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NEWSLETTERPHILIPPAFAWCETT

Issue: 488 - May 2020

2520
ANTS FOR
DYLAN

MATHS
MEETS
ARTS

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EDITOR-IN-CHIEF

COPYRIGHT NOTICE

Eleanor Lingham (Sheffield Hallam University)
newsletter.editor@lms.ac.uk

News items and notices in the Newsletter may
be freely used elsewhere unless otherwise
stated, although attribution is requested when
reproducing whole articles. Contributions to
the Newsletter are made under a non-exclusive
licence; please contact the author or
photographer for the rights to reproduce.
The LMS cannot accept responsibility for theaccuracy of information in the Newsletter. Views
expressed do not necessarily represent theviews or policy of the Editorial Team or London
Mathematical Society.

EDITORIAL BOARD
June Barrow-Green (Open University)
David Chillingworth (University of Southampton)
Jessica Enright (University of Glasgow)
Jonathan Fraser (University of St Andrews)
Jelena Grbić (University of Southampton)
Cathy Hobbs (UWE)
Thomas Hudson (University of Warwick)
Stephen Huggett
Adam Johansen (University of Warwick)
Aditi Kar (Royal Holloway University)
Mark McCartney (Ulster University)
Susan Oakes (London Mathematical Society)
Andrew Wade (Durham University)
Mike Whittaker (University of Glasgow)
Early Career Content Editor: Jelena Grbić
News Editor: Susan Oakes
Reviews Editor: Mark McCartney

ISSN: 2516-3841 (Print)
ISSN: 2516-385X (Online)
DOI: 10.1112/NLMS

NEWSLETTER WEBSITE
The Newsletter is freely available electronicallyat lms.ac.uk/publications/lms-newsletter.
MEMBERSHIPCORRESPONDENTS AND STAFF
LMS/EMS Correspondent: David Chillingworth
Policy Digest: John Johnston
Production: Katherine WrightPrinting: Holbrooks Printers Ltd
EDITORIAL OFFICE
London Mathematical Society
De Morgan House
57–58 Russell Square
London WC1B 4HS
newsletter@lms.ac.uk

Joining the LMS is a straightforward process. For
membership details see lms.ac.uk/membership.
SUBMISSIONSThe Newsletter welcomes submissions offeature content, including mathematical articles,
career related articles, and microtheses, frommembers and non-members. Submission
guidelines and LaTeX templates can be found at
lms.ac.uk/publications/submit-to-the-lms-newsletter.
Feature content should be submitted to theeditor-in-chief at newsletter.editor@lms.ac.uk.

Charity registration number: 252660

News items should be sent tonewsletter@lms.ac.uk.

COVER IMAGE

Notices of events should be prepared using thetemplate at lms.ac.uk/publications/lms-newsletterand sent to calendar@lms.ac.uk.

Sounds of Surface: an art installation by
Jenny Hibberd & Andrew Johnston
Photo credit: Dean Leavers

For advertising rates and guidelines see
lms.ac.uk/publications/advertise-in-the-lms-newsletter.

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CONTENTS

The latest from the LMS and elsewhere4

Reports from the LMS

14

The Hong Kong Mathematical Society

17

On being LMS President

18

The First Woman Senior Wrangler

212520 Ants for Dylan

28

Maths Meets Arts

31

Mathematics News Flash36

Maths and Mingle

38

Microthesis: Massey Products in
Moment–Angle Complexes

40

From the bookshelf

42

In memoriam

47

EVENTS

Latest announcements

51

CALENDARAll forthcoming events

52

NEWS
LMS BUSINESSFEATURESEARLY CAREERREVIEWS
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FROM THE EDITOR-IN-CHIEF
Well here we all are at the start of May — the sun isshining, the birds are singing and the world is so verychanged. Who knew that these changes were coming?
Who could have predicted that the whole world (almost)would come to a standstill — that people would stay athome, that universities would empty, that conferences
would cease, that lecturing would be via a screen, andexams via the internet?
We could fill this Newsletter with what has stopped,
what has been affected, and what has changed. But
let’s try and think of the positives — potentially, we
have more time to think; more time of quiet; more time
to reflect — although, perhaps, as Jonathan Fraser’s
article describes, this isn’t the case for all of us. While
many conferences have been cancelled or postponed,
lots of us have finally bitten the ‘online’ bullet (yes,I have been quite slow to the party) and can now
virtually connect as well as anyone. And what about

funding? Application deadlines are ongoing — see the
EPSRC New Horizons call — and, who could believe it,
you might actually have a leisurely amount of time toput your application together.
What else? Will we join the coronavirus fight? Will we
finally read that book that has been on the shelf fornearly a decade? Will we take the time to brush up onthat bit of maths (or heavens, statistics!) that could be
useful? Will we think about joining a Society committee— or write an article for the Newsletter? (Please do.)Will we reconnect with our hobbies (remember those?)and other interests? Have a look at Katrin Leschke’s
article for inspiration.
And if none of these suggestions appeal — remember— there is always the Riemann Hypothesis.
Eleanor Lingham
Editor-in-Chief

LMS NEWS

Caroline Wallace

decide on the allocation of users to channels. Caroline
remains a Chartered Engineer, and a member of the
IET and IMechE.
Caroline’s career then took her into the economic
regulation of telecoms networks, and from there to
public interest regulation in general. She regards it as
crucially important that such regulation is informed bytechnical expertise.

Caroline Wallace is the Society’s new Executive
Secretary. She was born in New Zealand, where at theage of 8 she noticed the non-uniform gradations on awedge-shaped rain gauge and was intrigued to know
how the spacings were worked out.
A decade later Caroline became a student at
Cambridge, graduating with a first in Engineering. She
worked for some time as an engineer. For example,
she was the radio network capacity manager for atelecoms company, where she used queueing theory to

For the last several years Caroline has been theStrategy Director at the Legal Services Board, which
has made her very familiar both with managing
multi-disciplinary teams and with the legal system.
Now she is very pleased to have come back into thescientific world, and started as the LMS Executive
Secretary on 6 April. She says, “I am very impressed by
the calibre of people at the LMS, and by its reputationand history. I am keen to make a contribution to theimportant work the Society does for mathematics.”
Stephen Huggett
LMS General Secretary

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Open Access publishing in the UK
The LMS is continuing to monitor and, whereappropriate, respond to changes within the academic
publishing landscape, including the recent increased
drive by funders toward Open Access outputs.
In April 2020, the Society responded to a consultation
on a draft Open Access policy for peer-reviewed
research articles and academic books that result fromresearch supported by UK Research and Innovation(UKRI). The Society’s response reflected its role as arepresentative of the UK mathematics community andas a publisher. Details of the UKRI Open Access Review
can be found on their website at tiny.cc/UKRI_OA.
Recently, a ‘read and publish’ agreement was struck
between Jisc (the UK’s research and education
non-profit organisation that arranges licences anddigital content agreements on behalf of higher
education institutes), and Wiley (the Society’s partnerin the production, hosting and distribution of six LMS
journals).
As well as being able to read all of Wiley’s journals,
researchers from 138 Jisc member institutions will be
eligible to publish Open Access in hybrid and fully
Open Access journals published by Wiley at no direct
cost to the author(s) or their institutions.

Eligible corresponding authors whose papers are
accepted for the Bulletin, Journal, Proceedings,
Transactions, Journal of Topology and Mathematika
will be automatically presented with the option of
publishing the version of record of their article Open
Access under a Creative Commons licence. Specificguidance can be found at tiny.cc/Wiley_Jisc.
Similar arrangements have previously been negotiatedbetween Jisc and a number of other academic
publishers. These include Cambridge University Pressand Institute of Physics Publishing, with whom theSociety also has publishing partnerships.
These new agreements, which are increasingly being
requested by and negotiated with international libraryconsortia, represent one way in which many publishers
are proposing to gradually transition towards Open
Access in a sustainable fashion.
The Society is currently preparing its response to a
consultation (which closes on 29 May) on a draft Open
Access policy for peer-reviewed research articles andacademic books that result from research supported
by UK Research and Innovation (UKRI).
John Hunton
LMS Publications Secretary

OTHER NEWSClay Research Fellows 2020
The Clay Mathematics Institute is pleased toannounce that Amol Aggarwal and Yang Li have been
awarded Clay Research Fellowships.
Amol Aggarwal will receive his PhD in 2020 from
Harvard University, where he has been advised
by Alexei Borodin. His research lies largely inprobability theory and combinatorics, as wellas their connections to mathematical physics,
integrable systems, and dynamical systems.
Yang Li received his PhD in 2019 from Imperial
College London, under the guidance of Simon
Donaldson and Mark Haskins. He has already madesignificant contributions to the understandingof Calabi–Yau metrics in complex differential
geometry and Riemannian manifolds withexceptional holonomy.

IMA Catherine Richards Prize
The best article in Mathematics Today (published by
the Institute of Mathematics and Its Applications)is awarded the Catherine Richards Prize each year.
Dr Mark McCartney (University of Ulster) was awarded
the prize for Fluids, Fluorescence and a Hat Full of Beetles
which was published to celebrate the bicentenary of
the birth of George Gabriel Stokes.
Mark McCartney is has published widely on the historyof mathematics. His most recent book is George Gabriel
Stokes: Life, Science and Faith, which covers the many
facets of Stokes’s life through a collection of essaysthat Mark co-edited with Andrew Whitaker and Alastair
Wood.
Mark is President of the British Society for the History ofMathematics, Reviews Editor of the LMS Newsletter, LMS
Librarian and Associate Editor of the International Journalof Mathematical Education in Science and Technology.

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2020 Abel Prize Winners Announced

Fellow of the Royal Society of
Edinburgh 2020
Beatrice Pelloni (Heriot-Watt University), a member ofthe LMS, the 2019 Mary Cartwright Lecturer and aformer member of Council, has been elected a 2020
Fellow of the Royal Society of Edinburgh.

May 12: Celebrating Womenin Mathematics

Hillel Furstenberg (left) and Gregory Margulis

The Norwegian Academy of Science and Letters
has awarded the 2020 Abel Prize jointly to Hillel
Furstenberg (Hebrew University of Jerusalem) and
Gregory Margulis (Yale University) ‘for pioneering theuse of methods from probability and dynamics ingroup theory, number theory and combinatorics’.
Furstenberg and Margulis invented random walktechniques to investigate mathematical objects suchas groups and graphs, and as a result introduced
probabilistic methods to solve many open problems
in group theory, number theory, combinatorics andgraph theory. A random walk is a path consisting of a
succession of random steps, and the study of random
walks is a central branch of probability theory.
Furstenberg has a deep technical knowledgeof diverse areas and has made insightful andsurprising connections between them. In particular,
he has made fundamental contributions to
the field of ergodic theory, which have had
far-reaching applications in number theory, geometry,combinatorics, group theory and probability. Of his
many accolades, Furstenberg won the Israel Prize in
1993, regarded as the top cultural honour in Israel,
and the Wolf Prize in 2007.
During his illustrious mathematical career, Margulis
has introduced many influential ideas, solved
long-standing open questions, and discovered deep
connections between different mathematical fields.
His signature approach has been to use ergodic
theory in unexpected and ingenious ways, which
has led to the creation of entirely new areas ofstudy. Throughout his career he has received several
significant awards, including the Fields Medal in 1978
at the age of 32 and the Wolf Prize in 2005.

Maryam Mirzakhani, the first female Fields medallist,
was born on 12 May 1977, and died prematurely on
14 July 2017. As a result of an initiative proposed by
the Women’s Committee of the Iranian MathematicalSociety, the World Meeting for Women in Mathematics
in Rio in 2018 agreed that 12 May should be chosen as
the day to celebrate women in mathematics aroundthe world.
Because of the COVID-19 crisis, 12 May 2020 will bevery different from that in 2019. But despite all the
restrictions, virtual events can still be organised — see
may12.womeninmaths.org for ideas.
In particular, the 12 May organisers have made anagreement with zalafilms for a free screening of theirbeautiful film Secrets of the Surface, about Mirzakhani
and her work. If you would like to watch the film, you
are invited to register at tinyurl.com/vysafy8. You will
receive a link by email for a free screening on a dateof your choice the date in the period 1 April to 15 May.
This link is personal to you and you are asked not toshare it outside the specific use you are requesting.

Mathematics and COVID-19
An interesting side-effect of the COVID-19 pandemic
has been the raising of the profile of mathematics
in the public eye. Suddenly, experts on mathematicalmodelling of the spread of infectious diseases suchas Neil Ferguson of Imperial College, London appeared
as key figures on news programmes, and the generalpublic has come to grapple with concepts of infection
ratios, exponents and second derivatives.
In parallel with the calls for retired medics, otherworkers and volunteers to help with the physicaldemands of fighting the pandemic, calls also went out
to mathematicians, data analysts, statistical modellers,
and the wider scientific community to join the fight at
the theoretical level: notably the Royal Society’s RAMP
initiative and the rapid response call from the Alan
Turing Institute, as well as the grassroots crowdfightcampaign.

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NEWS

The Gender Gap Project ReportThismulti-author
book
with
livelyillustrations, edited byMarie-Françoise
Roy
and Colette Guillopé,is the final report ofa three-year project(2017–19) funded by theInternational Science
Council and involving
elevenscientificpartner organisations, including the International
Mathematical Union (IMU). Marie-Françoise Roy, Chairof the IMU’s Committee for Women in Mathematics(CWM), was one of the initiators and a main driving
force throughout.
The main goal of this ambitious and extensive
project was to investigate the gender gap in STEM
disciplines from different angles, globally and acrossdisciplines. The authors carried out a global surveyof scientists, both male and female, with more than
32,000 responses; investigated the effect of genderin millions of scientific publications; and compiled
a list of best-practice initiatives which address thegender gap in Mathematical, Computing, and Natural
Sciences at various levels.
After detailed compilation of the results, theauthors conclude that the gender gap is very realin science and mathematics. The book presents
methodologies, insights, and tools that have beendeveloped throughout the project, as well as aset of recommendations for different audiences:
instructors and parents; educational institutions;scientific unions and other organisations responsible
for science policy.
The book is freely available and can be found fromthe Gender Gap webpage gender-gap-in-science.org,
together with a short summary booklet including thefull list of recommendations. The book can also be
downloaded directly from tinyurl.com/v9d8pmz. It
will shortly be published with an ISBN number and aprint on demand option.

Mathematics Subject
Classification 2020
The latest revision of the Mathematics Subject
Classification (MSC) has been published,replacing the 2010 Mathematics Subject
Classification (referred to as MSC2010). Searchable

versions are available from the zbMATH site:
zbmath.org/classification/ and the MathSciNet site:mathscinet.ams.org/mathscinet/searchMSC.html.
Mathematical Reviews (MR) and zbMATH collaborate
on maintaining the Mathematics Subject
Classification, which is used by these reviewing
services, publishers, funding agencies and others
to categorise items in the mathematical sciences
literature. It is a taxonomy created by and formathematical researchers. Every ten years, the two
editorial groups solicit input from the mathematics
community. For the current revision, we received
over 350 comments and suggestions from morethan 100 different people. MR and zbMATH carefully
considered this input from the community and used
it in the preparation of our joint revision of theclassification.
As anticipated, there are no changes at the two-digit
level, but several at the three-digit level, andhundreds at the five-digit level. Nine new three-digitclasses were added: 18M Monoidal categories andoperads; 18N Higher categories and homotopicalalgebra; 53E Geometric evolution equations; 57K
Low-dimensional topology in specific dimensions;
57Z Relations of manifolds and cell complexes with
science and engineering; 60L Rough analysis; 62R
Statistics on algebraic and topological structures;
68V Computer science support for mathematicalresearch and practice, and 82M Basic methods
in statistical mechanics. For five-digit classes, 113
classes were retired and 486 new classes wereintroduced. The new MSC contains 63 two-digitclassifications, 529 three-digit classifications, and6006 five-digit classifications.
There were some general changes. Descriptionsof classes were changed to be more useful whensearching online or via database interfaces. Previous
descriptions assumed the user was looking at afull list of the classifications, which would provide
context. An example of the limitation is a searchof MSC2010 for ‘optimisation’, which returns 18
matches, not counting essentially every class in
49 Calculus of variations and optimal control.
There were three classes named ‘Flow Control andOptimization’: 76B75, 76D55, and 76N25. The threedifferent contexts were incompressible inviscid fluids,
incompressible viscous fluids, and compressible
fluids and gas dynamics. Now they have descriptions
with more detail, as in 76B75 Flow control andoptimisation for incompressible inviscid fluids. There
were three classes just named ‘Optimization’ in theareas 74P Mechanics of deformable solids, 78M50

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Optics, electromagnetic theory, and 80M50 Classicalthermodynamics, heat transfer. Now they havedescriptions that include the context, as in 78M50
Optimisation problems in optics and electromagnetic
theory.
In previous versions of the MSC, there weresome ‘hyphen classes’ of the form XX–00 Generalreference works, XX–01 Introductory expositions,
XX–02 Research exposition, XX–03 History, XX–04
Software, and XX–06 Proceedings, conferences,collections, etc., along with other scattered hyphenclasses. The use of hyphen classes has been mademore uniform across the MSC, so that most two-digitclasses now have these five subclasses. Somehyphen classes would be redundant and are omitted,
such as the non-existent class 01-03, since thetwo-digit class for ‘History of Mathematics andMathematicians’ does not need a subclass for history.
The classes -08 for Computational methods forproblems from the parent class, -10 for Mathematicalmodeling or simulation for problems from the parent
class, and -11 for Research data for problems from theparent class have been added where appropriate. For
example, there is now the class 20-08 Computational
methods for problems from group theory and theclass 20-11 Research data for problems from grouptheory. An example of an omission of one of these
new hyphen classes for reasons of redundancy is
65 Numerical analysis, which does not need the -08
class for computational methods. Also, some classes
have alternatives to -08 with more detail, such as
the eight five-digit classes in the three-digit class 14Q
Computational aspects in algebraic geometry. The
hyphen classes -10 mostly occur for applied classes,
namely MSCs 70 through 94, as in 70-10 Mathematicalmodeling or simulation for problems from mechanics
of particles and systems.
The influences of data and computation on
the mathematical sciences are reflected in theclasses. In addition to the -08 classes, and notincluding classes from 03 (Mathematical logic andfoundations) or 68 (Computer science), there are
58 classes referring to computations, computational
methods, or computers. For instance, for MSC2020,
two new classes, 14Q25 Computational algebraic
geometry over arithmetic ground fields and 14Q30
Computational real algebraic geometry were added
to the three-digit class 14Q Computational aspects
in algebraic geometry, which had been added to
the MSC in 1991. Similarly, two new classes wereadded under 37M Approximation methods andnumerical treatment of dynamical systems: 37M21
Computational methods for invariant manifolds ofdynamical systems, 37M22 Computational methods
for attractors of dynamical systems and 37M25
Computational methods for ergodic theory. For
the -11 classes, examples of the types of dataenvisioned include statistical data, mathematical
tables, collections of mathematical objects and theirproperties, such as integer sequences (as found in
the OEIS, for instance), or databases of modular
forms or Calabi–Yau varieties. In addition to the -11
classes, there are 31 classes with specific instancesof data, including the new classes 62R10 Functional
data analysis, 62R40 Topological data analysis and
68P27 Privacy of data.
Mathematical Reviews and zbMATH are now using
MSC2020 as their classification schemes. We
welcome and encourage the community to also adopt
the MSC2020. It is available from msc2020.org in
PDF or TeX. A SKOS version will be available later.
The classification is jointly published by thetwo organisations under a Creative Commons
CC–BY–NC–SA license. Corrections to possible errors
in the new system can be submitted by email
to feedback@msc2020.org. All information about
MSC2020 is jointly shared by MR and zbMATH.
The editors and staff at Mathematical Reviews andzbMATH express their gratitude to the numerous
members of the community for their assistance inthis lengthy revision process.
Edward Dunne
Executive Editor, Mathematical Reviews

Mathematician Features in
The Times Daily Quiz
On 26 March 2020 the following question appearedin The Times Daily Quiz:
12 Which British mathematician (1916–2020)
proposed the partially tongue-in-cheek “Strong Law
of Small Numbers”?
The answer is someone mentioned in this Newsletter,
until very recently the oldest member of the LMS!
For the curious, the ‘Strong Law of Small Numbers’is described entertainingly — including 35 examplesof the law — in an article by this mathematician:tinyurl.com/vn3cro6.

