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NEWSLETTERETHICS IN
MATHEMATICS

Issue: 484 - September 2019

DRINFELD MODULESAND FACTORISATIONCOMPUTERSAND PROOF

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EDITOR-IN-CHIEF

COPYRIGHT NOTICE

Iain Moffatt (Royal Holloway, University of London)
iain.moffatt@rhul.ac.uk

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

EDITORIAL BOARD
June Barrow-Green (Open University)
Tomasz Brzezinski (Swansea University)
Lucia Di Vizio (CNRS)
Jonathan Fraser (University of St Andrews)
Jelena Grbić (University of Southampton)
Thomas Hudson (University of Warwick)
Stephen Huggett (University of Plymouth)
Adam Johansen (University of Warwick)Bill Lionheart (University of Manchester)Mark McCartney (Ulster University)
Kitty Meeks (University of Glasgow)
Vicky Neale (University of Oxford)
Susan Oakes (London Mathematical Society)
Andrew Wade (Durham University)
Early Career Content Editor: Vicky Neale
News Editor: Susan Oakes
Reviews Editor: Mark McCartney
CORRESPONDENTS AND STAFF
LMS/EMS Correspondent: David Chillingworth
Policy Roundup: John Johnston
Production: Katherine WrightPrinting: Holbrooks Printers Ltd
EDITORIAL OFFICE
London Mathematical Society
De Morgan House
57–58 Russell Square
London, WC1B 4HS
newsletter@lms.ac.uk
Charity registration number: 252660

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. For
membership details see lms.ac.uk/membership.
SUBMISSIONSThe Newsletter welcomes submissions of feature content, including mathematical articles, career related articles, and microtheses
from members and non-members. Submission guidelines and LaTeX templates can befound at lms.ac.uk/publications/submit-to-thelms-newsletter.
Feature content should be submitted to theeditor-in-chief at iain.moffatt@rhul.ac.uk.
News items should be sent tonewsletter@lms.ac.uk.

COVER IMAGE√
A formal proof that 3 is irrational. See thefeature starting on page 32. (Image courtesy of
Zak Buzzard.)

Notices of events should be prepared using the template at lms.ac.uk/publications/lmsnewsletter and sent to calendar@lms.ac.uk.
For advertising rates and guidelines see
lms.ac.uk/publications/advertise-in-the-lmsnewsletter.

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CONTENTSNEWSLMS BUSINESSFEATURESEARLY CAREERREVIEWS
OBITUARIESEVENTSCALENDAR

The latest from the LMS and elsewhere4

Reports from the LMS

13

The Importance of Ethics in Mathematics

22

Polynomial Factorisation using Drinfeld
Modules

27

Computers and Mathematics

32

The Belgian Mathematical Society

37

Microthesis: Hypergraph Saturation
Irregularities

38

From the bookshelf

40

In memoriam

45

Latest announcements

51

All upcoming events

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LMS NEWSLMS Elections andAnnual General Meeting 2019
Voting for the LMS Elections for Council andNominating Committee will open on 25 October 2019. The slate of candidates can be foundat lms.ac.uk/about/council/lms-elections and an
online forum for discussion is available at discussions.lms.ac.uk/lmselections.
In addition, members are to be asked to voteon changes to the LMS Standing Orders — theRoyal Charter, Statutes, and By-Laws. The current Standing Orders, together with the proposedchanges, can be found on the LMS website attinyurl.com/lmsstandingorders.
A vote on the proposed changes will be taken at
the Annual General Meeting, but LMS Council has
agreed that members who are unable to attend
will be able to place an online proxy vote from 25
October to 25 November, through the independent Electoral Reform Services, or if preferredby completing a hard copy proxy voting form thatwill be available from the LMS website.
Instructions for each of the ballots i.e. a) the
elections to Council and Nominating Committee
and b) by proxy on the changes to the Standing
Orders, will be sent to members by email or post
before the ballots open. Members are encouraged to check that their contact details are up
to date at lms.ac.uk/user.
This year the AGM will be held at GoodenoughCollege, Mecklenburgh Square, London WC1N 2AB
on Friday 29 November at 2.30 pm. Please notethe change of date to the end of the month andthe slightly earlier start time. All those attending the AGM will be required to register on theday. The registration desk will be open from 1.30
pm to allow enough time for members to verify
their details and receive voting cards prior to thestart of the meeting. The results of the Council and Nominating Committee elections will beannounced at the meeting, as will the results ofthe vote on the changes to the Standing Orders.
Fiona Nixon
Executive Secretary

Donating to the LMS
Making a donation to the LMS has just become easier!
There is now a ‘donate’ button on the main menu
bar near the top of the LMS webpage (lms.ac.uk),
enabling anyone to make an online donation to theSociety.
The LMS gives grants totalling in excess of £680,000
per year to support many mathematical activities.
By far the greatest part of our income currently
comes from publications, which one can view as away of recycling money from everyone’s work as
authors, editors and referees into the community.
However, as the publishing industry moves to various forms of open access, the income we get fromthis resource is likely significantly to diminish. The
threat to publications income, combined with thecurrent unpredictability of the financial sector, meanthat this aspect of our income has become crucial.
Additional donations will help the Society maintainand if possible increase its level of support for all of
its objectives.
In the past the Society has benefitted from donations
from many individuals. Two in particular stand out:
£1,000 from Lord Rayleigh in 1874, which supported
the printing of the LMS Proceedings and rescued theSociety from collapse, and the bequest of £50,000
by G.H. Hardy in 1963 which was completely transformative for the Society.
Alongside the suggested amounts for donations, youwill also see a line for a ‘De Morgan Donation’ of£1865 or more (no prizes for working out where thisnotable number comes from!) This is a new venture
which it is hoped will encourage anyone with theresources to do so to support the Society with asubstantial donation. In recognition of their support,
De Morgan Donors will from time to time receive
invitations to lectures and other special events.
For those in a position to do so, you are also encouraged to think about leaving a legacy to the LMS inyour will. Advice on how to do this can also be foundon the web page.
Of course, a donation can also be added when
you pay the annual membership fee. Whether yourdonation is large or small, the Society really does
value your support.

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Stop Press! DeMorgan@21
Unfortunately the big People’s Vote march in London recently changed its date to October 19th whichmeans that transport in and out of London is likely to
be very crowded on that day. Those planning to cometo DeMorgan@21 are therefore advised to reserve
train seats and hotels well in advance. We apologise
to anyone who was planning to go on the march.
Registration for this event has re-opened with space
for another 20 participants. These will be allocatedon a first-come-first-served basis. If you are interested please register at tinyurl.com/y4r935w9.

Survey of Mathematics Postdocs
The LMS has recently published the results of its 2017
Survey of Postdoctoral Researchers in the Mathematical Sciences in the UK. This survey was undertakenby the LMS Research Policy Committee in recognitionof the fact that there was little overall understandingof postdoctoral activity in the UK, in terms of thesize of the population, its distribution in subject area
and geographical terms, its origin and its source offunding.
The survey was conducted in autumn 2017 with acensus date of 31 October 2017. Individual heads ofdepartment were asked to supply information about
the postdoctoral researchers in their department.
They were asked for data about gender, nationality, home department, country of undergraduate
degree, field of interest and source of funding, withreassurances about the confidentiality of detailed
information. The initial request was followed up withseveral reminders and then collated and processedin 2018.
Geography: the survey reported 756 postdoctoral
researchers in total — a much larger figure than was
initially expected. A few departments did not participate in the survey, so the total number will in fact behigher than this figure. 25% of these were from theUK, and 43% were from the EU outside of the UK.

This proportion is remarkably uniform across the 32
departments with EU postdocs. The results showedthat there was a greater percentage of UK nationals
in the postdoc population in northern areas of the
UK than in southern areas.
Funding source: sources of funding were gathered
into 8 groups. The diversity of sources was rather
encouraging. As expected, EPSRC was by far thelargest single funding source at 34% (being twice as
big as any other category); the EU, other ResearchCouncils, and the postdoc’s own institution were theother main categories, at about 17% each. In severalof the larger departments the amount of EU andEPSRC funding was about the same. The effect oflosing EU funding is likely to be significant. The survey
also identified the proportion of funding to different
fields, both overall and per funding source. Generally,
the patterns here conformed to expectations.
Gender: the survey looked at gender balance both bydepartment and by field. The departmental balance
showed that the overall average percentage of female
postdocs was 23%. The survey identified a large variation, from 38% female in industrial mathematics
to 11% in PDEs and analysis, 10% in mathematicalphysics and integrable systems, and 6% in number
theory.
Postdocs and the REF: As expected there is astrong relationship between REF performance andnumbers of postdocs. It seems that there are typically very few postdocs in departments with lessthan 20 FTE REF returns, and beyond that range aratio of 1 postdoc per 2.4 research active staff look
like a rather general trend across all sizes.
The full results of the survey can be accessed attinyurl.com/yyrh7fnv. We are very grateful to departments for their cooperation in helping us to carryout this valuable exercise. It is intended to repeat thesurvey at regular intervals. Comments and queries
are of course very welcome.

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

John Greenlees
LMS Vice-President

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LMS Honorary Members 2019

tureship in Applied Mathematics for his leadership innumerical linear algebra, numerical stability analysis,
and communication of mathematics.
Dr Alexandr Buryak of the University of Leeds isawarded a Whitehead Prize in recognition of his outstanding contributions to the study of moduli of
curves and integrable systems.

LMS Honorary Members Ed Witten (left) and Don Zagier

At the Society meeting on 28 June 2019, the
LMS elected Professor Edward Witten (Institute forAdvanced Study, Princeton) and Professor Don Zagier(Max Planck Institute for Mathematics) as honorary
members of the Society.
Edward Witten occupies an unrivalled position
amongst contemporary mathematical and theoretical physicists. He has made profound contributions
to the development of contemporary physics, including topological quantum field theory, string theory,
M -theory and quantum gravity.

Professor David Conlon of the University of Oxford
is awarded a Whitehead Prize in recognition of his
many contributions to combinatorics. His particularexpertise is Ramsey theory, where he has made fundamental contributions to both the arithmetic andgraph-theoretic sides of the subject.
Dr Toby Cubitt of University College London isawarded a Whitehead Prize in recognition of his outstanding contributions to mathematical physics, inparticular the interconnections between quantum
information, computational complexity, and manybody physics.
Dr Anders Hansen of Cambridge University isawarded a Whitehead Prize for his contributions tocomputational mathematics, especially his development of the solvability complexity index and its corresponding classification hierarchy.

Don Zagier is an outstanding mathematician who
has made major contributions in number theory, particularly to the theory of modular forms, and in its
interactions with other areas of mathematics andmathematical physics.

Professor William Parnell of the University ofManchester is awarded a Whitehead Prize for highly
novel and extensive research contributions in thefields of acoustic and elastodynamic metamaterials
and theoretical solid mechanics, as well as excellencein the promotion of mathematics in industry.

2019 LMS prize winners

Dr Nick Sheridan of the University of Edinburgh isawarded a Whitehead Prize for his ground breaking
contributions to homological mirror symmetry andthe structure of Fukaya categories.

The Society extends its congratulations to the following 2019 LMS prize winners and thanks to all thenominators, referees and members of the Prizes
Committee for their contributions to the Committee’s work this year.
A De Morgan Medal is awarded to Professor Sir
Andrew Wiles FRS of the University of Oxford forhis seminal contributions to number theory and for
his resolution of ‘Fermat’s Last Theorem’ in particular, as well as for his numerous activities promoting
mathematics in general.
Professor Ben Green FRS, of the University of
Oxford, is awarded a Senior Whitehead Prize for his
ground breaking results in additive combinatorics,
analytic number theory and group theory.
Professor Nicholas Higham FRS, of the Universityof Manchester, is awarded a Naylor Prize and Lec-

The Berwick Prize is awarded to Dr Clark Barwickof the University of Edinburgh, for his paper On thealgebraic K -theory of higher categories, published in
the Journal of Topology in 2016, which proves that
Waldhausen’s algebraic K -theory is the universal
homology theory for ∞-categories, and uses this universality to reprove the major fundamental theoremsof the subject in this new context.
Dr Eva-Maria Graefe of Imperial College London
is awarded an Anne Bennett Prize in recognition ofher outstanding research in quantum theory and theinspirational role she has played among female students and early career researchers in mathematics
and physics.

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2019 LMS prize winners

Sir Andrew Wiles
De Morgan Medal

Senior Whitehead Prize

Ben Green

Nick Higham

Alexandr BuryakDavid Conlon

Anders Hansen

Nick Sheridan

William Parnell

Whitehead Prize

Whitehead Prize

Whitehead Prize

Naylor Prize

Whitehead Prize

Clark Barwick
Berwick Prize

Whitehead Prize

Toby Cubitt

Whitehead Prize

Eva-Maria Graefe
Anne Bennett Prize

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Ken Brown awarded the 2019 David
Crighton Medal
The LMS and IMA haveawarded the 2019 David
Crighton Medal to Ken
Brown, Professor of
Mathematics at the University of Glasgow, for
his seminal contributions to noncommutative algebra and for his
remarkable record of service and dedication to the
UK mathematics community.
Ken has the rare ability to apply subtle ring-theoretic
ideas to solve important problems in related areas.
In the 1970s he solved the zero-divisor question for
abelian-by-finite groups, introducing the key homological techniques which would form the basis of alllater major progress in this area; according to Formanek, he made “the most important and originalcontribution to the problem since Higman’s [1940]
work.” In the 1980s he introduced the class of homologically homogeneous rings: in the last five years,
these have been key in the study of noncommutative
geometry, derived categories, and moduli spaces. His
focus then shifted to the theory of quantum groupsand Hopf algebra where he harnessed the combination of Hopf algebras and homological algebra toconfirm important conjectures of Kac-Weisfeiler and
DeConcini-Kac-Procesi in representation theory. In
the last decade he has proved core results in several topics: Noetherian Hopf algebras; number theory through Iwasawa algebras; Poisson geometry in
Lie theory; and symplectic reflection algebras. Ken
wishes to acknowledge his many collaborators in his
research. He was elected a Fellow of the Royal Societyof Edinburgh in 1993.
Ken has mirrored his distinguished international
mathematical career with extraordinary service tothe UK Mathematical Sciences community. He sat on
the London Mathematical Society Council for almost
two decades, including terms as Vice-President from1997-99 and 2009-17. During this time he was instrumental in the development of the voice of the Council for the Mathematical Sciences, providing critical
input to consultations and leading a variety of task
forces, particularly helping highlight the important
issues that affect the Mathematical Sciences people
pipeline. Beyond this long-term involvement with the
LMS, Ken has made numerous further contributions
to the UK Mathematical Sciences.

Beyond this long-term involvement with the LMS, Ken
has made numerous further contributions to the UK
Mathematical Sciences, including as a member of the
Research Assessment Exercise (RAE) Mathematics
subpanel in 1996, as Vice-Chair of this panel in 2001,
and then as Chair of the RAE Pure Mathematics Subpanel in 2008. He was also a member of the Research
Excellence Framework (REF) expert advisory groupin 2008-09. He has been a member of the EPSRC
College since 1996 and served on the EPSRC Mathematical Sciences Strategic Advisory Team (SAT) firstas a member from 2013 to 2015 and subsequently
as its Chair from 2015 to 2017.
Read the full citation at tinyurl.com/y5ubvpg3.

Calderon PrizeLMS member Carola Schönlieb (University of Cambridge) has been awarded the 2019 Calderon Prize forher work in image processing and partial differential
equations. The Inverse Problems International Association awards the Calderon Prize to a researcher
under the age of 40 who has made distinguishedcontributions to the field of inverse problems broadly
defined. Carola is the first female mathematician toreceive this award. She won an LMS Whitehead prizein 2016 and is the current leader of European Womenin Mathematics.

Forthcoming LMS Events
The following events will take place in the next threemonths:
Prospects in Mathematics Meeting: 6–7 September, Lancaster (tinyurl.com/y4c9aaxk).
Midlands Regional Meeting: 11 September, Nottingham (tinyurl.com/y5vtaytx).
Popular Lectures: 19 September, University of Birmingham (tinyurl.com/hu58wjk).
DeMorgan@21: 19 October, London.
Joint Meeting with the IMA: 21 November, Reading(tinyurl.com/y4sdm74b).
Computer Science Colloquium: 13 November, London (tinyurl.com/cscoll19).
LMS/BCS-FACS Evening Seminar: 21 November,
London (tinyurl.com/yyc9oyse).
Graduate Student Meeting: 29 November, London(tinyurl.com/yy58t78v).
Society Meeting and AGM: 29 November, London(tinyurl.com/yy58t78v).
A full listing of upcoming LMS events can be foundon page 54.

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OTHER NEWS

Alan Turing honoured on new £50
banknote

Alan Turing banknote concept

An announcement made by the Bank of England on
15 July 2019 has confirmed that Alan Turing will be
the character to feature on the reverse side of thenew £50 banknote, which will come into circulationin 2021.
Alan Mathison Turing OBE FRS (1912–1954) was a mathematician, computer scientist, logician, cryptanalyst,
philosopher and theoretical biologist who was instrumental in formalising the concepts of algorithm andcomputation. Turing worked as a code-breaker during the second world war and is widely accreditedwith having helped bring an earlier end to the war.
The story of his life has had wide implications forchanges in political, legal and social attitudes towards
human diversity and homosexuality.
In his article ‘On computable numbers, with anapplication to the Entscheidungsproblem’ (submitted 28 May 1936, published in Proceedings of the
London Mathematical Society 42 (1937) 230–265;
tinyurl.com/y3vlud4j), Turing presented a first model
for a general-purpose computer, later to become
known as a ‘Turing machine’.
The London Mathematical Society welcomes thisgreat exposure and boost to the public appreciationof mathematics, and is delighted to have contributed
to the design of the banknote by giving approval andpermission for two mathematical excerpts from thisTuring article to be included on the new banknote.

The first excerpt is a table from page 240 which
provides a schema for succinctly representing Turingmachines. The table gives a complete descriptionof how to specify such machines and therefore canbe thought of as one of the first examples of a programming language.

The second excerpt, from page 241, is a sequenceof Turing machine transitions that helps explain how
to encode a Turing machine as a number. The more
modern analogue of what Turing describes is how
to take an abstract representation of a computer
program and convert it into a binary sequence of
0s and 1s so that it can be stored on a disc or in
the memory of a computer. The idea that a program
can be stored as a number, and used as data (by anoperating system) in order to execute the program,is hugely important.

Turing went on further in his article to describe large
classes of real numbers whose binary expansions
are computable by his machines; to describe a ‘universal machine’ that could serve the purpose of an
operating system; and to describe the theoreticallimits of his machines. Ultimately, Turing showed that
there can be no algorithmic method for determining
whether or not a given mathematical statement canbe proved in a certain axiomatic system. This provedthat David Hilbert’s famous Entscheidungsproblem
has no solution (which was also proved independently
by Alonzo Church).
Alan Turing is the second mathematician to appear
on a Bank of England banknote. A £1 banknote incirculation between 1978 and 1988 depicted Sir Isaac
Newton. Famous Britons have featured on the reverseof Bank of England banknotes since 1970.
Paul Shafer, University of Leeds
Ola Törnkvist, LMS Editorial Manager

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MATHEMATICS POLICY DIGEST

Brexit threat to researchand innovation
Royal Society President, Sir Venki Ramakrishnan
wrote a letter in July warning of the impact a
no-deal Brexit will have on research and innovation. More information and the letter are available attinyurl.com/yyas8lz3.

Exploring the workplace for LGBT+
physical scientists
The Institute of Physics, the Royal Society of Chemistry and the Royal Astronomical Society, have conducted a comprehensive survey to gather datafrom across the community — giving new insights
into the current workplace environment for LGBT+
physical scientists. The full report is available attinyurl.com/y38x9kxh.

New Director General, Industrial
Strategy and Innovation
Jo Shanmugalingam became Director General, Industrial Strategy, Science and Innovation at the Department for Business, Energy and Industrial Strategy(BEIS) on 15 July 2019. More information about the new
Director General is available at tinyurl.com/y2hch6cn.

Digest prepared by Dr John Johnston
Society Communications Officer

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

EUROPEAN MATHEMATICAL SOCIETY NEWS

From the EMS President

Funding for Mathematical Research

The last months have been full of activities. HereI would like to mention a few. A highlight was theaward of the Abel Prize to Karen Uhlenbeck, the firstfemale recipient, on 21 May. This was followed by a
day of talks related to her work. These can be viewed
at tinyurl.com/yx8p4v6f.

Volker Mehrmann (President of the EMS) writes: As
President of the European Mathematical Society I
would like to point out a very urgent and unfavourable
situation for the funding of mathematics in Europe.
The European Research Council (ERC) budget foreach discipline is allocated each year in proportion to
the number of proposals and the requested budget
received. It has been observed that, since the founding of the ERC, the budget for mathematics in thethree funding streams (advanced, consolidator, andstarting grants) has dropped to almost half, because
there are not enough applications.

The restructuring of the EMS publishing house isunderway. The new organization, in the form of alimited company owned by the EMS, was establishedat the end of March. The hiring process for the
new management is in progress. The new publishinghouse will immediately face a major challenge, with
the increasing importance of open access publishing,in particular Plan S of the European Commission. The
EMS has reacted to this plan, pointing the possible
consequences for small publishers and the mathematics community. See tinyurl.com/y54uw828.
Another major concern is research funding for mathematics in Europe, which is constantly decreasing
even within the European Research Council. See thenext item, appealing to EMS members to become
more active on all levels.

There may be several reasons for this decline inapplications, e.g. low acceptance rate, the feelingthat certain subfields of mathematics have small
chances, or the fact that for interdisciplinary researchof mathematics with other sciences it is difficult toget funding. Also in mathematics there are often
complaints that the maximal possible budgets are
too large.
All this is partially right, but not submitting applications leads to a vicious cycle, and further decline of

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mathematics funding. How can we counteract thisunfortunate development? First of all, there is no
reason to apply for the full possible budget if thisis not appropriate for a research project, smaller
proposals are very welcome, and second we mathematicians should be more self-confident in writing
proposals. It is not a wasted time, even if one is notfunded. In several European countries there is evenfinancial support for proposals that make it to thesecond round but do not get funded due to budget
restrictions.
It is very important that applications are encouraged throughout the mathematical community andthe EMS is planning to create an initiative to support applicants. So please distribute this information
within your community.
Volker Mehrmann
President of the EMS

Academia–industry roundtable
The EMS Applied Mathematics Committee notes with interest the minutes of an academia-industry roundtableorganised by the ministry of economy in Japan (March2019) (tinyurl.com/y3o54aud) which claims “we haveidentified the top three science priorities in order forJapan to lead the fourth industrial revolution and toeven go beyond its limits: mathematics, mathematics,
and mathematics!”

