contentsgravitational waves, etc.) which play a dominant role in multiple areas of physics:...

Page

Director’s Foreword

Brief Scientific Report on Programmes

Programme Participation

National Advisory Board and UK Mathematics

International Activity

Other Institute News

Public Understanding Events for 10th Anniversary Year

Newton Institute Publications

Young Scientists

Scientific Steering Committee

Scientific Policy Statement

Future Programmes

Management Committee

Hewlett-Packard Senior Research Fellow

Programme Reports:

Foams and Minimal Surfaces

Computation, Combinatorics and Probability

New Contexts for Stable Homotopy Theory

Computational Challenges in Partial Differential Equations

Nonlinear Hyperbolic Waves in Phase Dynamics and Astrophysics

Finances

APPENDICES

Please note that the following statistical information may be obtained from the Institute on request, orfrom http://www.newton.cam.ac.uk/reports/0203/appendices.html

1 Long-Stay Participants2 Chart of Visits of Long-Stay Participants3 Junior Members of the Newton Institute4 Nationality and Country of Residence of Participants5 Cumulative Frequency Graph of Ages6 Preprints Produced by Participants7 Papers Produced or in Preparation by Participants8 Seminars and Lectures9 Seminars Given Outside the Institute

Contents

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44

50


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The year under review began auspiciously with thecelebration, on 2 July 2002, of the tenthanniversary of the Institute. A rich programme ofmathematics, and a meeting of our correspondentsfrom many universities, was followed by myinaugural lecture as NM Rothschild & SonsProfessor. This was chaired by Sir Alec Broers, ourever supportive Vice-Chancellor, and we weredelighted to welcome Sir Evelyn and Lady deRothschild and our Honorary Fellow Dr DillFaulkes. The generosity of our several benefactors,individual and corporate, has been essential to theestablishment of the Isaac Newton Institute as oneof the foremost centres of excellence in themathematical world.

That excellence depends of course on the quality ofproposals, and on the willingness of goodmathematicians to define and organise effectiveprogrammes. It is good to report that we have anexcellent pipeline of forthcoming programmes,running now into 2006. I have every confidencethat we shall maintain the very high scientificstandard in our second decade, and I pay tribute toprogramme organisers who put so much into theirdemanding role.

Only by keeping up the excitement and quality ofour science can we expect the support of fundingbodies that have to be highly selective in theallocation of their funds. During the year we faceda stringent review by the Engineering and PhysicalSciences Research Council of the quality ofresearch and the cost effectiveness of the Institute.We emerged from this review with flying colours,and with an assurance of continued funding atleast until 2008.

This is crucial for our planning, because EPSRCand its sister research councils are almost the onlysources of funding for the continuing operation ofour programmes. Support for long termparticipants, and for essential support staff, comesalmost entirely from them. Shorter workshopsusually require separate funding, and this isincreasingly difficult to achieve. We need todevelop new funding streams to give confidence toprogramme organisers to arrange such events in

the knowledge that they can meet the expenses oftheir visitors.

The Institute exists to advance mathematics and itsmany applications, to the benefit of the whole UKscientific community. I have continued to tour thecountry, seeking ideas and criticism from univer-sities in all parts, and have been encouraged by theconstructive approach of those with whom I havediscussed our work. One particularly encouragingdevelopment has been a closer relationship withthe International Centre for Mathematical Sciencesin Edinburgh, under its new Director ProfessorJohn Toland. The ICMS and the INI arecomplementary in their missions, the formerconcentrating on much shorter events, andtogether they add great strength to UKmathematics.

Among the unsung heroes are those who refereeprogramme proposals, and the members of theScientific Steering Committee who maintainrigorous standards in judging which will result insuccessful meetings. This is a major undertaking,and I am grateful to those who take it on, as I amgrateful too for the work of members of theNational Advisory Board and of the ManagementCommittee.

What impresses our visitors, however, is the warmwelcome and friendly efficiency of the Institutestaff. I am proud of the way we are able to lookafter the finance, accommodation, secretarial,computing and library requirements ofparticipants, and the way in which inevitableproblems are cheerfully tackled. The pleasant andconstructive atmosphere of the Newton Institute,which is part of our reputation, owes everything tothe work of all the staff, to whom I record my deepgratitude.

31 July 2003

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For full scientific reports see pages 19 to 44.

Programme 47: Foams and MinimalSurfaces

Foams are familiar gas−liquid systems of wideimportance to industry, exhibiting many propertiesof interest to physicists, mathematicians, engineersand other applied scientists. In recent yearscomputation has given fresh impetus to the subject,since it enables us to explore the consequences ofidealised models for complex disordered foams byaccurately simulating them. Some of the problemshave direct bearing on potentially significant areasof application, such as in determining theproperties of metallic foams.

This programme was conceived as an opportunityto maximise the cross-fertilisation of ideas betweenmathematicians, physicists and allied disciplinessuch as chemical engineering. It was successful inattracting world-leading figures from all thesecommunities. The topics considered ranged fromrefined minimisation problems to physicalexperiments and their simulation, includingdrainage and rheology.

The communities which were brought togetherinteracted well, and many future collaborationsshould result which would not have emergedwithout such a stimulus.

Programme 48: Computation, Combinatorics and Probability

The relationships between computer science andmathematics have become more wide ranging andprofound, and this has been particularly evident inquantitative theoretical computer science,especially in the areas known as computationalcomplexity and the analysis of algorithms. The aimof this programme was to explore two particularlyfruitful interfaces, namely those involvingcombinatorics and probability theory. Within thebroad area delineated by the title, four themesreceived special emphasis: randomised algorithms,random graphs and structures, phase transitionsand probabilistic analysis of distributed systems. Aseries of three very well attended workshops

provided rather hectic periods of concentratedinteraction, interspersed with quieter interludes forreflection.

As a result of this broad scope, the researchachievements of the programme were varied intopic. The work was highly collaborative, andoften bridged two or more areas. The overallsuccess of the programme can be judged by thevolume and quality of the output.

Programme 49: New Contexts for Stable Homotopy Theory

The versatility of stable homotopy theory and itsassociated range of cohomological techniques hasmade it an important branch of mathematics.Recently there have been several fundamentaldevelopments which have been used to solve anumber of long-standing questions in areas such ascomplex algebraic geometry, number theory andelliptic cohomology.

This programme was structured around threeworkshops (one being a pedagogical NATOAdvanced Study Institute) and a number of relatedlecture series. It started and finished with intensiveworkshop activity, separated by a period forsustained research and collaboration amongst thelonger term participants.

The programme was the foremost internationalevent in the subject in 2002, and one of specialsignificance as it brought together several strandsof work at seminal stages of their development. Aremarkable collection of researchers from a widerange of areas took part, and many newcollaborations were born. The programme madesignificant progress in a number of areas,including:

• Algebra of rings up to homotopy • Elliptic and chromatic phenomena • Equivariant cohomology theories • Algebraic groups and cohomology theories • K-theory • Algebraic and motivic homotopy theory • Abstract homotopy theory

The programme organised, disseminated andconsolidated the existing state of the subjects.



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govern a broad spectrum of physical phenomena incompressible fluid dynamics, nonlinear materialscience, general relativity, etc. Such equationsadmit solutions that may exhibit various kinds ofshocks and other linear and nonlinear waves(propagating phase boundaries, fluid interfaces,gravitational waves, etc.) which play a dominantrole in multiple areas of physics: astrophysics,cosmology, dynamics of (solid−solid) materialinterfaces, multiphase (liquid−vapour) flows,combustion theory, etc.

In recent years, major progress has been made inboth theoretical and the numerical aspects of thefield, while the number of applications hasskyrocketed. Hyperbolic models arising inapplications often face serious mathematicaldifficulties related to the occurrence ofdiscontinuities, coordinate singularities, resonancebetween two or more wave speeds, elliptic regionsin phase space, etc. This programme set out tobuild new bridges between the most recentdevelopments in the general mathematical theoryand the areas of applications that are currentlymost active. In particular, five main themes wereaddressed:

• Nonclassical shock waves and propagatingphase boundaries

• Well-posedness theory of systems ofconservation laws

• Multidimensional hyperbolic problems• Computational methods for complex fluid

flows• Hyperbolic models in general relativity

Weekly seminars, short courses, workshops andconferences provided valuable opportunities forreviewing the state of the art for physical models,techniques of mathematical analysis andnumerical analysis. Many challenging problems inthe field were discussed.

One of the most unexpected and lasting outcomesof the programme was the launching of a newmathematical research journal, the Journal ofHyperbolic Differential Equations, devotedprecisely to the topics treated during thisprogramme.

Programme 50: ComputationalChallenges in Partial Differential Equations

The study of partial differential equations (PDEs)is a fundamental area of mathematics which linksimportant strands of pure mathematics to appliedand computational mathematics, with relevance tothe physical, natural and social sciences. Scientistsand mathematicians have naturally been led toseek techniques for the approximation ofsolutions. High speed computation has led toconsideration of very large systems of PDEs andthe development of appropriate numericalalgorithms and their analysis, both theoretical andapplied.

This programme focussed on some of the mostexciting and promising mathematical ideas, and onthose branches of PDE theory that provide asource of physically relevant and mathematicallyhard problems to stimulate future developments.The three main themes were:

• Cognitive algorithms: adaptivity, feed-backand a posteriori error control

• Hierarchical modelling: multi-scale mathe-matical models and variational multi-scalealgorithms

• Nonlinear degenerate PDEs and problemswith interfaces

The programme attracted a large UK participationowing to the current strength of the UK in thisfield. A total of nine events, from one-day meetingsto week-long workshops, drove the programmeforward. Significant collaborative work was donein several areas, including nonlinear PDEs, freeboundary problems, superconductivity, evolvingsurfaces, microstructure, adaptivity, multiscalealgorithms and wave propagation. Applications inengineering and in industry received specialattention.

Programme 51: Nonlinear Hyperbolic Waves in Phase Dynamics andAstrophysics

Quasilinear hyperbolic systems in divergence form,commonly called “hyperbolic conservation laws”,

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A total of 1114 visitors was recorded for2002/2003. This includes 240 long-stay part-icipants, each staying between two weeks and sixmonths (6 weeks on average), and 351 short-stayparticipants who stayed for two weeks or less.Within the five completed programmes there was atotal of 27 workshops (periods of intense activityon specialised topics) which attracted a further 297visitors to the Institute. There were many otherswho attended informally at lectures, workshops,Institute Seminars or other events. Within all theprogrammes, workshops and other activities,around 988 seminars were given in total at theInstitute during the year.

In addition to workshops, which serve to widenUK participation in programmes, programmeorganisers are encouraged to organise moreinformal special days, short meetings or intensivelecture series which can attract daily or short-termvisitors, so further opening the activities of theInstitute to the UK mathematical community.

The Institute also funds visits by programmeparticipants to other UK institutions to giveseminars, and 123 such seminars took place lastyear.


Long-stay participants Short-stay and workshop participants

Programme Long-stay Mean stay Short-stay Mean stay participants (days) participants (days)

Foams and Minimal Surfaces 27 23 37 7

Computation, Combinatorics 46 72 23 9 and Probability

New Contexts for Stable 38 58 41 10 Homotopy Theory

Computational Challenges in 57 41 125 5Partial Differential Equations

Nonlinear Hyperbolic Waves in 72 32 125 7Phase Dynamics and Astrophysics

Totals 2002/2003 240 45 351 7

The pie charts below show the percentages of long-stay and short-stay participants broken down bycountry of residence:

Rest of World7%

E Europe 4%

W Europe37%

USA15%

Cambridge2%

UK (non-Cambridge)

35%

W Europe29%

E Europe 4%

Rest of World11%

USA28%

Cambridge 8%

UK (non-Cambridge)

20%


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The following chart summarises the total figures for long- and short-stay participation since the Institutebegan:

0

5000

10000

15000

20000

25000

98/99 99/00 00/01 01/02 02/03

The median age for long- and short-stay participants combined in 2002/2003 is 40 years, with aninterquartile range of 33−48 years. The following chart shows the cumulative frequency of participantages:

More detailed statistics, including visit dates and home institutions of participants, and a complete list ofseminars and papers are given in the Appendices, available separately from the Institute or at

http://www.newton.cam.ac.uk/reports/0203/appendices.html

0

10

20

30

40

50

60

70

80

90

100

20 25 30 35 40 45 50 55 60 65 70 75 80Age

0

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600

800

1000

1200

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92/93 93/94 94/95 95/96 96/97 97/98 98/99 99/00 00/01 01/02 02/03

Other visitors

Short-Stay/Workshop

Long-Stay

Age

Perc

enta

ge

The following chart summarises the total number of person-days for long- and short-stay participantscombined:

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National Advisory BoardFollowing discussions with EPSRC, a NationalAdvisory Board (NAB) for the Institute wasestablished during 1999. Its remit is “To advise theDirector in all matters relating to the role of theNewton Institute as a National Institute for theMathematical Sciences.”

The membership, as at 31 July 2003, is given in thetable above. The overlap with the ScientificSteering Committee and Management Committeeis deliberate and intended to ensure goodcommunication with the Board. Some of the issuesaddressed by the NAB have been:

• The attendance of young UK scientists• The Institute’s strategy vis-à-vis its national

role, interdisciplinarity and outreach• Flexibility in the scientific programming to

respond to new developments• The interests of the EPSRC programmes

which contribute to the Institute• A focus on the Institute’s databases to enable

it to produce information on subjectcoverage, the geographical distribution ofparticipants and the status of participants

• The Institute’s mode of interface withindustry

Anyone with views about the national role of theInstitute is invited to make these known to anymember of the NAB.

Symposia ActivitiesThe Institute continues to maintain a list offorthcoming UK symposia, workshops, etc., in theMathematical Sciences. This list is maintained inconsultation with representatives of LMS, IMA,RSS, ICMS (Edinburgh) and the WarwickMathematics Research Centre. For details, see

http://www.newton.cam.ac.uk/symposia.html

UK correspondents

During 2000/01, at the suggestion of the NationalAdvisory Board, the Newton Institute established alist of correspondents in UK Universities to act as achannel of communication between the Instituteand the mathematical sciences community in theUK. During 2002/03, following furthersuggestions, the Director wrote to a number ofrelevant non-University institutions and learnedsocieties to invite them also to nominatecorrespondents. All correspondents are regularlyinformed about activities of the Institute, and it istheir responsibility to ensure that the informationis disseminated to relevant groups and individualswithin their institution, in both mathematics andother appropriate departments. Correspondentsalso provide feedback to the Institute. The namesof all the correspondents so far established can befound on the Institute website at

http://www.newton.cam.ac.uk/correspondents.html


Professor Sir Michael Berry FRS University of BristolProfessor J Brindley University of LeedsProfessor KA Brown FRSE University of GlasgowProfessor J Howie FRSE Heriot-Watt UniversityDr RE Hunt Deputy Director, Newton InstituteProfessor Sir John Kingman FRS Director, Newton InstituteProfessor MA Moore FRS University of ManchesterDr H Ockenden University of OxfordProfessor EG Rees FRSE University of EdinburghDr M Sheppard Schlumberger Cambridge Research LtdProfessor AFM Smith FRS (Chairman Queen Mary, University of London Professor DM Titterington FRSE University of Glasgow

Membership of the National Advisory Board as at 31 July 2003 was as follows:


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Universities and other relevant bodies not yetrepresented on this list are encouraged to provide asuitable nominee.

SeminarsLong-term participants in Newton Instituteprogrammes are strongly encouraged to visit otherUK institutions during their stay at the Institute,and many did so during 2002/2003 (see p 4). To promote this activity, the Institute coverson request the travel costs within the UK for anyoverseas participant.

The Institute has recently set up a register ofoverseas participants who are willing to travel toother UK institutions to give seminars. It is hopedthat organisers of seminar series will consult thisregister when planning their schedule of speakersand will contact potential speakers directly. Theregister can be found at

http://www.newton.cam.ac.uk/programmes/Speakers.html

Satellite WorkshopsThe Institute encourages organisers of longer (4- or6-month) programmes to cooperate with localorganisers in holding “satellite” workshops at UKUniversities and institutions outside Cambridge.

Satellite workshops are on themes related to theInstitute programmes, and involve a significantnumber of longer-term overseas participants fromthe Institute. They also, crucially, draw in andinvolve UK mathematicians and scientists whomight not otherwise be able to participatesubstantially in the Institute programme.

