS Newsletter June 2018

Dorin Andrica is professor at Babes-Bolyai
University in Cluj-Napoca, Romania, where
he received his PhD in 1992 with a thesis
on critical point theory with applications
to the geometry of differentiable submanifolds. His scientific interests are in Differential Topology (critical point theory andapplications, Morse theory with applications), Differential
Geometry, Number Theory, Problem Solving, Mathematics for competitions and Olympiads. He published morethan 170 research papers, 35 books in Romanian, English,
Arabic, Portuguese, Japanese, Korean, and proposed morethan 1000 original problems of various level of difficulty
to different mathematical competitions. Professor Dorin
Andrica is well known for his conjecture concerning the
gap of the consecutive primes, presently called “Andrica
Conjecture”.

one-dimensional non-local model. Next, and under different circumstances, a hyperbolic approach with a non-local
convection velocity is built up. The second part of thischapter deals with another application of Ohmic heatingprocess in a thermistor device.
Chapter 3 deals with an application arising in the process of linear friction welding applied in metallurgy. Next,a similar non-local model is derived for the hard-materialcase where the exponential nonlinearity is replaced byf(u) = (–u)p, for p = 1/a.
Chapter 4 discusses a degenerate non-local model
which is associated with the industrial process of resistance spot welding and the unknown $u$ represents thetemperature in the welding area.
Part II of this monograph deals with some applicationsof non-local models coming from biology. The authors areconcerned with the Gierer-Meinhardt system, an application arising in evolutionary game dynamics, biologicalphenomena arising in chemotaxis, and a mathematicalmodel in cell biology that describes the evolution of protein dimers within human cells.
The models developed in this book are based on
various laws of physics such as mechanics of continuum,
electromagnetic theory, and thermodynamics. For thesereasons, the arguments come from many areas of mathematics such as calculus of variations, dynamical systems,
integrable systems, blow-up analysis, and energy methods.
The book under review is mainly addressed to researchers and upper grade students in mathematics, engineering,
physics, economics, and biology.
Teodora-Liliana Radulescu is professorof mathematics at the “Fratii Buzesti” National College in Craiova, Romania. She received her Ph.D. in 2005 from Babes-Bolyai
University of Cluj-Napoca. Her researchinterests are problem solving and methods
in nonlinear analysis. She co-authored thebook “Problems in Real Analysis. Advanced Calculus on
the Real Axis”, Springer, 2009.

59

Personal Column

Personal Column
Please send information on mathematical awards anddeaths to newsletter@ems-ph.org.

AwardsThe Unione Matematica Italiana has awarded the following
prizes for 2017:
the Mario Baldassarri Prize to Andrea Seppi;
the Giuseppe Bartolozzi Prize to Andrea Mondino;
the Guido Castelnuovo Prize to Chiara Andrà, Nicola Parolini
and Marco Verani for their project BetonMath;the Stefania Cotoneschi Prize to Daniele Pasquazi;
the Bruno de Finetti Prize to Alessandro FoschiDavid Pérez García (Universidad Complutense de Madrid,
Spain) received the Prize Miguel Catalán 2017 in recognitionof his contributions in the area of quantum technologies.
The 2017 Stefan Bergman Prize was awarded to Bo Berndtsson(University of Gothenburg and Chalmers University, Sweden)
and Nessim Sibony (Université Paris-Sud Orsay, France) for
their many fundamental contributions to several complex variables, complex potential theory, and complex geometry.
David Alonso Gutiérrez (Universidad de Zaragoza, Spain)
received the 2017 Research Award from Real Academia deCiencias de Zaragoza.
The Richard-von-Mises-Prize was awarded to Marc Avila(Bremen, Germany) in acknowledgment of his scientificachievements in the area of applied mathematics and mechanics.
The Israel Mathematical Union awards the 2018 Anna andLajos Erdős Prize in Mathematics to Ronen Eldan (Weizmann
Institute, Rehovot, Israel) and the 2018 Haim Nessyahu Prize
to Sara Tukachinsky (Hebrew University, Jerusalem, Israel)and Eliran Subag (Weizmann Institute, Rehovot, Israel).
Assaf Naor (Princeton University, USA) has been awarded the2018 Frederic Esser Nemmers Mathematics Prize for “his profound work on the geometry of metric spaces, which has led tobreakthroughs in the theory of algorithms.”
The Prize Marc Yor 2018 has been awarded to Christophe Garban (Université Lyon 1, France), for his work in probability andstatistical physics. The prize was instituted by SMAI (Society
for Applied and Industrial Mathematics) and SMF (Société
Mathématique de France) and it is patronized by the French
Académie des sciences.
The Norwegian Academy of Science and Letters has decided
to award the Abel Prize for 2018 to Robert P. Langlands (Institute for Advanced Study, Princeton, USA) “for his visionary
program connecting representation theory to number theory”.
The CNRS (France) has awarded the médaille d’argent to
Grégory Miermont (UMPA), the médaille de bronze to Anne-60