Bernoulli Center (CIB)The CIB in Lausanne has announced the following schedule of research programmes: Dynamics with Structures(1 July – 31 December 2019); Functional Data Analysis (1January – 30 June 2020); Locally Compact Groups Acting
on Discrete Structures (1 July – 31 December 2020); Dynamics, Transfer Operators, and Spectra (1 January – 30
June 2021). For more details see http://cib.epfl.ch.

European Congress of Mathematics
The preparations for the 8th European Congress ofMathematics in Portorož, Slovenia, from 5 to 11 July
2020 are proceeding energetically. A full list of plenary
and invited speakers at for the 8th European Congressof Mathematics in Portorož, Slovenia, is available at
8ecm.si/program.
EMS News prepared by David Chillingworth
LMS/EMS Correspondent

OPPORTUNITIESEMS Prizes
Calls for nominations of candidates for ten European
Mathematical Society Prizes as well as The Felix Klein
Prize and The Otto Neugebauer Prize for the Historyof Mathematics are still open: details can be foundat the European Mathematical Society website.

Events within Europe and speakers with strong relations to European academic institutions will be given
preference. Closing date for applications is 30 September 2019. For further details and nomination forms go
to the EMS website and click on Scientific Activities.

EMS Call for Proposals

Maryam Mirzakhani Prize inMathematics: call for nominations

The European Mathematical Society (EMS) Meetings
Committee is calling for nominations or proposals forspeakers and scientific events in 2020. The EMS is
willing to provide support to cover the cost of EMS
Lecturers and Distinguished Speakers, and to give partial support to the organization of EMS Weekends andof EMS Summer Schools of high scientific quality andrelevance. The EMS is committed to increasing the participation of women in research in mathematics andits applications. Efforts to give opportunities to mathematicians of both genders and to integrating early
career mathematicians will be particularly appreciated.

In recognition of Dr Mirzakhani’s remarkable life andachievements, the National Academy of Sciences
has established a newly named Maryam Mirzakhani
Prize in Mathematics (formerly the NAS Award in
Mathematics, established in 1988 by the American
Mathematical Society in honour of its centennial).
The prize will be awarded biennially for exceptional
contributions to the mathematical sciences by a midcareer mathematician. Nominations for the inaugural
Mirzakhani Prize are due by 7 October 2019. For more
information see tinyurl.com/yykobras.

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Women in Mathematics and other
Diversity Events
A call for expressions of interest in running Women in
Mathematics Days, Girls in Mathematics Days and a
new event for 2019/20, Diversity in Mathematics Day,is now open. The deadline for submitting expressionsof interest is Friday 20 September 2019.
More details about Women in Mathematics Days areavailable at tinyurl.com/y4wy8px3, Girls in Mathematics Days at tinyurl.com/y44t4mdt, and Diversity
in Mathematics Day at tinyurl.com/y2a3pmbg.

ICME-14 Bursaries
The Joint Mathematical Council of the UK has
launched the ICME bursaries scheme, which is partly
funded by the LMS. The scheme will provide ninebursaries of £500 each to attend the 14th International Congress on Mathematical Education to beheld from 12 to 19 July 2020 in Shanghai, China(tinyurl.com/yyxbvxth).
The bursaries can fund travel, subsistence, childcare, registration fee or preparation of a presentation.
They cannot fund salary or related costs. Applications are encouraged from the full range of thoseworking in all areas of mathematics and mathematics
education, including teachers, researchers and postgraduate students who can demonstrate a benefit
from attending ICME-14.
Download the application form at tinyurl.com/yyagyo23.
Email the completed application form to Professor
Chris Budd, University of Bath (mascjb@bath.ac.uk).
The deadline is 30 November 2019.

Clay Research Fellowships:
call for nominations
The Clay Mathematics Institute calls for nominations for its competition for the 2020 Clay Research
Fellowships. Fellows are selected for their researchachievements and their potential to become leaders in research mathematics. All are recent PhDs,
and most are selected as they complete their thesis
work. Terms range from one to five years, with mostgiven in the upper range of this interval. Fellows are
employed by the Clay Mathematics Institute, which
is a US charitable foundation, but may hold theirfellowships anywhere in the USA, Europe, or elsewhere in the world. Nominations should be received
by 16 November 2019. To nominate a candidate see
tinyurl.com/y6stxs6o.

VISITS

Visit of Grigory Belousov
Dr Grigory Belousov (Plekhanov Russian University of Economics, Bauman Moscow State Technical University) will visit the UK from 1 to 17
November 2019. During his visit, he will give lectures at the University of Liverpool, Loughborough
University and the University of Edinburgh. His
research revolves around Singular del Pezzo Surfaces.
For further details contact Nivedita Viswanathan(Nivedita.Viswanathan@ed.ac.uk). Supported by an
LMS Scheme 2 grant.

Visit of Murray ElderDr Murray Elder (University of Technology Sydney,
Australia) will visit the UK from 4 November to 4
December 2019. His research interests include geometric group theory, complexity theory, automata
and formal language theory, enumerative combina-

torics and pattern-avoiding permutations. While in
the UK he will visit and give talks in Newcastle, Heriot
Watt and St Andrews Universities. For further details
contact andrew.duncan@ncl.ac.uk. Supported by an
LMS Scheme 2 grant.

Visit of Albert Visser
Professor Albert Visser (Universiteit Utrecht) willvisit the University of Cambridge from 7 to 29
October 2019. He is professor emeritus of logic
and a member of the Royal Netherlands Academyof Arts and Sciences; he is particularly well-known
for his research contributions on weak systemsof arithmetic. In addition to seminar presentations
in Cambridge, talks will be given at the Universities of Oxford (14 October) and Leeds (16 October). For further information contact Benedikt Löwe(b.loewe@dpmms.cam.ac.uk). Supported by an LMS
Scheme 2 grant.

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LMS BUSINESSLMS Council Diary —A Personal View
Council met at De Morgan House on Friday, 28 June, alittle earlier than usual, due to the LMS General Meeting being held that same afternoon. As usual, themeeting began with an update on the President’s activities since the last Council meeting, which included
attendance at the LMS Reps Day and attendanceat several of the Society’s committees. She also
reported that three members of the LMS received
honours in the Queen’s birthday list: Kenneth Brown
received a CBE, Peter Ransom an MBE, and Peter Donnelly received a Knighthood. Vice President Hobbs
reported from the European Mathematical Society
Presidents’ meeting, which was held in March 2019
in Berlin. She reported that diversity took up somepart of the discussion. With the calls for EMS prizes
now published, the LMS was encouraged to promote
the nominations process, particularly to address theneed for diversity of nominations. Council agreed
that the free EMS membership for PhD students
should be advertised.

were disadvantaged in comparison to those funded
elsewhere. Council agreed that this item should be
deferred for further discussion at the October 2019
meeting.
We also heard reports from the IT Resources Committee and from the Publication Secretary. Furthermore,
we received the Third Quarterly Financial Review. Following a report from the Prizes Committee Council
agreed that Professor Kenneth Brown be confirmed
as the Crighton Medal recipient in 2019.
We then decamped to the nearby Mary Ward Housefor the General Meeting and Aitken Lecture 2019.
Brita Nucinkis

Perigal Artefacts

The Education Secretary then reported that a working group had been set up in December 2018 toaddress the issue of the shortage of qualified mathematics teachers. It was felt that the group should beformalised as a sub-committee of Education Committee, which Council approved. Council also agreed
to extend the Teachers CPD to allow universities topropose events for HE teaching and learning events.
The General Secretary reported that the latest version of the document containing the proposedchanges to the Standing Orders was now available
to view on the website, having received informal
approval from the Privy Council.
The Chair of the Women in Mathematics Committee
reported on the foreword to the National Benchmarking Survey Report, which is the LMS’s sole contribution to the report; this foreword was approved.
Council also agreed on the LMS Statement on Diversity in Mathematics.
Council noted with pleasure that the Irish Mathematical Society had agreed to a Reciprocity Agreementwith the LMS.
There was some discussion regarding the amountgiven to individual LMS Undergraduate Bursaries —some Council members felt that it would be detrimental to the Society’s reputation, and the diversity of applicants, if recipients of LMS bursaries

The London Mathematical Society has been by given
by Daniel Miskow some items which once belonged
to Henry Perigal, an LMS member famous for his
proof of Pythagoras’s Theorem by dissection. The
items include a collection of index cards containing
diagrams from Euclid’s Elements, and a beautifullydecorated envelope made by folding paper. The envelope contains six cardboard triangles and five quadrilaterals, as well as three pieces of blueish paper,
carefully cut and folded. The members of the Library
Committee have been unable to deduce their purposes and would welcome elucidation from members who are welcome to contact the LMS Librarian(librarian@lms.ac.uk).
Mark McCartneyLMS Librarian

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LMS BUSINESSLMS Committee Membership
The detailed business of the LMS is run by about 23
committees and working groups, each usually consisting of about 10 people. Altogether this comes toa large number of people, to whom the Society isextremely grateful for this vital work.
It is Council’s responsibility to make the appointments to all these committees and to turn theirmembership over regularly, so that (a) the broadest
possible spectrum of our membership is represented,
and (b) the committees remain fresh and energetic.
Of course, when forming a committee, account has
to be taken of many things, such as maintaining subject and demographic balance, which means that on
a given occasion otherwise very strong candidates
may not always be able to be appointed.
So we are always looking for new people! Seelms.ac.uk/about/committees for a list of committees.
If you are interested, or would like to recommend a colleague, please contact James Taylor atjames.taylor@lms.ac.uk in order that Council canmaintain a good list of potential members of its various committees. It is not necessary to specify aparticular committee. If you would like to know what
is involved, you could in the first instance ask yourLMS Departmental Representative.
Stephen Huggett
General Secretary

Annual LMS Subscription 2019-20
Members are reminded that their annual subscription, including payment for additional subscriptions,for the period November 2019 – October 2020 is dueon 1 November 2019 and payment should be received
by 1 December 2019. Payments received after thisdate may result in a delay in journal subscriptions
being renewed.
LMS membership subscription rates 2019–20
Ordinary membership£88.00

US$176.00

Reciprocity£44.00

US$88.00

Career break or part-timeworking
Associate membership£22.00

US$44.00£22.00

US$44.00

Associate(undergrad)
membership£11.00

US$22.00

Access to LMS Journals
The Society offers free online access to the Bulletin,
Journal and Proceedings of the London MathematicalSociety and to Nonlinearity for personal use only. If
you would like to receive free electronic access tothese journals, please indicate your choices either
on your online membership record under the ‘Journal
Subscription’ tab or on the LMS subscription form.
The relevant publisher will then contact memberswith further details about their subscription.
Subscribing to the EMS and JEMS via the LMS
Members also have the option to pay their European
Mathematical Society subscription via the LMS andsubscribe to the Journal of the EMS. If you would
like to subscribe to the EMS and JEMS via the LMS,
indicate either on your online membership record
under the ‘Journal Subscription’ tab or on the LMS
subscription form.
Payment of membership fees for EWM
LMS members who are also members of European
Women in Mathematics may pay for their EWM fees
when renewing their LMS membership. You decide
your category of fees: high, normal, low. Indicate yourcategory of fee either on your online membership
record under the ‘Journal Subscription’ tab or on
the LMS subscription form. To join EWM, registerat tinyurl.com/y9ffpl73. It is not possible to join the
EWM through the LMS.
Online renewal and payment
Members can log in to their LMS user account(lms.ac.uk/user) and make changes to their contact
details and journal subscriptions under the ‘My LMS
Membership’ tab. Members can also renew their subscription by completing the subscription form andincluding a cheque either in GBP or USD. We do not
accept payment by cheques in Euros.
LMS member benefits
An LMS annual subscription includes the following
benefits: voting in the LMS elections, free online
access to selected journals, the bi-monthly Newsletter, use of the Verblunsky Members’ Room at De
Morgan House in Russell Square, London and use of
the Society’s Library at UCL. For a full list of member benefits, see lms.ac.uk/membership/memberbenefits.
Elizabeth Fisher
Membership & Grants Manager

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LMS BUSINESSLMS Grant Schemes
For full details of these grant schemes, and for information on how to submit an application form, visit
www.lms.ac.uk/content/research-grants.
MATHEMATICS RESEARCH GRANTS

holder. For those mathematicians going to their collaborator’s institution, grants of up to £2,000 are
available to support a visit for collaborative researchby the grant holder to a country in which mathematics could be considered to be in a disadvantaged
position.

The deadline is 15 September 2019 for applications
for the following grants, to be considered by the
Research Grants Committee at its October meeting.

Research Workshop Grants: Grants of between£3,000 - £5,000 are available to provide support for
Research Workshops held in the United Kingdom, theIsle of Man and the Channel Islands.

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

AMMSI African Mathematics Millennium Science
Initiative: Grants of up to £2,000 are available tosupport the attendance of postgraduate students
at conferences in Africa organised or supported by
AMMSI. Application forms for LMS-AMMSI grants areavailable at ammsi.africa.

Visits to the UK (Scheme 2): Grants of up to £1,500
are available to provide partial support for a visitor
to the UK, who will give lectures in at least three
separate institutions. Awards are made to the host
towards the travel, accommodation and subsistence
costs of the visitor. It is expected the host institutions will contribute to the costs of the visitor.
Joint Research Groups in the UK (Scheme 3):
Grants of up to £4,000 are available to support joint
research meetings held by mathematicians who havea common research interest and who wish to engage
in collaborative activities, working in at least threedifferent locations (of which at least two must bein the UK). Potential applicants should note that the
grant award covers two years, and it is expected thata maximum of four meetings (or an equivalent levelof activity) will be held per academic year.
Research in Pairs (Scheme 4): For those mathematicians inviting a collaborator to the UK, grants of up
to £1,200 are available to support a visit for collaborative research either by the grant holder to anotherinstitution abroad, or by a named mathematician
from abroad to the home base of the grant holder.
For those mathematicians collaborating with anotherUK-based mathematician, grants of up to £600 are
available to support a visit for collaborative research.
Collaborations with Developing Countries(Scheme 5): For those mathematicians inviting acollaborator to the UK, grants of up to £3,000 are
available to support a visit for collaborative research,
by a named mathematician from a country in whichmathematics could be considered to be in a disadvantaged position, to the home base of the grant

MATHEMATICS/COMPUTER SCIENCE RESEARCHGRANTS
Computer Science Small Grants (Scheme 7):
Grants of up to £1,000 are available to support visits
for collaborative research at the interface of Mathematics and Computer Science, either by the grant
holder to another institution within the UK or abroad,or by a named mathematician from within the UKor abroad to the home base of the grant holder.
Deadline: 15 October.
GRANTS FOR EARLY CAREER RESEARCHERS
The deadline is 15 October 2019 for applications forthe following grants, to be considered by the Early
Career Research Committee in November.
Postgraduate Research Conferences (Scheme 8):
Grants of up to £4,000 are available to provide partial
support for conferences held in the United Kingdom,which are organised by and are for postgraduate
research students. The grant award will be used tocover the costs of participants.
Celebrating New Appointments (Scheme 9):
Grants of up to £600 are available to provide partial
support for meetings held in the United Kingdom
to celebrate the new appointment of a lecturer at
a UK university.
Travel Grants for Early Career Researchers:
Grants of up to £500 are available to provide partial travel and/or accommodation support for UKbased Early Career Researchers to attend conferences or undertake research visits either in the
UK or overseas.

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LMS BUSINESSREPORTS OF THE LMS

Report: LMS Invited Lectures 2019

Professor Søren Asmussen

The LMS Invited Lectures 2019 were delivered byProfessor Søren Asmussen (Aarhus University) on
20-24 May at the International Centre for Mathematical Sciences in Edinburgh, on advanced topics inlife insurance mathematics. This course was basedon parts of Professor Asmussen’s forthcoming bookRisk and Insurance: A Graduate Text, co-authored with
Mogens Steffensen (University of Copenhagen).

Thursday 23 May, organised on behalf of the Applied
Probability Section of the Royal Statistical Society.
Professor Asmussen presented applications of phasetype distributions in life insurance, building from thegraduate-level material he discussed earlier in theweek to the state of the art in the area. Additional,
complementary presentations were given by Burak
Buke (University of Edinburgh), Ayalvadi Ganesh (University of Bristol) and Ronnie Loeffen (University ofManchester) on various aspects of the theory of
stochastic processes, with applications to financial,biological and other systems.
A relaxed schedule and the hospitable environment
offered by the ICMS allowed for plenty of interaction between participants at various career stages
throughout the meeting, and also time to enjoy thecity of Edinburgh in one of the better weeks ofweather this summer has given us.
Fraser Daly and Sergey Foss
Heriot-Watt University

Report: LMS Popular Lectures 2019

Professor Asmussen gave a fascinating account of
inhomogeneous Markov models, semi-Markov models, and their applications in life insurance. Topics
covered in detail included unit-linked insurance, interest rate and mortality rate modelling, and dividends
and bonuses. The course consisted of nine lectures
and two tutorials, and was complemented by an
opening lecture by Takis Konstantopoulos (Universityof Liverpool) and a closing lecture by Andrew Cairns(Heriot-Watt University).
Participants (particularly early-career researchers)
were given the opportunity to present some of
their own work. During the week we heard interesting presentations from Jason Anquandah (University of Leeds) on Optimal Stopping in a Simple
Model of Unemployment Insurance, Abdul-Lateef HajiAli (Heriot-Watt University) on Multilevel Monte Carlofor Efficient Risk Estimation, and Lewis Ramsden (University of Hertfordshire) on The Time to Ruin for a
Dependent Delayed Capital Injection Risk Model.
In addition, a workshop on Stochastic Processes in
Finance and Beyond was held on the afternoon ofPeter Higgins

On 26 June 2019 the Institute of Education, London
hosted the 2019 LMS Popular Lectures. The audienceenjoyed an evening of mathematics with a journey
through a life in mathematics and a look at the complexity of the human brain and some of the basic
principles at work in shaping our brains.
Peter Higgins (University of Essex) presented the firstlecture and took the audience through his personal

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LMS BUSINESS
mathematical journey. Going back to his roots andrecounting personal episodes that led to him becoming a professional mathematician. He went on to tell
the story of how he invented circular Sudoku. Higgins recognised from his work that the knowledge hehad accumulated could be applied to mathematicalproblems in this case relating to Burrows–Wheelertransform, which is an algorithm that rearranges character strings into runs of similar characters and isimportant in data compression.

Alain Goriely

The second lecture was presented by Alain Goriely(University of Oxford). The outer surface of the brain(cerebral cortex) is very convoluted (wrinkled) so thata maximum amount of gray matter (brain neurons)
can fit inside the skull, but how do these convolutions
emerge and how is the brain’s geometry related tofunction? Professor Goriely began by looking at therelative sizes of the brains of some famous scientists
and mathematicians, with some interesting results.
Einstein in particular had a small brain compared withBabbage, Helmholtz, Gauss and Dirichlet. He went on
to describe comparisons in the convolutions of Gauss
and De Morgan and discussed how the brain develops
its shape by using mechanics. Gauss had complex
convolutions and De Morgan had convolutions that,
although ’voluminous’, were less intricate. Professor
Goriely explained how different parts of the brain areconnected to each other in brain networks and how
neurodegenerative diseases develop. In essence heshowed how some of the basic principles at work inshaping the brain can be explained using geometry,scaling laws and network topology.
If you weren’t able to attend the London lectures,
there is another chance to attend the Lecture in Birmingham on Thursday 19 September. You can register
online at lms.ac.uk/events/popular-lectures.
John Johnston
LMS Communications Officer

Report: LMS Undergraduate
Summer Schools
The LMS Undergraduate Summer Schools are annual
two-week courses, which are held every summer at a
UK university. The aim is to introduce modern mathematics to the best UK undergraduates who will enter
their final year the following autumn, and encouragethem to think about an academic career in mathematics. The LMS Undergraduate Summer School consists of a combination of short lecture courses withproblem-solving sessions and colloquium-style talks
from leading mathematicians. 50 places are available
to students per year. Gwyn Bellamy, organiser of the2018 LMS Undergraduate Summer School at Glasgow,
reports on the LMS Undergraduate Summer School2018. The 2019 LMS Undergraduate Summer School
took place at the University of Leeds from 14–26 Julyand the next LMS Undergraduate Summer School
will take place at the University of Swansea from
12-24 July 2020. Nominations for places will open in
October 2019.
In July 2018, 53 extremely eager students from acrossthe United Kingdom descended on Glasgow for a
jam-packed two weeks of mathematical activities
which is the LMS Undergraduate Summer School.
Each mathematics department in the UK elects up to
three applicants to attend, and from these we selectapproximately 50 participants. Almost the entire costof the two weeks is covered by the LMS.
In the first week, lecture courses were given by Professor Mike Prest (University of Manchester) on The
Compactness Theorem, by Dr Robert Gray (Universityof East Anglia) on The Word Problem in Combinatorial Group and Semigroup Theory and by Professor
Shaun Stevens (University of East Anglia) on Local–
Global Principles in Number Theory. In the second
week, lecture courses were given by Dr Martina Lanini(Università di Roma Tor Vergata) on Introduction to
Schubert Calculus, by Dr Derek Harland (University ofLeeds) on Fun with Solitons and by Dr David Bourne(Heriot-Watt University) on Optimal Transport Theory.
As one would expect, the lectures were uniformly
fascinating and engaging, and a real pleasure to listen to. It was clear that the lecturers had spent agreat deal of time and energy in preparing their lectures. Moreover, each lecturer had prepared a seriesof exercises to go with the lectures. Together with
PhD and MSc students from the University of Glasgow, the lecturers guided the students through theexercises during the exercise sessions.