Costs for satellite workshops are typicallyapproximately £10,000 (excluding the overseastravel costs of Institute participants) and are sharedapproximately 50/50 between the Institute and thehost institution. Both EPSRC and LMS welcomeapplications from host institutions for grants tocover their share of the costs (subject to the usualreview procedures), and we are very grateful toboth organisations for the fact that all suchapplications have so far been successful.

Institutions interested in holding satelliteworkshops should contact either the organisers ofthe relevant programme or the Deputy Director, DrRE Hunt ([email protected]).

Institute SeminarsThe regular series of Institute Seminars, held onMondays during term-time, is intended to be ofgeneral interest and to attract a wide range ofmathematical scientists. Audio files of InstituteSeminars, with accompanying transparencies andstills, are published on the web at

http://www.newton.cam.ac.uk/webseminars/

This year’s seminars were:

• László Lovász (Microsoft Research),Global information from local observation

• Bill Dwyer (Notre Dame), Lie groups andtheir classifying spaces

• Jeffrey Steif (Chalmers), An overview ofcertain phase transitions

• Douglas Ravenel (Rochester), What iselliptic cohomology?

• Steve Lichtenbaum (Brown), Special valuesof zeta-functions

• Tom Pence (Michigan), Multifield theoryfor materials

• Charlie Elliott (Sussex), Mathematical models and computation of fluxpenetration in type II superconductors

• Carsten Carstensen (Vienna), Mathematicaland computational aspects of nonconvexminimisation problems allowing formicrostructure

• Constantine Dafermos (Brown), Progress inhyperbolic conservation laws

• Franco Brezzi (Pavia), Theoretical andnumerical questions related to thin elasticplates and shells

• Joel Smoller (Michigan), Cosmology of thelarge scale structure of the universe

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EMS

The European Mathematical Society (EMS) wasfounded in 1990 in Madralin, near Warsaw(Poland). The meeting which created the EMS washeld under the auspices of the EuropeanMathematical Council, chaired by Sir MichaelAtiyah before he came the first Director of theNewton Institute.

The purpose of the Society is “to further thedevelopment of all aspects of mathematics in thecountries of Europe.” In particular, the Societyaims to promote research in mathematics and itsapplications, as well as concerning itself with thebroader relation of mathematics to society. TheEMS acts as an intermediary betweenmathematicians and those in charge of politics andfunds in Brussels; the membership consists ofabout 50 mathematical societies throughoutEurope and around 2000 individual members whohave joined through their national societies. Moreinformation can be obtained from the Society’swebsite at http://www.emis.de/

The current Director of the Newton Institute, SirJohn Kingman, became President of the EMS inJanuary 2003. His term of office runs until the endof 2006.

ERCOM

ERCOM (“European Research Centres OnMathematics”) is a committee of the EMSconsisting of the directors of all the researchcentres and institutes throughout Europe whichhave a substantial visitor research programme inthe mathematical sciences. ERCOM was foundedin 1997 and meets annually. The current chair isProfessor M Castellet of the Centre de RecercaMatemàtica in Barcelona. The purposes ofERCOM are:

• to constitute a forum for communication and exchange of information between thecentres themselves and with EMS

• to foster collaboration and coordinationbetween the centres and with EMS

• to foster advanced research training on aEuropean level

• to advise the Executive Committee of the EMS on matters relating to activities of thecentres

• to contribute to the visibility of the EMS• to cultivate contacts with similar research

centres within and outside Europe

ERCOM is particularly concerned at present withthe EC’s Framework programmes, and is makingrepresentations to ensure that basic researchreceives sufficient support in both the current andfuture frameworks, and that the underpinningnature of mathematics in the sciences (and indeedmore widely) is properly recognised.

More information about ERCOM can be obtainedfrom its website at http://www.crm.es/ERCOM/or from the Newton Institute. European visitorsare particularly encouraged to pick up a leaflet.

EPDI

The European Post-Doctoral Institute forMathematical Sciences (EPDI) was founded in1995 by the Newton Institute, the Institut desHautes Études Scientifiques (Bures-sur-Yvette,France) and the Max-Planck-Institut fürMathematik (Bonn, Germany). Since then six morecentres have joined the group, in Leipzig, Vienna,Djursholm, Warsaw, Barcelona and Zurich.

Each year, EPDI offers five two-year grants toyoung European scientists who have recentlycompleted their PhDs, on condition that theyspend at least 18 months in a foreign country andbetween 6 and 18 months at one of the EPDIinstitutes. Competition is strong and the scientificquality is high. British applicants, who arecurrently under-represented, are particularlyencouraged.

Further information, including full conditions forgrant applications, can be found at the EPDI website, http://www.ihes.fr/EPDI/


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Director. In 1994 he moved to become Master of StJohn’s College, Cambridge, and Professor in theDepartment of Applied Mathematics andTheoretical Physics.

The Institute for Advanced Study was founded in1930 and is an independent, private institutionwith an academic membership of around 200 atany one time. The Institute’s research fields includemathematics, social sciences, historical studies andnatural sciences, and past faculty members includeAlbert Einstein, John von Neumann and KurtGödel.

Stokes Centenary Meeting

On Tuesday 18 March 2003 the Institute hosted ameeting to celebrate the centenary of Sir GeorgeGabriel Stokes (1819−1903). The day-long eventwas devoted to his life, his scientific work and hisinfluence on modern mathematics and science. Thespeakers were Professors Michael Hayes(University College Dublin), Keith Moffatt(Cambridge), Alastair Wood (Dublin CityUniversity), David Wilson (Iowa State), SirMichael Berry (Bristol) and Sir Michael Atiyah(Edinburgh). Following the talks themselves, anexhibition of Stokes memorabilia (includingoriginal letters and other documents) was held atPembroke College before dinner. The meeting wasextremely well attended.

The talks can be heard online at

http://www.newton.cam.ac.uk/webseminars/stokes/

Professor Goddard’s research is in string theoryand conformal field theory. He was elected aFellow of the Royal Society in 1989, and receivedthe Dirac Prize and Medal of the InternationalCentre for Theoretical Physics in 1997 inrecognition of his “farsighted and highlyinfluential contributions to theoretical physics”over many years. He received his CBE in 2002, thesame year in which he became President of theLondon Mathematical Society (as which he willnow stand down in November 2003). He is also aFellow of the Institute of Physics and of the RoyalSociety of Arts.

The Newton Institute was formally incorporated in1990, two years ahead of its official opening. Thefirst Director of the Institute, Sir Michael Atiyah,was also President of the Royal Society at the timeand so the then Dr Goddard, who had been pivotalin the establishment and development of theInstitute, was appointed as its first Deputy

Professor Peter Goddard

J Hedgecoe

Peter GoddardProfessor Peter Goddard CBE ScD FRS, one of theprime movers in the establishment of the IsaacNewton Institute, is to become the eighth directorof the Institute for Advanced Study, Princeton, on1 January 2004.

Professor Keith Moffatt delivering his lecture at theStokes Centenary Meeting

S Wygard

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episodes starting in January 2004. They will alsobe shown in a video entitled Journey throughCollege Mathematics being produced by ValenciaCommunity College in Orlando.

All available copies of the posters have now beendistributed or sold, despite several reprintings.However, the Institute still receives regular requestsfor copies, so an agreement has now been reachedfor the Mathematical Association to reprint theposters and sell them through its website. TheInstitute has also produced a pocket-sized bookletof the posters, which will be sold at the Instituteand via its website for a nominal £1.00 (pluspostage and packing) per booklet.

The posters can be seen on the web at

http://www.newton.cam.ac.uk/wmy2kposters/

where details of the booklet and the MathematicalAssociation reprinting can also be found.

Elements of Grace ExhibitionFrom 5 September to 31 October 2002 theInstitute hosted an exhibition by Canadian artistCatherine M Stewart (see front and back covers),with the financial assistance of the Canada Councilfor the Arts. The exhibition consisted of two suites:Elements of Grace is a collection of photo-etchingscombining diagrams from Principia Mathematicawith photographic details of the human body, andCopernican Notes is a set of multiple plate etchingscombining text and diagrams from DeRevolutionibus Orbium Cœlestium with images ofmoving figures.

The artist explained that “The use of the humanform in combination with these scientificreferences highlights the fact that it is through ourbodies that we experience reality and acquireknowledge about the physical world. In both setsof prints, the human body is visually linked to theeternal forces of nature.”

Catherine M Stewart kindly donated one of theworks, Deriving the Motions of Saturn and JupiterII from Copernican Notes, to the Institute, whereit is now displayed on the first floor.

National Science WeekDr Robert Hunt, Deputy Director of the Institute,gave a public lecture on Saturday 22 March 2003as part of National Science Week. In his lecture,entitled Maths at work in the real world, Dr Huntexplained how mathematics, despite sometimesappearing abstract and irrelevant to many people,is actually at work in our everyday lives. The talkfocussed on examples of mathematics in natureand on how pieces of technology which we take forgranted depend on mathematics for their veryexistence, and conveyed the excitement of applyingmathematics. The lecture was well attended andthe audience included many teenagers.

Dr Robert Hunt addresses the audience at his publiclecture entitled “Maths at work in the real world”

S Wygard

Posters in the LondonUnderground

The Maths in the Underground project, which randuring 2000 in the trains of the LondonUnderground, was described in the Institute’sAnnual Reports for 1999−2000 and 2000−2001but continues to generate considerable interest. Areprinting, kindly funded by EPSRC, resulted infree sets of the posters being distributed to everyUK secondary school courtesy of the OCRExaminations Board in September 2001. Amongstother film and TV appearances, the posters wererecently used as set dressing in the MGM filmAgent Cody Banks released in 2003, and will befeatured in the maths classroom in the next seriesof Grange Hill to be shown on BBC1 for 20

Public Understanding Events for 10thAnniversary Year

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Last year the Newton Institute celebrated its tenthanniversary as a national resource for research inthe mathematical sciences since it was opened in1992 by HRH the Duke of Edinburgh.

A variety of events took place in summer 2002 tomark the occasion, some academic and othersaimed at the general public. The academic eventshave already been described in last year’s AnnualReport. The public events were aimed at threeseparate age groups and were extensivelyadvertised within Cambridgeshire and the localarea, especially to schools, and more widely via theweb.

Each of the three public talks was well attended byaround 100 people including many teenagers andyounger children, and they were all enthusiasticallyreceived. The talks were:

• How Long is a Piece of String? for 8−11year olds, by Rob Eastaway (author of Whydo Buses Come in Threes? and How Long isa Piece of String?), Wednesday 3 July 2002

• Newton’s Apple (and other interestingmaths) for 11−16 year olds, by KjartanPoskitt (author of the Murderous Mathsseries of books), Saturday 29 June 2002

• Our Universe and Others for ages 16+ andthe general public, by John Barrow (authorof 15 popular books including Pi in the Sky,The Artful Universe and The Book ofNothing), Saturday 6 July 2002

More photographs from all three events togetherwith audio recordings of the complete talks can befound at

http://www.newton.cam.ac.uk/webseminars/10th_anniversary

Public Understanding Events for 10th Anniversary Year

Kjartan Poskitt in full flow during “Newton’s Apple (and other interesting maths)”

Rob Eastaway and his willing assistant during“How Long is a Piece of String?”

John Barrow presenting “Our Universe and Others”

S Wilkinson

S Wilkinson

S Wilkinson

Visitors trying to solve puzzles during the interval

S Wilkinson

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Papers and PreprintsOver 100 papers were produced or in preparationby participants at the Institute during 2002/2003 (acomplete list is given in Appendix 7). Many ofthese are included in the Newton Institute’sPreprint Series to which participants areencouraged to submit papers. A web page givingdetails of Newton Institute preprints is available at

http://www.newton.cam.ac.uk/preprints.html

Books arising from NewtonInstitute ProgrammesThe following titles were published during2002/2003:

M Beck (Ed.)Environmental Foresight and Models: A ManifestoElsevier, 2002473pp, ISBN 008044086X, (Hbk) £75.00

M Burger and A Iozzi (Eds.)Rigidity in Dynamics and GeometrySpringer, 2002492pp, ISBN 3540432434, (Hbk) £56.00

S Fomin (Ed.)Symmetric Functions 2001: Surveys ofDevelopments and PerspectivesKluwer, 2002288pp, ISBN 1402007736, (Hbk) £69.00

J Norbury and I Roulstone (Eds.)Large-Scale Atmosphere−Ocean Dynamics I:Analytical Methods and Numerical ModelsCambridge University Press, 2002400pp, ISBN 052180681X, (Hbk) £50.00

J Norbury and I Roulstone (Eds.)Large-Scale Atmosphere−Ocean Dynamics II:Geometric Methods and ModelsCambridge University Press, 2002394pp, ISBN 0521807573, (Hbk) £50.00

A complete list of books published as a result ofNewton Institute programmes is available at

http://www.newton.cam.ac.uk/inibooks.html


Young Scientists

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Young Scientists

The Institute holds a number of events each yearwhich are specifically targeted at young scientists.In 2002/2003 these events included:

• Euroworkshop on Randomised Algorithms

• Euroworkshop on Elliptic Cohomology andChromatic Phenomena

• Euroconference on Multiscale Modelling,Multiresolution and Adaptivity

• Euroconference on Hyperbolic Models inAstrophysics and Cosmology

• Workshop on Very High-Order NumericalSchemes for Conservation Laws

• Series of lectures entitled An Introduction toNumerical Methods

The following young scientists were recipients ofbursaries from the Cambridge PhilosophicalSociety in 2002/2003:

Computation, Combinatorics and Probability• A Hubenko (Memphis)• N Fountoulakis (Oxford)

New Contexts for Stable Homotopy Theory• N Ganter (MIT)• S Wuetrich (Berne)

Computational Challenges in Partial DifferentialEquations

• O Lakkis (Sussex)• V Styles (Milan)

Nonlinear Hyperbolic Waves in Phase Dynamicsand Astrophysics

• AO Arancibia (Manchester Metropolitan)• T Iguchi (Tokyo Institute of Technology)

The Institute recognises that junior researchershave much to contribute to and much to gain fromInstitute programmes and events. In order tomaximise the information available to juniorresearchers, and to facilitate their involvement inInstitute activities, we introduced in 1997 acategory of Junior Membership of the NewtonInstitute. To be eligible for membership you mustbe a Research Student or within 5 years of having

received a PhD (with appropriate allowance forcareer breaks) and you must work or study in a UKUniversity or a related research institution.

Junior Members receive regular advance inform-ation about programmes, workshops, conferencesand other Institute events via a Junior Members’Bulletin; detailed information about anyworkshops of an instructional or general naturelikely to be of special interest to young researchers;and information about suitable sources of fundingor support for visits to the Institute, whenavailable.

The Institute makes available some of its generalfunds specifically to support junior researchers inInstitute activities. Junior Members may apply forgrants from these funds. The types of involvementsupported include (but are not limited to)attendance at workshops, conferences, etc., andvisits of up to two weeks to work or study withlonger-term participants in the Institute’sprogrammes. Those wishing to become JuniorMembers should consult the Institute’s web site at

http://www.newton.cam.ac.uk/junior.html

Participants in the programme “ComputationalChallenges in Partial Differential Equations”

S Wygard

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more participants may be invited. Each prog-ramme is allocated a budget for salary support,subsistence allowances and travel expenses.

The following members of the Scientific Steeringcommittee stepped down at the end of their term ofservice on 31 December 2002:

• Professor RH Dijkgraaf (Amsterdam)• Professor CM Elliott (Sussex)• Professor WT Gowers FRS (Cambridge)

The following new members were elected:

• Professor S Abramsky (Oxford)• Professor SK Donaldson FRS (Imperial)• Professor BW Silverman FRS (Bristol)


The Institute invites proposals for researchprogrammes in any branch of mathematics or themathematical sciences. The Scientific SteeringCommittee (SSC) meets in April and October eachyear to consider proposals for programmes (of 4-week, 4-month or 6-month duration) to run two orthree years later. Proposals to be considered atthese meetings should be submitted by 31 Januaryor 31 July respectively. Successful proposals areusually developed in a process of discussionbetween the proposers and the SSC conductedthrough the Director, and may well be consideredat more than one meeting of the SSC beforeselection is recommended. Proposers may wish tosubmit a shorter ‘preliminary’ proposal in the firstinstance with a view to obtaining feedback fromthe SSC prior to the submission of a full ‘definitive’proposal.