Laure Dalibard (LJLL) and the médaille de cristal to Elisabeth
Kneller (Bibliothèque mathématique Jacques Hadamard).
The Rolf Schock Prize 2018 of the Royal Swedish Academy of
Sciences has been awarded Ronald Coifman (Yale University,USA) “for his fundamental contributions to pure and applied
harmonic analysis”.
The 2018 Wolf Prize for Mathematics has been awarded toAlexander Beilinson and Vladimir Drinfeld (both Universityof Chicago, USA) “for their groundbreaking work in algebraic
geometry, representation theory, and mathematical physics”.
Sir John M. Ball (University of Oxford, UK) is the winnerof the 2018 King Faisal Prize for Science for his outstanding
contributions to mathematics. He also received the European
Academy of Sciences Leonardo da Vinci Award 2018.
The 2018 Breakthrough Prize in Mathematics was awarded
to Christopher Hacon (University of Utah, USA) and James
McKernan (University of California San Diego, USA) for
transformational contributions to birational algebraic geometry, especially to the minimal model program in all dimensions.
Martin Branda, Jan Šaroch (both Charles University, Prague,
Czech Republic) and Michal Doucha, Václav Mácha (bothInstitute of Mathematics of the CAS) received in 2018 thePrize for Young Mathematicians awarded by the Czech Mathematical Society.
Mickaël Launay is winner of the 2018 award of the SMF. He
received the Prix d’Alembert for his contributions to the popularisation of mathematics.

DeathsWe regret to announce the deaths of:
Rudolf Gorenflo (20 October 2017, Berlin, Germany)
Klaus Keimel (18 November 2017, Darmstadt, Germany)
Jan-Erik Roos (15 December 2017, Uppsala, Sweden)
Robert Adol’fovich Minlos (9 January 2018, Moscow, RussianFederation)
Jean-Louis Koszul (12 January 2018, Grenoble, France)
Ulrich Dieter (25 January 2018, Graz, Austria)
Manfred Stern (1 February 2018, Halle, Germany)
Alan Baker (4 February 2018, Cambridge, UK)
Herbert Heyer (8 February 2018, Tübingen, Germany)
Thomas Friedrich (27 February 2018, Berlin, Germany)
John Roe (9 March 2018, State College, USA)
Stephen William Hawking (14 March 2018, Cambridge, UK)
Willi Törnig (2 April 2018, Darmstadt, Germany)
Miguel Ángel Revilla Ramos (2 April 2018, Valladolid, Spain)
Bernardo Cascales Salinas (5 April 2018, Murcia, Spain)

EMS Newsletter June 2018

FUNCTIONAL ANALYSIS
Theo Bühler & Dietmar A. Salamon, ETH, Zurich
Functional analysis is a central subject of mathematics with applications in many areas of geometry, analysis, andphysics. This book provides a comprehensive introduction to the field for graduate students and researchers. With 10
to 20 elaborate exercises at the end of each chapter, this book can be used as a text for a course on functional analysis
for beginning graduate students.
Graduate Studies in Mathematics, Vol. 191
Jul 2018 472pp 9781470441906 Hardback €86.00

LECTURES ON NAVIER-STOKES EQUATIONS
Tai-Peng Tsai, University of British Columbia
A graduate text on the incompressible Navier-Stokes system, which is of fundamental importance in mathematicalfluid mechanics as well as in engineering applications. The goal is to give a rapid exposition on the existence,uniqueness, and regularity of its solutions, with a focus on the regularity problem.
Graduate Studies in Mathematics, Vol. 192
Jul 2018 224pp 9781470430962 Hardback €86.00

A PROBLEMS BASED COURSE IN ADVANCED CALCULUS
John M. Erdman, Portland State University
This textbook is suitable for a course in advanced calculus that promotes active learning through problem solving. It
can be used as a base for a Moore method or inquiry based class, or as a guide in a traditional classroom setting wherelectures are organized around the presentation of problems and solutions. This book is appropriate for any student
who has taken an introductory course in calculus.
Pure and Applied Undergraduate Texts, Vol. 32
Jul 2018 365pp 9781470442460 Hardback €82.00

VOLTERRA ADVENTURES
Joel H. Shapiro, Portland State University,
Introduces functional analysis to undergraduate mathematics students who possess a basic background in analysis
and linear algebra. By studying how the Volterra operator acts on vector spaces of continuous functions, its readers will
sharpen their skills, reinterpret what they already know, and learn fundamental Banach-space techniques.
Student Mathematical Library, Vol. 85
Jun 2018 248pp 9781470441166 Paperback €54.00

Free delivery at eurospanbookstore.com/ams
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New books published by the

Individual members of the EMS, member
societies or societies with a reciprocity agreement (such as the American, Australian andCanadian Mathematical Societies) are entitledto a discount of 20% on any book purchases, ifordered directly at the EMS Publishing House.