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LMS BUSINESSIn addition to the lecture series, there were colloquium talks later in the afternoon on most days.
These were:• Arithmetic and Geometry of Conway Rivers by Professor Alexander Veselov• Chasing the Dragon: Tidal Bores in the UK and Elsewhere; Quantum and Hawking Radiation Analogies
by Sir Michael Berry• Faster than Fourier (pre)revisited: Vorticulture, Noise,
Fractals, Escape. . . by Sir Michael Berry• Unexpected Connections. . . by Dr Tom Leinster• A Tour of the Mandelbrot Set by Dr Holly Krieger• The Birch and Swinnerton-Dyer Conjecture: From theAncient Greeks to a 1 Million Dollar Problem of the
21st Century by Dr Alex Bartel• Mathematics, Magic and the Electric Guitar by DrDavid Acheson• Random Games with Finite Groups by Professor
Colva Roney-Dougal

Surprisingly, around 10 of the students managed tosolve all the problems.
There is a real hunger amongst undergraduate mathematics students in the UK to be exposed to researchlevel mathematics. The LMS Undergraduate Summer School is one of the few ways in which we
can effectively meet this need. Though it is, as one
can imagine, a great deal of work to organize an
LMS Undergraduate Summer School, it is also an
extremely rewarding experience and an effective way
to make a significant contribution to undergraduate
mathematics in the UK. For those LMS members ina position to apply to organize a summer school, I
would strongly encourage you to do so.
I sincerely hope that the summer school will have
had a positive influence on each student’s long-term
relationship with the mathematical sciences, and that
we’ll see many of them studying for a PhD in thenear future. I would also like to give my thanks to thelecturers and colloquium speakers for all their hard
work in preparing their talks, and their enthusiasm
in delivery.

As one can guess from the above list, these wonderful talks took the students on a journey through ahuge swath of exciting mathematics, all of which was
beautifully illustrated (and in one case accompanied
by live music!)
On the Sunday between the two weeks of lectures,
there was an organised bus trip to Balmaha on the
shores of Loch Lomond. All the students braved theweather to climb the nearby Conic Hill, where there
were spectacular views of the loch and nearby Ben
Lomond; at least this was the case until about half
way up, the summit itself was unfortunately shroudedin mist for the day. Afterwards we retired to the pubat the base of the hill for a well-deserved drink.
As organiser, by far the most enjoyable aspect of
the summer school was interacting with the participants. It was energizing being surrounded by somany bright, enthusiastic and hardworking young
people! They really gave it their all, and I think thisensured that they got the most out of the two weeks.
It was particularly gratifying to see them very quickly
bond, seeming to be old friends by the end of thefirst week. It was also a great experience to be in
exercises classes where the students were asking
questions throughout and racing to try and solve theproblems. To provide motivation for the students towork on mathematics questions (as it turned out,
this was not required at all!), a series of mathematicalproblems were set, with prizes for the best solutions.

Gwyn Bellamy
University of Glasgow

Report: LMS Northern Regional
Meeting and Workshop

Conference participants

An LMS Northern Regional Meeting took place inthe afternoon of Tuesday 28 May 2019, followed bya wine reception and dinner, and a Workshop on
Higher Dimensional Homological Algebra was held on
Wednesday 29 May 2019 at Newcastle University.

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LMS BUSINESSThe LMS Northern Regional Meeting and the workshop brought to the UK international experts in the
representation theory of finite dimensional algebras
to present their work on the most recent developments and the speakers included many of the mostrenowned experts in the area. The workshop also
gave early career mathematicians such as PhD students and postdocs an opportunity to present theirwork. In total there were nine speakers, of whom one
was a PhD student and three were postdocs.
There were two general talks at the Northern
Regional Meeting on the latest developments in thefield. These talks were given by Karin Baur (Universityof Graz and University of Leeds) on CM Modules for
Grassmannians and by Sibylle Schroll (University ofLeicester) on Geometric Models and Derived Invariants
for Gentle Algebras.

The workshop was organised by Ilke Canakci andPeter Jorgensen. The organisers thank the LMS
and the School of Mathematics, Statistics andPhysics at Newcastle University for financial support. The website of the workshop can be found at
tinyurl.com/y4qc2gwt.
Ilke Canakci
Newcastle University

Report: LMS Graduate Student
Meeting and the General Meetingof the Society & Aitken Lecture

The workshop was focused on Higher Dimensional
Homological Algebra. This area concerns d -cluster
tilting subcategories, which are structures found at
the cusp of algebra and combinatorics. They have arich algebraic structure, investigated by homologicalmethods, and there are many examples relating tohigher dimensional structures in combinatorics, suchas cyclic polytopes.
There were seven talks at the workshop, two of
which were expository talks by Steffen Oppermann(Norwegian University of Science and Technology)on (d + 2)-angulated cluster categories and by HughThomas (Universite du Quebec a Montreal) on Tropical
Coefficient Dynamics for Higher-Dimensional Cluster
Categories. Talks focusing on recent developmentsof higher homological algebra were by Francesca
Fedele (Newcastle University) on A (d + 2)-angulated
generalization of a theorem by Bruning, by Martin Herschend (Uppsala University) on Wide subcategoriesof d -Cluster tilting subcategories for higher Auslander
algebras, by Karin Marie Jacobsen (NTNU) on d-abelian
Quotients of d + 2-angulated Categories, by Gustavo
Jasso (Universitat Bonn) on Generalised BGP Reflection
Functors via Recollements and by Sondre Kvamme(Université Paris-Sud) on d -abelian Categories are d cluster Tilting.
The atmosphere during the meeting was very pleasant and convivial. The frequent coffee breaks, thewine reception and dinner followed by the Northern
Regional Meeting and the lunch at the workshop provided great opportunities for discussions, informal
exchanges of ideas and for establishing professional
contacts. There were 31 participants, 10 of whomwere PhD students.

Aitken Lecturer Bakh Khoussainov

The LMS Graduate Student Meeting and the General
Meeting of the Society & Aitken Lecture were held
on Friday 28 June 2019, a warm and sunny summer’sday, at Mary Ward House in London. The morning
session was kicked off by Dr Robert D. Gray, of the
University of East Anglia, whose lecture served as
an introduction to the material which would be presented in lectures later in the day. Ideas surrounding
finite presentations of algebraic structures, recursive enumerability, and notions such as the word
problem were all introduced. This lecture was followed by a number of graduate student presentations, where the broad selection of topics ranged
from climate models, modular representation theory, and polyhedral combinatorics, among others. By
popular vote, two graduate speakers were then eachawarded a prize: Aras Asaad, for a presentation on
using homology to detect computer generated faces,
and Carl-Fredrik Nyberg Brodda, for a presentation
on decision problems for groups and semigroups.
After the morning’s Graduate Student Meeting, theafternoon’s General Meeting was opened by Professor Caroline Series. While the morning meeting had

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LMS BUSINESS
seen a respectable audience, the room was quickly
filled up by new attendees at the beginning of theGeneral Meeting. Two new Honorary Members of theSociety were appointed, and the announcement of
the 2019 LMS Prize Winners followed. Following tradition, several new members of the Society werethen given the opportunity to sign their names in theMembers’ Book. To serve as an introduction to thelater Aitken Lecture, Dr Paul Shafer, of the Universityof Leeds, presented a thorough lecture on the topic
of computability, which served well to extend thematerial presented earlier in the day.
As could be expected on a warm day in June, theroom — which at this point had become filled to thebrim with mathematicians — had grown hot, to thepoint that a collective sigh of relief could be heard
when the windows were finally opened. Refreshedby this flood of fresh air, the audience members
all seemed sharpened and ready to enjoy the final
event of the General Meeting; this was the Aitken
Lecture, given by Professor Bakh Khoussainov of the
University of Auckland. A central idea therein, forwhich the audience was now more than prepared asa result of the earlier lectures, was the question of inwhich circumstances one can (or, indeed, cannot) findfinitely presentable expansions of a given algebra.
This question, and the context in which it is asked,
touches on a number of different topics, including
immune algebras, residual finiteness of algebras, andthe generalised Burnside problem for groups.

Paul Shafer

While many new definitions were planted in the mindsof the audience members during the course of thepresentation, there was no lack of prepared groundin which to place them owing to the earlier lectures;the end result was an excellent insight into this fascinating field of research laying at the intersectionof many different areas.
Finally, to finish off the day, a wine reception followed by a conference dinner were both held at the
Ambassadors Bloomsbury Hotel, rounding off a mostinformative and enjoyable day in June.

Carl-Fredrik Nyberg Brodda
University of East Anglia

Records of Proceedings at LMS meetings
Northern Regional Meeting at Newcastle University: 28 May 2019
The meeting was held at the Herschel Building, Newcastle University, as part of the workshop on Higher
Dimensional Homological Algebra. Over 30 members and guests were present for all or part of themeeting. The meeting began at 3.10 pm, with the Vice-President, Professor Cathy Hobbs, in the Chair.
There were 31 members elected to Membership at this Society Meeting. Four members signed theMembers’ Book and were admitted to the Society by the Vice-President during the meeting. There were
no Records of Proceedings signed at this meeting of the Society.
Dr Ilke Canakci introduced the first lecture, given by Dr Karin Baur (ETH Zurich/Leeds) on CM-Modules
for Grassmannians.
Dr Ilke Canakci then introduced the second lecture, given by Dr Sibylle Schroll (Leicester) on Geometric
Models and Derived Invariants for Gentle Algebras.
The Vice-President thanked the speakers for their talks, and further extended her warm thanks to
the local organisers, Dr Ilke Canakci and Professor Peter Jorgensen, for hosting such a well-structured
and interesting meeting, and additionally thanked the attendees for coming to the Northern Regional
Meeting.
Dr Ilke Canakci thanked the speakers, and invited all those present to attend a wine reception held in
the Penthouse of the Herschel Building. A Society Dinner was held following the wine reception in the
Herschel Building.

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LMS BUSINESS

Records of Proceedings at LMS meetings
General Meeting: 28 June 2019
The meeting was held at Mary Ward House, Tavistock Square, London. Over 60 members and visitors
were present for all or part of the meeting.
The meeting began at 3.30 pm with the President, Professor Caroline Series, FRS, in the Chair. On a
recommendation from Council, it was agreed to elect Professor Charles Goldie and Professor Chris
Lance as scrutineers in the forthcoming Council elections. The President invited members to vote, by ashow of hands, to ratify Council’s recommendation. The recommendation was ratified unanimously.
The President, on Council’s behalf, proposed that following people be elected to Honorary Membershipof the Society: Professor Ed Witten, of the Institute for Advanced Study at Princeton University andProfessor Don Zagier, of the University of Bonn. This was approved by acclaim. The President read ashort version of the citations, which would be published in full in the Bulletin of the London MathematicalSociety.
The President then introduced the General Secretary who gave a report on the review of the Charter,
Statutes and By-Laws.
The President then announced the awards of the prizes for 2019:
De Morgan Medal: Professor Sir Andrew Wiles (University of Oxford); Senior Whitehead Prize: Professor
Ben Green (University of Oxford); Naylor Prize & Lectureship in Applied Mathematics: Professor Nicholas
Higham (University of Manchester); Whitehead Prizes: Dr Alexandr Buryak (University of Leeds), Professor
David Conlon (University of Oxford), Dr Toby Cubitt (University College London), Dr Anders Hansen(University of Cambridge), Professor William Parnell (University of Manchester), and Dr Nick Sheridan(University of Edinburgh); Berwick Prize: Dr Clark Barwick (University of Edinburgh); Anne Bennett Prize:
Dr Eva-Maria Graefe (Imperial College London).
Fourteen people were elected to Ordinary Membership: Dr Alla Detinko, Professor Johny Doctor, Mr
Dalebe Gnandi, Dr Marina Iliopoulou, Dr David Kimsey, Dr Tomasz Lukowski, Dr Jesu Martinez Garcia,
Professor James Maynard, Mr Luthais McCash, Professor Monica Musso, Dr Louis Theran, Dr Sofya
Titarenko, Mr Bunonyo Wilcox and Dr Gordon Woo.
Twelve people were elected to Associate Membership: Mr Isarinade Ayodeji Felix, Mr Eduard CampilloFunollet, Mr Carl Dawson, Mr Fabio Ferri. Dr Maciej Matuszewski, Mr Dimitris Michailidis, Dr Isaiah Odero,
Mr Wasim Rehman, Mr Mark Scott, Dr Joni Ter äväinen, Mr Nicholas Williams and Dr Mehdi Yazdi.
Three people were elected to Reciprocity Membership: Mr Rizwan Kassamally, Dr Kwara Nantomah and
Dr Alan Sola.
Three members signed the book and were admitted to the Society.
The President announced the dates of the next two Society Meetings to be held on 11 September in
Nottingham as part of the Midlands Regional Meeting and on 21 November in Reading as part of the Joint
Meeting with the Institute of Mathematics and its Applications. The President also reminded membersthat the date of the Annual General Meeting had moved from 15 November to 29 November.
The President announced that, to celebrate the 21st Anniversary of the Society’s move into De Morgan
House, there would an event on 19 October 2019.
The President further announced that the Society was seeking to recruit a new Editor in Chief for the
LMS Newsletter.
The President introduced a lecture given by Dr Paul Shafer (University of Leeds) on An Introduction to
Computable Functions and Computable Structures.
Following a break for tea, the President introduced the Aitken Lecture by Professor Bakh Khoussainov(University of Auckland) on Semigroups, Groups, Algebras, and their Finitely Presented Expansions.
At the end of the meeting, the President thanked both speakers for their brilliant lectures.
The President also thanked Robert Gray (University of East Anglia), who gave the main talk at theGraduate Student Meeting in the morning. Aras Assad (University of Buckingham) and Carl-Fredrik
Nyberg Brodda (University of East Anglia) were also congratulated on winning the prizes for the best
Graduate Student talks.
After the meeting, a reception was held at the Ambassador Bloomsbury Hotel in the Enterprise Suite,
followed by a dinner at the Number Twelve restaurant in the Ambassador Bloomsbury Hotel.

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The Importance of Ethics in Mathematics
MAURICE CHIODO AND TOBY CLIFTON

Mathematics is useful because we can find things to do with it. With this utility comes ethical issues relating
to how mathematics impacts the world. Now, more than ever, we mathematicians need to be aware of these,
as our mathematics, and our students, are changing society. In the first of a two-part series on Ethics inMathematics, we address why, as mathematicians, we need to consider the ethics of what we do.
Mathematics and the world
We study one of the most abstract areas of humanknowledge: mathematics, the pursuit of absolute
truth. It has unquestionable authority. But, in somesense, absolute truths have absolutely no meaning.
The statement “2 + 3 = 5” is an absolute truth,
but what does it mean? Its meaning and utility
are added later when people who understand thestatement reconcile it with the physical world. It is
the mathematically-trained who interpret and apply
mathematics to the real world and thereby assign itmeaning; through this it becomes useful.
Indeed, it is clear that mathematics is one of themost useful and refined tools ever developed. Whensomething is useful, however, it can often also be
harmful; this can be either through deliberate misuseor ignorance. The humble knife provides an illustration of the principle; in order to use such a tool
responsibly, one must be made aware — often by
those who first introduce you to it — of the potential dangers. If a primitive tool like a knife can be
so useful and harmful at the same time, then what
is mathematics capable of? Mathematics has many
more applications, and by the same reasoning mustalso have a greater potential to do ill. As mathematicians we are seldom warned of this. Other disciplines
such as law, medicine, and engineering have, for along time, addressed the potential for harm within
their field. We, as the practitioners and wielders ofmathematics, need to be similarly aware, and adjust
our actions accordingly. Otherwise we can, and sometimes do, cause harm with our work. But how could
mathematics possibly be harmful, and what exactly
might this harm be?
In this article our emphasis is on the experiences ofpure mathematicians, although our arguments apply
equally to applied mathematicians, statisticians andcomputer scientists. Many of us (although certainly
not all) are motivated to study mathematics by its

beauty and intrinsic interest, rather than its applications in science and industry. It’s as though we are
studying a form of abstract art; far from real world
impact or considerations, and only fully appreciatedby a small number. Despite all this, government andindustry pay for our work; one suspects they don’t
just do this for the sake of our intellectual stimulation. If our work is completely abstract and detached— an art form, so to speak — then shouldn’t we seek
funding from those who fund abstract art? So whatmight be the value that science councils and industry
see in what we do? It is not only the mathematicalresults that we produce, but also the mathematicians we train. Our mathematics makes a difference,and our students go out and do real things with
their training. If it is the case that our work is being
funded because it has perceived impact, then surely
we should query and understand why we are being
paid to do it.
There is already much discussion of ethics in themathematical community. However, these discussions usually focus exclusively on issues within thecommunity. These are important and many of us are
already familiar with them: from improving diversity
and inclusivity, to widening participation in mathematics, to addressing instances of plagiarism andpublishing irregularities. These are pressing concerns.
Every discipline engages with such intrinsic ethicalissues. These, however, are not our focus in thisarticle. Mathematics is one of the few disciplines thatfails to address extrinsic ethical issues; those concerning how the community impacts wider society.
This particularly includes ethical implications relating to the applications of mathematics and work ofmathematicians. It is these extrinsic ethical issues
that we are trying to raise awareness of. Our concern
is not so much that mathematicians are deliberately
malign, but instead that they fail to recognise these
extrinsic ethical issues. Indeed, most of the mathematicians we have come across would baulk at thethought of acting unethically; the problem, instead, is“NLMS_484” — 2019/9/11 — 16:05 — page 23 — #23

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that many do not recognise that mathematical work
can have such an effect.
Some case studies
Having recognised that mathematics is useful
because it can be applied, and that with these applications come extrinsic ethical issues, we now consider two concrete examples: the global financial
crisis (GFC) of 2007–8 and targeted advertising.

as such it’s now possible to specify who doesn’t see
an advert. This allows advertising campaigns thatare selective, that contain adverts that contradict
each other, and that are impossible to externallyscrutinise. In short, adverts can now be used to manipulate individual people. This becomes particularly
dangerous when applied to political advertising. Using large data sets obtained through social media,
it is possible to profile the political persuasions andpreferences of an individual. Machine learning has
become the main tool of the trade here, and it is
the mathematically trained doing it [3]. These adverts can even deceive by appearing non-partisan.
For instance, one can send an advert saying “Voting
is important; make sure you vote” only to those who
might be inclined to vote for your party. Whatever
the strategy, these types of adverts are increasingly
prevalent, and it is thought that such tactics influenced the 2016 US election and UK referendum on
EU membership. It is we mathematicians who makeall this possible. Cambridge Analytica, one of theorganisations alleged to have been involved in suchadvertising, had a small team of no more than 100
data scientists [4], some of whom were trained mathematicians. Regardless of one’s political persuasion,
it is clear that this sort of work is deceptive anddangerous, and that mathematicians are enabling it.
Ultimately, it is mathematicians who make up partof the teams specifying how such targeting works
and carrying it out.

The GFC was one of the defining events that shaped
the modern global economy. Its repercussions havebeen felt around the world, with many suffering
a decline in living standards. The causes of the
GFC are complex, however, there is consensus thatmathematical work played a vital role. An important factor is thought to have been the misuse of
Collateralised Debt Obligations (CDOs). These saw
mathematicians pool large collections of interestbearing assets (mostly mortgages), then ‘cut the pool
into pieces’ to form a collection of interest-bearing
products. Mathematically these products had lessoverall risk and thus higher value than the original
assets. They were traded wildly. The mathematics
behind their construction is highly non-trivial, requiring stochastic calculus, differential equations, etc.
Research mathematicians, beginning with the workof Black and Scholes, and later Li, derived a modeland pricing formula for CDOs. Though it took a deep
understanding of mathematics to derive these models, only a more superficial understanding (at theundergraduate level) was required to apply and to
trade them. As a result, their users may not havefully appreciated their limitations or inner workings.
Mathematics — which by itself is sure and certain— seemed to explain their value, and so most were
happy. Unfortunately, some of the assumptions did
not hold. For example, the model assumed there
wasn’t tail dependence in the default risk of underlying assets, but there was; for instance when two
mortgaged houses were on the same street. In theend, the risk was not properly accounted for, andwhen house prices declined it led to the write-down
of $700billion of CDO value from 2007 to 2008. The
rest is history.

As a result of the pace and scale at which modern technology operates, through use of internet
connectivity and readily-available fast computation,
the consequences of the actions of mathematicians
are more quickly realised and far-reaching than everbefore. A mathematician in a big tech company canmodify an algorithm, and then have it deployedalmost immediately over a user base of possibly billions of people. Even on a smaller scale, we haveseen that a small number of mathematicians, despite
limited resources, can have a vast impact on theworld; targeted advertising exemplifies this.

Our next example is targeted advertising. Adverts
have always been placed so as to catch the eye of
their desired audience. However, now that people
possess portable internet-connected devices andsocial media accounts, it has become possible to target adverts at the individual level. Nowadays, thesecan be tuned to fit very specific demographics, andIf you model a physical system, such as gravity, thenyour model is falsifiable. If the model does not accurately reflect the physical system, then on application
it clearly fails — your rocket doesn’t launch properly. You know when a model was good because therocket makes it to the moon and back. Modelling afinancial system is more difficult, as the system is

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affected by the application of the model. A pricing
algorithm, if widely-used to buy or sell a product,
influences the market for the product in question.
How does a model model its own impact?
So now what happens if you are modelling the future
behaviour of people by predicting something like
‘How likely is a particular individual charged witha crime to reoffend with a serious offence, a nonserious offence, or not reoffend, in the next 24
months?’ Furthermore, what if that is being used todetermine what prosecution and sentencing mechanisms are applied to that person?1 If you predict thata person will reoffend seriously in 24 months, andthey don’t (after being released or acquitted), thenyou might observe that. But what if they are foundguilty and sentenced to 25 months, with the choiceof judicial process based on your prediction? How doyou test whether your prediction was correct? Now
we have a serious ethical issue: we are using mathematical reasoning to make decisions about people
that impact their lives, and in many of these cases
we can never know whether the decisions made weredesirable or appropriate. Is it right to use mathematics in such a way without careful reflection?
We now face an ethical dilemma. Do we limit ourselves to falsifiable claims, or do we allow ourselves
to make claims, make decisions, and initiate actions
that are unfalsifiable? We are of course entitled to do
the latter, however we should then bear in mind that
we have lost mathematical certainty. Furthermore, if
we do this, we should broaden our perspective andtraining so that we can incorporate as many aspectsof society as possible.
Concerns for the future
So what is on the horizon for mathematicians? Is it
sufficient to simply look at the above list of cases andavoid those specific actions or industries entirely?
Unfortunately not; new mathematics produces new
ethical issues every day. Such a future example maylie in alternative credit scoring. This is starting to bedone by new companies who lack access to standard
datasets that established credit-scoring agencies
have (such as financial records, bill payment history,etc.). They instead use different datasets such associal media profiles, in some cases requesting fullaccess to social media accounts by asking for login
credentials [2]. While this sounds undesirable to thepoint that most people will not be interested, it must
be remembered that some people will be sufficiently
desperate for credit that there will always be sometakers. These companies scrape an applicant’s social
media looking for actions they perceive to reflect
creditworthiness. These could include places the person visits, the hours they sleep, the ‘quality’ of thefriends they have, and so on. This approach is unfalsifiable, lacks proper regulation, and has the potential
to harm society since the extension of credit is amechanism of social mobility. If such a process, onethat is enabled by mathematically-trained people,
starts having negative impact, who is accountable?
Ultimately, we must live in the world that we and ourstudents create, and we must ponder whether there
is a sense in which we are partially responsible.