Further details of the call for proposals, includingguidelines for submission, can be found on theInstitute’s website at

http://www.newton.cam.ac.uk/callprop.html

The scientific planning and organisation of eachprogramme are the responsibility of a team ofthree or four Organisers (aided in some cases by anAdvisory Committee). The Organisers recommendparticipants in the programme, of whom up totwenty can be accommodated at any one time; theyalso plan short-duration workshops and conf-erences within the programme, to which many

Membership of the Scientific Steering Committee at 31 July 2003 was as follows:

Professor S Abramsky University of OxfordProfessor JM Ball FRS FRSE University of OxfordProfessor C Bernardi University of ParisProfessor SK Donaldson FRS Imperial College LondonProfessor Sir John Kingman FRS (Secretary) Director, Newton InstituteProfessor AJ Macintyre FRS FRSE University of EdinburghProfessor TCB McLeish University of LeedsProfessor MA Moore FRS University of Manchester Professor EG Rees FRSE (Chairman) University of EdinburghProfessor G Ross FRS University of OxfordProfessor BW Silverman FRS University of BristolProfessor J Stark Imperial College London Professor JR Whiteman Brunel University

Professor EG Rees, Chairman of the Scientific Steering Committee


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From its inception, it has been intended that theNewton Institute should be devoted to theMathematical Sciences in the broad sense. In thisrespect the Institute differs significantly fromsimilar institutes in other countries. The range ofsciences in which mathematics plays a significantrole is enormous, too large for an Institute ofmodest size to cover adequately at any one time. Inmaking the necessary choices, important principlesare that no topic is excluded a priori and thatscientific merit is to be the deciding factor.

One of the main purposes of the Newton Instituteis to overcome the normal barriers presented bydepartmental structures in Universities. Inconsequence, an important, though not exclusive,criterion in judging the ‘scientific merit’ of aproposed research programme for the Institute isthe extent to which it is ‘interdisciplinary’. Oftenthis will involve bringing together researchworkers with very different backgrounds andexpertise; sometimes a single mathematical topicmay attract a wide entourage from other fields.The Institute’s Scientific Steering Committeetherefore works within the following guidelines:

(a) the mixing together of scientists with differentbackgrounds does not per se produce a successfulmeeting: there has to be clear common ground onwhich to focus;

(b) each programme should have a substantial andsignificant mathematical content;

(c) each programme should have a broad base inthe mathematical sciences.

Research in mathematics, as in many othersciences, tends to consist of major breakthroughs,with rapid exploitation of new ideas, followed bylong periods of consolidation. For the NewtonInstitute to be an exciting and important worldcentre, it has to be involved with thebreakthroughs rather than the consolidation. Thismeans that, in selecting programmes, a maincriterion should be that the relevant area is in theforefront of current development. Since theInstitute’s programmes are chosen two to threeyears in advance, it is not easy to predict where the

front line will be at that time. The best one can dois to choose fields whose importance and diversityare likely to persist and to choose world leaders inresearch who are likely to be able to respondquickly as ideas change.

Although the novelty and the interdisciplinarynature of a proposed programme provideimportant criteria for selection, these must besubject to the overriding criterion of quality. Withsuch a wide range of possibilities to choose from,the aim must be to select programmes whichrepresent serious and important mathematicalscience and which will attract the very bestmathematicians and scientists from all over theworld. However, the Institute is receptive also toproposals of an unorthodox nature if a strongscientific case is made.

Although the Institute operates on a world-widebasis and contributes thereby to the generaladvancement of mathematical science, it must alsobe considered in the context of UK mathematics. Anatural expectation of all those concerned is thateach programme will be of benefit to the UKmathematical community in a variety of ways. Ifthe UK is strong in the field, UK scientists will playa major part in the programme; if the UK iscomparatively weak in the field, the programmeshould help to raise UK standards. Instructionalcourses, aimed primarily at younger researchersand research students, will play a vital role here.

Because of the wide base of support for theNewton Institute in the EPSRC and elsewhere, theInstitute’s programmes shall as far as possiblerepresent an appropriate balance between thevarious mathematical fields. In order to retain thebacking of the mathematical and scientificcommunity, the Institute will run programmes overa wide range of fields and, over the years, achievethis balance. Such considerations, however, aresecondary to the prime objective of having highquality programmes.

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Future Programmes

Future Programmes The diagram below shows the forthcoming programmes which have been selected by the ScientificSteering Committee. To participate in a workshop, registration is required. For longer-term participationin a programme, an invitation is usually required, and applications are best made to the programmeorganisers in the first instance. Further details of each of these programmes, including

• the scientific content and background• the names of the organisers• the names of those who have so far been invited to take part in the programme• contact details • dates, topics and information about workshops which will take place during the programme

can be found on the Newton Institute website at http://www.newton.cam.ac.uk/programmes/

Further information on how to participate in Newton Institute programmes can be found athttp://www.newton.cam.ac.uk/participation.html

JULJAN DECSEP

2003

2004

2005

Granular and Particle-Laden FlowsComputational Challenges in Partial Differential Equations

Interaction and Growth in Complex Stochastic Systems

Magnetohydrodynamics of Stellar Interiors

MagneticReconnection

Theory

Statistical Mechanics of Molecular and Cellular Biological Systems

Random Matrix Approaches in Number Theory Quantum Information Science

Model Theory and Applications to Algebra and Analysis

Developments in Quantitative Finance

Nonlinear Hyperbolic Waves in Phase Dynamics and Astrophysics

Spaces of KleinianGroups and Hyper-bolic 3-Manifolds

2006

Principles of the Dynamics of Non-Equilibrium SystemsSpectral Theory andPartial Differential

Equations

TBA

Noncommutative GeometryLogic and Algorithms

Pattern Formation in Large Domains

Global Problems in Mathematical Relativity


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Membership of the Management Committee at 31 July 2003 was as follows:

Dr A Bramley EPSRCProfessor J Brindley Co-opted at the discretion of the CommitteeProfessor WJ Fitzgerald Council of the School of TechnologyProfessor GR Grimmett Head of Department, DPMMSProfessor EJ Hinch FRS Trinity College Professor J Howie FRSE LMSDr RE Hunt (Secretary) Deputy Director, Newton InstituteProfessor PT Johnstone St John’s CollegeProfessor Sir John Kingman FRS Director, Newton InstituteProfessor PV Landshoff (Chairman) General Board Professor TJ Pedley FRS Head of Department, DAMTPProfessor EG Rees FRSE Chairman of Scientific Steering CommitteeProfessor Sir Martin Rees FRS Council of the School of Physical SciencesDr C Teleman Faculty of Mathematics

The Management Committee is responsible foroverall control of the budget of the Institute, andfor both its short-term and long-term financialplanning. The Director is responsible to theManagement Committee, which provides essentialadvice and support in relation to fund-raisingactivity, employment of staff at the Institute,appointment of organisers of programmes,housing, library and computing facilities, publicity,and general oversight of Institute activities.

Its aim is to facilitate to the fullest possible extentthe smooth and effective running of the visitorresearch programmes of the Institute and allrelated activities. The Committee is especiallyconcerned with the interactions between theInstitute and its funding bodies, particularly theUK Research Councils, Cambridge University, theCambridge Colleges, the London MathematicalSociety, the Leverhulme Trust and others. Itgenerally meets three times a year.

Staff of the Institute

The staff (full-time and part-time) of the Instituteat 31 July 2003 was as follows:

• Wendy Abbott, Director’s AdministrativeAssistant

• Amy Abram, Conference and ProgrammeAssistant

• Dr Mustapha Amrani, Computer Systems

Manager

• Tracey Andrew, Conference and

Programme Secretary

• Elsie Batcheler, Assistant Librarian

• Lynn Berry, Catering Assistant

• Jonathan Chin, Deputy Computer Systems

Manager

• Esperanza de Felipe, Housing Officer

• Louise Grainger, Receptionist

• Matt Hodson, Technical Assistant

• Dr Robert Hunt, Deputy Director

• Professor Konstantin Khanin, Hewlett-

Packard Senior Research Fellow

• Professor Sir John Kingman FRS, Director

• Doreen Rook, Clerk

• Christine West, Institute Administrator

• Sara Wilkinson, Librarian and Information

Officer (on leave)

• Stephen Williams, Senior Accounts Clerk

• Sarah Wygard, Acting Librarian and

Information Officer

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In the past year I have been working on threedifferent problems: Burgers turbulence inunbounded domains and KPZ scalings, rigiditytheory in one-dimensional dynamics and KAMtheory via renormalisations.

In a joint paper with J Bec (Nice) we started astudy of Burgers turbulence in unbounded domainsmore than a year ago. The statistical properties ofshocks and the velocity field are determined by thestructure of minimisers for a random time-dependent Lagrangian system with a potential ofthe form F(x)W(t), where W(t) is the white noise.It is interesting that fluctuations of minimisers forpotentials of this type are the same as for KPZpotentials F(x,t) which are white both in x and t. Inboth cases the scaling exponent for the spatialfluctuations of minimisers is equal to −2/3. However,the structure of minimisers seems to be muchsimpler in the first model which perhaps makes itmore tractable rigorously. Notice that the analysisof the statistics of minimisers is closely related tothe problems of first-passage percolation anddirected polymers. Potentials of the type F(x)W(t)are very interesting on their own, and it seems thatthey have not been studied before. We call suchpotentials quasi-stationary. Indeed in many casesthey behave like stationary time-independentrandom potentials. At the same time, in the casewhen F(x) is bounded, many properties of directedpolymers and minimisers in quasi-stationarypotentials are similar to the properties of thecorresponding objects for completely independentnon-stationary random potentials F(x,t). The mainaim of this ongoing research project is to studydirected polymers in quasi-stationary potentials indifferent regimes and settings.

In the area of rigidity theory the main efforts of ourjoint research project with A Teplinsky (Ukraine)were concentrated on analysis of smoothness forconjugacies between two circle diffeomorphismswith singularities. We consider both critical circlemaps and diffeomorphisms with breaks. It isassumed that both mappings have the sameirrational rotation number and the same local

structure of singular points, that is the same orderof critical point or the same size of break. Themain result in the case of critical circle maps can beformulated in the following way: two critical mapswith the same but arbitrary irrational rotationnumber are C1-smoothly conjugate to each other.In other words, the asymptotic metric properties oftheir trajectories are the same. It is quiteremarkable that the rigidity properties are strongerin the presence of singularities. Indeed, in the caseof smooth diffeomorphisms a similar result isknown to be false. The result will be proven in apaper which is still in progress but we hope tofinish in 2004. To the best of my knowledge this isthe first rigidity-type result which holds withoutany Diophantine restrictions on a rotation number.

The third topic, KAM theory via renormalisations,is not new. Renormalisations in this context wereused previously by R MacKay, K Khanin & YSinai, H Koch and others. Most of the results wereobtained in the two-dimensional case whichcorresponds to Hamiltonian systems with twodegrees of freedom. In this case the aim is toconstruct two-dimensional invariant tori withquasi-periodic trajectories on them. Using Poincaréfirst return maps the problem can be reduced toconstruction of invariant curves for two-dimensional area-preserving maps. In this case onecan use continued fractions to define a naturalrenormalisation scheme. However, in the multi-dimensional case the situation is morecomplicated. The new idea which we use in a jointwork with J Marklof (Bristol) and J Lopes-Dias(Instituto Superior Tecnico, Lisboa) is to use amulti-dimensional continued fraction algorithmbased on the geodesic flow in the space of lattices.This algorithm was proposed by J Lagarias about10 years ago but was not used in dynamicalsystems applications. In my view renormalisationbased on Lagarias’ algorithm is a very powerfultool which will be used intensively in dynamics inthe future.

Report from Professor Kostya Khanin


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29 July to 23 August 2002Report from the Organisers: A Kraynik(Sandia), H Stone (Harvard),E Terentjev (Cambridge), DL Weaire(Dublin)

BackgroundFoams are familiar gas−liquid systems of wideimportance to industry, exhibiting many propertiesof interest to physicists, mathematicians, engineersand other applied scientists. In mathematicalterms, their governing principle (at least inequilibrium) is the minimisation of surface area.The study of idealised models of foam thereforethrows up a long list of significant mathematicalproblems, some of them associated with the nameof Joseph Plateau and his classic 1873 book. Inrecent years computation has given fresh impetusto the subject, since it enables us to explore theconsequences of idealised models for complexdisordered foams by accurately simulating them.Some of the problems have direct bearing onpotentially significant areas of application, such asin determining the properties of metallic foams.

Programme OutlineThis programme was conceived as an opportunityto maximise the cross-fertilisation of ideas betweenmathematicians, physicists and allied disciplinessuch as chemical engineering. It was successful in

attracting world-leading figures from all thesecommunities. Particularly notable are ThomasHales, who in recent years has produced proofs ofclassic minimal conjectures (the Kepler problemand the honeycomb problem mentioned below),and Ken Brakke, originator and developer of theSurface Evolver software which is widely used inproblems of surface energy minimisation.

The schedule of topics moved progressively fromrefined minimal problems to physical experimentsand their simulation. At the outset, Hales outlinedhis strategy for the honeycomb and otherproblems. That the 2D honeycomb minimises linelength for cells of equal size and any shape hasbeen recognised for centuries, but proving this hasbeen intractable until now unless restrictions suchas straight edges are imposed. The general proof ofHales is what he calls an ‘engineering’ solution,patched together out of various inequalities whichrigorously entrap every possible case.

What about the 3D case, the so-called KelvinProblem? In 1887 Kelvin conjectured that surfacearea was minimised by a regular stacking of his‘tetrakaidecahedron’. His remarkable notebooks,in which some of his sudden insights into thisproblem are recorded, are stored in the CambridgeUniversity Library, and participants were kindlygiven permission to scrutinise them during theprogramme itself. His conjecture was overthrownby the computational discovery of theWeaire−Phelan structure in 1994, and a proof bySullivan and Kusner that this indeed has a lowersurface area. But the proof that it is an absoluteminimum remains to be found, and Halescommented (not very optimistically) on thisdifficult challenge.

Another challenge was thrown up by AndrewKraynik, in the course of describing extensivesimulations of large samples of disordered foam. Itappeared that in the case of bubbles of equalvolume, the lowest possible energy in such a bulkstructure is that of a regular dodecahedron,obeying Plateau’s laws for angles and curvature.However, Ken Brakke was able to show, byconformal transformation, that the regular

Left to right: S Cox, Secretary to the organisingcommittee, with organisers DL Weaire and A Kraynik

S Wilkinson

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dodecahedron is a saddle point, and that theenergy can be lower for asymmetric bubbles. Yetanother challenge concerns an upper bound on theshear elastic modulus of a 2D disordered foam. Inaddition, John Sullivan adduced a long list ofrather general conjectures about periodicity,pressures and other properties of equilibriumfoams. In some cases the physicists protested thatthe answers were ‘obvious’ and indeed were able toshow the value of physical insights.

Careful analysis of the statistics of the cellularpatterns that represent foams in two dimensionshave thrown up several intriguing correlations;these were reviewed by Schliecker, Delannay andothers. The case of three dimensions has provedmore difficult. For example, what is the 3Dequivalent of von Neumann’s law in 2D, whichrelates the growth rate of a cell to its number ofsides? Sascha Hilgenfeldt explained the relevanceof theorems of Minkowski which concern integralsof mean curvature of a cell. These lead to a 3Dgrowth law which plays the role of that of vonNeumann, if only in an average sense.

The phenomena of drainage (the motion of liquidthrough a foam) and rheology (which deals withthe flow of the foam) lie at the frontier of currentresearch. Howard Stone and others debatedcurrent ideas on drainage, for which an adequatephenomenology now distinguishes between foamswith effectively rigid or free surfaces. Rheology is anotoriously difficult subject to get to grips with,and the description of foam flow at finite shearrates remains far from settled. Reinhard Höhlerrepresented the current efforts to measurerheological properties reliably. While at theInstitute, Masao Doi was able to derive andpresent a constitutive model for viscous foamsthat is based on affine film deformation. Anotherdifficult question remaining to be resolved isconcerned with normal stress differences.

It is much too soon to say that a trulycomprehensive practical theory of foam behaviouris available. Many of its individual ingredients,mentioned above, remain poorly defined, and theircouplings (e.g., drainage and rheology, drainageand coarsening) are still to be properly understood.Often the results of experiments are rationalisedafterwards, based on the influences of the differentsurfactants used. This made for lively discussion.