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MATHEMATICS POLICY DIGEST

Mathematics in FE colleges report
A Nuffield funded report into mathematics education
across England’s further education sector has foundthat ‘major investment in leadership training andteacher professional development is required tomake more significant and sustained improvements’.
The report has been published as part of theMathematics in Further Education Colleges project(MIFEC). The report outlines how ‘a range of policies,
processes and people interact at different colleges
and across the FE sector to either support orhinder improvement in mathematics learning’. The
full report is available at tinyurl.com/wptncgl.

Parliamentary Select Committees
The House of Commons and House of Lords
Select Committees have been formed for thisParliament. The Chair of the Science and Technology
Select Committee is Greg Clark MP. The Chairof the Education Select Committee is RobertHalfon MP. The Chair of the Business, Energy andIndustrial Strategy Committee is Rachel Reeves
MP. The Chair of the House of Lords Science andTechnology Committee is Lord Patel. Full details of
all Committees, including membership, are available
at committees.parliament.uk.

Digest prepared by Dr John Johnston
Society Communications Officer

Note: items included in the Mathematics Policy Digest
are not necessarily endorsed by the Editorial Board or
the LMS.

HEILBRONN ANNUAL CONFERENCE 2020
Location: University of Bristol
Date: 10-11 September 2020
Website: https://eur.cvent.me/eGyG
The Heilbronn Institute for Mathematical Research welcomes eight distinguished speakers for its 2020
annual conference:
Maria Chudnovsky (Princeton)
Adam Harper (Warwick)Özlem Imamoglu (ETH Zurich)
Kurt Johansson (KTH Royal Institute of Technology)
Ailsa Keating (Cambridge)
Hendrik Lenstra (Universiteit Leiden)
Ulrike Tillmann (Oxford)
Ronald de Wolf (CWI and Universiteit van Amsterdam)
The conference will take place over two days and lectures will be accessible to a general audience ofmathematicians. See heilbronn.ac.uk/events for more information and to register.
Funding is available to support a limited number of UK-based PhD students and early career researchers.
We will be pleased to consider applications for funding to support care costs.
Email heilbronn-coordinator@bristol.ac.uk for further information.

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EUROPEAN MATHEMATICAL SOCIETY NEWS

zbMATH is now Open Access
Zentralblatt MATH (zbMATH) is the world’s mostcomprehensive and longest-running abstracting andreviewing service in pure and applied mathematics.
It is edited by the EMS, the Heidelberg Academyof Sciences and Humanities and FIZ Karlsruhe— Leibniz Institute for Information Infrastructure.
The Federal Republic of Germany has now agreed
to support FIZ Karlsruhe — Leibniz Institute intransforming zbMATH into an open access platform.
For more information on this important initiative see
tinyurl.com/ucgvbce.

St Petersburg Thematic
Programs 2021
The Leonhard Euler International Mathematical
Institute (EIMI) in St Petersburg has opened acall for programmes in 2021. It is expected that
two such programs will be hosted during thecalendar year 2021. Proposals should be sent toprograms@eimi.imi.ras.ru by 15 May 2020, anddecisions will be made before the end of that month.
For further details visit eimi.ru.

PI Day 2020
Most events for March 14 = PI Day = The International
Day of Mathematics had to be cancelled becauseof the coronavirus crisis. However, at the Wolfgang

Pauli Institute in Vienna, at very short notice, an
online event took place in the form of a 24-minutevideo in which the WPI director Norbert J. Mauser
introduced the Thumb times PI: PI-Day, where sevenmathematicians in quarantine in seven different
cities presented short messages for ‘PI day in theyear of corona’. A programme of this unusual event
can be seen at tinyurl.com/vs2lvrz and the video is
at tinyurl.com/yx63karr.

IMU Abacus Medal
The IMU has decided to accept the joint bid
from the University of Helsinki and the Simons
Foundation for the funding of the IMU Abacus Medal,
which will be awarded once every four years at
the International Congress of Mathematicians foroutstanding contributions in mathematical aspectsof Information Sciences. The IMU Abacus Medal is acontinuation of the Rolf Nevanlinna Prize that wasawarded from 1982 to 2018. For further details visit
tinyurl.com/yx3bovxo.
EMS News prepared by David Chillingworth
LMS/EMS Correspondent
Note: items included in the European MathematicalSociety News represent news from the EMS and are
not necessarily endorsed by the Editorial Board or the
LMS.

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OPPORTUNITIESLMS Research Grants: Callfor Applications
The application deadline for LMS Research Grants is
15 May. The LMS offers a number of grant schemes
to support conferences (up to £7,000 — Scheme 1),
visits to the UK (up to £1,500 — Scheme 2), Researchin Pairs (up to £1,200 — Scheme 4) and collaboration
with developing countries (up to £3,000 — Scheme
5). For more details visit tinyurl.com/ujj7ev6.

New Horizons Call
EPSRC invites proposals for adventurous research
projects across the breadth of pure and appliedmathematics, statistics and operational research.
Grants may last for up to 2 years and have an upper
limit of £200,000. Full details of the call, including thestreamlined application procedure, can be found at
tinyurl.com/r8n96yk. The deadline for applications
is 4 June.

VISITS

Visit of Sascha Troscheit
Dr Sascha Troscheit (University of Vienna) will
be visiting the UK from 27 July to 14 August2020. He will give lectures at the Universities ofBristol, Birmingham, and Glasgow. His work combines
aspects of Dynamical Systems, Fractal Geometry,and Probability Theory. His recent research activity isconcerned with determining the Assouad dimensionof fractal sets that are defined via some random
process. For further details contact Simon Baker(s.baker.2@bham.ac.uk). The visit is supported by an
LMS Scheme 2 grant.

ADVERTISE IN THE LMS NEWSLETTER
The LMS Newsletter appears six times a year (September, November,
January, March, May and July).
The Newsletter is distributed to just under 3,000 individual members,
as well as reciprocal societies and other academic bodies such as
the British Library, and is published on the LMS website at lms.ac.uk/
publications/lms-newsletter.
Information on advertising rates, formats and deadlines are at:
lms.ac.uk/publications/advertise-in-the-lms-newsletter.
Examples in this issue can be found on pages 9 and 15, and on the backpage.
To advertise contact Susan Oakes (susan.oakes@lms.ac.uk).

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The standing and usefulness of the Society depends upon the support of a strong
membership, to provide the resources, expertise and participation in the running of theSociety to support its many activities in publishing, grant-giving, conferences, public policy,
influencing government, and mathematics education in schools. The Society’s Council
therefore hopes that all mathematicians on the staff of UK universities and other similar
institutions will support mathematical research by joining the Society. It also very muchencourages applications from mathematicians of comparable standing who are working or
have worked in other occupations.
Benefits of LMS membership include access to the Verblunsky Members’ Room, free online
subscription to the Society’s three main journals and complimentary use of the Society’s
Library at UCL, among other LMS member benefits (lms.ac.uk/membership/memberbenefits).
If current members know of friends or colleagues who would like to join the Society, please
do encourage them to complete the online application form (lms.ac.uk/membership/onlineapplication).
Contact membership@lms.ac.uk for advice on becoming an LMS member.

CONFERENCE FACILITIESDe Morgan House offers a 40% discount on room hire to all mathematical charities and 20% to all
not-for-profit organisations. Call 0207 927 0800 or email roombookings@demorganhouse.co.uk
to check availability, receive a quote or arrange a visit to our venue.

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LMS BUSINESSLMS Council Diary — A Personal View
Council met at De Morgan House on Friday 31 January
2020. The meeting began with an update from the
new President on various matters, including thegenerous donation from the Liber Stiftung that is
supporting the new Emmy Noether Fellowships, witha possible research event for women in mathematics
also currently under consideration by the Womenin Mathematics Committee, and the exciting recent
announcement by government of a major funding
boost for mathematics.
The discussion then turned to the Sir Michael
Atiyah Memorial Projects. The President noted that
the Society is involved in both the funding andthe organisation of ‘The Unity of Mathematics: a
conference in honour of Sir Michael Atiyah’, to beheld at the Isaac Newton Institute in September
2020. Following the successful launch of the Atiyah
UK–Lebanon Fellowships in November last year,
which was attended by the Lebanese ambassador
to the UK, the Treasurer reported that the deadline
for the first round of applications was imminent.
Given the need to establish connections andmomentum for the Fellowships, a key objective inthe first instance was felt to be to send UK-based
mathematicians who are effective communicators to
Lebanon.
Open access issues of a number of flavours were
the focus of the Publication Secretary’s report on
Publications Committee business. As part of a groupof learned societies, the Society had contributed toa consultation on open access proposals that had
been put forward by the international consortiumof funding organisations cOAlition-S, intended toshake up the way that publications currently operate.
The UKRI Open Access Review Consultation was
expected to be launched in mid-February. Concern
was noted about possible financial implications formathematics of the increasing tendency of publishers
to address open access via large-scale, country-wideagreements and the way in which income from suchdeals is being allocated to individual publications.
Following the recent Standing Orders Review, theGeneral Secretary reported that it was hoped
the proposed amended Standing Orders would beconsidered at the February meeting of the Privy
Council, but this was yet to be confirmed. Underfinancial matters, in addition to the Treasurer giving
an overview of the First Quarter Financial Results,
Council heard from the Education Secretary abouta proposal to run two Mathematics Communication
and Outreach Training Days at De Morgan Houseeach year, and it was agreed to increase the budgetof the Education Committee to provide resources for
this.
There was a discussion about the invitation of the
Chair of the Women in Mathematics Committee toattend Council, which has been in place for someyears. It was agreed that the Chair of the Women in
Mathematics Committee would become a designatedposition for a Member-at-Large of Council with effectfrom the end of the current Chair’s term of office. We
also heard a report from the LMS Scrutineers on the2019 Council and Nominating Committee Elections,
noted annual reports from various committees, anddiscussed committee membership.
The meeting closed with the President noting that
this was the last Council meeting of the retiring
Executive Secretary, thanking her sincerely for her
years of service to the Society, and extending thebest wishes of everyone for the future.
Elaine Crooks
Member-at-Large

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LMS BUSINESSREPORTS OF THE LMS

Report: Probabilistic Coupling andGeometry Workshop

Speakers at the event

A workshop on Probabilistic Coupling and Geometry
was held on 9–10 December 2019 at the Universityof Warwick to mark Professor Wilfrid Kendall’s 65thbirthday.
Probabilistic coupling refers to the practice ofconstructing two (or more) probability measures
on a single measurable space in order to compare
them. Some of its main applications include
representing one process in terms of another, proving
stochastic comparison arguments, bounding the rateof convergence of Markov processes, and simulation
algorithms. Wilfrid Kendall has made significant
contributions in all of these areas and beyond. In
addition, he has played important roles within theprobability and statistics community: he was the
co-founder of the Academy for PhD Training in
Statistics (APTS), which for the last 13 years has
provided fundamental training courses for first year
PhD students across the UK; in 2011 he helped
to found the Applied Probability Section of theRoyal Statistical Society; and from 2013–2015 he was
President of the Bernoulli Society.
Some 45 people attended the workshop, which
was funded by an LMS Conference Grant, andco-sponsored by both the Bernoulli Society and theRoyal Statistical Society. The workshop’s principal
speakers were: Sayan Banerjee (University ofNorth Carolina), Julia Brettschneider (University of
Warwick), Professor Krzysztof Burdzy (University ofWashington), Elisabetta Candellero (Università degliStudi Roma Tre), Professor Huiling Le (University of
Nottingham), Professor Gareth Roberts (University of
Warwick), Professor Jeffrey Rosenthal (University ofToronto) and Giacomo Zanella (Università Bocconi).
Over the course of
the two-day workshop,
attendees were treatedto15
high-quality
talks.
These
wereeither delivered by
Wilfrid’s collaboratorsand colleagues, or they
concerned work inspiredby his research. Topics
Wilfrid Kendall
covered included change
points in dynamic networks; analysis of dead pixel
patterns in x-ray detectors; Archimedes’ principle;
Stein’s method; a number of perfect simulation and
MCMC techniques; stochastic control; card shuffling;an oil and water model; phylogenetic tree analysis;
random growth models; random walks with memory.
The mixture of theory and applications, and thebroad range of subjects, showed just how widely
Wilfrid has influenced the field of probability during
the last forty years. The proceedings concluded witha conference dinner sponsored by the Departmentof Statistics at the University of Warwick.
Further details about the workshop can be found at
tinyurl.com/wsk65.

Stephen Connor
University of York“"NLMS_488 (Katherine Wright’s conflicted copy 2020-04-20)"” — 2020/4/21 — 10:17 — page 15 — #15

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Fundamental Concepts In Modern Analysis
An Introduction To Nonlinear Analysis(Second Edition)

by Vagn Lundsgaard Hansen (Technical University of Denmark, Denmark)With Poul G Hjorth
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Dec 2019
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This book provides a self-contained and rigorous presentation of the mainmathematical tools needed to approach many courses at the last year ofundergraduate in Physics and MSc programs, from Electromagnetism to
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FEATURES

Hong Kong Mathematical Society
The
Hong
Kong
Mathematical Society
was established in
1979, which went alongwith the contemporary
development of tertiary education and researchin mathematics in Hong Kong. The University ofHong Kong, the first university, opened at the
beginning of the 20th century and there were very
few mathematicians at that time. After World War II,
Professor Yung Chow Wong joined what was then thesole university as Chair of Mathematics. Professor
Wong was a key promoter who established theHKMS and served as the Honorary President of theSociety. In 1963 a second university, The Chinese
University of Hong Kong, was established. Moremathematicians were recruited for undergraduateteaching and postgraduate training as well as doing
research. These two universities were the maintask force in mathematical research but there wereother post-secondary institutions with mathematics
departments. In 1979, the HKMS was founded andits first President was Dr Irving Tang from the Hong
Kong Polytechnic. In 1991 a third university, Hong
Kong University of Science and Technology, waslaunched and some other institutes became fully
accredited universities in Hong Kong. Mathematics
research in Hong Kong is now well-rounded, coveringvarious areas in pure and applied mathematics.
The
HKMS
has
six
Institutional
Members(currently
local universities) andis supported by agreat
numberofmathematicians working
in these and otherlocal institutions. Since
1982 the Society has
Prof Y.C. Wong (1970s)
been a member of theInternational Mathematical Union. The Society
aims to foster academic research and informationexchanges amongst colleagues and to provide
favourable conditions for mathematics education,
both in theory and applications. One of the regularsociety activities is the Annual General Meeting; thesix institutional members take turn to co-organise
this event in their campus. The AGM is a one-day

event, consisting of a distinguished lecture, plenary
lectures, parallel sessions of invited talks and award
presentation. The HKMS presents a Best ThesisAward and a Young Scholar Award annually at theAGM. The HKMS Best Thesis Award recognises
outstanding PhD theses (dissertations) in the fields
of pure and applied mathematics, and statistics. The
HKMS Young Scholar Award recognises outstandingachievements of young scholars who are working in
the institutions in Hong Kong and not exceeding theage of 45.
In order to foster exchanges of ideas, the HKMS hasorganised several important conferences. These include
the Southeast Asian Mathematical Society Conference in
1980, The First Asian Mathematical Conference in 1990
and the AMS–HKMS Joint Conference in 2000. The plenary
speakers of the AMS–HKMS Joint Conference in 2000
consisted of Jianshu Li, Thomas Liggett, Fang Hua Lin,
Ngai Ming Mok, Gilles Pisier, Mike Shub, Gang Tian andZhouping Xin. Moreover, the Society supports conferences
and workshops organised in local institutions, and every
year it holds a HKMS Distinguished Lecture Series which
is another signature event of the society.
In November 2019, the Hong Kong Mathematical Society
hosted a visit of a delegation from the London
Mathematical Society to build on existing links between
UK and Hong Kong mathematics. There was a shortmeeting of people from both societies. The LMS delegation
included the President Designate, Professor Jon Keating,and the Publications Secretary, Professor John Hunton,
together with two LMS Members, Professor Jens Marklof(University of Bristol) and Professor Gui-Qiang Chen
(University of Oxford). The participants from HKMS were
the President, Professor Tong Yang, Professor Member at
Large, Professor Ngai-Ming Mok, Treasurer, Professor Wei
Xiang, Public Relations Secretary, Dr Yuk Kam Lau, General
Secretary, Professor Tao Luo, Programme Secretary,
Professor Zhian Wang. The meeting was pleasant andrelaxed, and members took this valuable opportunity
to build mutual understanding and to explore possible
collaborations between the two societies. More fruitfuland long term cooperation is anticipated in future.
More information about the HKMS can be found on its
website at www.hkms.org.hk.
Tong Yang
President of the Hong Kong Mathematical Society

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FEATURES

On being LMS President
CAROLINE SERIES
Professor Caroline Series FRS was LMS President November 2017–November 2019. Below she shares some ofher experiences during her time in office.
My time as LMS President brought with it
opportunities and challenges, responsibilities anda fair sprinkling of headaches, along with many
privileges and not a few rewards. The LMS is awell-oiled machine which largely runs like clockwork,
so in addition to admiring the output of our many
committees, what does the President actually do?
Besides representing the Society to all sorts ofpeople at all sorts of functions, chairing numberlessmeetings and being present at as many of ourregional meetings as possible, the job of the President
as I understood it, is to fill in the gaps and take thelead on anything that doesn’t sit conveniently in anyof the well established categories.

two committee structure, which took shape as theStrategic and Implementation Committees under
Claire Craig and Bernard Silverman respectively. We
await their conclusions later this year.

