pdf - London Mathematical Society

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NEWSLETTER Issue: 489 - July 2020

INTERVIEWWITH THEPRESIDENT

COMPLEXECOLOGICALMETA-NETWORKS

BOB RILEYAND HISMATHEMATICS

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EDITOR-IN-CHIEF

Eleanor Lingham (She�eld Hallam University)[email protected]

EDITORIAL BOARD

June Barrow-Green (Open University)David Chillingworth (University of Southampton)Jessica Enright (University of Glasgow)Jonathan Fraser (University of St Andrews)Jelena Grbic (University of Southampton)Cathy Hobbs (UWE)Christopher Hollings (Oxford)Stephen HuggettAdam Johansen (University of Warwick)Aditi Kar (Royal Holloway University)Susan Oakes (London Mathematical Society)Andrew Wade (Durham University)Mike Whittaker (University of Glasgow)

Early Career Content Editor: Jelena GrbicNews Editor: Susan OakesReviews Editor: Christopher Hollings

CORRESPONDENTS AND STAFF

LMS/EMS Correspondent: David ChillingworthPolicy Digest: John JohnstonProduction: Katherine WrightPrinting: Holbrooks Printers Ltd

EDITORIAL OFFICE

London Mathematical SocietyDe Morgan House57–58 Russell SquareLondon WC1B [email protected]

Charity registration number: 252660

COVER IMAGE

Riley slice detail from the article ‘Robert (Bob)Riley and his Mathematics’ (page 25)

COPYRIGHT NOTICE

News items and notices in the Newsletter maybe freely used elsewhere unless otherwisestated, although attribution is requested whenreproducing whole articles. Contributions tothe Newsletter are made under a non-exclusivelicence; please contact the author orphotographer for the rights to reproduce.The LMS cannot accept responsibility for theaccuracy of information in the Newsletter. Viewsexpressed do not necessarily represent theviews or policy of the Editorial Team or LondonMathematical Society.

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.

MEMBERSHIP

Joining the LMS is a straightforward process. Formembership details see lms.ac.uk/membership.

SUBMISSIONS

The Newsletter welcomes submissions offeature content, including mathematical articles,career related articles, and microtheses, frommembers and non-members. Submissionguidelines and LaTeX templates can be found atlms.ac.uk/publications/submit-to-the-lms-newsletter.

Feature content should be submitted to theeditor-in-chief at [email protected].

News items should be sent [email protected].

Notices of events should be prepared using thetemplate at lms.ac.uk/publications/lms-newsletterand sent to [email protected].

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

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CONTENTS

NEWS The latest from the LMS and elsewhere 4

LMS BUSINESS Reports from the LMS 13

FEATURES Interview with the President 15

Stability in Complex EcologicalMeta-Networks 19

Robert (Bob) Riley and his Mathematics 25

The International Mathematical Union 29

Mathematics News Flash 31

EARLY CAREER Microthesis: The Erdős Primitive SetConjecture 33

REVIEWS From the bookshelf 35

OBITUARIES In memoriam 41

EVENTS Latest announcements 45

CALENDAR All forthcoming events 46

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4 NEWS

FROM THE EDITOR-IN-CHIEF

This issue’s cover image is a detailed view of theRiley slice which was the subject of Caroline Series’Presidential Address at the Annual General Meetinglast November. It is a computer-drawn image by BobRiley from the 1970s, where the black plus signsrepresent groups that correspond to two-bridge knotor link groups, and the red crosses correspond towhat Bob called Heckoid groups – as described byDavid Singerman in his article on Robert (Bob) Rileyand his Mathematics (page 25).

This leads to an idea that we have had for theNewsletter: Do you have an image that might beof mathematical interest to Members? Could youwrite a short item describing this mathematics,or its historical signi�cance? Our new initiative isto publish cover images submitted by Members,along with a short description of the mathematical

or historical signi�cance of the image. To beacceptable, submissions need to be visually, as wellas mathematically, interesting and of su�cientlyhigh resolution (at least 300dpi). The image alsoneeds to be suitable for the set-up of our frontcover: that is, it will be cropped to (width x height)178mm x 148mm, so the main part of the photoneeds to �t these dimensions. Permission to publishthe image must be secured (or granted) by theMember, and submissions should be accompaniedby a 200–500 word description. Photo credit willbe given on our inside front cover. Images anddescriptions are welcome at any time, and shouldbe sent to [email protected].

Eleanor LinghamEditor-in-Chief

LMS NEWS

LMS Response to Covid-19

At its April meeting, Council established the Society’sCovid Response Working Party. This working party hasbeen meeting remotely on a weekly basis. It has beendiscussing suggestions which members have made onhow the Society might best mitigate the effects ofCovid-19 on the UK mathematics community, followingan email from the Treasurer and the Chair of the EarlyCareers Research Committee.

A pressing concern of the working party is the effectthe pandemic will have on young researchers andhence on the people pipeline into our profession. Otherissues include the very considerable efforts staff inmathematics departments have had to and will haveto spend setting up their courses online, and the quitedifferent effects that working from home can create forindividuals, particularly those with caring responsibilities.It is clear that the societal effects of Covid-19 jeopardizethe ability of manymathematicians to carry out research,particularly for long uninterrupted periods.

As a result, the Society has exceptionally created anEmergency Covid Reserve Fund. This fund will be used toalleviate some of the most damaging effects of the virus

and to ensure that something positive should emergefrom this crisis.

The Society has reopened its Early Career Fellowshipsthis year, so that strong mathematicians who findthemselves in limbo between completing their PhDand moving to a postdoc position can be supportedduring these uncertain times. It is anticipated that upto 20 additional Fellowships will be awarded, treblingthe number this year. These will be available to beginby the start of the coming academic year and will beapportioned by the usual process to ensure quality.

The Society is taking the opportunity to support andextend the activities of its existing Scheme 3 ResearchGroups. Groups in similar areas of mathematics willbe encouraged to work together to produce shortonline courses as introductions to their own speciality.Presentations should be delivered by an early careerresearcher, but it is hoped that the whole Group will beinvolved in its production. A sum of up to £1,000 will bemade available to support each course, rising to up to£2,000 if two or three Groups collaborate.

A third area in which the working party believes ourcommunity needs support is to help people learnhow to exploit modern technology in order to deliver

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NEWS 5

online courses and tutorials. Together with the IMAand the RSS, the Society is supporting the TALMOinitiative (Teaching And Learning Mathematics Online),talmo.uk. The Society also hopes that its discussionforum, tinyurl.com/y734ztbo, will become a channelfor disseminating good practice, flagging problems andsharing experiences, not only about teaching and learningissues, but also about working from home, Equality,Diversity and Inclusion matters, and other matters thecurrent crisis brings to the fore.

Finally, the Society continues to be as flexible as possiblewith regard to grants already awarded. If an event or avisit has to be postponed then the funds will carry over,and it is recognised that the form in which the activitytakes place may need to be modified significantly.

Professor Iain GordonLMS Vice President

Professor Robert CurtisLMS Treasurer

A National Academy for theMathematical Sciences?

Should the UK have a National Academy for theMathematical Sciences? This was the topic of anonline open meeting hosted by the InternationalCentre for Mathematical Sciences (ICMS) on June11. The purpose was to bring together interestedmembers of the mathematical community to discussthe role and mission of such an Academy and tomap out possible ways forward. This was the �rsttime that there has been an open meeting on thisimportant topic, �rst mooted in the Bond Review of2018. With over 250 registered participants the eventwas something of a challenge to organise, but Zoom,as managed by ICMS and the INI, worked brilliantly.Despite the obvious limitations of online meetings,the upside was that considerably more people wereable to attend than might have come to the physicalevents which were being planned before the Covid-19lockdown.

Readers will recall that, following the Bond Review,the Council for Mathematical Sciences (CMS) set uptwo committees to consider its recommendationsand discuss how to take them forward. Thesewere the Strategic Committee (SC), chaired by ClairCraig (Provost of The Queen’s College, Oxford)and the Implementation Committee (IC), chairedby Sir Bernard Silverman. The SC was to discussways of attracting possible outside support and

funding, while the IC was to look into the variousrecommendations of the review in much more detail.Along the way, the task of developing the manysuggestions of the Bond Review was renamed theBig Maths Initiative (BMI).

One of the most important proposals in the BondReview was the setting up of a National Academyfor Mathematical Sciences. Although the UK hasa number of specialist professional and learnedsocieties, it lacks an overarching body with the abilitye�ectively to bring together its diverse parts, frompure mathematics through industrial and appliedmathematics to statistics and operational research,in an e�ective broad-based forum. The experienceof the Royal Academy of Engineering shows that asingle Academy is a hugely powerful way to build onthe existing specialist bodies, enabling them far moree�ectively to contribute their individual strengths tothe overall discipline. The Academy would act as anenabler, not a competitor, enhancing the work of theexisting learned societies and other groups.

The slides presented at the open meeting as astarting point for discussion were the outcome ofmuch hard work by members of the IC and SC.The meeting began with a short introduction fromSir Ian Diamond, Chair of CMS (and of course alsothe National Statistician), and then moved on to apanel chaired by John Pullinger, (SC Member and pastRSS President and National Statistician 2014–19).

After an overview by Claire Craig summarisingthe work to date, David Leslie (Edinburgh, ICMember) presented slides about the BMI’s reporton what the purpose and functions of an Academymight be. This was followed by Bernard Silvermanwith discussion of possible governance models.Finally Caroline Series (SC Member and past LMSPresident) introduced thoughts about the possiblenext steps, and everything was rounded up withsome inspirational words from Nira Chamberlain (SCMember and current IMA President).

Participants then split into 12 breakout groups (sadlythe technology could not accommodate more) forin-depth discussion on the presentations. Duringa co�ee break, six ‘synthesisers’ were tasked withsummarising the feedback from the breakout groupswith the aid of some fancy technology called MURALdesigned to mimic post-it notes. The summaries werepresented during a �nal general session, with theopportunity for the panellists to comment, while useof ‘chat’ allowed further discussion and questionsamong participants.

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6 NEWS

Among the points raised were:

(i) The need to ensure maximum diversity in allsenses of the word, especially taking accountof the hugely wide variety of practitioners ofmathematics of all kinds.

(ii) Whether the new body should be con�nedto elected fellows or be broadened into a‘membership body’.

(iii) The exciting possibilities around setting up a‘virtual academy’ without the need for costlypremises.

(iv) It was regretted that there had been verylittle consultation with bodies involved withmathematics education.

The documentation, which was sent to theparticipants in advance of the meeting, also includeda one-page document brie�y outlining the case for aNational Academy. Prepared on the advice of the SC,this is designed to show to busy policymakers and

potential funders. The general feeling of the day wasthat, with some provisos, the ideas outlined in thepresentations were on the right track and, as neatlysummarised by one of the synthesizers Chris Budd,the Academy initiative should go forward carpe diem,that is, ‘seizing the day’. There was also a consensusthat the community should take up the generouso�er of ICMS and INI jointly to provide resourcesto facilitate setting up an interest group to take theproject forward, taking account of the input andfeedback from the meeting.

The panel discussion was recorded and furthermaterial was sent out following the event, togetherwith opportunity for those who wished to expresswillingness to join an ‘interest group’ or otherwiseo�er their services. Material can be accessed via theBMI website tinyurl.com/bmimtg. It is possible thata second online event will be organised sometimelater this year.

Professor Caroline SeriesSC Member and past LMS President

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NEWS 7

First Atiyah Fellows Announced

Ahmad Sabra (left) and Mark Wildon

The LMS UK–Lebanon Atiyah Fellowship scheme wasset up in memory of Sir Michael Atiyah OM (1929–2019).The LMS is delighted to announce that the first two,for the academic year 2020–21, have been awardedto Professor Mark Wildon, Royal Holloway, Universityof London and Professor Ahmad Sabra, AmericanUniversity of Beirut (AUB), Lebanon.

Professor Wildon’s main research area is therepresentation theory of the symmetric and generallinear groups. He will be visiting the Centre forAdvanced Mathematical Sciences at AUB for fourmonths in 2021. He plans to work on the geometricstructure of a family of representations of thegeneral linear group GL(2,C) and on the analogousrepresentations defined over number fields and fieldsof prime characteristic. The characters of thesemodules are obtained using the plethysm product onsymmetric functions: this brings in a rich circle of ideasfrom algebraic combinatorics and connects two excitingareas of mathematics. Professor Wildon also hopes tovisit Beirut Arab University during his stay in Lebanonand to attend and give seminars at both universities.

Professor Sabra is interested in inverse problemsinvolving surfaces which achieve prescribed opticaltasks. He has studied the existence and uniqueness ofsuch surfaces as well as their regularity properties andthe stability of such optical systems. Sabra was awardedhis PhD from Temple University in 2015 and followinga postdoc in Warsaw has been an Assistant Professorat AUB since 2017. He will be visiting Dr Omar Lakkisat the University of Sussex for two months in 2020-21.The aim of their project is to apply Galerkin methods toconstruct numerical approximations of solutions to theequation that appears in far field refractor problems.

During his stay he also plans to visit other researchersin Bath, Edinburgh, Oxford and elsewhere.

Applications for Fellowships to be held in 2021–22 willopen in early September. More information is availableat https://tinyurl.com/tvweckc.

Philippa Fawcett Collection: Appealfor Book Donations

The Society is seekingdonations for one ofits special collections,the Philippa FawcettCollection, whichis housed in theVerblunsky Members’Room at De MorganHouse.

Members are welcome to access the Collection duringweekdays from 9.00 am – 5.00 pm once De MorganHouse reopens after the current closure due toCovid-19. The Collection is a wide-ranging library ofsome 200 books written by and about women whostudied or worked in mathematical subjects in thenineteenth and �rst part of the twentieth century,or earlier. Some are academic texts, others arediscourses on science, some are school textbooks,and there is a selection of biographies and referenceworks. A copy of the current catalogue can be foundon the Society’s website at tinyurl.com/y9hgxn9m.

The Collection was donated to the LMS by one of itsmembers, A.E.L. Davis, in the hope that it will be auseful resource to scholars of the history of womenin mathematics, as well as an inspiration to femalemathematicians of the future. Dr Davis named theCollection in honour of Philippa Fawcett, the first womanto come top in the finals examination, in 1890, of theMathematical Tripos at the University of Cambridge. Asmentioned in our feature article on Philippa Fawcett(May 2020 Newsletter), in those days, women could notbe ranked in the same list as men, so instead, Fawcettwas described as ‘above the Senior Wrangler’.

The Society is seeking donations from its membersof books either written by female mathematicianswho came to adulthood prior to the 1940s (classifiedas primary sources) or about female mathematicians(secondary sources) where the mathematician wasworking or studying mathematics in the 19th and early20th centuries. Dr Davis’ research discovered that therewere 2,500 women who graduated with an honoursdegree in mathematics from 1878–1940 in Britain and

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Ireland alone. The aim is to have a complete collectionof the works by women in mathematics up to 1950to highlight their contribution to the advancement ofmathematics. A target list of women has been drawnup by A.E.L. Davis and a copy can be found on theSociety’s website at tinyurl.com/ycfv2l3c.

Members who wish to make a donation of booksto the collection may contact Elizabeth Fisher:[email protected]; 020 7291 9973. As always, theSociety is grateful to its members for their supportand hopes to enhance the Collection for the benefit ofall.

OTHER NEWS

Postponement of Atiyah Conference

Owing to the di�culties and uncertainties caused byCovid-19, the decision has been taken to postponethe conference on the Unity of Mathematics inhonour of Sir Michael Atiyah which was to have beenheld in the Isaac Newton Institute this September.The conference will now be held in September 2021. Itis expected that registration will open in early spring2021. For further information and updates, pleasesee the conference website tinyurl.com/ybyp3qvv.

Brin Prize 2020

Corinna Ulcigrai

The ninth Brin Prize has been awarded to CorinnaUlcigrai, an LMS member, for her fundamental workon the ergodic theory of locally Hamiltonian �ows onsurfaces, of translation �ows on periodic surfacesand wind-tree models, and her seminal work onhigher genus generalisations of Markov and Lagrangespectra.

The Brin Prize in dynamical systems is awarded foran outstanding impact in the theory of dynamicalsystems or in related �elds. The Prize recognisesmathematicians who have made substantial impactin the �eld at an early stage of their careers.For further information about the prize visittinyurl.com/ycwbzsbd. Ulcigrai is the �rst female toreceive the Brin Prize which was launched in 2008.