Participants were joined throughout theprogramme by colleagues from differentdepartments in Cambridge. Several participantsalso made visits to these departments to discuss‘foam-inspired’ topics of mutual interest. Onseveral occasions we were joined by researchersfrom Schlumberger Cambridge Research. Also,Mike Ashby from the Department of Engineeringmade a special presentation on the mechanicalproperties of metal foams.

The programme was enlivened by occasionalglimpses of the history of the subject in the form ofbiographical sketches. Some of the personalitiesincluded Kelvin, Riemann, Minkowski and CyrilStanley Smith.

The typical schedule for a given day was for nomore than four half-hour talks or focussed one-hour discussions. Throughout the meeting thediscussion was lively and interactive and it was

The Weaire−Phelan structure, as visualised by John Sullivan


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particularly valuable that participants fromdifferent backgrounds generally attended all of thetalks. To allow participants to solicit feedback onproblems of current interest, we also held smallsets of short five-minute talks with five furtherminutes allotted for discussion. This provedparticularly successful.

Special Events

Surface Evolver ExtravaganzaThe workshop started with a Hewlett-Packard day,the ‘Surface Evolver Extravaganza’, in whichseveral of the world’s leading authorities on the useof Brakke’s Surface Evolver entertained us withvarious applications from triply periodic crystalstructures to knots. The day finished with apresentation by Brakke on the stability of soap filmjunctions, after which we studied protein foamsfrom a local brewery.

Soap Bubble Geometry ContestThe Soap Bubble Geometry Contest was anopportunity to educate the general public. FrankMorgan performed demonstrations and setquestions to test the audience’s knowledge of howsoap films interact. The contest was well attended,and the youngest participants, less than ten yearsold, showed great enthusiasm.

Presentation of SculpturePrior to the contest, John Sullivan presented theInstitute with a minimal surface sculpture (below)as a token of gratitude on behalf of theparticipants. The sculpture will be displayed in theInstitute building.

John Sullivan’s computer-designed ‘Minimal Flower 3’ surface sculpture

Frank Morgan presenting the “Soap Bubble Geometry Contest”

T A

ndrew

Outcome and AchievementsThe communities which were brought togetherinteracted well, and many future collaborationsshould result which would not have emergedwithout such a stimulus. Some such collaborationswere already active during the programme. KenBrakke kindly volunteered to conduct practicalsessions in the use of the Surface Evolver. Thesetutorials introduced several new users to theprogram and its applications.

It was particularly pleasing to observe the fruitfuldiscussions between mathematical scientists frommany different intellectual backgrounds. TheNewton Institute is ideally suited and designed tofacilitate such interactions. The general conclusionwas that two-phase mixtures of academics can beas interesting, profitable and unpredictable as themixture of gas and liquid that constitutes a foam.

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Computation, Combinatorics andProbability

29 July to 20 December 2002

Report from the Organisers: M Dyer(Leeds), M Jerrum (Edinburgh), PWinkler (Bell Labs)

Scientific BackgroundOver time, the relationships between computerscience and mathematics have become more wideranging and profound. The traffic in ideas has beenin both directions. On the one hand, computerscience has opened up new areas of study tomathematics; on the other, increasinglysophisticated mathematical methods have beenbrought to bear on problems in computer science.These developments have been particularly evidentin quantitative theoretical computer science,especially in the areas known as computationalcomplexity and the analysis of algorithms. Both ofthese areas are concerned with the resources (oftentime or space) required to achieve specifiedcomputational tasks; the rough distinction is thatthe former concerns itself with general phenomenasurrounding resource-bounded computation, whilethe latter probes the inherent difficulty of specificcomputational tasks.

The aim of this programme was to explore twoparticularly fruitful interfaces between computerscience and mathematics, namely those involvingcombinatorics and probability theory. Sincecomputers perform manipulations of finite

structures (bit patterns or data structures ofvarying complexity, depending on the level atwhich we view the computation) it is not at allsurprising that combinatorics is pervasive in thedesign and analysis of algorithms. However, theimportance of probability may come as a surprise,given that determinacy and predictability wouldseem to be desirable features of a real computer!Nevertheless the notion of randomisedcomputation has assumed an increasinglyimportant role, both in theory and practice.

Within the broad area delineated by the title, anumber of themes were to receive special emphasis.

• Randomised algorithms. This themeaddressed the design and analysis ofalgorithms that make random choices, andalso of deterministic algorithms when runon random instances. The theoretical basisof this study includes, for example, the studyof parameters connected with random walkson graphs − coupling time, expansion,multicommodity flow, mixing time, etc. −and the relationships between them.

• Phase transitions in statistical physics andcomputer science. This heading covers notonly finite random systems, but also (infinitesequences of) finite systems that exhibitphenomena akin to phase transitions. Anexample of the latter is the satisfiabilitythreshold for random instances of CNFBoolean formulas with k literals per clause.Particular attention was paid to theapparent link between phase transitions andcomputational tractability.

• Random graphs and structures. We wereconcerned here with finite randomstructures, of which the Erdös−Rényirandom graph model is the primal example.Algorithmic aspects received specialattention.

• Probabilistic analysis of distributed systems.The central concern was to study thebehaviour of distributed systems in theabsence of global control. The prime

Left to right: M Dyer and M Jerrum

S Wygard

Computation, Combinatorics and Probability


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example of such a system is the Internet,though one could also think of traffic onroad networks, for example. This is a verynew interdisciplinary area bringing togethereconomics, game theory (Nash equilibria ofagents in networks) and computer science.

Structure of the ProgrammeThe programme attracted a strong contingent ofnearly fifty long-term visitors. It began and endedwith very well attended periods of heightenedactivity built around the workshops. Between theserather hectic periods of concentrated interaction,there was a quieter interlude for collaborativeresearch projects. Momentum was maintainedthrough a programme of regular weekly seminars.We were fortunate in having J Steif (Chalmers) andL Lovász (Microsoft Research) on hand to depictsome of our interests to the wider world in theInstitute’s Monday Seminar series.

Workshops

Randomised AlgorithmsEuroworkshop, 5−16 August 2002

Organisers: M Dyer, M Jerrum and P Winkler

One of the appeals of randomised algorithms forthe mathematician is that whereas the algorithmsthemselves are often quite simple, their analysis

can be very challenging. Because of this, the studyof randomised algorithms has prompted one of themost profound and fruitful interactions betweencomputer science and mathematics. This workshopaimed to stimulate progress in this interdisciplinaryarea, by bringing together computer scientists withprobability theorists, combinatorialists and othersworking in relevant areas of mathematics.

The plan was that the first of the two weeks wouldbe concerned with mathematical foundations, forexample, random walks on graphs, concentrationof measure, etc. The second would deal with thedesign and analysis of algorithms which makerandom choices, including Monte Carlo andMarkov Chain Monte Carlo (MCMC) algorithmsand deterministic algorithms when run on randominstances. This plan was roughly followed, thoughavailability of speakers meant in practice that therewas some blurring of the theory/applicationdistinction.

A typical day was built around four talks of one-hour duration, a timetable that left plenty of timefor informal discussions. Amongst the speakersgiving talks of a more expository nature were: C Greenhill (Melbourne) who gave a tutorial onthe “Differential equations method”, an analyticaltool whose value was amply illustrated in anumber of other talks during the course of theworkshop; M Jerrum (Edinburgh) who surveyed

Two realisations and the nominal behaviour of a stochastic process arising in the analysis of a MAX-2 SAT heuristic

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the varied methods, many of them quite recent, forthe non-asymptotic analysis of rates ofconvergence to stationarity of Markov chains; ASokal (New York) who demonstrated howmethods from statistical physics can answer long-standing open questions from combinatorics (andhow combinatorics can in the agenda of statisticalphysics); and P Winkler who untangled for us theweb of mixing times (measures of convergence ofMarkov Chains).

Many of the “leading lights” of the field were atthe workshop: M Dyer (Leeds) introduced hissurprising discovery that dynamic programming isuseful for counting as well as optimisation; RKannan (CMU) described “blocking conduct-ance”, a new and more precise tool for boundingmixing times of Markov chains; and A Sinclair(UC, Berkeley) explained how Clifford algebrasmight one day provide a practical approach toestimating the permanent of a non-negativematrix.

Statistical Physics, and Random Graphs and

Structures. The two parts were to be united by thenotion of phase transition, broadly interpreted.Roughly speaking, Part 1 was to concentrate onphase transitions in infinite systems (e.g., the Isingmodel on 2), and Part 2 on “phase transitions” infinite structures (e.g., random graphs or randompartial orders). The interdisciplinary aspect wasevident, with computer scientists meeting withprobabilists, mathematical physicists (particularlyin Part 1) and combinatorialists (particularly inPart 2).

In the event there were about sixty officiallyregistered participants, and a significant additionalnumber (particularly from in or aroundCambridge) attended on an informal basis. Anyselection of talks from the workshop willnecessarily be somewhat arbitrary, but here is abrief summary of a few notable contributions ofmore general interest. G Grimmett (Cambridge)surveyed some open problems in statisticalmechanics, reported on recent progress on theirsolution and expressed optimism that they couldbe completely resolved in the next few years, in thelight of recent work on “stochastic Löwnerevolutions”. J Kahn (Rutgers) presented animportant new result (with D Galvin) that thehard-core lattice gas model in d exhibits a phasetransition at an activity that decays to 0 with d.(This has long been expected, but not proved.) M Krivelevich (Tel Aviv) gave a personal accountof the future of colouring problems on randomgraphs. S Janson (Uppsala) and P Tetali (GeorgiaTech.) described progress on concentrationinequalities for combinatorial applications; suchinequalities are an essential tool for research in thisarea. Finally, A Scott (UCL) drew a surprisingconnection between two seemingly unrelatedtopics: a theorem of Dobrushin regardinguniqueness of Gibbs measures, and the LovászLocal Lemma. (This was joint work with A Sokal.)

The meeting was supported by the MathFITinitiative of EPSRC and the London MathematicalSociety.

A configuration of the “hard core latticegas” model − a finite square fragment of aconceptually infinite random configuration

Random Structures Workshop, 26 August−6 September 2002 Organisers: M Dyer, M Jerrum and P Winkler

This two-week workshop was structured into two

complementary parts of equal duration:

Combinatorial and Computational Aspects of


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Topics in Computer Communication andNetworks Workshop, 16−20 December Organisers: M Dyer, L Goldberg, M Jerrum, F Kelly and P Winkler

This one-week meeting, partly supported by CiscoSystems Inc., attracted a greater than expectednumber of participants, suggesting that the topichas captured the imagination of the community.

A number of themes could be identified. One ofthese was models for the web graph. The WorldWide Web can be viewed as an ever-evolvinggraph, with web pages as nodes and links asdirected edges. This graph, though random in somesense, has very different properties to the usualErdös−Rényi random graph; for example, thevertex degrees obey a power law. A number ofparticipants, among them J Chayes (MicrosoftResearch) and C Cooper (King’s College, London),proposed and analysed graph models that couldpotentially explain these observed features.

A second theme was the “cost of dispensing withglobal control”. In other words, how far can aNash equilibrium (obtained when agents optimisetheir own utility) be from a globally optimalsolution imposed by centralised control? Thisquestion was taken up by T Roughgarden(Cornell) in the context of routing in networks.

The final theme in this non-exhaustive list was thecomplexity of computing Nash equilibria.Problems involving these equilibria are interestingand challenging from a computational viewpoint,because they do not seem to fall naturally into theusual complexity-theoretic classes such as “NP-complete”. V Vazirani (Georgia Tech.) presentedone of the first moves in the direction of positiveresults.

Outcomes and AchievementsThe programme was broad in scope, coveringwork in computer science, probability, statisticalphysics, graph theory and combinatorics.Consequently the research achievements of theprogramme were varied in topic. The work was

highly collaborative, and often bridged two ormore areas. We summarise the outputs below,alphabetically by first collaborator. Most haveresulted in one or more publications, and severalformed the topic of a seminar during theprogramme. The overall success of the researchprogramme can be judged by the volume andquality of the output. Further details of many ofthe items mentioned below may be found in thegrowing list of Newton Institute preprints, and indue course in a special issue of Random Structuresand Algorithms.

Bollobás collaborated with Brightwell on the paperHow Many Graphs are Unions of k-cliques?, andwith Simonovits on extending their theorem, withBalogh, concerning the number of graphs thatexclude a certain set L of subgraphs. (When L isthe singleton set containing the complete graph onk vertices, this is the Erdös−Kleitman−Rothschildtheorem.) Brightwell worked with Winkler onproblems at the interface between graph theoryand statistical physics. Asymptotics were obtainedfor the following interesting problem: When doesthe unique simple invariant Gibbs measure for thehard-core gas model on the Bethe lattice (i.e.,infinite regular tree) cease to be extremal?

Cryan worked with Dyer and Randall on thecounting problem for a cell-bounded version of thecontingency table problem, and with Dyer, Jerrumand Karpinski on polynomial time generation ofindependent sets in hypergraphs. Contingencytables are matrices of non-negative integers withgiven row- and column-sums, which arise instatistics; in determining whether an apparentcorrelation is statistically significant it is necessaryto be able to count and/or sample such tables.

Dyer completed work on a new approach usingdynamic programming to polynomial timeapproximate counting for the knapsack andrelated problems. This gives simpler, and moreefficient, algorithms than were known previously.He worked with Frieze and Vigoda on samplingfrom the Ising and hard-core models on “largegirth” graphs. Related work, undertaken withGoldberg and Greenhill, attempts generalisations

to (other) H-colouring (also known as graphhomomorphism) problems. Also, with Goldbergand Jerrum, he investigated alternative dynamicsfor sampling particle systems on “simple” graphs,in particular colourings. With Sinclair and Vigoda,he completed some work on the relationshipbetween spatial and temporal mixing in physicalsystems.

Fill continued collaboration with Janson on a seriesof papers giving refined analysis of the Quicksortalgorithm. Quicksort is a classical randomisedalgorithm for sorting items using pairwisecomparisons. Describing the distribution of thenumber of comparisons used in sorting n items is achallenging problem.

Goldberg, Martin and Paterson completed work onMarkov chains on three-colourings of therectangular grid in 2, and showed rapid mixing ofthe Glauber dynamics. Somewhat surprisingly, thequestion of rapid mixing for four to six coloursremains open, even though movements within thestate space appear then to be less constrained.

Greenhill worked with Rucinski (and Wormald) onapplying the “differential equation approach” torandom hypergraph processes. The idea, maderigorous by Kurtz, and introduced to thealgorithms community by Karp and Sipser, is toapproximate the evolution of a Markov chain on

d by the solution to a differential equation. As thesize of the state space increases, the approximationbecomes more exact.

Hubenko worked (with Gyárfás) on certain factorsof the n-cube. The particular feature of thesefactors is that they arise as induced subgraphs ofthe cube. The prototype is the “Snake in a box”(largest induced cycle in the n-cube) problem thatarises in coding theory.

Janson collaborated with Rucinski (andOleskiewicz) on large deviation inequalities forcounts of (not necessarily induced) subgraphs in arandom graph. He also worked with Chassaing onthe Brownian snake. Both of these investigationshave led to subsequent publications.

Járai worked with van den Berg on a “forest fire”

model, and (with Athreya) on the abelian sandpilemodel in d ; both of these are models of self-organised criticality.

Jerrum (with Son) continued investigations of thespectral gap and log-Sobolev constants of Markovchains. One outcome, with Tetali and Vigoda, is aremarkable decomposition theorem for the log-Sobolev constant which closely mirrors that knownfor spectral gap. He worked with Sinclair andVigoda on (non-trivially) extending their celebratedsolution of the permanent problem to the case ofnon-bipartite matchings. This work remains inprogress. He also completed a monograph,Counting, Sampling and Integration, which hassince been published by Birkhaüser.

Kahn, during a short visit, worked with Randalland Sorkin on phase transitions for three-colourings of the lattice d in high dimension d.The expectation, which is consistent with what hasbeen discovered so far, is that there should be sixGibbs states; a typical such state has the evensublattice coloured largely “red” and the oddsublattice largely “blue” and “green”. He alsoproved a conjecture of van den Berg.