April 2018 saw thelaunch of the Bond
Review. Shortly before
the launch, Philip Bondsuggested that we meetat De Morgan House.
Having discussed thereview and his ideas atsome length, he quite
unexpectedly
asked
whether I could supply
some fractal images
The Bond Review coversimilar to those in mybook Indra’s Pearls. Swallowing my astonishmentthat he had come across the book at all, I was
pleased to help and delighted to see our picturesdesigned into his glossy Review booklet.

Early on, I became involved in the ongoing revision of
the fifty-year-old LMS Charter, Statutes and By-Laws
which set out how the LMS is governed. The revision
had been started in 2014, but had been put on
hold during the planning for our 150th anniversarycelebrations. By the time I joined in, the major lifting
was largely done, however there remained a lotof details and some knotty points to consider. A
timetable had been drawn up which indicated that
if we worked hard, I might hope that my last act
as President would be to oversee the vote on the
final revision. And indeed this is what happened,
although not without a great deal of work on thepart of our General Secretary, Stephen Huggett, andgeneral prodding from me to keep things rolling along.
At first sight even reading the document seemed adaunting task. I don’t think anyone had ever paid
much attention to the first page – I noticed that we
had had a change of address since 1965 and at a laterstage our Editorial Manager Ola Törnkvist realisedthat we had also changed parish! Such were thedetails – I very much hope that the general thrust of
the revisions will prove as enduring a document as
the previous one. It was a good moment when our
homework came back from the preliminary scrutiny
by the Privy Council with essentially nothing further
to do.

The main question of course was, how should the
LMS respond? It was clear to me both that wemust in some way consider the Review’s many
proposals, and also that whatever finally emerged
as a community response should grow out ofand be endorsed by the CMS (Committee forMathematical Sciences). Numerous phone calls anddiscussions behind the scenes over the summerof 2018 led to setting up the LMS Bond working
group under past President Terry Lyons and theeventual emergence of the CMS proposal for a

The President is always invited to the Abel Prizecelebrations in Oslo, but I certainly had not expected
to receive an invitation to give ‘the speech’ at oneof the highlights of the proceedings: the grandbanquet attended by many Norwegian politicians
and dignitaries including the King. At that point thewinner had not even been decided, but, after somethought and with considerable trepidation, I accepted.
In the event the winner was Robert Langlands andI had to scratch my head and think back to mygraduate student days when many of my colleagues

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FEATURES
were working on the mysterious ‘Jacquet-Langlands’.
My speech (which I am proud to say managed to
weave in Pythagoras, Abel, Fermat, Selberg, Wiles andLanglands) went down well. The ceremonies weremagnificent and it was an unforgettable week.

coming up which presented the ideal opportunity
for a brainstorming session on what the LMS
might do to commemorate him. Discarding theidea of a statue in Russell Square and otherunlikely proposals, we eventually settled on two
ventures: the LMS would spearhead a major
conference on the Unity of Mathematics (seewww.newton.ac.uk/event/ooew02), and, as a more
lasting memorial, set up the Atiyah Fellowships (seewww.lms.ac.uk/grants/atiyah-uk-lebanon-fellowships)
for collaboration between the UK and Lebanon.
Approaches made by myself and our President
Designate, Jon Keating, rapidly secured core funding
for the conference, while working hard with ourTreasurer, Rob Curtis, and Jihad Touma, Director of
the Center for Advanced Mathematical Sciences at
the American University of Beirut, we were able toannounce the Fellowships by video-link to Beirut tocoincide with the 20th anniversary of CAMS, the day
after our AGM.

(l-r) Robert Langlands and King Harald V of Norway at theAbel Prize 2018 celebrations

The President also attends the ICM, thus in summer
2018 it fell to me to travel to Rio for the first everCongress in the southern hemisphere. Most evenings
merged into blur of receptions put on by all the larger
mathematical societies, mainly held in the same room
and with similar refreshments, largely alcoholic. Thus
I was able to do a lot of networking which I trust
was good for the Society. We had agreed that the
LMS would partially sponsor Marianne Freiberger and
Rachel Thomas of the online Plus magazine to assist
with our publicity, and I spent quite a bit of time
initiating contacts with various people, including our
newly elected honorary members, Peter Scholze andMaria Esteban, for them to interview. But my abiding
memory has to be of Michael Atiyah who had madethe journey all the way from Edinburgh to deliver
the Abel lecture. I organised a mini-reception for
Caucher Birkar to meet the LMS delegation and mostimportantly Michael, who promptly launched intoa lengthy discourse suggesting to Caucher further
directions for his work. Michael also came to a lunchfor ‘supporters of the IMU’, in which he held thefloor telling the IMU that it should focus on its main
objectives and not try to spread itself too widely– advice which I have often thought about, if notexactly followed, since.
I heard the sad news that Atiyah had died early inJanuary 2019. To hit the Newsletter deadline only
a few days hence, I spent the weekend writing ashort obituary. The biennial Council retreat was(l-r) Caucher Birkar, Caroline Series, Michael Atiyah andJune Barrow-Green at an ICM 2018 reception

Shortly before I took up office, Council was having
discussions around our IT provision, which was atthat time mainly handled by our IT Consultant,
Donald Eastwood, working on a part-time basis,
although always on call in case of emergency. There
were many conflicting demands on Donald’s time,and it seemed to me that in the long term thissituation was not sustainable. About the first major
first thing I did as President was to propose setting
up an committee to look into the matter. Fiona
Nixon and Donald prepared a very thorough review
of our entire IT operation, and with advice fromJohn Cremona (Warwick) and Keith Gillow (Oxford),
after only three meetings the new committee had
proposed moving a large part of our operation (email
and files) to the Cloud and had identified a company
Fitzrovia IT to manage this. Fortuitously, our new

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FEATURES
Head of Conferences and Buildings Andrew Dorward
had overseen a similar operation in his previous post;
moreover by mid-2019 Donald had decided that it
was time to be moving on to other things. So myforesight had paid off. The move took place oversummer and autumn 2019 and I rather doubt thatmany of our members were even aware that it was
happening. The icing on the cake was that – quite
contrary to my expectations – after the initial movethe entire cost of the operation will be less than
before.
The President is able to suggest minor changes andimprovements with remarkable ease. The supply
of hangers on the coat rack just inside the door
past Reception had dwindled to three very scruffy
specimens which bothered me every time I entered
the building. With some hesitation, I mentioned thisto Fiona – and lo and behold, next time I arrived
we had a full new set! Another quick win were
the wrappers on our Newsletter. In January 2018
my National Trust magazine arrived in a wrapper
boasting that it was made of compostable potato
starch. I asked the Newsletter editor Iain Moffatt
whether we might look into doing the same, and wastruly delighted when the change was implemented
in time for the very next issue.
It is the President’s job to do a fair amount of‘networking’ on behalf of the Society. One example
is a conversation I had with a member of the Abel
Prize committee which we entertained in London,
asking him how the AMS managed to have its
Council meetings with members scattered across thecontinent. ‘A good video-conference system’ was theanswer, ‘couldn’t do without it’. The UK is not theUSA, but still, this seemed to me such an obvious
idea that I pressed for it early on. Once again, the
LMS machine (this time in the person of the thenbuildings manager Dominic Clark) swung into action,and before I knew it the equipment was installed, firstin the Cartwright room and then in the Hardy room.
It has been invaluable, its worth being first provedon the day of a critical Prizes committee meeting inMarch 2018 when, one by one, the extremities of thecountry got cut off by heavy snow.
The DeMorgan House anniversary event
DeMorgan@21 was my idea. I was rather taken abackwhen some of the younger and non-British born
Council members queried ‘Why 21?’. However that
hurdle successfully surmounted, Council approved
the idea and I went ahead with a small planningcommittee of which the central and indispensable
member was of course Susan Oakes. Our meetings
and the event as it gradually evolved were among

the most enjoyable of all the things I did, and I
got considerable satisfaction when I saw how muchpeople appreciated this unique event.

Caroline Series presenting the changes to the Statutes atthe LMS Representatives Day 2019

I didn’t get around to doing as much environmentallyas I might have hoped, with so many other things
going on it always got pushed back. But I did initiate a
discussion at Council, the upshot of which is that wehave commissioned a full energy audit of De Morgan
House to look into what energy saving and otherimprovements we might make, and I think everyone
was pleased when staff made the suggestion that
hard copies of papers for meetings will henceforth
only be produced on request. We are about to switch
to a fully renewable electricity supply and I am looking
forward to what the audit may suggest.
Of course the President’s job is not completely
finished until he or she has delivered the Presidential
address, which happens immediately after handing
over the chain of office. It is not easy to do researchat the same time as fulfilling all the Presidential
duties, but working on the lecture pushed me toinvestigate some ideas which I am looking forward
to continuing in my ‘retirement’. Perhaps those who
set up this tradition were wiser than we knew.
Throughout these two crowded and eventful years
among the greatest rewards have been the pleasureof working with so many dedicated and remarkablepeople, the satisfaction of setting balls rolling, andof seeing initiatives come to fruition. I was very
honoured to have been asked to take on this role andI could not have done it without the support of many
people, not least the wonderful staff at De Morgan
House and most of all our Executive Secretary, Fiona
Nixon. Without doubt, it has been one of the highpoints of my career.

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FEATURES“The Accident of Being the First Woman
Senior Wrangler”
JUNE BARROW-GREEN
In 1890 Philippa Fawcett caused a sensation by scoring more marks in the Cambridge Mathematical Tripos
than any of the male students. The recent discovery of letters dating from 1887–1890, including a letter from
Fawcett herself, throws some new light on this famous incident.

Introduction
On the morning of 7 June 1890, the Cambridge Senate
House was abuzz with excitement. The results of
the Mathematical Tripos examination were due to beread out and it was anticipated that for the first timea woman, Philippa Fawcett of Newnham College, wasgoing to outshine the men. Fawcett’s cousin, Marion
Cowell, described the scene:

to Cambridge, what the triumph meant for her andhow it was perceived by the popular press of theday. The motivation for the article is the discovery
of some letters from Fawcett’s mother, together with
one from Fawcett herself, to Karl Pearson.3

Fawcett’s route to Tripos success
Born in 1868, Fawcett was the only child of Henry andMillicent Garrett Fawcett, both staunch supportersof women’s rights. Henry Fawcett was 7th wrangler
in 1856, apparently missing the senior wranglership
through becoming over-excited in the examination!
Although blinded in a shooting accident at the ageof 25, he became Professor of Political Economy
at Cambridge and was a successful campaigner
for university reform. Millicent Garrett Fawcett, asuffragist from an early age and a strong supporter ofhigher education for women, was one of the foundersof Newnham College.4“The gallery was crowded with girls and a
few men, and the floor of the building wasthronged by undergraduates as tightly packed
as they could be. The lists were read from thegallery and we heard splendidly. The men’snames were read first, the Senior Wrangler1
was much cheered. There was a good deal of
shouting and cheering throughout; at last theman who had been reading shouted “Women.”
The undergraduates yelled “Ladies,” and forsome minutes there was a great uproar. A
fearfully agitating moment for Philippa, it musthave been; the examiner of course, could not
attempt to read the names until there waslull. Again he raised his cap, but would not
say “ladies” instead of “women,” and quite
right, I think. He signalled with his hand forthe men to keep quiet, but he had to wait
some time. At last he read Philippa’s name,and announced that she was “above the
Senior Wrangler.” There was a great deal of
prolonged cheering; many of the men turned
towards Philippa, who was sitting in the gallery
with Miss Clough [principal of Newnham], andwaved their hats.”2

Fawcett was schooled at Clapham High School, andfrom the age of fifteen also received coaching inmathematics from George Atkinson, a wrangler andgood friend of her father’s.5 As preparation for
Cambridge, she then spent five terms at University
College London (UCL) studying both pure and appliedmathematics, with time spent at Bedford College —the first higher education college for women in Britain— in the study of chemistry. In undertaking suchrigorous preparation, Fawcett was following the pathof many successful male students before her. At thetime it was not uncommon to obtain a degree fromone of the London or Scottish universities before
embarking on the rigours of the Tripos.

The facts surrounding Fawcett’s triumph are
well-known [2] and Fawcett is justly commemorated,
including by the LMS who own a collection of booksby and about women mathematicians which is named
for her. In this article, I focus on Fawcett’s journey

Among Fawcett’s teachers at UCL, was Karl Pearson,then the Professor of Applied Mathematics, whotaught the advanced mathematics class. Judging by
an account in The New York Times, Fawcett made asingular impression on him:

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FEATURES“Mr Pierson [sic] tells the story how when he
first saw Miss Fawcett attending his advanced
class he said to himself: ‘Dear me, there’s
a poor little girl turned ambitious; of course
she won’t understand a word,’ when presently,
what was his astonishment to hear a question
from the ‘poor little girl’ which showed that
she understood to some purpose.”6
Since this story appeared after Fawcett’s success at
Cambridge, precisely how much truth there is in it
is hard to tell but certainly of the five students inthe class Fawcett was the only woman, even though
at the time UCL had a relatively high proportion of
female students. Rather remarkably, one of the otherfive was none other than George Bennett, the man
who would become the official senior wrangler in1890.

coaching, she wanted to secure the best available.
Her letter included a list of the Cambridge men
who had previously coached women, and she asked
Pearson which of them she should choose (Figure1). Pearson replied recommending someone not on
her list, Andrew Russell Forsyth, and offering tocontact him on her behalf. Forsyth, senior wrangler in
1881, had been elected a Fellow of the Royal Society
the previous year for work in function theory, andhe would go on to succeed Arthur Cayley as the
Sadleirian Professor in 1895. He was an ambitious
choice, particularly since there is no evidence that hehad previously done any coaching. When it became
apparent that Forsyth was going to take somepersuasion, Fawcett’s mother asked her friends the
Sidgwicks to intervene.8 They suggested instead sheshould try Robert Webb, senior wrangler of 1872 andone of the foremost coaches at the time. Webb was a‘possible’ on her list and Pearson had recommended
him too. But Webb was reluctant to coach women,
not, he claimed, because he was prejudiced againstthem but rather because he “apprehended someembarrassment from the presence of a lady in theatmosphere of freely expressed criticism,” which
he considered an essential part of his coaching
method.9 Since he preferred to coach in groups oftwo or three, it seems he believed, although without
verification, that women would be unable to standup to public scrutiny of their work.
In the event, when Fawcett arrived at Newnham witha scholarship in October 1887, she was coached byneither Forsyth nor Webb. Such was her reputationthat the leading coach of the previous generation,
Edward Routh — he had 28 senior wranglers to his
name, including Forsyth — had been enticed out of
retirement for a term especially for her. Afterwards
she was coached by William Hobson, senior wrangler
of 1878 and another of Routh’s men. (Hobson would
later succeed Forsyth in the Sadleirian chair after
Forsyth’s resignation in 1910 due to the scandal arisingfrom his love affair with Marion Amelia Boys, the wifeof the physicist Charles Boys.)

Figure 1. “Mathematical lecturers or coaches who areavailable for Newnham Students”, compiled by Millicent
Garrett Fawcett, 15 May 1887.

In May 1887, in preparation for Fawcett’s entry to
Newnham, her mother wrote to Pearson seeking a
recommendation for a mathematical coach.7 Since
success in the Tripos was dependent on such

Regardless of her performance at Cambridge, Fawcett
knew that as a woman she could not be awardeda degree (with its privileges and voting rights), nor
could her result be listed with those of the men.
At least she did have the right to sit the Tripos
examination, a right which had been only recently
granted to women as a consequence of the successof Charlotte Scott who in 1880 had been ranked
equal to the 8th wrangler, having previously gained
special permission to sit the examination. As for21

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women being awarded a degree at Cambridge, thatwould come only in 1948 when women were officiallyadmitted to the university, a month before Fawcett’s
death at the age of 80.

She could hardly have cemented her reputation moreemphatically.
The response to Fawcett’s success
The news of Fawcett’s success spread rapidly.
Congratulations came into Newnham from acrossthe university, including from Cayley,12 Newnham’s
first president and chair of the college council.
Fawcett’s mother received many letters and “literally
hundreds of telegrams” which showered in on her“like snowflakes in a storm”.13 Among the many lettersthat Fawcett herself must have received was one
from Pearson as her reply to him attests. Written
only four days after the results were announced, herletter gives an interesting insight into her feelings at
the time:“Thank you very much for your kind letter andfor the hope which you express that I may beable in the future to do some scientific work.
But I am sure you will agree with me that theattainment of a high place in the Tripos is noevidence of power to do original work.
I can only regret that the accident of being thefirst woman senior wrangler did not happen
to someone who was capable of such work.

Figure 2. Philippa Fawcett at Newnham.

Immediately after Fawcett’s result was announced,
there was of course great jubilation at Newnham.
Bunting was hung out, bells were rung, and in
the evening a celebratory dinner was held. Miss
Clough proposed a toast to Fawcett to which Fawcett
responded with a “pretty little speech” praising Miss
Clough and modestly remarking: “in these matters
the person ‘coaching’ was everything, the person‘coached’ nothing”, before proposing a toast to
Hobson.10 The celebrations continued long into theevening with a dance and fireworks. Later Hobson
told Fawcett’s mother that he had it on authority fromthe examiners that it had been a strong year, andthat her daughter “was 400 marks (or 13%) ahead of
the senior”.11
The following year, Fawcett, the only woman to sit
Part 2 of the Tripos — in contrast to Part I where
she was one of seventeen women — was placed in
Division 1 of the First Class. Only two men achieved
a similar distinction, one of whom was Bennett andthe other was Lawrence Crawford, the 5th wrangler.

