NEWSLETTER - London Mathematical Society

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 1 — #1 ii

ii

ii

NEWSLETTER Issue: 493 - March 2021

MATHEMATICS OFFLOATING-POINTARITHMETIC

RANDOMLATTICESIN THE WILD

MARRIAGES,COUPLES,MATHS CAREERS

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 2 — #2 ii

ii

ii

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)Robb McDonald (University College London)Adam Johansen (University of Warwick)Susan Oakes (London Mathematical Society)Andrew Wade (Durham University)Mike Whittaker (University of Glasgow)Andrew Wilson (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

One step in a Monte-Carlo simulation usingVisualyse Professional software. See page 24for further details.

Do you have an image of mathematical interestthat may be included on the front cover ofa future issue? Email [email protected] fordetails.

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.

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 3 — #3 ii

ii

ii

CONTENTS

NEWS The latest from the LMS and elsewhere 4

LMS BUSINESS Reports from the LMS 11

FEATURES Joining the De Morgan House Team — OneYear On 22

Cover Image: Monte Carlo Simulation 24

Notes of a Numerical Analyst 25

Mathematics News Flash 26

Penrose’s Incompleteness Theorem 27

The Mathematics of Floating-PointArithmetic 35

Random Lattices in the Wild: from Pólya’sOrchard to Quantum Oscillators 42

Marriages, Couples, and the Making ofMathematical Careers 50

EARLY CAREER The Mathematician’s Academic Journey 55

Microthesis: A Novel Algorithm for SolvingFredholm Integral Equations 57

REVIEWS From the bookshelf 59

OBITUARIES In memoriam 65

EVENTS Latest announcements 69

CALENDAR All forthcoming events 71

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 4 — #4 ii

ii

ii

4 NEWS

LMS NEWS

Annual Elections to LMS Council

The LMS Nominating Committee is responsible forproposing slates of candidates for vacancies onCouncil and vacancies on Nominating Committee itself.The Nominating Committee welcomes suggestionsfrom the membership.

Anyone who wishes to suggest someone for aposition as an Officer of the Society or as aMember-at-Large of Council (now or in the future)is invited to send their suggestions to ProfessorKenneth Falconer, the current Chair of NominatingCommittee ([email protected]). Please providethe name and institution (if applicable) of thesuggested nominee, their mathematical specialism(s),and a brief statement to explain what they could bringto Council/Nominating Committee.

It is to the benefit of the Society that Council isbalanced and represents the full breadth of themathematics community; to this end, NominatingCommittee aims for a balance in gender, subject areaand geographical location in its list of prospectivenominees.

Nominations should be received by 16 April 2021 inorder to be considered by the Nominating Committee.

In addition to the above, members may makedirect nominations for election to Council orNominating Committee. Direct nominations mustbe sent to the Executive Secretary’s office([email protected]) before noon on 1 September2021. For details on making a direct nomination, seelms.ac.uk/about/council/lms-elections.

The slate as proposed by Nominating Committee,together with any direct nominations received up tothat time, will be posted on the LMS website in earlyAugust.

New Editor-in-Chief sought forLMS Newsletter

The LMS seeks someone with broad mathematicalinterests, who is passionate about communicatingmathematics and supporting the Society, to becomeEditor-in Chief of the LMS Newsletter, starting inMay 2021.

The LMS Newsletter has several purposes. It aimsto provide a sense of identity, community, andconnection for the Society’s members. It is achannel for communicating the power, beauty andvalue of mathematics and mathematical researchby disseminating new mathematical ideas andinformation. It also seeks to make transparent theSociety and its workings.

The main duties of the Editor-in-Chief, who isultimately responsible to the Society’s Council, are:

• overseeing the commissioning of content for themathematical features section

• overseeing the sourcing of material for othersections

• signing off on the content and layout for eachbi-monthly issue

• chairing, leading and coordinating support from theNewsletter Editorial Board.

In addition to the Newsletter Editorial Board, theEditor-in-Chief also works closely with, and is furthersupported by, members of the Society’s staff.

This role is unremunerated, although reasonableexpenses will be paid. The role requires a timecommitment of approximately one day a week onaverage, year round.

Back issues of the Newsletter are available on theSociety’s website at bit.ly/36h6iUL.

For further information about this role, includingattributes sought and how to apply, pleasecontact the Executive Secretary Caroline Wallace [email protected]. Applications will close on1 April 2021.

Plan S Update

In September 2018 several research funders formedan international consortium, cOAlition S, to launchthe Plan S initiative. The intention of Plan Sis to require that scientific publications arisingfrom research supported by consortium membersmust be published immediately open access incompliant journals or platforms. The timeline forimplementation of these requirements varies byfunder (see bit.ly/2MuZ1d5), but in general only grants

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 5 — #5 ii

ii

ii

NEWS 5

awarded by members of cOAlition S beginning after1 January 2021 will include these requirements. Itshould be noted that UKRI will not be publishing theiropen access policy until Spring 2021.

For those subject to Plan S requirements the threemain ways compliance can be achieved are:

• Publication of the final typeset version of record(VoR) with a CC BY licence in an open access journalor platform registered in the Directory of OpenAccess Journals (DOAJ).

• Open access publication of the VoR with aCC BY licence in a hybrid journal coveredby a transformative arrangement approved bycOAlition S (only until the end of 2024).

• Publication of the VoR with restricted access in asubscription journal but at the same time depositingthe Author’s Accepted Manuscript (AAM) with a CCBY licence in an institutional or subject repository,with immediate open access. The AAM is thefinal author-created version of the manuscript asaccepted for publication by the journal, includingany changes made during peer review. The use of aCC BY licence is to permit others to distribute, remix,adapt, and build upon an article, even commercially,as long as they credit the original author(s).

If the costs of publication of the VoR open access aresupported through an article processing charge (APC)then funders will only pay this if the journal is fullyopen access or denoted a ‘transformative journal’.Alternatively, authors can comply by making their VoRopen access in journals covered by ‘transformativeagreements’ that have been negotiated between theirinstitution and a publisher and cover the cost of openaccess. A list of compliant venues can be found onthe cOAlition S website.

Whilst most mathematics journals impose norestriction on posting early versions of a paper toa repository such as arXiv, the standard licencessigned with publishers typically impose conditions ondepositing the AAM, including an embargo or use ofa non-commercial licence.

All of the Society’s hybrid journals offertransformative agreements to a growing number ofauthors, negotiated between our publishing partnersand academic institutions. This has resulted in growthin the proportion of papers where the version ofrecord is available open access in all of the journalsmanaged by the Society. For example, close to one

fifth of the articles published online in 2020 in theBulletin, Journal, Proceedings and Journal of Topologywere open access.

It should be noted that the amount of contentsupported by APCs is taken into account in settingsubscription prices (see bit.ly/2MrLbrX).

The Society also publishes a fully open access journal,the Transactions. Through the appointment of a newEditorial Board (as described in the January LMSNewsletter) the Society intends to offer authorsa high-quality publication venue for those whosefunders support open access publishing.

The publications landscape is changing rapidly.The Society is looking to adapt to the changingrequirements of authors and funders in a way thatallows its publication activities to continue on asustainable basis while enabling the Society to returnall surplus income to support mathematicians andmathematics research.

John HuntonLMS Publications Secretary

De Morgan Donations andOther Gifts

Launched in 2019, De Morgan Donations are gifts to theSociety of £1,865 or more. Named after the Society’sfirst President in 1865, Augustus De Morgan, thesedonations have started to play a significant role inhelping the Society in its support of the mathematicalcommunity.

Our funding of mathematical research, typically closeto £700,000 annually, is one particularly important wayin which the Society meets its objectives to promote,disseminate and advance mathematics. In 2020, theSociety drew on its reserves to invest over £120,000in a second round of LMS Early Career Fellowships, tomitigate some of the impact of the covid-19 pandemicon the academic careers of Early Career Researchers.This, together with a most generous donation from theHeilbronn Institute for Mathematical Research, enabledus to support 22 more Early Career Fellows than wenormally would have done.

Going forward, the Society continues to explorenew ways in which it can support members of themathematical community who have been significantlyaffected by the pandemic. For example, with a generousgift from the Liber Stiftung we are providing higher

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 6 — #6 ii

ii

ii

6 NEWS

levels of support to mathematicians whose work hasbeen impacted by caring responsibilities, and with onefrom Dr Tony Hill we are establishing the new LevellingUp Scheme to support pupils from under-representedbackgrounds studying A-level Mathematics.

The De Morgan Donations we have received so farhave made a highly significant contribution to theseinitiatives and we are most grateful to the donors fortheir generosity. We are also extremely grateful for themany other generous gifts and bequests we receive insupport of our charitable work.

De Morgan Donations can be made either vialms.ac.uk/content/donations#Donate or by contactingme ([email protected]) for further details.

The Society is also grateful to its many volunteers forthe time and energy they contribute to our variousactivities, especially at a time when many are so busy.

Jon Keating FRSLMS President

Atiyah UK–Lebanon Fellowship2021–22

Maciej Dunajski (left) with Michael Atiyah

The Atiyah UK–Lebanon Fellowships were set up in 2019as a lasting memorial to Sir Michael Atiyah (1929–2019)in the form of a two-way visiting programme formathematicians between the UK and Lebanon, whereSir Michael had strong ties.

The LMS is delighted to announce that the AtiyahFellowship for the academic year 2021–22 has beenawarded to Professor Maciej Dunajski of CambridgeUniversity.

Professor Dunajski’s research is in mathematical physics,in particular the interplay between differential geometry,integrable systems, general relativity and twistor theory.

His main research achievement is the solution (joint withRobert Bryant and Mike Eastwood) of the metrisabilityproblem posed more than 120 years ago by RogerLiouville. He plans to make one or two short visitsto the Centre for Advanced Mathematical Sciences atthe American University of Beirut (AUB), as well asNotre-Dame University-Louaize, where he will lecture ontwistor theory.

The photograph shows Professor Dunajski with MichaelAtiyah. It was taken in Trinity College during what mayperhaps have been Sir Michael’s last visit to Cambridgein the summer of 2017.

The 2020–21 Fellows were Professor Mark Wildon (RoyalHolloway, University of London) and Professor AhmadSabra (AUB). Professor Wildon will be visiting AUB andProfessor Sabra will be visiting the University of Sussexas soon as the covid-19 situation permits.

For further information about the Fellowships andinformation on how to apply, see bit.ly/39CDKra. Itexpected that applications for Fellowships to be held theacademic year 2022–23 will open in September 2021.

Privy Council Approval ofAmended LMS Standing Orders

At its meeting on 16 December 2020, the Privy Councilagreed to proposed amendments to the LondonMathematical Society’s Royal Charter and Statuteswhich, with the By-Laws, are known collectively as theStanding Orders.

The Society’s Council set up the Standing Orders ReviewGroup in 2014. It was felt that the original wordingof the documents, agreed in 1965 when the Charterwas granted, was no longer appropriate and in placeswas out of date, not least with the way the LMS nowoperated. Linguistic changes were required to removesexist and ageist wording, and there were severalrelatively minor procedural changes. The StandingOrders Review Group had been willing to be bold, butincreasingly realised in their deliberations that in generalthe documents had been extremely well drafted in1965. The proposed changes received close scrutinyby Council on several occasions and were subject toa consultation with the whole LMS membership in2018–19.

LMS members voted overwhelmingly to approve theproposed changes to the Standing Orders at the 2019Annual General Meeting. Following this, the changesto the Royal Charter and Statutes were formallysubmitted to the Privy Council in February 2020 (the

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 7 — #7 ii

ii

ii

NEWS 7

By-Law changes do not require Privy Council approval).Unfortunately because of the covid-19 pandemic, thePrivy Council did not consider such proposals fromany organisations at its meetings for some months.Liaison between the Society and the Privy CouncilOffice continued throughout 2020 and notification wasreceived in early December that the Privy Counciladvisors raised no objections to the proposals. HerMajesty the Queen approved the two Orders in Councilat the December Privy Council meeting in Windsor.The revised Standing Orders are now posted on theSociety’s website and are being implemented.

The Society would like to thank all those memberswho have been involved in the review of the StandingOrders, including especially those who sat as membersof the Standing Orders Review Group (Caroline Series,Simon Tavaré, Terry Lyons, Alexandre Borovik, JuneBarrow-Green, John Toland and Fiona Nixon) — and inparticular Stephen Huggett, who as General Secretaryensured that the process was carried out efficiently andin a robust manner, taking into account all viewpointsand ‘future proofing’ the Standing Orders to take theSociety forward.

Further information on the Standing Orders Review,including the rationale for the changes, can be foundat lms.ac.uk/about/lms-standing-orders-review.

Forthcoming LMS Events

The following events will take place in forthcomingmonths:

LMS Women in Mathematics Day: 24 March, online(tinyurl.com/y5uwol5f)

Society Meeting at the BMC–BAMC: 8 April, online(tinyurl.com/yarpowdo)

LMS Spital�elds History of MathematicsMeeting: 14 May, online (tinyurl.com/y3kpv6ye)

Midlands Regional Meeting and Workshop:2–4 June, Lincoln (tinyurl.com/y5vtaytx)

Summer General Society Meeting: 2 July, London

Invited Lecture Series 2021: August, online(tinyurl.com/y2gyehr4)

Northern Regional Meeting: 6–10 September,University of Manchester (tinyurl.com/yamy8uvq)

A full listing of upcoming LMS events can be foundon page 71.

OTHER NEWS

Su�rage Science Awards 2020

Jewellery designed by students at Central SaintMartins–UAL

For the participants, one of the most upliftingmoments of 2020 was the virtual celebration of the2020 Su�rage Science Mathematics and ComputingAwards. The Su�rage Science Scheme celebrateswomen in science for their scienti�c achievementsand for their ability to inspire others. It aspires to

encourage more women to enter scienti�c subjects,and to stay. It was founded in 2011 with awards to aninitial cohort of 11 women in Life Sciences, and in 2013the Scheme expanded with a new cohort of womenin Engineering and Physical Sciences. In 2016 theScheme expanded again to include a Mathematicsand Computing cohort.

The awards are beautiful pieces of jewellery designedby students at the art and design college CentralSaint Martins–UAL, inspired by the jewellery wornby the su�ragettes and by conversations withscientists. The jewellery designed for mathematicsand computing awards are a simple circular silverbracelet with a pearl bead that traces out the unitcircle, with the equation e ic + 1 = 0 inscribed onthe inside, and a brooch made of gold punchedtape encoding su�ragette messages. Every two years,awardees are asked to nominate their successors —and the jewellery pieces are passed on in a ceremony.

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 8 — #8 ii

ii

ii

8 NEWS

The 2020 ceremony was held online, and featuredvideo clips from the nominator–nominee awardpairs, as well as a panel discussion on diversityin mathematics, statistics and computing. Itwas inspiring to hear the reasons why eachwoman had chosen her successor, as well as theawardees’ visions for equality in the future oftheir disciplines. There was also a lively paneldiscussion, with discussion around concerns aboutthe disproportionate impact of covid-19 on women,as well as the relevance of the Black LivesMatter movement to mathematics, alongside positivestories.

From a personal point of view, this has been aspecial award, in that it aims to develop a supportivenetwork of women. Gwyneth Stallard was one of theinaugural mathematics awardees and loved wearingthe bracelet — she was delighted to pass it ontoEugenie Hunsicker who has done an amazing jobas her successor as Chair of the LMS Womenin Mathematics Committee. In turn, Eugenie haspassed it on to IMA and LMS Women and DiversityCommittee member Sara Lombardo. At the sametime, jewellery passed this year from Nina Snaith toApala Mjumdar and from Vicky Neale to Anne-MarieIma�don, as well as along lines of statisticians andcomputer scientists. We three authors are currentlyworking together with EPSRC on ways of improvingequality and diversity in relation to funding, withGwyneth and Sara acting as Diversity Champions forthe Mathematical Sciences Strategic Advisory Team.

More details and photos of the awardees and thejewellery can be found at su�ragescience.org.

Gwyneth StallardEugenie Hunsicker

Sara Lombardo

PROMYS Europe Connect 2021PROMYS Europe Connect, a challenging four-weekmathematics summer programme online, based at theUniversity of Oxford, UK, is seeking applications frompre-university students from across Europe (includingall countries adjacent to the Mediterranean) who showunusual readiness to think deeply about mathematics.

PROMYS Europe Connect is designed to encouragemathematically ambitious students who are at least 16years old to explore the creative world of mathematics.Participants tackle fundamental mathematicalquestions within a richly stimulating and supportive

online community of fellow first-year students,returning students, undergraduate counsellors,research mentors, faculty, and visiting mathematicians.

PROMYS Europe is a partnership of Wadham Collegeand the Mathematical Institute at the University ofOxford, the Clay Mathematics Institute, and PROMYS(Program in Mathematics for Young Scientists, foundedin Boston in 1989).

The programme is dedicated to the principle that noone should be unable to attend for financial reasons.Most of the cost is covered by the partnership and bygenerous donations from supporters. In addition, fulland partial financial aid is available, for those who needit.

The application form and application problem setare available on the PROMYS Europe websitepromys-europe.org. The closing date for applicationsis 14 March 2021, and students will need to allow timebefore the deadline to tackle the application problems.PROMYS Europe Connect will run online from 12 July to6 August 2021.

Sir Michael Atiyah Conference

The conference on the Unity of Mathematics inhonour of Sir Michael Atiyah, postponed from 2020,has been rescheduled to take place in the IsaacNewton Institute from 21–23 September 2021. It isthe intention of the organisers that this meeting willbe either a face-to-face or a mixed face-to-face andonline meeting, depending on circumstances nearerthe time. It is anticipated that registration will openin late spring. For details and updates please see theconference website newton.ac.uk/atiyah.

Assuming the conference can take place as planned,some accommodation will be available in MurrayEdwards College, and limited funding should beavailable, especially for early career researchers. Foradvance expressions of interest and noti�cationwhen registration opens, email Kathryn de Ridderat o�[email protected], using the subject line ‘SirMichael Atiyah Conference’.

A number of grants funded by the National ScienceFoundation are available to cover full expensesfor PhD students and postdoctoral researchersfrom the United States. Women and members ofother under-represented groups in the mathematicscommunity are particularly encouraged to apply. Torecord an advance expression of interest, pleasecontact Laura Schaposnik at [email protected].

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 9 — #9 ii

ii

ii

NEWS 9

The organisers are grateful to all the sponsors:the Clay Mathematics Institute, Heilbronn Institutefor Mathematical Research, London MathematicalSociety, National Science Foundation, Isaac NewtonInstitute, and Oxford Mathematics Department.

LMS Honorary Member ReceivesTop Honour

Professor Cheryl Praeger(University of WesternAustralia) has beenmade a Companion ofthe Order of Australiain the 2021 AustraliaDay Honours List for‘eminent service tomathematics and to

tertiary education, as a leading academic andresearcher, to international organisations, and as achampion of women in STEM careers’.

Professor Praeger has had a long and distinguishedcareer and has made signi�cant contributions tovarious areas of mathematics, including grouptheory, permutation groups, combinatorics andthe mathematics of symmetry. Her expertise ingroup theory and combinatorial mathematics hasunderpinned advances in algebra research andcomputer cryptography.

Professor Praeger has received many honours andawards during her career. She received the 2019Prime Minister’s Prize for Science, she was the �rstfemale President of the Australian MathematicalSociety (1992–94) and the Society now awards

the Cheryl E. Praeger Travel Awards to femalemathematicians. Professor Praeger became anHonorary Member of the London MathematicalSociety in 2014.

Visit tinyurl.com/y3h2xnbc for more information.

Clay Research FellowsThe Clay MathematicsInstitute has awardedthe 2021 ClayResearch Fellowshipsto Maggie Miller,Georgios Moschidis, LisaPiccirillo and AlexanderSmith. Clay ResearchFellowships are awarded

on the basis of the exceptional quality of candidates’research and their promise to become mathematicalleaders.

Maggie Miller obtained her PhD in 2020 fromPrinceton University, where she was advised by DavidGabai. She will be based at Stanford University.Georgios Moschidis obtained his PhD in 2018 fromPrinceton University, where he was advised byMihalis Dafermos. He will be based at PrincetonUniversity. Lisa Piccirillo obtained her PhD in 2018from the University of Texas at Austin, where shewas advised by John Luecke. She will be based atthe Massachusetts Institute of Technology. AlexanderSmith obtained his PhD in 2020 from HarvardUniversity, where he was advised by Noam Elkies andMark Kisin. He will be based at Stanford University.

For more information visit claymath.org.

EUROPEAN MATHEMATICAL SOCIETY NEWS

European Women in Mathematics

Owing to covid-19, the European Women inMathematics (EWM) Society held its General Assemblyonline on 6 July 2020. The next EWM Generalmeeting will be in Helsinki in 2022. At the virtualGeneral Assembly, Andrea Walther (HU Berlin) andKaie Kubjas (Aalto) were elected the new convenorsof EWM, succeeding Carola-Bibiane Schönlieb andElena Resmerita who served as convenors since2016. EWM has published an open letter to advocatea proactive policy to support current employees

in temporary positions and future job applicantsin Mathematics in light of the Corona Crisis: seehttps://tinyurl.com/yxkg9v5n.

EMS News prepared by David ChillingworthLMS/EMS Correspondent

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

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 10 — #10 ii

ii

ii

10 NEWS

MATHEMATICS POLICY DIGEST

UKRI Ethnicity Analysis of FundingApplicants and Awardees

In December 2020 UK Research and Innovation (UKRI)published detailed ethnicity data on funding applicantsand awardees, which highlighted disparities betweendifferent ethnic groups. These data form part of UKRI’songoing work to “increase equality and diversity inthe research and innovation system, through effective,evidenced intervention”. These data are available attinyurl.com/y4w5odtp.

Funding Boost for MathematicalSciences Institutes

Three of the UK’s leading research institutes will besupported to widen access to Mathematical Sciencesand support training through funding announcedby UKRI. The investment will allow the IsaacNewton Institute (INI), the International Centre for

the Mathematical Sciences (ICMS) and the HeilbronnInstitute for Mathematical Research (HIMR) to launchand expand a wide range of activities supportingeducation and training.

The funding is part of the £300 million governmentinvestment in the Additional Funding Programme forMathematical Sciences announced in 2020. The fundingwill be delivered by the Engineering and PhysicalSciences Research Council (EPSRC), part of UK Researchand Innovation, and the Royal Society over a five-yearperiod from 2020/21 to 2024/25. More details areavailable at tinyurl.com/y3sorkka.

Digest prepared by Dr John JohnstonSociety Communications Officer

Note: items included in the Mathematics Policy Digest arenot necessarily endorsed by the Editorial Board or theLMS.

OPPORTUNITIES

Forthcoming LMS Grant Schemes

Cecil King Travel Scholarships: The LMS administerstwo £6,000 travel awards funded by the Cecil KingMemorial Foundation for early career mathematicians,to support a period of study or research abroad,typically for a period of three months. One Scholarshipwill be awarded to a mathematician in any area ofmathematics and one to a mathematician whoseresearch is applied in a discipline other thanmathematics.

Applicants should be mathematicians in the UK or theRepublic of Ireland who are under the age of 30 at theclosing date for applications, and who are registeredfor a doctoral degree or have completed one within 12months of the closing date for applications. The LMSencourages applications from women, disabled, Black,Asian and Minority Ethnic candidates, as these groupsare under-represented in the UK or the Republic ofIreland mathematics.

To apply, complete the application form attinyurl.com/yarns982 and include a written proposalgiving the host institution, describing the intendedprogramme of study or research, and the benefits tobe gained from the visit.

The application deadline for applications is 31 March2021. Shortlisted applicants will be invited to interviewduring which they will be expected to make a shortpresentation on their proposal.

Interviews will take place in May 2021. Queries maybe addressed to Tammy Tran ([email protected]).In view of the low number of applications received inprevious rounds, there is a high chance of success inthis scheme.

Computer Science Small Grants (Scheme 7): thedeadline for applications in the next round is 15 April.The grants support a visit for collaborative researchat the interface of Mathematics and ComputerScience. More details at tinyurl.com/y7rbdhpn.

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 11 — #11 ii

ii

ii

LMS BUSINESS 11

Maximising your LMS Membership

International Connections with OtherMathematical Societies

In forthcoming issues of the Newsletter we aim toshine a spotlight on di�erent membership bene�ts.This month we would like to highlight to members theopportunities to connect internationally with othermathematical societies through your membership ofthe LMS.

Reciprocal Agreements with 21 InternationalMathematical Societies

The Society has reciprocal agreements with thefollowing mathematical societies, enabling LMSmembers to join those societies at a 50% discounton the full membership fee of each society if theyare not normally resident in the same country as thesociety. For example, a UK-based Ordinary Membercould join the Mathematical Society of Japan at theirreciprocal membership rate. For further informationabout these societies, see tinyurl.com/y6jwpg8q.

American Mathematical SocietyAustralian Mathematical SocietyBelgian Mathematical SocietyCanadian Mathematical SocietyDansk Matematisk ForenigDeutsche Mathematiker-VereinigungFinnish Mathematical SocietySociété Mathématique de FranceIndian Mathematical SocietyIrish Mathematical SocietyUnione Matematica ItalianaMathematical Society of JapanKoninklijk Wiskundig GenootschapNew Zealand Mathematical SocietyNigerian Mathematical SocietyNorsk Matematisk ForeningReal Sociedad Matemática EspañolaSingapore Mathematical SocietySouth East Asian Mathematical SocietySvenska MathematikersamfundetSwiss Mathematical Society

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 12 — #12 ii

ii

ii

12 LMS BUSINESS

In return, members of the above societies who arenot normally resident in the UK can join the LondonMathematical Society as a Reciprocity Member andreceive a 50% discount on the Ordinary membershipfee (other subscription rates such as AssociateMembership are already discounts on the OrdinaryMembership rate and the Society does not o�er‘double discounts’). If you have any queries, pleasecontact [email protected].

LMS members based in either Northern Ireland orthe Republic of Ireland and who are also members ofthe Irish Mathematical Society can choose whetherto be a Reciprocity Member of either the LMS or theIrish Mathematical Society.

Discounted membership of the EuropeanMathematical Society

Members of the London Mathematical Society canjoin the European Mathematical Society (EMS) ata 50% discount of the full EMS membership fee:currently €25.00 annually instead of €50.00. Furtherinformation about the European MathematicalSociety is available at euro-math-soc.eu/.

To join the EMS, members can contact the EMSdirect and mention their LMS membership oradd EMS membership to their LMS membershiprecord when signing in via the LMS website here:www.lms.ac.uk/user, or by completing and returningthe LMS subscription form. Payment for EMSmembership can also be made via the LMS and over150 LMS members already pay for EMS membershipalongside their LMS membership. If you have anyqueries, please contact [email protected].

LMS members who are students may be interestedin the EMS’ free membership for students whileLMS members who are aged 60+ may be interestedin the EMS’ lifetime membership. Further detailsabout EMS membership rates are available here:https://euro-math-soc.eu/individual-members.

Option to pay for membership of EuropeanWomen in Mathematics

Members of the London Mathematical Society whoare also members of the European Women inMathematics (EWM) association have the optionto pay for their EWM membership via their LMSMembership account and over 30 EWM membersalready pay for their EWM membership alongsidetheir LMS Membership.

To do so, you must �rst be a member of EWM:further details about EWM membership and howto join are available at tinyurl.com/y2ta5xh5. If theywish, EWM members can then add their EWMmembership to their LMS membership account eitherby logging in via the LMS website www.lms.ac.uk/useror by completing and returning the LMS subscriptionform. If you have any queries, please [email protected].