Karonski worked with T Luczak and Thomason onlocal irregularity strength of graphs, with Pittel onperfect matchings in a class of sparse randomgraphs, and with Krivelevich and Vazirani oninformation hiding in graphs and its potentialapplications to software watermarking.

Kendall studied problems arising in image analysisconcerning phase transitions in the Ising modeldefined on “augmented quad trees”, and the formof the resulting interfaces.

Lovász wrote a paper (with Vempala) on thegeometry of log-concave distributions: these occurwidely in practice, the most familiar being theGaussian distribution. A concrete outcome of theirstudy is a particularly efficient algorithm forsampling a point from a log-concave distribution.Lovász also collaborated on other work in discreteprobability and combinatorics.

M Luczak worked with T Luczak on phasetransitions in the cluster-scaled model of a random____

26



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graph, and with Winkler on showing that arandom walk on the non-negative integersproduces a uniform random subtree of the Cayleytree.

T Luczak and Simonovits collaborated ongeneralising results of Andrásfai−Erdös−Sósconcerning the minimum degree of a graph ofchromatic number k that contains no clique on kvertices. Pikhurko and Simonovits (with Füredi)investigated extremal hypergraph problems, andproved a conjecture of Mubayi and Rödl as to themaximum size of a 3-graph on n vertices thatexcludes a specific 3-graph on five vertices.

Randall continued her work on simulatedtempering, a general method that has beenproposed for speeding up simulations of “lowtemperature” systems. The idea is to introduce adynamics that is able to move between lowtemperature states and high temperature states,where mixing occurs more freely. Preliminaryresults, both positive and negative, have beenobtained.

Scott worked with Wilmer on a wide range ofproblems (extremal, algorithmic and complexity-theoretic) concerning classes of permutationswhich occur in large-scale genome changes.

Stougie proved a polynomial upper bound on theedge-diameter of transportation polyhedra. His

Phase diagram for the maximum number of satisfiable clauses in a random 2-CNF formula

initial results were improved during the prog-ramme in collaboration with van den Heuvel, andsubsequently further improved in collaborationwith van den Heuvel and Brightwell.

Van den Berg (with Brouwer) studied “self-destructive” percolation in which all sites in aninfinite cluster become vacant through somecatastrophe. With what probability do vacant sitesneed to be re-populated before an infinite clusterre-appears? They conclude the answer may belarger than one might suspect.

Van den Heuvel extended some of his own resultson cyclic and distant 2-colourings of planargraphs. Vigoda (with Hayes) developed a new toolfor analysing non-Markovian couplings, giving anapplication to sampling graph colourings. The ideashould have many other applications.

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Scientific BackgroundAlgebraic topology started in the late 19th centurywith the work of Henri Poincaré. In the beginningits objective was to study geometric objects, suchas smooth manifolds, which arise in connectionwith the differential equations of physics, byconverting the high-dimensional geometry intomore accessible algebra using various cohomologytheories. Stable homotopy theory is the ultimate context inwhich to perform the type of conversion fromgeometrical to algebraic data which Poincarébegan. The cohomology theories themselves arerepresented by geometric objects; only veryrecently have really satisfactory models beenfound, opening a wide range of new possibilitiesfor exploiting algebraic ideas in topology,geometry, number theory and physics. Theprogramme explored several of these new avenues.

Algebraic topology was applied with astoundingsuccess by Lefschetz, Hodge and others to tackleproblems in complex algebraic geometry. The

development of cohomology theories whichbehaved in positive characteristic like the classicalcohomology used by Lefschetz was a majorachievement of the 20th century. TheGrothendieck school flourished in the 1960s and1970s and left a legacy of powerful newtechniques, famous successes (such as Deligne’sproof of the Weil Conjectures) and a series ofmysterious cohomological problems concerningGrothendieck’s nebulous notion of a ‘motive’ (ormotif).

In the 1990s, techniques of homotopy theory wereadapted to the algebraic geometry context to formmotivic homotopy theory by Morel, Voevodskyand others. Results in algebraic geometry can beexploited through structures familiar fromalgebraic topology. These ideas led to theconstruction of motivic cohomology and toVoevodsky’s proof of the Milnor conjecture. Thisexploitation of motivic homotopy theory innumber theory and algebraic topology was spurredon by the programme, which also played animportant role in spreading the developing body ofknowledge. The motivic world is only one ofseveral new manifestations of stable homotopytheory which made important progress during theprogramme.

Another subject benefitting from the newstructures of stable homotopy theory is ellipticcohomology. This covers a large and rapidlygrowing field, involving a conjunction of algebra,geometry, analysis and topology, of a depth andpower unseen since the development of K-theory.The first elliptic genus to be widely studied wasintroduced by E Witten in connection withconformal field theory, as the equivariant index ofa putative signature operator on the free loopspace of a manifold. He provided a physical proofof its rigidity, establishing at the outset theconnection with differential geometry.

A number of attempts to find geometric cycles forthis theory were discussed and developed duringthe programme. So far the greatest success hasbeen through homotopy theory. Quillen’sdiscovery that the formal group law of complex


Back, left to right: VP Snaith, HR Miller Front: JPC Greenlees

S Wygard

2 September to 20 December2002

Report from the Organisers:JPC Greenlees (Sheffield), HR Miller(MIT), F Morel (Jussieu), VP Snaith(Southampton)


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cobordism is universal opened a new era foralgebraic topology, because the theory of formalgroups provides a remarkably accurate image ofthe stable homotopy category. It is natural to askwhether there is a functorial way to associate acomplex orientable cohomology theory to a formalgroup, lifting parts of the category of formalgroups to the category of spectra itself, not merelyto the homotopy category. New techniques fordoing this using the new models for stablehomotopy theory were discussed during theprogramme. One notable success was theconstruction by Hopkins and Miller of a lift of themoduli stack of elliptic curves to the stablecategory. This is an elliptic analogue of Atiyah’s K-theory with reality, and by taking the homotopyinverse limit of this diagram one obtains thespectrum tmf of topological modular forms. Newcalculations, variations and consequences of thisremarkable object were central to several parts ofthe programme.

This programme at the Newton Institute furtheredthis synergistic development by bringing togetherthe practitioners of stable homotopy in all itscurrent diverse disguises − from arithmetic tophysics to topological modular forms − toinvestigate possible new applications of theircurrent techniques.

Programme StructureThe programme was structured around threeworkshops and a number of related lecture series:it started and finished with intensive workshopactivity, separated by a period for sustainedresearch and collaboration amongst the longerterm participants. The programme opened inearnest with the very well attended two weekNATO Advanced Study Institute Axiomatic,enriched and motivic homotopy theory, whichprovided introductions to several of theprogramme themes. Many participants stayed onfor the workshop K-theory and arithmetic. DuringOctober and November, regular seminar series andthree Newton Institute Seminars continued thethemes these opened up. The final two weeks were

taken up with an EU workshop, Elliptic andchromatic phenomena, giving a suitably climacticfinish before Christmas.

Meetings and Workshops

Axiomatic, Enriched and MotivicHomotopy TheoryNATO Advanced Study Institute, 9−−20 September2002Directors: JPC Greenlees and IB ZhukovOrganisers: PG Goerss, JF Jardine, F Morel and VP Snaith

This first workshop was a two week NATOAdvanced Study Institute attended by 90participants from 17 countries. It was arrangedaround a series of 15 introductory minicourses byexperts, designed to make the remainder of theprogramme accessible to newcomers. These weresupplemented by a handful of stand-alone lectures,a problem session and a poster session. Severalnew collaborations were born during the ASI.

The ASI came about because of a remarkableconjunction in two related subjects. The subject ofstable homotopy theory has been transformed inthe last ten years by key technical advances makingdistant dreams into reality, but the fact that itsmethods have also been used in recent spectacularprogress in motivic homotopy theory was a quiteseparate development. The ASI brought togetherprincipal exponents of both themes, and it wasgratifying to see how effective it was atencouraging substantial two-way interactionbetween them. This mathematical cohesion isillustrated by the manner in which workshops inthe programme often referred directly to oneanother. For example, (i) the talks of Hornbostel,Jardine, Hesselholt, Kahn, Levine, Morel, Rognes,Snaith, Toen, Totaro, Weibel and Vishik from themotivic homotopy week of the NATO workshopwere all cited during the K-theory workshop; and(ii) a similar relation is true of lectures of Goerss,Madsen and Strickland which surfaced again in theEU workshop on elliptic cohomology andchromatic phenomena. Particularly notable andstimulating was the number of originators of

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results who were present, many of whom had nevergiven lectures in the UK before.

The objective of the ASI was to survey recentdevelopments and outline research perspectives instable homotopy theory, the homotopy theory ofstructured ring spectra and motivic homotopytheory. A major effort was made to encouragecommunication between those with experiencefrom the various different themes.

The lectures at the ASI highlighted the directions ofresearch in which major breakthroughs have beenachieved in recent years. These can be broadlygrouped into six areas:

• Abstract homotopy theory and models(Dwyer, Jeff Smith, Schwede, Strickland)

• Operads and structured ring spectra(McClure, May, Richter, Smirnov)

• Chromatic and geometric applications ofstructured ring spectra (Goerss, Madsen,Rognes)

• Foundations of motivic homotopy theory(Jardine, Weibel, Toen)

• Motivic homotopy theory (Hornbostel,Levin, Morel, Totaro, Vishik)

• K-theory (Bloch, Carlsson, Hesselholt,Snaith)

We are confident that the positive impact of thisASI will be felt in the mathematical community formany years to come.

K−Theory and ArithmeticWorkshop, 30 September − 4 October 2002

Organisers: S Lichtenbaum and VP Snaith

This workshop attracted over 80 participants andconcentrated on aspects of interplay betweenalgebraic K-theory, arithmetic and algebraicgeometry. Particular emphasis was placed uponapplications of the recently developed homotopytheory of geometric and motivic categories. Inaddition to lectures on current results, a number ofexpository lectures were scheduled to provideresearchers and graduate students in related areas

with an opportunity to learn about these newtechniques. Topics of current interest in this areainclude: Beilinson−Soulé conjectures, Bloch−Katoconjecture, Beilinson−Borel regulators, Kato−Parshin−Saito higher class field theory, Lichten-baum−Quillen conjecture, Milnor K-theory,motivic cohomology, Brumer−Coates−Sinnottconjectures, polylogarithms, modularity andspecial values of L-functions.

Elliptic Cohomology and ChromaticPhenomenaEuroworkshop, 9−20 December 2002Organisers: HR Miller and DC Ravenel

Elliptic cohomology is the third of a sequence ofnatural probes into the nature of high dimensionalobjects bringing together ideas from geometry,arithmetic and analysis. Ordinary cohomologydominated the first half of the twentieth century,and K-theory dominated the second half. As EdWitten has said, elliptic cohomology is a piece oftwenty-first century mathematics that happened tocommence in the twentieth. It has been most highlydeveloped within the homotopy theory, but earlydiscoveries were made by mathematical physicistsstudying conformal field theory. The formalstructure of elliptic cohomology is becoming wellunderstood, but its geometric foundations are stillmysterious. Hints have come from physics(conformal field theory), algebraic geometry(stacks and 2-vector spaces) and representationtheory (moonshine and vertex operator algebras),as well as traditional algebraic topology(structured ring spectra, equivariant cohomology).

This workshop attracted over 75 participants from13 nations, bringing together top researchers inmathematical physics, homotopy theory, algebraicgeometry and representation theory to investigatethis mysterious object. Significant progress wasreported, and discussions at the conferenceresulted in further progress.

The first week was focussed sharply on bringingtogether a wide range of perspectives on ellipticgenera and elliptic cohomology. There were 19talks during this first week. The quality of


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exposition was simply excellent. The high degree of

audience participation and the animated

conversation, often between new acquaintances,

made it clear that the hoped for cross-fertilisation

was occurring. While each talk represented a

different and important perspective, we might pick

out the following sessions as shedding the most

new light specifically on the meaning of elliptic

cohomology.

• Mike Hopkins described the homotopy

theoretical construction of elliptic

cohomology, along with the ensuing theory

of topological modular forms and the

generalized Witten genus from string

manifolds.

• John Rognes described joint work with Nils

Baas and others giving a 2-vector space

model for the Waldhausen K-theory of

Diagram relating the smash product and triangulation in a stablehomotopy category

topological K-theory, and gave evidence of

its relationship with elliptic cohomology.

• Stephan Stolz described his work with Peter

Teichner, providing an operator algebra

approach to the notion of string structure,

and a corresponding candidate for elliptic

objects.

• Charles Rezk described explicit formulae for

homotopy theoretic logarithms in terms of

power operations, resulting in a homotopy

theoretic extension of the theory of Hecke

operators.

• Vassily Gorbunov described the

construction of a geometric object, a sheaf

of vertex operator algebras, associated with

a string structure.

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Summary of the Hopkins−Mahowald 2-local calculation of the coefficient ring of the topological modular formspectrum tmf, with the dark part giving the final answer.

The part which does not come from connective real K-theory is 192-periodic, so the picture (which stops at the197th homotopy group) gives the complete answer. The spectrum tmf is central to the study of elliptic cohomology (final workshop) and its construction uses enriched homotopy theory (first workshop).

(With thanks to MJ Hopkins)

• Burt Totaro described the universalcharacter of elliptic cohomology in relationto complex projective varieties.

• Matthew Ando described how to explainthe phenomenon of Witten rigidity from theperspective of elliptic cohomology.

The second week of this workshop was morenarrowly homotopy theoretic, but the focus wasbroadened to encompass the next steps, beyondelliptic cohomology. From a homotopy theoreticperspective, ordinary cohomology, K-theory andelliptic cohomology form the first three of aninfinite sequence of probes into the nature of highdimensional geometry.

The following were among the highlights of the tentalks. Hans-Werner Henn described the status ofjoint work with Paul Goerss, Mark Mahowald andCharles Rezk on the structure of the sphere

spectrum from the elliptic perspective. JorgeDevoto described a speculative approach to ageometric theory of K3-cohomology. Andy Bakerdescribed the obstruction theory associated to theconstruction of ring structures on certain higheranalogues of K-theory. Doug Ravenel describedpart of the theory of abelian varieties which seemsrelated to these higher K-theories.

Outcomes, Achievements andPublicationsThe programme was the foremost internationalevent in the subject in 2002, and one of specialsignificance as it brought together several strandsat seminal stages of their development. Some ofthis will be reflected in the proceedings volume forthe NATO ASI Axiomatic, Enriched and MotivicHomotopy Theory, to be published by Kluwer.


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Here we summarise some of the work that tookplace during the programme and was directlyrelated to one of the main themes of theprogramme. Although there have been many newcollaborations, it is especially notable how manyparticipants benefitted from less formal interactionwith those of more distant expertise. Which ofthese contacts will lead to new collaborations, andwhat the fruit of these will be, we will only knowin a few years’ time.

Some of the areas where significant work was doneduring the programme are as follows. Inevitably,there are many omissions. There are alreadypreprints reflecting this work, several of them inthe Newton Institute preprint series.

• Algebra of rings up to homotopy (Galoisand étale maps, duality and finitenessconditions, centres)

• Elliptic and chromatic phenomena(localisation and comodules, smallresolutions of local spheres, 2-vectorbundles, generalised Mumford conjecture)

• Equivariant cohomology theories (ellipticcohomology, sigma orientation, rationaltorus-equivariant cohomology)

• Algebraic groups and cohomology theories(arithmetic Jacobians in the non-equivariantcase and rational Jacobians in theequivariant case)

• K-theory (Weil-étale topology for numberfields and zeta functions, additivedilogarithm, de Rham−Witt complex,Galois modules, the Vandiver conjecture)

• Algebraic and motivic homotopy theory(transfer functors, Balmer−Witt groups ofclassifying spaces, motivic stable homotopygroups, higher stacks)

• Abstract homotopy theory (operads andhigher categories, sequence operads andstable p-adic homotopy theory, Frankeequivalence, cubical sets, algebraiccobordism)

ConclusionThis programme brought together a remarkablecollection of researchers from a wide range ofareas including parts of homotopy theory, K-theory and elliptic cohomology. It organised,disseminated and consolidated the existing state ofthe subjects. Many new collaborations wereformed, new dreams born, and new agenda set.The beneficial effects of this programme willappear and ripen over the coming years.