Mathematicians Among New RoyalSociety FellowsThe Royal Society announced the appointment ofnew Fellows for 2020, among whom is LMS memberProfessor Jack Thorne (University of Cambridge).

Other mathematicians to receive the honour areProfessor Ehud Hrushovski (University of Oxford)and Professor Andrew Stuart (California Instituteof Technology). Others elected include computerscientist Professor David Harel (Weizmann Instituteof Science) and Professor Hugh Osborn (Universityof Cambridge). Professor Wendelin Werner (ETHZurich) is elected a Foreign Member. ProfessorsStuart and Thorne have both been recipients of theLMS Whitehead Prize, in 2000 and 2017, respectively.

The full list of new Royal Society Fellows is availableat tinyurl.com/y7x6elyf. The Fellows Directory, whichhas extended biographies of all Fellows, is availableat royalsociety.org/fellows/fellows-directory/.

IMU Committee for Women inMathematics NewsletterThe International Mathematical Union (IMU)has published its latest Committee forWomen in Mathematics (CWM) Newsletter:tinyurl.com/yad3ob3w. It features an interview withLMS Honorary Member, Professor Cheryl Praeger, anarticle about the launch of the UNESCO InternationalDay of Mathematics �rst held on 14 March 2020,and a number of personal testimonies of CWMambassadors from around the world on what thecurrent Covid-19 pandemic has meant for their livesas women in mathematics.

Working from home during a pandemic has beennoted as potentially impacting women scientistsmore than their male counterparts, and the CWMis collecting further testimonials on this. Finally,the now annual ‘May 12, Celebrating Womenin Mathematics’ event – this year conducted

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NEWS 9

online – is highlighted. This event is a jointinitiative of European Women in Mathematics, theAssociation for Women in Mathematics, AfricanWomen in Mathematics Association, Indian Womenand Mathematics, Colectivo de Mujeres Matemáticasde Chile and the Women’s Committee of the IranianMathematical Society with the date chosen as thebirthday of Fields’ Medallist Maryam Mirzakhani.

First Woman Appointed GreshamProfessor of Geometry

Professor Sarah Hart,who is Professorof Mathematics atBirkbeck, Universityof London has beenappointed GreshamProfessor of Geometry.The position is thoughtto be the oldest

mathematical chair in Britain and Professor Hartis the �rst woman to be appointed in its 423-yearhistory. She is Vice President of the British Societyfor the History of Mathematics and a keen mathscommunicator.

IDM 2021International Day of Mathematics (IDM) GoverningBoard has decided the theme for IDM 2021 on 14March will be ‘Mathematics for a Better World’. Thischoice is motivated in part by the present pandemicof Covid-19 and the role that mathematical sciencescan play in understanding the dynamics of epidemicsand proposing strategies to control them. Subthemesand explanations of the theme will be posted on thewebsite idm314.org.

To be kept informed of new developments, includingmore information on the IDM 2021 theme, registeron the website for the IDM Newsletter.

George Temple and Albert Green

Albert Green (left) and George Temple

As part of a project with Chris Hollings (Oxford) onthe Sedleian Professors of Natural Philosophy at theUniversity of Oxford I am currently researching thelives and work of two of the holders of the chair:George Temple FRS (1901–92) and Albert Green FRS(1912–99). I would be extremely grateful to hear fromany readers of the LMS Newsletter who may havemet, worked with, or been taught by either of thesemen, and are willing to share their memories withme. I would also be interested to hear from readerswho carried out any form of correspondence witheither man and would be willing to allow access to it.

Dr Mark McCartneyUlster University

[email protected]

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MATHEMATICS POLICY DIGEST

Royal Society President ElectSir Adrian Smith, former chair of the Council forthe Mathematical Sciences, has been con�rmed asPresident Elect of the Royal Society. He will takeup the post of President on 30 November 2020.Sir Adrian is a distinguished statistician and waselected a Fellow of the Royal Society in 2001. Hehas an outstanding academic record, as well as awealth of experience of working with governmentand leading word-class research institutes, suchas Queen Mary University of London, where hewas Principal (1998–2008); the University of London,where he was Vice-Chancellor (2012–18); and The AlanTuring Institute (2018 to present). More informationis available at tinyurl.com/y98wkxtu.

New UKRI Chief ExecutiveProfessor Dame Ottoline Leyser DBE FRS hasbeen appointed as the new Chief Executive of UKResearch and Innovation (UKRI). Professor Leyserstarted her new role in June 2020, succeedingSir Mark Walport. More information is available attinyurl.com/yb4p9uhx.

Wellcome Trust: Science EducationTracker 2019More than 6,400 young people in Years 7–13 (aged11–18) in schools and colleges across England havetaken part in a survey into attitudes towards andexperiences of science education and careers. Somekey �ndings include:

• Many young people don’t see science as relevantto their everyday lives or their future plans.

• Gender gaps are a major issue – both in the typesof science subjects that young people do and

don’t choose to study, and how they perceive theirability. Male students in Years 11–13 are more likelyto choose maths, physics and computer sciencesubjects, while female students are more likely tochoose biology, arts and social science subjects.Chemistry is more balanced by gender. Femalestudents in Years 10–13 are less likely than malestudents to rate themselves as good at maths (63%males, 51% females), physics (46% males, 28%females) and chemistry (42% males, 34% females).

• Students from disadvantaged backgrounds areinterested in science, but have fewer opportunitiesto engage with it — both inside and outside ofschool.

More information is available at tinyurl.com/ybsrp66p.

New UK Research Funding Agency

In its March 2020 Budget the government stated thatit would ‘invest at least £800 million’ in a ‘blue skies’funding agency (�rst announced in the December2019 Queen’s Speech), which would fund ‘high risk,high reward science’.

The House of Commons Science and TechnologyCommittee opened a formal inquiry into the natureand purpose of this new UK research funding agency.The deadline for written submissions was 30 June2020. More information on the progress of the inquiryis available at tinyurl.com/y6vf4sok.

Digest prepared by Dr John JohnstonSociety Communications O�cer

Note: items included in the Mathematics Policy Digestare not necessarily endorsed by the Editorial Board orthe LMS.

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EUROPEAN MATHEMATICAL SOCIETY NEWS

JMI Special Issue

The Journal of Mathematics in Industry, published by theEuropean Consortium for Mathematics in Industry, ispreparing a special issue on Covid-19. Manuscripts canbe submitted now, will be processed as they arrive,and published online as and when they are ready. Visittinyurl.com/y7y9khph for further details.

ERC ResignationThe President of the European Research Council,Mauro Ferrari, resigned on 7 April at the unanimousrequest of the 19 active members of the ERC ScientificCouncil. The statement from the Scientific Councilwith the background to the resignation can be seenat tinyurl.com/vhnvmdl. The EMS expresses its publicsupport to the ERC Scientific Council in its decision anddeclaration, and would like to thank EU CommissionerMariya Gabriel and Director General Jean Eric Pacquetfor standing strongly by ERC and for the leading rolethey have taken to facilitate new initiatives within EUResearch & Innovation in response to the Covid-19crisis.

8ECM RescheduledThe 8th European Congress of Mathematics 2020 hasbeen rescheduled because of to the Covid-19 pandemic.The new date for the Congress is 20–26 June 2021 inPortorož, Slovenia.

EMS PrizesTen EMS prizes are awarded annually to youngresearchers not older than 35 years, of Europeannationality or working in Europe, in recognition ofexcellent contributions in mathematics. The EMSprizewinners for 2020 are:

• Karim Adiprasito (Hebrew University ofJerusalem/University of Copenhagen)

• Ana Caraiani (Imperial College London)• Alexander Efimov (Steklov, Moscow)• Simion Filip (Chicago)• Aleksandr Logunov (Princeton)

• Kaisa Matomäki (Turku)• Phan Thành Nam (LMU Munich)• Joaquim Serra (ETH Zurich)• Jack Thorne (Cambridge)• Maryna Viazovska (EPFL, Lausanne)

The Prize Committee, consisting of Europeanmathematicians drawn from across the continent andrepresenting the diversity of mathematics, held theirdecisive meeting at De Morgan House on 23 January2020. The Chair of the Prize Committee, Martin Bridson,expressed the EMS’s sincere gratitude to the LMSfor the use of both the building and the excellentvideo-conferencing facilities.

The Felix Klein Prize is awarded “to a scientist,or a group of at most three scientists, underthe age of 38 for using sophisticated methodsto give an outstanding solution, which meets withthe complete satisfaction of industry, to a concreteand difficult industrial problem.” The 2020 FelixKlein Prize winner is Arnulf Jentzen (University ofMünster). The core research topics of his researchgroup at the University of Münster are machinelearning approximation algorithms, computationalstochastics, numerical analysis for high-dimensionalpartial differential equations, stochastic analysis, andcomputational finance.

The Otto Neugebauer Prize is awarded “for highlyoriginal and influential work in the field of historyof mathematics that enhances our understandingof either the development of mathematics or aparticular mathematical subject in any period and inany geographical region.” The 2020 Otto NeugebauerPrize winner is Karine Chemla (Université de Paris andCNRS). She has a particular interest in the mathematicsof ancient and medieval China.

David ChillingworthLMS/EMS Correspondent

Note: items included in the European MathematicalSociety News represent news from the EMS and are notnecessarily endorsed by the Editorial Board or the LMS.

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OPPORTUNITIES

Prospects in Mathematics 2021: Callfor Expressions of Interest

UK departments are invited to submit Expressions ofInterest to host the LMS Prospects in Mathematics2021 meeting to the Prospects in Mathematics SteeringGroup.

Up to £7,000 is available to support the annual two-dayevents (usually taking place in September) for final-yearmathematics undergraduates who are consideringapplying for a PhD after they have completed theircurrent studies. This includes funding to cover faresand accommodation for up to 50 students, traveland accommodation for speakers and subsistence forparticipants including a social event.

LMS Prospects in Mathematics Meetings should featurespeakers from a wide range of mathematical fieldsacross the UK who discuss their current research andopportunities available to prospective PhD students.

Prospective organisers should send an expressionof interest (maximum one A4 side in length)to the Prospects in Mathematics Steering Group([email protected]) by 15 September 2020 with thefollowing details:

• Department’s confirmation of support to host theLMS Prospects in Mathematics Meeting.

• Reasons to host the LMS Prospects in MathematicsMeeting.

• A provisional list of speakers who are representativeof the UK research landscape both geographicallyand scientifically.

• Speakers from under-represented groups should beincluded and women speakers should account for atleast 40% of the invited speakers.

• Confirmation that prospective organisers haveread and understood the terms and conditionsin the Guidelines for Organisers (available fromtinyurl.com/y9yn2ryo).

For further details about LMS Prospects in Mathematicsvisit: tinyurl.com/y9yn2ryo.

Ferran Sunyer i Balaguer Prize

Ferran Sunyer i Balaguer (1912–1967) was a self-taughtCatalan mathematician who, in spite of a seriousphysical disability, was very active in research inclassical mathematical analysis, an area in which heacquired international recognition. Each year, the FerranSunyer i Balaguer Foundation awards an internationalmathematical research prize in his honour, open to allmathematicians. It was awarded the first time in April1993.

The 2021 prize will be awarded for a mathematicalmonograph of an expository nature presenting thelatest developments in an active area of researchin mathematics, in which the applicant has madeimportant contributions. The monograph must beoriginal, unpublished and not subject to any previouspublication commitment. The prize consists of €15,000and the winning monograph will be published inBirkhäuser series Progress in Mathematics. The deadlinefor submission is 27 November 2020. For furtherinformation visit the website ffsb.iec.cat.

Early Career Fellowships 2019–20

Second Round of Applications: Deadline 10 July

Applications are now open for a second round of theLMS Early Career Fellowships 2019-20. Recognisingthat one impact of the Covid-19 pandemic on EarlyCareer Researchers is the unexpected turbulencein the job market and to support early careermathematicians in the transition between positions,the LMS o�ers a number of Early Career Fellowshipsof between 3 and 6 months to mathematicians whohave recently or will shortly receive their PhD. Theaward will be calculated at £1,000 a month and o�ersno travel allowance.

For further details and the online application form,visit tinyurl.com/ycfrpz4s. The application deadlineis: 10 July 2020. If you have any queries, please [email protected].

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LMS BUSINESS 13

LMS Council Diary –A Personal View

On Friday 24 April 2020, Council met viavideo-conference. The meeting began with thePresident extending a warm welcome to the newExecutive Secretary, Caroline Wallace, who gavemembers of Council a brief description of herbackground together with an update on activitiesduring the �rst weeks in her new role, and thankedmembers of the society and colleagues for the warmwelcome she had received.

After noting that the Publications Secretary andPublications Committee are leading the Society’sresponse to the UKRI Consultation on Open Access,which has the potential to impact signi�cantly on boththe Society’s publishing activities and the publishingopportunities for UK mathematicians, the Presidentmoved to his own business and introduced an itemon the impact to date of Covid-19 on LMS activities.Council heard that De Morgan House had closed inthe second week of March, and LMS sta�, and almostall tenants of De Morgan House, were now workingremotely. Inevitably, the closure of the building hadled to a loss of conference facilities income, andmany events that the Society had wished to eitherrun or fund had had to be cancelled, postponed, ormoved online. In particular, the Chair of the SocietyLectures and Meetings Committee reported that the2020 Hardy Lecture, due to have been given byPeter Sarnak, would have to be either postponed ormoved online. The Treasurer made the point thatthe unprecedented situation would justify a modestincrease in expenditure over the agreed annualbudget to enable the Society to resource activitiesin response to the Covid-19 situation. Followingdiscussion, it was agreed to form a working groupto consider the Society’s response to the pandemicand to make recommendations on how it can bestsupport the mathematical community.

Initial plans for the next strategic retreat of Council,due to take place in February 2021, were presented.

The value of the socialising that such retreats a�ordswas noted and led to the President proposing thatCouncil should hold monthly virtual social meetingsto facilitate the exchange of ideas in an attempt toreplicate the conversations over co�ee and lunchthat are necessarily absent when Council meetingsare held via video-conference.

Following presentation by the Treasurer of the HalfYear Financial Review and Indicative Operational Plans2020–21, the General Secretary reported that it wouldunfortunately not be possible to hold a physicalgeneral meeting in June, but that options to hold avirtual meeting instead were being explored. We alsodiscussed the list of ideas for Honorary Members andheard updates from several committees, including areport from Frank Neumann on the Mentoring AfricanResearch in Mathematics (MARM) Board, which hadagreed several mentorships. These are now to bepostponed until at least November, but it was notedthat technology could be useful in providing supportto colleagues in Africa and, further, that access toonline activities run by the LMS, which may increasein the current climate, would be welcome.

The meeting concluded with the President thankingeveryone for the constructive discussions during themeeting, which he felt had gone very well despitethe new virtual environment, and in particular theSociety Governance O�cer for having arranged thetechnical aspects of the meeting.

Professor ElaineCrooksElaine Crooks isan LMS CouncilMember-at-Large andthe Council diarist.She is a Professor ofMathematics at SwanseaUniversity.

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14 LMS BUSINESS

LMS Grant Schemes

For full details of these grant schemes, and forinformation on how to submit an application form, visitlms.ac.uk/grants.

Research Grants

The deadline is 15 September 2020 for applications forthe following grants, to be considered by the ResearchGrants Committee at its October meeting.

Conferences Grants (Scheme 1): Grants of up to £7,000are available to provide partial support for conferencesheld in the United Kingdom. Awards are made to supportthe travel, accommodation, subsistence and caring costsfor principal speakers, UK-based research students andparticipants from Scheme 5 eligible countries.

Visiting Speakers to the UK (Scheme 2): Grants of up to£1,500 are available to provide partial support for a visitorto the UK, who will give lectures in at least three separateinstitutions. Awards are made to the host towards thetravel, accommodation and subsistence costs of thevisitor. It is expected the host institutions will contributeto the costs of the visitor.

Research in Pairs (Scheme 4): For those mathematiciansinviting a collaborator to the UK, grants of up to £1,200are available to support a visit for collaborative researcheither by the grant holder to another institution abroad,or by a named mathematician from abroad to the homebase of the grant holder. For those mathematicianscollaborating with another UK-based mathematician,grants of up to £600 are available to support a visit forcollaborative research.