We hope this spotlight on the bene�ts ofLMS membership in supporting your internationalconnections has been helpful.

Elizabeth FisherMembership & Grants Manager

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 13 — #13 ii

ii

ii

LMS BUSINESS 13

LMS Council Diary —A Personal View

Council met via video conference for a relatively shortmeeting on the morning of Friday 20 November 2020,before the Annual General Meeting and Naylor Lecturelater that day. The meeting began with the President’sbusiness, including a report on the successful two-dayonline Black Heroes of Mathematics Conference, hostedjointly with the IMA and British Society for the Historyof Mathematics and facilitated by ICMS, which attractedover 250 participants, and an update on the ‘LevellingUp’ scheme pilot, generously supported by Dr TonyHill, which aims to support the provision of onlinetutoring for A-level Mathematics students who comefrom backgrounds that are under-represented in themathematics community. Other business includedthe Publications Secretary giving a brief update on

contract negotiations for the future production of theSociety’s main journals and reporting that Mathematikawould be made available free to LMS members from2022 onwards, as well as an update on CommitteeMembership. In a discussion about Society Membership,the Treasurer noted that a healthy number of newmembers were due to be elected at the AGM later thatday, in the recruiting of which the LMS Representativesin universities had played a significant role. There wasalso a discussion about the importance of voting inSociety elections and how this could be encouraged.

The meeting concluded with the President giving thankson behalf of Council to the outgoing Council Members,and wishing luck to those members up for election.

Elaine CrooksMember-at-Large

LEVELLING UP

This article forms the second of an ongoing seriesof updates about this new scheme where, thanks toa generous donation by LMS member Dr Tony Hill,the LMS is developing a new venture to support theprovision of online tutoring for A-Level Mathematicsstudents who come from backgrounds that areunder-represented in the mathematics community.

For students from under-represented groups it mayappear an insurmountable task to achieve thenecessary quali�cations to study for a STEM degreeat university. From the perspective of a Year 12BAME student attending an under-performing schoolfor example, who is hoping to study engineering atuniversity, there may be no well-de�ned route toachieving their goal. Students from the school mayonly rarely attain the necessary grades to considerattending university. Teachers may be too stretchedto provide additional academic or personal support.There may be pressure from peers to do otherthings rather than study and, if the student is the�rst person in their family to consider going touniversity, family members may not know how bestto o�er their support. The covid-19 pandemic hasalso provided huge new challenges with face-to-faceteaching curtailed, leaving the student feeling moreisolated, under pressure and unfocused. With highgrades required, they must perform well in A-levelmathematics to be accepted onto their preferredcourse. Without extra academic and aspirational

support, the student may not achieve their goal.But what if a programme were available to providetailored support to a student in this situation?

The Levelling Up Scheme is exactly the type ofprogramme that can provide the support requiredto help such students succeed and achieve theiracademic goals. The overall goal of the Schemeis to signi�cantly increase inclusion, by boostingstudents’ self-con�dence, raising their aspirationsand accelerating their academic attainment. The aimis to prepare students to apply for a place on a STEMdegree at university, and then to succeed once theyare there. The Scheme is also vitally important inhelping to provide a diverse pool of graduate talentwith the skills to contribute to the UK’s long termeconomic growth.

The pilot scheme will provide eighteen months ofacademic and pastoral support for students fromunder-represented groups, starting midway throughYear 12 and continuing until the end of Year 13.Students will have the opportunity to take part in avariety of specially designed online subject speci�ctutorials, in collaboration with Durham and Leicesteruniversities.

See more information at levellingupscheme.co.uk.

John JohnstonSociety Communications O�cer

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 14 — #14 ii

ii

ii

14 LMS BUSINESS

REPORTS OF THE LMS

Report: Annual General Meeting

The 2020 LMS Annual General Meeting was held onFriday 20 November with LMS President ProfessorJon Keating, FRS in the Chair. This was the year ofthe AGM with a twist: it was held online, throughthe Zoom video conferencing software. Hostingin cyberspace certainly had no adverse impactson attendance; the meeting attracted over 100attendees, none of whom were required to travelto London to take part. The meeting began withSociety business. First, a presentation from theVice-President, Professor Iain Gordon, on Societyactivities over the course of the year. This wasfollowed by a report from the Treasurer, ProfessorRob Curtis, who gave a summary of Society accountsfor the year. Other matters included votes on Societyresolutions, and announcement of the results ofelections to Council and Nominating Committee.

Next, congratulations were recorded for recipients ofLMS awards. Each year, the LMS awards a selectionof prizes for achievements in and contributionsto mathematics. These range from the SeniorAnne Bennett Prize, which is awarded for workin relation to advancing the careers of women inmathematics, to the Senior Berwick Prize, awardedfor an outstanding piece of research published bythe Society. 2020 was no di�erent, and attendeescongratulated 2020’s twelve LMS prize winners,and also the 276 mathematicians elected to LMSMembership this year.

To conclude Society business, the President,Professor Jon Keating, thanked all retiring membersof Council. Of particular note were the departuresof two long-standing members: Professors StephenHuggett and Rob Curtis stepped down as GeneralSecretary and Treasurer, having served eight and nine

year terms in those roles respectively. An admirablecommitment, and commendable to say the least.

At the conclusion of Society business, the meetingwas paused for a short break; even in virtualmeetings, co�ee breaks are always most welcome.The break was followed by the Naylor Lecture,given by Professor Nicholas J. Higham of theUniversity of Manchester — to whom the LMSawarded the 2019 Naylor Prize — on ‘TheMathematics of Today’s Floating-Point Arithmetic’.The overarching theme was an investigation into thereliability of low precision �oating-point arithmetic,with a particular emphasis on understanding theimplications for recent developments in the latesthardware implementing this arithmetic. ProfessorHigham — the author of an acclaimed book on thetopic — gave a fascinating and highly accessibletalk, frequently linking together historical workand current developments in the area, as well asregularly highlighting applications in, for example,deep learning.

It should be noted how smoothly this AGM ran. It iseasy to forget that running an event of this naturein an online setting is never simple, and requirescareful planning and preparation. Potential hurdles —of conducting formal votes virtually, and transitioningbetween presentations and speakers, for example— were made to look non-existent. The meetingwas, especially in light of current circumstances, aresounding success.

Matt StaniforthUniversity of Southampton

Report: LMS Graduate StudentMeeting

The LMS Graduate Student Meeting on 16 November2020 commenced with a lecture from Theo Maryof the Sorbonne entitled Mixed Precision Arithmetic:Hardware, Algorithms and Analysis. At the currentmoment in time, when our entire lives seem torevolve around computers, it is a welcome reliefto be reminded of their limitations. To a puremathematician like myself, it was fascinating to

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 15 — #15 ii

ii

ii

LMS BUSINESS 15

dive into the world of �oating-point arithmetic andconfront the fact that computers can only work withnumbers to �nite precisions; numbers, as I imaginethem, are Platonic ideals, not �nite sequences of bitson a computer.

Theo discussed the trade-o� between di�erentstandards for �oating-point arithmetic: the higherthe precision, the slower the speed. But he wenton to explain how this trade-o� can be overcomeby strategically deploying higher-precision arithmeticand lower-precision arithmetic at di�erent stages inthe calculation. By using higher-precision arithmeticwhere it counts, one can achieve both good accuracyand good speed.

A 32-bit �oating point number

A primary source of motivation for research intocomputer arithmetic is the application to deeplearning, where intensive computation is required.Consequently, much research is done in this �eldby big tech companies, who also implement newstandards for low-precision arithmetic in hardware.The chair of the session, Ian Short of the OpenUniversity, asked Theo the interesting question ofhow open these companies are when it comes toworking on this sort of research. Encouragingly, Theoresponded that they did have a good dialogue withseveral of these companies, though, naturally, someof their information remains strictly proprietary.

The meeting then turned to talks from GraduateStudents. A very healthy number of eighteen talkswere given. This was more than would be usualfor a graduate student meeting, showing that thereare advantages to the online format. In order toaccommodate such a large number, �ve breakoutrooms were required, each hosting four talks of 15minutes. The �ve rooms roughly corresponded tothe areas of algebra, combinatorics, analysis, �uids,and mathematical biology. It seems our geometrygraduate students must be rather shy.

I listened to the talks in the combinatorics room,where I also spoke. There was a rich range of talkshere, from Ramsey theory to set theory, along withmy own talk on polytopal combinatorics. Carl-FredrikNyberg Brodda �nished us o� with a very memorabletalk on his e�orts to uncover the origins of theB. B. Newman spelling theorem in combinatorialgroup theory. This was an exciting detective story of

missing PhD theses and the like, where everythingwas happily solved at the end.

Nicholas WilliamsUniversity of Leicester

Report: Black Heroes ofMathematics Conference

Growing up, we read about heroes in scienceand mathematics whose breakthrough ideas led toinnovations. Unnoticed by many is the fact that veryfew of these heroes are black or of black descent. Isthis so because black people have not contributedto innovations and discoveries?

A few years ago, I was a PhD student in mathematicsand I asked myself similar questions. This is becauseI needed to see role models who had blazed thetrail I was on who would inspire me. My quest ledme to an article: ‘Five Famous Black Mathematicians’by Hazel Lewis with thanks to Dr Nira Chamberlain(https://tinyurl.com/y32kka58). Katherine Johnsonwas a name that stood out in this article becausethe successful contributions she made at NASA indoing the calculations that sent the �rst Americanto space, beautifully captured in the movie HiddenFigures.

The Black Heroes of Mathematics Conference wasa virtual conference organised on Monday 26 andTuesday 27 October 2020 by the British Societyfor the History of Mathematics, the InternationalCentre for Mathematical Sciences, the Institute ofMathematics and its Applications and the LondonMathematical Society. The vision of the conferencewas “To celebrate the inspirational contributions ofblack role models in the �eld of mathematics”. Over250 participants from over 30 di�erent countriesattended all or part of the event online.

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 16 — #16 ii

ii

ii

16 LMS BUSINESS

Talks covered a range of technical and non-technicaltopics, with presentations by the followinginternational speakers: Dr Nira Chamberlain, DrAngela Tabiri, Dr Howard Haughton, Professor TannieLiverpool, Natalya Silcott, Dr Spencer Becker-Kahn,Professor Nkechi Agwu and Professor Edray Goins.At the end of each day, there was a panel discussionwith Professor Clive Fraser, Dr Joanna Hartley,Jonathan Thomas, Susan Okereke and the speakersfrom that day. Questions discussed included whatwe can do to increase the representation of blackpeople in mathematics, how can indigenous Africanmathematics be used to support the learning ofmathematics, why should mathematics teachers beinterested in Black History Month, and do we reallyneed black heroes of mathematics — to mention afew.

One of my favourite quotes at the conference wasby Dr Nira Chamberlain, “You do not need anyone’spermission to become a mathematician”. Otherquotes included: “Being the �rst is not somethingto be proud of, but is a calling to ensure that one isnot the last” — Dr Nira Chamberlain. Enthusiasticcomments from participants included “What anutterly brilliant, inspiring event — may there be manymore!” and “We hope that one day, we will live ina world where all children feel that maths belongsto them. Until then let us showcase the stories ofdiverse mathematicians so all of our students feellike they could be a mathematician.”

Videos of talks from the conference can be found athttps://t.co/MyGefYZ8Id?amp=1.

Angela TabiriAfrican Institute for Mathematical Sciences Ghana

Report: LMS Computer ScienceColloquium

The LMS Computer Science Colloquium washeld on Thursday 19 November 2020 online viaZoom with the topic of the colloquium beingAlgorithms, Complexity and Logic. A record numberof participants enjoyed a compelling series of talks.

The first speaker was Dr Anupam Das, from theUniversity of Birmingham, who gave a broad andbountiful talk across his wide interests in logic,taking in especially, proof complexity, computationalcomplexity and bounded arithmetic. Anchored

around these three disciplines, on a beautifully builtwhiteboard, he described the ways in which theareas interact, culminating in a seminal result ofBuss tying together polynomial time computation,the bounded arithmetic theory S 1

2 and induction onnotation over NP-predicates. He also explained therelationship of this bounded arithmetic theory tolength of proofs in the fundamental propositionalproof system called extended Frege.

The second speaker was Professor Nobuko Yoshidafrom Imperial College, London. She gave a talktitled Session Types: A History and Applications.She was introduced and hosted by one of theLMS Computer Science Committee members, DrOrnela Dardha. More specifically, Yoshida’s talkwas about: a history of (multiparty) session types;what are binary and multiparty session types;and several applications on multiparty sessiontypes, including: (3-1) runtime monitoring for largecyberinfrastructures; (3-2) robotics; (3-3) codegeneration in Go; and (3-4) inference of sessiontypes from Go code and verification by modelchecking. Her talk was followed by questions and alunch break where further interaction continued.

The next speaker was Dr Kitty Meeks from theUniversity of Glasgow, who discussed the interplaybetween certain decision problems and their exactand approximate counting versions. These wereintroduced in both monochrome and multicolouredversions. Of course, if one can solve exact counting,then also one can solve approximate counting; andthis latter is sufficient to solve the decision problem.The remaining interactions are more subtle. Forexample, it is known that an oracle for approximatecounting does not give rise to an efficient procedurefor exact counting (in both the multicolour andmonochrome regimes). The principal new resultof the talk was that, in the multicolour regime,an efficient procedure for the decision problemdoes give an efficient procedure for approximatecounting. This recent work (with Dell and Lapinskas)is especially remarkable.

The final speaker of the day was Dr IgorCarboni Oliveira from University of Warwick,who presented deep and fascinating questionsand results about the descriptive complexity offundamental notions like prime numbers. Startingfrom classical descriptive complexity as definedby Kolmogorov in 1963, he rephrased the questionwhether there are infinitely many Mersenne primesin terms of infinitely many primes of minimumdescription length. Time-bounded Kolmogorov

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 17 — #17 ii

ii

ii

LMS BUSINESS 17

complexity as introduced by Levin in 1984, canbe used to make assertions about the timecomplexity of deterministically generating n-bitprimes. Oliveira then turned to a randomisedanalogue of time-bounded Kolmogorov complexityintroduced by himself in 2019, which can beused to identify short and effective probabilisticprocedures that are likely to generate data-likeprime numbers. Using this notion, he explained thatthere are infinitely many primes admitting ‘short’and ‘effective’ probabilistic representations, andthat we cannot feasibly distinguish ‘structured’ from‘random’ strings. Throughout the talk he highlightedopen problems which demonstrate the fascinationand importance this line of research is offering

at the intersection of mathematics and computerscience.

The colloquium provided the audience withperspectives of many facets of algorithms,complexity and logic. The discussion after the talkswas lively. There was something for everyone totake away from the talks and discussion.

Ornela Dardha (University of Glasgow)Arnold Beckmann (Swansea University)

Charlotte Kestner (Imperial College London)Barnaby Martin (Durham University)

Prudence Wong (University of Liverpool)

The LMS Newsletter appears six times a year (September, November, January, March, May and July).

The Newsletter is distributed to just under 3,000 individual members, as well as reciprocal societies and other academic bodies such as the British Library, and is published on the LMS website at lms.ac.uk/publications/lms-newsletter.

Information on advertising rates, formats and deadlines are at: lms.ac.uk/publications/advertise-in-the-lms-newsletter.

To advertise contact Susan Oakes ([email protected]).

ADVERTISE IN THE LMS NEWSLETTER

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 18 — #18 ii

ii

ii

18 LMS BUSINESS

Records of Proceedings at LMS meetingsAnnual General Meeting and Society Meeting of the London MathematicalSociety: Friday 20 November 2020

The meeting was held virtually on Zoom, hosted by the International Centre for Mathematical Sciences.About 110 members and visitors were present for all or part of the meeting. The meeting began at3:00pm, with the President, Professor Jon Keating, FRS, in the Chair.

The President explained that, due to the covid-19 pandemic, the Society had adapted to adhere to UKsocial distancing measures in order to keep its members, guests and sta� safe. In doing so, there hadbeen an impact on the Society’s Governance in relation to its Standing Orders. Members and guestswere asked to:

• note Council’s decision to hold the Annual General Meeting virtually;• note that the Society was still using the old Standing Orders;• note that the virtual AGM technically breached the Standing Orders, and in particular that physicalvoting in person could not take place; and

• note that the Society had followed the Charity Commission’s guidelines on this issue and had informedthe Commission of our actions.

The Vice-President, Professor Iain Gordon, presented a report on the Society’s activities and thePresident invited questions.

The Treasurer, Professor Robert Curtis, presented his report on the Society’s �nances during the2019–20 �nancial year and the President invited questions.

The President introduced the members’ votes on four resolutions. As the voting was open to LMSmembers only, guests who were not Society members were placed in the ‘virtual’ waiting room for theduration of the vote, after which non-members were re-admitted to the meeting. This was in keepingwith the guidance from the Charity Commission that the Society should have a system in place toensure that only those eligible to vote could do so.

The minutes of the General Meeting held on 26 June 2020 had been circulated 21 days before the AnnualGeneral Meeting and members were invited to ratify the minutes by a completing an onscreen poll. Theminutes were rati�ed.

Copies of the Trustees Report for 2019–20 were made available and the President invited membersto adopt the Trustees Report for 2019–20 by completing an onscreen poll. The Trustees Report for2019–20 was adopted.

The President proposed Moore Kingston Smith be re-appointed as auditors for 2020–21 and invitedmembers to approve the re-appointment by completing an onscreen poll. Moore Kingston Smith werere-appointed as auditors for 2020–21.

The President introduced the fourth poll and advised members that, following a suggestion by theLMS Reps which was subsequently approved by Council in June 2020, the Society would be separatingthe Ordinary membership subscription rate into three tiers: low, middle and high, based on Members’annual professional salaries, as reported by the Members themselves. For the �rst membership year inwhich the new fee would be implemented (2021–22), the high rate would represent an increase of morethan 10% over the previous year’s rate for those Members a�ected. Statute 11 of the Society’s StandingOrders required the agreement of Members voting at a General Meeting where an increase of morethan 10% was proposed. An example of the tiered subscription rates would be:

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 19 — #19 ii

ii

ii

LMS BUSINESS 19

• Ordinary (high): £120.00 for members earning over £65,000• Ordinary (middle): £100.00 for members earning between £35,000 – £65,000• Ordinary (low): £80.00 for members earning up to £35,000.

Members at all tiers would retain the same bene�ts.

The President advised that Council was recommending the approval of the resolution to increase bymore than 10% the subscription rate for those Members paying a ‘high’ rate under the Society’s newthree-tiered subscription rate structure.

The President invited members to approve the resolution by completing an onscreen poll. The resolutionwas approved.

Guests were then re-admitted from the waiting room.

43 people were elected to Ordinary Membership: Babatunde Aina, Murat Akman, Anna Ananova,Eleftheriou Antonia, Ovidiu Bagdasar, Nicholas Baskerville, David Bate, Christopher Birkbeck, ChristophCzichowsky, Remy Dubertrand, Amit Einav, Josephine Evans, Goitom Fessahaye, Fernando Galaz-Garcia,Nicos Georgiou, Noel Giacometti, Andre Henriques, Nick Hills, Ashley Kanter, Tom Kempton, Dawid Kielak,Sergey Kitaev, Angeliki Koutsoukou-Argyraki, Robert Laugwitz, Brendan Masterson, Andrea Mondino,Pieter Naaijkens, Sarah Nagawa, James Newton, Davide Proment, Matthew Pusey, Yogendra KumarRajoria, Timothy Reis, Yue Ren, Hayley Ryder, Anuradha Sahu, Nicholas Simm, Ravendra Singh, TerrySoo, Pierpaolo Vivo, Graeme Wilkin, Julian Wilson and Andrew Wilson.

64 people were elected to Associate Membership: Edward Acheampong, Costanza Benassi, AndreaBoido, Alessio Borzì, Stephen Butler, Bromlyn Cameron, Giulia Carigi, Dimitrios Chiotis, Lily Clements,Andrea Clini, Daniel Cocks, Cristina Criste, Joshua Cromwell, Håvard Damm-Johnsen, Thibault Decoppet,Georgios Domazakis, Loki Dunn, Gilles Englebert, Sam Fearn, Samuel Fisher, Yanik-Pascal Foerster,Luca Gamberi, Wissam Ghantous, Katrina Gibbins, Tristan Giron, Sergio Giron Pacheco, Dewi Gould,Elena Hadjicosta, Grant Harvey, Anastasia Ignatieva, Philipp Jettkant, Panagiotis Kaklamanos, RajeshKaluri, Daniel Kaplan, Duncan Laurie, Michael Liedl, Abigail Linton, Joseph MacManus, Alexander Malcolm,David Malka, Grace El-Raphaella Matonga, Holly Middleton-Spencer, László Mikolás, Patrick Nairne, JohnNicholson, Chimezie Nnanwa, Shreenil Odedra, Chirag Pithia, Daniel Platt, Ellen Powell, Bhairavi Premnath,Michael Savery, Amit Shah, Thomas Sharpe, Eoin Simpkins, Julia Stadlmann, Matthew Staniforth, SamuelStark, Benedikt Stock, Angela Tabiri, James Taylor, Federico Trinca, Benjamin Ward and Finn Wiersig.

14 people were elected to Associate (undergraduate) Membership: Isobel Baddeley, Bhanu Banerjee,Matthew Bond, David Cawthorne, Max Durrant, William Evans, Rahul Gupta, Lloyd Hughes, Brian Judelson,Lynn Wei Lee, Yasmin Mussa, Glenn O’Callaghan, Qaisar Shah and Jiguang Yu.

12 people were elected to Reciprocity Membership: Daniel Asimov, Iyai Davies, Praveen Kumar Dhankar,Matt Insall, Shobha Lal, G. Muhiuddin, Ram Kripal Prasad, Margaret Readdy, Mansur Saburov, OnurSaglam, Subhrajyoti Saha and Jyoti Singh.

143 people were elected to Associate Membership for Teacher Training Scholars: Roba Abu Hantash,Aduragbemi Adebogun, Hasan Ahmad, Anees Ahmed, Kabir Akmal, Bayan Ali, Zahra Ali, Chris Allen,Henry Allen, Edward Antwi-Berchie, Joanna Aranowska, Dayana Arasarathnam, Thomas James Attrill,Jennifer Auger, Tayyib Azeem, Steven Barker, James Barlow, Ethan Barnes, Joseph Bashir, RebeccaBedford, Jamina Begum, Jessica Bradbury, Thomas Brasier, Francis Brooks-Tyreman, Hannah Buckley,Christan Butters, Milan Chrastina, James Colton, Etienne Corish, Samuel Corrigan, Nicholas Cossins,Stephen Cox, Joshua Cunningham, Edward Curr, York Deavers, Francis Denton, Louise Devaney, ConradDoggett, Thomas Donnelly, Ankita Dudani, Edet Efretuei, Christopher Ellis, David Errington, MehmetEsen, Olivia Faggi, Ibrahim Farooq, Ciprian Faur, Peter Fitzpatrick, Katherine Frewer, Oliver Green,

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 20 — #20 ii

ii

ii

20 LMS BUSINESS

Bethany Hall, Laura Hallmark, Mark Hamilton, Richard Hayes, Thomas Henshaw, Curtis Holmes, ChrisHolmes, Lawrence Holmes, Nicole Houston, Tara Howard, Grant Hubbard, Stuart Huntley, Joshua Jenkins,Sharmin Joarder, Joshua Johnson, Sarah Johnson, Steve Joiner, Emma Jones, Meggie Jordan, Rajdeep Josan,James Keable, Dasina Kerai, Azrah Khan, Charlotte Kingdom, Emily Larkin, Robert Little, Yun Liu, VictoriaLong, Ross Longden, Thomas Lowe, Muhammad Mamsa, Ethan Marks, Peter Mills, Elizabeth Morland,Simon Mullis, Angela Murphy, Mohammed Nasseri, Andrew Nolan, Alex Nuttall, Tim Oliver, FlorianaPacchiarini, Tilly Porthouse, Gareth Pugh, Katherine Purdy, Christopher Quickfall, Mohammed HabeebRabbani, Benjamin Raine, Zainab Razvi, Molly Reeve, Olivia Revans, Katy Rigby, Richard Robertson, KeithRochfort, Erin Rodrigues, Connor Rollo, Lucille Rostron, George Savage, Lauren Sealey, Eloise Sear, LucyShelley, Amber Shiels, Daniel Shipp, Callum Shreeve, Matthew Simcock, Chris Simpson, Alexander Sinclair,Joe Skerman-Gray, Sophie Slade, Sarah Slater, Edward Slater, Susannah Smith, Nicholas Stancill, ShaneSteele, Danny Sugrue, Megan Sullivan, James Taylor, Rachel Taylor, Benjamin Vickers, Beth Waghorn,Mark Walklin, Benjamin Walne, Kyle Ward, James Ward, Duncan Weaving, Joseph Webster, Toby Wehrle,Emma Wheeler, Francesca Williams, Luke Ryan Wilson, Crystal Wincy Wincent, Richard Winstanley, ChloeWong and Carina Yew-Booth.

No members signed the Members’ Book or were admitted to the Society. The President advised theaudience that, while the Society was unable to o�er the opportunity to members to sign it at thismeeting, the Members’ Book would once again be available for signing when face-to-face meetingscould be resumed.

The President invited non-members within the audience to join the Society and advised that detailsabout membership were on the Society’s website.

The President informed the audience that donations to the Society were most welcome and thatdonations, including to the De Morgan Donations scheme, could be made online.

The President invited members of the audience to congratulate the 2020 Prize-winners:

Pólya Prize: Professor Martin Liebeck (Imperial College, London)Senior Anne Bennett Prize: Professor Peter Clarkson (University of Kent)Senior Berwick Prize: Professor Thomas Hales (University of Pittsburgh)Shephard Prize: Regius Professor Kenneth Falconer (University of St. Andrews); Professor Des Higham(University of Edinburgh)Fröhlich Prize: Professor Françoise Tisseur (University of Manchester)Whitehead Prizes: Dr Maria Bruna (University of Cambridge), Dr Ben Davison (University of Edinburgh),Dr Adam Harper (University of Warwick), Dr Holly Krieger (University of Cambridge), Professor AndreaMondino ((University of Oxford), Dr Henry Wilton (University of Cambridge)LMS–IMA Christopher Zeeman Medal: Matt Parker

The certi�cates had been posted to the prize-winners.

The Scrutineer, Professor Chris Lance, announced the results of the ballot. The following O�cers andMembers of the Council were elected.

President: Professor Jon KeatingVice-Presidents: Professor Iain Gordon, Professor Catherine HobbsTreasurer: Professor Simon SalamonGeneral Secretary: Professor Robb McDonaldPublications Secretary: Professor John HuntonProgramme Secretary: Professor Chris ParkerEducation Secretary: Dr Kevin Houston

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 21 — #21 ii

ii

ii

LMS BUSINESS 21

Members-at-Large elected for two-year terms: Professor Peter Ashwin, Professor Anne-ChristineDavis, Professor Minhyong Kim, Professor Niall MacKay, Professor Anne Taormina, Dr Amanda TurnerMember-at-Large (Librarian): Dr Mark McCartney

Five Members-at-Large, who were elected for two years in 2019, have a year left to serve: ProfessorElaine Crooks, Professor Andrew Dancer, Dr Tony Gardiner, Dr Frank Neumann and Professor BritaNucinkis.