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Computational Challenges in PartialDifferential Equations

The most powerful and generally applicablealgorithms for the approximate solution of partialdifferential equations rely on the concept ofdiscretisation, whereby the PDE underconsideration is replaced by a finite-dimensionalproblem. The transition from the partialdifferential equation to the discrete model is a non-trivial mathematical problem, and the selection ofan appropriate finite-dimensional representation israrely a matter of arbitrary choice: physicalproperties behind the mathematical model (such asenergy and mass conservation, positivity, total-variation-boundedness, dispersion and dissipation)have to be borne in mind, as well as issues ofresolution of relevant scales, complete andguaranteed control of the discretisation error, inaddition to concerns about the efficiency andreliability of the resulting algorithm. The study ofthese mathematical questions represents the focusof the field of Numerical Analysis of PartialDifferential Equations.

Simultaneously, the development of high speeddigital computers has led to consideration of verylarge systems of PDEs in science and engineeringand to scientific demands for highly accurateapproximations. In addition, in recent years theapplied analysis of nonlinear partial differentialequations has progressed in parallel with theirtheoretical numerical analysis and the developmentof appropriate numerical algorithms, leading to avery important and intellectually rich area ofapplied mathematics referred to as ComputationalPartial Differential Equations (CPDEs).

This programme focussed on some of the mostexciting and promising mathematical ideas in thesefields, and those branches of PDE theory thatprovide a source of physically relevant andmathematically hard problems to stimulate futuredevelopments. The three main themes were:

• Cognitive algorithms: adaptivity, feed-backand a posteriori error control

• Hierarchical modelling: multi-scale mathe-matical models and variational multi-scalealgorithms

Computational Challenges in Partial DifferentialEquations

20 January to 4 July 2003

Report from the Organisers: M Ainsworth(Strathclyde), CM Elliott (Sussex), E Süli(Oxford)

Left to right: CM Elliott, E Süli and M Ainsworth

S Wygard

Scientific Background The study of partial differential equations (PDEs)is a fundamental area of mathematics which linksimportant strands of pure mathematics to appliedand computational mathematics. Indeed PDEs areubiquitous in almost all of the applications ofmathematics where they provide a naturalmathematical description of phenomena in thephysical, natural and social sciences.

Partial differential equations and their solutionsexhibit rich and complex structures. Unfor-tunately, closed analytical expressions for theirsolutions can be found only in very specialcircumstances, and these are mostly of limitedtheoretical and practical interest. Thus, scientistsand mathematicians have naturally been led toseek techniques for the approximation ofsolutions. Indeed, the advent of digital computershas stimulated the incarnation of ComputationalMathematics, much of which is concerned with theconstruction and mathematical analysis ofnumerical algorithms for the approximate solutionof PDEs.


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The parallel programme at the Institute, NonlinearHyperbolic Waves in Phase Dynamics andAstrophysics, had a number of activitiesconcerning computational aspects of hyperbolicand related systems. This led to several discussionsbetween participants of the two programmes andto mutual attendance of seminars and lectures.

Workshops and Conferences

Mathematical Challenges in Scientific and Engineering ComputationWorkshop, 20−24 January 2003Organisers: M Ainsworth (Strathclyde), CM Elliott(Sussex) and E Süli (Oxford)

In the opening lecture R Rannacher (Heidelberg)discussed the issues of error control and adaptivityin numerical computation, and presentedapplications to numerical approximation ofchemically reacting flows. These issues were takenup in the context of free boundary problems by R Nochetto (Maryland), who observed that manyof the difficulties revolve around identification ofan appropriate notion of residual for suchproblems, and showed how error control isaccomplished for a range of examples includingnonlinear degenerate parabolic equations, meancurvature problems and surface diffusion. The dayconcluded with an open discussion on the role ofcomputational mathematics in complex industrialsimulations chaired by J Ockendon (Oxford).

Problems involving interfaces formed the theme ofthe second day opened by O Pironneau (Paris VI),who described his recent work on the sensitivity ofshocks in solutions of Euler equations which areimportant in computations of flutter. A variety ofphysical problems where curvature-dependentevolution of interfaces plays a lead role, includingdiffusion-induced grain boundary motion, alongwith an overview of the principal mathematicalmodels was presented by CM Elliott (Sussex).Diffuse interface, or phase field, models and theirapplication to solidification were discussed by A Wheeler (Southampton), while the use of

• Nonlinear degenerate PDEs and problemswith interfaces

Structure of the ProgrammeApart from the nine events described below we hada seminar programme. Because of the large numberof talks during the events and also in the parallelprogramme, we restricted the number of seminarsto about one per week. Most of the long-termparticipants delivered a lecture or seminar duringthe programme.

There were 57 long-term (21 UK) as well as 125short-term (approximately 56 UK) participants.The meetings also attracted others who were notformally participants in the programme. The largeUK participation is a reflection of the currentstrength of and interest in the field in the UK. Thethree organisers M Ainsworth, CM Elliott and E Süli together with J Barrett were in residencethroughout the programme. Other participantsstaying for substantial periods, partially supportedby EPSRC fellowships, were C Carstensen, PB Monk, C Johnson, R Nochetto and C Schwab.The Rothschild Visiting Professor was F Brezzi.These contributions added greatly to the cohesionof the programme throughout the six months.

The Newton Institute’s Junior Membership schemesupported the participation of 11 youngresearchers. Other sources of funding for youngresearchers were the EU (for attendance at theEuroworkshop) and the EPSRC ComputationalEngineering Mathematics programme.

E Süli, W Dahmen and C Johnson

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simulation at the atomistic level was advocated byA Sutton (Oxford).

Atomistic and multi-scale modelling of materialswere the topic on the following day, where E Tadmor (Technion) discussed the problems ofdealing with widely varying length-scales found inmaterials science. He described the “quasi-continuum” approach whereby different locallevels of resolution are identified using an adaptiveapproach. B Engquist (Princeton) continued thetheme in his talk on the Heterogeneous MultiscaleMethod, in joint work with Weinan E (Princeton)who later described some of the new mathematicsunderlying the method. The importance of usingconstitutive laws at the molecular level washighlighted in the talk of T McLeish (Leeds), whogave two examples in polymer processing wheredifference in the structure of the polymer at themolecular level is responsible for completelydifferent macroscopic behaviour.

Thursday was devoted to interaction with theEPSRC Computational Engineering Mathematics(CEM) programme (which provided an additional£10,000 of financial support for the meeting) andconsisted of two lectures by overseas experts: R Glowinski (Houston) and T Chan (UCLA).These were followed by a number of short talksbased on projects supported through the CEMinitiative presented by the principal investigators,and a poster session comprised of other projectssupported under CEM. The day concluded withpresentations by EPSRC personnel from theengineering and high-performance computingprogrammes, paving the way for an open forumdiscussion on the topics raised.

The final day witnessed a change of pace as, firstly,A Quarteroni (EPFL, Lausanne) discussed theapplications and advantages of numericalmodelling of the cardiovascular system, a problemrequiring coupled, heterogeneous PDE models ofthe entire system. PB Monk (Delaware) gave anoverview of numerical techniques used in solvinginverse problems and described a new technique,the linear sampling method, developed incollaboration with D Colton. The meeting closed

with talks by R Davies (Aberystwyth) on PDEs inthe food industry and issues of mesh generation incomputational modelling by N Weatherill(Swansea).

The organisers prepared an article which appearedin the June 2003 issue of SIAM News thatdescribed the meeting and its context in the six-month programme at the Newton Institute.

Computational Challenges inMicromagnetics and Superconductivity Hewlett-Packard Event, 13−14 February 2003Organisers: M Ainsworth (Strathclyde), CM Elliott(Sussex) and PB Monk (Delaware)

One of the aims of this two-day meeting was toidentify the major applications in each area whereprogress in computational techniques for PDEs isneeded and will have a significant impact. Leadingexperts, including physicists, engineers, numericalanalysts and theoretical PDE analysts deliveredsurvey lectures on their area of expertise.

The speakers were selected with the aim to providea comprehensive overview of the current state ofthe field, with particular emphasis on theinteractions between theory, applications andpractice.

J Chapman (Glasgow) described experimentalresults in micromagnetism obtained using Lorentzmicroscopy showing the existence of free vorticesin magnetic permalloy films and reversalmagnetisation effects in multi-layer structures. T Schrefl (Vienna) discussed the different length-and time-scales that arise in micromagneticsimulations, and demonstrated the performance ofadaptive finite element schemes for some three-dimensional applications. A Prohl (Zürich)discussed the numerical analysis of microstructurewith applications to micromagnetism. G Carbou(Bordeaux) presented results on the existence,uniqueness and regularity of solutions of theLandau−Lifschitz equations. PB Monk (Delaware)concluded the day with a presentation on hisrecent collaborative work on the numericalmodelling using an eddy current micromagneticmodel.


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The second day focussed on mathematical modelsfor superconductivity. M McCulloch (Oxford), inhis talk Using PDEs to develop the application ofsuperconductors, described the importance ofcomputational methods for PDEs in engineeringapplications and in particular the use ofsuperconductors in electric motors. J Chapman(Oxford) discussed in his talk Macroscopic modelsof superconductivity a hierarchy of models fromthe Ginzburg−Landau equations to Bean’s criticalstate model. Q Du (HKUST) described numericalcomputations associated with Quantized vortices:from Ginzburg−Landau to Gross−Pitaevskii. V Styles (Sussex) discussed critical statecomputations in her talk A finite elementapproximation of a variational formulation ofBean’s model for superconductivity. Finally, L Prigozhin (Ben Gurion) gave an overview of hisresearch on the Solution of critical-state problemsin superconductivity.

Computational PDEs: Giving Industrythe Edge Joint Workshop with the Smith Institute, 4−5 March 2003Organisers: CM Elliott (Sussex), RE Hunt(Newton Institute), R Leese (Smith Institute),J Ockendon (Oxford) and E Süli (Oxford)

The challenges of industry call for newmathematics of the highest scientific quality. TheIsaac Newton Institute and the Smith Institute forIndustrial Mathematics and System Engineeringheld this joint workshop in order to bring togetherworld-class researchers and industrialists to workon real problems. The meeting presented sixcurrent Smith Institute projects involvingcollaboration between industry and Universitymathematicians in the areas of food, electro-magnetism and violent mechanics. The industrialcollaborators described the background and theproblems being addressed. The academicresearchers discussed the development ofmathematical models together with the resultingPDE problems. The Computational Overviewsessions illustrated some of the current numericalapproaches to the areas of interest, and there were

chaired discussions on the critical computational

issues in the projects. Topics considered included

bread crusting, scraped surface heat exchangers,

microwaving, inertia-dominated free surfaces,

droplet impact on water layers, electromagnetic

compatibility and shaped charge mechanics.

Multiscale Modelling, Multiresolutionand Adaptivity Workshop, 7−11 April 2003Organisers: M Ainsworth (Strathclyde),

W Dahmen (Aachen), C Schwab (Zürich), and

E Süli (Oxford)

This meeting was devoted to identifying new

algorithmic paradigms which will lead to

computationally affordable numerical algorithms

for the approximate solution of multiscale

problems, with focus on recent developments. This

was achieved by bringing together leading experts

from applied mathematics, scientific applications

and various branches of scientific computation,

who work on different aspects of multiscale

modelling. The meeting stimulated interactions

and cross-fertilisation between the various subject

areas involved, and resulted in an assessment by

leading researchers of the state-of-the-art in the

field, the identification of key problems and

obstacles to progress, and the indication of

promising directions for future research.

Fifteen leading scientists working in the field were

invited to present plenary lectures.

Participants at the workshop “Multiscale Modelling,Multiresolution and Adaptivity”

S Wygard

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M Ainsworth (Strathclyde) discussed the use ofhigh-order finite element methods for theapproximation of Maxwell’s equations. Hepresented families of hierarchic basis functions forthe Galerkin discretisation of the space H(curl; Ω).The conditioning and dispersive behaviour of theelements was discussed along with approximationtheory.

F Brezzi (Pavia) highlighted the fact that residual-free bubbles proved to be a powerful technique todeal with subscale phenomena and to have optimalstabilising effects. He also discussed, throughseveral examples of PDEs, the amount ofcomputational effort required to solve the subgridproblem.

C Canuto (Torino) used the properties of waveletbases to design an adaptive descent algorithm forsolving a convex minimisation problem in afunction space of Hilbert type. He proved theconvergence of the algorithm and discussed itsoptimality in the framework of best N-termapproximation results due to A Cohen, W Dahmenand R DeVore.

C Carstensen (Vienna) considered three applic-ations for a nonconvex minimisation problemmodel: the examples included phase transitions,optimal design tasks and micromagnetics. Throughthe three examples at hand, he showed thatrelaxation theory provides auxiliary problems(relaxed models) which are convex but notuniformly or even strictly convex. Their numericalsimulation in the quasiconvexified approach tocomputational microstructures is relatively easy;he recommended the use of this approach, sinceeven the Young measures generated in thenonconvex model can be recovered.

A Cohen (Paris) discussed the convergence analysisof a class of adaptive schemes for evolutionequations. These schemes combined adaptive meshrefinement with wavelets based on a frameworkintroduced by A Harten. The analysis mainlyfocussed on the difficult case of nonlinearhyperbolic conservation laws.

W Dahmen (Aachen) spoke on the adaptive

application of (linear and nonlinear) operators inwavelet coordinates. He showed that there are twoprincipal steps: first, the reliable prediction ofsignificant wavelet coefficients of the output ofsuch adaptive applications from those of the input,and second, the accurate and efficient computationof the significant output coefficients. Some of themain conceptual ingredients of such schemes werediscussed. Moreover, it was also highlighted howthis approach leads to adaptive solution schemesfor variational problems that can be shown to haveasymptotically optimal complexity.

R DeVore (South Carolina) discussed two resultsthat have proven to be important in the analysis ofconvergence rates for adaptive finite elementmethods. The first is how to bound the number ofadditional subdivisions needed to remove hangingnodes in newest vertex bisection. Such a result isnecessary for deriving computational complexityestimates in adaptive methods. The second result ishow to generate a near-best adaptive approxi-mation to a given function in linear time. Thisresult was used in coarsening routines to keepcontrol of the number of cells in adaptivelygenerated triangular partitions.

Weinan E (Princeton) gave an extensive survey ofvarious multiscale methods for a range of physicalapplications that exhibit multiscale behaviour, andthen discussed the numerical analysis of multiscalemethods for several classes of multi-physicsproblems.

At the conference dinner for “Multiscale Modelling,Multiresolution and Adaptivity”


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B Engquist (Princeton and Stockholm) talkedabout the heterogeneous multiscale method anddiscussed its application to the numerical solutionof stiff ordinary differential equations. He arguedthat the computational complexity of thesetechniques is, in many cases, lower than that oftraditional methods. He illustrated this pointthrough analysis and numerical examples.

T Hou (Caltech) showed that many problems offundamental and practical importance containmultiple-scale solutions. He introduced a dynamicmultiscale method for approximating nonlinearPDEs with multiscale solutions. The main idea is toconstruct semi-analytic multiscale solutions localin space and time, and use them to construct thecoarse grid approximation to the global multiscalesolution. He showed that the method provides aneffective multiscale numerical method forcomputing two-phase and incompressible Eulerand Navier−Stokes equations with multiscalesolutions. Finally, he introduced a new class ofnumerical methods to solve the stochasticallyforced Navier−Stokes equations, and demonstra-ted that the method can be used to computeaccurately high order statistical quantities moreefficiently than the traditional Monte Carlomethod.

C Johnson (Chalmers, Göteborg) presented aframework for adaptive computational modellingwith applications to reaction−diffusion in laminarand incompressible flow. He estimated, using aduality argument, the total computational error indifferent outputs a posteriori, with contributionsfrom both discretisation in space/time and subgridmodelling of unresolved scales. He consideredsubgrid models based on extrapolation or localresolution of subgrid scales, and presentedcomputational results for laminar and turbulentCouette flow.

R Nochetto (Maryland) proposed an algorithm forsaddle point problems consisting of two nestediterations, the outer iteration being an Uzawaalgorithm to update the scalar variable and theinner iteration being an elliptic AFEM for thevector variable. He showed linear convergence in

terms of the outer iteration-counter provided theelliptic AFEM guarantees an error reduction ratetogether with a reduction rate of data-oscillation(information missed by the underlying averagingprocess), and applied this idea to the Stokes systemwithout relying on the discrete inf-sup condition.He finally assessed the complexity of the ellipticAFEM, and provided consistent computationalevidence that the resulting meshes were quasi-optimal.