Collaborations with Developing Countries (Scheme 5): Forthose mathematicians inviting a collaborator to the UK,grants of up to £3,000 are available to support a visit forcollaborative research, by a named mathematician froma country in which mathematics could be consideredto be in a disadvantaged position, to the home baseof the grant holder. For those mathematicians going totheir collaborator’s institution, grants of up to £2,000are available to support a visit for collaborative researchby the grant holder to a country in which mathematicscould be considered to be in a disadvantaged position.

Research Workshop Grants (Scheme 6): Grants ofbetween £3,000–£5,000 are available to provide supportfor Research Workshops held in the United Kingdom, theIsle of Man and the Channel Islands.

AfricanMathematics MillenniumScience Initiative (AMMSI):Grants of up to £2,000 are available to support the

attendance of postgraduate students at conferences inAfrica organised or supported by AMMSI. Applicationforms for LMS–AMMSI grants are available from theAMMSI Administrator, School of Mathematics, Universityof Nairobi, P.O. Box 30197, GPO 00100, Nairobi, Kenya(email: [email protected] or [email protected];tel: +254 786 234 678).

The deadline is 15 December 2020 for applications underthe Joint Research Groups in the UK scheme (Scheme 3),to be considered by the ResearchGrants Committee at itsJanuary meeting. Grants of up to £4,000 are available tosupport joint research meetings held by mathematicianswho have a common research interest and who wish toengage in collaborative activities, working in at least threedifferent locations (of which at least two must be in theUK). Potential applicants should note that the grant awardcovers two years, and it is expected that a maximum offour meetings (or an equivalent level of activity) will beheld per academic year.

Maths/Computer Science Research Grants

The deadline is 15 October 2020 for applications forScheme 7 grants, to support visits for collaborativeresearch at the interface of Mathematics and ComputerScience either by the grant holder to another institutionwithin the UK or abroad, or by a named mathematicianfrom within the UK or abroad to the home base of thegrant holder. Grants of up to £1,000 are available.

Grants for Early Career Researchers

The deadline is 15 October 2020 for applications for thefollowing grants, to be considered by the Early CareerResearch Committee in November.

Postgraduate Research Conferences (Scheme 8): Grantsof up to £4,000 are available to provide partial supportfor conferences held in the United Kingdom, which areorganised by and are for postgraduate research students.The grant award is to be used to cover the costs ofparticipants.

Celebrating new appointments (Scheme 9): Grants ofup to £600 are available to provide partial support formeetings held in the United Kingdom to celebrate thenew appointment of a lecturer in mathematical sciencesat a UK university.

Travel Grants for Early Career Researchers: Grants ofup to £500 are available to provide partial travel and/oraccommodation support for UK-based Early CareerResearchers to attend conferences or undertake researchvisits either in the UK or overseas.

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FEATURES 15

Interview with the President

Eleanor Lingham interviews LMS President Jon Keating FRS on his life, work and thoughts about the Society.

When you meet someone new, how do youdescribe your work?

I say that I’m a mathematician, and if pressedfurther a mathematical physicist. I am interestedin many areas of mathematics and I like to thinkof myself as a generalist. I suppose what excitesme most is understanding connections betweendi�erent, seemingly unrelated areas of mathematics.Much of my work involves searching for, or exploitingsuch connections. For example, I am interested inusing connections between random matrix theoryand number theory to understand better thestatistical properties of the Riemann zeta-functionand �uctuations in the distribution of the primes.

I am also interested in applications of mathematics,and especially applications of modern mathematicalideas. Depending on who asks me, I might also talkabout teaching, which I enjoy, and perhaps eventhe work I do in supporting other people to domathematics; for example, at the Heilbronn Institutefor Mathematical Research and, of course, withcolleagues at the London Mathematical Society.

What is your area of mathematics?

I have interests in several areas. My �rst degreeand my PhD were in physics – my PhD advisorwas Michael Berry. He encouraged me to thinkthat theoretical physics and mathematics are soclosely intertwined as to be inseparable. It is muchmore interesting to understand what discoveries areenabled by new ideas, or what novel connectionsthey suggest, than to worry about whether thoseideas should be labeled as belonging to mathematicsor to physics. However, I suppose it is fair to say thatover the years my interests have moved closer tothe centre ground of mathematics.

I started out working in the area of QuantumChaos, where the focus is on quantum propertiesof classically chaotic systems in the semiclassicallimit, that is when the quantum wavelength isasymptotically small. I am still interested in thatarea, but in recent years I have moved moretowards random matrix theory. This is a beautifularea of mathematics that has an extremely broad

range of applications, including to complex quantumsystems, data science, high-energy physics, machinelearning, mathematical �nance, numerical linearalgebra, population dynamics, quantum informationtheory, and telecommunications.

Jon Keating. Photo credit: Chrystal Cherniwchan

Random matrix theory also has deep connectionswith other areas of mathematics, includingcombinatorics, integrable systems theory, numbertheory, representation theory, statistical mechanics,and stochastic analysis. I �nd these connectionsfascinating. I have a particular interest in linksbetween random matrix theory and number theory.It was conjectured by Montgomery in 1973 that thestatistical distribution of the zeros of the Riemannzeta-function coincides, asymptotically, with thestatistical distribution of the eigenvalues of randomunitary matrices.

This highly surprising conjecture is supported byextensive numerical and theoretical evidence and,assuming that it is true, it provides an extremelypowerful model for various statistical problems innumber theory. However, we still are far from reallyunderstanding why it might be true. I have spent agood deal of my research life working on this, andon other problems relating to random matrices.

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16 FEATURES

Are you working on anything particularlyexciting at the moment?

I hope so! I am certainly enthusiastic about workI am involved with currently. I am thinking aboutproblems relating to the extreme values taken bythe characteristic polynomials of random matrices,and what these might tell us about the extremevalues taken by the Riemann zeta-function and otherL-functions. In the case of the zeta-function andother L-functions, the size and frequency of theextreme values is a deep and longstanding mystery.We would like to model this using random matrixtheory, but the corresponding problem there isalso extremely subtle and challenging. Analysing itseems to require ideas from many other areas ofmathematics, and I am enjoying trying to put thepieces of the jigsaw together.

For several years I have also been interestedin number-theory-inspired questions relating topolynomials de�ned over �nite �elds. Here theconnection with randommatrix theory can be proved,in a certain limit, and so the analogues of formulaethat are conjectures in number �elds can, in thissetting, be proved, allowing one to go much further.

Finally, I have also recently been working on theasymptotics of the moments of the Riemannzeta-function. It is known that these are related tocorrelations in the values taken by the generaliseddivisor function, but we have been stuck for overtwenty years in sorting out how to utilise thisconnection in general, because we have been missingat least one key idea. The approach that worksfor the �rst few moments fails spectacularly forthe higher ones, and understanding why this is thecase has been a major puzzle. I have been workingwith Brian Conrey on this and we now believe wehave understood the problem and how to �x it.In fact, it turns out that ideas developed in thetheoretical physics literature in the 1990’s play acritical role, as do a di�erent set of ideas, due toManin, relating to Diophantine geometry. I am veryexcited by this. In collaboration with Henryk Iwaniec,Kannan Soundararajan and Trevor Wooley, we aretrying to establish a general picture of what is goingon. We have a long way to go, but so far the journeyis proving to be an interesting one.

What do you enjoy most about university life?

I love ideas, and so being in an environment whereI can learn new ideas and see them being createdis highly stimulating for me. Going to a �rst-rate

mathematics lecture is one of life’s joys; for manymathematicians this is every bit as important asmusic, �lm, theatre or literature.

I also like working with students. I have beenfortunate to have had some excellent studentsand have found working with them to be highlystimulating too. Universities are changing, and manyare becoming more commercial and corporate intheir outlook. This doesn’t sit well with me, but I�nd that academics themselves are still very muchmotivated by a sense of scholarship, service andcommunity, which certainly aligns with my ownphilosophy.

Where has your interest in the LMS stemmedfrom?

I have been a Member for many years, but my�rst active involvement began in the 1990s, whenthe Society introduced its regional structure and Ibecame the Regional Coordinator for the South-Westof England and South Wales. I suppose the thing thatattracted me in the �rst instance is that I feel theSociety’s values and ethos, its focus on scholarship,teaching and community, resonate with my own.

Earlier in my career I bene�tted enormously from thetime and energy others put into the mathematicalcommunity, and I like the fact that that spirit isstill alive and well. I admire people who contributeto the community, often behind the scenes andas volunteers, and I hope I can make a similarcontribution. I also �nd the history of mathematicsfascinating, and so am interested in societies like theLMS that have played such a central role in the past,as well as currently. The LMS is quirky, which is anaspect I like, and it re�ects well the individualisticnature of many of its members. I am, like manymathematicians, naturally drawn to eccentrics andobsessives, and the LMS is an excellent place to�nd them!

What do you see as the main challenges thatthe LMS is facing?

If you had asked me this �ve months ago, then Iwould have said that the main challenges were:

• Diversity: while we have made signi�cant progressin this area, there is more that we could be doing.We are fortunate to have received earlier this yeara most generous donation to support our work inthis area, one that we believe will make a majordi�erence to what we can achieve. I am personallyextremely interested in issues relating to widening

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FEATURES 17

participation. Previous generations in my family didnot go to university, and the (state comprehensive)school to which I went was not one that wouldbe considered academically strong. I know thedi�erence higher education can make to one’sopportunities and would like to see how we cancontinue to widen access.

• Income: the current Open Access Consultationcould potentially have an outcome which wouldlead to a large reduction in our income, and soin the charitable support that we are able to giveto the UK mathematics community. It is criticallyimportant that we continue to build sustainabilityinto all of our activities, and that we look for waysto diversify our income streams.

• International relations: with Brexit, it is moreimportant than ever to ensure that the LMS isoutward-looking and engaged with our internationalcolleagues and institutions.

• Working with Government: we currently have agovernment that is unusually supportive of science,and in particular of mathematics. We need to thinkhow we can make the most of this opportunity.The recent uplift in funding is a good example ofwhat can be achieved by working with them.

• Bond Review: the mathematical community hasbeen challenged regarding communication andknowledge exchange. We need to ask ourselvessome hard questions: Is it possible for oursomewhat inhomogeneous community to worktogether, despite there being di�ering agendas?How best can we cooperate with our sistersocieties? What is the most useful way for usto contribute to the debate around the possibleestablishment of an Academy for MathematicalSciences?

With the current Covid-19 pandemic, we now facenew and unexpected challenges. Inevitably the UKhigher education landscape will change. Teaching,learning and assessment is moving online. Studentnumbers will be a�ected; and so too will conferencesand research collaborations. Universities will change,and some may falter. We need to work out how toprotect the interests of the mathematical community,and how best to support our members.

What do you see as the main advantages of theLMS as a society?

Our openness – the LMS is a genuinely democraticorganisation that seems to me to represent rather

well the mathematics community in the UK, fromthe grassroots upwards. We are non-corporateand diverse. We have a long-standing history andreputation. We are relatively �nancially secure, andwe are supported by wonderful sta� at De MorganHouse.

If you could have met any Member of the LMS –who would you choose?

G.H. Hardy. I have been strongly in�uenced by hisand Littlewood’s mathematics. Moreover, Hardy ledan interesting life – one which wasn’t always easy –and I would like to know more about him as a person.His book Ramanujan: Twelve Lectures on SubjectsSuggested by His Life and Work is a favourite of mine.It is mathematically beautiful, but also underscoredby the moving story about their relationship. I wouldlike to ask him more about that collaboration. Finally,I would really like to thank him for his bequest tothe Society, which has had a signi�cant impact andhelped a large number of mathematicians over thelast seventy years.

G.H. Hardy c. 1927

Can you tell me a little about your life outsideof mathematics?

My family would say that I don’t have one! But Ido like to cook, and to swim; and I have a healthyobsession with cricket.

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18 FEATURES

What about work/life balance?

Again my answer may cause raised eyes in my family,but this is extremely important to me. It is di�cult tocombine travel, research, and teaching commitmentswith family life, but I have always tried to give thisthe highest priority. My wife, who is also an academicscientist, would be right to feel that she was moresuccessful in this than I have been, but I hope I wouldget reasonably good marks. One thing I am learningfrom the current Covid-19 crisis is that travel is lessessential than I suspect I once thought. I am stronglysupportive of colleagues getting the balance right,and am aware that this is increasingly di�cult asworkloads grow. We need to support the community

in this debate. Here the impact of the current publichealth crisis may not be helpful, as we are forced tomove more of our teaching online.

Do you have any message for our Members?

I would encourage all Members to engage with theSociety – either to become involved in the work ofthe LMS, or to communicate with us about the workthat we are doing. Share with us how you think wecan best help you, and tell us what we should bedoing more of. I’m sure that we all want to see oursubject and our community thrive, and contributingto the Society’s work is as good a way as I can thinkof to achieve that.

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FEATURES 19

Stability in Complex Ecological Meta-Networks

GAVIN M. ABERNETHY

What does it mean for a spatial meta-network to be stable? A model of coupled food webs assembled byecological and evolutionary processes is described, and used to examine what properties allow an ecosystemto withstand perturbations representing catastrophic habitat destruction.

Introduction

How would you answer the question: is the Amazonrainforest a stable ecosystem? We can model anecosystem by constructing a food web – a directedgraph representing the predatory relationships,where the nodes are species (or groupings thereof)and edges indicate a predator’s food sources.Twentieth century ecologists had assumed that foodwebs hold together because of their sheer complexityand interconnectedness. However, in 1972 the lateRobert May’s results on linear stability of randomgraphs [1] seemed to indicate that stability wasfavoured by low diversity (the number of speciesin the ecosystem), low connectance (the fraction ofpossible relationships that are realised), and weakerfeeding relationships. In other words, according toMay’s criterion, more complex food webs oughtto lack stability. As an inspection of the naturalworld shows that highly-complex ecosystems havecontinued to exist for some time, linear stability ofequilibria therefore must not be the most appropriatecriterion for an ecologist. Consequently, as modelsincrease in scale from low-dimensional populationdynamics models to complex adaptive networksof many species, which may not have populationsat an equilibrium, the notion of stability needsto be rephrased for a more practical analysis ofmodel food webs. Some contenders for this measureare given in the sidebar (for more see [2]). Inthe ecological context, adaptive foraging, allometry(using body-size scaling to a�ect energy transferand the feeding relationships), multiple speciessharing the same predators and prey, and employingrealistic non-linear functional responses have all beensuggested to enhance web stability in some regard.

Stability has inherent links to the non-randomstructure of the network, and in the case offood webs, this structure cannot be untangledfrom the evolutionary processes that have shapedit. To address this, researchers have developedeco-evolutionary models that assemble the networks

through both the evolutionary dynamics of speciationand extinction, and the ecological processes ofpredation, competition, reproduction and mortalitypresent in the classical population dynamicsapproach employed in mathematical biology courses.

Measures of stability

Linear stability analysis: the question ofwhether small disturbances from a dynamicequilibrium are dissipated or ampli�ed.

Node deletion stability: the fraction of speciesin the web which, when they alone are deleted,do not result in any further extinctions.

Community robustness: the fraction of specieswhich must be arti�cially deleted in sequenceto induce a total loss (including resultingsecondary extinctions) of 50% of all species.A food web lacking robustness would behighly sensitive, with the loss of one speciesprecipitating a signi�cant collapse.

Persistence: whether (or how many) speciescan endure the duration of the perturbation.

These models can construct complex food webs withrealistic distributions of species in di�erent trophicroles, and we use these as the basis of our stabilityexperiments. Furthermore, by coupling multiple foodwebs and allowing populations to migrate betweenthose adjacent, we expand our eco-evolutionary foodweb model to a spatially-explicit meta-communitymodel. Again, the question of stability can beappropriately “scaled-up”, so that we ask what theproperties are of patches (nodes in a graph of thespatial meta-network) whose loss or perturbationcauses the greatest damage to the whole system?By answering these questions, models of complexnetworks can be deployed in service of conservationand preserving geographic biodiversity.