The following were elected to the Nominating Committee for three-year terms: Professor Chris Buddand Professor Gwyneth Stallard. The continuing members of the Nominating Committee are: ProfessorKenneth Falconer (Chair), Professor I. David Abrahams, Professor Beatrice Pelloni, Professor Mary Reesand Professor Elizabeth Winstanley. One member of Council will also be nominated to the NominatingCommittee.

Professor Nicholas J. Higham, University of Manchester, gave the Naylor Lecture 2020 on The Mathematicsof Today’s Floating-Point Arithmetic.

Before closing the meeting, Professor Keating thanked the retiring members of Council and welcomedthe President Designate Professor Ulrike Tillmann, FRS.

Professor Keating also thanked the speaker at the Graduate Student Meeting on 16 November 2020Theo Mary (Sorbonne), and congratulated the winners of the Graduate Student Talk Prizes: CarmenCabrera-Arnau (UCL), Giulia Carigi (Reading), Carl-Fredrik Nyberg Brodda (UEA), Onirban Islam (Leeds),Raad Kohli (St. Andrews) and Gustavo Rodrigues Ferreira (Open University). The President thanked theother 12 graduate students who also gave talks.

The President thanked everyone who had worked to organise the online Annual General Meeting. ThePresident closed the meeting. There was no reception or Annual Dinner.

Records of Proceedings at LMS Meetings:Society Meeting at the Joint Mathematics Meeting 2021

This meeting was held virtually on Zoom, at the Joint Mathematics Meeting co-hosted by the AmericanMathematical Society (AMS) and the Mathematical Association of America (MAA). Over 15 members andvisitors were present for the LMS meeting session.

The Society meeting began at 5.00pm GMT on 7 January with the Publications Secretary, ProfessorJohn Hunton, in the Chair. Professor Hunton welcomed guests, thanked the organising parties, and thenintroduced Professor Tim Browning who spoke about the new Proceedings of the London MathematicalSociety. Professor Browning then introduced a lecture given by Professor Sarah Zerbes (UCL) on SpecialValues of L-Functions.

Professor Hunton concluded the meeting by thanking Professor Zerbes, the organisers and the meetingattendees on behalf of the LMS.

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 22 — #22 ii

ii

ii

22 FEATURES

Joining the De Morgan House Team — One Year On

CAROLINE WALLACE

LMS Executive Secretary Caroline Wallace re�ects on her �rst eventful year in the role.

I became ExecutiveSecretary of the LondonMathematical Societyin early April last year.Someone asked merecently what hadattracted me to the role.There were several partsto my answer.

First and foremost, I love mathematics. In particular,I have a great interest in how it can be put to useto improve our lives. This interest intensified when,while still at school, I read the wonderful bookMathematics and the Imagination by Edward Kasnerand James Newman. I am the proud inheritor ofmy mother’s 1952 edition. It was given to her asa prize when she was a young woman, as she lefther secondary school in the rural far north of NewZealand, and, unusually for a woman in that timeand place, headed to university.

Pursuing my interest in the applications ofmathematics, I gained a degree in engineering atthe University of Cambridge and I spent the firstpart of my career working in industry. I can honestlysay that the ‘real world’ power of mathematics wasevident to me every day.

The second part of my answer to the questionposed is that I was very impressed by the historyand the reputation of the London MathematicalSociety. It has a strong and clear mission: toadvance, disseminate and promote mathematicalknowledge. It is an organisation that has stoodthe test of time and that has always placed highvalue on the (still very topical) virtues of reasoningand research. Yet it also remains a relatively smallSociety where an individual can make a difference.

And this leads directly to the third part of my answerto the question posed. Not only does the Societyhave an incredible history, but it is also a charity.It seeks to do good in the world by supportingmathematical research and mathematicians and itpromotes equality of opportunity. I �nd it highly

motivating to work in an organisation that contributesin such a concrete way to the public good. Forexample, the Society’s involvement in the LevellingUp Scheme, enabled by the extremely generoussupport of our donor, Dr Tony Hill, has the potentialto increase the aspirations and the attainmentof A-level maths students from under-representedgroups across the country. You can read more aboutthis Scheme on page 13.

As I approach the end of my �rst year as theSociety’s Executive Secretary, it is clear to me thatI am very fortunate to have this role. As I notedearlier, the Society has a clear mission and it has astrong desire to achieve that mission. I lead a skilledand knowledgeable sta� team. There are strongrelationships amongst the sta� and between thesta�, the Council, the Membership and the widervolunteer community.

The book plate for mymother’s copy ofMathematics and theImagination

This is not to say that ithas all been plain sailing.I took up my role atthe Society two weeksafter the �rst covid-19lockdown began. While itwas somewhat temptingto focus in this article onthe impact of covid-19,I did not want thepandemic to dominatemy re�ections on my�rst year with theSociety.

This is nonetheless agood opportunity tonote that sta� recogniseand share the challenges

that the covid-19 pandemic has created for themathematics community. This includes amongstmany other things managing greater caringresponsibilities, limited workspace, altered workexpectations and looking after our physical andmental wellbeing. I continue to be impressed by

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 23 — #23 ii

ii

ii

FEATURES 23

and deeply grateful for the �exibility and resilienceof my colleagues as they cope with the suddenand disconcerting changes that the pandemic hasimposed on all of us.

I am also impressed by — and very proud to bepart of — the Society’s response to the pandemic.The Society has demonstrated its willingness tolisten to its Membership and to change rapidlywhat it does and how it does it. It has madeadditional Early Career Fellowships available and givenadditional funding to Research Groups to produceonline lectures. It has pushed back its deadlines forLMS prize nominations, it has moved its meetingsand events online and it has explored ‘virtual’exhibition stands at conferences where face-to-faceattendance is impossible. Recently, it has sought toimprove the signposting not just to its own but toother organisations’ funding opportunities. And I ampleased to say that there is more in the pipeline.

One of the many e�ects of the pandemic is thatit has greatly reduced the opportunities for me tomeet and get to know the Society’s Members. Ihave attended as many online Society meetings andevents as possible. Unfortunately, it is just not thesame as an ‘in person’ meeting, at which there wouldbe a chance over co�ee to introduce ourselves andtalk about the latest developments at the Society.But if you do see me at an online meeting and wouldlike to say hello, please just message me in the chatand hopefully we can arrange a separate call.

In the meantime, there are reasons to be hopefulas the vaccination programme is rolled out, as wecontinue to learn about the bene�ts of remoteworking, and as we plan for how we can retainthe bene�ts of remote working (not least, reducedenvironmental impact) in our new post-covid world.I am looking forward to my second year with theSociety!

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 24 — #24 ii

ii

ii

24 FEATURES

Cover Image: Monte Carlo Simulation

IAN FLOOD

We are seeing unprecedented demand forradio spectrum with the advocates of emergingtechnologies pressing for access to frequency bandsalready populated by established services. This leadsto studies of the radio interference environment withspectrum managers increasingly motivated towardsconsideration of mathematical models and spectrallye�cient solutions.

The cover image is a snapshot of one step in a MonteCarlo simulation using Visualyse Professional software.The image shows a somewhat abstract model ofinterference sources in a mobile network deploymentcentred on St Louis, Missouri; local terrain featurescan be seen. The yellow markers represent sourcesof interference in the city, and the blue markersare sources of interference in the rural environment.There is one source for each mobile network cell andthe black markers show the locations of cell centres.

At each step in the simulation, the locationof an interference source is random within itscell allowing for an extensive search of possibleinterference geometries. The model calculatesaggregate interference ΣI from all sources incidentto a single victim receiver at the centre of the city.ΣI can be expressed in decibels relative to a unit ofsignal power in a speci�ed bandwidth. The resultsfrom our simulation can be presented as a graph ofthe Complimentary Cumulative Distribution Function(CCDF).

In general, spectrum engineers are concernedwith evaluating interference in relation tointerference protection criteria which can takeseveral forms. Typical examples are an aggregateinterference-to-noise ratio ΣI /N expressed indecibels, or a simple threshold for aggregateinterference ΣIT . Considering an aggregateinterference threshold, if ΣI = ΣIT then our criterionis satis�ed exactly, but if ΣI > ΣIT then excessinterference is incident to the receiver. This maybe acceptable if the criterion is associated with aconstraint which allows the threshold to be exceededfor a speci�ed percentage of time and this constraintis satis�ed. We can easily use our graph of the

CCDF to test such criteria when working in thetime domain, but this has been the focus of somediscussion recently as many simulations attemptto model large-scale network deployments and,because of uncertainties, include some variability inthe deployment domain.

If the victim receiver and interferer are co-frequencyin this simulation, with all potential sources ofinterference switched on, the modeller will not besurprised to �nd excess interference at the receiver.

When an interference protection criterion isexceeded, the modeller may consider mitigation. Onemethod is to calculate the radius of a zone aroundthe victim receiver where interference sources areexcluded; this might be appropriate if the receiveris part of an important installation and at a �xedlocation.

Another approach is to introduce a frequencyseparation between interferer and victim receiver.Here, spectrum masks, characterising emissionsfrom the interfering transmitters and the response ofreceiver �ltering to incident signals, can be modelled.A convolution of these masks allows for the Net FilterDiscrimination to be calculated at discrete frequencyseparations; this is the discrimination, expressed indecibels, available at the victim receiver when theinterferer is o�set in frequency.

FURTHER READING

[1] J. Pahl, Interference Analysis, Modelling RadioSystems for Spectrum Management, Wiley,Chichester, UK, 2016.[2] NTIA, Interference Protection Criteria. Phase 1- Compilation from Existing Sources, NTIA Report05-432, U.S. Dept. of Commerce, Oct. 2005.

Ian FloodIan is a consultant with Trans�nite Systems, London.His work involves modelling spectrum sharingproblems. He is a Chartered Engineer and holds aPhD in graph-theoretic studies.

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 25 — #25 ii

ii

ii

FEATURES 25

Notes of a Numerical Analyst

At the Edge of Infinity

NICK TREFETHEN FRS

2n is bigger than n, and Cantor showed this is trueeven when n is in�nite. The theory is beautiful,and most of us know the basics. But we are easilycaught o� guard by �nite numbers when they arebig enough.

Take this equation adapted from [2], with (!) as awarning that equality does not actually hold:

1010428,000

= e 10428,000

. (!)

Of course the two numbers aren’t really equal — theirdi�erence is enormous. Yet they are indistinguishableif you regard the top exponent as known to justthree digits, for the number on the left is equal toexp(10428,000.36...). Or consider this one:(

1010428,000 )2

= 1010428,000

. (!)

This time the number on the left is equal to10ˆ(10428,000.30...). Evidently the familiar rules ofarithmetic break down, in a practical sense, whennumbers are huge, giving us principles like

n2 = n, 2n > n . (!)

Note how these formulae echo Cantor’s results fortrue in�nities, which we can write in shorthand as

∞2 = ∞, 2∞ > ∞.

For another curiosity at the edge of in�nity, let Abe an in�nite “random Fibonacci matrix” with zeroentries everywhere except ±1 (independent cointosses) on the �rst two superdiagonals, i.e., entriesa j ,j+1 and a j ,j+2 [3]. The spectrum Σ of A as anoperator on ℓ 2 is the closed disk |z | ≤ 2 (withprobability 1), which we can explain by noting that Acontains arbitrarily large regions where all the signsare equal. Yet spectral theory is missing somethingessential about A if we view it as a limit of matricesAn of �nite dimension n. In an inner region Σi ⊂Σ, roughly the disk |z | < 1.3, the resolvent norm‖(z − An)−1‖ grows exponentially as n → ∞, butin the remainder of Σ it grows only algebraically,as shown by the plot of log10 (‖(z − An)−1‖) inFigure 1 for a matrix of dimension 200. The crownof this “witch hat” is very tall (truncated raggedly

by �oating-point arithmetic), but the brim is �at. Ifa physical system were governed by such matrices,the spectrum measured in the lab would probablybe Σi , not Σ.

Figure 1. Random Fibonacci witch hat

Mathematics has a wonderful ability to reasonrigorously about idealisations. Sometimes it is goodto remember, however, that they are idealisations. Inmoral philosophy, the �eld of “in�nite ethics” drawsconclusions based on the supposition that there maybe in�nitely many worlds with in�nitely many sentientbeings, including a creature epsilon close to my ownself down to the home address and the children’snames [1]. Personally, I �nd it hard to believe thatanything is quite that in�nite.

FURTHER READING

[1] N. Bostrom, In�nite ethics, Analysis andMetaphysics, 10 (2011), 9–59.[2] S. J. Chapman, J. Lottes, and L. N. Trefethen,Four bugs on a rectangle, Proc. Roy. Soc. A, 467(2010), 881–896.[3] L. N. Trefethen and M. Embree, Spectra andPseudospectra, Princeton, 2005.

Nick TrefethenTrefethen is Professor of NumericalAnalysis and head of the NumericalAnalysis Group at the University ofOxford.

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 26 — #26 ii

ii

ii

26 FEATURES

Mathematics News Flash

Jonathan Fraser reports on some recent breakthroughs in mathematics.

It is a pleasure to begin by thanking Aditi Kar for initiating the ‘News Flash’ section of the Newsletter and forwriting many beautiful mathematical vignettes. I will do what I can to follow in her footsteps. The three papersdiscussed below establish deep results with easily understood statements. I hope the readership enjoys themas much as I did.

Flat Littlewood polynomials exist

AUTHORS: Paul Balister, Béla Bollobás, Robert Morris,Julian Sahasrabudhe and Marius TibaACCESS: https://arxiv.org/abs/1907.09464

A Littlewood polynomial is a polynomial whosecoe�cients are all either +1 or −1. Littlewoodconjectured in 1966 that there should exist constantsa,b > 0 such that for every n ≥ 2 there exists aLittlewood polynomial P of degree n such that

a√n ≤ |P (z ) | ≤ b

√n

for all z ∈ ℂ with |z | = 1. This paper con�rmsLittlewood’s conjecture and was published in Annalsof Mathematics in 2020. As the authors explain, themost challenging part of the proof was establishingthe lowering bound. Explicit polynomials satisfyingthe upper bound only had been constructed byShapiro and Rudin over 60 years ago.

Littlewood’s problem, and the solution describedabove, have implications in autocorrelation for binarysequences. Autocorrelation refers to the correlationof a signal with a delayed copy of the signal as afunction of the delay.

The group of boundary �xing homeomorphismsof the disc is not left-orderable

AUTHORS: James HydeACCESS: https://arxiv.org/abs/1810.12851

A group G is said to be left-orderable if it admits atotal order < such that for all f , g ,h ∈ G ,

f < g ⇔ h f < hg .

The integers under addition is the archetypalexample of a left-orderable group, and amore sophisticated example is the group ofhomeomorphisms of the unit interval which �xthe endpoints. The 2-dimensional analogue of thislatter example became notorious: is the group ofhomeomorphisms of the closed disk which pointwise�x the boundary left-orderable? This question was

posed in several esteemed circles (pun intended)including in a paper of Navas published in theProceedings of the ICM in 2018 and in the famousKourovka Notebook.

Hyde answered this question in the negative withan ingenious construction of a �nitely generatedsubgroup which is itself not left-orderable. Thisremarkable paper is only �ve pages long and waspublished in the Annals of Mathematics in 2019.

On the Lebesgue measure of the FeigenbaumJulia set

AUTHORS: Artem Dudko and Scott SutherlandACCESS: https://arxiv.org/abs/1712.08638

The Julia set of a polynomial P : ℂ → ℂ is theboundary of the set of points whose orbit underP remains bounded. Julia sets are typically intricatefractal sets. Dudko and Sutherland consider the Juliaset of the infamous Feigenbaum polynomial z ↦→z 2 + cF where cF ≈ −1.401155 . . . . This polynomialis especially di�cult to study due to the dynamicsassociated with the critical point: subtle behaviourwhich occurs only with delicate choice of cF . Themain result of this paper, published in InventionesMathematicae in 2020, is that the Lebesgue measureof the Feigenbaum Julia set is zero. This answersa famous open question in complex dynamics. Infact, the authors prove the stronger statement thatthe Hausdor� dimension of the Julia set is strictlyless than 2. The proof uses a computer programmeto rigorously establish that a certain condition issatis�ed.

Jonathan Fraser is a Reader ofMathematics at the University ofSt Andrews. His research interestscentre on fractal geometry. He ispictured with his son, Dylan, whomakes his second appearance in theNewsletter.

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 27 — #27 ii

ii

ii

FEATURES 27

Penrose’s Incompleteness Theorem

MIHALIS DAFERMOS

On the occasion of Roger Penrose’s 2020 Nobel Prize in Physics, I discuss his remarkable incompletenesstheorem and its legacy for understanding black holes and singularities in general relativity.

This past year, Roger Penrose was awarded the2020 Nobel Prize in Physics. The accompanyingpress release [21] cites his 1965 paper ‘Gravitationalcollapse and space-time singularities’ [13] below.

Penrose’s [13] as published in Physical Review Letters

On the surface, this is a very unusual citation fora Physics Nobel: Penrose’s [13] did not proposea new theory, formulate a new equation, ordiscover a new explicit solution, achievements whichphysicists more readily appreciate and celebrate.Despite appearing in Physical Review Letters, [13]is a quintessentially mathematical paper, sketchingthe proof of a theorem—indeed, a theorem ofpure geometry. Yet it is hard to exaggerate howprofoundly this theorem in�uenced the way all ofus—mathematicans, physicists and even the widerpublic—today understand general relativity.

In this article, I will try to introduce Penrose’sincompleteness theorem of [13] and its legacy to abroad mathematical audience. To set the stage, letme �rst describe brie�y the mathematical structureof Einstein’s celebrated theory of general relativity.

General relativity

General relativity postulates a uni�ed structure,a Lorentzian metric g de�ned on a 4-dimensionalmanifold M—spacetime—governing gravitation,inertia and what we perceive as time and geometry.

Lorentzian metrics g are the analogue of the morefamiliar Riemannian metrics, except that they have (in

4-dimensions) signature (−,+,+,+). This just meansthat suitable local coordinates (x0,x1,x2,x3) arounda spacetime point p ∈ M can be chosen such thatthe metric gp at p may be written as

gp = −(dx0)2 + (dx1)2 + (dx2)2 + (dx3)2.

While for Riemannian metrics, gp (v,v) = 0 wouldimply v = 0, in Lorentzian geometry, the set

Np = {0 ≠ v ∈ TpM : gp (v,v) = 0}

forms a double cone in the tangent space TpMwhich can be viewed as an in�nitesimal version ofMinkowski’s light cone of special relativity (whereunits have been chosen such that the speed of lightc = 1). We call such vectors v ∈ Np null vectors.

The cone bounds the set of so-called timelike vectors

Ip = {v ∈ TpM : gp (v,v) < 0}.

We always assume that it is possible to select adistinguished connected component ofNp dependingcontinuously on p . This de�nes the so-called futurenull cone N +p , which in turn bounds a connectedcomponent I +p of Ip . We refer to vectors v ∈ N +p asfuture null and v ∈ I +p future timelike.

These concepts have immediate physicalinterpretation: test particles traverse curves W(g) inspacetime whose tangent W′(g) is future timelike,i.e. W′(g) ∈ I +

W (g) . Such curves are called worldlines.The integral∫ g2

g1

√−gW (g) (W′(g), W′(g))dg

is known as proper time, the time of local physicalprocesses, like a human observer’s heartbeat. In thecase of proper time parametrisation, where g (W′, W′) =−1, the vector W′ is known as the 4-velocity. Ifthe test particles are ‘freely falling’, then theseworldlines W(g) must in fact be geodesics of g(de�ned just as in Riemannian geometry). Lightrays traverse future directed null geodesics of themetric, i.e. geodesics with W′(g) ∈ N +

W (g) . Since such

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 28 — #28 ii

ii

ii

28 FEATURES

geodesics are in general curved, general relativitypredicts the ‘bending of light’.

The interaction of gravitational �elds andmacroscopic matter in the theory is provided by theso-called Einstein equations. These were formulatedby Einstein [5] in November 1915 and take the form

Ric(g ) − 12Scal(g )g = 8cT. (1)

Here, Ric(g ) denotes Ricci curvature of g andScal(g ) denotes scalar curvature, both de�nedjust as in Riemannian geometry. (Recall that theRicci curvature is itself a certain average of thefull Riemann curvature tensor Riem(g ), whereasthe scalar curvature is simply the trace of theRicci curvature. The precise formulae are of noparticular relevance for this discussion.) The objectT on the right hand side of (1) is the so-calledstress-energy-momentum tensor of matter. We willsee an example of such a tensor later on. (Inwriting (1), we note that we have chosen units suchthat in addition to c = 1 the Newtonian gravitationalconstant satis�es G = 1.)

One should view the Einstein equations (1) as thegeneral relativistic analogue of the Poisson equation

Δq = 4cd (2)

describing the Newtonian gravitational potentialq generated by the total mass density d ofmatter. We note already, however, some fundamentaldi�erences: equation (2), at �xed t , can be consideredas a linear elliptic equation completely determiningq from d and appropriate boundary conditions atin�nity. Thus, in Newtonian theory, gravity is onlynon-trivial in the presence of matter. In contrast,equation (1) is non-trivial already where T = 0 globally,in which case it simpli�es as:

Ric(g ) = 0. (3)

These are the Einstein vacuum equations.Equations (3) in fact constitute a nonlinear hyperbolicsystem with a well-posed initial value problem.

Equations (1) must in general be supplemented withequations for matter �elds, which are in turn coupledto (1) via the stress-energy-momentum tensor T. Forall conventional matter however, T satis�es certainnon-negativity properties, just as the mass density d

in Newtonian theory satis�es d ≥ 0. The most basicof these properties is the statement that T(v,v) ≥0 for any null vector v, in which case the Einsteinequations (1) imply the following inequality:

Ric(v,v) ≥ 0. (4)

It is worth noting already that equation (3), and moregenerally inequality (4), encode geometric content,analogous to that encoded in positivity assumptionsconcerning Ricci curvature in Riemannian geometry.

Both the evolutionary pde point of view on (3) and thegeometrical point of view on (4) will be essential toour story. We are already getting ahead of ourselves,however. Let us �rst return to 1915!

The Schwarzschild solution and the problem of‘singularity’

The problem of ‘singularity’ plagued Einstein’s theoryessentially from its inception. The issue arosealready in connection with the �rst non-trivialsolution of (3) to be discovered—only weeks afterEinstein’s formulation of the equations—namely thatof Schwarzschild [19]. Let me brie�y describe thismetric and the issues it gave rise to.

In local coordinates (t ,r , \, q) the Schwarzschildmetric can be written as

g = −(1 − 2m/r )dt2 + (1 − 2m/r )−1dr 2

+ r 2 (d\2 + sin2 \dq2). (5)

With a little bit of computation, the keen reader canexplicitly check that (5) indeed satis�es the vacuumequations (3) for all values of parameter m ∈ ℝ. Notethat ifm = 0, the expression (5) simply reduces to the�at Minkowski metric of special relativity, expressedin spherical polar coordinates.

The early well-known triumphs of general relativity(explaining anomalous precession of the perihelionof Mercury, predicting the bending of light [11]) can beeasily deduced from (5), interpreting it as the vacuummetric outside a spherically symmetric star of massm and radius R, measured in appropriate units.

As often happens, however, together with triumphcame new problems! In particular, if m > 0, thenthe expression (5) de�ning the Schwarzschild metricseems to be inadmissible at r = 2m, where the metriccoe�cient gr r manifestly blows up.

In the context of actual stars as understood at thetime, the issue seemed academic: typical stars haveR � 2m in these units, so there is no problem if oneonly considers (5) for r > R. However, a prophetic1939 paper [12] by Oppenheimer and Snyder—a

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 29 — #29 ii

ii

ii

FEATURES 29

paper little understood in its time—showed that thequestion was not as academic as �rst seemed! Thepoint was that one should not restrict to static stars,but allow also collapsing ones.

We will discuss [12] in the next section. Brie�y, [12]constructs a spherically symmetric spacetime (M, g )solving (1) coupled to the equations for a perfect �uid,where the �uid is pressureless, and thus

T = d u♭ ⊗ u♭, (6)

where d denotes the rest mass density and u♭ the1-form dual to the �uid 4-velocity. The precise form gtakes in the support of (6) is not important here. Weremark only that outside the support, the solution isboth vacuum and spherically symmetric, and thus (byBirkho�’s theorem [11]) necessarily coincides with (5).The support of the matter has the interpretation ofa collapsing star. If g denotes the proper time of afreely falling observer W(g) on the boundary of thestar, then its radius R (W(g)) → 2m as g → gcriticalfor some gcritical < ∞. Thus, one does have to facethe limit r → 2m in (5) after all!

The paper [12] did not quite make explicit whathappens to freely falling observers W when or afterthey reach r = 2m. It turns out, however, that thishad in fact already been understood (at least inthe vacuum region) by Lemaitre [9], who explicitlyextended the metric (5) across r = 2m. It is inretrospect remarkable that this question caused asmuch confusion as it did, since it su�ces to de�ne

v = t + r + 2m log |r − 2m |, (7)

in which case the metric (5) transforms into

−(1−2m/r ) (dv )2+2dvdr +r 2 (d\2+sin2 \dq2). (8)

This metric is manifestly regular for all r > 0 and allv ∈ ℝ. We will discuss its behaviour at r = 0 later!

Even more astonishing than the initial confusion itself,however, was how long it took the correct solutionto become common knowledge in the physicscommunity! Indeed, as late as 1958, Lemaitre’s workwas being rediscovered, for instance by Finkelstein [6].This turned out to be quite fortuitous for our story,as it was from a lecture of Finkelstein in London thatPenrose was to learn about this extension [17].

Oppenheimer–Snyder à la Penrose

The starting point for Penrose’s seminal [13] isprecisely a lucid presentation of Oppenheimer–Snyder

collapse, including Lemaitre’s extension as he learnedit from Finkelstein. The geometry of the spacetimeis illustrated by a diagram reproduced here.

It is worth walking through this depiction: good visualaides are central to Penrose’s work!

Diagram of Oppenheimer–Snyder spacetime fromPenrose’s paper [13]

The hypersurface C 3 is a 3-dimensional spacelikeslice of spacetime M4

+, i.e. a hypersurface on whichthe induced metric is Riemannian; one may viewC 3 as representing space at an instant. There isan initial ball of matter on C 3 of radius R0 > 2m,and its world tube through spacetime is depicted(labelled ‘matter’). This is where the �uid energymomentum (6) is supported. Outside the support,M4+ is vacuum, and thus, described by the Lemaitre

extension (8) of Schwarzschild. Note level surfacesv = const depicted, with v increasing moving up.

As a Riemannian manifold, C 3 is as nice as canbe; its metric is complete and asymptotically �at.(Asymptotic �atness, the condition that the metricapproach Euclidean at large r , is the analogue ofthe boundary condition q→ 0 in Newtonian theorygoverned by (2).) Thus, we may view M4

+ as ‘evolving’from a physically admissible initial state. In moretechnical terms, C 3 is in fact a Cauchy hypersurface,which, in pde language, means that its data determine

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 30 — #30 ii

ii

ii

30 FEATURES

uniquely the spacetime (M4+, g ) as a solution of (1)

with (6), and (M4+, g ) is in fact the so-called maximal

Cauchy development of these data. We will return tothis issue later!