Do these ethical issues arise in academia?
But what about mathematicians working in academia;
are any of these ethical issues relevant to them? Consider a pure mathematician, a number theorist, say.
Suppose they develop an algorithm for fast factorisation. Should they publish it? If so, when, where, andhow? If not, what should they do? Should they havethought about it beforehand? We have asked manymathematicians this exact question, and a typicalresponse is “I would publish it on arXiv immediately.
It’s my right to publish whatever mathematical work
I do.” (Not all mathematicians give such a response,but many do.) When pressed on the consequencesof publishing such an algorithm in that way — for
instance the breaking of RSA encryption in a chaotic
manner and the ensuing collapse of internet commerce and the global economy that would follow —one explained “Well, it’s their fault for using RSA.
It’s not my problem.” Of course, responsible disclosure is a complicated topic, and one that is heavily
debated by security researchers. But with an example like this, ethics has crept into the world of thepure mathematics researcher in academia. If an area
as abstract as number theory is not ‘safe’ from ethical considerations, is there any mathematical workthat is? Can a pure mathematician hide from ethicalissues in academia? What about a statistician, or an
applied mathematician? Or do ethical questions arise
for all mathematicians regardless of where we do ourmathematical work?

1 The Harm Assessment Risk Tool (HART) developed by the Durham Constabulary, which uses random forest machine learning, is used tomake such predictions, and then determine if an accused criminal is to be offered the opportunity of going through the Checkpoint
program (tinyurl.com/y4vxrd77) which is an alternative to criminal prosecution aimed at reducing re-offending.

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Why management can’t guide us
Some mathematicians (academic or industrial) may
think that, since they are not directly involved in theapplication of their work, they need not consider its
extrinsic ethical implications. After all, we just do themaths, and so it’s ‘not our problem’. This oft-heldbelief is generally associated with the perception that
there exist people and structures above us (managers, supervisors, advisory boards, etc.) who willintervene to prevent us from doing anything that weought not to. We work on the abstract problems, they
worry about why. But can we rely on management to
do this effectively? Will they vet our work, to ensurethat its use is aligned with the values of society? Ateach stage of separation from mathematical work,some understanding of it is lost. It is difficult fora manager to understand all of the mathematicalwork we do, and its limitations when applied andused. It is the nature of management that managers
will only have partial knowledge of the work being
done. There would be no point in a manager reproducing the work of all of the people under them,and mathematics is such that if you don’t ‘do it for
yourself’ then there is a chance you may not fully
understand it. Given this fact, there is always an onus
on the individual mathematician to consider the ethical implications of what is being done. Of course, it
must also be considered that managers might haveother values, perhaps more aligned with the objectives of the organisation than of wider society. We
should understand and anticipate this.
Some managers may go so far as to try to manipulate us. For instance, if we voice objection at what
we have been asked to do, they may try to quash itwith the classic argument: “if you don’t do it, thensomeone else will”. At a first glance, this seems
convincing, however, it fails on two counts whenreferring to mathematical work. First, there are notthat many mathematicians in the world. We possessa unique set of skills and abilities, and it requires
years of training to produce a good mathematician,even when starting with someone who has the rightinterests and reflexes. Given the scarcity of mathematicians, this argument fails in practice. Moreover,as mathematicians we understand its contrapositive;
the original statement is equivalent to “If no-one else
does it, then you will”. This is, of course, absurd. The
argument has as the implicit underlying assumptionthat the task being requested will definitely be completed. If no-one else builds me a nuclear bomb, thenwill you? What we should really be considering here is
the argument “If you don’t do it, then someone elsemight”. True, someone else might, but they may not
be easy to find, or even exist at all. Now the powerof meaningful objection has returned to the mathematician. Whether you choose to take the pragmatic
perspective that there are not many mathematicians,or the logical perspective of the contrapositive, yourobjection means something. Some mathematicianstake this idea even further, and make a conscious
decision to take a seat at the table of power, effecting positive change from the managerial level. This
happens in various areas: in academia, in industry,and even in politics. This is discussed in more detailin [1] as ‘the third level of ethical engagement’.

Why the law can’t guide us
The problem extends beyond management. We may
think that the law provides a clear description ofwhat is and is not acceptable to society, and thereby
presumably what is and is not ethical. However, thismisses the point for several reasons. Firstly, the law
is not an axiomatised system; it is interpreted bycourts rather than by machines. This is a type of system with whose details mathematicians are generally
not familiar. Furthermore, there is the problem thatthe law will always lag behind technological development; we cannot expect lawmakers to have done ourmathematics before we do it ourselves. Additionally,
the processes by which laws are made are (deliberately) slow, requiring public consultation, votes, andimplementation periods. Consider the case of theGeneral Data Protection Regulation (GDPR). It started
to be written in 2011, only came in to effect in 2018,
and is thought by many to be already out of date.
Finally, it can be the case that lawmakers lack a fullunderstanding of the fine details of the subject athand. For example, a member of the UK Scienceand Technology Select Committee, Stephen Metcalfe,
declared at a public outreach event that “one solution to algorithmic bias is the use of algorithms tocheck algorithms, and the use of algorithms to checktraining data”. Ultimately, the law is not there to serve
as moral advice; there are plenty of immoral things
one can do that do not break any laws. As such, it is
not well-suited as a source of ethical advice.
Thus, if we can’t rely on management and we can’t
rely on lawmakers and regulators, then who can werely on? The answer is as obvious as it is difficult
to admit: ourselves. The only way mathematicians
can try to prevent their work from being used to do
harm is if they think about it themselves. No one
else can, so we must.

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A growing awareness of Ethics in Mathematics
Awareness that mathematicians need to consider extrinsic ethical issues is building in the community. In
2018 the head of mathematics at Oxford, Professor
Mike Giles, commented at a panel discussion event:“Cambridge Analytica is interesting from one point ofview in that, if you’d asked me 20 years ago whether
mathematicians at the PhD level needed to be exposed to ideas of ethics, I would have said ‘Clearly,that is irrelevant to mathematicians’. Now I reallythink that this is something we have to think about.
In the same way that engineers have courses looking
at ‘What it means to be a professional engineer’, and‘Ethics, and your responsibilities as an engineer’, I
think that is something that we have to think about
as mathematicians now.” Moreover, arxiv.org is currently revising the description of their mathematics
tag History and Overview to include “Ethics in Mathematics” as a sub-category.
As part of their formal training, few mathematicians
have ever been told about extrinsic ethics before.
Previous generations of mathematicians have evaded
this crucial point, and in the process have possibly letsociety down. It rests on the current and upcoming
generations to pick up this idea, before it’s too late.
Mathematicians always take a generation or more toaccept a new and fundamental idea about the natureof their subject; debates about the admissibility ofzero as a number provide such an example. We’reat a similar juncture again. Now some say “Surely
there’s no use in considering ethical issues in mathematics”, but by the time our students are professors
and industry leaders, they may well be saying “Ofcourse we should be considering ethical issues inmathematics!” But why hasn’t the mathematical community taken this on board already? Why wasn’t thisdone 100 years ago, by the likes of Gödel and Russell?
Two reasons come to mind. Firstly, the dangers wereless proximate, since much of today’s technology
simply did not exist. Secondly, every mathematics
undergraduate was already exposed to philosophy,
as it formed part of every university education. Thus,
exposure to Ethics in Mathematics, in its own right,
was less urgently needed.
So if Ethics in Mathematics has become so important
to mathematicians, then how might we teach it tothem in a relevant and useful way, without foistingan entire philosophy degree upon them? Disciplines
such as law, medicine, and engineering have longtaught their undergraduate students about extrinsic
ethics in their respective fields. In the next issueof the Newsletter, we’ll further explore why suchteaching has not yet occurred in mathematics, andoutline how one might go about giving such directedteaching of Ethics in Mathematics.

Acknowledgements
We wish to thank Piers Bursill-Hall and Dennis Müller
for their valuable discussions over the past few years,
and Dennis in particular for his comments and suggestions on previous versions of this article.

FURTHER READING[1] M. Chiodo, P. Bursill-Hall, Four levels of ethical engagement, Ethics in Mathematics Discussion
Papers, 2018/1 (2018).
[2] Systems and methods for using online social
footprint for affecting lending performance andcredit scoring, Patent No. US 8,694.401 B2 , 8 April
2014.
[3] Cambridge Analytica: how did it turn clicks into
votes?, theguardian.com, tinyurl.com/yy89e5vl,
May 6th 2018.
[4] Secrets of Silicon Valley, Series 1 Episode 2:
The Persuasion Machine, see recording at 30:17.
BBC documentary, 2017

Maurice Chiodo
Maurice is a postdoctoral research fellow
at the University ofCambridge, and lead
investigator of the Cambridge University Ethics
in Mathematics Project(ethics.maths.cam.ac.uk). He’s developing a teaching
programme and curriculum to teach mathematicians
about the ethical implications of their work.

Toby Clifton
Toby is a recent graduate of astrophysics,
and is the current president of the Cambridge
University Ethics inMathematics
Society(cueims.soc.srcf.net).

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Polynomial Factorisation Using Drinfeld Modules
ANAND KUMAR NARAYANAN

The arithmetic of Drinfeld modules has recently yielded novel algorithms for factoring polynomials over finite
fields; a computational problem with applications to digital communication, cryptography and complexity
theory. We offer a gentle invitation to these developments, assuming no prior knowledge of Drinfeld modules.
Factoring polynomials modulo a prime
Let Fq denote the finite field of integers modulo an
odd prime number q . Polynomials Fq [x] over Fq share
striking analogies with integers, yet we begin withan algorithmic distinction. While factoring integers
remains a notoriously difficult problem, factoring polynomials in Fq [x] is long known to be easy, at least
with access to randomness.
Polynomial factorisation over finite fields is not amere curiosity, but has many applications. In number theory, finite fields arise as residue fields ofglobal fields such as number fields. While determining the splitting of a prime in a number field,
one factors a polynomial defining the number field
modulo the prime. Several instances of polynomial
factorisation appear while factoring integers using
quadratic/number field sieve algorithms or while performing index calculus to compute discrete logarithms, both foundational problems in analysing thesecurity of cryptographic systems. In digital communication, polynomial factorisation aids the construction of certain error correcting codes (BCH andcyclic redundancy codes), structures vital to reliabletransmission of information in the presence of noise.
Let us recount a simple polynomial factorisation
algorithm. We are given a monic f(x) ∈ Fq [x] of
degree n whose factorisation into irreducible polyQ
nomials f(x) = im=1 pi (x) is sought. Assume f(x)is square free, that is the pi (x) are distinct. This iswithout loss of generality for there are algorithms torapidly reduce to this special case. It takes n log2 q
bits to write down f(x) and we seek algorithms thatrun in time polynomial in n and log q . Berlekamp was
the first to show there is a randomized polynomial
time algorithm, but we follow a different two step
process.

uct of irreducible polynomials of the same degree.
Q
By Fermat’s little theorem, x q − x = a ∈Fq x − a. To
extract the product of linear factors of f(x), take thegreatest common divisor gcd(x q − x, f(x)). To extract
products of degree two factors, degree three factors
and so on iteratively, look to the succinct expressionYdxq − x =p(x)p:deg(p)|dfor the product of monic irreducible polynomials p(x)of degree dividing d . At the d th iteration, with smallerddegree factors already removed, gcd(x q − x, f(x))
yields the product of degree d irreducible factors.
dCare in handling x q is required for its degree is expod
nential in n log q . All we need is x q mod f(x), easily
accomplished by a sequence of q th powers modulo
f(x), each performed by repeated squaring. Betterdstill, x q mod f(x) can be rapidly computed with afast algorithm to compose two polynomials modulo
f(x) in concert with q th powers. Kaltofen and Shoupdevised an ingenious improvement over this naive
iteration resulting, in a significant speed up. The
Kaltofen–Shoup algorithm implemented using themodular composition algorithm of Kedlaya–Umans
performs distinct degree factorisation with run time
exponent 3/2 in the degree n and 2 in log q .
Equal degree factorisation

Distinct degree factorisation

Distinct degree factorization leaves us with the problem of factoring polynomials all of whose irreducible
factors are of the same known degree d . All known
algorithms for this task with polynomial runtime inlog q are randomized. Even for the simplest caseof factoring a quadratic polynomial into two linear
factors, no unconditional deterministic polynomial
time algorithms are known. It is closely related to the
problem of finding a quadratic nonresidue modulo agiven large prime.

The first step, known as distinct degree factorisation,
decomposes f(x) into factors each of which is a prod-

The following randomized algorithm can be traced
backed to ideas of Gauss and Legendre. For a uni-27

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formly random a(x) of degree less than n,
q d −1!

gcd a(x) 2 − 1 , f(x)gives a random factorisation of degree d irreducible factors. This follows since raising a(x) to the(q d − 1)/2-th power modulo a degree d irreducible
polynomial results in either 1 or −1, depending onwhether a(x) reduces to a quadratic residue or not.
Remarkably, this computation can be performed withrun time exponent 1 in the degree n using an algorithm of von zur Gathen and Shoup implemented
with the aforementioned Kedlaya–Umans modular
composition.
In summary, the best known polynomial factorisation algorithms have run time exponent 3/2 in the
degree with the bottleneck being distinct degree factorisation. To lower this exponent is an outstanding
problem. In fact, to lower this exponent, it suffices
for there to be an algorithm that merely estimates
the degree of some irreducible factor.

Drinfeld modules and polynomial factorisation
The use of Drinfeld modules to factor polynomials
over finite fields originated with Panchishkin andPotemine [3]. Drawing inspiration from Lenstra’s
elliptic curve integer factorization, they recast therole of the group of rational points on random elliptic
curves modulo primes with random finite Drinfeld
modules.
We describe three Drinfeld module based algorithms
for polynomial factorisation. The first two weredevised in [2] and the third in [1]. The first estimates factor degrees using Euler–Poincaré characteristics in hopes of speeding up distinct degree
factorisation. The second is a Drinfeld analogue of
Lenstra’s algorithm, closely related to the aforementioned algorithm of Panchishkin and Potemine [3].
Our exposition begins with a short account of finite Drinfeld modules followed by Euler–Poincaré thecharacteristic and Frobenius distributions, important
ingredients in the first two algorithms. The third
algorithm involves Drinfeld modules with complex
multiplication with an analogue of Deligne’s congruence playing a vital role. It is also the fastest of thethree algorithms, with runtime complexity matching
the best known algorithms, in theory and practice.
The hope is, the rich arithmetic of Drinfeld modules
will inform new algorithms to beat the 3/2 exponent
barrier.

Finite Drinfeld modules
Drinfeld introduced the modules bearing his name asan analogue of elliptic curve complex multiplication
theory. He in fact called them elliptic modules. Drinfeld modules and their generalisations have playeda crucial role in the class field theory of function
fields and in proving the global Langlands conjecture
over function fields for G Ln . We settle for a concrete simple notion of Drinfeld modules sufficientfor our context. Throw the q th power Frobenius σinto Fq [x] resulting in Fq (x)hσi, the skew polynomial
ring with the commutation rule σ u(x) = u (x)q σ, for
all u(x) ∈ Fq [x]. A rank-2 Drinfeld module over Fq (x)is (the Fq [x] module structure on the additive groupscheme over Fq (x) given by) a ring homomorphism
φ : Fq [x] −→ Fq (x)hσi

x 7−→ x + gφ (x) σ + ∆φ (x) σ 2

for some gφ (x) ∈ Fq [x] and nonzero ∆φ (x) ∈ Fq [x].

To better understand the map, it is instructive tocompute by hand to where x 2 , x 3 and so on, getmapped. By design, b(x) maps to a polynomial in σwith constant term b(x),b(x) 7−→ φb := b(x) +2 deg(b)Xi =1

φb,i (x)σi .

Consider an Fq [x] algebra M . In our algorithms to factor f(x), M will often turn out to be M = Fq [x]/(f(x)).
One way to make an Fq [x] algebra M into an Fq [x]module is to retain the addition and scalar multiplication but simply forget the multiplication. The Drinfeld
module φ endows a new Fq [x] module structure to M
by twisting the scalar multiplication. For b(x) ∈ Fq [x]and a ∈ M , define the scalar multiplicationb(x) ? a := φb (a) = b(x)a +2 deg(b)Xi =1i

φb,i (x)a q ,where the arithmetic on the right is performed in
the Fq [x] algebra M . Let φ(M ) denote the new Fq [x]module structure thus endowed to M .
Euler–Poincaré characterisitic

Cardinality is an integer valued measure of the size ofa finite abelian group (equivalently, a finite Z-module).
A convoluted definition is to assign as the cardinalityof a cyclic group of prime order the corresponding
prime, and for cardinality of finite abelian groups that
sit in an exact sequence to be multiplicative. The

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FEATURES
Euler–Poincaré characteristic χ is an Fq [x]-valued
cardinality measure of a finite Fq [x] module definedcompletely analogously. For a finite Fq [x] module A,
χ(A) ∈ Fq [x] is the monic polynomial such that:• If A  Fq [x]/(p(x)) for a monic irreducible p(x),then χ(A) = p(x).
• If 0 → A1 → A → A 2 → 0 is exact, then χ(A) =χ(A1 ) χ(A2 ).
For the Fq [x] module φ(Fq [x]/(f(x))) featuring in
our algorithms, the Euler–Poincaré characteristic
χ(φ(Fq [x]/(f(x)))) has a simple linear algebraic interpretation: the characteristic polynomial of the linear
map φx on Fq [x]/(f(x)). In particular, it is a degree npolynomial that can be computed efficiently.
Frobenius distribution of Drinfeld modules

Let us put our newly defined Fq [x] modules and cardinality measure χ to use. Take an elliptic curve E
over the rational numbers and reduce it at a prime p.
The Fp -rational points E(Fp ) famously form a finite
abelian group with cardinality p + 1 up to an error
determined by the Frobenius trace tE,p . The Hasse
bound, considered the Riemann hypothesis for ellip√
tic curves over finite fields, asserts that |tE,p | ≤ 2 p.
Thereby,#(E(Z/(p))) = p + 1 −

tE,p
|{z}

.

√

√
−2 p ≤ ≤2 p

Gekeler established the following Drinfeld module
analogue. Take a Drinfeld module φ, a monic irreducible polynomial p(x), and consider the resulting
Fq [x] module φ(Fq [x]/(p(x))). Its Euler–Poincaré characteristic equals .
χ φ Fq [x] (p(x)) = p(x) + tφ,p (x) ,|{z}≤deg(p)/2

which is p(x) plus an error determined by the Frobenius trace t φ,p (x) of degree at most half that of p(x).
The analogy with the Hasse bound is striking. The
error in each case takes roughly half the number ofbits as the estimate.
Factor degree by Euler–Poincaré characteristic
Gekeler’s bound concerns Drinfeld modules φ at an
Qirreducible p(x). What happens at f(x) = i pi (x),our polynomial to factor? The multiplicativity of theEuler–Poincaré characteristic implies .
 Y  .
χ φ Fq [x] (f(x)) =χ φ Fq [x] (pi (x))
i

=

Y
i


pi (x) + tφ,pi (x) = f(x) + tφ,f (x),for some tφ,f (x) of degree at most s f /2, where s fdenotes the degree of the smallest degree factor off(x). Thus, we have an extension of Gekeler’s boundto reducible polynomials .
χ φ Fq [x] (f(x)) = f(x) + tφ,f (x),|{z}≤s f /2

implying f(x) and χ(φ(Fq [x]/(f(x)))) agree at the highdegree coefficients! The number of agreements tells
us information about the smallest factor degree.
For a randomly chosen φ, tφ,f (x) likely has degree
exactly bs f /2c (with probability at least 1/4). The
number of agreements does not merely bound but
determines the degree of the smallest factor. To
claim this probability, one needs to prove for a randomly chosen φ, the Frobenius traces corresponding
to the irreducibles of smallest degree do not conspireyielding cancellations. To this end, we seek equidistribution formulae for the Frobenius traces. Analogous
to elliptic curves, there is a correspondence between
the number of isomorphism classes of Drinfeld modules with a given trace and Gauss class numbers incertain imaginary quadratic orders. The latter can becomputed using analytic class number formulae.
An algorithm to estimate the degree of the smallest
degree factor of a given f(x) is now apparent. Picka Drinfeld module φ (by choosing gφ, ∆φ at randomof degree less than n). Compute the Euler–Poincaré
characteristic χ(φ(Fq [x]/(f(x)))) and count the number of high degree coefficients it agrees in with f(x).
Drinfeld module analogue of Lenstra’s algorithm
It is instructive to begin with Lenstra’s elliptic curve
integer factorisation algorithm before seeing its Drinfeld module incarnation. Pollard designed his p-1
algorithm to factor an integer that has a prime factor
modulo which the multiplicative group has smoothorder. But this smoothness condition is rarely met.
Lenstra recast the role of the multiplicative groupwith the additive group associated with a random
elliptic curve. If the integer has a prime factor modulo which the randomly chosen elliptic curve has

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FEATURES
smooth order, the algorithm succeeds in extractingthat factor. For a random elliptic curve, this smoothness condition is met with a probability depending
sub-exponentially on the size of the smallest prime
factor. Consequently, it is among the most popularalgorithms for integer factorisation, particularly as
an initial step to extract small factors.
Pollard’s p − 1 algorithm

Fix a positive integer B as the smoothness boundand denote by m, the product of all prime powers
at most B. Given an N to factor, choose a positive
integer a < N at random. Assume a is prime to N
for otherwise gcd(a, N ) is a nontrivial factor of N .
If N has a prime factor p with every prime power
factor of p − 1 at most B,
a m −1 = (a p−1 )m/(p−1) −1  0

mod p =⇒ p | a m −1

and gcd(a m − 1, N ) is likely a nontrivial factor of N .
The running time is exponential in the size of B.
For typical N , B needs to be as big as the smallest
factor of N and thus the running time is typicallyexponential in the size of the smallest factor of N .
Lenstra’s algorithm

Lenstra’s elliptic curve factorization algorithm factors
every N in (heuristic) expected time sub-exponential
in the size of the smallest factor p of N . A key insightof Lenstra was to substitute the multiplicative group(Z/pZ)× in Pollard’s p − 1 algorithm with the group
E(Fp ) of Fp rational points of a random elliptic curve
E over Fp . The running time depends on the smoothness of the group order |E(Fp )| for a randomly chosen
E. The Hasse bound guarantees that ||E(Fp )| − (p +√1)| ≤ 2 p and Lenstra proved that his algorithm runs
in expected time sub-exponential in the size of p assuming a heuristic on the probability that a random√√
integer in the interval [p + 1 − 2 p, p + 1 + 2 p] issmooth.
Drinfeld module analogue

In the ensuing Drinfeld module version, random elliptic curves will be recast with random Drinfeld modules to factor polynomials. As before, let f(x) ∈ Fq [x]denote the polynomial to factor. Pick a Drinfeld module φ at random and a random element α in theFq [x] module φ(Fq [x]/(f(x))). The order, Ord(α), of
α is the smallest degree polynomial b(x) ∈ Fq (x) thatannihilates α, that is φb (α) = 0. Extract and divide
away the linear factors of Ord(α),r(x) := Ord(α)/ gcd(Ord(α), x q − x)and apply the Drinfeld action φr (α) on α. It is likelythat φr (a) is a zero at some but not all factors pi (x)of f(x) and gcd(φr (a), f) gives a random factorisation.