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20 FEATURES

Model equations

Feeding scores:

Si ,j = max{0,

10∑m=1

10∑n=1

V im ,jn ×1

1.5√2c

exp(−(ri − r j − 3)2

2(1.5)2

)}(1)

Competition scores:

Ui ,k = c + qi ,k ×1 − c

0.6√2c

exp(−(ri − r j )2

2(0.6)2

)(2)

Foraging e�orts:

fi ,j =gi ,j∑

k ∈Ki gi ,k(3)

Functional response:

gi ,j =Si ,j fi ,jN j

bN j +∑k ∈P j Ui ,kSk ,j fk ,jNk

(4)

Population dynamics:dNidt

= −2 e−0.25ri Ni + _Nisi

n∑j=0

gi ,j s j −n∑k=1

Nk gk ,i (5)

Model

The model we shall be using is as follows [3]. Wede�ne a species i by which ten discrete binary traits{im |m = 1, . . . ,10} it possesses, and its body-sizesi which is a continuous variable converted to alog scale as ri = log(si ). Each possible trait p israndomly assigned a non-negative score Vp ,q againstevery other trait q . Then the potential feeding scoreSi ,j of species i on species j is determined bythe sum of the scores of each pair of their traits,scaled by the probability density function of anormal distribution of relative body-size di�erencescentered at three. This means that the most e�ectivepredatory relationship is upon a prey with body-sizeexp(−3) that of the predator (1).

Any pair of species i ,k who utilise the same preyexperience a base level of competition c = 0.6, whichis then increased by an amount that is modi�edby the similarity of the species: linearly with thefraction of shared traits qi ,k , and according to anormal distribution probability density function ofrelative body-size di�erences centered at zero (2).Thus, if two very di�erent species use the sameresource they experience weaker competition andcan feed with greater e�cacy. In real ecosystems,members of the same or related species are likelyto consume the same parts of their prey, while forexample a �ying-fox that feeds on eucalyptus nectar

will experience less competition with the koala thatprimarily eats the leaves of the same plant.

At a given timestep, each local population of eachspecies must decide what available prey it is goingto feed upon, and how to allocate the proportion ofits hunting e�ort among them. To do this, a pair ofequations for the foraging e�orts fi ,j of species ion j (3), and the corresponding functional responsesgi ,j (4) that govern the actual transfer of energy fromprey to predator, are updated between each iterationof the population dynamics, so that species aregradually able to adjust their strategies in response tochanging conditions and the success of their previouse�orts. Here b = 0.005 controls the e�ectiveness ofpredation in the ratio-dependent functional response,Ki is the set of current prey of species i , and P j isthe set of current predators of species j .

All local populations are then updated according toODE (5), with terms accounting for loss due to naturalmortality, gains due to feeding where the ecologicale�ciency _ = 0.3 controls the �ow of biomassto the next trophic level, and losses to predators.If the model allows multiple patches, populationsmay then move to adjacent patches on the gridaccording to the processes described in the secondbox. Two mechanisms of movement, henceforthreferred to as di�usive and adaptive migrationrespectively, are proposed. Both are designed to

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FEATURES 21

generate meta-communities with distinct local foodwebs, as if we allow all species to move by di�usionbetween any neighbouring patches (even at low rates)typically the food webs become highly synchronisedand overall biodiversity is constrained [4]. This rateis also allometrically scaled, so that creatures with alarger body-size are able to move faster.

The above mechanisms describe the ecologicaldynamics of the model. The evolutionary dynamicsthen consist of occasionally, after many iterations ofthe population dynamics, introducing a new speciesinto the global ecosystem by mutation of an existingspecies. A parent species is selected from theensemble, and a child introduced with minimumpopulation size. This new species has one of its tentraits randomly exchanged with another choice, andits body-size is uniformly selected from within 20%of the parent’s body-size. A model simulation beginswith a single species, and in each patch a unique

resource that can never move, mutate or be fullydepleted, then over time these rules assemble acomplex meta-community of hundreds of specieswith di�erent ecosystems in each patch.

Model communities

We will consider results from three sets ofmodel communities. First, a single food web (withno spatial implementation) can be constructedover 110,000 speciation events and we canexamine patterns of robustness in these individualecological communities. Then we shall consider twometa-communities, assembled on 6 × 6 spatial grids,using the two di�erent movement mechanisms (7-8).As these simulations are far more computationallyexpensive, we shall only use 10,000 speciationevents for these scenarios. The four patches in thetop-right corner of the meta-network constructedwith di�usive movement are illustrated in Figure 1.

Movement in the meta-network

Let N ti ,x ,y denote the population of species i in patch (x ,y) during the t th ecological timestep. Then

movement between patches is implemented by

N ti ,x ,y ↦→ N t

i ,x ,y +xmax∑j=1

ymax∑k=1

X j ,k ,x ,y `i ,j ,k ,x ,yNti ,j ,k −

xmax∑j=1

ymax∑k=1

Xx ,y ,j ,k `i ,x ,y ,j ,kNti ,x ,y (6)

where X j ,k ,x ,y = 1 if the patches ( j ,k ) and (x ,y) are connected and distinct, and zero otherwise. Theparameter `i ,j ,k ,x ,y denotes the fraction of the local population of species i in patch ( j ,k ) that migratesto patch (x ,y).

We contrast two choices for this migration:

(1) Trait-gated di�usive migration: we associate all links between adjacent patches with 100randomly-selected traits

`i ,j ,k ,x ,y = max{1, 10−4

sis0×M j ,k ,x ,y (i )

}×D−1j ,k . (7)

Here, s0 = 1 denotes the body-size of the resources, and D j ,k the degree of patch ( j ,k ). M j ,k ,x ,y (i )returns the number of the traits of species i associated with the link between patches (x ,y) and( j ,k ) and so scales the total migration by how well adapted the species is to traverse this link.

(2) Adaptive migration: we allow any species to traverse any link, but only when they have experienceda decline in their local population

`i ,j ,k ,x ,y =

max

{1, 0.03 sis0 ×

N t−1i ,j ,k−N

ti ,j ,k

N t−1i ,j ,k

}×D−1j ,k , if N t−1

i ,j ,k > N ti ,j ,k

0 otherwise.(8)

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22 FEATURES

Meta-community: Trait-gated di�usive migration on a 6 × 6 lattice

Figure 1. Four local food webs in a corner of the 6 × 6 meta-network with 458 total biodiversity (and 36 uniqueresources), assembled over 10,000 speciation events from a single initial species. Purple denotes the resource,green a basal species (feeds only upon the resource), red a top predator, and yellow otherwise. Line thickness isproportional to feeding e�ort, with arrows from predator to prey, and vertical height corresponds to trophic level.Node radius is proportional to population size on a logarithmic scale.

Stability of an individual food web

First, a result already known to theoretical ecologists[5] is con�rmed: when we consider food websexisting on a single isolated patch, network stabilityin the sense of community robustness (see box1) is positively correlated with connectance of thefood web (Figure 2), that is, when proportionallymore of the potential feeding relationships in thenetwork are actively realised [3, 6]. (Connectanceof the food web is measured by L/S (S − 1), whereS is the number of species and L the number offeeding links.) In such cases, species tend to bemore �exible and less dependent on a few essentialnon-resource prey for their survival. This can beachieved in a relative sense by reducing the size ofthe community, or by allometric scaling in�uencingthe structure of the web to a more pyramidalshape (Figure 3), so that more species feed on

the resource which is excluded from being deleted.

Figure 2. Robustness of individual food webs against theirconnectance. Red indicates our model, with green andyellow indicating two choices of parameters in a versionthat lacks the in�uence of body-size e�ects.

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FEATURES 23

Figure 3. Robustness of individual food webs against theirbiodiversity. Red indicates our model, with green andyellow indicating two choices of parameters in a versionof the model that lacks the in�uence of body-size e�ects.

Stability of a meta-network

To investigate the stability of our two evolvedmeta-communities of many species occupyingmany patches, let us model the e�ect of habitatdestruction by subjecting them to either individual,or sequences of, perturbations by removing patchesfrom the meta-network, and observing the resultingimpact on the biodiversity of the global ecosystem.

Figure 4. E�ect of disrupting a single patch in themeta-community with di�usive migration. Red indicateseliminating the local population, while green denotes theresult of displacing them to neighbouring patches. Theshaded area indicates a loss of more species than therewere in the disturbed patch.

We contrast two forms of disturbance: either thelocal populations of the a�icted patch are instantlyeliminated, or they are uniformly dispersed amongstall neighbouring patches.

First, we con�rm the obvious hypothesis that globalbiodiversity loss due to eliminating the occupantsof a given patch is strongly correlated to the priordiversity of that perturbed patch (Figure 4, red),although even this depends on the rules governingdynamics and especially species movement. Both ofthe movement mechanisms we have employed giverise to fairly isolated local food webs in each patch,with most species occupying only one or two patchesby the time of the experiment. If a more liberaldi�usion mechanism was permitted, the food webswould co-evolve as highly synchronised, and therewould consequently be little e�ect of eliminatingthe local population of any patch [4]. However, thecurrent model shows a more interesting responseto displacement (Figure 4, green). In Figure 1, patch(6,5) has only two resident species compared tothe 24 in its neighbour (5,5). Displacing these 24yields an overall loss of 25 species, but if the twospecies of (6,5) are forced to emigrate, the endresult is a quite disproportionate 13 extinctions! Asone of the pair have another population elsewhere,if there was a need to temporarily clear that patch ofwildlife it seems that no conservation e�ort wouldhave been better than one that carelessly introducesnon-native species into the complex ecosystem ofpatch (5,5). Of course, a better (though costly)approach would attempt to transfer the two speciesto a suitable distant habitat. This captures thehighly destructive potential of real invasive speciessuch as the introduction of rattus rattus to islandcommunities, although the overall role of invasionsin continental extinctions is more contentious [7].

Figure 5. Sequentially perturbing random patches in themeta-community with di�usive migration.

When we consider sequences of disrupting thelocal populations of randomly-selected patches,displacing the a�ected individuals is about asdamaging to the global ecosystem as eliminating

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them for around the �rst 20 events (Figure 5).Subsequently, although it continues to be limiting,the severity is reduced compared to killing thea�ected populations and a reasonable number ofnon-resource species can survive inde�nitely. If weextend our concept of robustness of a network tode�ne the “meta-robustness” of the system as thenumber of patches we must disrupt to induce a 50%or greater loss of biodiversity, the model with thedi�usive migration mechanism (7) yields 18.9-26.0for elimination and 23.5-36.5 for displacement fromten trial sequences. This overlap is still presentwhen using adaptive migration (8), but with valuesincreased to 21.8-34.0 and 31.2-45.3 respectively(Figure 6), so we can say that this system is “morerobust” to patch deletion of both kinds.

Figure 6. Sequentially perturbing random patches in themeta-community with adaptive migration.

The stability of the ecosystem in the sense ofpersistence exhibits a stronger dependence on and areversal regarding the migration rules. When adaptivemovement is employed, an initial meta-network withgreater diversity and complexity is assembled, but itis unable to endure sequences of random eliminationof patches inde�nitely (Figure 6, blue), while a modelusing constant low-level di�usion may do so but withvery few survivor species (Figure 5, blue). This can beunderstood in terms of “tall” vs. “wide” play in howspecies make use of the spatial network – adaptivemovement causes initial wide play as species spreadout to �nd better pastures at the beginning of thesimulation, but subsequently settle on tall strategiesto make the most of the best patch they �ndthemselves in. This allows them to �ourish for atime and the patches become relatively isolated, thuswhen a neighbouring patch experiences catastrophe,the local populations cannot take advantage of it andsimply await the destruction of their own habitat. Onthe other hand, a di�usive mechanism essentially

causes all species to always have a “slightly-wide”strategy, which is not so bene�cial initially but doesallow them to re-colonise destroyed patches and soensure that some species persist.

As we are witnessing currently, mathematical andcomputational models do indeed have potentiallysigni�cant in�uence on public policy when it comesto human health, wellbeing and the economy. Moresophisticated ecological and evolutionary modelling,such as spatial models which explore the optimalsize and placement of nature reserves, will beneeded to inform responsible human stewardshipof the environment and mitigate the impact onbiodiversity of advancing urbanisation, deforestationand exploitation of natural resources.

FURTHER READING

[1] R.M. May, Will a large complex system bestable?, Nature 238 (5364) (1972) 413–414, 08.[2] M. Pascual, J.A. Dunne, From small tolarge ecological networks in a dynamic world,in: Ecological Networks: Linking Structure toDynamics in Food Webs, OUP, 2006, pp. 3–24.[3] G.M. Abernethy, Allometry in aneco-evolutionary network model, EcologicalModelling 427 (2020) 109090.[4] G.M. Abernethy, M. McCartney, D.H. Glass,The role of migration in a spatial extension ofthe webworld eco-evolutionary model, EcologicalModelling, 397 (2019) 122–140.[5] J.A. Dunne, R.J. Williams, N.D. Martinez.Network structure and biodiversity loss in foodwebs: robustness increases with connectance,Ecology Letters, 5(4) (2002) 558–567.[6] G.M. Abernethy, M. McCartney, D.H. Glass, Therobustness, link-species relationship and networkproperties of model food webs, Communicationsin Nonlinear Science and Numerical Simulation, 70(2019) 20–47.[7] C. Bellard, P. Cassey, T.M. Blackburn, Alienspecies as a driver of recent extinctions, Biologyletters, 12(2) (2016).

Gavin AbernethyGavin is a Lecturerin Engineering Math-ematics at She�eldHallam University. Hiscurrent interests arein ecological modellingand nonlinear dynamics.He enjoys growing chilli

peppers, frying prawns, and playing Animal Crossing.

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Robert (Bob) Riley and his Mathematics

DAVID SINGERMAN

Robert Riley was a US mathematician who spent many years at the University of Southampton. He combinedhis mathematical and computing skills to discover the Riley slice, which was the subject of Caroline Series’LMS Presidential Address. Here his PhD supervisor, friend and colleague describes his mathematical work.

Introduction

Robert (Bob) Riley was a US mathematician who spentmost of the years 1968–1980 at the University ofSouthampton. During these years he made someremarkable discoveries which we will explore later.

Bob Riley

First, some background:Bob was born inNew York City in1935. In 1957 hestudied at CornellUniversity andafter graduating, heproceeded to MIT forgraduate studies innumber theory. However,he did not get on toowell with some of thealgebraic geometry, andgave up his course, and went to work for a computingcompany. This was in the early days of computing,and the result was that he became expert in theusage of computers, which would become valuablefor later mathematical pursuits.

In 1966, we �nd Bob in Amsterdam. Why did he gothere? I �rst heard that this was to escape beingdrafted to go and �ght in the Vietnam War. But morerecently, an old university friend of Bob told me thathe had fallen madly in love with a Dutch girl, andwent to Holland to marry her. A more romantic storywhich is more likely to be true, but I never againheard about this Dutch girl.

While in Amsterdam, Bob developed an interest inknot theory [5]: “On settling in Amsterdam in October1966 I wrote o� to virtually everyone publishingin knot theory for their reprints and preprints. Irecall with gratitude that R.H. Fox and H. Seifertwere especially generous. An unassuming little paperby Fox written in Utrecht some 20 miles away,took my fancy.” Fox was looking at representationsof a knot group (the fundamental group of the

complement of the knot in the three-sphere) intoA5. Now A5 is isomorphic to PSL(2,5) and thisled Bob to study parabolic representations of knotgroups into PSL(2,F ), where F is a �eld. Theseare representations such that meridians in the knotgroup map to parabolic matrices, that is matriceswith trace ±2.

The Riley Slice

H.B. (Brian) Gri�ths, a professor from Southampton,met Bob in Amsterdam and invited him to take atemporary position at Southampton, which he didin 1968. I started working at Southampton in 1970,after getting my Ph.D. at Birmingham under thesupervision of Murray Macbeath. I worked on topicsrelated to Fuchsian groups, discrete subgroups ofPSL(2,ℝ), with particular interest in their quotientRiemann surfaces, so I did have an interest in discretematrix groups and their quotients. On my �rst dayat Southampton I was taken to meet with Bob, andthe �rst thing he said to me was:

“Consider the group generated by the matrices(1 10 1

)and

( 1 0c 1

), where c is a complex number. When is

this group discrete and free?”

At the time I did not realise the importance ofthis question. It led to the Riley slice which manymathematicians have been investigating in recentyears.