May I take this opportunity of thanking you forthe great help and stimulus that your teaching
was to me.”14
Fawcett’s remark about the relationship between
a high place in the Tripos and the ability to
do original work, does of course have truth toit, but it is nevertheless the case that manyhigh-ranking wranglers did go on to do original
work, including Bennett who spent his career inCambridge and in 1914 became an FRS [4]. So how
should one read Fawcett’s remark? Did she really
believe it of herself or was it another example ofher characteristic modesty alongside her crediting
her success entirely to Hobson? Suffice it to say,comparable remarks from high-ranking wranglers of
the period are conspicuously absent. In the event,
Fawcett was largely correct in her self-analysis. Her
only contribution to research was a single paper on
the motion of a helicoid in a fluid [5]. This was theresult of the year she spent on a research scholarship
at Newnham immediately after completing Part 2 of
the Tripos. Following this, she taught for some years
at Newnham and in South Africa, before making a
successful career in educational administration in
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research and that at the time, apart from teaching,
the opportunities for women mathematicians were
few and far between, her career choice might notseem so unexpected. And yet Fawcett had won
the respect of many male colleagues at Cambridgeand elsewhere, some of whom would surely havesupported her in research — as Cayley had donewith Charlotte Scott — had she wished to continue.15
Perhaps her first taste of research had put her off;
perhaps she found that she preferred teaching. But
perhaps also she was disheartened by the publicclamour surrounding her, peppered as it was withconflicting attitudes towards women mathematicians.
It is no surprise that there was enormous popularinterest in Fawcett’s achievement. During the 19th
century the Tripos results commanded attention far
beyond the bounds of Cambridge, with the senior
wrangler reaching close to celebrity status. Fawcett
had not only surpassed the senior wrangler, achieving
what many had believed impossible, but she camefrom a well-known family with female emancipationin their veins. From London to Glasgow and fromNew York to Melbourne, “The Lady Senior Wrangler”
was news. Newspapers concentrated mostly on facts,
giving details of her family background, schoolingand so on, sometimes embellishing their reports withremarks about her steadfastness and composure.
Popular periodicals were different. As well asreporting facts — and those not always accurately— they were vehicles for social comment, and thephysical appearance of women mathematicians was
not a subject to pass them by. Ten years earlier, whenCharlotte Scott had been the focus of their attention,
they were concerned that women with mathematicalability might develop “a character for deficiency infeminine gentleness”, that is if they were not already“women of the masculine type”.16 In Fawcett’s case,it was the fact that this didn’t apply to her, that she
had nothing of the bluestocking about her, that wasconsidered newsworthy.17
Aside from stereotyping appearances, periodicals
continued to propagate the myth that women
engaging in mathematics or science had little roomor strength to do, let alone be good at, anythingusually expected of a woman.18 Fawcett again bucked
the trend, as the Ladies Pictorial pointed out:
“Exceptional interest will naturally be felt in
the portrait of Miss Fawcett which we giveupon this page (Figure 2). As an interesting
fact it may be mentioned that the gown
seen in the photograph was made entirely byMiss Fawcett, proving conclusively that it isquite possible to unite in one woman practicaldomestic virtues and the highest intellectual
attainments. Every stitch in the pretty tucked
bodice was worked by the clever fingers of thelady who is now virtually the Senior Wrangler
for 1890.”19
Thus, it was alright for women to be good atmathematics so long as they were also good at
sewing!
The Penny Illustrated took a different tack, they usedtheir column on Fawcett to offer advice to “clever
women” on how to procure a husband:“Men admire clever women, but they do not
like women who are priggish, who dogmatise,
who lay down the law. She will be the cleverest
woman of all who, with her brilliant andcultivated intellect, retains all her pretty andcoaxing ways. Depend upon it that, even if
she were able to “square the circle” or to
add another dimension to space, she wouldfind that a pretty pout or a big tear on herlong eyelashes would have far more powerover the tyrant man than her mathematicalgenius, and that, no matter how cordially he
may admire and praise the clever woman, softand gracious creatures and graceful kittenish
ways will subdue him far more quickly thanthe most brilliant exercise of brain power.”20
The idea that mathematics and marriage wereincompatible in a woman was discouragingly popular.
Only the year before Fawcett’s success, the science
writer and novelist Grant Allen had remarked:“Out of every hundred women, roughly
speaking, ninety-six have husbands provided
for them by nature, and only four need to go
into a nunnery or take to teaching the higher
mathematics [6].”Allen made no secret of his reactionary views on thehigher education of women. That these views were
widely known and supported is all too evident froma report on Fawcett in the Bristol Mercury:“In the regularity and self-control whichhave governed her work throughout, Miss
Fawcett has offered a complete contrast to
the conventional picture of the lady student,

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whom Grant Allen and others have loved
to depict as a hysterical, overwrought, andnervous being who after a few years of morbid
application prepares to sink into an earlygrave.”21
Predictably, Fawcett’s success was fodder for thesatirical periodicals. The weekly magazine Fun
published a cartoon showing the victorious Fawcett
standing tall with a vanquished man crouching inthe shadows (Figure 3).22 In the top right corner,
blowing a trumpet and holding a suffragist wreath,is a curious amalgam of a man and a woman. The
figure appears to be allegorical but maybe it would
have been obvious to the magazine’s audience who
exactly is being lampooned.23 At any rate, it is hard to
imagine it being an image likely to appeal to women.
Furthermore, in the caption Fawcett is defined byher father, an all too common trope.

tone in 1894 when there was only one woman ranked
equivalent to a wrangler (Figure 4).24

Final Remarks
Philippa Fawcett is rightly celebrated for her
remarkable performance at Cambridge. Her
achievement unequivocally demonstrated theequality of mathematical potential in women andmen, adding impetus to the movement for women
to be awarded degrees in Cambridge, even if it was
not enough for the movement to succeed at thetime.25 It was a hard battle to win, and the press
response towards Fawcett’s success shows the deep
entrenchment of culturally adverse attitudes towards
women in mathematics. From the point of view ofFawcett herself, although her mother had wanted
her to make a career as an astronomer, physicist,
lighthouse designer or engineer, or at the very least
do something no woman had done before, such as
her aunt, Elizabeth Garrett Anderson, had done, theexperience left her with no such aspiration.26
Since
the
formaladmission of womento Cambridge in 1948,
there have been women
who have done well in
the Tripos and gone on
to have distinguishedcareers in mathematics,
although they are far
fewer in number thanmight have been hoped.
And in 72 years, there
has been only one
Figure 4. Punch cartoon.
female senior wrangler,
Ruth Hendry in 1992. Like Fawcett, Hendry chose notto remain in mathematics [8].

Figure 3. Ladies First – The Senior Wrangleress “A Worthy
Daughter of a Worthy Sire” Miss Fawcett demonstrates
the superiority of her sex, and defeated man takes a backseat.

Fun’s older rival, Punch, had plenty to offer by way ofcomment and cartoons on both Fawcett and womenstudents in general. Although it treated Fawcett withrespect — their cartoon of her shows “Seniora
Fawcett” walking victoriously between two lines ofbowing men – it adopted an unmistakably demeaning

FURTHER READING[1] M. G. Fawcett, What I Remember, T. Fisher
Unwin, 1924.
[2] S. Siklos, Philippa Fawcett and theMathematical Tripos, Newnham College, 1990.
[3] A. Warwick, Masters of Theory. Cambridge andthe Rise of Mathematical Physics, University of
Chicago Press, 2003.
[4] C. M. Neale, The Senior Wranglers of the
University of Cambridge, F. T. Groom & Sons, 1907.

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FEATURES[5] P. Fawcett, Note on the motion of solids in aliquid, The Quarterly Journal of Pure and Applied
Mathematics 24 (1893), 231–258.
[6] G. Allen, Plain Words on the Woman Question,
Popular Science Monthly 36, December 1889.
tinyurl.com/s96y4xe[7] R. Strachey, Millicent Garrett Fawcett, John
Murray, 1931.
[8] R. Hendry, Where are they now? Ruth Hendry(1989): the only known female Senior Wrangler inHistory. The Bridge, Queens’ College Cambridge(Spring 2016), 4. tinyurl.com/y6wk99kw

June Barrow-Green
June Barrow-Green isProfessor of Historyof Mathematics at
the Open University.
Her current researchinterests are in thehistory of dynamicalsystems, 19th century geometric surface models,
and the history of women in mathematics. When
not buried in an archive or in an art gallery she canbe found on a bicycle or running wherever her feettake her.

Notes
1 Students in the first class were called wranglers, with the top student being known as the senior wrangler, anappellation which persists even today.
2 [1, p. 143].

3 Pearson is known today for his work in mathematical statistics and in eugenics.
4 Newnham was founded in 1871 and moved to its current site in 1875.
5 Atkinson, 12th wrangler in 1856, was at Trinity Hall together with Henry Fawcett.
6 New York Times, 24 June 1890.
7 Millicent Garrett Fawcett to Karl Pearson, 15 May 1887. Karl Pearson Papers, General Correspondence 11/1/6/8.

University College London Special Collections.

8 Henry Sidgwick was Professor of Moral Philosophy and one of the cofounders of Newnham.
9 Bristol Mercury, ’The Lady Senior Wrangler’ 10 June 1890, p. 2. For a description of Webb’s coaching methods,

see [3, pp. 247–249].

10 The Ipswich Journal, Above the Senior Wrangler, 14 June 1890, p. 5.
11 [1, p. 141].
12 The Australasian, Miss Fawcett’s triumph at Cambridge, 19 July 1890, p. 135.
13 Extracts from some of these letters are given in [1, p. 140–2].
14 Philippa Fawcett to Karl Pearson, 11 June 1890. Karl Pearson Papers, General Correspondence 11/1/6/9. University

College London Special Collections. Pearson’s letter to Fawcett no longer exists.

15 Charlotte Scott was a successful research mathematician, but she spent her career in the United States

where she was the founding professor of mathematics at Bryn Mawr. Grace Chisholm Young, who was only
three years behind Fawcett at Cambridge, began her research career in Göttingen under Felix Klein.
16 The Spectator, 7 February 1880.
17 The Graphic, The Cambridge Mathematical Tripos, 21 June 1890, Issue 1073, p. 692.

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18 At the most extreme, it was believed that engaging in such study would compromise a woman’s ability tohave children.

19 Ladies Pictorial, 14 June 1890, pp. 920–921.
20 Penny Illustrated, The World of Women: Miss Fawcett, 21 June 1890, p. 394.
21 Bristol Mercury, ’The Lady Senior Wrangler’ 10 June 1890, p. 2.
22 Fun, Ladies First – The Senior Wrangleress, 18 June 1890, p. 261.
23 Despite the reference to him, it can’t involve Henry Fawcett, who had died in 1884, since he didn’t have along beard. Could it represent Millicent Garrett Fawcett and Henry Sidgwick, the joint founders of Newnham?
24 Punch, ‘The Girton Girl, B.A.’, CVI, 23 June 1894, p. 297.
25 In 1897 a proposal to recognise women’s degrees but with no share of university government was defeatedby 1713 votes to 662.

26 Elizabeth Garrett Anderson was the first female doctor to qualify in England [6, p. 205].

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2520 Ants for Dylan
JONATHAN M. FRASER
I discuss an entertaining problem which arose in the context of singing my son to sleep.

Ants go marching...
I’m writing this from my in-laws home in Harrison,
New York, during the coronavirus pandemic. My son,
Dylan, is asleep at the moment (afternoon nap) andmy wife, Rayna, is working in the next room. Dylan
is 15 months old and, although he is far from theworst sleeper, his sleeping habits and patterns arechallenging. Since our very early days with him I
have sung him to sleep (benefiting from the factthat he is too young to tell me not to). It used tobe nursery rhymes (Baby Beluga, Ants Go Marching,
Twinkle Twinkle etc), but more recently I have driftedback towards Leonard Cohen. I digress. One suchsong I’m very fond of is Ants Go Marching. It’s a sweetsong about ants marching in various formations. The
first verse is as follows:
The ants go marching one by one, hurrah, hurrah!
The ants go marching one by one, hurrah, hurrah!
The ants go marching one by one,the little one stops to suck his thumb,and they all go marching down,
to the ground,to get out of the rain.
The song then repeats with “two by two” and adifferent activity for the little one. This continues
all the way until ten and the little one stops to say
the end! One evening as I was singing this song to
Dylan in the dark of his room, in a state of pseudodelirium, it occurred to me that there must be a lotof ants if they are able to walk in so many different
formations. Assuming, for example, that “three by
three” really means rows of three with no ants
left out, the smallest number of ants required tosing this song is 2520. At first sight, this might seem
surprisingly large. However, when I teased my familywith this riddle the following day I tended to get an
even larger number as an answer. The most common
answer I received was 3628800 — oddly specific?
Certainly this number is a solution, but it is far fromthe smallest. The number of ants must be divisible
by one, two, three, all the way up to ten, and themost obvious way to construct such a number is tomultiply them all together (10 factorial, written 10!):
10! = 10 × 9 × · · · × 3 × 2 × 1 = 3628800.
OK, I admit that nobody (myself included) was able
to give me the number 3628800 off the top of theirheads, but the idea of multiplying the numbers one
to ten together was the most common answer given.
So, why is this solution such an enormous overshoot?
The reason is that there is a dramatic duplication ofeffort in multiplying the numbers one to ten together:you don’t need six, because you already have twoand three (and 6 = 2 × 3) and you don’t need ten
because you already have two and five, and so on.
So perhaps we only need the prime numbers lessthan ten? But 2 × 3 × 5 × 7 = 210 is not nearly enough.
The problem this time is that we sometimes need aparticular prime more than once. For example, to get
four we need two twice (4 = 22 ), and the less said
about eight the better! This line of thinking leads us
to the answer: take the product of all the primes lessthan ten raised to the largest power we will need.
This yields the optimal solution:
23 × 32 × 5 × 7 = 2520.
One way to think of “the largest power we will need”
is the largest number we can raise a given prime towhilst keeping it less than ten. For example, 23 = 8,
32 = 9, but 52 = 25, which is too big.

Finding a formula
Just as any sleep deprived mathematician would do,rather than rest with the solution to the ‘ants gomarching problem’, I instead began to think of amore general problem. What is the smallest numberof ants required to sing n verses of the song, forany given positive integer n? (In order to sing nverses, the ants must be able to march in rows ofone, two, three, all the way up to n.) In the previous
section, n = 10. This is much more typical of the sortof problem mathematicians actually consider. The
solution for a fixed n is just a calculation after all,but if we can understand how the solution behaves

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as n changes then we gain a deeper understanding.
Moreover, when n is small the problem is especiallysubject to “edge effects” to do with primes andprime powers near n (for example, when n = 10,
32 is a relatively large prime power which ‘makes
the cut’, whereas 11 just misses out). Therefore, themost elegant question to ask is what happens tothe solution as n becomes very large? Of course it
will grow, but how does it grow? What surprised me
about this problem was that the answer is remarkably
simple: the function grows like e n .
Before I go on, I should point out that all I am reallyasking for here is the lowest common multiple ofthe numbers 1 through n, that is, lcm(1, . . . ,n). Thisis well-understood and the mathematics which I am
about to discuss goes back a long time. That said,
I had fun thinking about it for myself in the darkof Dylan’s bedroom. Therefore, perhaps the reader
would also prefer to think about it for themselves?

Challengen: if pi is small, then mi will be correspondingly large.
(Certainly, pimi > n/pi .) This means
p 1m1 p 2m2 . . . pkmk
should be roughly equal to n k , where k is the numberof primes no larger than n. We understand fromour first interactions with the primes that there areinfinitely many of them in total, but that they become
more sparse as we look further up the real line. The
number of positive integers no larger than n is clearlyn, whereas the number √of perfect squares no larger
than n is only around n, which is much smaller
than n. The prime number theorem, a cornerstone ofnumber theory, tells us that the number of primes nolarger than n is actually much closer to the number of
positive integers than the number of perfect squares.
In fact, the number of primes no larger than n isroughlyn.
log n
Therefore the solution of the n by n ants marchingproblem should be roughlynWithout reading ahead, can you justifylcm(1, . . . ,n) ≈ e ?
n

You can decide what sort of approximation ispossible here but, for concreteness, can you
deduce that
log lcm(1, . . . ,n)→1n
as n → ∞? That is, log lcm(1, . . . ,n) is
asymptotic to log(e n ) = n.
The reasoning presented in the opening section
generalises and it is ‘obvious’ that the solution islcm(1, . . . ,n) = p 1m1 p 2m2 . . . pkmk
where p 1 , p 2 , . . . denote the primes written inascending order, pk is the largest prime no larger
than n, and mi is the largest integer such that pimi ≤n. This is not a particularly aesthetic solution andthe challenge lies in trying to express it in a moredigestible form. In what follows, take yourself backto Dylan and the ants and permit me to be a little
imprecise. More precision will follow in the next
section.
Each term pimi in the above product is no larger thann (by definition), but it seems reasonable to think
that, when n is very large, it will be roughly equal ton log n = e n .
So that’s that! Let’s check the original problem, which
considered only 10 ants. Unfortunately,e 10 = 22026.4 . . .
which is a rather disappointing overshoot, althoughmuch better than 10! (Recall the correct answer
is only 2520.) So what is the problem? The final
approximation we used came from the prime number
theorem, so let’s start there. We had alreadyestablished that the solution should be roughly n k ,where k is the number of primes no larger than n.
The primes less than 10 are 2,3,5, and 7, and so thisformula would predict an answer of
104 = 10000
which is also a large overshoot, but not quite as bad
as before. Does this mean that the prime number
theorem is wrong? Certainly not. The prime number
theorem does not claim thatnlog nis precisely correct, but only that this is a good
estimate for very large values of n. Small values are
subject to “edge effects” of the type I discussed at
the start of this section and 10 is certainly far too
small a number. A bigger problem with my roughcalculations in this section is the approximation pimi =n, which is often much too large. For example, if
n = 10 and pi = 5, then mi = 1 and pimi = 5 is muchsmaller than 10.