The cones depicted in the diagram are preciselythe future light cones N +p at each spacetime pointp . Drawing these cones allows us to pick out bysight worldlines and trajectories of light rays (suchcurves are collectively known as causal). Note thatas depicted in the diagram, the N +p tilt inwards (withrespect to the r coordinate) compared with �atMinkowski space, tilting more and more as r → 0.

Considering the radius R (v ) of the support of thematter (6) as a function of v , one may moreover inferfrom the diagram that there is aVcrit (not labelled) forwhich R (v ) → 0 as v →Vcrit, while for v > Vcrit, thespacetime is vacuum and thus completely describedby (8). At least for v > Vcrit, one sees easily that thefuture null cones N +p on r = 2m are in fact tangent tor = 2m, with all other null directions pointing insider ≤ 2m, i.e. w(r ) ≤ 0 for all w ∈ N +p , where w(r )just denotes the action of vectors on functions bydi�erentiation. For r < 2m, we have w(r ) < 0 forall w ∈ N +p . Thus, if W(g) is a future causal curveand r (W(g0)) < 2m, v (W(g0)) > Vcrit then r ′(W(g)) <0 for all g ≥ g0, and so r (W(g)) < 2m. It followsin particular that there is a non-empty region ofspacetime which cannot send signals to far awayoutside observers for which r is large. This regionwould later be named the black hole region [11].

Finally, we notice that all worldlines which enter theblack hole region eventually reach r = 0. An easycomputation with (8) reveals that this in fact happensin �nite proper time, that is to say, given any worldlineW(g) parametrised by proper time g, then thereexists a gmax such that r (W(g)) → 0 as g → gmax.Similarly, future null geodesics entering the interior ofthe black hole reach r = 0 in �nite a�ne time, whilethere is the marginal case of those remaining for alla�ne time on the boundary, along which eventuallyr = 2m. This boundary came to be known as theevent horizon. Examples of null geodesics lying onthis horizon are depicted on the diagram.

We now �nally turn to discuss r = 0. The spacetimeM cannot be extended to include r = 0, at least notin a suitably regular fashion, as is clear by computingthe Kretschmann scalar K , a contraction of thetensor Riem ⊗ Riem, which equals K = 48m2r −6

and thus diverges as r → 0. This thus representsa singularity, where physical quantities diverge, andpresumably, general relativity itself breaks down.

Let us note already that there is another way ofsaying that something is ‘wrong’ with spacetime,without explicitly talking about the ‘singularity’ atr = 0. We say a future causal geodesic (i.e. onewith W′(g) ∈ I +

W (g) ∪ N+W (g) ) is future complete if it

can be extended to be de�ned on [g0,∞), otherwise,future incomplete. Geodesics with r (W(g)) → 0 asg → gmax are incomplete in view of the above, andsince it contains such geodesics, M4

+ is itself said tobe future causally geodesically incomplete.

In the above example, geodesic incompletenessseems intimately tied both with the black hole regionand with the presence of the r = 0 ‘singularity’.Future causal geodesics in Oppenheimer–Snyderturn out to be future incomplete if and only if theyapproach r = 0, in fact if and only if they enterthe interior of the black hole region. In general,however, it is trivial to come up with spacetimes withfuture incomplete causal geodesics but which are inno reasonable sense ‘singular’ nor have black holeregions: one can just remove appropriate sets fromMinkowski space. We shall return to this issue later!In the meantime, let us introduce Penrose’s theorem.

Trapped surfaces and incompleteness

The Oppenheimer–Snyder spacetime depicted aboveis all well and good, but at the end of the day, it is justa single explicit solution of the equations (1)—and avery symmetric one at that. Moreover, the singularbehaviour it exhibits corresponds to r → 0, andit is natural to expect that behaviour there isvery sensitive to perturbation away from symmetry.Indeed, on the eve of the appearance of [13], thegeneral belief was that all this strange causal andsingular behaviour—largely misunderstood in anycase—was an artefact of symmetry [17].

The key to address, and �nally frustrate, thisexpectation was a profound new concept, geometricin nature: that of a closed trapped surface.

Brie�y, a closed trapped surface is a compactspacelike 2-surface T 2 (without boundary) such thatits area element at every p ∈ T 2 is in�nitesimallydecreasing in both future null directions orthogonalto T 2. (Compare with usual spheres in Minkowskispace decreasing in one and increasing in the other.)A more precise de�nition of this is given in the box.

In Oppenheimer–Snyder spacetime, any surface S 2

of constant (v ,r ) with r < 2m lying in the vacuum

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 31 — #31 ii

ii

ii

FEATURES 31

region of the spacetime is in fact a closed trappedsurface. (This is clear since w(r ) < 0 in this regionforw ∈ N +p .) An example of such a surface is labelledon the diagram. Let us already note, moreover,that the presence of such a surface is stable toperturbation of spacetime. This follows trivially fromthe compactness of S 2 and the fact that trappednessis de�ned by strict inequalities.

Trapped surfaces

If T 2 is a spacelike 2-surface (i.e. its inducedmetric is Riemannian), then for all p ∈ T 2 onecan de�ne two unique future null normals Land L to the tangent space of T 2 at p . WecallT 2 trapped if tr (X ,Y ) ↦→ g (∇X L,Y ) andtr (X ,Y ) ↦→ g (∇X L,Y ) are both negative,where X ,Y are tangent to T 2. Note that thenull geodesics generated by L and L span thetwo components of boundary of the causalfuture of T 2, and thus this de�nition can beinterpreted as saying that the area element ofT 2 is in�nitesimally decreasing as it is �owedalong either component of the boundary.

Given the notion of closed trapped surface, Penrose’stheorem is incredibly simple to state, and, as it turnsout, not so di�cult to prove (see the Box at the endof the article), by methods of global geometry:

Theorem 1 (Penrose’s incompleteness theorem [13]).If (M, g ) is su�ciently smooth, admits a non-compactCauchy hypersurface, contains a closed trapped surfaceand satis�es the inequality (4) for all null v, then it isfuture-causally geodesically incomplete.

Closed trapped surfaces thus are ‘surfaces of noreturn’ which, once present, ensure incompleteness!

Note how the theorem’s assumptions and conclusionare indeed exhibited for Oppenheimer–Snyderspacetime (M4

+, g ) itself. In particular, the curvatureinequality (4) for null v follows from the Einsteinequations (1) and the de�nition (6), in view of thefact that the �uid 4-velocity u satis�es u ∈ I +p .

As remarked already, geodesic incompleteness inthe special case of Oppenheimer–Snyder seems tobe intimately connected to both black holes andsingularities. It is thus tempting to interpret thistheorem as predicting these. (Indeed, the traditionalname for the above theorem, which we have avoidedhere for reasons we shall return to later, is the

Penrose singularity theorem.) As we shall see, thisinterpretation is not in fact correct, and the truesituation is far more interesting. Before trying toexplain, let us introduce the evolutionary point ofview, which will be essential for what follows.

The evolutionary point of view

The true signi�cance of Penrose’s theorembecomes apparent by interpreting it in an explicitlyevolutionary context.

The precise language in order to do this was not infact available in 1965, but was clari�ed a few yearslater in a paper [1] of Choquet-Bruhat and Geroch,which introduced the notion of the maximal Cauchydevelopment. This important concept is explainedfurther in the box. Brie�y, given appropriate initialdata for (1) on a 3-manifold C 3, this is the biggestspacetime (M, g ) of (1), together with the matterequations, admitting C 3 as a Cauchy hypersurface.With this, one may now talk of a unique spacetime(M, g ) which is ‘predicted’ by general relativity frominitial data, resolving the ambiguity of domain, that,as discussed earlier, would allow for many trivialexamples of geodesically incomplete spacetimes.

The maximal Cauchy development is the object towhich one should apply Penrose’s incompletenesstheorem. We may in particular state the following:

Corollary 1. For all initial data su�ciently close to dataon C 3 in Oppenheimer–Snyder collapse, the resultingmaximal Cauchy development (M, g ) will still be futurecausally geodesically incomplete.

In deducing the above, we have used also thefact that the presence of a closed trappedsurface is stable not just to perturbation ofspacetime but to perturbation of initial data, bygeneral Cauchy stability arguments. Thus, geodesicincompleteness of the maximal Cauchy developmentis an inescapable prediction of the theory, followingfrom assumptions expressible on initial data alone,robust to perturbation. Incompleteness cannot beavoided by perturbing the initial data.

Note that there are pure vacuum solutions like (8)which contain closed trapped surfaces. A moreinteresting question, however, is whether closedtrapped surfaces can form in vacuum (3) frominitial data which don’t initially contain trappedsurfaces, just like Oppenheimer–Snyder at the initial

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 32 — #32 ii

ii

ii

32 FEATURES

hypersurface C 3. This was shown to be true in [3].As in Corollary 1, it then follows that all initial datasuitably close to those of [3] again lead to anincomplete maximal Cauchy development.

The initial value problem and themaximal Cauchy development

For convenience, let us restrict to the vacuumequations (3), although similar considerationsapply for a wide class of Einstein mattersystems. A vacuum initial data set is atriple (C 3, g ,K ) with C 3 a 3-manifold, g aRiemannian metric on C 3 and K a symmetric2-tensor, such that g and K satisfy thevacuum constraint equations. (These are thenontrivial relations arising for the �rst andsecond fundamental forms from di�erentialgeometry if C 3 were a spacelike hypersurfaceof a Lorentzian manifold satisfying (3).) Thetheorem of Choquet-Bruhat and Geroch [1]states that given such a smooth (C 3, g ,K ),there exists a unique smooth (M, g ) whichadmits C 3 as a Cauchy hypersurface, satis�esthe Einstein vacuum equations (3), and ismaximal in the sense that any other suchspacetime isometrically embeds in (M, g )preserving C 3. (The statement that C 3 is aCauchy hypersurface is simply the statementthat all inextendible causal curves of M

intersect C 3 exactly once.)

The cosmic censorship conjectures

We have remarked already that Penrose’s theoremis often misinterpreted as saying that black holesgenerically form or that ‘singularities’ (in the senseof local physics breaking down) generically arise.This is presumably because geodesic incompletenessin Oppenheimer–Snyder seems to be directlyconnected with both of these features.

What Penrose actually proved, however, is perhapseven more profound than what the press release [21]says. For it is fair to say that his theoremchanged our very ‘value system’. Whereas before,Oppenheimer–Snyder looked like the ultimatepathology, which would hopefully disappear onceperturbed, Penrose showed us that we should notjust tolerate black holes and singularities, but thatwe should in fact hope that black holes form, and

what’s more we should hope for singularities insidethem—and the stronger those are, the better!

Why? Because, as we shall see, the alternative thathis theorem allows is even worse.

Let us �rst understand the alternative to thepresence of a black hole. We need look no furtherthan negative mass Schwarzschild, i.e. the metric (5)with m < 0. Here, r = 0 can again be viewedas a singular boundary, but now one which isvisible to outside observers. We say the spacetimepossesses a ‘naked singularity’. In contrast to theOppenheimer–Snyder case, it would appear that onewould need to go beyond general relativity to describeobservations accessible to far-away observers.

Fortunately, the evolutionary point of view allowsus to exclude the particular example of negativemass Schwarzschild outright, because it does notin fact arise as a maximal Cauchy developmentof complete asymptotically �at initial data C 3. Butwho is to say that there do not exist spacetimeswith naked singularities that do arise from suchdata, even perhaps from small perturbations ofOppenheimer–Snyder data as in Corollary 1?

The conjecture that naked singularities should notoccur, or at least should generically not occur, isPenrose’s original ‘cosmic censorship’ [16].

The conjecture can be nicely re-formulated [7] in theevolutionary context with the help of yet anotherfundamental concept introduced by Penrose, thatof ‘future null in�nity’ [14], typically denoted as I+,which under suitable circumstances can be attachedas a conformal boundary of spacetime. ConsideringI+ is extremely useful for formulating the laws ofgravitational radiation, but it can also serve as astand-in for the role of far-away observers whende�ning black holes. For instance, one can de�ne theblack hole region as M\ J − (I+), where J − denotescausal past, although one should also impose thatI+

itself is complete [7], loosely related to the statementthat far-away freely falling observer worldlines befuture complete. In this language, the de�ning featureof a ‘naked singularity’ is that information from therewould arrive at I+ at �nite a�ne time, rendering I+

incomplete. The modern formulation [2] of Penrose’sconjecture, adapted to the evolutionary setting, is

Conjecture 1 (Weak cosmic censorship). For genericasymptotically �at initial data for (3) (or moregenerally (1) coupled to suitable matter), the maximalCauchy development possesses a complete I+.

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 33 — #33 ii

ii

ii

FEATURES 33

The word ‘weak’ is traditional, meant to distinguishthis statement from the later ‘strong’ Conjecture 2.The genericity assumption turns out to be necessaryeven for vacuum (3) in view of recent examples [18] ofnaked singularities, related to the previous sphericallysymmetric [2]. A highly nontrivial symmetric toyversion of Conjecture 1 has been proven [2]. Thegeneral problem, however, remains completely open!

The other ‘good’ feature of Oppenheimer–Snyderis perhaps even more di�cult at �rst sight torecognise as good: as remarked earlier, all incompleteobservers in Oppenheimer–Snyder spacetime fallinto the singularity r = 0. Moreover, not only dothey fall into the singularity, but an actual physicalobserver with arms and legs would be destroyed atr = 0, torn apart by in�nite tidal deformations [11].

While this is hardly ‘good’ for that poor observer, onecan argue that it is very ‘good’ for general relativityas a classical physics theory! For perversely, it givesthe theory an attractive epistemological closure.Observers either live forever or are torn apart. Eitherway, their future as classical observers is completelydescribed and determined by the theory—for as longas it makes sense to talk about them.

On the other hand, let us contemplate avery di�erent situation. Imagine that there existincomplete observers who encounter no singularity.The theory doesn’t say what happens to them, butsurely something must. What determines this?

Remarkably, this phenomenon is precisely whatoccurs in the celebrated Kerr solution, a 2-parameterfamily of vacuum metrics generalising (5). See [11].Here the maximal Cauchy development of(2-ended) asymptotically �at initial data satis�es theassumptions of Theorem 1. It is thus incomplete, butit is extendible smoothly as a Lorentzian manifold,in fact as a vacuum solution, in fact it is extendibleso that all incomplete observers may live anotherday in the extension. These extensions fail howeverto admit the initial hypersurface as a Cauchyhypersurface. The boundary of the maximal Cauchydevelopment in such an extension is thus known asa Cauchy horizon. This notion is due to Hawking [8].This strange situation can be understood betterconsidering the solution’s so called Penrose diagram,an in�uential way of representing the geometry ofspacetimes which is beyond the scope of this article.See [14].

In a certain sense, Cauchy horizons can be viewed as‘worse’ than singularities. The Kerr case is extremelypernicious in that not a single incomplete observer

encounters anything that would even suggest thatthe regime of classical relativity has been exited.So it is a spectacular and seemingly inexplicablefailure of the predictability of the theory, in no wayaccompanied by singularity. It was again Penrosewho discovered a possible way out, noticing thatKerr’s Cauchy horizon is subject to a blue-shiftinstability [15]. This led him to put forth his‘strong cosmic censorship’, which in its evolutionaryformulation [2] is the conjecture that, for genericinitial data, Cauchy horizons should not occur, i.e.

Conjecture 2 (Strong cosmic censorship). Forgeneric asymptotically �at initial data (3) (or moregenerally for (1) coupled to suitable matter), themaximal Cauchy development is inextendible as asuitably regular Lorentzian manifold.

(Note that as stated, Conjecture 2 is not in fact‘stronger’ than Conjecture 1.) To make Conjecture 2precise, one must specify how ‘suitably regular’should be interpreted. E�ectively, this corresponds tospecifying how ‘singular’ the boundary of spacetimeshould be. The strongest formulation would have itthat the boundary is so singular so that all incompleteobservers are torn apart, just as in Schwarzschild,providing the de�nitive closure described above.This would correspond to the C 0 formulation ofConjecture 2, where ‘suitably regular’ just means‘continuous’. Unfortunately, this version is in factfalse [4]. A weaker formulation is proposed in [3]and there are some positive non-trivial results [10]for a symmetric toy version. As with Conjecture 1,however, a positive resolution of a suitable versionof Conjecture 2 remains completely open!

In conclusion, Penrose’s theorem may not imply thatblack holes form or even true ‘singularities’ develop,but it very much taught us to live with black holesand singularities—indeed, to love them. Black holesare not themselves the singularity, but they are whatprotects us from singularity, and singularity in turnis what protects us from a much more dangerouskind of incompleteness associated with loss ofpredictability. The wide acceptance of black holes,now central both in astronomy and even popularculture, ultimately stems from this. It is di�cult toimagine a more impactful contribution to generalrelativity—a more ironic reversal—arising from theproof of a mathematical theorem of pure geometry.With [13], our view of gravitational collapse irreversiblychanged, and the resulting weak and strong cosmiccensorship conjectures will undoubtedly remain themain source of inspiration for further progress inclassical general relativity for many years to come.

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 34 — #34 ii

ii

ii

34 FEATURES

FURTHER READING

[1] Y. Choquet-Bruhat, R. P. Geroch. Global aspectsof the Cauchy problem in General Relativity. Comm.Math. Phys. 14 (1969), 329–335.[2] D. Christodoulou, On the global initial valueproblem and the issue of singularities, Class.Quantum Grav. 16 (1999), A23.[3] D. Christodoulou, The Formation of BlackHoles in General Relativity, EMS, 2009.[4] M. Dafermos, J. Luk, The interior of dynamicalvacuum black holes I: The C 0-stability of the KerrCauchy horizon, Preprint 2017, arXiv:1710.01722.[5] A. Einstein, Die Feldgleichungen derGravitation, Sitz. Ber. Kgl. Preuss. Akad. d. Wiss.(1915), 844–847.[6] D. Finkelstein, Past-Future Asymmetry of theGravitational Field of a Point Particle, Phys. Rev.110 (1958), 965–7.[7] R. Geroch, G. Horowitz, Global structure ofspacetimes, in: General Relativity, CambridgeUniversity Press, 1979, pp. 212–293.[8] S. Hawking, The occurrence of singularitiesin cosmology. III. Causality and singularities, Proc.Royal Soc. A 300 (1967), 187–201.[9] G. Lemaitre, L’Univers en Expansion,Publication du Laboratoire d’Astronomie et deGéodésie de l’Université de Louvain 9 (1932),171–205.[10] J. Luk, S.-J. Oh, Strong cosmic censorship inspherical symmetry for two-ended asymptotically�at initial data I., Ann. of Math. 190 (2019) 1–111.[11] C. W. Misner, K. S. Thorne, J. A. Wheeler,Gravitation, W. H. Freeman, 1973.[12] R. Oppenheimer, H. Snyder, On continuedgravitational contraction, Phys. Rev. 56 (1939)

455–459.[13] R. Penrose, Gravitational collapse andspace-time singularities, Phys. Rev. Lett. 14 (1965)57–59.[14] R. Penrose, An analysis of the structure ofspace-time, Adams Prize Essay, Cambridge, 1968.[15] R. Penrose, Structure of space-time, in: BatelleRencontres: 1967 Lectures in Mathematics andPhysics, W. A. Benjamin, 1968, pp. 121–235.[16] R. Penrose, Gravitational collapse: The role ofgeneral relativity, Riv. Nuovo Cim. 1 (1969), 252–276.[17] R. Penrose, lecture in memory ofStephen Hawking’s birthday, 8/1/2021,https://www.youtube.com/watch?v=3sRwjf0cML0.[18] I. Rodnianski, Y. Shlapentokh-Rothman,Naked Singularities for the Einstein VacuumEquations: The Exterior Solution, Preprint 2019,arXiv:1912.08478.[19] K. Schwarzschild, Über das Gravitationsfeldeines Massenpunktes nach der EinsteinschenTheorie, Sitz. Ber. Kgl. Preuss. Akad. d. Wiss. (1916),189–196.[20] J. M. M. Senovilla, D. Gar�nkle, The 1965Penrose singularity theorem, Class. Quantum Grav.32 (2015) 124008.[21] https://www.nobelprize.org/prizes/physics/2020/press-release/.

Mihalis Dafermos

Mihalis Dafermos holdsthe Lowndean Chair ofAstronomy & Geometryin the University ofCambridge and is aProfessor at PrincetonUniversity.

The proof of Penrose’s incompleteness theorem

The proof of Theorem 1 can be thought of as an ingenious adaptation of ideas from the Bonnet-Myerstheorem to the setting of Lorentzian geometry in the presence of a closed trapped surface. We sketchthe proof here. One considers the so-called causal future J + (T ) of the closed trapped surface T .If spacetime admits a Cauchy hypersurface C 3, then one can show that the boundary B of this setcanonically projects to C 3. The variational theory of null geodesics, however, yields that B is coveredby null geodesic segments none of which may extend beyond its �rst focal point to T . The curvatureassumption (4) implies on the other hand that null geodesics emanating from T must develop focalpoints if they can be extended to arbitrary a�ne time g. Thus, if these null geodesics are futurecomplete it follows that every null geodesic emanating from T develops a focal point, whence onecan show that B is compact without boundary, whence it cannot project to the non-compact C 3. Thiscontradicts the null geodesic completeness. See also Penrose’s Adams Prize essay [14]. There are manyextensions of this result, starting from work of Hawking [8]. See the survey [20].

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 35 — #35 ii

ii

ii

FEATURES 35

The Mathematics of Floating-Point ArithmeticNICHOLAS J. HIGHAM

Floating-point arithmetic is ubiquitous in computing and its implementation in evolving computer hardwareremains an active area of research. Its mathematical properties di�er from those of arithmetic over the realnumbers in important and sometimes surprising ways. We explain what a mathematician should know about�oating-point arithmetic, and in particular we describe some of its not so well known algebraic properties.

Floating-point arithmetic has been in use for overseventy years, having been provided on some ofthe earliest digital computers. For the �rst halfof that period there was tremendous variationin �oating-point formats and in the ways thearithmetics were implemented. Some �oating-pointarithmetics could produce anomalous results andit was di�cult or impossible to write programsthat were portable, i.e., produce similar results ondi�erent computer systems with little or no change.

The 1985 ANSI/IEEE Standard for Binary Floating-PointArithmetic [9] provided binary �oating-point formatsand precise rules for how to carry out arithmeticon them. Carefully designed over several years bya committee of experts, it brought much-neededorder to computer arithmetic and within a few yearsvirtually all computer manufacturers had adopted it.

From a mathematical perspective we can ask severalquestions about a �oating-point arithmetic.

• What mathematical properties does it havecompared with exact arithmetic?

• What sort of mathematical structure is it?

• How can we understand the accuracy ofcomputations carried out in it?

First, we need to de�ne the set of numbers underconsideration. A �oating-point number system isa �nite subset F = F (V,t ,emin,emax) of the realnumbers ℝ whose elements have the form

x = ±m × V e−t+1. (1)

Here, V is the base, which is 2 on virtually all currentcomputers. The integer t is the precision and theinteger e is the exponent, which lies in the rangeemin ≤ e ≤ emax. The signi�cand m is an integersatisfying 0 ≤ m ≤ V t − 1. To ensure a uniquerepresentation for each nonzero x ∈ F it is assumedthat m ≥ V t−1 if x ≠ 0, so that the system isnormalised.

The reason for the “+1” in the exponent of (1), whichcould be avoided by rede�ning emin and emax, is forconsistency with the IEEE standard. The standardalso requires that emin = 1 − emax.

The largest and smallest positive numbers in thesystem are xmax = V emax (V − V 1−t ) and xmin = V emin ,respectively. Two other important quantities are u =12 V

1−t , the unit roundo�, and n = V 1−t , the machineepsilon, which is the distance from 1 to the nextlarger �oating-point number. See the box for a simpleexample of a �oating-point number system.

A toy �oating-point number system

This diagram shows the nonnegative (normalised) numbers in a binary �oating-point number systemwith t = 3, emin = −2, and emax = 3. Note that the �oating-point numbers are equally spaced betweenpowers of 2 and the spacing increases by a factor of 2 at each power of 2. Here, the unit roundo� isu = 0.125 and the machine epsilon n = 0.25; these are the distances from 1 to the next smaller andnext larger �oating-point number, respectively.

0 0.5 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 10.0 12.0 14.0

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 36 — #36 ii

ii

ii

36 FEATURES

The system F can be extended by includingsubnormal numbers, which have the minimumexponent and m < V t−1, so that they are notnormalised; they �ll the gap between 0 and xmin withnumbers having a constant spacing V emin+1−t . TheIEEE standard includes subnormal numbers.

We assume throughout the rest of this article thatF is a binary system (V = 2), and that it followsthe IEEE standard by including the special numbers±∞ and NaN (Not a Number). The numbers ±∞obey the usual mathematical conventions regardingin�nity, such as ∞ + ∞ = ∞, (−1) × ∞ = −∞, and(finite)/∞ = 0. A NaN is generated by operationssuch as 0/0, 0 ×∞, ∞/∞, (+∞) + (−∞), and

√−1.

We also assume, again following the IEEE standard,that the results of the elementary operations ofaddition, subtraction, multiplication, division, andsquare root are the same as if they were carried outto in�nite precision and then rounded back to F , andthat rounding of x ∈ ℝ to F is done by mapping tothe nearest �oating-point number, with ties brokenby rounding to the �oating-point number with a zerolast bit. We denote the operation of rounding by fl.With a standard abuse of notation, fl(expr), whereexpr is an arithmetic expression, is also used todenote the result of evaluating expr in �oating-pointarithmetic in some speci�ed order.

With the inclusion of ∞ and NaN, F is a closednumber system: every �oating-point operation onnumbers in F produces a result in F .

Algebraic properties

The real numbers form a �eld under additionand multiplication. It is natural to ask whatsort of mathematical structure �oating-pointnumbers form under the elementary (�oating-point)arithmetic operations. To investigate this questionwe will explore some basic algebraic properties of�oating-point arithmetic.

Let a,b ∈ F . By de�nition, fl(a + b) and fl(b + a)are equal, as are fl(a ∗ b) and fl(b ∗ a). However,with three numbers the usual rules of arithmeticbreak down: fl((a + b) + c )) is not necessarily equalto fl(a + (b + c )) and fl((a ∗ b) ∗ c ) is not necessarilyequal to fl(a ∗ (b ∗ c )). In other words, �oating-pointaddition and multiplication are not associative. For

example, in our toy system fl(0.25 + (8.0 − 7.0)) =1.25 but fl((0.25 + 8.0) − 7.0) = fl(8.0 − 7.0) = 1.0.Similarly, fl(a ∗ (b + c )) is not necessarily equal tofl(a ∗ b + a ∗ c ), so the distributive law does not hold.

If a > b > 0 then fl(a + b) > a need not hold. Thereason is that b may be so small that a is unchangedafter adding b and rounding. Indeed fl(1 + x) = 1 forany positive �oating-point number x < u .

Does the equation x ∗ (1/x) = 1 hold in �oating-pointarithmetic? The following result of Edelman says thatit may just fail to do so [6, Prob. 2.12].1

Theorem 2. For 1 < x < 2, fl(x ∗ (1/x)) is either 1or 1 − n/2

A closely related question is which �oating-pointnumbers are possible reciprocals of x ∈ F . Muller[10] showed that when 1/x ∉ F there are twopossibilities.

Theorem 3. The only z ∈ F that can satisfyfl(x ∗ z ) = 1 are min{ y : y ≥ 1/x , y ∈ F } andmax{ y : y ≤ 1/x , y ∈ F }.