A brief outline of why this is the case follows. As withgroups, the order of an element divides the cardinality of an Fq [x] module. That is, Ord(α) divides theEuler–Poincaré characteristic of φ(Fq [x]/(f(x))). In
fact, with high probability, Ord(α) equals the Euler–
Poincaré characteristic
Ord(α) =Yi
pi (x) + tφ,pi (x) .

If the factors on the right have/don’t have a linear factor independently and roughly uniformly at random,
then the algorithm yields a random factorisation. This
is indeed the case!
The factors on the right lie in the short intervals
Ii centred at pi (x) with interval degree boundedby deg(pi )/2. The Frobenius trace distribution assures a certain semi-circular equidistribution ofpi (x) + tφ,pi (x) in the short interval Ii . Remarkably,
unlike over integers, factorisation patterns in the
short intervals Ii are unconditionally proven to berandom enough. In summary, for random φ, the factorisation patterns of each pi (x) + tφ,pi (x) is like thatof a random degree deg(pi ) polynomial. Further, they
are independent.
The computation of Ord(α) dominates the runtime
and can be performed efficiently through linear
algebra. This is in stark contrast to integers, wherefinding the order of an element in the multiplicative
group modulo a composite appears hard.

Drinfeld modules with complex multiplication:
Our last algorithm is distinguished in that it samples
from Drinfeld modules with complex multiplication.
A Drinfeld module φ has complex multiplication if its
endomorphism ring
EndFq (x) (φ) ⊗Fq [x] (Fq (x))

is isomorphic to a quadratic extension of Fq (x). A typical Drinfeld module has an endomorphism ring onlyisomorphic to Fq (x). Complex multiplication is therare case where the Drinfeld module is more symmetric than typical. As with reducing elliptic curves overrational numbers at primes, Drinfeld modules canbe reduced at irreducible polynomials. The reduction

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FEATURES
is deemed supersingular if the endomorphism ringis noncommutative, and ordinary otherwise. Every
Drinfeld module with complex multiplication has theremarkable feature that the density of irreducible
polynomials where it is supersingular is roughly half.
To factor a given polynomial f(x), the strategy is
to choose a random Drinfeld module with complex
multiplication. Using explicit formulae, construct a
Drinfeld module φ with complex√ multiplication by
the quadratic extension Fq (x)( x − c ) with c ∈ Fq
chosen at random. Then attempt to separate out theirreducible factors of f(x) where φ is supersingularfrom the ordinary. This likely results in a random
factorisation of f(x), which is recursively factored toobtain the complete factorisation.
To separate the supersingular factors, we look to the
Hasse invariant, an indicator of supersingularity. The
Hasse invariant h φ,p ∈ Fq [x]/(p(x)) of φ at an irreducible p(x) vanishes if and only if φ is supersingularat p(x). For the chosen φ, we construct a polynomialthat is a simultaneous lift of Hasse invariants at all
irreducible polynomials of degree at most that of f(x).
The efficient construction of this lift relies criticallyon a Drinfeld module analogue of Deligne’s congruence due to Gekeler. The common irreducible factorsof this lift and f(x) are precisely the irreducible factors of f(x) where φ is supersingular. The GCD of f(x)and the lift separates out the supersingular factors
from the ordinary, as desired.
Hasse Invariants and Deligne’s congruence:

For a Drinfeld module φ with defining coefficients(gφ, ∆φ ), we now construct the aforementioned liftof Hasse invariants. Consider the sequence rφ,k (x) ∈Fq [x] of polynomials indexed by k starting with
rφ,0 (x) := 1, rφ,1 (x) := gφ (x) and for m > 1,
q m−1rφ,m (x) := gφ (x)rφ,m−1 (x)q m−2m−1−(x q −x) ∆φ (x)rφ,m−2 (x).
Gekeler showed that rφ,m (x) is the value of the normalized Eisenstein series of weight q m − 1 on φ andestablished Deligne’s congruence for Drinfeld modules, which ascertains for any p of degree k ≥ 1 with∆φ (x) , 0 mod p that h φ,p = rφ,k (x) mod p(x).

Hence rφ,k (x) is a lift to Fq [x] of the Hasse invariantsof φ at not just one but all irreducible polynomialsof degree k . Further, rφ,k (x), rφ,k +1 (x) are both zeroprecisely modulo the supersingular p(x) of degree at
most k . Since a factor of f(x) is of degree at mostn, takegcd(rφ,n (x), rφ,n+1 (x))

as the Hasse invariant lift.
To show the algorithm indeed works, it remains todemonstrate that with constant probability, our random choice of φ with complex multiplication yielding
a random factorization of f(x). By complex multiplication theory and Carlitz reciprocity, this probability
is identified with splitting probabilities in a certain hyperelliptic extension of Fq (x), and duly bounded. The
overall runtime is dictated by the time taken to compute the Hasse invariant lift. An intricate algorithm
for this task is devised in [1] using a fast procedure
to compute its defining recurrence. Remarkably, theruntime exponent matches the best known factorisation algorithm and is comparable in practice tothe fastest existing implementations. In light of this,thorough further investigation of Drinfeld module
inspired polynomial factorisation is warranted!
Acknowledgements
This work was supported by the European Union’s
H2020 Programme (grant agreement #ERC-669891).
FURTHER READING[1] J. Doliskani, A. K. Narayanan, É. Schost, Drinfeld Modules with Complex Multiplication, Hasse
Invariants and Factoring Polynomials over Finite
Fields, J. Symbolic Comput., to appear.
[2] A. K. Narayanan, Polynomial factorization overfinite fields by computing Euler–Poincaré characteristics of Drinfeld modules, Finite Fields Appl. 54(2018) 335–365.
[3] A. Panchishkin and I. Potemine, An algorithm
for the factorization of polynomials using elliptic
modules, Constructive methods and algorithms in
number theory, p. 117. Mathematical Institute of
AN BSSR, Minsk, 1989 (Russian).

Anand
Kumar
Narayanan
Anand is a research scientist at the Laboratoired’Informatique de Paris
6, Sorbonne Université.
His research interests
are in the algorithmic
aspects of number theory guided by applications incryptography, coding theory and complexity theory.
Anand was born in India and in his spare time enjoys
wandering jungles and mountains.

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1

Computers and Mathematics
Computers and mathematics
KEVIN BUZZARD
Kevin Buzzard
Mathematicians currently use computers to do tedious calculations which would be unfeasible to do by hand.
Abstract.
Mathematicians
currently
to do tedious
which
would
be unfeasible
In the future,
could they behelpinguse
us computers
to prove theorems,
or to calculations
teach students
howto write
proofs? to doby hand. In the future, could they be helping us to prove theorems, or to teach students how to write proofs?
Mathematics from the future
Mathematics from the future
Take a look at the following piece of computer code.
Take a look at the following piece of computer code.
lemma continuous_iff_is_closed{f : α → β} :continuous f ↔ (∀s, is_closed s →is_closed (f −1 ’ s)) :=hassume hf s hs, hf (−s) hs,
assume hf s, by rw [←is_closed_compl_iff,←is_closed_compl_iff]; exact hf _i
ThisThis isis aa proof
proof thatthat aa function
function isis continuous
continuous ifif andand
only
only if
if thethe preimage
preimage ofof everyevery closed
closed set
set isis closed.
closed.
It
It is
is written
written in
in aa computer
computer language
language called
called Lean,
Lean, aaprogramming
programming language
language developed
developed byby Leonardo
Leonardo dede
Moura
Moura at
at Microsoft
Microsoft Research.
Research. There
There isis aa “:=”“:=” somesomewhere
where inin thethe middle
middle ofof thethe code;code; thethe statement
statement ofofthetheoremisbeforethistoken,
andthe theorem is before this token, and itit isis basicallybasically
human-readable.
human-readable. The
The proof
proof ofof thethe theorem
theorem isis afterafter
thetoken,
andisincomprehensible
if
you
do
the token, and is incomprehensible if you do notnotknow
computerwithLeaninknow how
how totospeak
speakLean.
Lean.A A
computerwith
Leanstalled
would
have
no
troublecheckingthat
thisinstalled would have no trouble checking that this aisvalid
a validproof,
proof,andanda ahuman
humanmathematician
mathematician who
who can
can
speak
Lean
would
vouch
for
the
fact
that
the
speak Lean would vouch for the fact that the proof
proof
isis justjust thethe obvious
obvious proof.
proof.
We
We seesee fromfrom this
this example
example thatthat computers
computers can
can now
now
checkproofsofsimpletheorems
inpuremathematcheck proofs of simple theorems in pure mathematics,
ics, asas longlong asas youyou areare preparedprepared toto learn
learn how
how toto talk
talk
to
them
in
their
own
specialised
language.
Infact
to them in their own specialised language. In fact thisthishas
has been
been thethe casecase forfor decades,
decades, andand evidence
evidence has
has
appeared more recently that computers can checkappeared more recently that computers can checkquite complicated proofs. One observation which I
quite complicated proofs. One observation which I
find particularly striking is that even though thesefind particularly striking is that even though thesecomputer proof verification systems give us a very
computer proof verification systems give us a very
cool new way of doing mathematics, which also has
cool new way of doing mathematics, which also hasapplications to teaching and outreach (indeed I now
applications to teaching and outreach (indeed I now
use it for both), the number of people in mathematuse it for both), the number of people in mathematics departments across the UK who can use this sort
ics departments across the UK who can use this sortof software is extremely small. Moreover, most ofof software is extremely small. Moreover, most ofthem are doing research in foundational areas suchthem are doing research in foundational areas suchas category theory or type theory rather than moreas category theory or type theory rather than more
“mainstream” areas in pure mathematics such as“mainstream” areas in pure mathematics such as
analysis, topology and so on. Were software like thisanalysis, topology and so on. Were software like thistoto bebe adopted
adopted byby aa broader
broaderclassclassofofmathematicians,
mathematicians,
we
we might
might seesee aa future
future where
where these
these systems
systems start
start
toto becomebecome useful
useful forfor aa broader
broaderclassclassofofresearchers
researchers
too.
too.
In
In thisthis article
article we
we will
will seesee anan overview
overviewof
ofwhy
whythesethesesystems
systems exist
exist andand whatwhat they
they arearecurrently
currentlycapable
capableof.
are getting
getting better,better,faster,
faster,and
andsmarter
smarterevery
evof. They
They areery
I believe
is onlya matterofyear,year,
and and
I believethatthat
it is itonly
a matterof timetime
mathematicans
willforced
be forced
sitandup anduntil until
mathematicans
will beto sittoup
take
take
notice.
Note howeverthat computersnotnotice.
Note howeverthat computerswill not will
be provbe
theorems
by themselves
time
ingproving
theorems
by themselves
any timeanysoon.
All soon.
of theAllof the mathematics
I haveLean myself
ismathematics
I have donein doneLean inmyselfis mathemathematics
which
I
could
have
done
(in
many
cases
matics which I could have done (in many cases more
more
paper.
difference
easily)easily)with withpen penand andpaper.
TheThekeykeydifferenceofofcourse
is
that
with
the
pen
and
paper
method,course is that with the pen and paper method,
I could
Ihave
couldmadehave slips,
madeorslips,
or forgotten
to everything
check evforgotten
to check
erything
which
needs
checking;
Lean
will
make
sure
that needs checking; Lean will make sure everything
everythingiscorrect
beforeitwillcompileyour
proof.
is correct before it will compile your proof. There
There
are typecheckers
other languages
are typecheckers
writtenwritten
in otherinlanguages
that can
that
can
check
Lean
proofs,sothechances
errorscheck Lean proofs, so the chances of errorsofslipping
slipping
throughthesystemnowadaysareremote–through the system nowadays are remote — cercertainly
an
order
of
magnitude
less
than
the
chancetainly an order of magnitude less than the chancethat
thataahuman
humanwill
willmakemakeaamistake.
mistake.As
AsI Igetgetolder,older,I Ifind
find
II valuethiscertainty
more.
Perhaps
this
is
some
value this certainty more. Perhaps this is somekindkindofof mid-life
mid-life crisis—but
crisis — butLean
Leanisisfree
freeand
andopenopensource
sourcesoftware,
and
hence
much
cheaperthansoftware, and hence much cheaper than buying
buyingaa
Ferrari.
Ferrari.

Proofs
Proofs andand computer
computer programs
programs
Most of us now rely on computer code to do all
Most of us now rely on computer code to do all sorts
sorts of things. Bugs in code can cause chaos, alof things. Bugs in code can cause chaos, allowing
lowing hackers into systems or just making stuff
hackers into systems or just making stuff stop workstop working. It is hence not surprising that softwareing. It is hence not surprising that software develdevelopers have made tools which can check comopers have made tools which can check computer
puter programs for bugs. But what is a computer
programs for bugs. But what is a computer program?
program? It is just is a sequence of statements folIt is just is a sequence of statements following alowing a precise syntax, which performs a certain
precise syntax, which performs a certain task whentask when the statements are interpreted within thethe statements are interpreted within the framework
framework of a particular programming languageof a particular programming language (say, Python(say, Python or C++). A mathematical proof is also a

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FEATURES
or C++). A mathematical proof is also a sequenceof statements following a precise syntax, which performs a certain task when the statements are interpreted within the logic and language of pure mathematics. This observation is more than an analogy
and goes back to Kolmogorov.

expressed concern that this part of the argument
was not human-checkable. They have now all beenformally verified by computer.

What mathematics is formalized?

The system Coq was the system used in George
Gonthier’s formal proof of the four colour theorem,
and Gonthier also led the team which in 2013 formalized (again in Coq) the proof of the Feit–Thompson
theorem that every group of odd order is solvable.
These two theorems are of different natures. The
four colour theorem is another example of a theorem where the standard argument ends up having
to go through thousands of tedious case checks, but
this is not what the Feit–Thompson proof looks like;the latter proof is a long group-theoretic argument
for which Thompson recieved a Fields Medal in 1970.
The modern mathematical proof of the odd ordertheorem is written up in two books by Peterfalvi and
Bender–Glauberman, both published in the LMS lecture note series [1, 3]. The Coq proof was created bya team of 15 people over a period of 6 years, during
which time they formalized the 328 pages of researchlevel mathematics in the two books as well as all theprerequisites. Perhaps surprisingly, over two thirdsof the code base is the prerequisites for the books
rather than the books themselves. The prerequisites
covered basic results in the representation theoryof finite groups, Galois theory, some commutative
algebra, and a little number theory. Moreover, mostof the people involved did not work anywhere near
the full six years that the project lasted. Ultimately
there were under five lines of computer code corresponding to each line of the book on average, a figure
which perhaps better reflects the effort, although it
should be said that writing a correct line of code in
one of these languages can take a long time.

What however can we prove in practice in Lean or inthese other systems? Which results are already available to us? Let us start at the top, and go throughsome of the traditional breakthrough results.

The work of Hales et al., and Gonthier et al., are hence
evidence that these computer systems are now capable of handling very long and complex proofs about
relatively simple objects such as finite groups or
spheres.

The systems HOL Light and Isabelle/HOL were usedin the formally verified proof of the Kepler conjecture (“HOL” stands for Higher Order Logic). The formal verification was done by a team led by Tom
Hales. Hales’ 2017 talk at the Big Proof workshop
at the Newton Institute is available on the INI website (see tinyurl.com/y4avuzj9) and tells the interesting story of how he ended up having to formalize
the paper proof which he and Ferguson had constructed in the late 1990s. The argument involved, atsome point, checking thousands and thousands ofnonlinear inequalities, and the human referees had

In 2016, Ellenberg and Gijswijt established a newupper bound for the cap set problem, resolving thecap set conjecture. The cap set conjecture is a combinatorial question about the growth of the size ofa certain sequence related to an n-dimensional version of the card game “Set”, as n increases. The
Ellenberg–Gijswijt proof was published in the Annalsof Mathematics. In 2019 Sander Dahmen, Johannes
Hölzl and Rob Lewis formalized the proof in Lean.
Dahmen is a number theorist; Hölzl’s backgroundis in computer science and Lewis’s is in logic. The
mathematical proof was remarkable in the sense

Initially it was observed that certain kinds of mathematical proofs (namely, proofs using something called“constructive mathematics”) really are exactly thesame thing as certain kinds of computer programs.
As a consequence, the tools developed by computerscientists to verify that code has no bugs can also
be used to show that a proof in constructive mathematics has no gaps or errors.
However, often mathematicians do not work purely
in constructive mathematics. One key difference
between a proof and a program is that proofs areallowed to use non-constructive things such as theaxiom of choice, which can be viewed in this optic
as a function for which there is no algorithm, i.e.,
no program. But some of the newer computer proof
checking tools (such as Lean) have been designedwith mathematics in mind, and have simply added
things like the axiom of choice as an axiom of thesystem. This means that tools like Lean can now
check “normal” mathematical proofs. In short, if you
can prove something using classical logic and ZFC set
theory (which is what most pure mathematicians justcall “mathematics”) then you can, in theory, prove it
in Lean.

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that many of the methods used were elementary,
making the result a very good candidate for formalisation. The Lean proof is modern mathematics being
formalized in a realistic time frame.
The next example I will speak about here is of adifferent nature; it is the work of Sébastien Gouëzel
and Vladimir Shchur from [2]. The Morse Lemma
is a result about the Hausdorff distance between ageodesic and a quasi-geodesic with the same endpoints in a Gromov-hyperbolic space. In 2013 Shchurgave a quantitative (i.e., with explict constants) proofof the lemma, which was published in the Journalof Functional Analysis. Gouëzel decided to formalize
the proof in Isabelle/HOL, and during the process hefound a serious error missed by both author andreferee. A completely new argument was found byGouëzel and Shchur to fix the problem, and Gouëzel
formalized the new proof in Isabelle/HOL. The result
was written up as a joint paper which was publishedin the Journal of Functional Analysis earlier this year.
Again this is modern mathematics being formalized
in a realistic time frame, but this time the objects
are more complex.
Last year I got interested in exactly how complex theobjects handled by this sort of software could be.
Perfectoid spaces were discovered by Peter Scholze
in 2011 and his applications of the theory to fundamental questions in the Langlands program andarithmetic geometry won him a Fields Medal in 2018.
Earlier this year Patrick Massot, Johan Commelin andmyself formalized the definition of a perfectoid spacein Lean. The formalisation process took around ten
months in total, and we wrote well over 10,000 lines
of code in order to formalize this definition, involving(for example) formally proving many of the theorems in Bourbaki’s “Topologie Générale”. We haveas yet proved nothing at all about perfectoid spaces
and furthermore cannot yet produce any non-trivial
examples, because we still need to formalize the general theory of discrete valuations on finite extensionsof the p-adic numbers. Formalising this theory would
be an ideal problem for a PhD student who wanted
to learn the material in an innovative way, however
it would be far less efficient in terms of time.

Undergraduate level formalized mathematics
As well as these “complex” formalisations, there areformalisions of much of the pure material which istypically taught in a UK undergraduate mathematics

degree. Formalising undergraduate level mathematics
is a very different ball game. It takes far less time,and can be worked on by undergraduates who arelearning about the subject for the first time. All of
the systems I have mentioned so far contain a lotof undergraduate-level mathematics, but one cannot
easily port theorems from one system to another(the problem is analogous to porting computer programs from one language to another), so when new
systems come along these libraries need to be built
from scratch. Each system contains some, but not all,of the pure mathematics component of a typical UK
undergraduate degree; the union of all the systems
might well contain most of it nowadays, however
because of the portability issue no one system hasaccess to all of it.
I started learning Lean by formalising the questions
and solutions in Imperial’s first year “introduction to
proof” course, a course which I was lecturing at thetime. I typically got stuck several times per question,
I asked hundreds of questions in the Lean chatroom on Zulip and they were answered promptly andpolitely, often by computer science PhD students andpost-docs. When I had got the hang of things a little
more, I started encouraging Imperial’s undergraduates to join in, and we have made good progress.
Over the last two years, mathematics undergraduates at Imperial have been formalising results which
they have been taught during their degree, or learnt
independently. For example the theorem of quadratic
reciprocity, Sylow’s theorems, the fundamental theorem of algebra, matrices, bilinear maps, the theoryof localisation of rings, the sine, cosine and exponential functions, tensor products of modules and manyother things have been implemented by Imperial’s
mathematics undergraduates in Lean. Chris Hughes
and Kenny Lau in particular are two students who
have formalized many results from our undergraduate curriculum. There is more to come; we areworking on the Galois theory course and after that
we will start on the representation theory course.
One possible future project would be to turn someof these formalisation projects into some sort offormally verified book or website; imagine a book on
Galois theory which was guaranteed to contain noerrors, and which every proof which the reader did
not follow could be “unfolded” more and more until
the penny dropped.
In 2018–19 I also supervised BSc student Anca
Ciobanu, who formalized the basics of group cohomology in Lean, and MSc student Ramon Mir who
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Lean. I was very surprised to discover that schemes
had not been done in any other formal system. As
a result it seems that Mir’s work is of a publishable
standard. Nowadays we see students typing up theirlecture notes in LaTeX; it would be wonderful to see,in the future, students typing up their course notesin Lean or one of these other systems.

Using Lean for teaching.

just as knowledge of LaTeX now is. Another purpose
is an ongoing project to completely digitise the puremathematics part of Imperial’s undergraduate curriculum.
I have also used Lean for outreach, showing
schoolkids how to prove things like 2 + 2 = 4 fromthe axioms of Peano arithmetic and then raising thequestion of how much more mathematics can bedone like this.