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Something a little more mathematicalrows of one, two, three, all the way up to n, there
must be roughly e n of them.

The most succinct way to describe lcm(1, . . . ,n) is
via Chebyshev functions. These functions play animportant role in number theory and describe the
distribution of the primes. The second Chebyshev
function is defined byÕ𝜓(x) =log p
p k ≤xwhere the sum is taken over all prime powers p k ≤x. Here x is any positive real number but you canthink of it as being the integer n that we’ve been
considering. For example,
𝜓(10) = 3 log 2 + 2 log 3 + log 5 + log 7
where the coefficients come from the fact that there
are 3 power of 2 less than 10 (21 , 22 , 23 ), 2 power of
3 less than 10 (31 , 32 ) and so on. Recalling that mi is
the largest integer such that pimi ≤ n,ÖÖe 𝜓 (n) =e log p =p
p k ≤xFigure 1. A plot of log lcm(1, . . . ,n) (in red) and n (in blue)
for n = 1, . . . , 100.

p k ≤x=

p 1m1 p 2m2 · · · pkmk

= lcm(1, . . . ,n).
Therefore the function 𝜓 precisely describes the
problem we are considering. Moreover, precise
asymptotics for 𝜓(n) are known, which allow us
to partially justify the guessed formula from theprevious section. It turns out that 𝜓(n) ∼ n, which
formally means
𝜓(n)
lim
= 1,
n→∞ n
that is, 𝜓(n) is asymptotic to n. This does not directly
yield an asymptotic for lcm(1, . . . ,n) but it does
demonstrate
log lcm(1, . . . ,n) ∼ nwhich can be interpreted as a solution to our problem.
In order for a group of ants to be able to march in

FURTHER READING[1] T. M. Apostol. Introduction to Analytic
Number Theory, Springer Undergraduate Texts in
Mathematics, 1976.
[2] G. H. Hardy and E. M. Wright, An Introduction
to the Theory of Numbers, Oxford Science
Publications, 5th Edition, 1979.

Jonathan Fraser
Jonathan is a Reader
in Mathematics at the
University of St Andrews.
It should be obvious
from this article that heis not a number theorist.
In fact, he usually spends
his time studying fractal geometry. He is also an
Editor of this Newsletter. He is pictured here with
Rayna and Dylan (who is presumably looking at someants).

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Maths Meets Arts
KATRIN LESCHKE
When starting our project Maths Meets Arts, we asked whether arts can help to explain mathematics toa general audience. But soon other questions arose: Is mathematics art? Are mathematicians creative? Iscreativity taught in a mathematics degree?

Motivation and first steps
When asked by non-mathematicians about
my work in pure mathematics, quite often
I have difficulties explaining the results.
In my research areaofsurfacetheory,computergraphics
enable me to show
imagesoreven
3D printed models
of surfaces I workon.
However,thedeeper beauty of a
Figure 1. A 3D print ofmathematical surface
the Chen-Gackstattersurface, a minimal torus
often appears in the eyeof a mathematician from its context in mathematics.
This additional information, how results and methods
fit into a broader theory, is harder to disseminate.
Our starting question in our project Maths Meets
Arts was to investigate whether arts can help tocommunicate this deeper meaning of mathematics tonon-experts. Can we illustrate how mathematicians
work, what problems they work on, which methods
they use and what results they obtain? We used an
open question in minimal surface theory, the finite
topology conjecture (see text box) as a starting point
for our collaboration.
Minimal surfaces occur in nature as soap films:
surfaces on a wire loop dipped in soapy water
minimise the area with the given boundary, and thenormal curvatures of curves on the minimal surface
will average to give mean curvature zero. Minimal
surfaces also appear in various other settings: forexample in a biological context, where triply periodic
minimal surfaces describe the structure of photonic
crystals of butterfly wing scales and their opticalproperties [1]; or in cosmology, where the apparent
horizon, a quasi-local version of the event horizon,
is a minimal hyper-surface (thus linking the theoryof black holes to minimal surfaces and Plateau’s
problem [2]).
Since their introduction by Lagrange [3], minimal
surfaces in Euclidean 3–space have been extensively
studied in differential geometry. The classicaltheory flourished through contributions of leading
mathematicians including, amongst others, Catalan,Weierstrass, Enneper, Schwarz, Bonnet, Serret,
Riemann and Plateau. One important reason for thehuge success of the theory is its connection tocomplex analysis: conformally parameterised minimal
surfaces from a Riemann surface into Euclidean
3-space are exactly the real parts of holomorphic
null curves in complex 3-space. Therefore, tools fromcomplex analysis can be used, for example, to study
the behaviour at the ends of the surface.
Minimal surfaces played an important role inintroducing computer based experiments togeometry: J. Hoffman developed computer programs
to visualise Costa’s surface based on work of
Hoffman and Meeks. This made it possible to perform
mathematical experiments to obtain insights onproperties of the Costa surface, see [4, 5] for details.
These observations could later [6] be used to prove
results rigidly, for example, the embeddedness of
the Costa surface and its symmetries. Despite thenumerous powerful tools of the theory, there arestill many unsolved problems in this topic, such as
the finite topology conjecture.
Since minimal surfaces have a physical interpretation
and allow artists a visual approach, we chose a topic
for our arts project from this field. We selected thefinite topology conjecture as it lies at the core of
my current research as leader of a Leverhulme Trustinternational network on Minimal surfaces: integrable
systems and visualisation (www.le.ac.uk/miv) whileat the same time being a problem which can be
explained to the artists visually. Using a conjecture
serves as a way to demonstrate how research

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questions in mathematics arise and what modern
mathematical research looks like.

The finite topology conjecture
The finite topology conjecture is an open
question by Hoffmann and Meeks [7] inthe area of minimal surfaces. A complete
and embedded minimal surface of finite
total curvature is given by an embeddingf : M = M̄ \ {p 1 , . . . , p n } → ℝ3 withvanishing mean curvature, where M̄ is acompact Riemann surface, and p 1 , . . . , p n
are the ends of the minimal surface. The
assumption of finite total curvature ensuresthat the ends are well-behaved and are ofplanar or catenoidal type.
An immediate question is: how many ends
can occur depending on the genus of M̄ ?
The finite topology conjecture states thatany complete, embedded minimal surface ofgenus g ≥ 1 of finite total curvature has at
most g + 2 ends.
In the case of genus g = 0 the finite topology
conjecture is true: using a deformation on the
Weierstrass data, López and Ros [8] showedthat the only complete, embedded, finite total
curvature minimal surfaces of genus g = 0
are the catenoid and the plane, thus showingthat in this case the number of ends is lessthan or equal to 2.

During a three month residency funded by the
Wellcome Trust, three artists — Chloe Aligianni,
Lee Allatson and Jenny Hibberd — achieved anunderstanding of the contents of the conjecture,
why it is believed to be true, how mathematicians
approach a question and what the obstructions are to
fully answering it. The first outcomes of this ongoingproject were showcased by Lee Allatson and Jenny
Hibberd, who were supported with music technology
by Andrew Johnston, at the Artists’ performanceevening on 5th September 2019 at the AttenboroughArts Centre in Leicester. The event, which included a
Q&A session, was part of an international workshop
on minimal surfaces and was also open to the generalpublic.
A second public artist performance evening featuring
all four artists, was hosted at the Henry Wellcome

Atrium and Lecture Theatre in Leicester on Thursday
3rd October 2019. The event was part of the Welcome
Back activities for students returning to Leicester at
the start of term but again it was also attended by
the general public.
Following on from thesuccess of this artist
residency project, theUniversity of Leicesterisnow
supportinganinterdisciplinary
research team called the
Maths meets Arts Tiger
Team (MmATT), including
Figure 2. Maths meetsmathematicians, artists,
Arts Tiger Teamcomputerscientists,
anthropologists and teaching experts. The aim of
MmATT is to create new art and expand on theprevious collaboration, to investigate further how
the communication between scientists, artists andthe general public evolves, to explore ways to include
creativity into teaching and to showcase our work in
a Maths Meets Arts Festival in the summer.
New projects are arising from various areas ofmathematics and computer science. For example, we
aim to create a game which explains representation
theory of algebras, and explore mathematicalsurfaces as a virtual reality adventure.
What did we learn?
The first open call for artists to participate inthis project was a nerve-wracking moment in
the project. Due to time constraints, we had
to have a very short turn-around, and it was
unclear to me whether we would be able tofind artists who were willing to engage with amathematician and work on a joint project, rather
than responding to research only: the final output
needed to communicate modern mathematics tonon-experts. To my surprise, the response of theartistic community was overwhelming! In a firstworkshop I presented the underlying ideas of the
project, and there was a lively discussion withlots of in-depth questions about geometry which
demonstrated the willingness and capacity of theartists to engage with the material.
Unfortunately, we only had funding to work with threeartists. The selected artists came from a variety ofarts backgrounds — a drum set player, a dancer
and choreographer and a performer, writer and31

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musician. Over the duration of the arts residencies
I met with each of the artists about once per
week to discuss their project, each developing theirown understanding of various aspects of the finite
topology conjecture.
As in any interdisciplinary collaboration, there weredifficult moments, for example when the artistic view
clashed with my educational motivation, or whenconcepts were consistently misunderstood and there
was a fear that this could not be resolved.
However, this challenging process was amazing andenriching for all of us: we discovered a lot ofsimilarities between how mathematicians and artists
work and that, albeit with a different background,the same curiosity drives our work. In particular,
maybe to the surprise of the artists, the similarity of
exploring different routes, experimenting with ideas
and using intuition in our work shows that creativity
is a key aspect of mathematical research.

collaboration, we created a new “language” tocommunicate which allowed both sides to be equal
participants in the process.
The final presentations were well attended byacademics, students and general public. Highlights
for me included the following (tinyurl.com/yx6ajmaq):• a non-mathematician was reluctantly entering
the installation and within seconds engaged in
discussions about how the sound and the shapeof the surface are related,• a colleague and her daughter could experience
the shape of a surface by the sounds of theperformance,• the laughter at the clever linking of mathematicallanguage and “normal” life observations,• the delight of a maintenance worker at the dance
rehearsals, returning for the main performance.

Figure 4. f: countless deformations

Figure 3. Infinity core. Photo credit: G. Sian

In this collaboration I played both the role of a teacher
in explaining concepts, methods and motivation,and that of a learner: I had to learn to understandthe point of view of an artist but also to reshape
my own ways of explaining and demonstrating
mathematical concepts. Deprived of formulae andstandard explanations, I needed to learn new ways
to explain and explore mathematics. Through ourAs mentioned before, the first performances werepart of an international workshop on minimal
surfaces and included a Q&A session with artists.
The feedback from mathematicians after the event
ranged from “but can the artists really understandwhat we are doing” to “I will steal this idea!”. It was
clear from the discussion though that the researchcommunity indeed feels a need to explain to a
general audience what we are doing and why wethink our work is important, and that art might be
the appropriate tool to engage a new audience.
The second performance was also attended by
students. Building on students’ feedback we started
to investigate further how art can be used forinnovative approaches to teaching and learningof mathematics. In a first workshop, groups ofthree students worked with each artist to explore
mathematical concepts such as continuity or randomvariables.

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Observingtheconversations between
the students and artists,
it became clear thatboth sides found it
challenging to find acommon language toexplain topics. At the
Figure 5. Student art
same time, all enjoyedthe challenge, worked hard towards a final output
and improved their understanding of mathematicalconcepts. In a feedback session after the event, allstudents agreed that this kind of exercise would
have been very beneficial at the beginning of theirstudies. In particular, the setup helped students tolose their fear of making mistakes, and to learn tobe more playful with questions.

proposes an open dialogue between the languagesof algebra, dance, sound and motion sensing whilelooking for physical manifestations of mathematicalprocesses behind minimal surface creation. One
dancer, one musician and one digital artist co-exist ina space where members of the audience are able toshape the choreography by choosing mathematical
propositions which performers use as improvisation
formulas. The choreography, sound and light are thus
put together as pieces of a puzzle with audience
members experiencing the process of compositionand synthesis while having the time to explore key
mathematical ideas around minimal surface theory.
The work seeks to re-think and re-imagine thebody as brain and bring to life the workbook of amathematician.

The artists
Lee Boyd Allatson (tinyurl.com/t5u2n8t) is a drum
set player from Leicestershire, specialising in free
improvisation. He performed Infinity Core: At theepicentre of planes and curvatures; feeling minimal
environments within a vast sense of scape. An
opportunity for meditation, where tone, texture,
pitch and duration become attributes of surface
geometry. Communicating the “. . . deeper beauty ofmathematical surface. . . ” by appealing to the senses
over the mind.

Figure 7. f: countless deformations

Figure 6. Infinity Core

Chloe Aligianni (www.chloealigianni.com) is achoreographer and dance artist who makes
movement-based works for site specificenvironments and conventional theatre spaces.
She performed f: countless deformations which

Jenny Hibberd (tinyurl.com/vh8vcn4) is an artist,
performer, writer, musician and workshop facilitator.
She presented an art installation with Andrew
Johnston sounds of surface which explored the Costa
surface through light, sound and shape: ever thought
you could listen to a shape? 3D printed, tracked withproximity sensor technology then transformed into
audio data, you can physically experience the soundsof a Costa surface.

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surfaces. Communications of the ACM, 31 (6), pp.
648-661, 1988.
[6] D. Hoffman, W. H. Meeks III. A complete
embedded minimal surface with genus one, three
ends and finite total curvature. J. Diff. Geom. 21,
pp. 109-127, 1985.
[7] D. Hoffman, W. H. Meeks III. Embedded
minimal surfaces of finite topology, Ann. Maths 131,
pp. 1-34, 1990.
[8] F. J. López, A. Ros. On embedded complete
minimal surfaces of genus zero. J. Diff. Geom. 33(1), pp. 293-300, 1991.
Figure 8. Sounds of surface

Acknowledgements
FURTHER READING[1] K. Michielsen and D.G. Stavenga. Gyroid
cuticular structures in butterfly wing scales:
Biological photonic crystals. J. R. Soc. Interface 5,
pp. 85-94, 2008.
[2] M. Eichmair. The Plateau problem for
marginally outer trapped surfaces. J. Diff. Geom.
83 (3), pp. 551-584, 2009.
[3] J. Lagrange. Essai d’une nouvelle méthode pourdéterminer les maxima et les minima des formules
intégrales indéfinies. Miscellane Taurinensia 2, pp.
173-195, 1760.
[4] D. Hoffman. The computer-aided discovery of
new embedded minimal surfaces. Math. Intel. 9(3), pp. 8-21, 1987.
[5] M. J. Callahan, D. Hoffman, J. T. Hofmann.
Computer graphics tools for the study of minimal

We would like to thank the Leverhulme Trust, theWellcome Trust and the University of Leicester for
their support of this work.

KatrinLeschke

Katrin is a Reader
in mathematics attheUniversityofLeicester. Her main
research interests are
in differential geometry
and visualisation. Katrin was born in Berlin and hastaught in Germany, US and UK. Not being an artist,she still enjoys literature, theatre, movies and music.

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Mathematics News Flash
Aditi Kar reports on new path-breaking developments in mathematics from the past few months.

As the Covid-19 outbreak changes our lives as never before, mathematical modellers and statisticians cometogether to test the efficacy of social distancing. In the meantime, mathematicians grind on and we can reporton several very exciting announcements.

Ringel’s Conjecture: An Asymptotic Solution
AUTHORS: Richard Montgomery, Alexey Pokrovskiy and Benny Sudakov
ACCESS: https://arxiv.org/abs/2001.02665
A
graph
G
is
decomposable
into
copies of a graph H
2
if the edges of G can
7
be partitioned intoedge-disjoint subgraphs
36
isomorphic to H . For
instance, the complete
45graph on 3 or 7 vertices can be decomposed
into triangles while the one on 4 vertices cannot.
Graph decomposition problems have long captivatedmathematicians and inspired a formidable bodyof research. A favourite open question for graphtheorists is Ringel’s conjecture from 1963: ‘thecomplete graph on 2n + 1 vertices can be
decomposed into copies of any tree (a graph without
circuits) on n vertices.’ Montgomery, Pokrovskiy
and Sudakov announced an asymptotic solution
to Ringel’s conjecture in February this year. Their
affirmative solution for large n is remarkable in that
it works for trees of any valency.
0

81

A simplified version of the strategy can be describedas follows: one starts with a specific colouring of theedges of the complete graph K2n+1 , called the near
distance colouring (ND colouring) with n colours. For
instance, the ND-coloured K 5 has only blue edges
on the boundary pentagon with red on all the inneredges. One then finds a copy of a ‘rainbow tree’ withn edges in the coloured graph: all edges in a rainbow
tree bear different colours. When n = 2, there is
only one 2-tree up to isomorphism, namely, the pathwith two concatenated edges. Any subgraph madeof one red edge and one blue edge emanating fromthe same vertex of the ND-coloured K5 provides anembedding of the rainbow 2-tree. Observe now thatcyclically shifting the chosen vertex to every otherone step at a time, covers the entire K5 with copies
the chosen rainbow 2-tree.
Finding embeddings of every rainbow n-tree in the
ND-coloured K2n+1 is no mean feat and therein lies
the principal innovation by the authors.

Connes Embedding Conjecture is False
AUTHORS: Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, and Henry Yuen
ACCESS: https://arxiv.org/abs/2001.04383
Connes Embedding Conjecture (CEC) is a longstanding problem from the theory of von Neumann
algebras. Formulated by Alain Connes in the 1970s,
the CEC was found to be connected to a hostof other open problems and questions, originating
in apparently unrelated areas of mathematics. A

positive solution to the CEC would imply among
other things, that every countable discrete groupis hyperlinear. A negative solution to the CEC
was announced in January by Ji, Natarajan, Vidick,
Wright, and Yuen, researchers working in quantum
complexity theory.

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For some, it came as a surprise! Quantum complexity
theory arose in the realm of quantum computerswhich were inspired by principles of quantum
mechanics. Computational complexity is studied
using complexity classes: each class is defined as aset of (decision) problems that can be solved subject
to some given constraints. For instance, the class P
is the set of problems that can be solved by a Turingmachine in polynomial time. Complexity theorists
are interested in how different classes relate to eachother.