R Rannacher (Heidelberg) presented a systematicapproach to error control and mesh adaptivity inthe numerical solution of optimisation problemswith PDE constraints. By the Lagrangianformalism the optimisation problem wasreformulated as a saddle-point boundary valueproblem which was discretised by a Galerkin finiteelement method. The accuracy of thisdiscretisation was controlled by residual-based aposteriori error estimates. The main features of thismethod were illustrated by examples from optimalcontrol of fluids and parameter estimation.

C Schwab (ETH Zürich) showed that ellipticproblems with multiple scales in bounded domainsin d can be reformulated as elliptic problems witha single scale in high-dimensional domains.Standard finite element methods for such problemsexhibit poor accuracy due to lack of scale-resolution. Schwab showed how a two-scale finiteelement method in a higher dimensional space canbe developed to overcome this problem. He furthershowed how sparsification of the finite elementspace and hierarchical finite element spaces can beused to reduce computational complexity.

J Xu (Penn State) reported on some recent studieson using multigrid ideas in grid adaptation. Theresults presented included gradient and Hessianrecovery schemes by using averaging andsmoothing (as in multigrid), interpolation errorestimates for both isotropic and anisotropic gridsand multilevel techniques for global gridmovement and local grid refinement.

The meeting included a further 12 (contributed)talks, mostly by young researchers (under the age

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of 35) working in the field, and four one-hourcoordinated discussion sessions (on Adaptivity,Wavelets and multiresolution for PDEs, Multiscalemethods for PDEs, and on Multigrid andmultilevel methods); the meeting was attended by79 participants from academia and industry. Thediscussion sessions were productive, lively, and, ona number of occasions, quite controversial. Theworkshop was a very successful event in bringingtogether leading researchers working in the field.

INTERPHASE 2003: Numerical Methodsfor Free Boundary Problems Conference, 14−18 April 2003Organisers: CM Elliott (Sussex), J Barrett (Imperial), G Dziuk (Freiburg) and R Nochetto(Maryland)

INTERPHASE 2003 was the latest in a series ofmeetings on numerical methods for free boundaryproblems. The title is a word-play on the ideas thatfree boundary problems may be interfaces, thatmany applications involve multiple phases and thatthe interdisciplinary nature of the workshop bringstogether computational mathematicians, analysts,modellers and scientists. The meeting was heldover four days in which there were 21 talks andmuch time for discussion. There were 57participants including well known experts in thefield and many young researchers. The topicsranged from image processing, differentialgeometry, phase transitions and material science,cosmology, mesh generation and flow in porousmedia, to fluid drops.

Numerical Analysis of Maxwell Equations on Non-Smooth DomainsWorkshop, 30 April 2003Organisers: M Ainsworth (Strathclyde) andC Schwab (Zürich)

The numerical analysis of Maxwell equations onnon-smooth domains poses a number ofcomputational challenges due to the variationalsetting of the problem and the presence of edge andcorner singularities in the solutions. M Dauge(Rennes) gave two survey talks on the structure of

the singularities in three dimensions. M Costabel(Rennes) compared various alternative variationalsettings for the time-harmonic Maxwell equationsand outlined recent theoretical results on theconvergence of Maxwell eigenvalues using aregularised variational formulation. D Boffi beganby surveying the theory for the numericalapproximation of eigenvalues for non-coerciveoperators, and then showed how the generaltheory applies to the Maxwell equations andhighlighted the importance of the discretecompactness property.

The First European Finite Element Fair Workshop, 8−9 May 2003 Organisers: M Ainsworth (Strathclyde),C Carstensen (Vienna), CM Elliott (Sussex),C Schwab (Zürich) and E Süli (Oxford)

The European Finite Element Fair (EFEF) was theinaugural meeting of what is planned to become anannual series of completely informal smallworkshops throughout Europe with equal initialconditions for each speaker. The idea of EFEF is toprovide a platform for high-level discussions oncurrent research on finite element approximation,in the broadest sense, of PDEs. The format is basedon a long series of such meetings held in the USA.A few, but strict, rules were applied in order todistinguish it from existing workshops andminisymposia in the field.

• Provided that he or she was presentthroughout the meeting, each participantwas invited to talk.

Participants at the workshop “INTERPHASE 2003”

S Wygard


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• Based on the number of speakers the timeavailable was divided into slots for the talks.The order of the speakers was determinedthrough random choice, by drawing namesout of a hat. Speakers could not request aspecific time to talk.

• Each speaker had to introduce himself orherself, the title and topic, and was expectedto leave sufficient time, within the allocatedtime-slot, for discussion. Speakers had toprepare a talk that could be trimmed tovarious lengths.

There were 34 participants at the meeting.

Partial Differential Equations andComputational Material Science Spitalfields Day, 13 May 2003Organisers: CM Elliott (Sussex) and M Luskin(Minneapolis)

Three lectures of a general nature were presentedand were well attended by Institute participantsfrom both parallel programmes and academicsfrom within and without Cambridge. The aim wasto present exciting new areas of research inmaterials science requiring significant compu-tational mathematics based on PDEs.

A follow-up day on 14 May 2003, entitled PartialDifferential Equations and ComputationalMaterial Science and Solid Mechanics, was morespecialised and focussed on the interactionbetween the micro and macro scales.

Mathematics of Finite Elements andApplicationsSatellite Conference at Brunel University, 21−24 June 2003Organiser: JR Whiteman (Brunel)

The Newton Institute played a prominent role inthe latest of this long-standing series ofconferences. The Rothschild Visiting Professor, F Brezzi, gave the Isaac Newton Lecture:Stabilising sub-grids and their construction. TheDirector of the Institute, Sir John Kingman, gavethe after-dinner speech and there was a Newton

minisymposium running throughout theconference. In addition the following long termparticipants in the programme gave invitedlectures: M Ainsworth, C Carstensen, C Schwaband E Süli.

Outcome and AchievementsAt the date of printing there are over 35 preprintsassociated with the programme in the NewtonInstitute Preprint Series. The slides, overheads andcomputer presentations of a large number oflectures in the programme may be found on theweb pages of the Institute. There was a great dealof interaction and collaborative research duringthe programme and we discuss this below undernon-mutually exclusive headings.

Nonlinear PDEs and Free Boundary Problems

Diffuse Interface and Phase Field Methods

Z Chen, A Schmidt and R Nochetto madesubstantial progress in completing a project onAdaptive FEM for diffuse interface models. D Kessler, R Nochetto and A Schmidt derived aposteriori error bounds to use in Error control forthe Allen−Cahn equation. J Barrett and CM Elliottbegan a project to derive fully discrete stableschemes for Cahn−Hilliard fluids entitled Finiteelement approximation of Cahn−Hilliard, Navier−Stokes and Hele−Shaw systems. J Barrett, HGarcke and R Nürnberg developed an approach tosurface diffusion with elasticity via the coupling ofelasticity equations to a degenerate Cahn−Hilliardequation.

Geometric Flows, Surface Diffusion and LevelSet Methods

K Deckelnick and CM Elliott completed a projecton Uniqueness and error analysis for Hamilton−Jacobi equations with discontinuities. Thisinvolves viscosity solutions and has applications tosolving the level set eikonal equations indiscontinuous media. There was significant activityassociated with surface diffusion: K Deckelnick, GDziuk and CM Elliott derived an algorithm andproved optimal order error bounds for Fullydiscrete second order splitting finite element

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methods for anisotropic surface diffusion forgraphs; G Dziuk and CM Elliott developed levelset methods; and P Morin and R Nochettoimproved their algorithm for parametric evolutionof surfaces by introducing volume preserving meshregularisation. R Davies and CM Elliott derived alevel set model for the proving of shaped breadrolls. O Lakkis and R Nochetto completed theirwork A posteriori error analysis for the meancurvature of graphs. K Mikula completed a longmanuscript, Computational solution, applicationsand analysis of some geometrical evolutionequations, with a major application orientedtowards image processing.

Nonlinear PDEs

R Nochetto, A Schmidt, K Siebert and A Veesercollaborated on a posteriori error analysis forSemilinear elliptic PDEs with free boundaries. J Barrett, C Schwab and E Süli initiated a majorproject on the analysis and numerical analysis ofkinetic models for polymeric fluids. J Barrett, X Feng and A Prohl collaborated on the numericalanalysis of Erickson’s model for nematic liquidcrystals.

Pani collaborated with Süli on the error analysis ofdiscontinuous Galerkin finite element methods andderived optimal error bounds in the L2 norm. Healso contributed to the Newton Institute workshopat the MAFELAP conference.

Superconductivity

L Prigozhin made substantial progress on his workPenetration of nonuniform magnetic field fluctu-ations into type II superconductors and the AClosses. CM Elliott, D Kay and V Styles developedan analysis of a degenerate Stefan formulation of acritical state model in Finite element analysis of acurrent density−electric field formulation of Bean’smodel for superconductivity.

Evolving Surfaces

G Dziuk and CM Elliott developed an elegantcompact formulation for transport and diffusionon evolving surfaces. This has applications to, forexample, the transport of surfactants on fluidsurfaces. A finite element algorithm based on

evolving triangulations was developed togetherwith stability and error analysis.

Microstructure

C Carstensen developed an approach to theconvergence of adaptive finite elements fordegenerate convex problems.

Adaptivity and A Posteriori Error Control

R Nochetto, E Bäensch, O Lakkis, P Morin, KSiebert, A Schmidt and A Veeser continued theirvery active collaboration on the theory andimplementation of adaptive finite elementmethods. Areas where significant progress hasbeen made include error control for theAllen−Cahn equation, adaptive finite elementmethods for diffuse interface models, a posteriorierror analysis for mean curvature on graphs andconvergence of adaptive Raviart−Thomas finiteelement methods.

W Dörfler and M Ainsworth developed a new aposteriori error estimator based on thenonconforming Crouzeix−Raviart element for theStokes equations.

C Schwab, P Houston and E Süli devised a newregularity estimation strategy for hp-adaptive finiteelement methods, E Süli and J Robson developedthe a posteriori error analysis of finite elementapproximation of non-Newtonian flows of power-law type, while B Guo and M Ainsworth collabo-rated on the a posteriori error analysis of h- andhp-version finite element methods.

B Guo completed a paper on the hp-version of theboundary element method.

F Brezzi, LD Marini and E Süli developed an erroranalysis of the discontinuous Galerkin finiteelement method for hyperbolic problems whichsheds new insight on the dissipation properties ofthese methods.

P Houston and TJ Barth developed a new post-processing technique for adaptive finite elementapproximations of hyperbolic problems.

C Carstensen and A Veeser collaborated on theconvergence of adaptive finite element methods


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and developed a general technique of proof ofconvergence.

ADC Hill worked on the a posteriori error analysisof finite element methods for nonlinear parabolicPDEs.

Multiscale and Multilevel Algorithms

W Dahmen, A Cohen and R DeVore continuedtheir collaboration on multiscale methods and thestability of solutions to conservation laws inmetrics other than the usual ones. W Dahmen alsostarted a new collaboration with Kunoth duringtheir stay at the Institute on the a posteriori erroranalysis of wavelet-based methods in theframework of dual-weighted residual typeestimators.

F Brezzi made considerable advances on the erroranalysis of a class of multiscale finite elementmethods based on residual-free bubbles. W McLean and M Ainsworth continued theircollaboration on multilevel preconditioners forfirst-kind boundary integral equations with weaklysingular kernels.

R Duran and R Rodriguez collaborated on thedevelopment and analysis of numerical methodsfor folded elastic plates.

P Plechac and MAK Katsoulakis collaborated oncoarse-graining of multiscale discretisationalgorithms, while R Scheichl worked on thenumerical linear algebra of PDEs which arise fromfluid flows in porous media.

Computational Wave Propagation

M Costabel and M Dauge completed a manuscripton the regularity of solutions of Maxwell’sequations in non-homogeneous media, and, incollaboration with D Boffi, made significantprogress on proving the discrete compactnessproperty for higher order edge elements.

IG Graham completed a paper on the computationof diffraction coefficients in high frequencyacoustic scattering. PB Monk completed amanuscript on the ultra-weak variationalformulation for wave propagation in elastic media

and another on computational micromagnetics,and initiated a project on discontinuous Galerkinmethods for Maxwell’s equations with P Houston. M Ainsworth completed analyses of the dispersiveand dissipative properties of higher order elementsfor standard and discontinuous Galerkin methodsand initiated a new collaboration on this topicwith PB Monk. D Silvester developed software thatwill form part of his forthcoming book with A Wathen and HC Elman, and initiated a newcollaboration with M Ainsworth and PB Monk onpreconditioners for the Helmholtz equation.

Published Books

During the course of the programme, C Johnson completed a major book onmathematical analysis (for Springer-Verlag) inwhich constructive and computational/algorithmicaspects play a vital role, and E Süli completed theundergraduate textbook An Introduction toNumerical Analysis (Cambridge University Press,2003).

ConclusionBased on the reaction of the participants, theorganisers feel that this has been an extremelyexciting and fruitful programme in which a greatdeal has been achieved and in which a substantialnumber of new projects and collaborations hasbeen initiated. We wish to express our sinceregratitude to all members of staff of the IsaacNewton Institute for their help and supportthroughout the programme, and for providingsuch a friendly, pleasant and stimulating workingenvironment.

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Nonlinear Hyperbolic Waves in PhaseDynamics and Astrophysics

27 January to 11 July 2003 Report from the Organisers: CM Dafermos (Brown), PG LeFloch(CNRS), EF Toro (Trento)

for mathematicians is to comprehend theproperties of nonlinear waves and theirrelationships with the dynamics of many physicalphenomena.

The activities during the programme wereorganised around five main themes describedbelow.

Nonclassical Shock Waves andPropagating Phase BoundariesThe first period of concentration (January–February) was centred around the concept of akinetic relation, which determines the propagationof undercompressive waves such as those arising inphase transition dynamics. This idea was putforward by material scientists and appliedmathematicians in recent years. The mathematicalmodeling of such problems and the theoretical andnumerical aspects of the kinetic functions wereboth discussed. Several series of lectures were givenby long-term participants, on continuummechanical models of evolutionary structuraltransformations (Knowles) and on diffusive−dispersive singular limits for systems ofconservation laws when both viscosity andcapillarity effects play a role (Bedjaoui).

Mathematical Aspects of the Dynamics ofPhase TransitionsSpitalfields Day, 10 February 2003Organisers: CM Dafermos and PG LeFloch

This event was supported by the LondonMathematical Society, and was an opportunity toreview the state-of-the-art in the field and toinitiate several collaborations between long-termparticipants. The meeting covered a broad range oftopics, including the mathematical analysis of thekinetic relation (Asakura, Corli, Sablé-Tougeron),the dynamics of phase boundaries in solids(Berezovski, Knowles, Pence, Thanh) and thediscretisation of continuous models (Friesecke,Shearer, Truskinovsky).