In the 2018–19 academic year, I taught Imperial’s
introduction to proof course again, this time using
Lean live in the lectures for some of the proofs. For
example, I proved that a composite of surjective functions was surjective in Lean, I proved various results
about equivalence relations in Lean, and I proved
Cantor’s diagonal argument in Lean. I wrote the code
live during the lectures, projecting my laptop onto
the screen, but I also circulated “traditional” proofs
in the course notes afterwards. All of this was done
under the watchful eye of Athina Thoma, a specialist
in mathematical education, who interviewed and surveyed the students and is writing up her conclusions
for a forthcoming publication with Paola Iannone,
another such specialist based in Loughborough. Thisapproach was completely new to both me and thestudents, and I note with interest that their performance on the (conceptually complicated) question
about binary relations on the end of year exam wasdone very well. Of course one has to be careful here;
not all the students will want to see mathematics
being done with computers, and of course thesestudents should not be disadvantaged by such an
approach. However I did think it was important thatthe students learnt, as early as possible, what a logical argument looked like, and how a logical argument
is something which can be explicitly checked by analgorithm. This is not how mathematics is taught inschools but it seems to me to be a crucial message
which an introduction to proof course in a mathematics department should convey. Patrick Massot isalso using Lean in the mathematics department at
Orsay, in his introduction to proof course.

Currently, the only thing that Lean can offer the
researcher is a “certificate” that any theorem they
have formalized in Lean is indeed correct. Many
humans feel that they do not need this certificate. I
find it interesting that more and more mathematics
relies on unpublished work; talk to any pure mathematician in any big area and they will know of sometheorems which rely on results in the “grey literature”(a letter or an email from X to Y) or even results
which are not in the literature at all (“a forthcoming
paper”, sometimes announced years ago and which
never came forth). Humans are good at forgetting
to go back and fill in these details. Lean will not
let you forget. The computer-generated red squiggly
line underneath your theorem reminds you that yourproof is not complete, even if your work relies on
results announced by others and hence “it is notyour job to complete it”. During our work formalizing
perfectoid spaces, it got to the point where I knew
“we had done it”, but until the definition compiled weknew we had not convinced Lean. Another issue isthe fact that formalising the proof of a big theorem
like Fermat’s Last Theorem, given the state of theart, would surely take more than 100 person-years.
Getting money for this would involve a grant proposal
which no grant agency would fund, and which would
yield a result which no mathematician would careabout anyway — we all know Fermat’s Last Theorem
is true, because we know the experts have accepted
the result.

On Thursday evenings during term time I run a Leanclub at Imperial called the Xena Project (with associated blog at xenaproject.wordpress.com), where
undergraduates learn to formalize the mathematics
which they are learning in lectures, and any othermathematics of interest to them. One purpose of the
project is to make knowledge of formalisation moreprevalent amongst the undergraduate community,

Because Lean is offering something that pure mathematics researchers do not generally want (a fully
formalized verification of their theorems) at a pricewhich researchers generally will not pay (the time
taken to learn how to use the software, and then thetime taken to formalize their own proofs), in practice
one has to rule this out as a viable practical application of these tools at this point in time. Things might

Using Lean to help with research?

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change in the future, but rather than speculate on
how good computers can or will become at understanding natural language (i.e., arXiv preprints), letme simply observe that currently they are nowhere
near good enough, and instead turn to other things.
Tom Hales has suggested a completely different useof these tools, and has won a multi-million dollar
Sloan grant to implement his idea, which is called
FABSTRACTS (Formal Abstracts). Hales suggests that
we should create a mathematical database containing
not proofs but formalized statements of theorems,
cross-referenced to the traditional literature (journal
articles etc). He proposes that Lean be used for thisdatabase, and indeed that is why I chose Lean. The
reason this idea is a game-changer is that if all thedefinitions are there, then formalising a statementof a new theorem can be easy. The fact that Leancan handle the definition of a perfectoid space issurely evidence that it can handle any pure mathematical definition which we can throw at it. There
will also be a natural language component to theproject; the current plan is that theorems will bepresented in a controlled natural language, i.e., inhuman-readable form, understandable to a mathematician with no Lean training. One can imagine in
the near future such a database being constructed,
initially concentrating on the areas of speciality ofthe mathematicians already involved. What would be
the potential uses of such a database?
The first obvious use would be search. Say a
researcher is wondering what is known about a mathematical statement. They formalize the statement,
look it up in the database, and get either a perfect hit(a reference to the literature) or perhaps some partial
results (several references, perhaps to variants ofthe statement proved under certain hypotheses). If
we can train PhD students and post-docs to learn
enough about Lean to state their results, then thisdatabase could grow quickly.
The second obvious use would be AI. Already computer scientists are trying to develop tools which
can actually do mathematics better than humans.
They want to train their AI on databases — butthese databases are few and far between, and several of the databases which are used in practice are
databases of solutions to gigantic logic puzzles whichbear no relation to modern mathematics. I believethat it is up to us as a community to explain tocomputer scientists what we are actually doing, in aformat which is more useful to them than tens of
thousands of arXiv preprints where we use words
like “normal” and “complete” to mean five different things and where we write in the idiomatic andbizarre English language, thus adding another layerof obfuscation to the material. Formalisation solves
this problem — it makes us say what we mean in aclear and coherent manner. This is what these people
need to make tools for us — and we are not giving
it to them. Mathematicians do not seem to have any
desire to formalise a proof of Fermat’s Last Theorem,
however computer scientists are desperate to train
their AI’s on a formalised proof, to see how muchfurther they can take things. If any mathematician
reading this decides that they would be interested intaking up the challenge of formalising the statementof one of their theorems in Lean, then they can findme at the Zulip lean chat.
Computer scientists do not understand a lot of what
we as mathematical researchers do, or how we do it.
Let’s tell them in their own language, and see what
happens next. Computer scientists have developed
tools which can eat mathematics, but until thesetools get more self-sufficient they will need to behand-fed. It will be very interesting to see what thesetools become as they grow.

FURTHER READING[1] H. Bender and G. Glauberman, Local Analysis
for the Odd Order Theorem, Cambridge University
Press, 1994.
[2] S. Gouezel and V. Shchur, A corrected quantitative version of the Morse lemma, J. Funct. Anal.
277 (2019) 1248–1258.
[3] T. Peterfalvi, Character Theory for the Odd Order Theorem, Cambridge University Press, 2000.

Kevin Buzzard
Kevin Buzzard is analgebraic number theorist and a professor ofpure mathematics at Imperial College London.
He tweets occasionally
at @XenaProject.

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Reciprocal Societies:
Belgian Mathematical Society
The Belgian Mathematical Society (BMS) was
founded on 14 March 1921
(π-day!) at the Free University of Brussels; sucha choice of date is perhaps an early testimony
to the celebrated Belgian
sense of surrealism. Of the nine founders of thesociety, the best known are Théophile De Donder,
Lucien Godeaux and Alfred Errera. The aim of thesociety was the same in those days as it is now:
to contribute to the development and diffusion of all forms of mathematics in Belgium.
[The BMS] is concerned with mathematics,
pure and applied, in the broadest sense. It
will try to establish a permanent link between
secondary school and university.
At the beginning, andover the course of several decades, the society
organized monthly meetings during which an
extremely varied arrayof subjects were considered. Indeed, we findlectures on mathematical physics (with e.g. De
Donder), astrophysics
A founding father Lucien(Lemaître), algebraic geGodeaux (engraving by J.
ometry (Godeaux), analyBonvoisin 1947, courtesysis (De la Vallée Poussin,of the Royal Library of
Lepage),engineering
Belgium).
(van den Dungen), mathematics of insurance, and secondary school mathematics (with A. Mineur, sharply remembered bygenerations of Belgian school children for his treatise
on descriptive geometry). Amongst foreign speakers,
let us just pick two curios. On top of various lectures
on integration, Lebesgue gave a talk in 1925 on rulerand compass constructions. In 1922, Millikan gave
a lecture, comparing his ideas on the electron withthose of Planck and De Donder.
From 1947 onwards, and under the impetus of topologist Guy Hirsch, the BMS started editing its own

journal called the Bulletin de la Société Mathématiquede Belgique. The Bulletin underwent several transformations in the following decades and merged withthe journal Simon Stevin in 1994. It is now addressed
to a generalist audience, with a broad editorial board
covering a wide range of fields. Since 2003, it is available electronically via Project Euclid and all issuesof the new Bulletin older than five years are madeavailable with free access.
Attendance at the monthly meetings eventually dwindled and the habit was abandoned in the seventies.
In the eighties an annual meeting was launched, withgreat success at first. However, after a few years,
attendance diminished as the pertinence of such a
non-specialized national meeting found less priority
in the already hectic lives of academics. Since thosedays, the BMS has diversified the array of mathematical events in which it takes part: co-organizing joint
meetings with other National Mathematical Societies(including a joint meeting with the London Mathematical Society organized in Brussels in 1999); organizing joint meetings with the high school teachers’
associations; organizing the bi-annual “Ph.D. Day”
during which Ph.D. students or young researchers
have the opportunity to present their work throughposters and short communications; “thematic afternoons” during which the works of primed mathematicians (Fields medalists, Abel prize winners, . . . ) are
explained broadly by experts of the different fields.
The BMS also funds the Lucien Godeaux prize and isa sponsor of several mathematical initiatives aimed
at younger audiences.
On π-day 2021 our Society will celebrate its 100th
anniversary. The various mutations that it has gone
through were necessary and helped the Society toserve the mathematical community over time andremain relevant to the 21st century Belgian mathematical community.
Yvik SwanBMS President
Editor’s note: the LMS and the BMS have a reciprocity
agreement meaning members of either society may
benefit from discounted membership of the other.

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EARLY CAREER RESEARCHER
Microtheses and Nanotheses provide space in the Newsletter for current and recent research students tocommunicate their research findings with the community. We welcome submissions of micro- and nano-theses
from current and recent research students. See newsletter.lms.ac.uk for preparation and submission guidance.

Microthesis: Hypergraph Saturation Irregularities
NATALIE C. BEHAGUE

A graph is F -saturated if it doesn’t contain a copy of a graph F but adding any edge creates a copy of F . The
maximum number of edges an F -saturated graph can have is well-studied and called a Turán number. Our
topic here is the minimum number of edges an F -saturated graph can have, which behaves quite differently.
What is saturation?

Asymptotics

A graph is F -free if it does not contain a copy of
the graph F . Given a fixed graph F and a positive
integer n, Turán’s extremal number, ex(F, n), is the
maximum number of edges an F -free graph G on nvertices can have (see figure 1). What if we were to
ask about the minimum number of edges?

Turán’s Theorem and the Erdó´s–Stone Theorem tellus that for a (non-empty) graph F ,If we just replace the word ‘maximum’ with ‘minimum’ in the definition above we get a trivial function,
since a graph with no edges will be F -free. But noticethat in any maximal F -free graph, adding any edge
creates a copy of F . This inspires the following definition: we say a graph G is F -saturated if it does notcontain a copy of F as a subgraph but adding anyedge creates a copy of F .

1
ex(F, n) = 1 −+ o(1)
χ(F ) − 1

!

!
n
,
2

where χ(F ) is the chromatic number of F . In particular, ex(F, n)/ n2 converges to a limit as n tends toinfinity.
Can we say anything similar about the saturation number? Kászonyi and Tuza [2] proved thatsat(F, n) = O (n). Tuza [5] went on to conjecture thatfor every graph F the limit limn→∞ sat(F,n)
exists.
n

Now we have an equivalent definition of Turán’s
extremal number:ex(F, n) = max{e (G ) : |G | = n and G is F -saturated}

and if we replace max with min we get the saturation
number:sat(F, n) = min{e (G ) : |G | = n and G is F -saturated}.
ex(F, n) =
F =

$

n2
4

%

sat(F, n) = n − 1

Forbidden families
The definition of saturation can be extended to families of graphs. For a family F of graphs (called aforbidden family), a graph G is F-saturated if it does
not contain any graph in Fas a subgraph, but adding
any edge creates a copy of some graph F ∈ F as a
subgraph of G . We define the saturation number inthe same way as before:

Figure 1. Extremal and saturation numbers for the triangle

sat (F, n) = min{e (G ) : |G | = n and G is F-saturated}.

For example, if F is the triangle then the maximal
2
triangle-saturated graph has b n4 c edges. In contrast,
the minimal triangle-saturated graph is a star, which
has only n − 1 edges.

For a family of graphs F we have sat (F, n) = O (n)[2], just as we did for single graphs. However, thegeneralisation of Tuza’s conjecture to finite familiesof graphs is not true, as shown by Pikhurko in [4]:

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EARLY CAREER RESEARCHER
Theorem 1. There exists a finite family F of graphs
such that sat (F, n)/n does not tend to a limit as ntends to infinity.
In Pikhurko’s construction, the graphs in F depend
on some fixed constant k . For n divisible by k , one
can construct an F-saturated graph on n vertices
that uses relatively few edges. For n not divisible
by k , there is no such ‘nice’ construction and an
F-saturated graph on n vertices is forced to contain many extra edges. Pikhurko asked whether asimilar construction was possible for families ofhypergraphs.

Saturation for r -graphs
An r -graph H , often referred to as an r -uniform
hypergraph, is a pair, (V, E), of vertices and edges
where the edge set E is a collection of r -element
subsets of the vertex set V . Note that a 2-graph isexactly a graph.
The definitions of saturation and saturation numbers transfer immediately to the context of r -graphs— the only difference is that the edges contain r
vertices rather than 2.
For a family F of r -graphs, it was shown by Pikhurko[3] that sat (F, n) = O n r −1 when the family contains only a finite number of graphs. This leads tothe following generalisation of Tuza’s conjecture tor -graphs, first posed by Pikhurko [3].
Conjecture 1. For every r-graph F the limit
limn→∞ sat(F,n)
exists.
n r −1
As in the 2-graph case we can further generalise thisconjecture by replacing the single r -graph F with afinite family of r -graphs F. Pikhurko’s question now
becomes “Does there exist a family F of r -graphs
such that sat (F, n)/n r −1 does not converge to a limit
as n tends to infinity?” I have shown in [1] that theanswer to this question is yes.
Theorem 2. For all r ≥ 2 there exists a family F ofr -graphs and a constant k ∈ N such thatO (n)
if k | nsat (F, n) =  Ω n r −1  if k - n,n)does not converge.
In particular, satn(Fr −1

The proof uses a similar approach to Pikhurko’s
construction for graphs. The forbidden family used

contains two types of r -graph: a pair of intersecting complete r -graphs; or some disjoint completer -graphs together with a ‘bridge’ edge between them.

Figure 2. The family F of r -graphs for r = 5 and k = 7

An example can be seen in figure 2, where the vertices surrounded by a blue line represent a copy of(r )the complete r -graph on k vertices Kk , and verticesgrouped by a black line represent a bridge edge.
Tuza’s conjecture itself remains open. It seems difficult to reduce the family above to a single hypergraphwhile keeping the non-convergent behaviour: I have
only been able to reduce the family to one of size
four. It could be that such a large gap in the asymptotics is not possible for a single hypergraph, andone must look at the coefficients.

FURTHER READING[1] N. C. Behague, Hypergraph Saturation Irregularities, Electron. J. Combin. 25 (2018) 2–11.
[2] L. Kászonyi, Z Tuza, Saturated graphs with minimal number of edges, J. Graph Theory 10 (1986)
203–210.
[3] O. Pikhurko, The minimum size of saturated
hypergraphs, Combin. Probab. Comput. 8 (1999)
483–492.
[4] O. Pikhurko, Results and open problems on
minimum saturated hypergraphs, Ars Combin. 72(1994) 111–127.
[5] Z. Tuza, Extremal problems on saturated
graphs and hypergraphs, Ars Combin. 25 (1988)
105–113.

Natalie

BehagueNatalie Behague is a
PhD student at Queen
Mary University of London under the supervision of Robert Johnson,
researching problems in
extremal combinatorics. She enjoys boardgames,baking, and playing tag rugby quite badly.

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The Princeton Companion to Applied Mathematics
by Nichloas J. Higham (editor), Princeton University Press, 2015,£77, US$ 99.50, ISBN: 978-0691150390
Review by David I Graham

The first question thatcomes to mind whenreviewing a book likethis is “Why?” A Google
search for “Applied Mathematics” finds 382 million pages related to thesubject, many of which
will be very detailed andwith access to animations and relevant computer code as well as links torelated work. The editors are, of course, well awareof this and try to answer the question themselves. In
the Preface they claim that the distinguishing featureof the book is that it is “self-contained, structured
reference work giving a consistent treatment of thesubject”.
The book is in eight parts and runs to almost 1000
pages. There is certainly a serious attempt to be selfcontained, with the first part containing the longest
articles in the book, defining basic language andterms from coordinate systems through calculus up
to operators and stability. The second part thenbriefly reminds the reader of essential concepts fromasymptotics to wave phenomena. These are arranged
in alphabetical order, occasionally meaning that there
is no natural flow from one contribution to the next.
The structure becomes more free-form as we getfurther into the book. This is inevitable given thegreat variety in topics covered. With over 160 authors
contributing to the articles, consistency was always
going to be difficult to achieve and there is considerable variation in the later parts, especially in the levelof detail and follow-up information. One contribution
in Part V has a reference list of one item, namely abook written by the contributor himself. Conversely,
the exemplary contribution on financial mathematics
in Part V provides not only an excellent reference
list but also a discussion of what to look for in eachof the references. Overall, the reference lists seemto be generally good. In terms of self-containedness,some of the contributions in the later parts requireprior knowledge not fully detailed in the introductory
parts, though more detailed investigations utilising
the reference lists should mean an interested reader
would be able to fill the gaps.
Returning to the material covered, Part IV is reallythe heart of the book. It describes in considerable
detail (over 400 pages), forty “Areas of Applied Mathematics” including straightforward choices such as
various flavours of mechanics, differential equations
and numerical methods but also less obvious areas
such as algebraic geometry. There is a very readable
contribution on “Symmetry in Applied Mathematics",
which starts from the symmetries of plane figures
and goes as far as symmetry breaking, with muchdiscussion related to the various symmetries seen in
Taylor–Couette fluid flow between co-rotating cylinders. The author confesses that the article “barely
scratches the surface”. A typical example for thispart is the ten page contribution on “Fluid Dynamics”,
which rattles along at great pace, covering everythingfrom 2-dimensional streamlines through flight aerodynamics up to flow instability — enough material tofill a decent course module. Similarly, the nineteen
page contribution on “Numerical Linear Algebra andMatrix Analysis” is a comprehensive collection of themain results relating to matrix computations andnotes some useful ‘must have’ references. I have toconfess that some of the other contributions arerather dry for my taste, though the articles generally
represent excellent starting points for further investigation — which is one of the great strengths of thebook as a resource.
Parts V (“Modelling”), VI (“Example Problems”) andVII (“Application Areas”) together give us 64 different
examples, averaging at around five pages in length.
Several of these — including a contribution on “Sport”
that strangely covers only sailing, rowing and swim-

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ming — are related to fluid dynamics, meaning that
there is some repetition between articles. The article
on “A Symmetric Framework with Many Applications”
outlines a nice example of a unifying viewpoint forminimisation problems of various kinds. As a computational applied mathematician, I welcomed thecontributions in Part VII related to historical developments in programming languages (and the confusion
to which different conventions can lead) and thefuture in terms of high-performance computing.

My final thoughts are that this book is an excellent
resource for any mathematics departmental library.
The articles cover a vast array of different application of mathematics and are generally well-writtenwith useful reference lists. In particular, the book represents an excellent launch point for individuals suchas project students looking for an area of Applied
Mathematics to investigate in greater depth.

A particularly interesting feature of the book is thelast Part (VIII), which offers some “Final Perspectives”.
I found many of these contributions to be extremely
thoughtful and useful, including advice on how toread Mathematics articles or to write articles or evengeneral interest books. As an author of reasonably
large fortran codes myself, I found the sections on“Reproducible Research” and “Experimental Mathematics” to be especially thought-provoking.

David is a Senior Lecturer in Applied Mathematics at the Universityof Plymouth. His main
research interests are
in developing and using
numerical methods for
fluid dynamics. David is a keen footballer and, in
decreasing order of competence he also plays guitar,banjo, ukulele and bouzouki.