Here are some mathematical preliminaries: a Kleiniangroup is a discrete subgroup of PSL(2,ℂ), the groupof complex Möbius transformations. As Poincaréobserved, these groups have an action on hyperbolic3-space ℍ3. To see this, we think of this space asthe set of points (z ,t ), where z ∈ ℂ, t > 0. Thehyperbolic metric is de�ned by:

ds 2 =|z |2 + t2t2

.

Elements of PSL(2,ℂ) map circles to circles in thecomplex plane. Now each Möbius transformation is

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a product of an even number of inversions in circles.We extend these by looking at the inversions in theupper-half of the spheres. This gives an action ofPSL(2,ℂ) on upper-half 3-space, which is our modelof hyperbolic 3-space. More importantly, it givesPSL(2,ℂ) as the orientation-preserving isometrygroup of ℍ3. The quotient space ℍ3/Γ is then ahyperbolic 3-manifold. Indeed, if M is a completehyperbolic 3-manifold, then there exists a Kleiniangroup Γ such that M = ℍ3/Γ. A Kleinian groupΓ may act discontinuously on some region of theextended complex plane ℂ. The maximal openset in the extended complex plane where Γ actsdiscontinuously is called the ordinary set of Γ and isdenoted by Ω. We can then extend M to (ℍ3∪Ω)/Γwhich is M with its conformal boundary.

Figure 1. Riley slice. Photo credit: Y. Yamashita

In Figure 1, there is a white region with a fractalboundary which we call D. This is the Riley slice. Thepoints c which lie in the exterior of D correspondto free discrete groups, but there are points insideD marked in the diagram where the groups are stilldiscrete but not free.

Let us look at part of D in more detail: in Figure 2, theblack plus signs represent groups that correspond totwo-bridge knot or link groups. (All two-bridge knot orlink groups appear [1]). The red crosses correspond towhat Bob called Heckoid groups. These are Kleiniangroups with non-empty ordinary set generated bytwo non-commuting parabolic elements and whichcontain elliptic elements (elements of �nite order). Allthese points correspond to non-free discrete groups.Also included are the Hecke groups at the points±4 cos2 c

n . These are groups isomorphic to a freeproduct C2 ∗Cn . The whole Riley slice goes from −4to +4 on the real axis and from −2i to +2i on theimaginary axis.

Figure 2. Riley slice detail

The Riley slice picture (Figure 3) was constructedby Bob with the aid of a computer in Southampton.Riley’s early employment as a computer programmercertainly paid o�! Figure 3 shows the part of the Rileyslice in the �rst quadrant of the complex plane. It isdated 26 March 1979. When Bob left Southampton, hegave a copy to David Chillingworth who then passedit onto Caroline Series at Warwick. She then passedit onto John Parker, who was a postdoc at Warwickat the time. In his lecture on the complex Riley slice,John Parker [4] said “I have a computer printout inmy o�ce in Durham which is dated 1979. It was ona huge piece of paper produced by one of thoseprinters with an arm and a pen.”

Figure 3. Riley slice by Bob Riley

Figure 1 is a rather beautiful representation of thewhole slice by Yasushi Yamashita. Luckily, John kepthis diagram of the Riley slice, which as far as Iknow was the only version available in the UK. Thisdiagram has proved useful for Caroline Series who

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FEATURES 27

has written on this topic, and it was the subjectof her LMS Presidential Address [8] in November2019. The only other versions were held by Bob. TheJapanese mathematician Masaaki Wada picked up acopy from Bob when he visited Binghamton muchlater. This appeared on page VIII of [2]. After theRiley slice diagram they write: “Riley’s pioneeringexploration of groups generated by two parabolictransformations. The computer-drawn picture hasbeen circulated among the experts and has inspiredmany researchers in the �eld of Kleinian groups andknot theory. This speci�c copy of the picture wasobtained directly from Professor Riley when M. Wadavisited SUNY, Binghamton in February 1991.”

Hyperbolic structure on the �gure eight knotcomplement

In 1973, Bob made what was possibly his mostimportant discovery: the hyperbolic structure on the�gure eight knot complement. Before we describethis let us return to some personal history: in October1972 his fourth temporary position at Southamptoncame to an end, and Bob found himself with alarge pile of computer output and no prospect offurther employment. He managed to get a six-monthappointment at Strasbourg, and after the summervacation of 1973, he returned to Southampton wherethe mathematics professors granted him the use ofan o�ce and all university facilities except for theuse of the computer, which was heavily overloaded.

The result was that Bob had to read about Kleiniangroups, and in particular Poincaré’s theorem onfundamental polyhedra. This made more progresspossible and he looked again at the parabolicrepresentation \ of the group of the �gure eight knot.This is a two generator group and its image underthe parabolic representation is the group generatedby

(1 10 1

)and

( 1 0l 1

), where l = − 1

2 (1 +√−3) is a

cube root of unity. Note that this point correspondsto one of the black plus signs in the Riley slice!

As Bob wrote [5]: “This group is obviously discreteand only its presentation was in doubt.” He could�nd the presentation because he knew Poincaré’stheorem: “I remember my surprise at �nding thisp-rep is faithful. The �rst version of my accountin [6] was received by the editors on 30 November1973 and it didn’t mention the orbit space ℍ3/cK \

because I had not even thought of it.”

By constructing a fundamental domain for cK \,Bob proved that the orbit space is the �gure eight

knot complement and so he had now obtained ahyperbolic structure on this knot complement! Evenin his revised version of [6] he did not make muchof a fuss about the hyperbolic structure. He just hada corollary stating that the identi�ed fundamentaldomain is homeomorphic to the knot complement.I remember Bob telling me of his result one daywhen we went walking on Southampton Common.As I had experience of quotients of Fuchsian groupsgiving hyperbolic Riemann surfaces, I knew what hehad done, although I did not fully understand thethree-dimensional topology.

Thus by 1975, Bob had a major result but nopermanent academic job. The reason was that hehad left MIT without getting his Ph.D. and so in1975, I formally took him on as a research student.Of course, I did not need to give him a researchproblem; he had many of his own. I also proposedthat we apply to the Science Research Council (SRC– a forerunner of ESPRC) to support a hyperbolicproject at Southampton. As Bob wrote “the plan wasto time the submission of the proposal so that thereferee would be at the summer 1975 conferenceon Kleinian groups at Cambridge where I wouldpublicise hyperbolic structure. Whether or not theplan worked, the Kleinian groupies liked my examples,especially because these examples pointed up theimportance of their own work. The SRC did fund theproject generously, ultimately for four years, from1976 to 1979.” In 1980, Bob was awarded a Ph.D. forhis thesis “Projective representations of knot groups”[7] which was examined by David Epstein from theUniversity of Warwick.

William Thurston

An interesting part of the Riley story is his connectionwith Bill Thurston. In 1982, Thurston produced hisseminal work Three dimensional manifolds, Kleiniangroups and Hyperbolic Geometry [9]. This was themain work which led to Thurston being awardedthe Fields Medal in 1982. Thurston acknowledgeshis debt to Riley. His Theorem 2.3 tells us whichthree-manifolds have a hyperbolic structure. As acorollary he states that if K ⊂ S 3 is a knot, thenS 3 \ K has a geometric structure, if and only if Kis not a satellite knot. It has a hyperbolic structure,if and only if it is not a torus knot. Thurston wrote:“This corollary was conjectured by R. Riley based onhis construction of a number of beautiful exampleswith the aid of a computer. His work gave me a bigimpetus to prove Theorem 2.3.”

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Bob �nally met Thurston in Warwick in 1975. FromBob’s account [5]: “On hearing my name, a tall mansprawled over three chairs sprang up. He said hisname was Bill Thurston, that he wanted to meet me,and that for about a year he had been working on ageneral conjecture which included everything I wasdoing. The shock was immense.”

Binghamton

Now that Bob had �nally got a Ph.D. he could seriouslyapply for a permanent position. In 1980, he receiveda grant to work with Thurston in Boulder, and then in1981 obtained a permanent position at Binghamton,New York. He would often come back to England tovisit. One reason was that he could then go for cyclingtours of Scotland. He would keep his bicycle in mygarage, and cycled from Southampton to Scotland(a distance of at least 430 miles), and then cycle upScottish hills. It was because he had trouble climbingthese hills that he sought medical attention for hisheart. Tragically, following successful heart surgery,he died from complications in Binghamton on 4 March2000. He was 64 years old.

At Binghamton he had become close to Matt Brinand Ross Geoghegan. Matt Brin was an experton the Richard Thompson group which many inSouthampton were also interested in, and so we hadan extended visit from Matt. We found that someof us were close to Bob at di�erent times in hiscareer. We also knew that Bob had written a personalaccount of his discovery of hyperbolic structures sowe prepared this document for publication [3, 5].

We end with some remarks of Ross Geoghegan fromRiley’s obituary: “Riley holds a a unique positionin American mathematics. In the circle of ideas inwhich he was expert he really was an innovator.His were very original ideas that have had a largeimpact on mathematics in the last couple of decades.Personally Riley was a character. He was one of a kind,slightly eccentric, delightful company. There was justone Riley. He was interested in Science, particularlyPhysics, and was an Anglophile who listened to theBBC World Service on his short-wave radio twice aday. He wasn’t English but he loved all things English.”

Bob regarded himself as a 19th centurymathematician with the added advantage that hewas pro�cient in the use of computers. One of hisprized possessions was a letter of rejection from avery reputable British journal saying that they “nolonger publish 19th century mathematics”!

Acknowledgements

I would like to thank John Parker, Caroline Series andDavid Chillingworth for their helpful comments.

FURTHER READING

[1] K. Akioshi, K. Ohshika, J. Parker, M. Sakuma,H. Yoshida, Classi�cation of non-freeKleinian groups generated by two parabolictransformations, arXiv:09564v1.[2] H. Akiyoshi, M. Sakuma, M. Wada, Y. Yamashita,Punctured torus groups and 2-bridge knot groups,Springer Lecture Notes in Mathematics, 2007.[3] M. Brin, G.A. Jones, D. Singerman, Commentaryon Robert Riley’s article “A personal account of thediscovery of hyperbolic structures on some knotcomplements”, Expositiones Math., vol 31 issue 2,2013, 99–103.[4] J. Parker, A complex Riley slice, Lecture from“Geometries, surfaces and representations offundamental groups” conference, Maryland 2016.[5] R. Riley, A personal account of the discovery ofhyperbolic structures on some knot complements,Expositiones Math., vol 31 issue 2, 2013, 104–115.[6] R. Riley, A quadratic parabolic group, Math.Proc.Camb. Phil Soc. 77, 1975, 281–287.[7] R. Riley, Projective representations of linkgroups, Ph.D. thesis, University of Southampton,1980.[8] C. Series, All about the Rileyslice, LMS Presidential Lecture 2019,homepages.warwick.ac.uk/ masbb/PresLecture.pdf.[9] W.P. Thurston, Three dimensional manifolds,Kleinian groups and Hyperbolic Geometry, Bulletinof the AMS, vol 6, number 3, 1982, 357–380.

David Singerman

David is an EmeritusProfessor of Mathe-matics at the Universityof Southampton. Hisoriginal research wasin Fuchsian groups andRiemann surfaces, but

he then moved to the study of dessins d’enfants(maps on Riemann surfaces). His other interests aremusical, with a particular interest in the music ofAnton Bruckner. For exercise he likes to run (slowly).

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FEATURES 29

The International Mathematical Union

On 20 September 1920,during the InternationalCongress of Mathematicians(ICM) in Strasbourg, France,representatives from Belgium,Czechoslovakia, France,Greece, Italy, Japan, Poland,Portugal, Serbia, the United

Kingdom, and the United States signed the statutesfor the International Mathematical Union (IMU),electing C.J. de la Vallée Poussin (Belgium) asPresident and W.H. Young (UK) as Vice President.Thus the IMU came into being. Perhaps surprisingto the modern mind, the initial statutes were to bevalid for a period of just twelve years.

This was in the aftermath of the First World Warand it was the time for the creation of scienti�cunions (the International Union of Biological Sciences(IUBS) and the International Union of Pure andApplied Chemistry (IUPAC) were founded in 1919;the International Union of Pure and Applied Physics(IUPAP) in 1922). Mathematicians tried to heal thewounds from the war and to promote internationalcollaboration in mathematics. The main vehicleto accomplish this was to provide the scienti�cframework for future ICMs being organised under itsauspices1.

C.J. de la Vallée Poussin,the �rst IMU President

The �rst decade of theIMU proved to be achallenging one, andwith political tensionsunresolved, the IMUwas dissolved duringthe ICM in Zürich in1932. Paradoxically, itwas during a periodwithout an existing IMU,namely at the ICM inOslo in 1936, that themost coveted prize inmathematics, the FieldsMedal, was introduced.

After the Second World War attempts were made tore-establish the IMU and in 1952 the modern IMU wascreated as an international non-governmental and

non-pro�t scienti�c organisation. IMU’s objectivesare: to promote international cooperation inmathematics; to support and assist the InternationalCongress of Mathematicians and other internationalscienti�c meetings or conferences; and to encourageand support other international mathematicalactivities considered likely to contribute to thedevelopment of mathematical science in any of itsaspects, pure, applied, or educational.

W.H. Young, the �rst IMUVice President

These objectivesdrive the currentIMU and its activities.Membership hasincreased considerablyand today the IMU hasaround 90 membercountries, which fareswell compared toother internationalunions, but remainsnoticeably behind the193 members2 of the United Nations.

The quadrennial ICMs remain a focal point of the IMU.The practical demands of organising an ICM havebecome enormous and the �nancial commitmentsare huge – and this responsibility remains with thelocal organisers. On the other hand, the IMU isresponsible for the scienti�c content: the StructureCommittee decides the disciplinary sessions (relativesize and content), the Program Committee selectsall speakers, and the various prize committeesdetermine the recipients of all IMU awards. Withincreased and justi�ed attention regarding diversity(geography, gender, mathematical discipline, etc.),and the problems associated with con�icts of interestand unconscious bias, it is no small task to composeall these committees.

In addition to the ICMs, the activity of the IMU isfocused on the �ve commissions and committees:

The Commission for Developing Countries (CDC)runs a plethora of programs for mathematicians indeveloping countries. In addition to funding directlyfrom membership dues, the CDC receives generousdonations from the winners of the BreakthroughPrizes, the Simons Foundation, the Abel Board,

1The �rst ICM had been hosted in Zürich in 1897. To this day, the hosting of the Congresses remains one of the main focuses of the IMU.2Albeit with a di�erent de�nition of ‘member’.

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and individual donations from several mathematicalsocieties, including the LMS (thanks!).

The International Commission on MathematicalInstruction (ICMI) focuses on activities related tomathematics education. Being older than the IMU(established at the ICM in Rome in 1908), it runs itsown quadrennial congresses, the ICMEs, as well asits own General Assembly.

The International Commission on the History ofMathematics (ICHM) is a joint union between theIMU and the Division of History of Science andTechnology of the International Union of Historyand Philosophy of Science and Technology, bringingtogether mathematicians interested in history withhistorians interested in mathematics.

The Committee on Electronic Information andCommunication (CEIC) serves as an advisorycommittee to the IMU regarding a �eld that has been,and still is, in an amazingly rapid transition, namelythat of how we communicate and publish.

The Committee for Women in Mathematics (CWM)is the most recent committee. Having secured asubstantial grant from the International ScienceCouncil (ISC) on the Gender Gap, the committee hasvery quickly become an active and e�ective additionto the IMU’s activities.

A major turning point for the IMU was the decisionof the General Assembly in 2012 to accept thegenerous o�er of Germany to support a permanentIMU Secretariat in Berlin, located in the heart ofthe city and hosted by the Weierstrass Institutefor Applied Analysis and Stochastics (WIAS). Thishas transformed the IMU completely, giving the IMUstability and robustness. The annual support fromGermany exceeds the sum of all membership dues.In addition to providing a safe IMU Archive, the factthat the IMU has a stable o�ce makes it possible todisperse and receive funds worldwide, in a settingwhere the measures against money-laundering arebecoming ever more complicated.

Mathematics, like all science, has developedphenomenally over the course of the last century.