EuropeanMathematicalSociety

Timothée Marquis (Université Catholique de Louvain, Louvain-la-Neuve, Belgium)
An Introduction to Kac–Moody Groups over Fields (EMS Textbooks in Mathematics)
ISBN 978-3-03719-187-3. 2018. 341 pages. Hardcover. 16.5 x 23.5 cm. 48.00 Euro
The interest for Kac–Moody algebras and groups has grown exponentially in the past decades, both in the mathematical andphysics communities, and with it also the need for an introductory textbook on the topic.
The aims of this book are twofold:- to offer an accessible, reader-friendly and self-contained introduction to Kac–Moody algebras and groups;- to clean the foundations and to provide a unified treatment of the theory.
The book starts with an outline of the classical Lie theory, used to set the scene. Part II provides a self-contained introduction
to Kac–Moody algebras. The heart of the book is Part III, which develops an intuitive approach to the construction and fundamental properties of Kac–Moody groups. It is complemented by two appendices, respectively offering introductions to affinegroup schemes and to the theory of buildings. Many exercises are included, accompanying the readers throughout their journey.
The book assumes only a minimal background in linear algebra and basic topology, and is addressed to anyone interested inlearning about Kac–Moody algebras and/or groups, from graduate (master) students to specialists.

Alexander V. Kosyak (National Academy of Science of Ukraine, Kiev, Ukraine)
Regular, Quasi-regular and Induced Representations of Infinite-dimensional Groups (EMS Tracts in Mathematics
Vol. 29)
ISBN 978-3-03719-181-1. 2018. 587 pages. Hardcover. 17 x 24 cm. 98.00 Euro
Almost all harmonic analysis on locally compact groups is based on the existence (and uniqueness) of a Haar measure. Therefore,
it is very natural to attempt a similar construction for non-locally compact groups. The essential idea is to replace the nonexisting Haar measure on an infinite-dimensional group by a suitable quasi-invariant measure on an appropriate completion of
the initial group or on the completion of a homogeneous space.
The aim of the book is a systematic development, by example, of noncommutative harmonic analysis on infinite-dimensional(non-locally compact) matrix groups. We generalize the notion of regular, quasi-regular and induced representations for arbitrary
infinite-dimensional groups. The central idea to verify the irreducibility is the Ismagilov conjecture. We also extend the Kirillov
orbit method for the group of upper triangular matrices of infinite order.
In order to make the content accessible to a wide audience of nonspecialists, the exposition is essentially self-contained and very
few prerequisites are needed. The book is aimed at graduate and advanced undergraduate students, as well as mathematicians
who wish an introduction to representations of infinite-dimensional groups.

Non-Linear Partial Differential Equations, Mathematical Physics, and Stochastic Analysis. The Helge Holden
Anniversary Volume (EMS Series of Congress Reports)
Fritz Gesztesy (Baylor University, Waco, USA), Harald Hanche-Olsen (The Norwegian University of Science and Technology,
Trondheim, Norway), Espen R. Jakobsen (The Norwegian University of Science and Technology, Trondheim, Norway), Yurii I.
Lyubarskii (The Norwegian University of Science and Technology, Trondheim, Norway), Nils Henrik Risebro (University of Oslo,
Norway) and Kristian Seip (Norwegian University of Science and Technology, Trondheim, Norway) , Editors
ISBN 978-3-03719-186-6. 2018. 502 pages. Hardcover. 17 x 24 cm. 98.00 EuroThis volume is dedicated to Helge Holden on the occasion of his 60th anniversary. It collects contributions by numerous scientists
with expertise in non-linear partial differential equations (PDEs), mathematical physics, and stochastic analysis, reflecting to alarge degree Helge Holden’s longstanding research interests. Accordingly, the problems addressed in the contributions deal
with a large range of topics, including, in particular, infinite-dimensional analysis, linear and nonlinear PDEs, stochastic analysis,
spectral theory, completely integrable systems, random matrix theory, and chaotic dynamics and sestina poetry. They represent
to some extent the lectures presented at the conference Non-linear PDEs, Mathematical Physics and Stochastic Analysis, held at
NTNU, Trondheim, July 4–7, 2016 (https://wiki.math.ntnu.no/holden60).
The mathematical tools involved draw from a wide variety of techniques in functional analysis, operator theory, and probability
theory.
This collection of research papers will be of interest to any active scientist working in one of the above mentioned areas.

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