Perhaps surprisingly, these two possible z cansimultaneously give equality, so a �oating-pointnumber can have two �oating-point reciprocals. Infact, of the 24 positive numbers in the toy system,eight have two �oating-point reciprocals; for example,y = 0.625 and y = 0.75 both satisfy fl(1.5 ∗ y) = 1,and these are the two nearest �oating-point numbersto 1/1.5 = 2/3.

Now consider the computation n ∗ (m/n), where mand n are integers. If m/n is a �oating-point numberthen fl(n ∗ fl(m/n)) = fl(n ∗ (m/n)) = fl(m) = m, asno rounding is needed. Kahan proved that the sameidentity holds for many other choices of m and n [4,Thm. 7].

Theorem 4. Let m and n be integers such that|m | < 2t−1 and n = 2i + 2 j for some i and j . Thenfl(n ∗ fl(m/n)) = m.

The sequence of allowable n begins 2,3,4,5,6,8,9,10,12,16,17,18,20 (and is A048645 in the On-LineEncyclopedia of Integer Sequences), so Theorem 3covers many common cases. Nevertheless, theequality does not hold in general.

It can be shown that fl(√x2

)= |x | for x ∈ F , as long

as x2 does not under�ow (round to zero) or over�ow1We give a minimal set of references in this article. Original sources can be found in the references cited.

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 37 — #37 ii

ii

ii

FEATURES 37

(exceed the largest element of F ), but fl((√x)2) = |x |

is not always true (by the pigeonhole principle) [6,Prob. 2.20].

Rounding (to nearest) is monotonic in that for x ∈ ℝand y ∈ ℝ, the inequality x ≤ y implies fl(x) ≤ fl(y).As a result, it is easy to show that

x ≤ fl(x + y

2

)≤ y .

While this result holds for base 2, the computedmidpoint can be outside the interval for base 10.

The inequality fl(x/

√x2 + y2

)≤ 1 always holds,

barring over�ow and under�ow [6, Prob. 2.21].Although this fact may seem unremarkable, insome pre-IEEE standard arithmetics this inequalitycould be violated, causing failure of an attemptto compute one of the angles in a right-angledtriangle with shortest sides of lengths x and y asacos

(x/

√x2 + y2

).

The following result of Sterbenz guarantees thatsubtraction is exact for two numbers that are atmost a factor 2 apart.

Theorem 5. If x and y are �oating-point numberswith y/2 ≤ x ≤ 2y then fl(x − y) = x − y (assumingx − y does not under�ow).

This result is notable because inaccurate results areoften blamed on subtractive cancellation. It is notthe subtraction itself that is dangerous but the wayit brings into prominence errors already present inthe numbers being subtracted, making these errorsmuch larger relative to the result than they were tothe arguments.

Finally, we note that a NaN is unique among elementsof F in that it compares as unordered (includingunequal to) everything, including itself. In particular,a statement “if x = x” returns false when x is a NaN.This is why some programming languages provide afunction to test for a NaN (e.g., isnan in MATLAB).

We conclude that �oating-point arithmetic is arather strange mathematical object that does notcorrespond to any standard algebraic structure.These examples could make one pessimisticabout our ability to carry out reliable numericalcomputations. Fortunately, these peculiar featuresof �oating-point arithmetic are not a barrier toits successful use or to deriving satisfactory errorbounds, as we now illustrate.

Experimenting with di�erent�oating-point arithmetics

It is instructive to run experiments in�oating-point arithmetics based on di�erentparameters t , emin, and emax.

We used the MATLAB function chopa [8]for this purpose. This function rounds singleor double precision numbers to a speci�edtarget format (limited to emin = 1 − emax)and supports several rounding modes andother options. A library CPFloat o�ers similarfunctionality for C [3].ahttps://github.com/higham/chop

Error analysis

If we want to understand the e�ects of roundingerrors on a �oating-point computation then we needto analyse how the individual rounding errors interactand propagate. A natural way to try to do this is tode�ne “circle operators” ⊕, , ⊗, and � by

x ⊕ y = fl(x + y), x y = fl(x − y),x ⊗ y = fl(x ∗ y), x � y = fl(x/y),

and then rewrite the expressions being evaluated interms of these operators. For example, consider theevaluation of the cubic polynomial p = ax3 + bx2 +cx + d by Horner’s rule as p = ((ax + b)x + c )x + d(using 6 operations instead of the 8 required if weexplicitly form x3 and x2). We would then write thecomputed p as

p =((a ⊗ x ⊕ b) ⊗ x ⊕ c

)⊗ x ⊕ d .

However, we cannot easily simplify this expressionbecause the circle operators do not satisfy theassociative or distributive laws.

The right way to do error analysis is to obtainequations in terms of the original operators andindividual rounding errors. We need the result that[6, Thm. 2.2]

x ∈ ℝ =⇒ fl(x) = x (1 + X), |X | ≤ u , (2)

where u is the unit roundo�. Since fl(x op y) isde�ned to be the rounded exact value, it follows thatfor op = +,−,∗,/ we have

fl(x op y) = (x op y) (1 + X), |X | ≤ u . (3)

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 38 — #38 ii

ii

ii

38 FEATURES

This is the standard model of �oating-pointarithmetic used for rounding error analysis. Note thatit does not fully characterise �oating-point arithmeticbecause (2) does not fully characterise rounding:for some x , two di�erent �oating-point numbers ysatisfy y = x (1 + X) with |X | ≤ u . It is possible to usemore re�ned models of �oating-point arithmetic thatmore fully re�ect the de�nition of rounding, whichtends to give results with slightly smaller constantsat the cost of a much more complicated analysis. Themain purpose of a rounding error analysis, though,is to gain insight into accuracy and stability ratherthan to optimise constants.

For our cubic example, we can use the model to write

p =(( (ax (1 + X1) + b

)(1 + X2)x (1 + X3) + c

)× (1 + X4)x (1 + X5) + d

)(1 + X6)

= ax3 (1 + X1) (1 + X2) (1 + X3) (1 + X4) (1 + X5) (1 + X6)+ bx2 (1 + X2) (1 + X3) (1 + X4) (1 + X5) (1 + X6)+ cx (1 + X4) (1 + X5) (1 + X6) + d (1 + X6),

where |Xi | ≤ u for all i . This expression is rathermessy, but we can rewrite it as

p = ax3 (1 + \6) + bx2 (1 + \5)+ cx (1 + \3) + d (1 + \1), (4)

where the \i are bounded by the following lemma [6,Lem. 3.1].

Lemma 1. If |Xi | ≤ u and di = ±1 for i = 1: n, andnu < 1, then

n∏i=1

(1 + Xi )di = 1 + \n ,

where|\n | ≤

nu1 − nu =: Wn .

Applying the lemma to (4), we obtain

|p − p | ≤ W6 ( |a | |x |3 + |b | |x |2 + |c | |x | + |d |), (5)

which is a concise and easily interpretable errorbound, with constant W6 = 6u +O (u2).

With careful use of the lemma, the profusion of 1+Xiterms that arise in a rounding error analysis can bekept under control and manipulated, using the usualrules of arithmetic, into a useful bound.

Fused multiply-add operation

Since the 1990s some processors haveprovided a fused multiply-add (FMA) operationthat computes x + y ∗ z with just one roundingerror instead of two, so that

fl(x + y ∗ z ) = (x + y ∗ z ) (1 + X), |X | ≤ u .

The motivation for an FMA is speed, as itis implemented in such a way as to takethe same time as a single multiplication oraddition.

When an FMA is used the number of roundingerrors in a typical computation is halved. Ourcubic polynomial can be evaluated with threeFMAs, giving

p =(( (ax + b

)(1 + X1)x + c

)× (1 + X2)x + d

)(1 + X3)

= (ax3 + bx2) (1 + \3) + cx (1 + \2)+ d (1 + \1),

which is more favourable than (4).

Although it generally brings improvedaccuracy, an FMA can also lead to someunexpected results.

If we compute the modulus squared of acomplex number from the formula

(x + iy)∗ (x + iy) = x2 + y2 + i(xy − yx)

then the result is real, because fl(xy) = fl(yx).But if an FMA is used in evaluating xy−yx thenthe imaginary part may evaluate as nonzero.

Similarly, if the discriminant b2 − 4ac of aquadratic is nonnegative then the computedresult is guaranteed to be nonnegative by themonotonicity of �oating-point arithmetic, butwith an FMA the result can be negative.

Error analysis strategy

Even with the use of Lemma 1, rounding error analysiscan be tedious, and it is natural to ask whether it canbe automated. Can we harness a computer to carry

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 39 — #39 ii

ii

ii

FEATURES 39

out the necessary manipulations? For focused classesof algorithms some progress has been made [1],but in general the task is di�cult or impossible toautomate. The reason is that the hardest part of anerror analysis is deciding what one wants to prove.For the evaluation of the cubic we obtained a bound(5) on the error in the computed p , known as theforward error. Along the way we obtained (4), which isa backward error result: it shows that the computedp is the exact result for a polynomial with perturbedcoe�cients a (1+\6), b (1+\5), c (1+\3), and d (1+\1),and it bounds the size of the relative perturbationsby W6.

In general, for a computation y = f (x), where xand y are vectors (say), we have three measures oferror for the computed y :

• forward error: ‖y − y ‖/‖y ‖,

• backward error:

min{‖Δx ‖‖x ‖ : y = f (x + Δx)

},

• mixed backward–forward error: the smallest n forwhich there exist Δx and Δy such that

y + Δy = f (x + Δx), ‖Δx ‖‖x ‖ ≤ n ,

‖Δy ‖‖y ‖ ≤ n .

Depending on the problem, any one of these errorsmay be the best one to bound in a roundingerror analysis, or perhaps the only one that it isfeasible to bound. Determining the right approachand working out how to achieve a result that isreadable, understandable, and insightful can bedi�cult.

Backward error analysis was developed byJ. H. Wilkinson in the 1950s and 1960s [5]. It hasthe attractive feature of decoupling the numericalstability properties of an algorithm from theconditioning of the underlying problem (its sensitivityto perturbations in the data).

What is quite remarkable is that despite the strangebehaviour of �oating-point arithmetic illustratedabove, it is possible to carry out rounding erroranalysis of a wide variety of algorithms and obtainuseful results.

Recovering the error

The sum s = a +b of a,b ∈ F is not in generalin F , so the computed sum s = fl(a + b) maybe inexact.

However, the error e = s − s is in F , and for|a | ≥ |b | it can be computed (exactly) as [11,Sec. 4.3.1]

e = b − (s − a),

so thata + b = s + e .

This computation is known as Fast2Sum.Let us denote it by [s ,e ] = Fast2Sum(a,b).(There are other, more complicated, ways ofcomputing e that do not require |a | ≥ |b |.)

Of course, if we try to form fl(s + e ) thenwe will just obtain s , because s is the best�oating-point representation of a+b . However,in a sequence of operations we can add theerror from an earlier operation into a lateroperation, where it can potentially have ane�ect.

An important usage of Fast2Sum is incompensated summation, proposed by Kahanin 1965, which computes

∑ni=1 xi by

1 s = x12 e = 03 for i = 2: n4 t = xi + e5 [s ,e ] = Fast2Sum(s ,t )6 end

For standard recursive summation thecomputed s satis�es

|s − s | ≤ cnun∑i=1

|xi | +O (u2)

with cn = n−1, whereas the computed s fromcompensated summation satis�es the samebound with cn = 2 (even though compensatedsummation does not sort the arguments ofFast2Sum). For large n, this reduction in theconstant makes a signi�cant di�erence.

The 2019 revision of the IEEE standardincludes so-called augmented arithmeticoperations for addition, subtraction, andmultiplication, which (like Fast2Sum) returnboth the computed result and the error in it.

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 40 — #40 ii

ii

ii

40 FEATURES

Probabilistic analysis and stochastic rounding

A numerical computation with n × n matrices usuallyhas a rounding error bound proportional to cnu withcn growing at least linearly. Traditionally, numericalcomputations have been done in single precisionarithmetic or double precision arithmetic, with unitroundo�s u of order 10−8 or 10−16, respectively.Hence nu � 1 for practical problems.

However, half precision arithmetic is now increasinglyavailable in hardware, with u of order 10−3 for theb�oat16 format and 10−4 for the IEEE half precisionformat. In these arithmetics, nu = 1 for quitemodestly-sized problems, and in these cases an errorbound proportional to nu provides no information.

Mixed-precision algorithms

It is becoming common for computer systemsto o�er half precision, single precision, anddouble precision �oating-point arithmetics inhardware, possibly with quadruple precisionarithmetic in software. In designing algorithmswe wish to exploit the speed of execution oflower precision arithmetic while ensuring thatenough higher precision is used to deliver aresult of the desired accuracy. Rounding erroranalysis, parametrised by the unit roundo�sfor the di�erent precisions, helps to identifysuitable algorithms.

Traditional rounding error bounds, such as thoseabove for Horner’s rule, are worst-case bounds.As Stewart observes [12], “To be realistic, we mustprune away the unlikely. What is left is necessarilya probabilistic statement.” The idea of obtainingprobabilistic rounding error bounds by modellingrounding errors as random variables is not new, buta rigorous treatment producing bounds valid forany dimension has only recently been developed,by Connolly, Higham, and Mary [2], [7]. This analysisproves that under the assumption that the roundingerrors are mean independent random variables ofmean zero, error bounds with constants

√f (n)u

hold with high probability in place of worst-casebounds f (n)u .

A form of rounding called stochastic rounding hasrecently been �nding use in deep learning and otherareas. It rounds a number lying between two adjacent

�oating-point numbers a < b to a with a probabilityproportional to the distance to b , and conversely for b .Stochastic rounding is somewhat worse behaved thanround to nearest vis-à-vis its algebraic properties forindividual operations. However, the random natureof the rounding is bene�cial. It can be shown [2]that the rounding errors from stochastic roundingare random variables satisfying both the meanindependence and the mean zero assumptions, sothat the

√f (n)u bounds hold unconditionally. This

means that stochastic rounding can provide moreaccurate results than round to nearest for largeproblems.

As a simple example, we computed∑104i=1 xi in IEEE

half precision arithmetic, where xi is 1/i roundedto half precision with round to nearest. The sumcomputed with round to nearest had relative error2.7 × 10−1, whereas the minimum, mean, andmaximum errors over ten sums computed withstochastic rounding were 2.2 × 10−3, 1.2 × 10−2,3.0 × 10−2, respectively. In this example, roundto nearest su�ers from stagnation, whereby thesmallest terms cannot change the computed partialsum. By contrast, stochastic rounding gives all termsa nonzero probability of increasing the sum, and infact it does so in just the right way to ensure that theexpected value of the computed sum is the exactsum [2].

Outlook

The provision of half precision �oating-pointarithmetic in hardware is motivated by machinelearning, where its greater speed is proving bene�cialdespite its lower accuracy. Half precision can alsobe exploited in general scienti�c computing, butrounding error analysis is needed to determinewhether su�ciently accurate results are beingcomputed.

An example of how half precision arithmeticcan be harnessed to great e�ect is the HPL-AIMixed Precision Benchmark2, which is one of thebenchmarks that the TOP500 project uses to rankthe world’s most powerful supercomputers. Thisbenchmark solves a double precision nonsingularlinear system Ax = b of order n using an LUfactorisation computed in half precision and it re�nesthe solution using iterative re�nement in doubleprecision. As of November 2020, the world record

2icl.bitbucket.io/hpl-ai

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 41 — #41 ii

ii

ii

FEATURES 41

execution rate for the benchmark is 2.0 ExaFlop/s(2 × 1018 �oating-point operations per second,where most of the operations are half precisionones) for a matrix of size 16,957,440, which wasachieved by the Fugaku supercomputer in Japan. Fora successful benchmark run, the relative residual‖Ax − b ‖/(‖A‖‖x ‖ + ‖b ‖) of the computed x mustbe no larger than a threshold that is about 10−8

in this case. So after approximately 2n3/3 ≈ 3 ×1021 �oating-point operations, Fugaku’s computedsolution x had a small residual, which is a testamentto e�ectiveness of �oating-point arithmetic giventhat each half precision operation has a relative errorof order 10−4.

Despite �oating-point arithmetic having somestrange mathematical properties, seventy yearsof experience show that it usually works wellin practice, and it is supported by rigorousmathematical analysis—both worst-case andprobabilistic. With hardware implementations of�oating-point arithmetic evolving constantly andnew algorithms regularly being developed, interestingmathematical questions will continue to arise overthe coming years.

Acknowledgements

I thank Michael Connolly, Massimilano Fasi, SvenHammarling, Theo Mary, Mantas Mikaitis, and SrikaraPranesh for suggesting improvements to a draftof this article. This work was supported by theRoyal Society and Engineering and Physical SciencesResearch Council grant EP/P020720/1.

FURTHER READING

[1] P. Bientinesi and R.A. van de Geijn.Goal-oriented and modular stability analysis.SIAM J. Matrix Anal. Appl. 32 (2011) 286–308.[2] M.P. Connolly, N.J. Higham, and T. Mary.Stochastic rounding and its probabilisticbackward error analysis. SIAM J. Sci. Comput.,2021. To appear.[3] M. Fasi and M. Mikaitis. CPFloat: a Clibrary for emulating low-precision arithmetic.MIMS EPrint 2020.22, Manchester Institutefor Mathematical Sciences, The University ofManchester, UK, Oct. 2020.

[4] D. Goldberg. What every computer scientistshould know about �oating-point arithmetic.ACM Comput. Surv. 23 (1991) 5–48.[5] S. Hammarling and N.J. Higham.Wilkinson and backward error analysis.https://nla-group.org/2019/02/18/wilkinson-and-backward-error-analysis/,Feb. 2019.[6] N.J. Higham. Accuracy and Stability ofNumerical Algorithms. Society for Industrial andApplied Mathematics, Philadelphia, PA, USA, 2nded., 2002.[7] N.J. Higham and T. Mary. A new approachto probabilistic rounding error analysis. SIAM J.Sci. Comput. 41 (2019) A2815–A2835.[8] N.J. Higham and S. Pranesh. Simulating lowprecision �oating-point arithmetic. SIAM J. Sci.Comput. 41 (2019) C585–C602.[9] IEEE Standard for Binary Floating-PointArithmetic, ANSI/IEEE Standard 754-1985.Institute of Electrical and Electronics Engineers,New York, 1985.[10] J.-M. Muller. Some algebraic properties of�oating-point arithmetic. In: P. Kornerup (ed.)Proceedings of the Fourth Conference on RealNumbers and Computers (2000), pp. 31–38.[11] J.M. Muller, N. Brunie, F. de Dinechin, C.P.Jeannerod, M. Joldes, V. Lefèvre, G. Melquiond,N. Revol, and S. Torres. Handbook ofFloating-Point Arithmetic. Birkhäuser, Boston,MA, USA, 2nd ed., 2018.[12] G.W. Stewart. Stochastic perturbationtheory. SIAM Rev. 32 (1990) 579–610.

Photo credit: RobWhitrow

Nicholas J. Higham

Nick is Royal SocietyResearch Professor andRichardson Professorof Applied Mathematicsin the Department ofMathematics at theUniversity of Manchester.His current researchinterests include mixed

precision numerical linear algebra algorithms. Heblogs about applied mathematics at https://nhigham.com/. Nick shudders to recall that in somepre-IEEE standard computer arithmetics one couldhave fl(1.0 ∗ x) ≠ x for a �oating-point number x .

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 42 — #42 ii

ii

ii

42 FEATURES

Random Lattices in the Wild: from Pólya’s Orchardto Quantum Oscillators

JENS MARKLOF

Point processes are statistical models that describe the distribution of discrete events in space and time.Applications are everywhere, from galaxies to elementary particles. My aim here is to convince you that thereis an exotic but interesting class of point processes — random lattices — that have fascinating connectionswith various branches of mathematics and some basic models in physics.

So what is a random lattice? First of all, a lattice indimension one is any non-trivial discrete subgroupof the additive group of real numbers ℝ. (Non-trivialmeans anything but the group of one element.) Theadditive group of integers ℤ is an example and, upto rescaling by a constant factor, it is in fact the onlyexample. Now in order to turnℤ into a random object,let us translateℤ by a real number U to obtain the setS(U) = ℤ +U, and then view U as a random variableuniformly distributed in the unit interval [0,1]. Thechoice of the unit interval is natural since U and U+1will lead to the same shifted lattice S(U). With this,S(U) becomes a random set, which we take (for thepurposes of this discussion) to be synonymous withrandom point process. One can check that S(U) isa translation-stationary random point process, i.e.,S(U) + t has the same distribution as S(U) for everychoice of t ∈ ℝ — a simple consequence of the factthat U is assumed to be uniformly distributed in [0,1].A random point process describes the probabilityof �nding k points in a given set B . In the presentsetting, for B a bounded interval of length |B | andinteger k ≥ 0, we have that

ℙ

(|S(U) ∩ B | = k

)= max

(1 −

��k − |B |�� , 0) .It is not di�cult to see that the expected number ofpoints in B is |B |, which means that the process hasintensity one — compare this with the correspondingprobabilities for a Poisson process!

The above construction has produced a simpleinstance of a point process in ℝ. Independentsuperpositions of one-dimensional randomly shiftedlattices explain for example the limiting gapdistribution of the fractional parts of the sequencelog n, with n = 1,2,3 . . . [14]. But the fun really startsin dimension two!

Poisson process

A homogeneous Poisson process withintensity one in ℝ can be realised asa sequence of random points wherethe distances between consecutive pointsare independent random variables withan exponential distribution. That is, theprobability that a gap is larger than s is e−s .It follows that the probability of having kpoints in the interval B is given by the Poissondistribution

|B |kk !

e−|B | .

Two-dimensional random lattices

To construct a two-dimensional random lattice, webegin with the integer lattice ℤ2. We could proceedas before and de�ne a random point process in ℝ2

by shifting ℤ2 randomly by a vector ", say, uniformlydistributed in [0,1]2. This is �ne, but there is amore interesting avenue. Unlike in dimension one, wehave a non-trivial group of linear volume-preservingtransformations acting on ℝ2. We can use this action,rather than the group of translations as above, torandomise ℤ2 and thus produce a two-dimensionalrandom lattice with a fundamental cell of volume one.Here is how it works. We represent elements in ℝ2

as row vectors x = (x1,x2). A linear transformation isthen represented by real matrix multiplication fromthe right,

x ↦→ x(a bc d

)= (ax1 + cx2,bx1 + dx2).

Volume is preserved if and only if the determinanthas modulus one, that is |ad − bc | = 1. We will only

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 43 — #43 ii

ii

ii

FEATURES 43

need to consider the case where also orientation ispreserved, which means ad − bc = 1. Such matricesform a group, which we will label as SL(2,ℝ). Lstands for linear and S for special (referring to theunit determinant). To produce our �rst example of arandom lattice in ℝ2, consider the sheared lattice

P1 (u) = ℤ2(1 u0 1

).

Note that, P(u + 1) = P(u), and it is thereforenatural to consider u as a random variable uniformlydistributed on [0,1]. This turns P(u) into a randomset, a random point process. A similar constructionis possible for the randomly rotated lattice

R1 (q) = ℤ2(cos q − sin qsin q cos q

)where q is uniformly distributed in [− c

2 ,c2 ]. It is a

fact that any matrix inM ∈ SL(2,ℝ) can be uniquelywritten as a product of a shear, stretch and rotationmatrix

M =

(1 u0 1

) (v1/2 00 v−1/2

) (cos q − sin qsin q cos q

)where u is real, v is real and positive, and −c < q ≤c. This is known as the Iwasawa decomposition ofSL(2,ℝ), and provides a parametrisation of SL(2,ℝ)in terms of (u ,v , q). It follows that any choice ofrandom elements (u ,v , q) yields a random latticeℤ2M . The above examples of randomly sheared orrotated lattices are simply special cases! But is therea particular natural choice of probability measure for(u ,v , q) that plays the role of a uniform measure?One could start with u uniformly distributed in[0,1] and q uniformly distributed in [− c

2 ,c2 ], as

above — but what is a natural uniform probabiltymeasure on the positive axis for v? The answeris highly non-trivial, but has a beautiful geometricinterpretation. The key to the solution is the modulargroup Γ = SL(2,ℤ), where now all matrix coe�cientsare restricted to integers. It is a discrete subgroupof SL(2,ℝ) and in fact precisely the subgroup ofall W ∈ SL(2,ℝ) such that ℤ2W = ℤ2. This meansthat M and WM lead to the same lattice ℤ2M , andwe can therefore restrict our attention to only onerepresentative of the coset ΓM = {WM | W ∈ Γ}. Aconvenient set of such representatives is for examplegiven by

F=

{(u ,v , \) ∈ ℝ3 | − 1

2 < u < 12 ,

u2 + v2 > 1, v > 0, − c2 < q < c

2

}

(we should also include about half of the boundary).This set is called a fundamental domain of theΓ-action, just as the unit interval is a fundamentaldomain of the ℤ-action on ℝ. The most naturaluniform measure on F is obtained from the Haarmeasure of SL(2,ℝ), restricted to Fand normalisedas a probability measure. Explicitly, this Haarprobability measure is

`F =3c2

du dv dqv2

.

Geometers will have spotted the intriguing similaritywith formulas from hyperbolic geometry: The groupSL(2,ℝ) acts on the upper complex halfplane ℍ =

{g ∈ ℂ | Im g > 0} by Möbius (fractional linear)transformations

g ↦→ ag + bcg + d , M =

(a bc d

).

The Möbius transformation for M as in the Iwasawadecomposition maps i to u + iv , and thus the Möbiusaction really comes from group multiplication inSL(2,ℝ). In fact, we can identify Γ\ SL(2,ℝ) withthe unit tangent bundle of the modular surfaceΓ\ℍ, where the angle \ = −2q parametrises thedirection of the tangent vector at the point g =

u + iv . With this identi�cation, the Haar probabilitymeasure `F becomes the natural invariant measurefor the geodesic and horocycle �ows for the modularsurface.

Haar probability measure

If (x1,x2,x3) is a uniformly distributed randomvector in the unit cube (− 1

2 ,12 )

3, then

(u ,v , q) =(sin( c3x1),

cos( c3x1)12 − x2

, cx3

).

is a random element in F distributedaccording to the Haar probability measure `F.

A key property of Haar measure on SL(2,ℝ) isthat it is invariant under left and right multiplicationby its group elements. This implies that (usingthe invariance under right multiplication) for Mdistributed according to `F, the random latticesℤ2M and ℤ2M g have the same distribution forevery element g ∈ SL(2,ℝ). In other words, therandom point process ℤ2M is SL(2,ℝ)-stationary!The process is, however, not translation-stationarysince the origin is always realised. But even with

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 44 — #44 ii

ii

ii

44 FEATURES

the origin removed, the random process ℤ2M \ {0}is not translation-stationary (as the formulas belowwill show). Nevertheless, Siegel’s famous mean valueformula (published in 1945) shows that its intensitymeasure is the standard Lebesgue measure dy .

Siegel’s mean value formula

Motivated by questions in the geometryof numbers, Siegel proved that for anymeasurable function f : ℝ2 → ℝ≥0,∫

F

( ∑x ∈ℤ2M \{0}

f (x))d`F =

∫ℝ2f (y ) dy .

Siegel’s formula in fact works for latticesin arbitrary dimension d . In 1998 itwas generalised by Veech to generalSL(d ,ℝ)-stationary point processes in ℝd .(Veech in fact proved it for a more generalclass of random locally �nite Borel measuresin ℝd ).