Looking back at the �rst ICMs, completely dominatedby males from Europe and North America, andcomparing them to the audience at the amazingopening ceremony of the 2018 ICM in Rio de Janeiro,attended by people of all creeds and colours fromacross the world, one realises how internationalmathematics has become. Indeed, for the �rst time,the Executive Committee of the IMU has membersfrom all continents. This is a promising developmentthat bodes well for the future as we embark on oursecond century.

Helge HoldenSecretary General of the IMU

References

D.J. Albers, G.L. Alexanderson, and C. Reid. AnIllustrated History 1893-1986, Springer, 1987.G.P. Curbera. Mathematicians of the World, Unite!The International Congress of Mathematicians — AHuman Endeavor, A.K. Peters, 2009. (Taylor & Francis)C.D. Hollings and R. Siegmund-Schultze. Meetingunder the Integral Sign? The Oslo Congress ofMathematicians on the Eve of the Second World War,American Mathematical Society, 2020.C.E. Kenig. The International Mathematical Union(IMU) at 100. Notices of the American MathematicalSociety 67 (3) 404—407 (2020).O. Lehto. Mathematics Without Borders, a History ofthe International Mathematical Union, Springer, 1998.

The references above give a multifaceted viewof the IMU. On the occasion of the centenary,the IMU has solicited Norbert Schappacher towrite a book to appear next year on the IMU’shistory, generously supported by the Klaus TschiraFoundation. Furthermore, the book MathematicalCommunities in the Reconstruction after the Great War(1918–1928): Trajectories and Institutions (L. Mazliak, R.Tazzioli, eds.), containing a chapter by G.P. Curberaon W.H. Young, will appear this autumn.

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Mathematics News Flash

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

The world remains in the grip of Covid-19; nevertheless there is much to celebrate in the world of mathematics.We report on the resolution of several famous conjectures: the Schinzel-Zassenhaus from number theory, thenon-sliceness of Conway’s knot, the analytic description of the di�usion equation representing lattice randomwalks in �nite space having discrete parameters and �nally, a non-linear dynamical model that promises tosave lives at sea.

A Proof of the Schinzel-Zassenhaus Conjectureon Polynomials

AUTHORS: Vesselin DimitrovACCESS: https://arxiv.org/pdf/1912.12545

In the 1960s, Schinzel and Zassenhaus conjecturedthe existence of an absolute constant c > 0 suchthat for any algebraic integer U ≠ 0 which is not aroot of unity, we have

max{ |Ui | | 1 ≤ i ≤ n } > 1 + cn.

Here, the Ui ’s are the conjugates of U. A disproof ofthe above conjecture implies an a�rmative solutionto the Lehmer conjecture (1933) in Number Theory.

Vesselin Dimitrov has proposed a proof of theSchinzel Zassenhaus conjecture. He shows that forany integer polynomial P (x) ∈ ℤ[x] with constantterm 1 and degree n, there is always the followingdichotomy: either P (x) is a product of cyclotomicpolynomials or at least one of the complex rootslies outside the disc |z | ≤ 2

14n . Consequently, the

conjectured absolute constant can be taken to belog 24 .

The Conway Knot Is Not Slice

AUTHORS: Lisa PiccirilloACCESS: https://arxiv.org/abs/1808.02923

Lisa Piccirillo’s much talked-about proof appeared inthe Annals of Mathematics earlier this year. News�ashtakes this opportunity to celebrate her achievementas a graduate student and also to salute the late JohnConway, after whom the mathematical protagonistof her work is named. The world sadly lost Conwayto Covid-19 in April.

In mathematics, a knot is an embedding of atopological circle S 1 into the 3-sphere, consideredup to continuous deformations. A knot is trivial if itbounds a disc in S 3. Fox extended this notion to 4Dthrough concordance: a knot is trivial in concordanceif it bounds a smoothly embedded disc in the 4-ballB4. A knot is slice if it bounds a smoothly embeddeddisc in B4 but topologically slice if it bounds a locally�at disc in B4.

The Conway Knot. Photocredit: Victoria Dixon

John Conway discovereda pair of knots, eachwith 11 crossings, one ofwhich came to be knownas the Conway knot. TheConway knot remained amystery for many years– Freedman showed thatboth of Conway’s knotswere topologically slice.However, all moderntechniques failed to

verify whether the Conway knot was slice or not.Piccirillo’s result established that the Conway knotis not slice. This completed the classi�cation ofslice knots with under 13 crossings and gave the�rst example of a non-slice knot which is bothtopologically slice and the positive mutant of a sliceknot.

Exact Spatiotemporal Dynamics of Con�nedLattice Random Walks in Arbitrary Dimensions

AUTHORS: Luca GiuggioliACCESS: tinyurl.com/ya5qd8oa

A lattice random walk is a stochastic processcomprising a random path traversed on a lattice. Itis commonly referred to as Pólya’s walk when the

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32 FEATURES

steps occur to any of the nearest-neighbour sites.The random movement is modelled via a di�usionequation and in real life applications, the equation isset in �nite space. Finding the analytic descriptionof the space-and-time dynamics of such a con�nedrandom walk with discrete parameters has remainedan open problem for more than a century.

A de�nitive solution to this has now been foundby Bristol mathematician Luca Giuggioli. Using a setof analytic combinatorics identities with Chebyshevpolynomials, Giuggioli developed a hierarchicaldimensionality reduction of the di�usion equationto �nd the exact space and time dependence ofthe occupation probability for con�ned Pólya’s walksin arbitrary dimensions. The �ndings are directlyrelevant to a vast number of applications such asmolecules moving inside a cell, animals foraging forresources in their home ranges, robots searching ina disaster area, and humans passing information ora disease.

Non-linear dynamics to trace castaways at sea

AUTHORS: Serra, M., Sathe, P., Rypina, I. et al.ACCESS: https://doi.org/10.1038/s41467-020-16281-xEvery year, hundreds of people lose their lives insea accidents. The chance of �nding a survivor fallsdrastically after the �rst six hours as rescue teamsbattle ocean currents, coastal tides and unfavourable

weather conditions. An international research teamled by George Haller, Professor of Nonlinear Dynamicsat ETH Zurich has used tools from dynamical systemstheory and ocean data to develop a new algorithmthat predicts where objects and people �oating inwater will drift. Results of the study, which promisesto save lives at sea, have recently been publishedin Nature Communications (Search and rescue atsea aided by hidden �ow structures. Nat Commun11, 2525; 2020.)

Rescue teams already use dynamical models topredict the trajectory of �oating objects butinaccuracies arise from missing data or uncertaintiesin parameters of tidal behaviour, weather forecasting,et cetera. The mathematical methods developed byHaller’s research team can trace special curves usinginstantaneous ocean data. These curves, which theycall TRAPs (Transient Attracting Pro�les) enable moreprecise planning of search routes than is currentlypossible. The new system has been tested in twoexperiments located o� the north-eastern coastof the US. Buoys and mannequins thrown into thecoastal waters near Martha’s Vineyard were foundto gather faithfully along the identi�ed TRAPs.

Dr Aditi KarAditi is Senior Lecturer of PureMathematics in Royal HollowayUniversity. Her research lies inGeometric Group Theory.

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EARLY CAREER RESEARCHER 33

Microtheses provide space for current and recent research students to communicate their �ndings with thecommunity. We welcome/invite submissions – see newsletter.lms.ac.uk for guidance, and award authors withLMS associate membership for one year.

Microthesis: The Erdős Primitive Set Conjecture

JARED DUKER LICHTMAN

A subset A of ℤ>1, the set of integers greater than 1, is primitive if no number in the set divides another.Erdős proved in 1935 that the sum of 1/(n logn) for n running over a primitive set A is universally boundedover all choices of A. In 1988 he asked if this universal bound is attained for the set of prime numbers. In thismicrothesis, I describe some recent progress towards this conjecture.

On a basic level, number theory is the study of theset of integers ℤ. Maturing over the years, the �eldhas moved beyond individual numbers to study setsof integers, viewed as uni�ed objects with specialproperties. A set of integers A ⊂ ℤ>1 is primitive ifno number in A divides another. For example, theintegers in a dyadic interval (x ,2x] form a primitiveset. Similarly the set of primes is primitive, along withthe set ℕk of numbers with exactly k prime factors(with multiplicity), for each k ≥ 1. Another exampleis the set of perfect numbers {6,28,496, ..}, thatis, those equal to the sum of their proper divisors,which has fascinated mathematicians since antiquity.

We de�ne

f (A) :=∑n∈A

1n log n

,

and let p be prime. After Euler’s famous proof ofthe in�nitude of primes, we know

∑p 1/p diverges,

albeit “just barely”, with∑p≤x

1p∼ log log x .

On the other hand, we know∑p

1p log p converges

(again “just barely”) and we may compute f (ℕ1) =∑p

1p log p ≈ 1.6366. In 1935 Erdos generalised this

result considerably, proving f (A) < ∞ uniformly forall primitive sets A. In 1988 he conjectured that themaximum is attained by the primes ℕ1:

Conjecture 1. f (A) ≤ f (ℕ1) for any primitive A.

Since 1993 the best bound has been f (A) < 1.84,due to Erdős and Zhang [3]. Recently, Pomeranceand I [6] improved the bound to the following:

Theorem 1. f (A) < e W ≈ 1.78 for any primitiveA, where W is the Euler-Mascheroni constant. Furtherf (A) < f (ℕ1) + 0.000003 if 2 ∈ A.

Primitive from perfection

In modern notation, a number n is perfectif f(n) = 2n where f(n) = ∑

d |n d is thefull sum-of-divisors function. Similarly n iscalled abundant if f(n)/n > 2, and de�cientif f(n)/n < 2.

Paul Erdős (1913–1996)

Since f(n)/nis multiplicative,one sees thatperfect numbersform a primitiveset, along withthe subset ofnon-de�cientnumbers nwhose divisorsd | n are allde�cient.

It is a classical theorem that non-de�cientnumbers have a well-de�ned, positiveasymptotic density. This was originally provenwith heavy analytic machinery, but Erdosfound an elementary proof by using primitivenon-de�cient numbers (this density is nowknown ≈ 24.76% [4]). His proof led him tointroduce the notion of primitive sets andstudy them for their own sake.

This typi�ed Erdos’ penchant for provingmajor theorems by elementary methods.

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34 EARLY CAREER RESEARCHER

One fruitful approach towards Conjecture 1 is to splitup A according to the smallest prime factor, that is,for each prime q we de�ne

Aq := {n ∈ A : n has smallest prime factor q }.

We say q is Erdős strong if f (Aq ) ≤ f (q ) for allprimitive A. Conjecture 1 would follow if every primeis Erdős strong, since then f (A) = ∑

q f (Aq ) ≤f (ℕ1).

Unfortunately, we don’t know whether q = 2 is Erdősstrong, but we do know now that the �rst 108 oddprimes are all Erdős strong, [6]. And remarkably,assuming the Riemann Hypothesis, over 99.999973%of primes are Erdos strong, [7]!

A conjecture of Banks & Martin

In 1993, Zhang proved f (ℕk ) < f (ℕ1) for each k >

1, which inspired the following by Banks and Martin[1]:

Conjecture 2. f (ℕk ) < f (ℕk−1) for each k > 1.

They further conjectured that, for a set of primes Q,

f(ℕk (Q)

)< f

(ℕk−1 (Q)

), for each k > 1,

where A(Q) denotes the numbers in A composed ofprimes in Q. Banks and Martin managed to prove thisconjecture in the special case of su�ciently “sparse”subsets Q of primes.

This result, along with Conjectures 1 & 2, illustratesthe general view that f (A) re�ects the primefactorisations of n ∈ A in a quite rigid way. Beautifulthough this vision of f may be, it appears realityis more complicated. Recently in [5], I preciselycomputed the sums f (ℕk ) (see Fig. 1), and obtaineda surprising disproof of Conjecture 2!

Theorem 2. f (ℕk ) > f (ℕ6) for each k ≠ 6.

Figure 1. Plot of f (ℕk ) for 1 ≤ k ≤ 10, [5].

I also proved limk→∞ f (ℕk ) = 1, however muchabout this data remains conjectural. For instance,the sequence { f (ℕk )}k ≥6 appears to increasemonotonically (to 1), and the rate of convergenceappears to be exponential O (2−k ), while onlyO (k Y−1/2) is known.I hope this note illustrates Erdős’ conjecture spawningnew lines of inquiry. For example, researchers arenow studying variants of the problem in function�elds Fq [x]. Also, in forthcoming work [2] we manageto prove Conjecture 1 for 2-primitive A, that is, aset where no number in A divides the product of 2others.

The full Erdős primitive set conjecture has remainedelusive, but working towards it has led to interestingdevelopments. In the words of Piet Hein:

“Problems worthy of attackprove their worth by �ghting back.”FURTHER READING

[1] W. Banks, G. Martin, Optimal primitive sets withrestricted primes, Integers 13 (2013), #A69, 10 pp.[2] T.H. Chan, J.D. Lichtman, C. Pomerance, Ageneralization of primitive sets and a conjecture ofErdős, submitted. arXiv:2003.12166[3] P. Erdős, Z. Zhang, Upper bound of∑1/(ai log ai ) for primitive sequences, Proc.

Amer. Math. Soc. 117 (1993), 891–895.[4] M. Kobayashi, On the density of abundantnumbers, PhD thesis, Dartmouth College (2010).[5] J.D. Lichtman, Almost primes and theBanks–Martin conjecture, J. Number Theory 211(2020), 513–529.[6] J.D. Lichtman, C. Pomerance, The Erdősconjecture for primitive sets, Proc. Amer. Math. Soc.Ser. B 6 (2019), 1–14.[7] J.D. Lichtman, G. Martin, C. Pomerance, Primesin prime number races, Proc. Amer. Math. Soc. 147(2019), 3743–3757.

Jared DukerLichtman

Jared is a DPhil studentat the University ofOxford under thesupervision of JamesMaynard. His researchinterests lie in number

theory, especially problems of analytic andcombinatorial �avour. Jared is originally from theUnited States, and enjoys sports of all kinds. Hewas converted to maths from his native math, butmaintains that football ≠ soccer.

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REVIEWS 35

The Universe Speaks in Numbers: How ModernMaths Reveals Nature’s Deepest Secrets

by Graham Farmelo, Faber & Faber, 2019,£20, US$30, ISBN: 978-0571321803

Review by Noel-Ann Bradshaw

Graham Farmelo hasopened my eyes to theworld of theoretical physicsand, more importantly,its links with puremathematics. As a teenagerI loved maths, music andart. I was interested inchemistry and history, butphysics left me cold; partlybecause I had enduredthe same teacher for �veyears and she had failed to

inspire me. Since embarking on my academic career, Ihave become somewhat embarrassed by my aversionto all things physics-related, and consequently I sawthe opportunity to review this book as a way ofaddressing this.

In true academic style Farmelo begins each chapterwith a summary of what will be covered, but theneach chapter unfolds into a beautiful historicalaccount of the development of ideas and therelationships between those who created them. Itreminds me of tracing the genealogy of intertwinedfamily trees, showing where they overlap, cometogether and then separate for a time before comingback to create powerful new dynasties.

This book clearly depicts the development ofmathematical physics, starting with a brief overviewof the very early history of the subject. As anenthusiastic mathematical historian I enjoyed readingabout Aristotle’s rejection of Plato’s view, thatmathematics was fundamental to understandingscience and the world. And also then about theimpact of mathematical giants such as Euclid, Kepler,Galileo and Newton, who paved the way for the likesof Laplace and Maxwell (said by one of his teachers to

practice maths with “exceeding uncouthness”). Thisjourney and progression of ideas is important formathematicians and physicists to understand and,in my opinion, should be taught in schools where thesubjects are still presented very separately.

Farmelo’s style of writing particularly brings alivethe work of Maxwell, demonstrating how he linkedelectromagnetism and optics with his fascination fortopology and knots. It is clear that he believed inthe importance of mathematics for understandingthe universe, whereas his friend and collaborator,Thomson, is described as seeing mathematics as aservant of physics rather than a guide.

Next on the scene is Einstein, who is reportedas realising that advanced mathematics was aphysicist’s most valuable tool. There follows adelightful account of the relationship between Hilbertand Einstein as they race to complete the theory ofgravity, coming at it with very di�erent backgrounds.For me the book really starts to come alive with theentry of Dirac and his vision for the future of thesedistinct, but linked, disciplines.

