He eliev ed passionately in the imp ortance of mathematics with strong links to reallife situations and in system of mathematical education in whic the solution of problems tak es precedence er the generation of theory He will remem ered for his ork ID: 83284
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JohnMichaelHammersley JOHNMICHAELHAMMERSLEY21March1920|2May2004ElectedFRS1976ByGeo reyGrimmettandDominicWelshCentreforMathematicalSciences,UniversityofCambridge,CambridgeCB30WBMertonCollege,OxfordOX14JDJohnHammersleywasapioneeramongstmathematicianswhode edclassi cationaspureorapplied;whenintroducedtoguestsatTrinityCollege,Oxford,hewouldsayhedid\dicultsums".Hebelievedpassionatelyintheimportanceofmathematicswithstronglinkstoreal-lifesituations,andinasystemofmathematicaleducationinwhichthesolutionofproblemstakesprecedenceoverthegenerationoftheory.Hewillberememberedforhisworkonpercolationtheory,subadditivestochasticprocesses,self-avoidingwalks,andMonteCarlomethods,and,bythosewhoknewhim,forhisintellectualintegrityandhisabilitytoinspireandtochallenge.Quiteapartfromhisextensiveresearchachievements,forwhichheearnedareputationasanoutstandingproblem-solver,hewasaleaderinthemovementofthe1950sand1960store-thinkthecontentofschoolmathematicssyllabuses.FAMILYBACKGROUNDJohnHammersleywasborntoacouplewithstronginternationalconnections.Hismother,Marguerite(neeWhitehead),wasbornon29June1889inMoscow,whereherfatherThomaswasengagedintheexportandsaleofcotton-spinningandothertextilemachineryfromLancashire.Attheageof14,shewassenttoboardingschoolinEngland,thusescapingthedicultiesanddeprivationsfacedbyherbrothers,anddocumentedin[P2],asaconsequenceoftheRevolutionof1917,whentheBolsheviksdeclaredallforeignassetstobeownedbytheRussianpeople.Theirpropertywasseized,andtheirfamiliesretreatedtoLondonviaMurmanskin1918.Earlyon1January1920,John'suncleGeorgewashauledoutofbedbythesecretpolice(theCheka)andinterrogatedoveraperiodofthreeweeksintheLubianka,sleepingonabareconcrete\rooratsub-zerotemperatures.George'sbrother,Alfred,managedtoextricateGeorgefromthelabourcamptowhichhehadbeenmoved,onthegroundsthathewasabouttodie.Hesurvived,however,andheandAlfredcaughtatrainthatsamedaytotheFinnishborder.John'sfather,GuyHugh,wasbornon5March1883,thesecondsonofafashionableLondongynaecologistwho,whenGuywas14,collapsedanddiedintheprimeoflife,leavinghisfamilyinstraitenedcircumstances.Guyhadtoleaveschool,andhetookajobasanoceboyattheLondonoceoftheUnitedStatesSteelCorporation.BythetimeofJohn'sbirthin1920,GuyhadworkedhiswayuptobeinchargeofthebranchoceinGlasgow.Therewereupsanddownsinhiscareer,occasionedbytimesofretrenchmentandrecessionintheUnitedStates.GuyandMargueritemovedbacktoEngland,andhewasmaderedundantaround1925.HefoundworkastheLondonmanagerfortheYoungstownSteelandTubeCompany,andlateras