Nonlinear Hyperbolic Waves in Phase Dynamics andAstrophysics

Left to right: PG LeFloch, EF Toro, CM Dafermos

S Wygard

BackgroundQuasilinear hyperbolic systems in divergence form,

commonly called “hyperbolic conservation laws”,

govern a broad spectrum of physical phenomena in

compressible fluid dynamics, nonlinear material

science, general relativity, etc. Such equations

admit solutions that may exhibit various kinds of

shocks and other linear and nonlinear waves

(propagating phase boundaries, fluid interfaces,

gravitational waves, etc.) which play a dominant

role in multiple areas of physics: astrophysics,

cosmology, dynamics of (solid−solid) material

interfaces, multiphase (liquid−vapour) flows,

combustion theory, etc. In recent years, major

progress has been made in both the theoretical and

numerical aspects of the field, while the number of

applications has skyrocketed. Hyperbolic models

arising in applications often face serious

mathematical difficulties related to the occurrence

of discontinuities, coordinate singularities,

resonance between two or more wave speeds,

elliptic regions in phase space, etc. The challenge


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Well-Posedness Theory ofSystems of Conservation LawsRecent developments on the general theory ofentropy solutions to one-dimensional hyperbolicsystems of conservation laws were covered duringthis second concentration period. Over the two-week period 10–21 March, a total of six series oflectures was offered on:

• Fundamental concepts of the theory ofshock waves and their connection withthermodynamics and continuum mechanics(Dafermos)

• Conservation laws with diffusion or relax-ation (Marcati, Serre)

• Vanishing viscosity solutions (Bressan)

• Well-posedness theory for general hyper-bolic systems (LeFloch)

• Boltzmann, Euler and Navier−Stokes equa-tions of gas dynamics (Liu)

Mathematical Theory of HyperbolicSystems of Conservation LawsWorkshop, 24–28 March 2003Organisers: CM Dafermos and PG LeFloch

This workshop was very successful. It attracted alarge and enthusiastic participation from many ofthe worldwide experts in the field and provided a

special opportunity to investigate current trends on

the mathematical theory of shock waves and

conservation laws. All contributions consisted of

one-hour presentations, allowing plenty of time for

questions and discussions during the breaks. Two

poster sessions were also organised. The main

themes treated during the week were:

• General properties of hyperbolic systems

arising in continuum physics (Godunov,

Brenier, Ferapontov)

• Kinetic models from mathematical biology

and discrete velocity Boltzmann equations

(Perthame, Tzavaras)

• Well-posedness theory of systems of cons-

ervation laws, especially existence theory for

general flux-functions and classical and

nonclassical entropy solutions (Iguchi,

LeFloch, Trivisa, Liu)

• Models of diffusive relaxation approx-

imations (Marcati, Serre)

• Vanishing viscosity approximations (Bres-

san)

• Euler and Navier−Stokes equations includ-

ing results on critical thresholds in Eulerian

dynamics (Tadmor)

• Asymptotic behaviour (Nishibata)

Participants at the workshop on “Mathematical Theory of Hyperbolic Systems of Conservation Laws”

S Wygard

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Multidimensional Hyperbolic ProblemsIn April, the programme concentrated on multi-

dimensional aspects of hyperbolic conservation

laws, including computational methods with

applications to multiphase flows. Several series of

three lectures each were given on multidimensional

Riemann problems and related issues (Chen,

Keyfitz) and on mixed equations of steady

transonic flow dynamics (Morawetz). The

equations of transonic flow are particularlychallenging: change of type of the equations,degeneracy at the sonic line and the appearance offree boundary problems which describe shockinterfaces. Building on her earlier results on mixedequations for transonic flow, CS Morawetzpresented new existence theorems for viscousflows, including an application of Noether’stheorem to the Tricomi equation. PD Lax’sinspiring lectures presented an illuminatingoverview on the zero-dispersion limit for theKorteweg−de Vries equation.

Shock wave on a wedge: numerical results

Shock wave on a wedge: experimental results


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Multiphase Fluid Flows and Multi-Dimensional Hyperbolic ProblemsWorkshop, 31 March–4 April 2003Organisers: J Ballmann, PG LeFloch, R LeVequeand EF Toro

This workshop attracted a large internationalaudience of about 80 participants from all over theworld. The talks fell within three categories: first,multidimensional hyperbolic problems includingmixed models for transonic flows (Morawetz,Chen), self-similar solutions to the Riemannproblem in two space dimensions (Keyfitz) and theasymptotic stability of non-planar Riemannsolutions (Frid); second, numerical methods,especially the evolution Galerkin schemes (Noelle)and the positive schemes for systems ofconservation laws (Lax); third, a large variety ofapplications, especially the simulation of cavitationprocesses (Ballmann), the propagation of interfaces(Nikiforakis, Greenberg, Marquina), wavestructure for elastic−plastic flow (Menikoff) andvarious problems from fluid dynamics (Toro,Drikakis, Falle, Peregrine).

Computational Methods forComplex Fluid FlowsDuring May, there were two main concentratedperiods of activity. A short course (LeVeque, Shu,Toro) was organised over four days on numericalmethods for hyperbolic conservation laws. Thecourse was aimed primarily at young engineers andscientists within various disciplines in whichmodern numerical methods for hyperbolic systemsare used. Support for young UK scientists wasprovided via the Institute’s Junior Membershipscheme. There were about 45 participants, 30 ofwhom were junior scientists and the remaindersenior participants. The contents of the courseincluded both basic theoretical aspects ofconservation laws and notions on numericalmethods for hyperbolic equations (Riemannsolvers, high-order schemes). To illustrate thepotential applicability of the methods threelectures were offered on traffic flow modelling

One of PD Lax’s lectures in progress

(Shu), environmental fluid dynamics (Toro) andastrophysics (LeVeque).

Very High-Order Numerical Methods forHyperbolic Conservation LawsWorkshop, 27–30 May 2003Organisers: N Nikiforakis and EF Toro

This four-day workshop was about currentresearch on approaches for constructing numericalmethods of very high-order accuracy for solvinghyperbolic conservation laws. The emphasis wason algorithms that allow high accuracy (thirdorder and above) in both space and time, for one-and multi-dimensional problems and for problemsincluding additional algebraic source terms(balance laws) or higher order derivatives(viscosity, capillarity). The talks in the workshopfell within three categories: algorithm design,numerical analysis and applications. A centralobjective of the workshop was to include all majorcurrent approaches for hyperbolic equations,aiming at bringing together various schools ofthought. Topics studied included: finite volumemethods, total variation diminishing, essentiallynon-oscillatory schemes and variants, adaptivefinite-element methods, discontinuous Galerkinmethods, multi-dimensional upwinding, adaptivemeshes and applications. The workshop enjoyedthe participation of about 40 scientists fromseveral countries. All contributions consisted ofone-hour informal presentations, allowing plentyof time for questions and lively discussions.

S Wygard

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Hyperbolic Models in General RelativityThe last period of the programme (June–July) wasdevoted to hyperbolic aspects of Einstein’s fieldequations of General Relativity. Several majoraspects of the mathematical theory were covered.This period started with short advanced courses onthe Cauchy problem for the Einstein vacuumequations (Klainerman), shock wave solutions ofthe Einstein equations (Temple), linear hyperbolicequations in a black hole geometry (Finster) andthe Dirac sea and the principle of the fermionicprojector (Finster again). S Klainerman and IRodianski presented a new conjecture on theoptimal regularity of the solutions to the Einsteinequations. The programme included also someactivity on the interaction of gravity with otherforce fields, described by Einstein’s equationscoupled to the Yang−Mills, Maxwell, and Diracequations (Smoller, Finster). These coupled, highlynonlinear equations have been shown to admitsolutions with surprising properties.

Recent Activity on Numerical General RelativityHewlett-Packard Day, 2 June 2003 Organisers: PG LeFloch, and JM Stewart

The speakers discussed recent numerical work inGeneral Relativity, addressing central issues relatedto cosmology, the collision of two black holes, thegeneration of gravitational waves and shock

waves, etc. The themes included the most recentprogress in numerical relativity concerning thebinary black hole inspiral and merger (Hawke,Seidel), recent advances in numerical relativisticmagnetohydrodynamics motivated by applicationsin astrophysics (Marti) and the Geroch reductionand the issue of axisymmetry in numericalrelativity (Stewart).

Hyperbolic Models in Astrophysics andCosmologyEuroconference, 23–27 June 2003Organisers: CM Dafermos, PG LeFloch, JA Smoller and JM Stewart

This meeting was devoted to the Einstein fieldequations of General Relativity and covered boththe general mathematical theory and variousapplications in astrophysics and cosmology. Itbrought to Cambridge the front-line specialists inmathematical general relativity. A large number ofyoung researchers from the European Communityparticipated in this event and the conferencefostered exchanges between several researchcentres of excellence working on general relativity,including Cambridge, Golm, Oxford, Princeton,Southampton and Trieste. There were about 20lectures of one hour each plus two poster sessions,leaving plenty of time for lively discussionsbetween participants.

A presentation of the most recent research onexistence theory for the Einstein equations wasprovided by Friedrich, Klainerman, Reula and

Participants at the workshop on “Multiphase Fluid Flows and Multi-Dimensional Hyperbolic Problems”

S Wygard


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reviewing the state of the art for physical models,techniques of mathematical analysis andnumerical analysis. Many challenging problems inthe field were discussed. Significant work wascarried out by the participants and new bridgesarose between the most recent developments in thegeneral mathematical theory of shock waves andthe areas of applications that are currently mostactive.

For instance, the programme helped to increase theinterest of the applied mathematics community inrelativistic fluid models. The lectures by HFriedrichs and JM Stewart were very helpful toclarify the most important issues in GeneralRelativity and initiate exchanges between the long-term participants.

Progress was also made on nonclassical travellingwaves with viscosity and capillarity terms and theconcept of nucleation and its connection with thestability or instability of interfaces in phasetransition dynamics.

One of the most unexpected and lasting outcomesof the programme was the launching of a newmathematical research journal, the Journal ofHyperbolic Differential Equations, devotedprecisely to the topics treated during thisprogramme. Almost all of the members of theeditorial board were long-term participants, andmany participants have already submitted paperswritten at the Institute for publication in theJournal.

Rodianski. The other themes were:

• Late-time behaviour of solutions (Rendall)

• Gravitational waves and shock waves(Griffiths, Stewart, Temple)

• Interaction of gravity with other force fields(Smoller, Linden)

• Recent progress on numerical methods ingeneral and special relativity (Bona, Font)

• Geometric aspects of Lorentzian manifolds(Zeghib)

• The interior of charged black holes and theproblem of uniqueness in general relativity(Dafermos)

• Many other topics such as black holespacetimes and scattering problems (Komis-sarov, Lindblad, MacCallum, Mason,Melnick)

Outcome and AchievementsThe programme proved very popular and broughtto Cambridge a large number of short- and long-term participants, including both young and seniorresearchers. All participants were very favourablyimpressed by the working conditions offered by theNewton Institute and many would have liked tocome for a longer period.

Weekly seminars, short courses, workshops andconferences provided valuable opportunities for

Lecture by CS Morawetz

S Wygard

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Finances

2001/2002 2002/2003Year 10 Year 11

Income

Grant Income − Revenue 1,247,710 1,230,928Grant Income − Workshop 377,796 226,421Trust Fund Income 78,975 183,904Donations − Revenue 604 512Interest on Deposits 31,901 31,525General Income 5,066 7,661Housing 10,938 7,239

Total Income 1,752,990 1,688,190

Expenditure

Scientific Salaries 331,912 362,158Scientific Travel and Subsistence 410,712 395,477Scientific Workshop Expenditure 288,281 181,483Other Scientific Costs 16,701 22,113Staff Costs (Payroll) 290,285 348,294Staff Costs (Contract and Agency) 18,789 21,738Computing Costs 57,118 58,689Library Costs 11,577 11,459Building − Rent 221,258 227,896Building − Repair and Maintenance 13,042 23,494University Overheads 30,597 15,650Consumables 28,113 27,370Equipment − Capital 16,819 41,996Equipment − Repair and Maintenance 4,819 5,000Publicity 12,253 6,540Recruitment Costs 3,761 2,519

Total Expenditure 1,756,037 1,751,876

Operating Surplus / (Deficit) (3,047) (63,686)

Transfer (to) / from Building Capital Fund (4,964) 0Transfer (to) / from Reprovision 21,829 49,291

Total Surplus / (Deficit) 13,818 (14,395)

Finances

(13 months)

Accounts for July 2002 to July 2003 (Institute Year 11)

Finances

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1. Accounting PeriodThe Institute’s financial year has been amended from 1 July−30 June in previous years to 1 August−31 July in future, to be in line with the University Finance System. This has resulted in the current year2002/2003 being of 13 months’ duration.

2. Grant Income −− Revenue This breaks down as follows:

2001/2002 2002/2003Year 10 Year 11

EPSRC/PPARC Salaries 344,992 314,598EPSRC/PPARC Travel and Subsistence 298,112 297,743Trinity College (Isaac Newton Trust) 100,000 100,000Hewlett-Packard 115,000 115,000PF Charitable Trust 0 19,700CNRS 25,668 0Leverhulme 79,513 79,287LMS 20,000 20,000Cambridge Philosophical Society 2,250 2,000Daiwa Anglo-Japanese Foundation 3,074 0Royal Society 0 2,314University of Cambridge (Staff) 37,843 22,390University of Cambridge (Rent) 221,258 227,896University of Cambridge (Equipment) 0 30,000

Total 1,247,710 1,230,928

3. Grant Income −− WorkshopSee note 8 below.

4. Trust Fund IncomeThis breaks down as follows:

Rothschild − Visiting Professors 25,655 28,207Rothschild − Director 53,320 155,697

Total 78,975 183,904

An additional late stipend payment of £31,793 for 2001/2002 has been included in the reported figuresfor 2002/2003.

5. HousingThis figure represents the net balance of income and expenditure, and is lower than in 2001/2002 mainlyowing to reduced income from rental properties in the second half of 2002.

6. Scientific SalariesThis includes stipends paid to EPSRC/PPARC Fellows, Rothschild Visiting Professors, the Hewlett-Packard Senior Fellow, the Director and the Deputy Director.

7. Scientific Travel and SubsistenceThis includes expenditure incurred by programme participants, Junior Members and the Hewlett-PackardSenior Fellow.

Notes to Accounts

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Finances

8. Scientific Workshop ExpenditureThis figure is lower than in 2001/2002 owing to the reduced level of workshop activity planned byprogramme organisers. The grant income for workshops is also, therefore, correspondingly lower.

9. Other Scientific CostsThis includes costs relating to meetings of Institute committees, Institute Correspondents’ expenses andthe travel expenses of overseas participants who visit other UK institutions to give seminars during theirstay (see p 7).

10. Building −− RentThis is the rental for both the main and Gatehouse buildings, and is covered by a grant from theUniversity of Cambridge. The University also pays for all gas, electricity and rates, which have not beenincluded.

11. Building −− Repair and MaintenanceThis includes rewiring costs related to the Library refit, provision of additional computer storage spaceand work on the car park.

12. Equipment −− CapitalPurchase of security access and monitoring equipment, new Library furniture, kitchen equipment andshelving for the Gatehouse storeroom. Much of this is covered by a one-off Equipment grant from theUniversity of Cambridge and by a transfer from the Reprovision fund.

13. PublicityThis includes costs for the production of the Annual Report, Institute brochures and any other outsourcedprinting. This budget has fallen since 2001/2002, as more of the Institute’s publicity is now performedelectronically when appropriate.

14. Donations in KindCOMSOL Ltd donated their FEMLAB software for a six-month period for use by participants of theComputational Challenges in Partial Differential Equations and Nonlinear Hyperbolic Waves in PhaseDynamics and Astrophysics programmes.

Over 4,100 books and journals have been donated to date by a large number of publishers and individualmembers of the mathematical community.

Finances

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Cumulative Financial Grants and Donations

SERC/EPSRC/PPARCTrinity College (Isaac Newton Trust)NM Rothschild and SonsAnonymous DonationHewlett-PackardDill Faulkes FoundationLeverhulme TrustSt John’s CollegeNATOEuropean UnionUniversity of Cambridge (excluding rent grant)Le Centre Nationale de la Recherche ScientifiqueRosenbaum FoundationPF Charitable TrustLondon Mathematical SocietyClay Mathematics InstituteGonville and Caius CollegePrudential Corporation plcInstitute of PhysicsBritish Meteorological OfficeNuffield FoundationTSUNAMIDaiwa Anglo-Japanese FoundationAFCU (Hamish Maxwell): $50kAFCU (Anonymous Donation): $50kEmmanuel CollegeJesus CollegeBritish AerospaceRolls RoyceCambridge Philosophical SocietyCorporate Members (FIN programme)British GasDERAMagnox ElectricPaul Zucherman TrustThriplow TrustSchlumbergerBank of EnglandWellcome TrustBenfield GreigNERCUnileverApplied Probability Trust

£10,571k £2,510k £2,083k£1,065k£1,065k £1,000k

£855k£750k £728k£736k£573k £435k £330k£240k £225k £152k£100k£100k£68k£64k£57k£40k£36k £32k£32k£30k£30k £25k£25k£25k£22k£20k£20k£20k£20k£18k£17k£15k £15k£10k£10k£10k£10k

over 16 yearsover 12 yearsover 10 years

over 10 years

over 12 yearsover 5 yearsover 10 yearsover 11 years over 11 years over 10 yearsover 7 yearsover 3 yearsover 12 years

over 4 yearsover 7 years

over 4 years

over 6 years

over 10 years

over 3 years

This table includes sums agreed but yet to be received.

contentsgravitational waves, etc.) which play a dominant role in multiple areas of physics:...

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