Decision problems in this field are often describedusing formal language theory. A formal language
is said to be recursively enumerable (RE) if it is arecursively enumerable subset of the set of all words
in the alphabet, or equivalently, if all valid strings ofthe language are recognised by a Turing machine.
Ji, Natarajan, Vidick, Wright, and Yuen announced aproof that the class RE is equivalent to the moremysterious MIP* class of languages. RE=MIP* gives
a negative solution to Tsirelson’s problem which inturn is known to give a negative solution to the CEC.

G. Baumslag’s problem on ℚ-groups
AUTHORS: Andrei Jaikin-Zapirain
ACCESS: http://matematicas.uam.es/andrei.jaikin/preprints/baumslag.pdf
Around 1968, G. Baumslag was studying the structureof groups, in which extraction of n-th roots is
uniquely possible for every positive integer n. The
quintessential example of such a group is the
additive group of the rationals and Baumslag called
these groups ℚ-groups. Evidently, a free ℚ-group is
obtained by repeatedly, freely adding n-th roots to afree group and the resulting intermediate quotients.
Baumslag asked if the limiting object of this inductive

process, namely, the free ℚ-group is residually
torsion-free nilpotent (RTFN).
This decades-old technical problem in the theoryof groups has recently been resolved by Andrei
Jaikin-Zapirain. A group is RTFN if every non-identity
element survives in a torsion-free nilpotent quotientof the original. RTFN groups are known topossess a rich structure and hence the interest.
Jaikin-Zapirain’s proof uses his theory of mod-p
L 2 -Betti numbers for the free pro-p group.

People can handle the truth!
AUTHORS: Anne Marthe van der Bles, Sander van der Linden, Alexandra L. J. Freeman, and David J. Spiegelhalter
ACCESS: https://www.pnas.org/content/early/2020/03/17/1913678117
Who do we believe in these uncertain times? The
good old Times, or lines read hastily in the Evening
Standard? Our dilemma intensifies as Twitter and therest of social media begin to expand into the news
world. Experts and journalists have always presumed
that hiding the uncertainty of data preserves publicfaith. Statistical ambiguity is presented to readers
as ‘absolute’ data. However, recent research bymathematicians collaborating with psychologists at
the University of Cambridge has shown that “people
can handle the truth”! The team, supported by the
Nuffield Foundation, wanted to see if they could get
people much closer to the statistical truth by being

honest about what we don’t know, like the exact
number of Covid-19 cases in the UK.
In October 2019, the research group worked with the
BBC to conduct a field experiment. In a BBC News
article, figures were either presented as usual or withsome uncertainty like a verbal caveat or a numerical
range. Readers were asked to complete a brief survey.
The survey results bolstered observations from theirfour laboratory experiments. The findings, publishedin the Proceedings of the National Academy of
Sciences on 23rd March, will persuade scientists andjournalists to be more candid about the limits ofhuman knowledge.
Mathematics News Flash prepared by Dr Aditi Kar

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Maths and Mingle: a Monthly Gathering Organisedby PhD Students, for PhD Students
LILY CLEMENTS, SAM HUGHES, MATTHEW STANIFORTH
The ever-evolving PhD environment in the mathematical sciences encourages a high level of interdisciplinary
networking, skill-development, and collegiality. In this article, we introduce Maths and Mingle: a novel undertaking
at the University of Southampton with the purpose of achieving exactly these aims.

What is Maths and Mingle?
Maths and Mingle is a seminar series taking
place in the School of Mathematical Sciencesat the University of Southampton. We are a
multi-faceted, monthly get-together of graduate
students, where attendees from across the school
meet to address goals underpinning the postgraduate
programme in the mathematical sciences. We provide
a platform for the enhancement of generic key skills,
whilst simultaneously encouraging presentation of
students’ own research, within a welcoming, friendly
atmosphere. Importantly, we strive to achieve theseaims within a streamlined, efficient framework, whilst
appealing to students and remaining adaptable tofluctuating needs.

Many graduate students are indeed highly skilled ina plethora of areas which fundamentally underpin
postgraduate study in the mathematical sciences.
Progression through a PhD carries with it aninherent, continual development of skills such asteam-working, communication and presentation,and self-organisation, to an advanced level. This
happens via various channels, for example throughregular meetings with supervisors and colleagues,
attendance of conferences, and presentation of workto peers.
Despite this, when applying for jobs, many
PhD students do not sufficiently recognise theirproficiency in these fields, since they were not trainedin them in a formal context. One of the goals of Maths
and Mingle is to provide an opportunity for students
to not only develop recognition of their achievementsin these skill areas, but also to fine-tune them.
Maths and Mingle also provides a platform forstudents to invite speakers highly experienced inareas unrelated to the mathematical sciences, suchas underlying university structures and topics inwider society, to pass on their knowledge. In thisway, students can be kept abreast of changes, forexample, to data management regulations, health andsafety requirements, and even the concrete influenceof political issues on their future prospects — either
in or outside of academia.

Students mingle before a talk

The dynamic current progression of the graduate
environment in the mathematical sciences is enhancedby the continual evolution of expectations of funding
bodies, universities, and students themselves. One
aspect of this broadening of outlook is that increasing
numbers of PhD students are moving into jobs outsideacademia, where there is an expectation of a keen
grasp of a variety of professional skills.

Another objective of Maths and Mingle is tohave a positive impact on student welfare; toprovide an open, welcoming environment in which
students can socialise, encouraging inclusiveness anddiscouraging isolation. Consider the progression fromundergraduate to graduate study: this transition issignificant, accompanied by necessary changes inmindset and workflow. Maths and Mingle aims toease the psychological load of this transition, by
providing an opportunity for students to discussmatters affecting them.

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Our framework
The framework of our individual meet-ups is not
set in stone, but over time a regular pattern has
formed which works well. For the first 15 minutes
or so whilst everyone arrives, we relax, chat, and(crucially!) mingle. During this period, we bring out
our monthly puzzles — often taking the form of acryptic crossword set by the organisers. This is also
a time in which we can welcome our speakers, andset up any relevant presentation software for them.

a priority, as is coordination with leaders of otherregular events. In terms of our calendar, we convene
approximately monthly, aiming for ten to twelve
meet-ups per year. When organising the specificdates and times of these sessions, it is important that
we work closely with other school and faculty-level
event organisers; not only to avoid potential clashes,
but also to seize opportunities to widen our scope,joining forces and hosting cross-faculty events where
possible.
Our choice of topics for skills and research talks, andrespective speakers, is taken based on the factors ofcurrent relevance and utility to our audience, alongwith general potential interest. It is here in particularthat having student organisers is of great benefit.

Where we are today

Our monthly crossword can take a bit of team work!

Business then begins in earnest with our talks. First,
a 20–30 minute skills talk given on the topic forthis seminar. We follow this with a very short break,and then continue with our student research talk,where a student from one of the research groups
in our department gives a 20–30 minute talk on
their on-going work. Each of these talks is followed
by a 5–10 minute period of questions, and general
open discussion on/around the subject at hand. Thiscan often lead to quite a bit of a digression, and we
actively encourage this!
After a second brief interlude, we finish with our
PGR forum; the mathematical sciences postgraduate
representatives host an open discussion — overpizza — about matters affecting postgraduates inthe school. This not only provides an opportunity forstudents to raise their own concerns, but also allows
our representatives to give feedback and updates
on issues raised at previous meet-ups.
On the level of a more global organisational format,
we have come to the conclusion that flexibility is

Maths and Mingle is a vibrant, busy seminar series,
regularly attracting around 40 attendees. We haverecently drawn interest in attendance from students
outside the school, as well as from staff in runningjoint events at faculty level. Our generic skills
training sessions host expert speakers both internaland external to the university, where topics range
from the opportunities for work in international
mathematical consultancy firms, to issues aroundequality and diversity in mathematics. One of ourongoing pursuits is to formalise this training, so that
we receive formal external recognition. Running since
2018, we are now into our third year, and anticipate
further evolution and improvement going forwards.

The organisers
Matthew Staniforth, Lily Clements,
and Sam Hughes are second
year PhD students in the Schoolof Mathematical Sciences at the
University of Southampton. They
took over the organisation of
Maths and Mingle at the start of2019. Whilst their research interests
differ wildly (algebraic topology,
statistics, and geometric grouptheory, respectively), they connect
on a much more fundamental level:
their enjoyment of discussing Maths
and Mingle meetings at the pub.

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Microtheses provide space for current and recent research students to communicate their findings with thecommunity. We welcome submissions — see newsletter.lms.ac.uk for guidance.

Microthesis: Massey Products in Moment–Angle
Complexes
ABIGAIL LINTON

Massey products detect the "tangled" structure of topological spaces. They detect that the Borromean rings arelinked, which the linking number cannot. In algebraic topology and symplectic geometry, Massey products are
obstructions to formality and Kähler structure but Massey products are algebraically complicated to compute.
We use combinatorics to calculate these difficult operations.
I work with moment-angle complexes Z K , which arespaces built by gluing unions of products of discs
and circles according to the relation of simplices in a
simplicial complex K . These objects arise naturally
in symplectic and complex geometry, topology andcombinatorics. A simple example of a moment-angle
complex is the 3-sphere, S 3 , when K is the disjoint
union of two points. If K is a triangulation of a sphere,
then Z K is a manifold. In general Z K is complicatedbut the key is that the combinatorics of K encode asurprising amount of topological data about Z K .

Z K are encoded in K . Suppose K is the simplicial
complex below, built on 3 pairs of vertices {1, 2},{3, 4} and {5, 6}. Each disjoint pair carries a class
𝛼 1 , 𝛼 2 , 𝛼 3 of Z K respectively. The 2-Massey producth𝛼 1 , 𝛼 2 i in Z K is carried by 1-dimensional cycles in
K restricted to the vertices 1, 2, 3, 4. Since there isno cycle in this restriction, h𝛼 1 , 𝛼 2 i is trivial. Similarly
there is no 1-cycle on K restricted to 3, 4, 5, 6, so
h𝛼 2 , 𝛼 3 i is trivial too. This implies that the 3-Massey
product h𝛼 1 , 𝛼 2 , 𝛼 3 i in Z K is defined.
4

Massey products via combinatorics
Massey products are higher operations (ondifferential graded algebras, such as H ∗ (Z K )), whichmeans to define an n-Massey product h𝛼 1 , . . . , 𝛼 n i
we need the (n − 1)-Massey products h𝛼 1 , . . . , 𝛼 n−1 iand h𝛼 2 , . . . , 𝛼 n i to be defined and trivial. Unlike
many operations, a Massey product is a set, whichis called trivial if it contains 0. If it contains morethan one element, we say the Massey product has
indeterminacy. What makes it difficult to compute a
Massey product is having to check not only that it isdefined, but also that each element is non-zero.
The Borromean Rings: any
two rings are not linked, but
all three together are. This isdetected by a 3-Massey
product.

It wasn’t originally clear if moment-angle complexes
could have non-trivial Massey products, but the firstexamples were constructed by Baskakov in 2003 [1].
Let us see an example of how Massey products in

1

6

2

3

5

A simplicial complex K
such that Z K has
non-trivial triple
Massey product withindeterminacy.

Also the fact that the restriction of K on 1, 2, 3, 4 isdisconnected implies that this Massey product has
indeterminacy. In this example we can show thath𝛼 1 , 𝛼 2 , 𝛼 3 i is non-trivial by checking each element.
This is one of the first examples of a non-trivial
Massey product in Z K with indeterminacy [5].
Massey products and edge contraction
If Z K has a non-trivial Massey product and L
is a simplicial complex that can be continuously
deformed to K by contracting edges of L, one result
from my PhD is to construct a new Massey productin Z L [6]. Suppose L is the simplicial complex on
9 vertices below. By contracting the coloured boldedges in L, we obtain K in the example above. So

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EARLY CAREER RESEARCHER
by our result, Z L has a non-trivial 3-Massey productwith non-trivial indeterminacy.
4

2

5
8

9

7
3
1

6

A full subcomplex L ⊂ KP , when P is a truncatedoctahedron

We can use L to give an example of a non-formal
manifold, that is, a manifold with a non-trivial Massey
product. Consider KP = 𝜕(P ∗ ), the boundary ofthe dual of a truncated octahedron P . The vertices
of KP are the 9 vertices of L with the 4 othervertices shown above, plus an additional vertex off
the page that is connected to 1, 2, 4, 6. A truncatedoctahedron P is a 3-dimensional polytope whose
faces are squares and hexagons, so the vertices of
KP have valency either 4 or 6. The moment-angle
complex ZP = Z KP is a (polytopal moment-angle)
manifold. Since L sits inside KP , the Massey productin Z L lifts to a Massey product in ZP . In general it
is difficult to find examples of non-formal manifolds,
but our result shows that ZP is non-formal.
This result can construct many new and known
examples of Massey products. For example we use it
to give an alternative proof of Massey products in ZP
for Pogorelov polytopes P by Zhuravleva [7] from theclassification of lowest degree 3-Massey products inmoment-angle complexes due to Denham & Suciu[4] and Grbić & Linton [5].
Massey products and star deletion
The categorical product of two simplicial complexes
is the join, K1 ∗ K 2 = {𝜎 ∪ 𝜏 | 𝜎 ∈ K1 , 𝜏 ∈ K2 }.
This induces a product in moment-angle complexes
Z K1 ∗K2 = Z K1 × Z K2 . Another result from my PhD
is to construct a non-trivial n-Massey product bystarting with the join of n simplicial complexes Ki andcarefully cutting simplices to trivialise lower products[6]. For example the boundary of a square K is
the join of two disjoint points {1, 2} and {3, 4}. The
moment-angle complex Z K is S 3 × S 3 , which has a
non-trivial (2-Massey) product that is supported on
the cycle in K . We break this cycle and trivialise theproduct by deleting the edge {2, 3}.

1

3

1

3

4

2

4

2

There is no restriction in this construction on n or
Ki , so we can construct higher Massey products on
torsion elements. For example by deleting simplices
in ℝP 2 ∗ S 0 ∗ S 0 , we construct a 3-Massey producton a torsion element, and this Massey product has
indeterminacy. Also we can detect many non-formal
manifolds ZP by finding a (full) subcomplex of KP
that edge contracts to a simplicial complex built
by this construction. For example we use this to
give an alternative proof that ZP is non-formal
for all permutahedra P , which was first shown by
Buchstaber & Limonchenko [2].
FURTHER READING[1] I. V. Baskakov, Triple Massey products in thecohomology of moment-angle complexes, Uspekhi
Mat. Nauk 58 (2003) 199–200.
[2] V. M. Buchstaber, I. Yu. Limonchenko, Massey
products, toric topology and combinatorics ofpolytopes, Izv. Ross. Akad. Nauk Ser. Mat. 83 (2019)
3–62.
[3] V. Buchstaber, T. Panov, Toric topology,
American Mathematical Society, 2015.
[4] G. Denham, A. I. Suciu, Moment-angle
complexes, monomial ideals and Massey products,
Pure Appl. Math. Q. 3 (2007) 25-60.
[5] J. Grbić, A. Linton, Lowest-degree triple Massey
products in moment-angle complexes, to appear
in Uspekhi Mat. Nauk, arXiv:1908.02222v2.
[6] J. Grbić, A. Linton, Non-trivial higher Massey
products in moment-angle complexes, Preprint
2019, arXiv:1911.07083.
[7] E. G. Zhuravleva, Massey products in thecohomology of the moment-angle manifolds
corresponding to Pogorelov class polytopes, Mat.
Zametki 105 (2019) 526–536.

Abigail Linton
Abi is an EPSRC
Doctoral Prize SeniorResearch Assistant at
Southampton, where
she finished her PhD
under the supervisionof Jelena Grbić in 2019.
Her research interests are in algebraic topology andcombinatorics. In her spare time Abi enjoys singing
in a choir, running and learning to play the guitar; so
far only two of these at the same time.

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The Maths of Life and Death — why maths is(almost) everythingby Kit Yates, Quercus, 2019, £20, ISBN: 978-1787475427
Review by Neil Saunders
In Daniel Kahneman’sbook Thinking Fast andSlow [3], he details
many examples andpsychological studies
revealing that manyof us rely on the
maxim “what you see
is all there is” inour decision making.
Furthercomplicating
this process are theinteractions, or lack thereof, of our “two systems”
of thought: “system 1” being our fast and automatic
response mechanism, good for making quick andinstinctive decisions; and our “system 2” being
responsible for the more effortful and purposefulmental deliberations. He argues that having awell-equipped and functioning “system 2” that canintervene in the immediate, but often misguided,conclusions of “system 1” is a necessary component
for making better decisions in our everyday lives.
An underlying thread in the many examples ofthe book is that we humans are quite bad at
thinking statistically (and more broadly, without
proper training, mathematically) as this is the workof “system 2”.
While it is clear to the LMS readership that one of the
key tools in building a robust “system 2” for making
better decisions in an increasingly complex world is
a competent literacy in mathematics, our duty as
researchers and educators to spread this message as
far and wide as possible will remain pressing.
Kit Yates’ wonderfully lucid new book The Maths of Lifeand Death is a fantastic contribution to this endeavour.
It skilfully makes the case for the importance ofmathematical literacy in our society, highlighting theindelible role it plays in every part of our lives.
The Maths of Life and Death has something foreveryone in and around the world of mathematics.
If you are a practising mathematician or educatorneeding real-world examples to illustrate a point to yourstudents or peers, then this book provides a rich buffetof examples. On the other hand, for anyone who might
have been put off by mathematics in the past, yet still
appreciates its importance, the book welcomes andquickly engrosses its reader into the many different
facets of mathematics and their applications.
The most striking feature of this book — a book onmathematics — is that there isn’t a single equation in
it. There are no overly long explanations into any deep
mathematical theories and no demands on the reader
to have a sophisticated mathematical background.
Yates accomplishes his task of bringing mathematics tolife by vividly recounting real-life stories in an engaging
and vibrant way. He underscores the importance ofmathematics by often recounting, in devastating detail,the tragic consequences an ignorance of mathematics
can wreak on people’s lives. The book offers acorrective to Kahneman’s “what you see is all there
is” maxim, widening the reader’s vision to see thatmathematics is literally everywhere.
The book comprises seven chapters canvassing awide range of topics and the underlying mathematicalprinciples behind them. After recounting the power of
exponential growth in Chapter 1, in phenomena likean algal bloom or a nuclear reaction, Yates powerfully
demonstrates the role of statistics in the medical
arena. Of particular resonance are his sections on‘false positives’ in diagnostic tests and how these evenconfuse seasoned medical practitioners from time totime: the central message being that even if a result ofa diagnostic test has come back positive, the likelihoodthat the patient actually has the disease being tested
for can still be very small.
Chapter 3 contains some sobering accounts of
how ignorance of statistics, simultaneously combinedwith its misuse, has resulted in some heartbreaking
miscarriages of justice. The case of Sally Clark, themother who was wrongfully convicted of murdering her
two infant children largely on the statistically misguided(yet extremely convincing to a statistically ignorantjury) testimony of the expert witness, is particularly
devastating.