One challenge is now to work out the probability

ℙ

(|ℤ2M ∩ B | = k

)for a given Borel set B . This turns out to be moredi�cult than one would think, despite the explicitand simple form of the Haar probability measure.The problem is the domain of integration! Let usspecialise to the case of lattice points in a strip.

Lattice points in a strip

Consider the lattice ℤ2M restricted to the verticalstrip

Zw ,R =(w −R,w +R

)× (0,∞),

the green strip in Figure 1. For simplicity (and becauseit’s all that is needed for our applications below) weassume that −R < w < R, so that the vertical axisinteresects Zw ,R . We can now look for the latticepoint in the strip with the lowest height, i.e., with thesmallest positive x2-coordinate. For typical latticesthis point will be unique, and we will denote it by q .

It is remarkable that, for any given lattice ℤ2M , thereare at most three possible choices for q : the twobasis vectors r ,s of ℤ2M with minimal height in thelarger vertical strip between −2R and 2R (see Figure

1), and their sum r + s . This fact, and its link to thefamous three gap theorem for circle rotations, isexplained in [15]. This pretty observation enables usto calculate the distribution of the minimal heightvector q [12].

s

r

r + s

−2R w −R 0 w +R 2R

Figure 1. The two linearly independent lattice vectors withlowest and second-lowest heights in the vertical stripbetween −2R and 2R form a basis. One can show that atany vertical strip of width one (in green) contains at leastone of the three points, and hence the minimal heightvector q is either r , s or r + s .

Distribution of the lattice point withminimal height

If ℤ2M is a Haar random lattice, then theminimal height vector q = (q1,q2) in Zw ,Ris distributed according to the probabilitymeasure Kw ,R (q )dq with density Kw ,R (q1,q2)given by

6c2H

(1 +

q−12 −max(|w |, |q1 −w |

)−R

|q1 |

)

where H (x) =

0 if x ≤ 0x if 0 < x < 11 if 1 ≤ x .

The density Kw ,R (q ) evidently depends on thechoice of w , which proves that the randomprocess ℤ2M \ {0} is not translation-stationary. TheSL(2,ℝ)-stationarity of our random lattice implieson the other hand that all distribution functions mustbe invariant under a simultaneously scaling of thehorizontal and vertical directions by factors of _ > 0

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 45 — #45 ii

ii

ii

FEATURES 45

and _−1, respectively. And indeed, the invariance

K_w ,_R (_q1,_−1q2) = Kw ,R (q1,q2)

is consistent with the explicit formula above.

If one is only interested in the height q2 of q but notits direction, simply integrate over q1 ∈ [w−R,w+R].The result of this integration can be found in [12,Eq. (26)]. There is nothing to prevent us to furtheraverage over w , thus providing the distribution ofthe minimal height for a randomly shifted strip. Theresult of this second integration is as follows.

Distribution of minimal height on average

For a Haar random lattice ℤ2M the minimalheight of a lattice point in the strip Zw ,R ,on average over w , is distributed accordingto the probability measure PR (q2) dq2 =

2R P (2Rq2) dq2, with P (s ) given by (see alsoFigure 2)

6c2×

1 (s ≤ 1)1s + 2

(1 − 1

s

)2log

(1 − 1

s

)− 12

(1 − 2

s

)2log

���1 − 2s

��� (s > 1).

The �rst moment is∫ 10 sP (s )ds = 1. There is,

however, a heavy tail: for s large, we have

P (s ) ∼ 4c2s−3.

So already the second moment diverges! Comparethis with the exponential distribution in Figure 2,which we would have obtained for minimum heightpoints from a Poisson point process with unitintensity, in a strip of unit width.

0.5 1.0 1.5 2.0 2.5 3.0

0.2

0.4

0.6

0.8

1.0

Figure 2. The exponential density e−s (blue) vs. P (s ) (red).

Let us now discuss two natural examples wherethese distributions can be found in the ‘wild’. The�rst describes visibility in Pólya’s orchard or —equivalently — the free path length in the periodicLorentz gas, and the second the energy levelstatistics for quantum harmonic oscillators.

Figure 3. The author in a perfectly periodic orchard: Apoplar plantation near Pordenone, Italy.

rw s/r

Figure 4. Intercollision �ight of a particle in the Lorentzgas with scatterers of radius r . The free path length s ismeasured in units of 1/r and the the exit parameter w inunits of r .

Pólya’s orchard and the Lorentz gas

Pólya asked how far one could see in a forest, if alltree trunks had the same radius r and were either (a)randomly located or (b) planted on a perfect periodicgrid. The same question arises in the study of thefree path length for the two-dimensional Lorentz gas,

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 46 — #46 ii

ii

ii

46 FEATURES

where in the simplest setting a particle moves alongstraight lines in an array of spherical scatterers, seeFigure 4. Let us here focus on the periodic setting,where the trees/scatterers are centered at points ofℤ2. What is the visibility, or free path length, withthe observer at a given tree looking in direction(− sin q,cos q)? Is there a limit distribution when ris small and \ random?

Figure 5. Left: A ray of length s/r in direction(− sin q,cos q) intersecting k tree trunks of (small) radiusr . Right: A rectangle containing k lattice points pointingthe same direction, same length and width 2r .

Figure 6. Left: The con�guration in Figure 5 (right) rotatedclockwise by q. Right: The con�guration on the leftrescaled in the horizontal and vertical directions byfactors of r−1 and r , respectively. The rectangle has nowwidth 2 and height s .

The number of tree trunks of radius r intersectinga ray of length s/r is the same as the number oflattice points in a rectangle of width 2r and lengths/r , see Figure 5. Now let’s rotate the whole picture

clockwise as in Figure 6 (left). The rectangle is nowvertical, and instead of the lattice ℤ2 we have therotated lattice

R1 (q) = ℤ2(cos q − sin qsin q cos q

),

which we have met before. Finally, we stretch thepicture as described in Figure 6 (right), and obtainthe rectangle of height s and width 2 — ther -dependence is gone! On the �ipside, the underlyinglattice has now transformed to the r -dependentlattice

Rr (q) = ℤ2(cos q − sin qsin q cos q

) (r −1 00 r

).

The visibility, or free path length, can now beexpressed as the minimal height of lattice points inthe strip Zw ,1, where w describes the o�set of theray relative to the center of the initial tree trunk; seeFigure 6. (For example w = 0 means the ray emergesfrom its centre as in Figure 5.) The condition |w | < 1ensures we are sitting somewhere on the tree trunk.In the context of the Lorentz gas, the fact that theminimal height can only take three values asw variesis known as Thom’s problem, in turn a close variantof Slater’s problem. The key fact we will now use isthe following:

Randomly rotated lattices

If q is a uniformly distributed random variablein [− c

2 ,c2 ], then the random lattice Rr (q)

converges in distribution to the Haar randomlattice ℤ2M as r → 0.

This statement is a consequence of theequidistribution of large circles in the homogeneousspace Γ\ SL(2,ℝ). The convergence implies that thelimit distribution for the minimal height vector qin the lattice Rr (q) restricted to the strip Zw ,1 isgiven by the density Kw ,1 (q ), and the correspondingdistribution of the free path length is P1 (s ) = 2P (2s ),see Figure 7. Note that if we had measured visibilityin units of the diameter 2s rather than radius r , thelimit distribution would be P (s ).

In the case of the Lorentz gas, P1 (s ) was in fact�rst found by the physicist Dahlqvist [3] in 1997,and only in 2007 established rigorously by numbertheorists Boca and Zaharescu [1], who employedanalytic methods based on continued fractions andFarey sequences. The density Kw ,1 (q ) plays animportant role in describing particles in transport inthe periodic Lorentz gas, and in 2008 was calculated

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 47 — #47 ii

ii

ii

FEATURES 47

independently by Caglioti and Golse [2] by continuedfraction techniques, and by Strömbergsson andthe author [12] via random lattices. The principaladvantage of the latter method is that it works inany dimension [13] and even extends to aperiodic,quasicrystalline point con�gurations! Now, on to thesecond ‘real-world’ appearance of random lattices.

0.2 0.4 0.6 0.8 1.0 1.2 1.4

0.5

1.0

1.5

2.0

Figure 7. The distribution of free path for the periodicLorentz gas with scatterers of radius r = 10−8, sampledover 6000 initial conditions. Theoretical curves are theexponential density 2e−2s (blue) vs. P1 (s ) = 2P (2s ) (red).The data was computed using the algorithm in [9].

Quantum oscillators

In quantum mechanics, the energy levels of boundstates can only take speci�c discrete (‘quantized’)values. One of the simplest and most fundamentalquantum systems with a purely discrete spectrum isthe harmonic oscillator. In two space dimensions, itsenergy levels are given by

Em,n = (m + 12 )~l1 + (n + 1

2 )~l2

where m,n = 0,1,2, . . . run through the non-negativeintegers. The quantities l1,l2 are positive reals,the oscillation frequencies and ~ denotes Planck’sconstant. If we measure energy in units of ~l2, wehave the simpler expression

nm,n = (m + 12 )u + (n +

12 ), u =

l1

l2.

Of particular signi�cance are the spacings betweenenergy levels, as they determine the emissionspectrum of the system. After a little thought, youcan convince yourself that the spacings betweenconsecutive levels nm,n in the interval [E ,E + 1) arethe same as the gaps between the fractional partsbm of the sequence mu , where m = 0, . . . ,N − 1 and

N is number of nm,n in [E ,E + 1). The three gaptheorem mentioned earlier thus implies that we havethe same phenomenon for the energy levels for aharmonic oscillator, at least for intervals of lengthone. A numerical illustration of this fact is given inFigure 8.

0.5 1.0 1.5 2.0 2.5 3.0

1

2

3

4

5

6

7

Figure 8. The gap distribution for the fractional parts ofnu , with n = 1, . . . ,50000 and u = c.

One can show, however, that the distribution in Figure8 will not converge as N becomes large. The onlyhope to see a limit is to introduce a further averageover u . Using the approach in [15], we can expressthe gap between bm and its nearest neighbour to theright as the minimal height of all lattice points in thestrip Zw ,1/2 (of width one), with w = m

N −12 and the

lattice

PN (u) = ℤ2(1 u0 1

) (N −1 00 N

).

As in the case of randomly rotated lattices, also herewe have a limit theorem.

Randomly sheared lattices

If u is a uniformly distributed random variablein [0,1], then the random lattice PN (u)converges in distribution to the Haar randomlattice ℤ2M as N →∞.

This fact is based on the equidistribution of longclosed horocycles on Γ\ SL(2,ℝ), which was provedby Zagier in 1979 in the case of the modular surface,and for more general discrete subgroups Γ by Sarnakin 1981. The most powerful extension of results ofthis type (as well as the rotational averages usedfor Pólya’s orchard) is due to Ratner in the early1990s [16]. It describes equidistribution of unipotentorbits on quotients Γ\G whereG is now a general Lie

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 48 — #48 ii

ii

ii

48 FEATURES

group. (Horocycles are special examples of unipotentorbits.) Recent breakthroughs that build on Ratner’swork include the deep measure classi�cation andequidistribution theorems for moduli spaces by Eskin,Mirzakhani and Mohammadi. For an introductionto dynamics on homogeneous spaces and theirrelevance in number theory I recommend theexcellent textbook by Einsiedler and Ward [4].

0.5 1.0 1.5 2.0 2.5 3.0

0.2

0.4

0.6

0.8

1.0

Figure 9. The gap distribution for the fractional parts ofnu , with n = 1, . . . ,2000 and u sampled over 2000randomly chosen points in [0,1]. Theoretical curves arethe exponential density e−s (blue) vs. P (s ) (red).

By the same reasoning we used earlier forPólya’s orchard, the convergence of randomlysheared lattices to Haar distributed random latticesestablishes the convergence of the gap distributionfor the fractional parts of mu . The one di�erence iswe now sum overw = m

N −12 (m = 0, . . . ,N −1) rather

than integrate — but this discrete average can betreated as a Riemann sum which approximates theRiemann integral for N large. We can conclude thatthe gaps between fractional parts on mu , and thusthe energy level spacings for quantum oscillators,have the same limit distribution as the free pathlength in the periodic Lorentz gas! Figure 9 comparesnumerical data with the theoretical prediction.

The explicit form of the level spacing distribution forquantum oscillators (in Figure 9) was �rst establishedby Greenman [7] in 1996, following previous workon the problem by Berry and Tabor (1977), Bohigas,Giannoni and Pandey (1989), Bleher (1990-91), Pandeyand Ramaswamy (1992), Mazel and Sinai (1992); see[10] for details and references. Greenman’s paperpredates Dahlqvist’s and Boca and Zaharescu’s workon the Lorentz gas; and perhaps more remarkably,the likeness of the two distributions seems to havebeen overlooked even in the recent literature [17]!That the two are the same is evident of course bysimply staring at the explicit formulas, and perhaps

no surprise given the similarity of their arithmeticsetting. The beauty of using lattices is that wehave a conceptual understanding of why the limitdistributions must coincide: random rotations andrandom shears both converge to the same Haarprobability measure — a non-trivial fact!

Other applications

We can construct random lattices that are not onlySL(2,ℝ)-stationary but also translation-stationaryas follows. Take the randomly shifted lattice ℤ2 + "with " uniformly distributed in the unit square [0,1]2(recall our construction in dimension one), then applya linear transformation to obtain the random a�nelattice

(ℤ2 + "

)M with M distributed in F with

respect to Haar measure. This point process is nowtranslation-stationary and it has intensity one. Infact, also its second moment coincides with thatof a Poisson point process; again a consequenceof Siegel’s mean value formula [5, App. B]. In 2004,Elkies and McMullen [6] proved that the limitinggap distribution for the fractional parts of

√n, n =

1,2,3, . . . can be derived via a random a�ne lattice.The proof uses equidistribution of certain nonlinearhorocycles, which is a consequence of Ratner’smeasure classi�cation theorem. The distributionfound by Elkies and McMullen also describes thelimiting distribution for directions in a �xed a�nelattice [13].

Random lattices appeared in the probabilityliterature in Kallenberg’s disproof of the Davidsonconjecture [8] on the classi�cation of line processeswhich have (almost surely) no parallel lines.The counterexamples to the conjecture wereconstructed using two-dimensional random a�nelattices restricted to a vertical strip, where eachlattice point represents a line via the standardlinear parametrisation. This is particularly impressiveas Kallenberg was unaware of Siegel’s classicalconstruction in the geometry of numbers, as clari�edby Kingman; see the quote at the end of Kallenberg’spaper.

Other examples where random lattices play animportant role are the value distribution of quadraticforms, such as in Margulis’ proof of the Oppenheimconjecture, the Hall distribution describing thegaps between Farey fractions, random Diophantineapproximation, diameters of random Caley graphsof abelian groups, the Frobenius problem, hittingtimes for integrable dynamical systems, deviations of

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 49 — #49 ii

ii

ii

FEATURES 49

ergodic averages of toral translations, etc. And howabout random lattices in non-Euclidean settings?

But these are stories for another day!

Take home message

• Random lattices are important pointprocesses with connections to ergodictheory, geometry, number theory,combinatorics, probability and physics.

• The level spacing distribution of a quantumoscillator equals the free path distributionof the periodic Lorentz gas.

Acknowledgements

Much of the material presented here is based on jointwork with Andreas Strömbergsson, whom I thankfor our longstanding collaboration. I am grateful toAtahualpa Kraemer and David Sanders for sendingme the data for Figure 7. The author’s research iscurrently supported by EPSRC grant EP/S024948/1.

FURTHER READING

[1] F.P. Boca and A. Zaharescu, The distributionof the free path lengths in the periodictwo-dimensional Lorentz gas in the small-scattererlimit, Commun. Math. Phys. 269 (2007), 425–471.[2] E. Caglioti and F. Golse, The Boltzmann-Gradlimit of the periodic Lorentz gas in two spacedimensions. C. R. Math. Acad. Sci. Paris 346 (2008),no. 7-8, 477–482.[3] P. Dahlqvist, The Lyapunov exponent inthe Sinai billiard in the small scatterer limit.Nonlinearity 10 (1997), no. 1, 159–173.[4] M. Einsiedler and T. Ward, Ergodic theory witha view towards number theory. Graduate Texts inMathematics, 259. Springer-Verlag London, Ltd.,London, 2011.[5] D. El-Baz, J. Marklof and I. Vinogradov, Thedistribution of directions in an a�ne lattice:two-point correlations and mixed moments. Int.Math. Res. Not. IMRN 2015, no. 5, 1371–1400.[6] N.D. Elkies and C.T. McMullen, Gaps in

√n mod

1 and ergodic theory. Duke Math. J. 123 (2004), no.1, 95–139.

[7] C.D. Greenman, The generic spacingdistribution of the two-dimensional harmonicoscillator. J. Phys. A 29 (1996), no. 14, 4065–4081.[8] O. Kallenberg, A counterexample to R.Davidson’s conjecture on line processes. Math.Proc. Cambridge Philos. Soc. 82 (1977), no. 2,301–307.[9] A. Kraemer, N. Kryukov and D. Sanders,E�cient algorithms for general periodic Lorentzgases in two and three dimensions. J. Phys. A 49(2016), no. 2, 025001, 20 pp.[10] J. Marklof, The n-point correlations betweenvalues of a linear form. With an appendix byZ. Rudnick. Ergodic Theory Dynam. Systems 20(2000), no. 4, 1127–1172.[11] J. Marklof, Distribution modulo one andRatner’s theorem. Equidistribution in numbertheory, an introduction, 217–244, NATO Sci. Ser. IIMath. Phys. Chem., 237, Springer, Dordrecht, 2007.[12] J. Marklof and A. Strömbergsson, Kinetictransport in the two-dimensional periodic Lorentzgas. Nonlinearity 21 (2008), no. 7, 1413–1422.[13] J. Marklof and A. Strömbergsson, Thedistribution of free path lengths in the periodicLorentz gas and related lattice point problems.Ann. of Math. (2) 172 (2010), no. 3, 1949–2033.[14] J. Marklof and A. Strömbergsson, Gapsbetween logs. Bull. Lond. Math. Soc. 45 (2013), no.6, 1267–1280.[15] J. Marklof and A. Strömbergsson, The threegap theorem and the space of lattices. Amer. Math.Monthly 124 (2017), no. 8, 741–745.[16] D.W. Morris, Ratner’s theorems on unipotent�ows. Chicago Lectures in Mathematics. Universityof Chicago Press, Chicago, IL, 2005.[17] G. Polanco, D. Schultz and A. Zaharescu,Continuous distributions arising from the threegap theorem. Int. J. Number Theory 12 (2016), no.7, 1743–1764.

Jens Marklof

Jens is Professor ofMathematical Physicsand Dean of Scienceat the University ofBristol. He is currentlyworking on problemsin the kinetic theory ofgases, quantum chaos,pseudo-randomness,and the distribution ofautomorphic forms.

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 50 — #50 ii

ii

ii

50 FEATURES

Marriages, Couples, and the Making ofMathematical Careers

DAVID E. DUNNING AND BRIGITTE STENHOUSE

By considering instances of mathematicians who have worked closely with a spouse or partner, we o�erhistorical perspectives on gender and work-life balance in mathematical research. We aim to use historyto open space for re-imagining how collaboration, home-life, and labour �t together in the mathematicalcommunity today.

The home life of mathematics

Though mathematicians are often imagined asthe quintessential solitary researchers, many havemanaged the daily routines of a mathematicalcareer through partnership with a spouse whowas intimately involved in their working life. Whilstmarriage is certainly not the only, nor even the mostcommon form that collaboration can take, it doeso�er an especially clear window on the unstableboundaries dividing labour into the intellectual andthe domestic, the masculinised and the feminised, orthe credited and the unacknowledged. As historiansof mathematics, we suggest that by looking at howsuch categories were made, sustained, and changedin the past, we can not only deepen our historicalunderstanding but also support more equitablemathematical practice in the present.

A focus on collaboration is part of a broader trendin history of science scholarship which has soughtto unravel the myth of the ‘lone genius’, thatheroic, solitary — and usually white, male, European— individual who is celebrated as the sole mindbehind innumerable discoveries. This is perhapsbest encapsulated by Isaac Newton’s so-called annusmirabilis or ‘Year of Wonders’, a period of intenseproductivity when he escaped from Cambridge toWoolsthorpe Manor during the Great Plague of1665–6; it was here that he seemingly ‘invented’calculus out of nothing, revolutionising physics andmathematics. However, this narrative sidelines andundervalues the work that had already been doneby mathematicians such as Pierre de Fermat, RenéDescartes, or Isaac Barrow on the problems of �ndingtangents and quadratures. Futhermore it rendersinvisible the extensive network of mathematicianswho corresponded with each other on such topics,and of which Newton was a part. These writtenexchanges could be facilitated by formal bodies, such

as learned academies and societies, but just as oftenwere part of personal correspondence.

Thus our need to understand individualachievements in their wider intellectual and socialcontext should not end at the boundary of o�ciallyrecognised scholarly activity. The importance ofscienti�c knowledge production in the ‘domesticsphere’ — that is at home, in private, or throughinformal exchange — has been well treatedin literature on women in science. Until veryrecently women were unable to access the ‘public’institutions which have long been privileged asknowledge-making spaces: universities, scienti�cacademies, or research laboratories. Only by lookingbeyond these spaces have historians recognisedthe many creative ways women found to participatein scienti�c endeavours. Ineligible to study at theÉcole Polytechnique in 1794, Sophie Germain enteredinto correspondence with Joseph-Louis Lagrangeunder the pseudonym Antoine-Auguste Le Blanc inorder to get a copy of his lecture notes to study.Germain subsequently situated herself within a widernetwork of mathematical correspondents, perhapsmost notably Carl Friedrich Gauss, and althoughshe never directly published her work on Fermat’sLast Theorem it was certainly read by Adrien-MarieLegendre who explicitly attributed a result to her ina memoir he presented to the Académie des Sciencesin 1823 [2].

To bring the collaboration that takes place withina household to the foreground is then to unitethese two currents in historical research, viewingcollaboration and domesticity together. Historians ofscience have studied collaboration between marriedcouples and other domestic partners, but so far welack a study dedicated to collaborative couples inthe history of mathematics. Collaborative couplesin mathematics, however, present a special case in

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 51 — #51 ii

ii

ii

FEATURES 51

Figure 1: Participants at the 1950 ICM. HUPSF International Congress of Math (BP1), Harvard University Archives.

that so many kinds of mathematical practice arepossible without any sort of specialised equipmentor facilities; there need be no di�erence betweendomestic space and the space of mathematicalresearch. At times this fact has made mathematicalwork more accessible to women than other formsof scienti�c contribution, though that access hasnot meant their work was regarded in equal orungendered terms. Rather instances of mathematicalcollaborative couples provide us a window on thecomplex gendered terrain of collaboration within amarriage.

At home, the lines between the kinds of laboura couple divvied up among themselves and thosewhich they delegated to servants, secretaries, orextended family, position mathematicians in a widerstructure of class and familial relations. For DorothyVaughan, the transition from school teacher toprofessional mathematician was contingent on herwider family providing childcare when she moved 137miles away from her children to take up a job atLangley Research Centre, part of the United StatesNational Advisory Committee for Aeronautics, in 1943.Vaughan’s life and career is treated in Margot LeeShetterley’s book Hidden Figures, and the 2016 �lm ofthe same name. Living through the global pandemicin 2020 has certainly underscored the relationshipbetween gender, class, and caring responsibilities,with the greatest reduction in time available forresearch being felt by female scientists with youngdependents [5].

Couples and careers

We have so far emphasised domestic settings,but a couple’s collaborative activity is certainlynot limited to the home. Many couples have

worked together to construct a shared network ofmathematical acquaintances via letter writing or,more recently, through attendance at internationalmeetings, congresses, and conferences — sites atwhich it can be impossible to separate mathematicalfrom purely social exchange. O�cially, womenoften attended such conferences as spouses andtherefore do not turn up on the list of participants,but nevertheless engaged with the mathematicalcommunity in a meaningful way. Indeed the Women’sCommittee of the 1950 International Congress ofMathematicians in Cambridge, Massachusetts wasmade up of the wives of the organisers, and oversawsome of the social activities at the conference whichwere vital to international exchange. Thus women,including many who were not mathematiciansthemselves, helped sustain the professional networksthat made international mathematical researchpossible.

Exclusion from formal membership in such networks,however, was often one of the tactics used byelite scientists to contain the perceived threatto their professional status represented by risinggender, sexual, or racial diversity in science.Heterosexual couples who were also colleaguescan serve as useful comparative illustrations ofthe di�erential obstacles women faced, evenwhile their marriages also sometimes o�eredstrategies for circumventing those obstacles. Themathematical logician, psychologist, and activistChristine Ladd-Franklin completed the requirementsfor a PhD in Mathematics at Johns Hopkins Universityin 1882. But the university — employing anotherincreasingly common tactic for hindering women’sscienti�c activity — drew the line at actually awardingdegrees to the few women it grudgingly permittedto become students. Her husband Fabian Franklin’sscienti�c career, however, o�ered them stability

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 52 — #52 ii

ii

ii

52 FEATURES

and even the opportunity for them both to spenda sabbatical year in Europe. Ladd-Franklin spentthis time working in the labs of Georg Müller inGöttingen and Hermann von Helmholtz in Berlin.Franklin left academia for journalism in 1895, whereasLadd-Franklin remained an active, highly regardedscholar into old age, but she never had access tothe academic positions and resources he had had athis disposal. In 1926 she �nally received the PhD shehad earned 44 years earlier.

The so-called ‘two-body problem’, where bothpartners are early career researchers on theacademic job market, continues to create tensionfor those hoping for work-life balance. The likelihoodof both partners successfully �nding work inthe same geographic location is often decreasedfurther when their research is in the same or verysimilar �elds. According to the documentary �lmby George Csicsery, Secrets of the Surface: TheMathematical Vision of Maryam Mirzakhani, suchconsiderations even in�uenced the career trajectoryof Fields Medallist Mirzakhani, who was married tomathematician Jan Vondrák.

In the case of couples who have collaborated evenmore closely, working together on the �nest details oftheir research, the distinction between cooperationand exploitation can be slippery. A challenge forhistorical interpretation arises in cases of joint workappearing under a single (usually male) name, anarrangement that may or may not have been mutuallyagreeable depending on each partner’s interpretationof their own role. The most well-known case inmathematics is that of Grace Chisholm Young andWilliam Henry Young. In 1895, aged 27, ChisholmYoung was awarded her doctorate in mathematicsat Göttingen University, and between then and1929 the Youngs published over 200 mathematicalpapers. They collaborated closely throughout thistime, however only 13 papers were published jointly,and only 18 were published under Chisholm Young’sname alone. At a time when there were very fewpaid positions for women to teach or researchmathematics (and even fewer for married women), itseems that it was more bene�cial economically forthem as a household to attribute the work solely toWilliam Young.

The terms of a collaboration, however, do not alwaysremain amiable. When Mileva Mari threatened herex-husband Albert Einstein with revealing the extentof their collaboration on work published under hisname, his chilling response was to point out that noone would believe her:

“You made me laugh when you began tothreaten me with your memories . . .Whena person is completely insigni�cant, thereis nothing else to tell such a person but toremain modest and silent. This is what I adviseyou to do.” [1, p. 241].