Dirac proposed a new theory: the beauty ofmathematics. As someone who has recently beentold by a teacher not to mention the b-word intalks to school children because apparently thisturns them o� the subject, I was delighted toread that he mentioned the beauty of mathematicsseventeen times in his talk on the relationshipbetween mathematics and physics at the RoyalSociety of Edinburgh in 1939 – a talk which iscrucial to Farmelo’s book. Here he urged theoreticalphysicists to learn a lot of advanced mathematics,concluding that “big domains of pure mathematicswill have to be brought in to deal with the advancein fundamental physics.”

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The history and theories discussed by Farmelodevelop swiftly with much of the bookunderstandably focussed on more recentdevelopments, some of which are less easyfor a non-physicist to understand. However,understanding the detailed theory is not essentialas it is the accounts of the relationships andcollaborations between the likes of Dyson andFeynman, Penrose and Hawking, Weyl and Wignerthat are fascinating and thought-provoking. Idefy any pure mathematician not to be movedas Farmelo’s account becomes personal. Hedescribes his conversations with Atiyah, Dyson,Langlands, Uhlenbeck and other modern daygreats, demonstrating not only the link betweenmathematics and physics but also the desperatenecessity for the two disciplines to collaborate andwork together. As I am personally aware that themathematical modeller needs ideas and problemsfrom industry in order to perfect his/her craft, Iappreciated seeing this echoed by Uhlenbeck, whois quoted as saying how “Research mathematiciansneed physicists’ ideas.”

I believe this is an important book that shouldbe read by both mathematicians and physicists. Itchallenges, but yet is sympathetic, to the di�erenthistories, backgrounds and indeed prejudices of thetwo disciplines. Farmelo sometimes presents his ownopinion but much more frequently uses the words,

actions and works of others to put his point across.What makes his call for intimate collaboration morepowerful is the acknowledgement that he was not ofthis opinion when he started out in his career. Overtime, his experience has shown him the importanceof working together to further developments inareas such as String Theory, Supersymmetry andthe discovery of the Higgs particle.

In my opinion this book should be on the readinglist for every mathematics and physics A-levelstudent and every new undergraduate of both thesedisciplines. Wherever their interests in these subjectscurrently lie, they should be made aware of theoverlap of mathematics and physics, the power ofcooperation and where the sharing of ideas can lead.

Noel-Ann Bradshaw

Noel-Ann Bradshaw isHead of Computing andDigital Media at LondonMetropolitan University.She recently surviveda brief foray into theworld of Data Science at

Sainsbury’s Argos in 2018. In her spare time sheenjoys the company of her two cats, skiing, makingcocktails, sunbathing, learning Italian and takingholidays that combine as many of these as possible.

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REVIEWS 37

Di�erence Equations for Scientists and Engineeringby Michael A. Radin, World Scienti�c, 2019, £50, US$58, ISBN: 978-9811203855

Review by Mark McCartney

Di�erence equationshave always beenvery much the poorrelations of di�erentialequations. They are lesslikely to be includedin the undergraduatecurriculum at any level ofdepth, and while there isa constant �ow of textson di�erential equations,there is merely a drip

of texts to be found on their �nite di�erencecousins. (And yes, dear reader, that was a somewhatweak attempt at a mathematical joke.) This lack ofcoverage is a pity, because di�erence equationsare not only interesting in their own right, but arealso pedagogically helpful in, amongst other things,building basic skills in pattern spotting; developingalgebraic con�dence; and highlighting the fact thatmathematical modelling in discrete time and/orspace can be very powerful techniques.

In Difference Equations for Scientists andEngineering Michael A. Radin provides astudent-centred introduction to the subject. Hestates in the introduction that the aim of thebook is to provide ‘repetitive type examples toenhance the understanding of the fundamentalsof difference equations and their applications’(p. 3). This repetition of examples is most clearlyfound, as might be expected, in the exercisesections at the end of each chapter. In totalwithin the confines of just six chapters there areover 400 questions (with solutions for all theodd numbered ones at the end of the book). Thequestions test students’ engagement with thematerial and provide good consolidation of thekey ideas in each chapter.

But the chapters themselves also have arepetition of examples, with the author graduallyleading the student from the concrete to theabstract. Thus, for example in Chapter 2, on firstorder linear difference equations, we start withthe concrete xn+1 = 4xn , gradually building by

example all the way up to xn+1 = anxn + bn . Forthe coefficients an and bn solutions arising fromperiod 2 and 3 sequences are studied beforemoving on to general odd and even periods 2k +1 and 2k . At first glance this may seem muchtoo slow, after all the general solution of xn+1 =anxn + bn can be easily enough found, but theauthor’s methodical technique means that as thestudent reads through the chapter she gains aclear grasp of the possible behaviours of thisgeneral solution. It is a style which is repeatedthroughout the book as it moves on to chapters onfirst order nonlinear difference equations, secondorder linear and nonlinear difference equations,and finally a chapter introducing higher order andcoupled systems.

As might be expected from the title, pepperedthrough the book are examples of differenceequations which appear in applied mathematics(e.g. the Beverton-Holt model, the logistic andRicker maps and the Rulkov model) but theemphasis of the book is firmly on the methodsrather than the models. Although chaos gets amention, it is only a brief one. Indeed it is toobrief to be helpful to the reader. For example, theclassic Feigenbaum diagram for the logistic mapis presented but not explained, apart from statingthat it ‘evokes chaos as period doubling’ (p. 118).

I have to admit that the book has some quirksof style which jarred with me, but the only oneI feel was substantive enough to allow me tomention it in a review is that axes on graphs werehabitually not labelled. This is an omission, whichas any student taught by me knows, I considerto be punishable by death. (Unfortunately myemployer feels that, aside from the inevitablehuman rights issues involved, such a response isan overreaction.)

However, my personal bugbears aside, I thinkRadlin has done a nice job in producing a textbookwhich provides a learner friendly introductionto difference equations. It would suit as a coretext for a first year course in the topic, aimed,

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38 REVIEWS

as the title suggests, at physical science orengineering undergraduates. The student who isprepared to work through the book will get a goodgrounding in basic techniques and gain a feel forthe possible behaviours of standard equations.He will also be given some indication of theusefulness and potential complexity of discretesystems in modern science and engineering. It isthus a pity that the hefty £50 price tag is likelyto put him off the idea of purchasing a copy.

Mark McCartney

Mark McCartney lecturesin mathematics at UlsterUniversity. His wife andchildren suspect thatduring the COVID-19lockdown he wasresponsible for the

regular disappearance of substantial quantities ofchocolate biscuits from the kitchen cupboard. Hecategorically denies this.

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REVIEWS 39

Figuring Fibers

edited by Carolyn Yackel and Sarah-Marie Belcastro,American Mathematical Society, 2018, US$40.00, ISBN: 978-1-4704-2931-7

Review by Julia Collins

It is always wonderful tosee another book publishedthat celebrates and exploresthe connections betweenmathematics and �bre arts.I am of the belief that itis impossible to do craft(of any kind) without animplicit understanding of

mathematics, and that mathematics can beunderstood more deeply when explored throughcraft. Figuring Fibers presents eight chapters thatuncover more of these connections. Some startwith a craft project and investigate the mathematicsarising from it; others start with a mathematical ideaand use this to generate a craft project. In the bestchapters, craft and mathematics develop together.

Figuring Fibers is the third such book edited bySarah-Marie Belcastro and Carolyn Yackel, followingin the successful footsteps of Making Mathematicswith Needlework and Crafting by Concepts. Althoughit is beautifully presented, with colourful pictures andphotographs, this book is not for the casual reader.Each of the chapters arose from presentationsgiven at the American Mathematical Society’s JointMathematics Meetings: they are therefore aimed atan audience with a serious mathematical background.At the same time, the crafting aspect of eachchapter assumes that the reader has a workingfamiliarity with �bre arts: knitting, crochet andquilting. Beginners in either �bre arts or mathematicswill likely �nd this book to be too technical forcomfort.

The book begins with two di�erent overviews ofthe chapters: one aimed at mathematicians and oneaimed at crafters. This is an excellent idea and helpsthe reader get into both mindsets needed for thebook. It also allows the audience to quickly �nd theprojects that will interest them most, whether itis picking an area of maths they appreciate or atype of craft they enjoy. The mathematical ideasrange over: fractals and space-�lling curves, graph

theory, topology, linear congruences (solving systemsof equations in modular arithmetic), knot theory,polyhedra and tessellations, and hyperbolic geometry.Craft-wise, three of the chapters use needleworkand quilting techniques, three use knitting and twouse crochet. (British readers should be wary thatthe crochet chapters use US terminology, so ‘singlecrochet’ (US) means ‘double crochet’ (UK).)

My favourite project is Chapter 2 by Kyle Calderhead,‘Gosper-like fractals and intermeshed crochet’. Thisis the chapter in which the mathematics and the�bre-art develop most naturally side by side, withoutone or the other dominating. Inspired by the Hilbertcurve, a space-�lling curve generated iteratively ina fractal way, the author develops a new fractalspace-�lling curve to �t within a hexagonal gridinstead of a square one. He proves that there is onlyone such feasible tiling before going on to design anew hexagonal-intermeshing crochet technique toimplement the pattern. Both the mathematics andthe crochet are carefully and e�ectively explained,without using overly technical language yet withoutleaving out any details. As the author explains, theproject is a great example of how “discoveries inboth mathematics and �bre arts are made that mightnever have happened otherwise”.

Chapter 4, by Mary D. Shepherd, is inspired by aquilting pattern called Snake in a Hollow Maze andis another great example of craft and mathematicsmutually inspiring one another. A Truchet tileis a square tile whose motif consists of twoquarter-circles joining midpoints of adjacent sides.When these tiles are arranged into a grid, thequarter-circles join up to create a winding ‘snake trail’around the grid. A Snake in a Hollow Maze patternresults when the snake trail is a single connectedpath starting and ending on the edges of the quilt.Historically the instructions on how to create thiswere a carefully guarded secret, passed down fromquiltmaker to quiltmaker. But no longer! This chapterprovides an algorithm for generating such a patternfrom any random initial con�guration of the tiles. The

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40 REVIEWS

�nal result is some novel mathematics that is easyto follow, and which inspires more questions thatare likely to keep both mathematicians and quiltersthinking for a while. Readers who are not quilters(like me) can also explore the ideas using printedtiles.

In Chapter 6, Sarah-Marie Belcastro wishes to knittorus knots (those that can be drawn on the surfaceof a torus) using a particular con�guration of circularknitting needles. The constraints generated by thistechnique necessitate a mathematical investigationto �nd out which knots can be knitted in this way.While the mathematics is technical, it is not verydeep, and therefore even a relative newcomer toknots and braids should be able to follow it. Oncethis problem is resolved, the chapter concludes withseveral methods for knitting torus knots, creatingboth the beautiful image shown on the front coveras well as some more practical bracelets and cowls.

Chapter 7, by S. Louise Gould, will appeal to thosereaders intrigued by tessellations and polyhedra. Themathematics in this chapter is not new, but it iswell-explained and provides an excellent introductionto the idea of ‘triply-periodic polyhedra’. First, thereader is introduced to regular and semiregulartessellations, Platonic and Archimedean solids. Justas these tessellations and solids are each composedof gluing together regular polygons with the samecon�guration at each vertex, so it is possible to createin�nite polyhedra with the same constraints but withnegative curvature at each vertex. That is, the sumof the angles at each vertex is greater than 360◦. The�nal geometric object being crafted in this chapteris a model made of regular pentagons, arranged

�ve around each vertex, which has translationsymmetries in three independent directions. Detailedinstructions are provided for how to construct sucha model from cloth, but those readers not adeptwith a sewing machine will appreciate the additionalinstructions for a model made from card.

The remaining chapters, while having interestingstarting points in terms of either the mathematicsor the craft, did not, to me, suitably balance the twoaspects. Some chapters were too mathematicallytechnical with the craft appearing as an afterthought,while in others the mathematics was too trivial to beof interest.

Overall this is a beautiful, intriguing and interestingbook with plenty of ideas to explore and create, solong as the reader approaches it with patience andcuriosity. I hope it will inspire more mathematiciansto express their ideas in craft, and inspire craftersto investigate the mathematics in their projects.

Julia Collins

Julia Collins works asa Lecturer at EdithCowan University in Perth(Australia) with scholarlyinterests in mathematicseducation, outreach andknot theory. Her latest

popular maths book, Numbers in Minutes, has justbeen published. When not teaching or writing, Juliawill be found knitting mathematical objects or hikingin the Australian bush, accompanied by her trustysheep Haggis.

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OBITUARIES 41

Obituaries of Members

John Horton Conway: 1937 – 2020John Horton Conway, whowas elected an HonoraryMember of the LondonMathematical Society in2015, died on 11 April2020, aged 82. He wasawarded the LMS BerwickPrize in 1971 and the �rstLMS Pólya Prize in 1987.

Rob Curtis writes: John was one of the most celebratedBritish mathematicians of the latter half of the 20thcentury. It would be di�cult to think of anyone whohas made such substantial contributions to so manybranches of Mathematics as he did.

Conway matriculated at Gonville and Caius College,Cambridge in 1956. As an undergraduate he wasalready undertaking mathematical research, oftenin collaboration with Mike Guy, for instance �ndingall solutions to Piet Hein’s SOMA cube. They drew agraph whose vertices were the 240 distinct solutions,joined if one could be obtained from the otherby withdrawing two pieces, twisting and replacingthem. The resulting SOMAP had a single connectedcomponent and an isolated vertex. He proceeded toa PhD under the number theorist Harold Davenportand set about solving the Waring problem expressingevery integer as a sum of �fth powers. Conwayproved the result using a combination of analytictechniques for large integers together with the adhoc methods for which he would become famousfor the small integers. But by this time his interesthad moved on to logic and in�nite numbers, leadingto a thesis on trans�nite numbers.

In 1964 Conway became a University Lecturer andFellow at Sidney Sussex College, Cambridge. I wellremember him striding across the College lawns inthe snow, long hair and gown �owing out behindhim, followed by his wife and a retinue of youngdaughters. It was during his time at Sidney that JohnMcKay introduced Conway to John Leech’s recentlydiscovered 24-dimensional lattice, and suggestedthat it might have an interesting group of symmetries.Indeed it did, and in 1968 in an epic piece of workConway constructed this new simple group andfound that it contained two further new simplegroups as subgroups. The paper announcing this

discovery was a highlight of Volume 1 of the Bulletinof the LMS. This dramatic event motivated Conwayto produce a reference book devoted to the �nitesimple groups and some years later, with co-authorsRobert Curtis, Simon Norton, Richard Parker andRobert Wilson, the Atlas of Finite Groups appeared.It has since become a standard reference book forany mathematician whose work involves symmetriesof �nite con�gurations.

The Leech lattice had been discovered in connectionwith sphere-packing problems. For instance, whatproportion of n-dimensional space can be occupiedby non-intersecting n-dimensional spheres of thesame size? This took Conway into the world of codingtheory and sphere-packing and eventually to a longcollaboration with Neil Sloane which resulted in themonumental Sphere-Packing, Lattices and Groups, anessential companion for a sojourn on a desert island.

By this time Conway had acquired internationalfame. He had moved back to Caius College in 1970having resigned from Sidney when he consideredthat behaviour in connection with the election ofthe new master was underhand, and in 1971–2 hetook up a sabbatical at the California Institute ofTechnology. During his time there he collaboratedwith David Wales in winning the race to constructthe Rudvalis group. He also gave talks aroundthe States on The Game of Life which had beeninvented in the common room at Cambridge whereConway was a �xture, and which had acquireda cult following, occupying millions of hours ofcomputer time around the world. He used to worrythat he would be best remembered for Life butthat apparently frivolous pastime has now gainedscienti�c respectability as an example of a universalcellular automaton. When it appeared in MartinGardner’s column on recreational mathematics inScienti�c American, to which he had contributed onmany occasions, Conway o�ered a prize of $50 toanyone who could produce a con�guration whichcould be shown to grow inde�nitely. This was won byBill Gosper of MIT who produced a ‘glider gun’ and,happily for Conway as it was a substantial sum ofmoney at that time, Scienti�c American paid the bet!