JohnMichaelHammersley(1920{2004)3EuropeanmanagerfortheBethlehemSteelCompanyfollowingtheDepressionintheUSA.MargueriteandGuyweremarriedin1914,andtheironlysontosurvivechildbirth,JohnMichael,wasbornon21March1920.EDUCATIONThefollowingextractsfromsomeautobiographicalnotespresentaninterestingac-countofJohn'slifepre-Oxford,aswellasinsightintohischaracter.IattendedakindergartencalledtheWatersideSchoolinBishopsStortfordfrom1925to1929.Itwasrunbyaheadmistress,MissBlandford,anditgavemeanexcellentstartinreadingandwritingandarithmetic.Inmylastyear,herfather,MrBlandford,gavemeanintroductiontoLatinandalgebra.In1929IwassentasaboardertoBembridgeSchoolontheIsleofWight.Thiswasaschoolwithprogressiveideasaboutteachingartsandcraftsandcarpentrybutlittleemphasisonanythingacademic:afteracoupleoftermsatBembridge,mypar-entsweredissatis edwithwhatIwasbeingtaughtandIwassentinsteadtoamoreconventionalpreparatoryschool,StrattonParknearBletchley,whereIremainedfrom1930to1934.ThemanwhotaughtmathematicsatStrattonPark,MrPilliner,almostputmeo thesubjectbyaskingmehowmanybluebeansmade ve.WhenIfailedtoan-swertheconundrum,hesaidtheanswerwas5andIwasafool:butIhadalreadydismissedthisastooobvioustobecorrect(andinretrospect,thecorrectanswerisprobablysomethinglike5[bluebeans] 1).However,mymathematicalfortunesweresavedshortlyafterthisincidentbythearrivalatStrattonParkofanotherteacherofmathematics,GeraldMeister.HehadbeenahousemasteratSedberghSchool,wheretherewasaconventionthathousemastershipscouldonlypersistfor15years.Whenhis15-yearstintwascomplete,hedecidedtotryhishandatpreparatoryschoolteach-ingandtookupresidenceatStrattonParkandremainedthereforacoupleofyears,afterwhichhetaughtatWellingtonCollegeandnextattheDragonSchoolinOxford.DuringhistimeatStrattonParkhegavemeasolideducationinmathematicsandalikingforthesubject.ThiscoveredplentyofEuclideangeometry(includingsuchtopicsasthenine-pointcircle)andalgebra(Newton'sidentitiesforrootsofpolynomials)andtrigonometry(identitiesgoverninganglesofatriangle,circumcircle,incircle,etc),butnocalculus.Duetohishelp,IgotascholarshiptoSedbergh.IwasatSedberghSchoolfrom1934to1939.Thereitwastraditionalinthosedaysforthebrighterboystobeshovedontheclassicalside,andinmy rstyearIwasputintheClassicalFifthform,whereIcompletedtheSchoolCerti cateinclassics(theequivalentoffourO-levelstoday)andthenattheendofmy rstyearintotheLowerSixthClassical.However,LatinandGreekdidnotinterestme,andafteronetermintheLowerSixthClassicalIwasallowedtomigratetotheUpperSixthModerntolearnsomescience.IhadsomeexcellentteachinginphysicsfromLenTaylor,andinchemistryfromCharles[sic]Mawby1.MymathematicsmasterwasSydneyAdams(subsequentlyheadmasterofBancroftsSchool).Hisknowledgeofmathematicswas1N.JamesMawby JohnMichaelHammersley(1920{2004)4verysound,butdidnotextendmuchbeyondwhatwasappropriatetoschoolteaching:Irecallbeingpuzzledthatacontinuousfunctionmightbenon-di erentiableeverywhere;andalthoughhewasabletocon rmthis,hecouldnotexhibitaspeci cexampleforme.IpassedHigherCerti cate(theequivalentofA-leveltoday)inmathematics,physics,andchemistryinthesummerof1937,butIdidnotgainadistinctioninmathematics.IsatthescholarshipexaminationforEmmanuelCollege,Cambridge,inDecember1937,andalsoforNewCollege,Oxford,inMarch1938,withoutsuccessinbothcases.However,IwasawardedaMinorScholarshiptoEmmanuelCollegeatasecondattemptinDecember1938.CAMBRIDGEIwenttoCambridgeasanundergraduatein1939.Thewarhadjuststarted,andmanyundergraduatesincludingmyselfpresentedourselvestoenlistattheSenateHousewhichservedasarecruitingstationinCambridge.Atleastasfarasthisrecruitingstationwasconcerned,therewasnotmuchevidenceatthattimeofmakingwartimeuseofpeoplewithpotentialscienti