The exploitation of collaborators arising from anunequal power dynamic is still extremely relevanttoday and of course not con�ned to partnerships.PhD students and post-doctoral researchers facechronic job instability whilst being reliant on thesupport and collaboration of supervisors whenpreparing their work for publication. This is furthercomplicated by the widespread sexual harassmentwhich persists at universities in the UK. The 2018NUS Report on sta�-student sexual misconduct inhigher education found that 41% of the 1535 studentswho responded to the survey had experiencedsexual misconduct from sta�, with postgraduatesmore likely to have experienced misconduct thanundergraduates. Students were also more likely tohave experienced sexual misconduct from universitysta� if they were women, and more again if theyidenti�ed as gay, queer, or bisexual. [6, pp. 8–9].

Given that the division of labour within a coupleis so often governed by prevailing inequities in thesociety in which they live, it is no surprise that malemathematicians have tended to more easily receivecredit, compensation, and prestige than their femalepartners. By favouring William Henry Young’s name,the Youngs adopted a highly successful strategy ina publishing landscape that was not of their owndesign.

But we also �nd examples of cooperative e�ortsto prioritise a woman’s mathematical career, suchas the case of Mary Somerville (née Fairfax) andher husband Dr. William Somerville. Ineligible fora university education or for election to a learnedsociety as a woman, Somerville’s access to themathematical knowledge circulating in these spaceswas highly restricted. As a ‘clubbable’ gentlemenwith interests in natural history and mineralogy,her husband, on the other hand, was elected amember of numerous learned societies includingthe prestigious Royal Society of London. He activelysupported Somerville in her studies and scienti�cwriting by borrowing books from libraries on herbehalf, soliciting information from other societymembers, either in person at meetings, or vialetter correspondence, and liaising with her publisherduring the production of her books [7]. Dr. Somervilleseems to have had no interest in mathematical

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 53 — #53 ii

ii

ii

FEATURES 53

research or in cultivating a reputation for himselfas an eminent scientist. The importance of thisdisinterest was noted by geologist Charles Lyell in1831 when he wrote the following:

“had our friend Mrs. Somerville been marriedto La Place, or some mathematician, weshould never have heard of her work. Shewould have merged it in her husband’s, andpassed it o� as his.” [3, p. 325]

While emphasising the role domestic partnershipshave played in mathematical work, we should notneglect the converse in�uence that mathematicalcareers can exert on a given couple’s way of buildinga life together. In his survey of collaboration ofqueer couples in the sciences, Opitz suggests that“the ethos of professional respectability claimeda signi�cant role in shaping the dynamics of[queer] collaborative partnerships” [4]. That is tosay, scientists curated an image of themselvesand their relationships in order to conform withscienti�c practice of the time, whether that was asequal partners sharing expertise, or one partnerbeing positioned as a researcher and the otheras a domestic helpmate. This in turn a�ected thedynamics of the relationship itself, for examplewhether the partners desired or were able toachieve cohabitation. Moreover, the lived experienceof a queer scienti�c couple was, and is, heavilyin�uenced by social factors, such as the need to avoidharassment and discrimination in the workplace.

Paying attention to mathematicians’ marriages alsoreveals ways that a mathematical career continuesto be shaped and reimagined after an individual’sdeath. After Bernhard Riemann’s death, his widowElise Riemann played an active role in the productionof his Collected Works, while Emilie Weber helpedbuttress the friendship of Heinrich Weber andRichard Dedekind as they edited the publication.Similarly, Mary Everest Boole asserted quite an activevoice in the commemoration of her husband, thelogician George Boole, whom she survived by halfa century. After his death she published proli�callyon mathematical and logical pedagogy intertwinedwith religious issues, developing a mystical (andoften mystifying) interpretation of George’s work.In light of his well-documented reticence to speakpublicly about his own religious beliefs, along withthe temporal distance between his career as anauthor and hers, it is di�cult to discern which ofher ideas he shared. But whereas sexist dismissalsof Mary’s admittedly eccentric views were once

common, scholarly consensus now rightly recognisesher as a generally reliable witness to the morepersonal manifestations of George’s thought. Todayhis contributions are better remembered throughthe lens of the information-theoretic interpretationdeveloped by Claude Shannon in the mid-twentiethcentury. (Claude and his wife Betty Shannon, acomputer at Bell Labs, o�er another example of amathematically collaborative marriage.) But Mary’se�orts to shape the commemoration of George’slegacy stand as an insightful body of work, o�eringa useful reminder that the meaning of a person’scareer is not �xed at the time of their death, anddoes not belong to the deceased alone.

Figure 2. A letter from Augustus De Morgan to Dr.Somerville, sending Bailly’s History of Astronomy for “MrsSomerville”. Bodleian Library, Somerville Collection, Dep.c. 370, MSD-3 126, reproduced courtesy of the Principaland Fellows of Somerville College.

The work of mathematics, past and present

Mathematical research — as the readers of theNewsletter will hardly need reminding — is work.When we understand the history of mathematicsas the history of a particular kind of work, it is

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 54 — #54 ii

ii

ii

54 FEATURES

clear that a full picture must include the relatedand interdependent kinds of labour that togetherform the context in which people make their livesas mathematicians. Such a historical perspective inturn compels us to recognise the seemingly mundanequestions around various divisions of labour asmeaningfully intrinsic to the work of mathematics inthe present.

In suggesting marriages as a focal point, we certainlydo not mean to overlook the many workers of diversekinds who have not been part of a mathematicalcouple; this is just one line of historical inquiryamong many. We call attention to it as a particularlyilluminating one: given the feasibility of doingmathematics at home, and the paper-based practicesso often constitutive of mathematical knowledge,studies of collaborative couples stand to o�er muchinsight to the history of mathematics. Moreover, suchstudies naturally look beyond ‘lone geniuses’ anddestabilise the history of mathematics as presentedin university courses, namely as a body of knowledgesteadily unearthed through the conjecturing andproving of theorems by the individuals after whomthey are named.

To organise mathematical work in a particular way, tothe advantage or disadvantage of particular people,has always been part of the making of mathematicalcareers. But the great diversity of ways this processhas played out in the past illustrates the contingencyof any given arrangement, and hence the possibilityof re-imagining how collaboration, domesticity, andlabour �t together in the mathematical communitytoday.

To �nd out more...

We encourage readers to attend theforthcoming workshop Marriages, Couples, andthe Making of Mathematical Careers, supportedby the LMS and the British Society for theHistory of Mathematics, to be held online29–30 April 2021.

For more details and free registration pleasevisit mathmarriages.wordpress.com.

FURTHER READING

[1] P. Gagnon, Who Cares About Particle Physics?Making sense of the Higgs Boson, the Large Hadron

Collider and CERN, Oxford University Press, 2016.[2] E. Kaufholz-Soldat, N. M. R. Oswald (eds),Against All Odds: Women’s ways to MathematicalResearch Since 1800, Springer, 2020.[3] Mrs Lyell (ed), Life Letters and Journals of SirCharles Lyell, Bart., Volume 1, John Murray: London,1881.[4] A. Lykknes, D. L. Opitz, B. Van Tiggelen (eds),For Better or For Worse? Collaborative Couples inthe Sciences, Birkhäuser, 2012.[5] K. R. Myers et al., Unequal E�ectsof the COVID-19 pandemic on scientists,Nature Human Behaviour, 2020.doi.org/10.1038/s41562-020-0921-y.[6] National Union of Students,Power in the Academy: Sta� SexualMisconduct in UK Higher Education, 2018.www.nusconnect.org.uk/resources/nus-sta�-student-sexual-misconduct-report.[7] B. Stenhouse, Mister MarySomerville: Husband and Secretary,The Mathematical Intelligencer, 2020.doi.org/10.1007/s00283-020-09998-6.

David E. Dunning

David E. Dunning is aPostdoctoral ResearchAssociate in theHistory of Mathematicsresearch group at theMathematical Institute ofthe University of Oxford.

His current book project examines the rise ofmathematical logic through the lens of notation,exploring both technical and social aspects ofsymbolic systems. He hails from Philadelphia and isan ardent dog person.

Brigitte Stenhouse

Brigitte Stenhouse is aPhD student in Historyof Mathematics atthe Open University,UK. Her thesis looksat the work of MarySomerville (1780–1872),

and considers questions around translations;di�erential calculus in early-19th-century westernEurope; and gendered access to knowledge. Herfavourite way to unwind is to go splashing in thesea with her �ve-year-old nephew — the colder thebetter!

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 55 — #55 ii

ii

ii

EARLY CAREER RESEARCHER 55

To Ithaca

MARINA ILIOPOULOU

“As you set out for Ithaka / hope your road is a long one, / full of adventure, full of discovery....” 1. After ten yearsof postgraduate experience, Marina Iliopoulou re�ects on the mathematician’s academic journey — bumpy,constant and exciting.

In February 2019, after four years of PhD and nearlysix years of postdoctoral positions, I enthusiasticallyassumed my �rst permanent appointment, asLecturer in Pure Mathematics at the University ofKent. Starting a new life in a new place, I was eager tobring my mathematical friends over, to discuss ourwork and show them around beautiful Canterbury. InFebruary 2020, this became a reality: using my LMSCelebrating New Appointments grant, I organised acosy one-day conference at the University of Kent.

The meeting featured four specialist talks on recentadvances in harmonic analysis, and surpassedmy expectations, attracting about 25 people fromaround the UK. I was particularly happy to seethat there was a lot of mathematical interaction,even between participants who had not met before.The experience made me feel at home at Kentand renewed my connection to the UK harmonicanalysis group, fuelling me with further excitementfor upcoming collaborations. Now, after severalmonths, the memory of the meeting is even morespecial, marking the last time our harmonic analysisgroup met, before coronavirus changed everything.

Having forgotten to take a photo of the meeting, I provideone of the venue: SMSAS, University of Kent

As mathematicians, we primarily aim to create newmathematics. This goal largely shapes our lives.Before taking on permanent positions, we take onyears of training (in my case, a decade) of PhDstudy and postdoctoral work, close to experts aroundthe world. Solving a mathematical problem can takea lot of time — even years — and requires dailydedication and deep concentration. Being so focusedwithout getting disheartened is not always easy. Welike being productive, but unfortunately performingmathematical research o�ers no guarantee of results— at least not when the problem is worth it. Formany of us, however, hunting down the truths behinddi�cult questions is reward in itself, motivating usto lead this, often uncertain, life.

My own mathematical journey started in my hometown, with an undergraduate degree in mathematicsat the University of Athens (Greece). My professorsthere were truly inspirational — and, even thoughI had not the remotest idea what academic life islike, I knew well enough that I loved puzzles andwanted advanced mathematics to stay in my life.My lecturers advised me to do a PhD abroad. I stillremember my surprise when I was told that gettinga PhD requires proving new theorems — somehowuntil then I had assumed that all maths had alreadybeen created, by people long dead. So, even thoughI had never thought of leaving Greece (or wantedto), I applied for postgraduate programmes abroad.I was exceptionally lucky to be accepted for a PhDat the University of Edinburgh, to work on harmonicanalysis under the supervision of Tony Carbery.

My four PhD years in Edinburgh were the happiestof my life. Tony was a wonderful supervisor, whorespected my personal taste in mathematics andgave me problems that I truly cared about —combinatorial at the time. He also granted meabsolute intellectual freedom, trusting that I wouldask for guidance if I needed to. This was exactly whatI needed to be creative. Research became intertwined

1The �rst lines of "To Ithaca" by C. P. Cavafy

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 56 — #56 ii

ii

ii

56 EARLY CAREER RESEARCHER

with a care-free life, and was constantly in my mind.In fact, I came up with the �nal piece of the solutionof my �rst problem — a piece I was missing formonths — after returning from a party at 4am. Imention this not to endorse heavy drinking, but todemonstrate that we always think our maths, andthat a regular 9am–5pm o�ce schedule doesn’tnecessarily get our ideas �owing.

At the start of my PhD, an unexpected breakthroughin harmonic analysis (not induced by myself!)made the �eld one of the most fertile in modernmathematics. In particular, harmonic analysis aims tounderstand the interaction of waves. Mathematicianshad long been trying to understand this interactionvia toy problems (including combinatorial questions,such as the ones I worked on during my PhD). As I wasstarting my PhD, such combinatorial problems wereshown to have a deep algebraic nature. Since then,this algebraic behaviour has been systematicallyexploited, leading to major advances in geometricand analytic problems that in the recent past hadbeen considered untouchable. For us who work inthe �eld, these are exciting times to be alive. I quicklybecame eager to work on the original harmonicanalytic problems that gave rise to the combinatorialones I was focusing on, and to contribute a little tothis wave of progress.

I achieved this during my postdoctoral positionsover the next six years, in Birmingham, MSRI andUC Berkeley. Inspired by the vision of my mentors(such as Jon Bennett in Birmingham and MichaelChrist in Berkeley), I started realising that being asuccessful researcher means much more than justsolving problems. It also means developing a tastefor what is interesting; seeking connections betweendi�erent mathematical areas; and creating questionsthat matter. I started adopting this way of thinking,and creating research plans and proposals of myown.

All these years of e�ort and travelling had theirgood and bad moments, and naturally shaped meand my personal life. There are successes anddisappointments: for every paper I have produced,I can provide a sizeable list of problems that I havefailed to solve, despite trying very hard. Often workhas been very hectic. For example, during my �rstsemester at Berkeley, I somehow managed to impose

upon myself the tightest travel restrictions shortof house arrest: I was spending so much time onteaching preparations that I didn’t get a chance towalk a single step west of my �at (I only had timeto go to the university and the supermarket, whichsadly were both east).

Not knowing where our next job will be, or even ifwe will manage to secure one, despite so many yearsof hard work, can be stressful, and can seriouslyhinder our personal life. However, it is also exciting,because, truly, anything can happen. It is a life fullof travel and experiences, anticipation and strongexcitement. Our love for research gives us energyand con�dence, and can take us very far from wherewe started, to destinations that we never imagined.

Some destinations where mathematics has taken me

My long-anticipated permanent job gave mecertainty and relief. Naturally, it comes with otherresponsibilities, apart from research and teaching,which I am still learning to balance. And while itmeans the end of care-free research-oriented years,the search for new questions and ideas never ends.This search has the power to make every momentinteresting.

Marina Iliopoulou

Marina is a Lecturer inPure Mathematics atthe University of Kent.She is interested in theinterface of harmonicanalysis, incidencegeometry and additive

combinatorics. She also loves singing, but herneighbours prefer when she quietly does maths.

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 57 — #57 ii

ii

ii

EARLY CAREER RESEARCHER 57

Microtheses and Nanotheses provide space in the Newsletter for current and recent research students tocommunicate their research �ndings with the community. We welcome submissions for this section fromcurrent and recent research students. See newsletter.lms.ac.uk for preparation and submission guidance.

Microthesis: A Novel Algorithm for SolvingFredholm Integral Equations

FRANCESCA ROMANA CRUCINIO

Fredholm integral equations of the �rst kind are the prototypical example of ill-posed linear inverse problems.They model, among other things, reconstruction from noisy or delayed observations and image reconstruction.My PhD project explores the use of Monte Carlo methods to solve these integral equations.

Fredholm Integral Equations

Fredholm integral equations of the �rst kind

h (y) =∫

g (x ,y) f (x) dx , (1)

are linear integral equations in which the functionf is the unknown and g ,h are given. Theygeneralise linear systems of equations to thein�nite-dimensional setting and describe thedistortion caused by g on the function f .

Solving (1) corresponds to reconstructing f fromits distorted version h. In the simplest case, thedistortion g models addition of noise to the signal f ,which has to be reconstructed from its noisy versionh, a task known as deconvolution.

Fredholm equations �nd applications in medicalimaging, where f corresponds to an image whichis reconstructed from data provided by tomographyscanners. In epidemiology, (1) links the incidencecurve of a disease to the observed number of cases.

Regularisation

Fredholm integral equations (1) are generally ill-posedand stable solutions can be found minimising adistance between the h and the right-hand-side of (1).We consider regularised solutions f which minimisethe Kullback-Leibler distance∫

h (y) log(

h (y)∫g (x ,y) f (x) dx

)dy , (2)

with additional constraints to ensure smoothreconstructions of f .

To minimise (2), we resort to iterative techniqueswhich, given an initial guess, reduce (2) sequentiallyuntil a �xed point is reached and the reconstructionof f stops improving.

Computational Considerations

Standard approaches to regularisation requirediscretisation of the domain of f , restricting theirapplications to low-dimensional scenarios, and makestrong assumptions on the regularity of f . Often,knowledge of an analytic representation of h isrequired.

Monte Carlo methods are a class of simulation basedtechniques which approximate a (density) function fthrough a set of samples. These algorithms providea stochastic discretisation of the domain of f whichcan be applied in high-dimensional scenarios and canbe naturally implemented when only observationsfrom h are available.

Interacting Particle Methods

Interacting particle methods are a class of MonteCarlo methods which approximate a probabilitydensity through a population of (weighted) samplesevolving over time. My PhD project considers aparticular family of interacting particle methods,sequential Monte Carlo (SMC).

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 58 — #58 ii

ii

ii

58 EARLY CAREER RESEARCHER

Figure 1. SequentialMonte Carlo

In SMC, a populationof weighted samplessequentially undergoesrandom mutations,which are weightedso that mutationsthat produce �tterindividuals are morelikely to survive(selection). A newpopulation is originatedby replicating �ttermutations, while theother individuals dieout; see [2] for a moredetailed account.

We use SMC to approximate the �xed point ofthe iterative scheme and show that the estimatorswe propose enjoy good asymptotic properties:as the discretisation gets �ner they converge toa regularised solution of the integral equation.Currently, we are exploring the use of McKean-Vlasovstochastic di�erential equations to approximate thefunction f minimising a penalised version of (2).

Image Reconstruction

Given the blurred image in the �rst panel of Figure 2we can reconstruct the corresponding clear imageby solving a 2D Fredholm integral equation.

Blurred Image Iteration 5

Iteration 20 Iteration 100

Figure 2. Given the blurred image, h, we iterativelyreconstruct the original image, f . The function gdescribes the motion which caused the blur.

Reconstruction of cross-sections of the brain fromthe noisy measurements provided by positronemission tomography (PET) scanners is one of themost relevant applications of Fredholm integralequations. These reconstructions are used to analyse

internal biological processes to detect medicalconditions such as schizophrenia, cancer, Alzheimer’sdisease and coronary artery disease.

The algorithm reconstructs the reference imagein the �nal panel of Figure 3 by re�ning thereconstruction until a �xed point is reached.

Iteration 1 Iteration 5 Iteration 10

Iteration 20 Iteration 100 Reference Image

Figure 3. A cross-section of the brain is reconstructedfrom the data given by a PET scanner.

Acknowledgements

This work was supported by funding from the EPSRCand MRC OXWASP Centre for Doctoral Training(EP/L016710/1).

FURTHER READING

[1] F. R. Crucinio, A. Doucet, A. M. Johansen, AParticle Method for Solving Fredholm Equations ofthe First Kind, Preprint 2020, arXiv:2009.09974[2] P. Del Moral, Feynman-Kac Formulae, Springer2004

Francesca RomanaCrucinio

Francesca is a PhDstudent at theDepartment of Statisticsof the University ofWarwick, supervised byAdam M. Johansen and

Arnaud Doucet. Her main research interest is inMonte Carlo methods, with a particular focus onparticle methods and their theoretical properties.Outside of research, she enjoys travelling, good foodand board games.

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 59 — #59 ii

ii

ii

REVIEWS 59

Mage Merlin’s Unsolved Mathematical Mysteries

by Satyan Linus Devadoss and Matthew Harvey, MIT Press, 2020, £20.00,US$25.00, ISBN: 978-0262044080

Review by Troy Kaighin Astarte

I was summoned toCamelot, where the greatmage Merlin told me ofsixteen mysteries...

This is a pleasantand satisfying littlebook, perfect for theaesthetically-inclinedmathematician’s co�eetable. It is short (I readit cover-to-cover inforty-�ve minutes) and

beautifully presented.

The book opens with a scan of the landscape ofmathematics, as seen by the authors. They say thatmost people think of mathematics like a mountain: asolid base of well-known topics, tapering up throughlayers of increasing complexity, to the rarefied andmysterious peaks of unsolved problems. The authorspropose that instead, mathematics should be seen likean ice-cream cone: a palatable, if mundane, conicalbase, growing steadily more tasty as one movesupwards towards the delicious, downwards-tricklingfrozen treat of mathematical mystery. The metaphoris perhaps a little odd as most people do not begineating ice-cream cones from the point, but servesto illustrate that unsolved maths can be seen asreachable and desirable.

A new metaphor is swiftly employed after the firstfew pages, which sticks throughout the book: ournarrator is Maryam, a young mathematician namedafter Fields Medallist Maryam Mirzakhani, who is adistant descendant of the legendary Merlin. A bookof tales written by Merlin himself has been handeddown to Maryam, and she has picked out sixteenmathematical puzzles from the book. While unsolvedby Merlin or anyone else since, Maryam offers usa tantalising glint of hope that we might be able tosolve them by reminding us of Andrew Wiles’ famoussolving of Fermat’s Last Theorem.

It is these ‘mysteries’ which constitute the majorityof the book. They cover a number of areas inmathematics. From geometry, we have a puzzle aboutarranging smaller squares to cover a larger one; ingraph theory, we ponder the relationship betweenedges and vertices in thrackles; and in number theory,we wonder whether there are infinitely many twinprimes. Many of the puzzles will be well-known to themathematical reader; others may be less so.

It is in the presentation of the puzzles that thebook shines. Each mystery is described by Merlinin a double-page spread with beautiful typography,lovely illustrations, and a short in-universe storyin which characters from Arthurian lore displayperverse attachments to particular mathematicalconcepts. (Guinevere insists that on her daughters’prime-numbered birthdays, red candles be lit.) Everytale ends with Merlin sighing that “even with hispowers of magic and logic” he is never able to solvethem.

Each puzzle is preceded by a short introduction tothe mathematical concept handwritten in characterby Maryam, and a longer discussion follows each,written (as far as I can tell) in the voice of Devadossand Harvey. The discussions explain the problemin modern mathematical terminology and examinerelated concepts. They include some proven resultsrelated to the puzzle and explain who solved them;some pointers are given about the direction one mightgo to solve it. The puzzles are all clearly explained,and, if inspired, one could easily start work on themright away.

So, who is the book for? Despite its playful framing,it is probably not a book for young children to readthemselves — the explanation sections are a littleadvanced for anyone younger than secondary-schoolage. The puzzles themselves are very accessible,though, and one can certainly imagine young childrenenjoying listening to the stories and talking about howto think about the puzzle. Personally, I have always

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 60 — #60 ii

ii

ii

60 REVIEWS

struggled with puzzles, tending to feel stupid thatI can’t solve them; but at least here, knowing thebest mathematical minds of generations hadn’t solvedthem, there was no expectation that I should! Ifound the most fun in trying to work out the mathsconcepts behind each puzzle and beginning to developa strategy for their solution.

I think this book would do its best work on a coffeetable. I imagine most mathematically-minded peoplewould enjoy reading quickly through it when firstbought, and then dipping into it occasionally later. Theexperienced mathematician is unlikely to find anythingnew here, except perhaps the motivation to startthinking about one of the problems. It might also serveto interest a young relative caught by the illustrationsand start a conversation about maths.

One thing that an older reader might be prompted todiscuss is the role of computers in mathematics. Manyexplanations mention that computer-assisted projectshave helped get some way towards solving puzzlesbut not provided complete proofs. It is a shame, then,that there is no discussion of the four-colour theorem,which was significant and contentious for being along-standing conjecture whose computer-producedproof was too complex for a human to comprehend.1

Well then, should you buy the book? I think so! If you havethe disposable income, twenty pounds on this is rathernice. Think of it as a nice art piecewith a funmathematicalflavour that could prompt some good discussions.

FURTHER READING

[1] Appel, Kenneth, and Wolfgang Haken. ‘Thesolution of the four-color-map problem.’ Scienti�cAmerican 237.4 (1977): 108-121.[2] MacKenzie, Donald A. Mechanizing proof:computing, risk, and trust. MIT Press, 2001.

Troy Astarte

Troy K. Astarte is aresearcher at NewcastleUniversity. Their mainresearch interest ishistory of computerscience and thedemilitarised zones

between computing, mathematics, and logic. Troycomes from Lancaster (UK) and regularly leads asmall team of enthusiastic problem-solvers throughimprovisational and creative challenges (we playDungeons & Dragons).

1See [1] for the initial publication; a good discussion of the socio-philosophical implications of computers in proof is [2]. Chapter 4 of thatbook specifically deals with the four-colour theorem.

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 61 — #61 ii

ii

ii

REVIEWS 61

Fundamentals of Graph Theory

by Allan Bickle, American Mathematical Society, 2020, US$85.00,ISBN: 978-1-4704-5342-8

Review by Claire Cornock

The author presentsstandard topics thatyou would expect tosee within a graphtheory book. Theseinclude Eulerian graphs,Hamiltonian graphs,trees, algorithms (e.g.to �nd a minimumspanning tree), planarand non-planar graphs,

colour theorems and bipartite graphs. The contentsextend far beyond this list, including more advancedtopics such as generalised graph colourings.

This book is fairly expensive, but you certainly get alot for your money. Amongst the 336 pages, there is alarge number of definitions and theorems, over 1,200exercises and a long list of references to other sources.There are nine main sections, each with an average ofover 20 pages and an average of 125 exercises, withfurther information and questions in the appendices.

My background is within Pure Mathematics, withlimited knowledge of graph theory. I am familiarwith some of the basic methods and conceptswithout any of the depth. I found that parts of thebook were very straightforward to follow, particularlywhen de�nitions and results were backed up withexamples and/or diagrams. I had di�culty with someconcepts that I had not encountered before, but allthe information is there to persist with learning thematerial. The longer you spend with the book, theeasier it is to follow.

This book aims to be appropriate for a rangeof audiences, which is ambitious for anypublication. This includes undergraduates onMathematics-related degrees (e.g., ComputerScience), Mathematics undergraduates (with limitedprior exposure to proof), more experiencedMathematics undergraduates and Mathematicspostgraduate students. There is a very detailed guidefor using the book which is confusing at �rst glance,

but the detail includes handy information on whichsections are needed before each part of the book,and contains recommendations for the content oflectures for each of the four student groups. Thereseems to be a lot of content for a series of lectures,but this is a useful guide for a lecturer to use.

I believe that the book is most suitable for lecturersand PhD students. It is a great reference book foranyone who wants to study the subject furtherto work towards some of the unsolved parts ofgraph theory. It can be used by more experiencedundergraduate Mathematics students, but withcaution. If they focus only on the parts thatcorrespond to their studies, this is a great book foradditional reading. It would be particularly good forthose students studying graph theory within their�nal year project, under the guidance of a lecturer. Ido not think this book is suitable for students whoare not studying a Mathematics course. There is alot of information that is not relevant and it would bedi�cult to pick out the parts that are, as the morecomplex results are alongside the more basic ideas.

The book is really well thought out. For example, theorder of the topics has been carefully considered.At the end of each section, there is a list of topicsthat relate to the ideas that are presented. There isa very good section in the appendices on generalproof. This includes techniques, examples and lotsof exercises, with some linked to graph theory.

The best feature of this book is the extensive setof exercises. These are conveniently presentedfor each subsection, rather than listed together.Understandably there are no answers, but thequestions are such a valuable resource regardlessof this.