David Graham

The Turing Guide
by Jack Copeland, Jonathan Bowen, Mark Sprevak and Robin Wilson,
Oxford University Press, 2017, £19.99, US$ 29.95, ISBN: 978-0198747833
Review by David Glass

For several decades
after his tragic anduntimely death in 1954
at the age of just 41, very
little was known about
Alan Turing’s importantwork at Bletchley Park
during the second world
war. However, as thefull scale of his achievements in codebreaking
and the relevance of the wartime effort to the historyof computing became clearer, Turing’s reputation
has increased dramatically. By providing a very wideranging and yet accessible account of Turing’s life

and work, The Turing Guide is an excellent contribution to this development and the growing literature
on Turing.
The book consists of forty-two chapters divided
into eight sections, with the first section providing
biographical material. The first chapter provides abrief sketch of Turing’s life, with a helpful timelineof key events, while in the second chapter, entitled‘The man with the terrible trousers’, Turing’s nephew,
Sir John Dermot Turing, provides a unique family
perspective. The third chapter is a compilation of
extracts from papers and reminiscences of the latePeter Hilton, who worked with Turing for the last 12
years of Turing’s life. According to Hilton, Turing was

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REVIEWSa ‘warm, friendly human being’, ‘an approachable
and friendly genius’ and he gives a sense of Turing’s‘quirky and infectious sense of humour’. In addition
to providing insights into Turing’s early life, schooldays, and working life, the first section also discusses
his conviction for homosexuality, the terrible chemical treatment he had to undergo as a result, thesubsequent apology from Gordon Brown in 2009 androyal pardon in 2013, as well as the circumstances ofhis death. As for whether Turing committed suicide,‘We shall most probably never know.’Those interested in the history of computing andthe codebreaking work at Bletchley Park will finda lot of useful material in sections two to four. In
Chapter 6, Jack Copeland provides an introduction
to Turing’s greatest contribution: the universal Turingmachine. He also addresses the origin of the conceptof the stored program computer, an issue which isfurther explored in Chapter 20 in the context of thedevelopment of Baby, the first electronic universal
stored program computer, in 1948. Chapters 7 and 37
provide very readable introductions to Turing’s important work on Hilbert’s famous Entscheidungsproblem,
while Chapter 24 explores congruences between Turing’s work and the earlier ideas of Charles Babbage
and Ada Lovelace.
The Enigma machine is the subject of Chapter 10 with
Turing’s bombes (special purpose electromechanicalcomputers used to decipher Enigma messages) the
subject of Chapter 12. Both of these chapters provide helpful explanations about how the machines
worked while Chapter 14 explores Tunny, the encryption machine used by Hitler later in the war. Turing’s
contribution, as well as that of Bill Tutte, Max Newman and Tommy Flowers, including the essential roleof Flowers’ Colossus computer, are all discussed. A
number of people who worked as codebreakers or
operated Colossus or the bombes provide fascinating
accounts of their time at Bletchley Park.
Section five explores Turing’s contributions to artificial intelligence and the mind. These chapters include
discussions of topics such as the Turing Test, his
use of heuristic search techniques in the bombe, his
early work on connectionist models, child machines,
his contribution to cognitive science, and his chessprogram. Particularly interesting is Diane Proudfoot’s
discussion of Turing’s concept of intelligence (Chapter 28). She argues that Turing thought of intelligenceas an emotional concept that is partly in the eye ofthe beholder and she also argues against the common view that Turing was a behaviourist. Perhaps
the most surprising contribution is a chapter on theparanormal (Chapter 32), which provides context toTuring’s equally surprising comments about the powers of extra-sensory perception in his famous article
in Mind in 1950.
Sections six and seven look at Turing’s contributions to biology and mathematics, with the former
providing an introduction to his fascinating work on
morphogenesis in which he used reaction-diffusion
equations to model the development of biologicalstructures. In Chapter 33, Margaret Boden explores
the influence of Turing’s work on research in selforganisation, artificial life, and structuralism in biology. Interestingly, she argues that Turing’s innovativeideas were ‘out of line with the biological orthodoxy’of neo-Darwinism and associates his approach withthat of the structuralist D’Arcy Thompson. In Chapter 35, Bernard Richards describes how he carriedout work to validate Turing’s theory by showing how
solutions to his equations in three dimensions could
explain the shapes of marine creatures known asRadiolaria. He had arranged to meet Turing on 8 June
1954 to show him his intriguing findings, but sadly
Turing died the previous day.
Chapter 36 provides a very helpful overview of Turing’s work on the central limit theorem, group theory
and the Riemann hypothesis, while other chapters inthis section explore further topics such as Turing’s
work on randomness, which ‘anticipated by nearly
thirty years the basic ideas behind the theory ofalgorithmic randomness’. In Chapter 38 (and also 13),
Edward Simpson, who worked as a cryptanalyst at
Bletchley Park (and after whom Simpson’s paradox
is named), provides a very interesting and insightful
explanation of Banburismus, a process for reducing
the work the bombes had to do to decipher Enigma
messages. He also gives a nice account of Turing’s
use of Bayes factors and logarithmic scoring to weighevidence.
The final section has a very interesting chapter
on whether the universe is computable. This wideranging chapter covers topics such as computability,
Turing’s thesis (or the Church–Turing thesis) and misconceptions about it, and the physical computability
thesis. It ends by arguing that Turing was not committed to the idea that the universe or even the brain
is computable. The final chapter highlights Turing’s
impressive legacy both within science and beyond.
The Turing Guide is extremely informative, highly readable, and well produced with many photographs anduseful figures to aid exposition (though some colour

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would have been nice). The preface states the bookwas ‘written for general readers, and Turing’s scientific and mathematical concepts are explained inan accessible way’. This has been achieved withgreat success. However, those working in a rangeof fields will also benefit a lot from articles written
by experts and pointers to the extended literature.
In other words, it does exactly what a good guide
should do. Alan Turing’s legacy has grown enormously
in recent years and The Turing Guide will serve tofurther enhance understanding of his remarkableachievements.

David GlassDavid Glass is a seniorlecturer in the Schoolof Computing at Ulster
University. He carriesout research in artificial
intelligence, philosophyof science, and the modelling of complex systems. He enjoys football andrugby (watching not playing) and has an interest in
the relationship between science and religion.

Anxiety and the Equation:
Understanding Boltzmann’s Entropy
by Eric Johnson, MIT Press, 2018,£18, US$ 22.95, ISBN: 978-0-26203-8614
Review by Andrew Whitaker

Nobody today would
question that Ludwig
Boltzmann was one of
the greatest of physicists. It could be said
that he provided theroute from nineteenthcentury physics, which
was broadly macroscopic
in nature, and whereatoms were treated withconsiderable suspicion(at least by many physicists, less so by chemists,
but even more so by a number of philosophers) andon to a physics largely based on atoms and theirconstituents, with theoretical ideas centred on thequantum. His main work used three considerably
overlapping sets of ideas. The first was the existenceof atoms and their centrality in any discussion ofphysics. In particular this implied the extensive useof probability arguments, and especially the idea thatthe macrostate, or macroscopic state of the systemthat we actually meet, will be one corresponding

to an enormous number or microstates, which areparticular states of the individual particles.
The second set of ideas is related to the famous
Boltzmann distribution. Given a fixed amount ofenergy to be divided between a number of particles,
this distribution tells us, with good probability, thenumber or particles having various values of energy
at equilibrium.
The last idea clarified the idea of entropy. While in thenineteenth century this was looked on as a macroscopic function of the thermodynamic variables ofthe system, for example the entropy of a gas could
be expressed as a function of its pressure and volume, for Boltzmann its definition was statistical. The
relevant and famous equation is S = k logW , where
S is the entropy of the system, W is the multiplicityof the appropriate macrostate, and k , of course, isalways known as Boltzmann’s constant. The ideaof this equation is undoubtedly due to Boltzmann,
who discussed all the ideas at great length, though
Eric Johnson points out that the actual equation and43

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indeed ‘Boltzmann’s constant’ itself were in fact written down by Max Planck!
The enormous importance of Boltzmann’s work, so
obvious today, was unfortunately not at all clear
to a number of extremely influential scientists andphilosophers, and disagreements were expressed
increasingly aggressively from around 1890, whenBoltzmann was in his mid-forties. The three chief
antagonists were Ernst Mach, Planck and Ernst Zermelo, who was Planck’s assistant.
As an arch-positivist, Mach was diametrically
opposed to any idea of atoms, and clashes with Boltzmann were inevitable. Boltzmann hated them. In 1895
Mach became Professor of the History and Philosophy of Sciences at Vienna, where Boltzmann had
become Professor of Theoretical Physics the year
before. Boltzmann’s unease was such that he moved
to Leipzig in 1900, returning to Vienna only in 1902,
after Mach’s retirement due to illness.
Planck was (until his dramatic conversion) the lastof the important exponents of classical thermodynamics, following in particular Rudolf Clausius andHermann von Helmholtz. As such he detested Boltzmann’s statistical approach to matters of heat flow
and particularly the use of probability in physics.
Zermelo concentrated on the nature of entropy. A
crucial point in classical thermodynamics was thatentropy never decreases. In certain very special circumstances it might remain the same, but otherwise
it increases thus providing an arrow of time. On thestatistical definition, on the other hand, the entropyof very small systems would often decrease, but
as the size of the system increased the probabilityof entropy decreasing became smaller and smaller,
becoming infinitesimally small for systems of macroscopic size but never exactly zero. Zermelo plagued
Boltzmann on this point
It is well known that Boltzmann tragically hanged
himself in September 1906. The community of physicists has traditionally absolved itself of blame forthis event, providing as evidence his various illnesses, both physical, including enormously diminished vision, asthma, chest pains and headaches, andmental, depression and in particular what Johnson
characterises as anxiety. Readers of Johnson’s slim
book may judge for themselves to what extent physicists contributed to these illnesses. Johnson provides
detailed analysis of relatively small systems demonstrating Boltzmann’s main achievements. He studies
the probabilities of students distributing themselves

over the two halves of a lecture room, atoms over
the two halves of a volume. He divides 7 units ofenergy over 4 atoms and calculates the resulting distribution. He studies entropy by analysing different
numbers of bedbugs moving between the two halvesof an auditorium. For just 4 bedbugs the entropy
initially moves towards what might be called an ‘equilibrium value’ but clearly that terminology is hardly
appropriate because it continues to fluctuate wildly
about this value. For 1000 bedbugs we see similar
behaviour but the fluctuations are far smaller. In
both cases, though, there are clearly periods overwhich the entropy decreases. All Johnson’s analysis isextremely helpful to anybody wishing to understandBoltzmann’s achievements.
He also presents a series of vignettes of Boltzmann’slife — some amusing, such as his walking down themain street accompanied by a cow, purchased toprovide his children with fresh milk, but more oftensad, including his disastrous manoeuvers attempting
to take up a chair in Berlin without informing his
current employer, the Austrian education minister,and a study of his self-destructive concern abouta student lecture to be given the following day, alecture he should have been able to give practically in
his sleep. He even provides a ‘happy ending’ in which
Boltzmann dies at peace, soothed by his daughter
playing the Moonlight Sonata. Sadly, of course, thiswas not to be.
Ironically, as Johnson points out, it was at precisely
this period that Boltzmann’s main ideas were being
justified. It was argued, long before Einstein’s detailed
analysis in 1905, that Brownian motion, the irregularmotion of particles in a fluid, was due to bombardment by atoms; and more specifically, Planck’s epochmaking study of black-body radiation in 1900, which
introduced the quantum theory, was based soundly
on Boltzmann’s probabilistic arguments, which Planck
had long excoriated. In repentance, as Johnson mentions, Planck twice proposed Boltzmann unsuccessfully for the Nobel Prize for Physics. For Boltzmann
this was all too late. This book is an interesting andthoughtful account of Boltzmann’s life and work.

Andrew Whitaker
Andrew Whitaker is
Emeritus Professor of
Physics at Queen’s University, Belfast, andChair of the Instituteof Physics History ofPhysics group.

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Obituaries of Members
Ernst Sondheimer: 1923 – 2019
Ernst Sondheimer, who
was elected a member ofthe London Mathematical Society on 28 June
1974, died on 9 June 2019,
aged 95.
Julian Sondheimer writes:
Ernst was born in 1923
in Germany. In 1937 thefamily emigrated to London, and Ernst started his
new life in England at University College School in
Hampstead. In 1942 he went up to Trinity College
Cambridge and remained there as research student
and fellow until 1952. In 1946 he became a British
citizen and in 1950 he married Janet Matthews, a history fellow at Girton College, Cambridge. In 1951 he
was appointed as a lecturer at Imperial College andin 1955 he became a reader at Queen Mary College.
At the age of 37, in 1960, Ernst was appointed Professor of Mathematics at Westfield College in the
University of London, where he remained for 22 years
until his retirement in 1982. For most of this period
he was Head of Department and he was instrumental
in creating a very successful mathematics research
department. Ernst cared deeply about his students
and staff and is fondly remembered by them.
Earlier, as a young researcher in the field of mathematical physics, his main achievement was a seminaland groundbreaking paper in 1948 with G.E.H. (Harry)
Reuter (also a German refugee) on the anomalous
skin effect. This was in the area of the optical properties of metals and involved the difficult solutionof an integro-differential equation. Later Ernst coauthored the books Green’s Functions for Solid State
Physicists and Numbers and Infinity.
Towards the end of his time at Westfield, in 1981,
Ernst and his family were hard hit by the death of
his younger brother Franz, the eminent chemist (aFellow of the Royal Society, Franz was distinguishedfor his work on the synthesis of natural products).
After retiring from Westfield College, Ernst revelledin his two passions of mountains and flowers, taking
up the editorship of the Alpine Journal and devoting
himself to his wonderful garden at his home in North
London. At age 88, an unexpected difficulty with a
medical procedure left him in a frail condition, but

this did not stop him continuing to receive visitors,
attend concerts and read widely. He never forgot his
gratitude to University College School and Trinity College Cambridge, for their support given to a refugee
from Nazi Germany in difficult times.

Gerald Gould: 1925 – 2019
Gerald Gould, who waselected a member ofthe London Mathematical Society on 18 March
1954, died on 23 February 2019, aged 93.
David Edmunds and Des
Evans write: Gerald wasborn in Bermondsey,
London, on 9 September 1925. From Wilson’s Grammar School he won an open scholarship to study
physics at Christ’s College, Cambridge, an event
celebrated by the school with a day off. After graduating, he served his military service working for
Tube Alloys, the code name for the top security
research and development programme to develop
nuclear weapons. That done, Gerald enrolled for amathematics degree in Birkbeck College, Universityof London. There he came under the influence ofLionel Cooper, and after graduating started a PhD
under Cooper’s supervision. In 1950 Cooper was appointed to the chair in what was then the University
College of South Wales and Monmouthshire (nowCardiff University), and Gerald went with him; he wasto spend the rest of his career in Cardiff. Apart fromhis mathematics, Gerald had many interests and talents. He was an accomplished musician, being highly
regarded and sought after by orchestras as a timpanist, bassoonist and flautist. Bridge was anotherpassion at which he excelled, winning a worldwide
competition in 2006 involving 1,000 players on fourcontinents. In 1954 Gerald developed TB and spenttime in an isolation hospital. He claimed that tocontend with the boredom, he learned to smoke,
became a master of Meccano, and learned Russian.
He would continue to smoke with obvious pleasure
until 1984, and his knowledge of Russian had a big
part to play in the contribution he made to themathematical community; he went on to act for the
LMS as English Edition Editor of Sbornik, and translated into English various Russian books, notably twothat appeared in the Encyclopaedia of Mathematical
Sciences series: Dynamical Systems by D.V. Anosov
and General Topology III by A.V. Arhangel’skii.

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Gerald’s early work evolved from his PhD on
integration spaces, his principal contribution toresearch being perhaps his paper Integration Over
Vector-Valued Measures which appeared in the Proceedings of the LMS. This introduced a type of integral (the Gould integral) of a bounded real-valued
function with respect to a finitely additive set function taking values in a Banach space. Gerald had agreat appetite for new areas of mathematics andwas able to absorb challenging new material withenviable ease. Much of his research was generated by problems and flaws discovered through his
extensive reading. His later work on test-function
spaces and various aspects of spectral theory areexamples of this.
Outside his mathematics and his many interests,
Gerald had a rich family life. He is survived by his
beloved wife Enid, his children Nina and Ben andthree grandchildren. They will miss him sorely, as
will his friends and colleagues.

John Marstrand: 1926 – 2019
John Martin Marstrand,
who was elected a member of the London Mathematical Society on 18
June 1953, died on 29
May 2019, aged 93.
Kenneth Falconer writes:
Although he took his
DPhil at Oxford, John
was essentially supervised by Cambridge Professor
Abram Besicovitch. His thesis contained a numberof remarkable results, notably ‘Marstrand’s projection theorems’ on the Hausdorff dimensions of orthogonal projections of subsets of the plane ontolines. Apart from generalisations by Pertti Mattila andRobert Kaufman, this attracted little attention until the late 1970s when Mandelbrot popularised andunified the notion of fractals. It was then realisedthat John’s work was just what was needed for the
study of the geometry and dimensions of fractals,
and his projection theorems became the prototype
for the now flourishing area of fractal geometry. Indeed, the paper based on John’s thesis that appearedin the LMS Proceedings in 1954 was highlighted in2015 as one of six ‘landmark’ papers published in
the Proceedings during its first 150 years.
After short periods in Cambridge and Sheffield,
John moved to the University of Bristol in 1958

where he was awarded a personal chair and spent
the rest of his career. He was challenged by difficult questions and, although he wrote only 17
papers, virtually all of them contain deep andclever ideas. He continued to work in geometricmeasure theory, relating rectifiability of sets (i.e.
whether they are ‘curve-like’ or ‘surface-like’) to
local measure densities. But he had a broad interest across mathematical analysis. He constructeda clever counter-example to Khinchin’s conjectureon the uniform distribution of certain sequences
modulo 1. He was fascinated by variants of the
Kakeya set and curve packing problems and hesolved the problem of ‘the minimal comfortable living quarters for a worm’ by showing that given any
plane set of zero area (Lebesgue measure) there
are arbitrarily short smooth curves that cannot
be mapped into the set by a similarity (or, much
more generally, an analytic) transformation. Outstanding was his geometrical proof of the ‘circle
packing conjecture’, of fundamental importancein harmonic analysis: given a plane set E of zeroarea, every circle with centre x intersects E inzero perimeter length, apart from a set of centres x of zero area. In a probabilistic direction,
John, together with Geoffrey Grimmett, answeredin 1990 the major outstanding question on site
percolation on lattices in 3 or more dimensions.
John was a highly regarded and engaging lecturer.
Once, lecturing on ‘random walks’, he jumped onto
the front bench, blindfolded himself and stepped
forwards and backwards at random until he eventually fell off the end of the bench! He was very
careful in all his duties, often spending hours
on tasks others would take a few minutes over.
Once when examining a PhD thesis his attention
to detail resulted in a five and a half hour viva!
Outside mathematics, John went through a rangeof passionate interests, from fast sports cars to
extreme skiing. In his 40s he took up fell-racing
and in 1982 became British over-50s fell-running
champion. He regularly participated in the Ben
Nevis Race and on several occasions won the over40s or over-50s sections, with his fastest time in1983 an incredible 1 hour 48 minutes 54 seconds.
John was a kind and considerate person and isremembered by his former colleagues as a ‘loveable eccentric’. He was modest about his achievements but his determination and insight when hebecame absorbed by a project led to some quite
remarkable work of enduring quality.

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Christopher Hooley: 1928 – 2018
Christopher Hooley, who
was elected a member ofthe London Mathematical Society on 28 June
1974, died on 13 December 2018, aged 90.
Roger Heath-Brown andThomas and Adam
Hooley write: Professor
Christopher Hooley, FRS, FLSW was one of the leading analytic number theorists of his day. He won
Cambridge University’s Adams Prize in 1973, and theLMS Senior Berwick Prize in 1980. He was electeda Fellow of the Royal Society in 1983, and was aFounding Fellow of the Learned Society of Wales. He
gave a plenary lecture at the 1983 ICM in Warsaw.
Christopher Hooley did his undergraduate degree at
Corpus Christi College, Cambridge, and continued
there with a PhD under Albert Ingham. His careerthen took him to Bristol, to Durham, and finally to
Cardiff, where he served as Head of the School ofMathematics from 1988 to 1995.
He was the author of nearly 100 papers. The best
known of these concerns Artin’s conjecture on primitive roots. The conjecture proposes that primes witha given primitive root have a definite density, andgives a formula for what the density should be. In
particular it would follow that 10, for example, is aprimitive root for infinitely many primes p, so that thedecimal expansion for 1/p would recur with periodp − 1. Hooley’s work, in 1967, proved the conjecture
in full, assuming the Riemann Hypothesis for a classof Dedekind zeta-functions. These would have beenhighly abstruse objects to analytic number theorists
in those days, and even now it seems remarkablethat they could have any bearing on such a concrete
problem about decimal fractions.
Many of Hooley’s papers made use of estimates for
exponential sums originating in the work of Weil and
Deligne. Thus for example, he was able to establish
the Hasse Principle for non singular integral cubic
forms in 9 or more variables, by using information
about the associated exponential sums. Anotherrecurrent theme was the introduction of novel sieve
methods, and his Cambridge Tract “Applications ofsieve methods to the theory of numbers” showcases
the impressive variety of sieve tools he developed.
Modern analytic number theory owes a great deal

to these ideas. In particular the use of exponential
sums, as pioneered by Hooley, is now ubiquitous.
Hooley’s writing was well known for his love of sucharcane phrases as ‘the dexter side of the antepenultimate equation’. Equally however his papers
were a model of clarity and accuracy. They were often
couched in seemingly old-fashioned terms, both linguistically and mathematically, but they never failed
to enlighten.
Christopher Hooley had many interests, most ofwhich were pursued with vigour. From early on asa pupil at Abbotsholme School during the Second
World War he developed a love not only of mathematics, but also of classical, military, and naval history. In
years to come he would delight in discussing the history of western and eastern Roman empires with hiscolleagues from the history department at Cardiff,
who would ruefully admit they were out of their comfort zone, using that standard excuse ’not my period’!
— Hooley counted all European history of the twomillennia A.D. as his period. This interest went alongwith a love of antiques and especially of collectingWest Country friendly society brasses. Many days
were spent scouring antique shops and following up
leads to remote places in Somerset during the 1970s
and 80s. His wife, Birgitta, denied pets as a child in
Sweden, made sure that the family always had plentyof dogs — at one time six, sometimes cats as well:this interest led to another hobby, that of travellingto terrier shows in the summer months, from which
Birgitta, Christopher, and their two sons, Thomasand Adam, often returned proudly bearing coloured
rosettes for their dogs’ successes. Supportive as afather he was nonetheless quick to spot and point
out inaccuracies of memory in the reminiscences of
his sons at his 90th birthday party, held in Bristol
just four months before he died in December 2018.
Christopher’s grandfather had taught carpentry at
Macclesfield and on inheriting his tools Christopher set about using them to restore or completely
remake a large number of period sash windows in
the family home in Somerset, often at some considerable height. Once done, he turned his hand topanelling, lath and plaster, and similar skills with great
success.
The practical side was also present when it came to
motoring. The first family car, a 1950s’ Wolseley 6/80,
was kept and later given a thorough restoration in
the 1980s. Christopher showed no fear in creating
an entire wiring loom from scratch!

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These interests and his family life provided the counterpoint to Hooley’s academic career. At times unconventional, for instance researching deep into the nightwith a beloved whisky and cigarette alongside an
eclectic mixture of steam railway magazines, Dante,
Milton, and paperback thrillers, he had a ready wit
and excellent sense of humour. He will be remembered with fondness by colleagues and family alike.