Conway’s other great passion was mathematicalgames and in 1982 he produced, together withElwyn Berlekamp and Richard Guy, Mike’s father,the wonderful Winning Ways for your MathematicalPlays. Some of the games in this book, such as thewell-known ‘dots and boxes’, had been around formany years, but many more, such as ‘sprouts’ and‘hackenbush’ were invented by Conway himself. It

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is indeed ironic that the three authors of this veryspecial book have all died within a few months ofone another. The analysis of these games involvesa great deal of serious mathematics and it is noexaggeration to say that for Conway, ‘Mathematicsis a Game and Games are Mathematics’.

However, the discovery for which Conway wouldmost like to be remembered is that of what DonaldKnuth christened surreal numbers, which combinedthe approaches of Dedekind and Cantor to producea rich system of �nite, in�nite and in�nitesimalnumbers with many remarkable properties. It isa striking fact that in 1968–69, the year Conwayreferred to as his annus mirabilis, he had discoveredthe three �nite simple groups named after him,The Game of Life and the surreal numbers. Conwaycontinued to be immensely productive and original.Again John McKay came on the scene, pointing outthat the smallest degree of an irreducible complexrepresentation of the Monster group is just one lessthan a coe�cient in the Fourier expansion of themodular function J. This innocent observation led tothe Conway–Norton theory of Monstrous Moonshineand to the Monstrous Moonshine conjecture for theproof of which Conway’s student Richard Borcherdswas awarded the Fields Medal.

By the late 1970s Conway’s personal life had becomechaotic and both his marriages to Eileen, with whomhe had four daughters (Susie, Rosie, Ellie and Annie)and Larissa Queen, with whom he had two sons(Alexander and Oliver) ended in di�cult divorces.It does seem that he was so utterly committed tomathematics that he could not a�ord to let humanrelationships interfere with his research. Some ofthose closest to him undoubtedly paid a price forhis brilliant work. He had moved to Princeton withLarissa in 1986 to take up the von Neumann chairand, following this break-up, he married his thirdwife Diana, with whom he has a son Gareth. But theinventive brilliance continued, not least in the FreeWill Theorem, which he proved together with SimonKochen in 2004.

Working closely with Conway was a joy and aninspiration. When dealing with some new structureor concept he would hone and hone the notationuntil he felt it conveyed all the information requiredof it as concisely as possible. I often came into workin the morning to �nd that the language with whichI had become familiar had been ditched overnightto be replaced by a more elegant version. Despitehis being rather sni�y about combinatorics for itsown sake, feeling that every mathematician should

possess the required skills, he was a consummatemaster of the art as is evident in so much of hiswork. His lectures, both to undergraduates and toresearch seminar audiences, were refreshing andspontaneous. In the early days he invariably lecturedwithout notes, o� the top of his head, spendingjust a few moments in contemplation before going‘on stage’, although I understand that in later yearshe prepared his lectures meticulously. Indeed, atone time he had a cult following among Cambridgemathematics undergraduates who founded theConway Appreciation Society.

Such is his international fame that he already has aprize-winning biography written about him: Geniusat Play, the curious mind of John Horton Conway bySiobhan Roberts.

Conway received many honours during hisdistinguished career. Apart from his LMS prizes, hewas awarded the Nemmers Prize in Mathematics in1998 and the Leroy P. Steele Prize for MathematicalExposition in 2000. He was elected a Fellow of theRoyal Society in 1981.

John Conway was my mentor, inspiration,co-researcher and friend. I shall miss himenormously.

Freeman J. Dyson: 1923 – 2020Professor Freeman J.Dyson, who was electeda member of the LondonMathematical Society on17 March 1943, died on28 February 2020, aged96. He was elected LMSHonorary Membershipin 2000.

Michael Th. Rassias writes: Freeman J. Dyson was oneof the world’s most famous and vocal scientists. Wewere all saddened by his passing, as – even at his96 years – he seemed unstoppable, with his mostrecent book having been published in 2018. Honoredand humbled to be surrounded by such pillars ofscience as a visiting researcher at the Program inInterdisciplinary Studies of the Institute for AdvancedStudy, Princeton, over the last years, I had the greatprivilege of meeting Dyson around 2015. Since then,I had the opportunity to spend some time with him,hoping to absorb some of his wisdom. Inspired by hisaccomplishments, I was always being carried away by

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OBITUARIES 43

his beautiful narrations of the numerous interestingevents of his life.

Dyson, born at Crowthorne in Berkshire, England, wasan American theoretical physicist and mathematicianwhose academic stature had reached that of ahistorical �gure of science, long before his passing.

At the age of 17, in 1941, he arrived at Trinity College,Cambridge, as an undergraduate at a period whenHardy, Littlewood, Besicovitch and other alreadyfamous professors were there, with whom he becamepersonal friends: “Especially with Besicovitch, whowas the owner of the billiard table”, he smilinglypointed out raising his �nger. In 1945 he obtained hisBA in Mathematics from Cambridge University, forthe period 1945–46 he was an Instructor at ImperialCollege and in 1947 he went to Cornell Universityas a graduate student, where he worked with HansBethe and Richard Feynman. Subsequently, for theperiod 1948–49 he was a Member at the Institutefor Advanced Study, Princeton, and for the period1949–51 he was a Research Fellow at the Universityof Birmingham. He then became Professor at CornellUniversity where he remained until 1953. Surprisingly,he was made Professor at Cornell notwithstandingthe fact that he did not have and never actuallyobtained a PhD. Throughout his career he was aharsh critic of the PhD system, which he stronglybelieved should be abolished. In a discussion wehad, he said that he considered himself “lucky tohave been educated in England at a time when thePhD was not required as an entrance ticket to anacademic career”. He was very much bothered by thefact that the current rigid and lengthy PhD system isone of the main reasons why talented women dropout of academic careers.

In 1953, Dyson became a permanent Professor atthe Institute for Advanced Study, Princeton, wherehe remained throughout the rest of his career.

Dyson has made numerous, profound and versatilecontributions in a broad spectrum of subjects ofMathematics and Physics. Among his most importantcontributions is the uni�cation of the three versionsof quantum electrodynamics invented by RichardFeynman, Julian Schwinger and Shin’ichiro Tomonaga,all three of whom were awarded the Nobel Prize inPhysics in 1965. His work and lectures on Feynman’stheories played a decisive role in making themunderstandable to physicists of the time and thisvery much helped Feynman’s work being acceptedby the academic community. Dyson’s work on thissubject impressed J. Robert Oppenheimer – who

was at the time the Director of the Institute forAdvanced Study, Princeton – and had an impacton him being o�ered a permanent position there.Curiously, despite his stellar accomplishments, Dysonwas never awarded the Nobel Prize. He somehowmissed his chance. On this subject, he humorouslysaid with his playful character that it is better forpeople to ask you why you did not get a Nobel prize,rather than why you actually did.

In 1958, at the age of 35, he was a member ofthe design team under Edward Teller for a smalland really safe nuclear reactor called TRIGA usedthroughout the world in hospitals and universities forthe production of medical isotopes. Some of thesereactors are still in use, sixty years later.

Teller, Feynman, Hardy, Littlewood, Besicovitch, Gödel,and many other legendary names were just a fewof the people entangled with Dyson’s spectacularacademic life. I must admit that I often caught myselfbeing mesmerized by the surreality of discussingwith a living piece of history when I immersed myselfin one of the two opposite armchairs in his o�ce atthe Institute for Advanced Study.

Throughout his career, Dyson had the characteristicpassion to delve into the exploration of problemsthrough which Mathematics can be usefully applied.His span of scienti�c interests and his everlastingappetite for research quests and the pursuit of thetruth, had lead him to investigate problems not onlyin Mathematics, Physics and their interconnections,but also to other fascinating subjects, such asAstrobiology.

During his career, he had been bestowed with aplethora of awards and distinctions. However, noaward bestowment provided him with greater joythan that of unraveling the mystery and beauty ofNature.

Jan Saxl: 1948 – 2020Jan Saxl, who waselected a member of theLondon MathematicalSociety on 19 March 1976,died on 2 May 2020,aged 71.

Martin Liebeck writes:Jan Saxl was bornand grew up in Brno,

Czechoslovakia. He began his university studiesin 1966 at Masaryk University in Brno, but his life

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44 OBITUARIES

changed dramatically in the summer of 1968, whenhe was away on holiday in the UK. On the trainback home, he learned of the Soviet invasion ofhis country. He got o� the train in Frankfurt anddecided not to return, instead making his way toBristol, where he had relatives. Jan continued hisstudies at Bristol University, graduating in 1970. Hewas not able to return to his native country until1988.

Jan then moved to Oxford to do a DPhil under thesupervision of Peter Neumann, �nishing in 1973 with athesis entitled Multiply Transitive Permutation Groups.Oxford was an exciting place to do research in algebraat that time, and Jan’s student contemporariesincluded Cheryl Praeger, Peter Cameron, GarethJones, Rosemary Bailey, Donald Taylor, Steve Smith,Derek Holt and Jonathan Hall, all of whom wenton to distinguished academic careers. After theDPhil, Jan spent a year at the University of Illinoisat Chicago Circle, and was then awarded ResearchFellowships, �rst at Hertford College Oxford, andthen Downing College Cambridge. He was appointedas a lecturer at Glasgow University in 1978, butreturned to Cambridge in 1979, where he spentthe remainder of his career, eventually retiringin 2015 as Professor of Algebra and Fellow ofGonville and Caius College. Jan married Ruth Williams,another Cambridge mathematician, in 1979, and theirdaughter Miriam was born in 1980.

Jan was a leading �gure in algebra for over 40 years,publishing around 100 papers and books on a widevariety of topics: permutation groups, �nite simplegroups, maximal subgroups, representation theory,probabilistic group theory, algebraic groups, algebraiccombinatorics, and applications to other areas suchas number theory, Galois theory and model theory.He was tremendously collaborative in his research,publishing with 55 di�erent co-authors, and holdingvisiting appointments at Chicago, Perth, Rutgers,Princeton, Jerusalem and Caltech. Some of thesecollaborations were very long-lasting, particularlythose with Cheryl Praeger, Bob Guralnick and me.

Let me describe brie�y just one of the many themesof Jan’s research. Starting in the 1980s, many grouptheorists worked on developing the theory of thesubgroup structure of the �nite simple groups. Jan

was at the forefront of this work, and one of theo�shoots was his classi�cation (together with CherylPraeger and me) of the maximal factorizations ofthe simple groups – that is, expressions G = ABof a simple group G as a product of two maximalsubgroupsA and B . These factorizations have provedto be fundamental to many applications. For example,together with Bob Guralnick and others, Jan usedthem in an ingenious way to solve problems goingback to Dickson, Schur and Carlitz in the theory ofexceptional polynomials over �nite �elds – these arepolynomials which induce permutations on in�nitelymany �nite extensions of the �eld of coe�cients;later on, they extended their theory to rationalfunctions over number �elds, and found a beautifulconnection with elliptic curves.

Jan served as editorial adviser for the LMS journalsfrom 1998–2001; he was also an editor for the Journalof Algebra and the Mathematical Proceedings of theCambridge Philosophical Society for many years. Heorganised two memorable LMS Durham Symposiain 1990 and 2001, and many other high pro�leconferences. He supervised ten Cambridge PhDs, andwas an inspiring and dedicated teacher of hundredsof undergraduates.

Orienteering and skiing were two of Jan’s favouritepastimes, and he also had a deep love of classicalmusic, particularly opera. He combined warmth andgenerosity with an irresistible self-deprecating wit,and was wonderful company. He is deeply missed byhis many friends, colleagues and students.

Jan is survived by his wife Ruth, daughter Miriam, andgranddaughters Maya and Eva.

Death NoticesWe regret to announce the following deaths:

• Professor Peter J. Bushell of University of Sussex,who died on 26 May 2020.

• Professor Mark H.A. Davis of Imperial CollegeLondon, who died on 18 March 2020.

• Dr Kirill C.H. Mackenzie of University of She�eld,who died on 2 May 2020.

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EVENTS 45

Covid-19: Owing to the coronavirus pandemic, many events may be cancelled, postponed or movedonline. Members are advised to check event details with organisers.

LMS Meeting

LMS Northern Regional MeetingUniversity of Manchester, 7–11 September 2020

Website: tinyurl.com/yamy8uvq

This conference is held in celebration of the 60thbirthday of Bill Crawley-Boevey. The representationtheory of quivers has developed through a stronginteraction of general theory and the investigationof examples. Crawley-Boevey has had a majorin�uence on the �eld, his work exemplifying thisinteraction as well as the entwining of �nite- and

in�nite-dimensional representation theory. Quiversand their representations appear in many parts ofmathematics, in physics and in an increasing varietyof applications, and this range will be represented bythe speakers.

Supported by the LMS, EPSRC, the Clay MathematicsInstitute, the Heilbronn Institute and the Alexandervon Humboldt Foundation.

LMS Meeting

LMS Prospects in Mathematics (online)10–11 September 2020, hosted by the University of Bath

Website: tinyurl.com/yaqtyzmm

All �nalist mathematics undergraduates who areconsidering applying for a mathematics PhD in 2021are invited to attend this meeting, which will featurespeakers from a wide range of mathematical �eldsacross the UK who will discuss their current research

and what opportunities are available to you. Toregister, send an email headed ‘Prospects 2020Application’ to [email protected] with thestatement “I am on track academically to begin PhDstudies in 2021” and evidence of your predicteddegree classi�cation. The registration deadline is 15August 2020.

RSC Student Conference

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

The Research Students’ Conference is returningto the University of Nottingham for a fourth timesince it began in 1980. This conference is for PhDstudents based in Probability and Statistics �elds,organised by PhD students every year in the UK. Ifyou’re a student and interested in speaking at theconference, complete the validation questionnairewhen booking your place. Supported by an LMS EarlyCareer Research grant.

IMA Induction Course for New Lecturersin the Mathematical Sciences 2020

Location: Isaac Newton Institute, CambridgeDate: 16–17 September 2020Website: tinyurl.com/t5ryxgg

This course, designed by members of themathematics community, is suitable for anyonenew to or with limited experience of teachingmathematics/statistics in UK higher education.Session leaders will have signi�cant experience ofteaching in the mathematical sciences, and delegateswill have the opportunity to discuss their own ideas,challenges and experiences.

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Covid-19: Owing to the coronavirus pandemic, many events may be cancelled, postponed or movedonline. Members are advised to check event details with organisers.

Society Meetings and Events

September 2020

7-11 Northern Regional Meeting, Conferencein Celebration of the 60th Birthdayof Bill Crawley-Boevey, University ofManchester

10-11 Prospects in Mathematics Meeting,University of Bath

October 2020

29 EMS/EdMS/LMS Meeting, ICMS, Edinburgh

November 2020

19 Computer Science Colloquium, London19 Society Meeting and AGM, London

December 20205 Midlands Regional Meeting, Lincoln

Calendar of EventsThis calendar lists Society meetings and other mathematical events. Further information may be obtainedfrom the appropriate LMS Newsletter whose number is given in brackets. A fuller list is given on the Society’swebsite (www.lms.ac.uk/content/calendar). Please send updates and corrections to [email protected].

July 2020

21-24 RSC Student Conference, University ofNottingham (489)

24-30 27th International MathematicsCompetition for University Students,Blagoevgrad, Bulgaria (487)

27-31 Integrable Probability Summer School,online event (488)

September 2020

10-11 Heilbronn Annual Conference 2020,online event (488)

16-17 IMA Induction Course for New Lecturersin the Mathematical Sciences, IsaacNewton Institute, Cambridge (489)

July 2021

7-9 22nd Galway Topology Colloquium,University of Portsmouth (488)

12-16 New Challenges in Operator Semigroup,St John’s College, Oxford

12-19 14th International Congress onMathematical Education Shanghai, China

14-16 IMA Modelling in Industrial Maintenanceand Reliability Conference, Nottingham(486)

20-26 8th European Congress of Mathematics,Slovenia

August 2021

16-20 IWOTA, Lancaster University (481)

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September 2021

19-24 8th Heidelberg Laureate Forum,Heidelberg, Germany

21-23 Conference in Honour of Sir MichaelAtiyah, Isaac Newton Institute,Cambridge (487)

July 2022

24-26 7th IMA Conference on Numerical LinearAlgebra and Optimization, Birmingham(487)

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