My favourite section was the one on Hamiltoniangraphs. Really interesting facts are presented,such as the connection with puzzles. There is aparticularly nice example of a Hamiltonian graph,which contains a very detailed description of howa Hamiltonian cycle was found. The applications

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 62 — #62 ii

ii

ii

62 REVIEWS

include the Knight’s Tour and visiting differentcities. I was unfamiliar with the use of tournamentsin voting theory and found this part especiallyinteresting. I also liked the sections where thehistorical background is provided. For example, agood account of the historical developments ispresented within the section on the Four ColourTheorem.

Real-life examples are used to motivate someof the topics, which include road networks, datastorage and sporting fixtures. I especially liked howgraph theory is introduced at the start with theconsideration of social networks. Results movefar beyond the practical, motivating examples andare studied mostly from an abstract perspectiveand regarding specific graphs. There is extensiveconsideration of when certain conditions hold.Proofs are presented for most results, andreferences are generally provided when they are

not. The book was especially good at highlightingareas that had unsolved or partially solvedproblems. It is made clear when results are onlyknown for certain cases. This makes the bookespecially useful for the more advanced students.

Claire Cornock

Claire is a PrincipalLecturer at She�eldHallam University. Shestudied SemigroupTheory for her PhD andnow researches teachingand learning pedagogy.

Claire is known for teaching with Rubik’s cubes tohelp her students’ understanding of abstract ideaswithin group theory.

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 63 — #63 ii

ii

ii

REVIEWS 63

Number Theory: A Very Short Introduction

by Robin Wilson, Oxford University Press, 2020, £8.99, US$11.95,ISBN: 978-0198798095

Review by Zachary Walker

Robin Wilson’s NumberTheory: A Very ShortIntroduction is aconcise book but veryinformative, aimed atreaders unfamiliar withthe subject. It is part ofa series of ‘Very ShortIntroductions’ knownfor being of a veryhigh quality, this bookcertainly lives up to that

reputation. It is only 150 pages and split into ninechapters. Despite its relatively in-depth content giventhe target audience, it is a digestible short read.

The introductory chapter provides a fantasticlist of questions to provide motivation for theinvestigation of the topic. These questions coverboth real world and abstract ideas, which onewould expect to capture the attention of thewide audience the book is intended for. Themajority of the book is spent going over manystandard results of the field, but this does notseem academic or dry. Particularly in the secondhalf, Wilson goes on to include applications ofsome of the ideas covered, a few of these appearmore contrived than others but succeed in keepingmomentum throughout the book. The crescendoof the chapter is a look at some of the mostwell-known unsolved problems and recent resultswhich can be explained well with the materialintroduced.

As former president of the British Society forthe History of Mathematics, Wilson unsurprisinglytakes the opportunity to include some fascinatingreferences to the timeline of number theory.An impressive span of Eratosthenes through toAndrew Wiles. The context of how number theorywas developed helps to justify interest in thesubject. Even if one was familiar with the mathsin this book then I think simply seeing how ideaswere developed could make it an interesting read.

Wilson starts with explanations that seem to presumelittle prior knowledge, going over congruences, factorsand Euclid’s algorithm. These first ideas are explainedvery well but as the ideas become more complicatedthe fine details of the proofs are generally omitted,which could be frustrating or confusing to someonereading into the subject for the first time. I do notthink this is a significant issue but it does leavesome ambiguity as to whom the book is intendedfor. While there are a set of official questions set out,Wilson is not constrained to these and is constantlyusing questions to point to theorems as a solutionto them. Knowledge of which results were likely tobe shown did not spoil the anticipation of resolvingthe problems.

So many of the examples could be picked out butI found the short section about Charles Dodgson’smethod for determining the day of the week of anydate especially satisfying. On one level this is anamusing party trick, but I think it is more than that.Wilson demonstrates that the simple tools that havebeen explained can be used to provide a solution tosomething, such that there is a sense of an underlyingorder, although it is not clear what order that is. Thisreveals something of the beauty of numbers andmathematics in general.

The structure of call and response between theproblems and solutions being developed by revealingmore results from number theory sets up expectationfor the reader, which is broken in the final chapters.After hearing about the Goldbach conjecture onealmost expects Wilson to introduce a new idea,building on the rest of the book to solve theconjecture, but the lack of an answer is far fromdissatisfying. The cliffhanger of open problems is notonly exciting but gives relevance to the subject byshowing how far away it is from being a complete field.Unrealistic as it may be, I think leaving the reader ina place where they are so desperate for resolutionthat they attempt to find the answer themselves istestament to this book being an inspiration.

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 64 — #64 ii

ii

ii

64 REVIEWS

In case the reader missed it, the �nal short chaptersummarises all the questions covered and givessuccinct answers to all of them. It is in reading thesolutions to these problems that one realises thatthe maths developed goes beyond the questionsthemselves to provide a powerful framework.

Overall, this book is a good introduction to thenumber theory but does an even better job of gettinga reader excited enough about the subject that Ithink they would want to pursue it further. Wilsonhas impressively captured the essence of the topic.

Zachary Walker

Zachary Walker isan undergraduate inhis third year at TheQueen’s College, Oxford.This year he has chosento study algebra and thehistory of maths. When

he is not trying to �nish problem sheets, he enjoysplaying the cello.

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 65 — #65 ii

ii

ii

OBITUARIES 65

Obituaries of Members

Peter M. Neumann: 1940–2020

Photograph by Veronika Vernier (2007). © MathematicalInstitute

Peter Neumann, who was elected a member of theLondon Mathematical Society on 17 December 1964,died on 18 December 2020, aged 79.

Cheryl Praeger and Martin Liebeck write: PeterNeumann was the �rst son of the well-knownmathematicians Bernhard and Hanna Neumann, whocame to the UK from Germany in the 1930s. Peter wasborn in Oxford on 28 December 1940, where Hannawas working on her DPhil while Bernhard servedwith the Pioneer Corps in the British Army. Afterthe war, Bernhard and Hanna obtained universityappointments in Hull, where Peter grew up beforegoing to Queen’s College Oxford in 1959. During1961–62, while still an undergraduate, Peter joinedhis parents on their sabbatical year at the CourantInstitute in New York, and began mathematicalresearch on varieties of groups. This resulted in his�rst research paper in 1962, the ‘3N’ paper writtenjointly with his parents, and in 1964, the ‘B+3N’paper published jointly also with Gilbert Baumslag.Peter started his DPhil at Oxford in 1963 under thesupervision of Graham Higman, and by the time he�nished in 1966, he had published �ve more researcharticles, and had been awarded a Tutorial Fellowshipat Queen’s, to be followed a year later by a universitylectureship. He remained at Oxford for his entirecareer, retiring in 2008.

Peter’s lifelong contribution to mathematics in the UKand worldwide was monumental and wide-ranging:through his research in algebra and the history ofalgebra; his supervision of over 40 doctoral students,many of whom went on to have distinguishedacademic careers; his extensive service to the

London Mathematical Society; and his enormouscontribution to mathematics education.

Let us �rst brie�y discuss Peter’s research. Peterwas a leading �gure in algebra for over 50 years,publishing around 100 papers and books on awide range of topics: varieties of groups, solublegroups, group enumeration, permutation groups, andalgorithms in computational algebra. Each of hispublications is beautifully crafted, and its place inmathematics carefully thought out and explained,together with insightful comments on where furtherwork might lead. His work was highly in�uential, andwe are just two of his many bene�ciaries. Peter wasalso a great collaborator, publishing with 38 di�erentco-authors, and holding visiting positions at manyplaces around the world. Peter described himself asa ‘mathematician historian’ and wrote extensively onthe history of algebra, including his comprehensivebook on the mathematical writings of Évariste Galois,published in 2011. Peter’s contributions to researchand scholarship were recognised by the LondonMathematical Society with the award of the SeniorWhitehead Prize in 2003, and by the British Societyfor the History of Mathematics which established theNeumann Prize in 2009 in his honour.

Peter’s service to the London Mathematical Societywas very extensive: he was Publications Secretary(1967–72); Journal Editor (1976–79); Bulletin BookReview Editor (1979–81); Bulletin Editor (1979–84)and Monographs Editor (1999–2003). He was alsoan O�cer of the Society, holding the position ofVice-President from 1990–92. The LMS honouredPeter not just with the Senior Whitehead Prize, butalso with the joint LMS–IMA David Crighton Medal in2012.

Peter also made an enormous contribution tomathematics education in the UK. He was thefounding Chairman of UK Mathematics Trust (UKMT),serving from 1996 to 2005. The Trust works withhundreds of volunteers across the UK to organisecompetitions promoting problem solving and teamwork and other mathematical enrichment activitiesfor schoolchildren. During his period as chairman,Peter led UKMT in taking on the staging of the 2002International Mathematical Olympiad. For his servicesto mathematics education, Peter was awarded anOBE in 2008.

Peter loved music and was a �ne violin and violaplayer. Before his stroke in early 2018 he wouldfrequently cycle long distances to meetings. In July2018, Peter moved to a care home on Cumnor

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 66 — #66 ii

ii

ii

66 OBITUARIES

Hill. He continued to solve The Guardian crypticcrosswords regularly — with numerous Facetimediscussions with his wife Sylvia throughout thepandemic lockdown when visitors were not allowed.Peter was a wonderfully generous, warm and wiseperson, and is deeply missed by his many friends,colleagues and students.

Peter is survived by his wife of 58 years, Sylvia, theirsons David and James and daughter Jenny, their tengrandchildren, and their �rst great-grandchild Isaac,born 4 October 2020.

Gordon D. James: 1945–2020Professor Gordon James,who was elected amember of the LondonMathematical Societyon 10 May 1985, diedon 5 December 2020,aged 74. Professor Jameswas LMS Journal Editor1989–93.

Rob Curtis writes: Gordon’s natural talent formathematics �rst became apparent at EastbourneCollege where he was taught by Eric Laming, aninspirational teacher who became a �rm friend of thefamily for many years. From Eastbourne, Gordon wona scholarship to Sidney Sussex College, Cambridge,where he was tutored by John Conway, and wherehe obtained a First Class degree in 1967 followedby Distinction in Part III of the Tripos the followingyear. He was then taken on as a research studentby John Thompson, the pre-eminent �nite grouptheorist of the day, and wrote a PhD thesis on themodular representations of the Mathieu group M24for which he was awarded a Smith Prize for his �rstyear research.

It was during Gordon’s Part III year whilst we shareda house in Cherry Hinton that he met Mary, hiswife-to-be, and they married in 1971. Shortly afterreceiving the PhD in 1972, he was elected to aFellowship at Sidney Sussex, a post he held until1985 when he moved to Imperial College, London.He was very soon promoted to a Readership in1986 and then to a Professorship in 1989, whenhe delivered an inaugural lecture entitled ‘Whatthe Hecke Algebras?’, being unable to resist thepun on the area of mathematics in which he hadbecome an international expert. Indeed, from thesporadic groups, Gordon’s consuming interest hadshifted to the representation theory of the symmetric

groups. In 1975 he had spent his sabbatical leavein Canada and visited G. de B. Robinson, himselffamous for his contributions to the representationsof the symmetric groups, and Gordon proceededto extend the delightful and highly combinatorialclassical theory to modular representations. Heproduced two books on this work, one joint withAdalbert Kerber, putting the whole theory on arigorous foundation. He then became interestedin developing an analogous theory for the generallinear groups and, together with Richard Dipper,introduced the concept of q -Schur algebras. Hiscollaboration with Dipper, Andrew Mathas and othersproduced a body of signi�cant results during thisperiod and posed tantalising conjectures whichhave led to further important developments in thearea. Gordon’s ground-breaking book on unipotentrepresentations of the �nite general linear groupswas awarded the Adams Prize in 1981.

Besides these advanced research monographs,Gordon, together with Martin Liebeck, produced ahighly regarded and popular undergraduate text onthe representation theory of �nite groups.

During his time at Imperial, Gordon served as Headof Pure Mathematics from 1991–97 and supervised8 PhD students. He was highly respected as adedicated, unsel�sh and sympathetic member of thedepartment.

Sadly, in 2002 Gordon was diagnosed withParkinson’s disease and a few years later had totake early retirement through ill health. He and Maryretired to the Yorkshire Dales and Gordon determinedto keep the disease at bay by walking miles overthe wonderful moorland. I myself have struggled tokeep up with him as he went up over those hills likea gazelle, although he was already less sure-footedgoing downhill. On one occasion I recall we were bothwinched down into the remarkable Gaping Gill cavern,300 feet below ground, halfway to the IngleboroughPeak. Gordon fought the disease with passion andfortitude but inevitably it caught up with him, and bythe end he struggled to keep his balance. Throughoutthis ordeal Mary was a constant and indefatigablesupport to him.

Gordon had many interests. He was a �ne chess andbridge player, although his fondness for a ‘psych’ onespade opening bid could mislead his partner as muchas his opponents! It was also not unknown for himto play a game in which hands traditionally contain�ve cards, the standard Cambridge ante being onetenth of a penny. After retirement Gordon threw

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 67 — #67 ii

ii

ii

OBITUARIES 67

himself energetically into Yorkshire village life andsoon became a hugely valued member of the localcommunity.

Gordon was a �ne mathematician, a superb colleagueand a loyal friend; he is survived by his wife Mary,their two children Elizabeth and William, and �vegrandchildren.

A.E.L. Davis: 1928–2020

A.E.L. Davis with Antonin Švejda (left) and Igor Janovský(right). Photograph from ntm.cz

Ann Elizabeth Leighton Davis, who was elected amember of the London Mathematical Society on 15January 1988, died on 23 November 2020, aged 92.

Snezana Lawrence writes: A.E.L. Davis (who alwayspreferred this form of address), a mathematicalhistorian, began her academic career with a thesis onKepler, which she completed in 1981 at Imperial College,University of London. Her thesis, ‘A MathematicalElucidation of the Bases of Kepler’s Laws’, establishedher as a foremost scholar on Kepler. This took heron to be an active member of the InternationalAstronomical Union (IAU) and the British Society ofthe History of Mathematics in the years to come.

Davis became a Vice-Chair of IAU’s Commission onJohannes Kepler twice in her lifetime, and was theleading scholar on Kepler until her passing. She wasa productive and energetic historian of mathematics,in more general terms, too. Her greatest outputin the history of mathematics was certainly hercompilation of the online archive named after her,The Davis Historical Archive: Mathematical Womenin the British Isles, 1878–1940, part of the largerMacTutor History of Mathematics archive at StAndrews University (bit.ly/39yhQ8h). The archive liststhe names of all women graduates in mathematics,around 2500 in total, from the twenty-one colleges

and universities that educated women in the givenperiod. As she went about her work on compilingthe archive, Davis collected about two hundred bookswritten by or about the women whose lives andcareers she investigated. This collection is diverse,including academic but also school-books, discoursesor biographies. She donated the collection to theLondon Mathematical Society, under the name of‘Philippa Fawcett Collection’ (bit.ly/3cyTtJx).

The Philippa Fawcett Collection is now one of the LMS’sSpecial Collections, and is housed in De Morgan House.Both the collection and the insistence on calling itafter Fawcett testifies to the generosity of spirit aswell as her unwavering efforts to promote the workof women in mathematics and to record and inspirefuture female mathematicians. It also tells somethingabout Davis’ own regard for the rights of women:Fawcett was ‘above the Senior Wrangler’ at Cambridgein 1890, and a daughter of a noted suffragist MillicentFawcett.

I met A.E.L. Davis many times at the BSHM meetings;her approach to life and to the history of mathematicswas refreshing, piercing, and inspiring. A fearlesslyindependent woman, she was always genuinelyinterested in others’ work and stories, and rarely spokeabout herself — I wish I had had more time to askher many more questions about her own life.

What I know is scarce and does not do justice to suchan important and productive historian of mathematics.For many years she worked as an Associate Lecturerfor the Open University (1989–2004), and towardsthe end of her life became an Honorary ResearchAssociate of University College London and anHonorary Visiting Fellow at the Mathematical SciencesInstitute, Australian National University. It is inAustralia that Davis died last year; she will be sorelymissed amidst the historians of mathematics of UK,and in our global community.

Robin J. Chapman: 1963–2020Dr Robin Chapman,who was elected amember of the LondonMathematical Society on19 June 1987, died on 18October 2020, aged 57.

Peter Cameron writes:Robin was born in May1963 in Swansea. He

attended Dynevor Comprehensive, where he won

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 68 — #68 ii

ii

ii

68 OBITUARIES

a Postmastership to Merton College, Oxford. Aftertaking one of the top Firsts in his year, he wentto Cambridge to do Part III and was accepted as aPhD student to work with Martin Taylor. He followedMartin to Manchester, completing his PhD in 1987.After a Junior Research Fellowship at Merton College,he took a position at Exeter University where heremained for the rest of his career, though heretained great a�ection for Oxford.

Robin was a very able undergraduate. When theGalois theory lecturer listed the subgroups ofthe symmetric group of degree 4 and asked thestudents to find polynomials realising each as aGalois group, Robin’s comment was “You missedone”. His tutorial partner Peter Kronheimer tookpride in finding the shortest and most elegantanswer to any problem; at first Robin simply foughtthe problem into submission, but as his stature asa mathematician grew he found he was capableof shorter and more beautiful arguments. Later inhis career, his co-author Patrick Solé praised theelegance of his work, when (for example) he provedby hand the equivalence of two constructions ofthe Leech lattice.

I greatly valued the information he provided on hisweb page, which included short and efficient proofsof various ‘folklore’ results such as Bertrand’spostulate and the characterisation of orders forwhich every group is abelian. Others shared thisopinion, and he was often cited on MathOverflowand StackExchange.

Robin was a mathematician first and foremost,but his interests were very wide indeed. PeterKronheimer played French horn in a wind-quartet;his quartet perfomed Ligeti’s Six Bagatelles, andPeter was surprised to find that not only did Robincome to the performance, but he could expoundon the work and its place in Ligeti’s oeuvre. Thisknowledge stood him well in Mastermind, where hereached the final in 2005: his special subjects inthe heats and final were The Life and Music of IgorStravinsky, One Foot in the Grave and The ScienceFiction Novels of Philip K. Dick. It is said that the pubquiz machine in the students’ bar at Manchesterhelped fund his studies there.

Robin’s mathematical interests lay in discretemathematics and number theory. One thinghe is remembered for is his “evil determinantproblem”, subsequently solved by Maxim Vsemirnov.He published 50 papers, was on the editorialboard of two journals, and organised the BritishCombinatorial Conference in 2011.

After the opening of the Heilbronn Institute forMathematical Research in 2005, Robin split histime between there and Exeter, doing collaborativemathematical research supporting the work ofGovernment Communication Headquarters. Heworked with the UK Olympiad team, and both TonyGardiner and Imre Leader write warmly of him.

Robin took great joy from mathematics and broughtjoy to many friends. He is survived by his brotherand family.

Death NoticesWe regret to announce the following deaths:

• Patrick D. Barry, Professor Emeritus of UniversityCollege Cork, who died on 2 January 2021.

• Colin J. Bushnell, Emeritus Professor at King’sCollege London, who died on 1 January 2021.

• Walter Forster, formerly of University ofSouthampton, who died on 17 January 2021.

• Robin L. Hudson, formerly of LoughboroughUniversity, who died on 12 January 2021.

• Brian H. Murdoch, formerly Erasmus SmithProfessor at Trinity College, Dublin, who died on 9December 2020.

• Stephen Pride, formerly of University of Glasgow,died on 21 October 2020.

• Tommy A. Whitelaw, formerly of the University ofGlasgow, who died on 21 January 2021.

Biographical MemoirsMemoirs of Michael Atiyah (bit.ly/39CeJMP),Christopher Hooley (bit.ly/3ap16ja), Frank Bonsall(bit.ly/36xSYvr) and Edward Fraenkel (bit.ly/2MM0yLP)have recently appeared in Biographical Memoirs ofFellows of the Royal Society.

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 69 — #69 ii

ii

ii

EVENTS 69

Early Career Mathematicians’ SpringConference 2021

Location: OnlineDate: 13 March 2021Website: tinyurl.com/y2povzt3

This IMA conference will interest mathematiciansearly in their career, in academia and industry,students of mathematical sciences, as well as thosewith an interest in the subject. It will featureplenary talks from distinguished speakers coveringa wide range of subjects, as well as networkingactivities. The Invited Speakers include Mihaela Rosca(DeepMind and UCL) and Nick Higham (University ofManchester).

LMS Women in Mathematics Day2021

Location: OnlineDate: 24 March 2021Website: tinyurl.com/y5uwol5f

This event, open to mathematicians of all gendersand from all backgrounds, aims to promoteinterest and careers in mathematics for women.In addition to talks, the event will include a paneldiscussion and a poster competition open to womenmathematicians at undergraduate, postgraduate andearly career levels. The deadline for registration is21 March 2021, 16:00. Register your attendance attinyurl.com/y6bvdfg8.

LMS Meeting at the JointBMC–BAMC 2021

Location: OnlineDate: 8 April 2021Website: tinyurl.com/yarpowdo

This event was originally scheduled for 2020 and waspostponed owing to covid-19. The meeting will beginwith Society business, followed by an LMS lectureby Ciprian Manolescu (Stanford). Further details andupdates on the meeting can be found on the website.

LMS Spitalfields History ofMathematics Meeting

Location: OnlineDate: 14 May 2021Website: tinyurl.com/y3kpv6ye

This event, held by the LMS and UCL SpecialCollections, will celebrate the launch of the EducationalTimes Digital Archive. Talks will have a mathematicalhistorical focus, and will include a presentation fromUCL about their work on the collections.

Korteweg-de Vries Equation, Toda Latticeand their Relevance to the FPUT Problem

Location: University of LincolnDate: 26 May 2021Website: https://wp.me/PcBUF5-6

This meeting aims to highlight aspects of integrablesystems theory applied to near-integrable many-bodydynamical systems. Postgraduate and final yearundergraduate students are particularly encouraged toapply. Participation is open to final year students, earlycareer researchers and academics.

Dynamics and Geometry Summer School

Location: University of BristolDate: 21 June–2 July 2021Website: tinyurl.com/s5hn592j

Dynamics and Geometry are two intertwinedareas of mathematics that have seen revolutionarybreakthroughs in recent years. In this summer schoolworld-leading experts will speak about some of thesedevelopments, alongside problem sessions and otheropportunities for discussion and interaction.

Modelling in Industrial Maintenance andReliability

Location: OnlineDate: 28 June–2 July 2021Website: tinyurl.com/IMAMIMAR

This conference is the premier maintenance andreliability modelling conference in the UK and buildsupon a very successful series of previous conferences.It is an excellent international forum for disseminatinginformation on the state-of-the-art research, theoriesand practices in maintenance and reliability modelling.

Research Students’ Conference inPopulation Genetics

Location: University of WarwickDate: 21–23 July 2021Website: tinyurl.com/y9fjp36b

This conference is aimed at young researchersinterested in mathematical and statistical aspectsof population genetics, including coalescent theory,stochastic processes in population genetics,computational statistics and machine learning forgenomics.

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 70 — #70 ii

ii

ii

70 EVENTS

Young Geometric Group Theory X

Location: Newcastle UniversityDate: 26–30 July 2021Website: conferences.ncl.ac.uk/yggt2021

The aim of this YGGT conference is to bringtogether young researchers in geometric grouptheory, post-docs and graduate students. It will allowthem to learn from one another and from seniormathematicians invited to give tutorial courses andlectures in several branches of geometric grouptheory. Supported by an LMS Conference grant.

Young Functional Analysts’ Workshop

Location: Lancaster UniversityDate: 12–14 August 2021Website: tinyurl.com/yce6j3gy

This is an event aimed at early-stage researchers(PhD students and postdocs) in functional analysisand related areas. It is a great opportunity tobring researchers with shared interests together andprovides the opportunity for participants to presenttheir own work in front of a supportive and interestedaudience.

Scaling Limits: From Statistical Mechanicsto Manifolds

Location: CambridgeDate: 1–3 September 2021Website: statslab.cam.ac.uk/james60

This workshop, postponed from 2020, is in honour ofJames Norris’ 60th birthday. There will be 16 invitedtalks covering: Random growth processes and SPDEs;Yang-Mills measure; Limits of random graphs, randomplanar maps, and fragmentation processes; Markovchains, interacting particle systems and �uid limits;Di�usion processes and heat kernels. A workshopdinner will be held at Churchill College.

Heilbronn Annual Conference 2021

Location: Heilbronn InstituteDate: 9–10 September 2021Website: tinyurl.com/y63vvoar

The Annual Conference of the Heilbronn Institutefor Mathematical Research is the Institute’s �agshipevent. The eight invited speakers are: Caucher Birkar,Jon Brundan, Ana Caraiani, Heather Harrington, GilKalai, Peter Keevash, Jeremy Quastel and TatianaSmirnova-Nagnibeda. They will deliver lecturesintended to be accessible to a general audience ofmathematicians.

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 71 — #71 ii

ii

ii

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

March 202124 LMS Women in Mathematics Day (online)

(493)

April 2021

8 Society Meeting at the joint BMC–BAMC2021 (online) (493)

May 2021

14 LMS Spital�elds History of MathematicsMeeting: Educational Times DigitalArchive Launch, London (493)

June 2021

2-4 Midlands Regional Meeting andWorkshop, Lincoln

22 Society Meeting at the 8ECM, Portorož,Slovenia

July 2021

2 General Meeting of the Society, London

September 2021

6-10 Northern Regional Meeting, Conferencein Celebration of the 60th Birthdayof Bill Crawley-Boevey, University ofManchester

January 2022

4-6 South West & South Wales RegionalMeeting, Swansea

Calendar of Events

This 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].

March 2021

13 Early Career Mathematicians’ SpringConference 2021 (online) (493)

14 International Day of Mathematics (491)

30-31 Mathematics in Defence and Security IMAConference (online) (492)

April 2021

6-9 British Mathematical Colloquium andBritish Applied Mathematics Colloquium2021 (online) (492)

20-23 Mathematics of Operational Research(online) (492)

29-30 Marriages, Couples, and the Making ofMathematical Careers (online) (492)

May 2021

26 Korteweg–de Vries Equation, Toda Latticeand their Relevance to the FPUT Problem,University of Lincoln (493)

ii

“NLMS_493” — 2021/3/9 — 12:09 — page 72 — #72 ii

ii

ii

June 2021

20-26 8th European Congress of Mathematics,Portorož, Slovenia (492)

21-2 Jul Dynamics and Geometry SummerSchool, University of Bristol include (493)

28-2 July Modelling in Industrial Maintenance andReliability (online) (493)

July 2021

7-9 Nonlinearity and Coherent Structures,Loughborough University (492)

12-16 New Challenges in Operator Semigroups,St John’s College, Oxford (490)

19-23 Rigidity, Flexibility and Applications,Lancaster University (492)

21-23 Research Students’ Conference inPopulation Genetics, University ofWarwick (493)

26-30 Young Geometric Group Theory X,Newcastle University (493)

August 2021

12-14 Young Functional Analysts’ Workshop,Lancaster University (493)

16-20 IWOTA, Lancaster University (481)18-20 Young Researchers in Algebraic Number

Theory III, University of Bristol (492)

September 2021

1-3 Scaling Limits: From Statistical Mechanicsto Manifolds, Cambridge (493)

9-10 Heilbronn Annual Conference 2021,Heilbronn Institute (493)

16-17 Statistics at Bristol: Future Results andYou 2021, Heilbronn Institute

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

21-23 Conference in Honour of Sir MichaelAtiyah, Isaac Newton Institute,Cambridge (493)

July 2022

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