Noel G. Lloyd: 1946 – 2019
Professor Noel Lloyd,
who was elected a member of the London Mathematical Society on 15
February 1973, died on 7
June 2019, aged 72.
Alun Morris writes: Noelsadly died after a prolonged fight against
prostate cancer which he met with his usual quiet
composure. This followed a full life distinguishinghimself both as a mathematician and in his later
years as a senior university administrator and in
public life.
Noel was born in Llanelli, Carmarthenshire. Afterattending Llanelli Grammar School, where his father
was a mathematics teacher, he entered Queens’ College, Cambridge in 1965 having obtained an Entrance
Scholarship. He graduated with a B.A. in 1968 witha first class in each year and an MA in 1972. On
graduating, he worked for his PhD in a research area
which was not too fashionable at the time, ordinary
differential equations, under the supervision of Sir
Peter Swinnerton-Dyer. His PhD thesis dealt with acase of van der Pol’s equation not considered by
J.E. Littlewood and Mary Cartwright. In addition tonumerous college prizes he also received the prestigious Rayleigh Prize. He proceeded to a Research
Fellowship at St John’s College, Cambridge. After two
years in that position he was enticed to a Lectureshipin Pure Mathematics at what was then the University
College of Wales, Aberystwyth (now Aberystwyth University), which he took up on 1 January 1975. At thattime, it was not anticipated that the university wasappointing its future Vice-Chancellor — certainly notby him. He did not need to be persuaded; he hadlong expressed his wish to come to his father’s almamater. His research work flourished in his chosenarea of applied analysis or non-linear analysis. He
established a strong school in that area supervising
a number of good PhD students. Among numerous

publications at that time was his book on Degree
Theory in the Cambridge Tracts in Mathematics series; this book was particularly well received andbecame an established text and was widely consulted. During the 1980s his interests moved towards
the study of the second part of Hilbert’s Sixteenth
Problem on the number and relative configurationof limit cycles of polynomial systems in the plane.
This is acknowledged to be a very difficult area of
research but it is internationally recognised that heand his research group have been responsible for
substantial developments, and his loss is strongly
felt amongst researchers in this area. Their progress
involved innovative use of computer algebra, as wellas analytic approaches.
In the 1980s Noel was a member of the Mathematics
Committee of the SERC where his detailed knowledge and his meticulous fairness came to the fore.
Also, in 1986–90 he was Joint Editor-in-Chief withme of the Journal of the LMS — a dramatic period
for the journal with an overnight need to move theprinting to the Cambridge University Press. He wasalso a long-serving Editor of the Mathematical Proceedings of the Cambridge Philosophical Society. He
was promoted to Senior Lecturer, Reader and thenin 1986 to a Personal Chair. Although at that point
not heavily involved in administration, he was highly
respected and was for a short time Head of the
Department of Mathematics, Dean of the Faculty of
Science and then Pro Vice-Chancellor, then Registrar
and Secretary of the University and finally in 2004
its Vice-Chancellor. He held that position with great
distinction until 2011. In 2010 he was appointed CBEfor Services to Higher Education in Wales and in 2011a Fellow of the Learned Society of Wales.
The many tributes that have appeared after his
untimely death are an indication of the esteem he
was held as Vice-Chancellor. He was meticulous in
his preparation, quietly firm and fair. He was universally regarded as a generous and compassionate man
and a person of great integrity, respected equally by
his academic colleagues and fellow administrators.
During his tenure as Vice-Chancellor he was Chairof Higher Education Wales and Vice-President of
Universities UK. He also served on many other UK
university bodies. On his retirement, his talents werein great demand. He became an independent member of the Silk Commission established by the UK
government to look at the future of devolution in
Wales. He also served as a lay member of the Judicial
Appointments Commission. For six years he was
Chair of Fair Trade Wales.

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Throughout his life, his church was central to his
life. He served not only his church with distinction
but also his denomination and other religious causes
both locally and nationally. Above all, he was a brilliant organist; as one tribute said, students would
see their VC relaxing on a Sunday morning at theorgan playing his favourite Bach. Noel married Dilys
in 1970; next year would have been their golden wedding. They had a son Hywel and daughter Carys andtwo grand-daughters Sioned and Catrin.

Stuart Brian Barton: 1952 – 2019
Dr Stuart Barton MD,
DPhil, who was elected amember of the London
Mathematical Societyon 10 November 2017,
passed away in hospitalin Cambridge on 12 April
2019, aged 66.
Phil Rippon writes: As ayoung person at Oldham Hulme Grammar School,
Stuart excelled at mathematics and physics, but
was advised that medicine was a more appropriate
career. Accordingly, he read medicine at Balliol College, Oxford, and took a DPhil there in physiology
in 1977, working much of the time in Stony Brook,
New York with Professor Ira Cohen. They had been
friends at Oxford and in fact published together arather mathematical paper on transmitter releasefrom nerve terminals (appearing in Nature in 1977).
Stuart then began a career as a GP in the UK andalso qualified as a consultant physician though hechose to remain a GP.
In 1993, Stuart changed direction and became alecturer at the University of Liverpool, specialising in evidence based medicine, where he could
bring into play his love of mathematics (especially statistics). He became a leader in that field,
publishing many articles, with co-authors such as
Professor Tom Walley CBE.

In 1998, Stuart again changed direction and built
on his research experience to found and edit a
BMJ publication Clinical Evidence, which assessed
the clinical significance of medical treatments anddrugs. In 2002 Stuart returned to GP practice inLondon. With Dr Elizabeth Robinson, he took over afailed GP practice in the Kings Cross area and they
worked together there for 8 years providing excellent health care to some of the most disadvantaged
communities in London.
Stuart retired from GP practice in 2010, in order todevote much of his time to mathematics, his firstacademic love. He began to study with the Open
University, completing a first class degree in mathematics in 2017. His tutors speak highly of his assignment work, which was always of excellent quality
and often contained material that went well beyond
what the assignments officially required. After moving to live in Cambridge, he regularly attended theOpen University’s maths tutorials there and also
weekend schools, where he greatly enjoyed meetingother people with similar interests, and is remembered by tutors as one of the brightest and liveliest
students. His partner Elizabeth Robinson reportsthat he read widely about the subject, and wasespecially influenced by the life of Paul Erdó´s, as
described in the book The Man Who Loved Only
Numbers by Paul Hoffman. Like many mathematicians Stuart would enjoy long walks (with their dog
Mabel), while thinking about mathematics problems.
Stuart then embarked on an MSc in Mathematics
at the Open University, his fascination with number
theory (and especially Goldbach’s conjecture) leading him to study modules based on Tom Apostol’sbook on analytic number theory. It was at this timethat he joined the LMS, confessing to me (as his LMS
proposer) that his dream was to add publications inpure mathematics to his publications in medicine
and biology. His tutors say that he would surely haveachieved this dream in time, but sadly it could not
happen. In April 2019, he suffered a brain aneurysm
and died in hospital some days later. He is survived
by his partner Dr Elizabeth Robinson, and by his
two adult children Jonathan and Andrew.

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

Mathematics Access Grid
Instruction and Collaboration

Students from universities outside the MAGIC
consortium can subscribe to MAGIC and joincourses, including assessment, for a small termlyfee. If you are a PhD supervisor or postgraduate
tutor, then the courses can provide low cost accessto high quality courses for your students.
2017CEMPS032

Online Advanced
Postgraduate
Courses inMathematics

MAGIC is consortium of 21 universities that runs
a wide range of PhD level lecture courses in pureand applied mathematics using video conferencing
technology. The lectures are streamed over the
web allowing students to interact in real time withcourse lecturers. Lectures are recorded so thatstudents can use them later.

Details of all the courses MAGIC provide can be found at: https://maths-magic.ac.uk

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EVENTSLMS Meeting

LMS Midlands Regional Meeting & Workshop
University of Nottingham, 11 September 2019
Website:tinyurl.com/y4vkvhld

Lectures are aimed at a general mathematical audience. The meeting forms part of the Midlands
Regional Workshop on Zeta Functions in Number
Theory and Mathematical Physics (11–13 September2019). Limited travel support is available: email christian.wuthrich@gmail.com for details. Visit the website for further details and to register for a place.
A Society Dinner will be held after the meeting at
the Victoria in Beeston, at a cost of £25.00 including
drinks. To reserve a place, email Chris Wuthrich.

Words in Finite and Profinite Groups

A Day of Lie Theory in Kent

Location:
Date:
Website:Location:
Date:
Website:University of Lincoln
16 September 2019tinyurl.com/y6xpjmgy

This is the ninth meeting of the Research Group
Functor Categories for Groups (FCG). The FCG Research Group is supported by an LMS Joint ResearchGroups in the UK Scheme 3 grant. Limited funding
is available for PhD students.

LMS Popular Lectures
Location:
Date:
Website:University of Birmingham
19 September 2019lms.ac.uk/events/popular-lectures

The LMS Popular Lectures are free annual events,
open to all, which present exciting topics in mathematics and its applications to a wide audience. The
speakers for the 2019 Popular Lectures are Professor Peter Higgins (University of Essex) and Professor
Alain Goriely (University of Oxford).

University of Kent
18 September 2019tinyurl.com/yy5m8uwn

Lie Theory is an old but pervasive subject in mathematics, due to its central role in understanding
symmetry. The meeting will discuss new research
directions led by Chris Bowman, Maud De Visscher,Radha Kesar and Stephen Donkin.

Novel Approaches to the Multi-Armed
Bandit Problem
Location:
Date:
Website:Imperial College London
25–26 September 2019tinyurl.com/y5n95ccn

This conference will bring together researchers frommathematics, statistics and computer science working on the multi-armed bandit problem. Register for
free online. Details of the submission deadline areon the webpage.

Sir Michael Atiyah: Forays into Physics
25 October 2019. Isaac Newton Institute, Cambridge
Website:tinyurl.com/yxhlkalx

This meeting is to celebrate Sir Michael’s contributions to physics. Confirmed speakers are Sir Roger
Penrose, Nigel Hitchin, Paul Sutcliffe, Matilde Mar-

colli and Nick Manton. Register to participate in themeeting. Early registration is advised due to limited
number of participants. The next day is the memorial service for Sir Michael at Trinity College Chapel
at 2.30.

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EVENTS

Edge Days 2019: Fano Varieties, Cone
Singularities and their Links

Young Researchers in Algebraic Number
TheoryLocation:
Date:
Website:Location:
Date:
Website:University of Edinburgh
4–8 November 2019tinyurl.com/y3xj764z

The workshop will focus on three closely relatedclasses of objects: Fano varieties, log-terminal singularities and links with a positive Sasakian structure.
It aims to bring together experts from singularity theory, birational geometry and mathematical physics.
Supported by an LMS Conference grant.

LMS Computer Science Colloquium
Location:
Date:
Website:De Morgan House, London
13 November 2019tinyurl.com/cscoll19

The theme of this event will be ‘Mathematics of Security’. The event is aimed at PhD students and post-docs,
although others are welcome to attend. It is free for students and £5 for others. Limited funding for travel isavailable. Register at tinyurl.com/cscoll19.

LMS/BCS-FACS Evening Seminar
Location:
Date:
Website:De Morgan House, London
21 November 2019tinyurl.com/yyc9oyse

Professor Marta Kwiatkowska (University of Oxford)
will give a talk titled When to Trust a Self-Driving Car....
The seminar is free of charge; to register your interest, email Katherine Wright, Society & Research Officer: lmscomputerscience@lms.ac.uk. See the website for an abstract and speaker biography.

University of Warwick
6–8 November
tinyurl.com/YRANT2019

The conference is aimed at PhD and post-docs working in or interested in algebraic number theory andarithmetic geometry. All participants are encouraged
to give talks. There will be three keynote talks, given
by Jack Thorne (University of Cambridge), Lynne
Walling (University of Bristol) and Sarah Zerbes (UCL).

Category Theory and its Applications
Postgraduate Conference
Location:
Date:
Website:University of Leicester
18–19 November 2019tinyurl.com/y5epeut4

This conference brings together postgraduate students in category theory and its applications topresent and discuss their research. It also features
three talks from established researchers.

LMS–IMA Joint Meeting:Mathematics of Planet Earth
Location:
Date:
Website:University of Reading
21 November 2019tinyurl.com/yxddzjqw

This joint event is run by the LMS and the Instituteof Mathematics and its Applications. The meeting is
free to attend. Further details and how to registerare available on the website. The lectures are aimed
at a general mathematical audience.

LMS Meeting

LMS Graduate Student Meeting
29 November 2019: 9.30 am – 2.00 pm, Goodenough College, Mecklenburgh Square, London WC1N 2AB
Website:lms.ac.uk/events/meeting/agm

The meeting will include student presentations of
their current work, with a prize awarded for thebest talk. This meeting is intended as an introduction to the AGM later in the day. Travel grants
of up to £100 are available for students who at-tend both the Graduate Student Meeting and theLMS AGM. The AGM will be followed by a wine reception. The Society’s Annual Dinner will be heldat 7.30 pm at Goodenough College. The cost of
the dinner will be £58.00, including drinks; email
AnnualDinner_RSVP@lms.ac.uk to reserve a place.

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EVENTSLMS Meeting

Annual General Meeting of the LMS
Goodenough College, Mecklenburgh Square, Holborn, London WC1N 2AB, 29 November 2019; 3.00 – 6.00 pm
Website:lms.ac.uk/events/meeting/agm

The speaker will be Marc Lackenby (Oxford). The
LMS President, Caroline Series (Warwick) will givethe presidential address. The meeting will include
the presentation of certificates to all 2019 LMS prize
winners and the announcement of the annual LMS
election results.

The meeting will be followed by a reception, which
will be held at Goodenough College, and the Society’s
annual dinner at 7.30 pm. The cost of the dinner willbe £58.00, including drinks. To reserve a place at thedinner, please email AnnualDinner_RSVP@lms.ac.uk.
For further details about the AGM, please contact
Elizabeth Fisher (lmsmeetings@lms.ac.uk).

PDE Models for Cancer Invasion

Probabilistic Coupling and Geometry

Location:
Date:
Website:Location:
Date:
Website:Queen’s University Belfast
6 December 2019tinyurl.com/y63v62hg

University of Warwick
9–10 December 2019tinyurl.com/y4tzewh7

This workshop will focus on mathematical models
for cancer invasion as well as their analytical andnumerical treatment, and aims to present recent
developments in this area. Supported by an LMS
Scheme 9 grant awarded to Anna Zhigun.

This workshop will bring together researchers fromdiverse fields that use coupling arguments, and will
be of interest to those working on adaptive MCMC,
probability and geometry, statistical shape analysis,
and perfect simulation.

Computational Complex Analysis

Integrability, Algebra and Geometry

Location:
Date:
Website:Location:
Date:
Website:

INI, Cambridge
9–13 December 2019tinyurl.com/y3bmuc5u

University of Glasgow
13 December 2019tinyurl.com/y4k4fxr3

The aim of the workshop on computational complex
analysis is to bring together numerical and appliedmathematicians to focus on: (a) new and existing
methods, (b) software tools, and (c) various application areas. Deadline for applications: 8 September.

The workshop will bring together people working in
integrable systems from various perspectives. Speakers are: Jenya Ferapontov (Loughborough), Marta
Mazzocco (Birmingham) and Daniele Valeri (Glasgow).
Supported by an LMS Scheme 9 grant.

British Postgraduate Model Theory
Conference 2020LMS South West & South Wales
Meeting & Workshop

Location:
Date:
Website:Location:
Date:
Website:University of Leeds
8–10 January 2020
tinyurl.com/bpgmtc20

This meeting aims to bring together young researchers interested in model theory. It will feature amini-course, invited talks by established academics,
and contributed talks by postgraduate researchers.
Supported by the LMS, the University of Leeds andthe British Logic Colloquium.

University of Bristol
15 January 2020
tinyurl.com/y37pked7

This meeting forms part of the South West & South
Wales Regional Workshop on Interactions between
Geometry, Dynamics and Group Theory on 16–17
January. Speakers are Martin Bridson FRS (Oxford),
Corinna Ulcigrai (Bristol) and Yves Benoist (Paris). Partial funding is available.

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Society Meetings and Events
November 2019

September 2019
6–7 Prospects in Mathematics Meeting,
Lancaster
11 Midlands Regional Meeting, Nottingham
19 Popular Lectures, University ofBirmingham

13
21
21
29
29

Computer Science Colloquium, London
LMS/BCS-FACS Evening Seminar, London
Joint Meeting with the IMA, Reading
Graduate Student Meeting, London
Society Meeting and AGM, London

January 2020

October 201919 DeMorgan@21, London

15 South West & South Wales Regional
Meeting, Bristol

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

September 2019
2 Applications of Mathematics in Ecology,
University of Kent (483)
2 One Day Function Theory Meeting, De
Morgan House, London (483)
4-6 Inverse Problems from Theory to Application, University College London (483)
5-6 Recent Trends in Stochastic Analysis andPartial Differential Equations, Universityof Chester (483)
5-6 Water Waves — Mathematical Theory& Applications, University of Plymouth(483)
5-7 British Logic Colloquium 2019, Universityof Oxford (483)
6-7 LMS Prospects in Mathematics Meeting,
Lancaster (484)8-11 Proof Society Summer School 2019,
Swansea University (483)
9-11 Curve Counting Theories and Related Algebraic Structures, University of Leeds(483)

9-13 Graph Complexes in Algebraic Geometry
and Topology, University of Manchester(482)
9-13 The Geometry of Derived Categories, University of Liverpool (483)9-13 The Complex Analysis Toolbox: New Techniques and Perspectives, INI, Cambridge(482)
11 LMS Midlands Regional Meeting, Nottingham (484)11-13 Proof Society Workshop 2019, Swansea
University (483)
11-13 Groups and Representation Theory, Conference in Memory of Kay Magaard, University of Warwick (483)12-13 Flood Risk Conference, Swansea University Bay Campus, Wales (483)
16 Random Matrices and Applications, University of Sussex (483)
16 Functor Categories for Groups, Universityof Lincoln (483)
16 Words in Finite and Profinite Groups, University of Lincoln (484)16-18 British Topology Meeting, Warwick (483)

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ii

17 Branching Processes and their Applications, University of Sussex (483)
18 Randomness, Symmetry and Free Probability, University of Sussex (483)
18 A Day of Lie Theory in Kent, Universityof Kent (484)18-19 Philip Maini’s 60th Birthday Workshop,
Mathematical Institute, Oxford (483)
18-20 Southampton–Bielefeld Geometric Group
Theory Meeting, University of Southampton (483)
18-21 Lattice Polytopes, with a View towards
Geometry and Applications, ICMS, University of Edinburgh (483)
19 LMS Popular Lectures, University of Birmingham (484)25-26 Novel Approaches to the Multi-Armed
Bandit Problem, Imperial College London(484)
29-4 Oct Clay Research Conference and Workshops, Mathematical Institute, Oxford(482)
30-4 Oct Structure Preservation and General Relativity, INI, Cambridge (482)

October 201910 Dependence Modeling and its Applications in Insurance and Finance, HeriotWatt University (483)7-11 Dirac Operators in Differential Geometry and Global Analysis, Bedlewo, Poland(482)
19 DeMorgan@21, De Morgan House, London (483)25 Sir Michael Atiyah: Forays into Physics,
INI Cambridge (484)28-1 Nov Complex Analysis in Mathematical
Physics and Applications, INI, Cambridge(483)

November 2019
4-8 Edge Days 2019: Fano Varieties, Cone Singularities and their Links, University of
Edinburgh (484)

6-8 Young Researchers in Algebraic Number
Theory, University of Warwick (484)
13 Computer Science Colloquium 2019, De
Morgan House, London (484)18-19 Category Theory and its Applications
Postgraduate Conference, University ofLeicester (484)21 Joint LMS Meeting with the IMA, Reading(484)
21 LMS/BCS-FACS Evening Seminar, De Morgan House, London (484)29 Graduate Student Meeting, London (484)29 Society Meeting and AGM, London (484)December 2019
6 PDE Models for Cancer Invasion, Queen’s
University Belfast (484)9-10 Probabilistic Coupling and Geometry
Workshop, University of Warwick (484)9-13 Computational Complex Analysis, INI,
Cambridge (484)13 Integrability, Algebra and Geometry, University of Glasgow (484)January 2020
8-10 British Postgraduate Model Theory Conference 2020, University of Leeds (484)15 South West & South Wales Regional Meeting, Bristol (484)

July 2020
5-11 8th European Congress of Mathematics,
Portorož, Slovenia
12-19 14th International Congress on Mathematical Education Shanghai, China

August 2020
17-21 IWOTA 2020, Lancaster University (481)

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ii

100 YEARS OF MATH MILESTONESThe Pi Mu Epsilon Centennial Collection
Stephan Ramon Garcia, Pomona College & Steven J. Miller, Williams College
This book is an outgrowth of a collection of 100 problems chosen to celebrate the 100th anniversary of theundergraduate math honor society Pi Mu Epsilon. Each chapter describes a problem or event, the progress
made, and connections to entries from other years or other parts of mathematics. Put together, these
problems will be appealing and accessible to energetic and enthusiastic math majors and aficionados ofall stripes.
Jul 2019 581pp 9781470436520 Paperback £59.50

EXPLORING CONTINUED FRACTIONS
From the Integers to Solar Eclipses
Andrew J. Simoson, King University
There is a nineteen-year recurrence in the apparent position of the sun and moon against the backgroundof the stars, a pattern observed long ago by the Babylonians. In the course of those nineteen years the
Earth experiences 235 lunar cycles. Suppose we calculate the ratio of Earth's period about the sun to the
moon's period about Earth. That ratio has 235/19 as one of its early continued fraction convergents,
which explains the apparent periodicity. Exploring Continued Fractions explains this and other recurrent
phenomena - astronomical transits and conjunctions, lifecycles of cicadas, eclipses - by way of continued
fraction expansions.
Dolciani Mathematical Expositions, Vol. 53
Jun 2019 371pp 9781470447953 Hardback £51.50

MAA Press

A FIRST JOURNEY THROUGH LOGIC

Martin Hils, Westfälische Wilhelms-Universität Münster & François Loeser, Sorbonne Université
Presents mathematical logic to students who are interested in what this field is but have no intention of
specialising in it. The point of view is to treat logic on an equal footing to any other topic in the mathematicalcurriculum. The book starts with a presentation of naive set theory, the theory of sets that mathematicians
use on a daily basis. Each subsequent chapter presents one of the main areas of mathematical logic: first orderlogic and formal proofs, model theory, recursion theory, Gödel’s incompleteness theorem, and, finally, theaxiomatic set theory.
Student Mathematical Library, Vol. 89
Sep 2019 195pp 9781470452728 Paperback £54.50

THE MATH BEHIND THE MAGIC
Fascinating Card and Number Tricks and How They Work
Ehrhard Behrends, Freie Universität
Translated by David Kramer
A magician appears able to banish chaos at will: a deck of cards arranged in order is shuffled - apparently
randomly - by a member of the audience. Then, hey presto! The deck is suddenly put back in its originalorder! Magic tricks like this are easy to perform and have an interesting mathematical foundation. In this rich,colourfully illustrated volume, Ehrhard Behrends presents around 30 card tricks and number games that areeasy to learn, with no prior knowledge required.
Jun 2019 208pp 9781470448660 Paperback £29.95

Free delivery at eurospanbookstore.com/ams
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