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Chapter 4 contains some humorous and infuriating
examples of how statistics are often misused byunscrupulous media organisations and advertising
corporations in creating false narratives. It serves as areminder that there are lies, damned lies and statistics
and that one must always have the sceptical dial turned
up to extra-sensitive when hearing claims that soundtoo good to be true.
After an entertaining Chapter 5 on the history of thedevelopment of our number systems and how they
have often let us down, Yates concludes the book withtwo chapters largely on mathematical algorithms andtheir use (and misuse) in decision-making, automatedmachines and disease modelling. It’s here where wereally see why the term ‘almost’ is put in brackets inthe subtitle of the book, as Yates is as eloquent as
he is convincing in arguing that while mathematics ispervasive and powerful, we shouldn’t blindly allow it
to control our lives. The algorithm is “as strong as
its weakest assumption” and we should “be careful
not to get carried away. . . by attempting to extend it
beyond its remit”. This perhaps further strengthens
the argument for including an ethics component to theteaching of mathematics in our institutions.
In Daniel Dennett’s book From Bach to Bacteria and Back:
The Evolution of Minds [2], he quotes a maxim of one of
his former students: “you can’t do much carpentry withyour bare hands and you can’t do much thinking withyour bare brain”. Kit Yates has written an enthrallingand enlightening account of how mathematics is anecessary tool for furnishing brains with better thinking
capacities. Its engaging and lucid style makes it readily
accessible for all readers regardless of the level of theirmathematical background.
FURTHER READING[1] D. Kahneman, Thinking Fast and Slow, Penguin,2012.
[2] D. C. Dennett, From Bacteria to Bach and Back:
The Evolution of Minds, Allen Lane, 2017.

Neil Saunders
Dr Neil Saunders isa Senior Lecturer inMathematics at the
University of Greenwich.
His research is inalgebra and geometricrepresentation theory.
He was born in Australia and completed both his
undergraduate and postgraduate studies at The
University of Sydney. He’s a keen cricketer andmusician.

Geometry & Art: How Mathematics transformed
Art during the Renaissanceby David Wade, Shelter Harbor Press, 2017, £16.99, US$24.95,
ISBN: 978-1627951050
Review by Snezana Lawrence
Geometry & Art: How
Mathematics TransformedArtduringtheRenaissance,isabeautifully
produced
book
with
many
beautiful illustrations.
A
hardback
hefty
folio is reminiscent of
the now disappearing
coffee-table book format, and its content is aninvitation for an occasional as well as more concerted
effort by the reader. Certainly it is one of thosebooks you would like to be placed next to your most
comfortable reading chair so that you can remind
yourself often of how some of the most beautiful
images of the Renaissance art relied so heavily on
geometry. This reliance, of course, has many layers.
One is the actual knowledge of geometry, another,for example, the beauty of geometrical objects lent
to visual arts. The book we have in front of us isimportant in another way too: it leads its readers
to understand more of geometry themselves, andtherefore enhance their own enjoyment of art which
had such a close connection with geometry and themathematical and scientific progress of the times.

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The book is divided into five sections: The divine
beauty of geometry, The revival of the Latin West,
Geometria and the Northern Renaissance, Polyhedra
and the creative imagination, and The continuing
influence of Geometria. Its opening statement
gives a clear aim: to begin to trace the common
threads of invention and imagination which wereso purposefully joined in the Renaissance art:
‘The history of Art is intimately linked to the historyof Ideas, and both have common ground in thehuman capacity for invention and imagination’.
This sets the tone for what is to come and provides
a common thread from the history of geometry tomodern times, with a heavy emphasis on the periodleading up to the Renaissance, this period itself, andits aftermath. You will find here diverse subjects
and topics: regular and semi-regular polyhedra, thetheory of optics, the neo-Platonic influence on bothWestern and Islamic arts, invention and the usesof perspective, and the portrayals of astronomical
investigations in paintings of the time, to namesome.
Each part of the book has individual chapters
ranging from seven to seventeen. If we assume
that the number of these shows the emphasis
more clearly, then The Revival of the Latin West(seventeen chapters) and Polyhedra and the creative
imagination (twelve) are the sections that are mostimportant. The chapters are short, succinct, andwell-illustrated. Each chapter has an emphasis box
in which the most important concepts, histories,
and personalities are explained. The narrative isclear, well-written, and with many details to get theinterested reader to search the sources further.
But unlike Pedoe’s [1] or Heilbron’s [2] books Wade
does not contain direct references. This is probably
the weakest link in the book for a scholarly reader— but certainly makes the reading for a novice oran art enthusiast quicker. A bibliography is given at
the end of the book pointing those wanting to learn
more to the main sources the author consulted,
and through which he based his findings.

The images are plentiful and beautiful. But unlike, forexample, Martin Kemp’s now classic The Science of Art[3], this volume does not give image references next to
the images themselves. These are to be found at theback of the volume, and are quite hard to distinguishor find.
Perhaps these are comments which are not relevant
for the public for which the book seems to be aimed.
Its aim is to educate and entertain, not the scholar or
academic, but the popular and interested public. Suchpublic will enjoy the text and images as a starting point
for meditations and thinking about the links between
geometry and art. Sure, readers may need to do somemore research themselves to either confirm or learn
more of what they find in this volume, but enjoy they will,
and feast on this book weaved from the many common
threads between art, mathematics, and history.
FURTHER READING[1] D. Pedoe, Geometry and the Visual Arts, DoverBooks, St Martin’s Press, New York, 1976.
[2] J.L. Heilbron, Geometry Civilized, History, Culture,
and Technique, Clarendon Press, Oxford, 1998,
2000.
[3] M. Kemp, The Science of Art: Optical Themesin Western Art from Brunelleschi to Seurat, Yale
University Press, New Haven and London, 1990.

Snezana Lawrence
Snezana Lawrence isa Senior Lecturer inthe Department ofMathematics at the
University of Middlesex,
London, England. She
has published on thehistory of mathematics and its relationship to
education, and her main focus area within this fieldis the history of geometry, as well as the perceptionof mathematics and mathematicians in mathematics
education and popular culture.

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From Servant to Queen: A Journey throughVictorian Mathematics
by John Heard, Cambridge University Press, 2019, £34.99, US$44.99,
ISBN: 978-1107124134
Review by Alex D.D. Craik†This book is notintendedasacomprehensive surveyof Victorian mathematics,
about which muchhas been written inrecent years. Instead,
it focusses mainly on
the development of
attitudes towards puremathematics and its
rise to become an
independent subject free of association withapplications to ‘the real world’. In other words, Heard
charts the transition from ‘Servant’ to the appliedsciences then dominant in Britain, towards ‘Queen’:
an independent professional group concernedwith abstract ideas and their intrinsic beauty. In
Heard’s words, “Their success did not come easily,
because they had adopted a philosophy concerning
mathematics that was opposed to the prevailing
utilitarianism of the Victorians, and consequently
they became the target of considerable criticism
concerning the value of their activities” (p.1).
After an introductory chapter Setting the Scene,
Chapter 2 The Legacy of Newton addresses thestate of British mathematics in the eighteenth
century, highlighting the dominance of Cambridge inmathematical instruction, with its emphasis on thecompetitive Tripos examinations. But the preference
for Newton’s fluxions and fluents over Leibniz’s
superior differential notation for the calculus led toa too insular view in which continental achievements
were disregarded. The few who espoused continental
methods of analysis are identified as Woodhouse,
Ivory, Babbage, Herschel and Thomas Knight. (To† We note with sadness that Alex Craik died on 17 November 2019.

this list might be added William Wallace and William
Spence.) The Royal Commissions of 1850 into Oxford
and Cambridge castigated Oxford for its neglectof mathematics; but regarding Cambridge “On theimportance of [mathematical] studies it is impossible
to dwell too strongly”. The chapter concludes witha few random examples of mathematics that may
be regarded as abstract, including quaternions,matrices and Boole’s Laws of Thought; and with somecomparisons with the study of classical literature.
The long Chapter 3 is devoted to the London
Mathematical Society, from its foundation with
Augustus De Morgan as president until the 1920s.
Suffice to say that it became the leading society
for pure mathematicians, who were kept abreast ofrecent British work by its lectures and Proceedings.
But Heard is critical of the narrow view that prevailed:“it did little to engage with the non-mathematicaluniverse, but rather immured itself in splendid
isolation, keeping the world at bay as it preserved
the engrained traditions of British mathematics”. A
harsh view, but one supported by the Society’s lackof involvement with other bodies.
Chapter 4, entitled The Pure Mathematician as Herohighlights the work of James W.L. Glaisher, Henry
Baker, Henry J.S. Smith (of Oxford), and Percy
McMahon. Less is said about Arthur Cayley andJames Joseph Sylvester, rightly described as Britain’s
leading pure mathematicians, both being the subjectsof full recent biographies: one learns only thatthey conducted research primarily on matrices andinvariant theory, and that their work was supersededby continental advances. Also mentioned is Andrew
Russell Forsyth, who broke with British insularity towrite an influential work on complex analysis. It isdisappointing that Heard does not engage with these

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various research topics: what, for example, is Cayley’s
invariant theory about? The absence of any mentionof William Burnside’s work on group theory is also
surprising. And why were the pure mathematiciansso insular when the applied mathematicians werenot?
Chapter 5, Mathematics in an Aethereal World, leaves
the pure mathematicians behind to outline theremarkable accomplishments of British appliedmathematicians and natural philosophers at thistime, including Airy, Kelvin, Stokes, Maxwell and Tait.
But this is something of a digression and much has
been written elsewhere. Chapter 6, Apologias forPure Mathematics discusses the professionalisationof Victorian society, but concludes that “puremathematicians could not have constituted a social
profession”. Conflicting views were advanced asjustifications for pure mathematics. According toHenry John Rose, an editor of the Encyclopaedia
Metropolitana, its value lay in providing “storesfor future use” in natural philosophy; but WilliamThomson complained to Hermann Helmholtz that
Cayley and his like should “take their part in theadvancement of the world” instead of devoting
their skill to “pieces of algebra which possiblyinterest four people in the world”. Further such
criticisms came from Airy and T.H. Huxley. A reply
to the latter from Sylvester emphasised that puremathematics required imagination and creativity,“constantly invoking the aid of new principles, newideas, and new methods. . . ”. Citing Cayley as “theDarwin of the English school of mathematics”, hecompared the apprehension of new mathematicalobjects with the discovery of new species of living
things, and likened the pure mathematician to an
explorer of an abstract world. But, according to Heard,“by the late 1870s. . . pure mathematicians. . . needed
to present their discipline in a completely new way”.
This new way is outlined in Chapter 7, Embracing
Beauty. Walter Pater’s essays on aesthetics rejected
the notion that the value of a work of art lay
in the moral truth that it conveyed, and claimed
that beauty could be found anywhere. Adopted byartists and writers, including J.M. Whistler and Oscar
Wilde, this became the influential, but sometimes
lampooned, Aesthetic Movement. By 1883, the Irish
mathematician George Salmon, in a celebratory piece
about Cayley, described him as a “great artist” who

had lived “a life devoted to the contemplation of
beauty and truth. . . the fact that it is a very limited
circle which is capable of appreciating the beautyof the work done by a great mathematician should
not prevent men from understanding that it is likethe work done by a poet or painter. . . ”. Salmon’s
article closely reflects the views of Pater and theaesthetes. Later examples include Bertrand Russell’s1907 statement that “Mathematics, rightly viewed,
possesses not only truth, but supreme beauty. . . ”.
The idea that the function of pure mathematics wasto pursue truth and beauty raised the question ofwhat pure and applied mathematics had in common.
Sir George Darwin, an expert on ocean tides, inan address to the 1912 International Congress ofMathematicians, admitted that “when I gaze on someof the papers written by men in this room I feelmyself much in the same position as if they were
written in Sanskrit”. Further, he believed that appliedmathematicians had come to be thought inferior
to pure; yet “If our methods are often wanting inelegance and do but little to satisfy that esthetic
sense. . . yet they are honest attempts to unravel thesecrets of the universe in which we live”.
Chapter 7 ends with an appreciation of theachievements of Hardy and Littlewood in firmly
establishing the status of pure mathematics in
Britain and rescuing it from its insularity. A
final brief Epilogue completes the book. Readers
may be disappointed to find little about actual
mathematics in this book; but it interestingly charts
the evolution of attitudes towards pure mathematics,
and the growing confidence of pure mathematicians
themselves to present themselves as searchers fortruth and beauty, rather than mere servants ofscience.

Alex Craik
Alex
Craik
was
Emeritus
Professorof Mathematics at St
Andrews
University.
Before turning to history,
his research concerned
fluid mechanics. He also
enjoyed playing classical guitar and travelling in
Greece.

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Obituaries of Members
Vernon Armitage: 1932 – 2019
Dr Vernon Armitage,
who was elected amember of the London
Mathematical Society on
15 December 1955, diedon 12 December 2019,
aged 87. Dr Armitage was
Obituaries Editor of the
LMS Bulletin (1973–96).
John Parker writes: Vernon Armitage was born on 21 May
1932 in Settle, Yorkshire. His father was a signalmanon the Settle to Carlisle railway and Vernon retained
a passion for steam trains all of his life. Another ofVernon’s passions was mathematics, and especiallyteaching mathematics. He loved to be there when
students understood any new concept, and in turn thestudents appreciated his care and love for the subject.
Vernon studied at University College London where
he was awarded a Mathematics BSc in 1953 and a
PhD, supervised by Harold Davenport, in 1956. He
then spent a year at Cuddesdon College Oxford
before teaching mathematics at Pontefract HighSchool and Shrewsbury School. In 1959 Vernon
became a Lecturer in Pure Mathematics at Durham
University and was invited by fellow mathematician
Syd Holgate, the Master of the newly founded Grey
College, to become a fellow of the college. In 1967
Vernon returned to London as a Senior Lecturer
at Kings College before becoming the first Shell
Professor of Mathematics Education at the Universityof Nottingham in 1970. Vernon was back in Durham in1975 as Principal of the College of St Hild and St Bede,which was then a Church of England college formedby the merger of two teacher training colleges.
Vernon navigated the college through the processof joining Durham University in 1979. He was heldin great affection by many generations of students
in Hild-Bede, who gave him the nickname ‘Prin’.
When he retired in 1997, the students renamed thecollege bar ‘The Vernon Arms’ in his honour. Vernon
continued teaching and supervising student projects
in the Department of Mathematical Sciences until
2016 with the title Honorary Senior Fellow. During thistime Vernon published a book in the LMS Student
Texts series on Elliptic Functions, which combined amanuscript by Eberlein, which he edited, and material
he had taught in Durham.

On a personal note, it was while I was taking Vernon’s
third year course on Elliptic Functions that I was
inspired to do a PhD. I was greatly honoured that,when I returned to Durham ten years later, I was
given this course to teach. Many years later, I asked
Vernon for help with non-Archimedean metric spaces
and the resulting research paper was Vernon’s last.
He will be greatly missed.
Martin Taylor writes: After his thesis, Vernon’smathematical interests widened to include curves
over finite fields and especially their zeta functions.
He was fascinated by the interplay of ideas between
such curves and number theory, and this came into
sharp focus with his study of the constants in thefunctional equation of L-functions known as root
numbers.
His knowledge of root numbers played a pivotal
role in the formulation of the Fröhlich conjecture.
Fröhlich and Martinet had been studying the Galois
structure of the rings of integers of tame Galois
extensions with Galois group H8 ; and Vernon had
been thinking about the root numbers in just thissituation. Serre joked that it would be wonderful
if the two were connected. Vernon checked someroot numbers, and his calculations matched perfectlywith the Galois side. Fröhlich went on to prove theresult for H8 -extensions, and then to formulate aconjecture for general Galois groups.
Vernon also worked in the field of mathematicaleducation and during the 1960s he helped withthe development of the School Mathematics Project(SMP). But, over time, he became unhappy as
mathematical education moved towards academic
theory.
Paul Cohen had once said to David Williams “Thereis only one thing sacred in this world and thatis the Riemann Hypothesis”. Vernon was greatly
inspired by this comment, and he devoted aconsiderable amount of time in trying to prove the
Riemann Hypothesis by utilising symmetry properties
associated with the equation of heat flow. Vernon
drew great encouragement from his conversations
on the matter with both David Williams and Enrico
Bombieri. The Riemann Hypothesis is considered tobe the ‘Holy Grail of Mathematics’ and, at least for
Vernon, like the Holy Grail, it was sought but neverfound.
After his retirement, Vernon collaborated with AliceRogers on Gauss sums and Quantum theory, andhe greatly enjoyed this project. Indeed, Vernon had

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always been inspired by using ideas from physics innumber theory.
To conclude, Vernon was once asked in an interview
how he would like to be remembered. He answered
simply “As a good man”. Well, he was certainlythat — with much room to spare! But he will
also be remembered as an inspiring and elegant
mathematician who could communicate his ideas
with wonderful charm and clarity.

S. James Taylor: 1929 – 2020
Professor S. James
Taylor, who was electeda member of the London
Mathematical Society on
17 June 1954, died on
22 January 2020, aged
90. Professor Taylor
was General Secretaryand Journal Secretary
1962–65; General Secretary and Proceedings
Secretary 1967–71; and Proceedings Secretary
1980–82.

Working with James was an entirely pleasantexperience. His deep knowledge, his insistence onhigh scholarly standards, and his ability to explain
complex concepts, were salutary. He was a scholar
and a gentleman. Another mathematician who
enjoyed his collaboration was Paul Erdó´s, with whom
James wrote several papers. James was one of the‘bankers’ who facilitated the wanderings which formany years Erdó´s made across the globe.
Peter Giblin writes: In 1964 James Taylor was appointed
to the Chair of Mathematics at Westfield College,
University of London (he was later Dean of Science)and in 1975 he moved to a Chair of Pure Mathematics
at the University of Liverpool. He succeeded Terry
Wall as Head of the then Department of Pure
Mathematics in 1979, retiring for the first time in1983. Much of James’s considerable energy during
this period was devoted to projects to improve
the quality of teaching in the university and also
in schools. He served as Chair and as Treasurerof the Joint Mathematical Council, but his greatest
contribution in Liverpool was his instrumental rolein the founding in about 1976 of ‘Mathematical
Education on Merseyside’ (MEM), an organisation
which continues to flourish today.

John Kingman writes: After a childhood in Nigeria,
James Taylor, an Ulsterman of missionary parents,
received his mathematical education in Belfast andCambridge. His PhD, under the supervision of A.S.
Besicovitch, was on Hausdorff measures, which canbe used to measure the size of complicated sets suchas the fractals popularised by Mandelbrot. Taylor
realised that another interesting class of sets weregenerated by the sample paths of random processes
such as those modelling Brownian motion. He wasable to prove many subtle results, by combining thehard analysis of the Cambridge school with a deepprobabilistic intuition.

James was also active in the already venerable
Liverpool Mathematical Society (LivMS, founded1899), serving as its President 1975–76. The first‘VI form Lecture’ in a series started by Jameswas given by his friend Sir Hermann Bondi. James,
together with the late Ian Porteous, worked very
hard to involve local school teachers and the
local authorities in planning and carrying out MEM
initiatives, including secondary schools’ ‘Challenge’
competitions sponsored by local industry andcommerce (these are still running) and recreational
mathematics events.

The interest he developed in probability theory drew
him in the early 1960s into the Stochastic Analysis
Group, a small group of British probabilists who
were trying to introduce into the UK the modern
approach due to A.N. Kolmogorov which was alreadyaccepted on the Continent and in the USA. Britain
had espoused a more ‘applied’ approach in which
probabilities were manipulated, often to good effect,with little concern for the underlying mathematicalbasis. The Group decided that a textbook wasneeded, that would develop basic probability theory
as a branch of measure theory, and Taylor wasa natural choice for author. In the event he andI collaborated on an Introduction to Measure andProbability, published in 1966.

Over the years, many more members of staff(including myself), from all three Liverpool
Universities — including now Liverpool John Moores
and Liverpool Hope — have been recruited to carryon the good work of MEM and LivMS, new ventures
including a more advanced VI form ‘Open Challenge’
competition, the LivMS FunMaths Roadshow, MEM
Masterclasses and a continuing series of schools
lectures by well-known speakers. There was evenfor a time a ‘Radio Challenge’ competition hosted bya local radio station, Radio City. But it was James’s
initiative, persuasiveness, imagination and foresightwhich started and nurtured the crucial early phasesof all this important outreach into local schools andthe local community. From Liverpool James and his

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OBITUARIES
wife Maureen moved in 1984 to the University ofVirginia at Charlottesville. James (predictably) became
Head of Department and remained in Charlottesville
until his second and final retirement to Sevenoaks
in 1996. James is survived by Maureen, whom hemarried in 1955, their four children, 17 grandchildren
and six great-grandchildren. His life’s work was
greatly influenced by a strong Christian faith, andhe combined this with his mathematical interests
in making many trips to China, including after his
second retirement.

Eric W. Wallace: 1931 – 2018
Eric Wallace, who waselected a member of the
London MathematicalSociety on 17 December
1964, died on 23 June
2018, aged 87.
Chris Lance writes: Eric
was the youngest of six
children of William andCatherine Wallace in Anfield. His father was a marble
polisher and his mother was a tram conductor
until she married. Eric was always very proud of
his Liverpool roots. From Anfield Road Elementary
School he passed the 11 plus into Alsop High School,
where he was grateful for an excellent education. He
was the only member of his family to continue tohigher education, living at home while a student at
the University of Liverpool as he could not afford
accommodation elsewhere.
His mathematical talent impressed his teacher
A.G. Walker (a future LMS President, 1963–65),
who took him on as a PhD student. He gained
his doctorate in 1956 with a dissertation On theClassification of Lie Algebras. He then, apparently
at Professor Walker’s suggestion, chose to do his
National Service in the Navy, where he achieved
the rank of lieutenant despite a lack of nauticalskills, by teaching mathematics and the art ofsemaphore signalling to the naval recruits. In 1956,
while stationed in Portsmouth, he married Anita
Fisher and they set up home in the Navy’s marriedquarters.
In 1958, with his National Service completed, Eric took
up a post in the University of Leeds, where he spent
his whole career as a Lecturer and later a SeniorLecturer in the Department of Pure Mathematics until
his retirement in 1996, interspersed with sabbaticalteaching visits in New York (SUNY Buffalo), Illinois andMichigan. He published several papers on Lie algebras
and associative algebras, and served a three-year
term as Head of Department. His three PhD students
in Leeds, John Fountain, David Towers and Mike
Holcombe, all went on to successful academic
careers. He also acted as an Open University Tutor
in Pure Maths for a number of years.
Eric and Anita were the social heart of the
Department, and were largely responsible for its
warm and friendly atmosphere. In the 1960s and1970s, when universities had many young lecturers
with growing families, Eric was among a small group
who each year organised a grand Christmas party
for the many children of the Maths and Computing
departments. Professors and parents provided funds,
and Santa Claus (possibly in the form of Eric himself)
had presents for the children. In later years Eric andAnita continued to give many parties at their large
house in Pool-in-Wharfedale.
Anita developed severe health problems, and Eric
cared for her devotedly for several years until her
death in 2003. But he found happiness again with
Moira Wright, whom he married in 2008, and they
continued the tradition of hospitality with dinnerparties for Eric’s colleagues, most of whom had
retired by then.
In 2018, Eric had a severe stroke which left him almost
completely incapacitated. His last weeks in hospitalwere difficult, but he was cheered by visits from his
old colleagues and friends. He is much missed by
Moira, his daughter Rachel and seven grandchildren.
His son Jonathan died in 2017.

Richard Guy: 1916 – 2020
Richard Kenneth Guy,
who was elected amember of the London
Mathematical Society
on 17 March 1960, diedon 9 March 2020, aged
103, being the oldest
member of the LMS.
Kenneth Falconer writes:
Richard Guy was born in Nuneaton in 1916, hisfather having survived the Gallipoli campaign. From
Warwick School he gained a scholarship to Gonvilleand Caius College, Cambridge, to read Mathematics.
After graduating he worked as a school teacher,interrupted by war service as an RAF meteorologist,
before becoming a lecturer at Goldsmith’s College,

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OBITUARIES
then a teacher training college, in 1947. A desire tosee more of the world took him to the University of
Malaya in Singapore in 1951 and in 1962 he went on
to the new Indian Institute of Technology in Delhi. In
1965 he moved to the newly founded University ofCalgary where he remained for the rest of his life.
Richard did not start serious research until around1960 when he made progress on some problems
given to him by Erdó´s. He started working on
combinatorics and graph theory, his interests
broadening into number theory and combinatorial
game theory — all areas where he made significant
contributions.
He wrote more than 100 papers with over 50
collaborators including John Conway, Paul Erdó´s,Martin Gardner and Donald Knuth. A problem-solver,he thrived on ‘intuitive’ problems which are easily
explained to the non-expert but which may require
sophisticated or ingenious proofs. He authored
two books in the Springer Unsolved Problems in
Intuitive Mathematics series, on number theory andon geometry. As well as publishing in research
journals he inspired mathematicians at all levels with
expository accounts, clever proofs and unsolved
problems in publications such as the American
Mathematical Monthly and Mathematics Magazine.
Richard’s best known contributions, whilst not his
deepest mathematically, have an endearing appeal.
He invented the ‘glider’, a period-4 configuration thattravels across the grid in Conway’s Game of Life. He
designed a ‘unistable’ polyhedron with 19 faces which
when made of a uniform material always rolls onto
the same face. He formulated the ‘Strong law of small
numbers’: that there are not enough small numbers
to meet the demands placed on them, illustrating thislaw with examples drawn from across mathematics.
Richard became increasingly interested in
combinatorial games and an extensive projectwith Elwyn Berlekamp and John Conway evolved.
Their two volume (four in the second edition)
classic Winning Ways for your Mathematical Plays
is totally unique in content and style and appeals
as much to enthusiastic amateurs as to researchmathematicians. The first volume covers thegeneral theory of partisan games, and the second
relates this to a cornucopia of combinatorial games
and puzzles. Richard’s wit and clarity of exposition
are readily apparent, as indeed they are throughout
his work.
In 1940 Richard married Louise Thirian who shared
Richard’s passion of mountaineering and together
they climbed many peaks in the Rockies and acrossthe world. They had a close association with the
Alpine Club of Canada who named a hut near Montdes Poilus after them. Louise was supportive of
all Richard’s activities and welcomed the manymathematicians who came to stay. As a 90thbirthday present for Richard she set up the Richard
and Louise Guy Popular Lecture Series at the
University of Calgary. After she passed away in2010 Richard raised considerable sums for charity
in her memory.
Although he nominally retired in 1982, Richard
was at his desk in the Calgary mathematics
department almost daily until shortly before his
death. He never took a doctorate but in 1991
Calgary awarded him an Honorary Doctor of Laws,‘out of embarrassment’ Richard claimed. His final
book The Unity of Combinatorics, co-authored with
Ezra Brown, will be published by the Mathematical
Association of America in May.

Death Notices
We regret to announce the following deaths:• Professor John H. Conway, FRS, Princeton University,
who died on 11 April 2020.
• Professor Freeman J. Dyson, FRS, of Princeton
University, who died on 28 February 2020.
• Mr Ronald C. Ledgard, who died on 12 November
2019.
• Dr R. Michael F. Moss, St Andrews University, whodied 5 November 2019.
• Professor Peter Vámos of Exeter University, whodied on 17 March.

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EVENTS

COVID-19: This information is provided so that Members are aware of events that were planned
before the coronavirus pandemic. Many of these events have now been cancelled, postponed or
moved online. Members are advised to check event details with organisers.

Beyond the Boundaries: New Directions in
Financial and Actuarial Mathematics

LMS/Gresham College Annual
Lecture 2020

Location:
Date:
Website:Location:
Date:
Website:The Rose Bowl, Leeds
18–20 May 2020
tinyurl.com/rpmuxxm

Museum of London
28 May 2020
tinyurl.com/scoxac8

The conference will bring together leading experts
and promising early career researchers from a variety
of mathematical backgrounds interested in theinterplay between theoretical and applied researchin financial and actuarial mathematics. At the coreof the conference: stochastic control, robust finance,
pathwise stochastic analysis, credit and default risk,
risk management, machine learning. Registration
deadline: 10 May 2020.

The first and only female Fields Medalist, Maryam
Mirzakhani, left an astonishing mathematical legacy
at her untimely death in 2017. The lecture, given by
Holly Krieger, will explain the lasting contributionsof her work to our understanding of the world
and give a glimpse into Maryam’s imaginative andhands-on approach to mathematics. No reservations
are required for this lecture. It will be run on a ‘firstcome, first served’ basis.

Bioinformatics Awareness Day (BAD)— Single Cell RNAseq Hackathon

22nd Galway Topology Colloquium

Location:
Date:
Website:University of Leeds
3–5 June 2020
tinyurl.com/ukturrr

Location:
Date:
Website:University of Portsmouth
6–8 July 2020
tinyurl.com/tpcomxe

Bioinformatics Awareness Day (BAD) Single Cell
RNAseq Hackathon will provide researchers withthe opportunity to gather, discuss, and implement
ideas/projects during intensive and productive coding
sessions in bioinformatics. The topics will be aligned
to challenges proposed by the local applied statistics
and mathematics and biomedical communities, withparticular focus on Single cell RNAseq data, together
with training on a selection of new methodologies.

These colloquia provide postgraduates, early career
researchers in topology, and seasoned academics
with an opportunity to engage in an open exchangeof ideas and results. Topology is interpreted
broadly and includes set-theoretic topology, algebraic
topology, continua theory, topological dynamics,
and topological data analysis, as well as crossfertilisation between topology and category theory,order theory, metric space theory, and analysis.
Early-bird registration: 5 June.

RSC Student Conference

Integrable Probability Summer School

Location:
Date:
Website:Location:
Date:
Website:University of Nottingham
21–24 July 2020
tinyurl.com/u8vy5u5

The Research Students’ Conference is returning
to the University of Nottingham for a fourth timesince it began in 1980. This conference is for PhD
students based in Probability and Statistics fields,
organised by PhD students every year in the UK. If
you’re a student and interested in speaking at theconference, complete the validation questionnaire
when booking your place. Supported by an LMS Early
Career Research grant.

University of Oxford
27 July–7 August 2020
tinyurl.com/y6lm7wtw

The domain of integrable probability seeks tounderstand universal phenomena through exactly
solvable probabilistic systems of algebraic andrepresentation theoretic origin. This CMI–HIMR school
will feature several mini-courses and research talks
on methods to find and analyse such solvablesystems, as well as on more probabilistic approaches
to expand the universality classes around them. For
graduate students or early career researchers who
work in this area or nearby fields.

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ii

COVID-19: This information is provided so that Members are aware of events that were planned
before the coronavirus pandemic. Many of these events have now been cancelled, postponed or
moved online. Members are advised to check event details with organisers.

Calendar of Events
This calendar lists Society meetings and other mathematical events. Further information may be obtained
from the appropriate LMS Newsletter whose number is given in brackets. A fuller list is given on the Society’s
website (www.lms.ac.uk/content/calendar). Please send updates and corrections to calendar@lms.ac.uk.

May 2020
1-3 UKACM 2020 Conference, Loughborough
University (486)
4-7 Partial Differential Equations and Fluid
Mechanics, University of Bath (487)
5-7 Swinnerton-Dyer Memorial Workshop,
Newton Institute, Cambridge (487)7 Burnside Rings for Profinite Groups,
Lancaster University (487)
6-9 Joint BMC/BAMC Meeting, University ofGlasgow (485)11-13 Heilbronn Distinguished Lecture Series
2020, University of Bristol (487)11-15 Methods for Random Matrix Theory& Applications, LMS Research School,
University of Reading (486)18-20 Wales Mathematics Colloquium 2020,
Gregynog Hall, Tregynon (487)
18-20 Beyond the Boundaries: New Directionsin Financial and Actuarial Mathematics,
The Rose Bowl, Leeds (488)18-22 Counting Conjectures and Beyond, INI,
Cambridge (486)18-22 Integral Equations and Operator Theory
Workshop, University of Reading (487)28 Joint LMS/Gresham College Annual
Lecture 2020, Museum of London (488)June 2020
1-5 Mathematical Physics: Algebraic Cycles,
Strings and Amplitudes, INI, Cambridge(486)
3-5 Bioinformatics Awareness Day — Single
Cell RNAseq Hackathon, University ofLeeds (488)8-10 Young Researchers in Mathematics 2020,
University of Bristol (487)15-19 Tropical Geometry, Berkovich Spaces,
Arithmetic D-modules and p-adic Local
Systems, Imperial College, London (487)12 Hanna Neumann Day, University of Hull(487)
24-26 7th IMA Conference on Numerical Linear
Algebra and Optimization, Birmingham(487)

July 2020
5-11 8th European Congress of Mathematics,
Portorož, Slovenia (486)
6-8 22nd Galway Topology Colloquium,
University of Portsmouth (488)12-19 14th
International
CongressonMathematical Education Shanghai, China
14-16 IMA Modelling in Industrial Maintenanceand Reliability Conference, Nottingham(486)
20-24 Point Configurations: Deformations andRigidity, LMS Research School, University
College London (487)21-24 RSC Student Conference, University of
Nottingham (488)24-26 7th IMA Conference on Numerical Linear
Algebra and Optimization, Birmingham(487)
27-7 Aug Integrable Probability Summer School,
University of Oxford
24-30 27th
International
Mathematics
Competition for University Students,
Blagoevgrad, Bulgaria (487)27-7 Aug Integrable Probability Summer School,
University of Oxford (488)August 2020
3-7 New Challenges in Operator Semigroup,
St John’s College, Oxford
17-21 IWOTA 2020, Lancaster University (481)

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A CORNUCOPIA OF QUADRILATERALSClaudi Alsina, Universitat Politècnica de Catalunya & Roger B. Nelsen, Lewis & Clark College
Collects and organises hundreds of beautiful and surprising results about four-sided figures. The bookcontains hundreds of challenging four-sided problems. Instructors of number theory, combinatorics, analysis,
and geometry will find examples and problems to enrich their courses.
Dolciani Mathematical Expositions, Vol. 55
MAA Press
Feb 2020 304pp 9781470453121 Hardback £56.50

FUNDAMENTALS OF GRAPH THEORY

Allan Bickle, Pennsylvania State University Altoona
Graph theory is a fascinating and inviting branch of mathematics. The goal of this textbook is to present thefundamentals of graph theory to a wide range of readers. The has author included the shortest, most elegant,most intuitive proofs for modern and classic results while frequently presenting them in new ways.
Pure and Applied Undergraduate Texts, Vol. 43
Apr 2020 336pp 9781470453428 Hardback £81.50

MEETING UNDER THE INTEGRAL SIGN?
The Oslo Congress of Mathematicians on the Eve of the Second World War
Christopher D. Hollings, Mathematical Institute and Queen’s College, University of Oxford &Reinhard Siegmund-Schultze, University of Agder
Examines the historically unique conditions under which the International Congress of Mathematicians took
place in Oslo in 1936. Relying heavily on unpublished archival sources, the authors consider the different goalsof the various participants in the Congress, most notably those of the Norwegian organizers, and the Nazi-led
German delegation.
History of Mathematics, Vol. 44
Apr 2020 362pp 9781470443535 Hardback £114.00

THE UNITY OF COMBINATORICS

Ezra Brown, Virginia Polytechnic Institute and State University & Richard K. Guy, University of Calgary
Combinatorics, or the art and science of counting, is a vibrant and active area of pure mathematical researchwith many applications. The Unity of Combinatorics succeeds in showing that the many facets of combinatorics
are not merely isolated instances of clever tricks but that they have numerous connections and threads
weaving them together to form a beautifully patterned tapestry of ideas.
Carus Mathematical Monographs, Vol. 36
MAA Press
May 2020 259pp 9781470452797 Hardback £62.50

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