cquali cation.Afterabriefmedicalcheck-up,Ifoundmyselfinfrontofatrestletableoppositeadon,disguisedintheuniformofasergeant,andthefollowingconversationensued.Sergeant:Doyouwanttojointhenavy,thearmy,ortheairforce?Me:Isupposeitshouldbethearmy|IwasintheOTCatschool.Sergeant:Whichregimentdoyouhaveinmind?Me:Ihavenoidea.IhavejuststartedtoreadmathematicshereinCambridge:isthereanyuseformathematicsinthearmy?Sergeant:No,thereisnouseformathematicsinthiswarandinanycaseyouareonlyanundergraduate.TheserviceshavetakenjustthreeprofessionalmathematiciansfromCambridge,oneforthenavytotellthemaboutun-derwaterexplosions,onefortheairforcetoexplainstellarnavigation,andIwasthethird.Mymathematicaljobistoaddupthedailytotalsofrecruitsforthenavy,thearmyandtheairforcerespectively.Iwonderwhothe`sergeant'was,maybeanumbertheorist.Ofcourse,hewaswrong2aboutthewartimeusesofscience,includingmathematics,andaboutthenumberofscientistsandmathematiciansrecruitedfromCambridge,butIdidnotknowaboutthatuntilmuchlater.Inthemeantime,waitinguntilIwaseventuallycalledup,IhungaroundinCambridgeprettyidly.IremembertutorialsfromStoneley,whotaughtmehowtoexpressr2 insphericalpolarcoordinatesbutnotmuchelse;andalsotutorialsfromP.W.Woods,whosefavouritesubjectwasthetwistedcubic.Pupilswouldstrivetokeephimo thetwistedcubicforaslongaspossiblebyaskinghimquestionsonotherbitsofpuremathematics,butoncehewaslockedonthetwistedcubicafterthe rsttenminutesofatutorial,therestofthetutorialwasaforegone2HewasmoreorlessinagreementwithG.H.Hardy,[H2],whofeltitplainthat\therealmathematicshasnodirectutilityinwar",but,whenasking\doesmathematics\dogood"inwar?",founditprobablethattechnicalskillkeepsyoungmathematiciansfromthefront,therebysavingtheirlives. JohnMichaelHammersley(1920{2004)5conclusion.IwasluckytogetaThirdClass3inthepreliminaryexaminationsinmathematicsintheEasterTermin1940,beforebeingcalledupformilitaryserviceintheRoyalArtillery.WARTIMESERVICEDespitetheassertionbytherecruiting`sergeant'intheSenateHouseinCambridgethatmathematicswasofnomilitaryinterestinwartime,Ididlater ndusesforitwhenservingintheRoyalArtilleryinconnectionwithanti-aircraftgunnery.Anair-craftisahigh-speedmovingtarget,whose\rightpathisdetectedandfollowedbyradar.Tohitatargetoneneedstopredicthowfartheaircraftwillhavemovedinthelapseoftimebetweenthegunbeing redandtheshellreachingit.Thiscalculationwasperformedbyapieceofcomputinghardwarecalledapredictor.Thereweretwosortsofsta ocerswhowereexpectedtohaveanenhancedtechnicalknowledgeofanti-aircraftequipment:theywererespectivelycalledInstructorsinGunnery(I.G.s)andInstructorsinFireControl(I.F.C.s)BoththeI.G.andtheI.F.C.hadtechnicalexper-tiseinthethreecomponents(radar,predictor,gun)ofthislinkage;buttheirparticularprovincesoverlappedinthesensethattheI.G.sspecializedinthegun{predictorpair,whiletheI.F.C.sspecializedintheradar{predictorpair.TheSchoolofAnti-AircraftArtillery(S.A.A.A.)wassituatedonthePembrokeshirecoastatManorbier;andtheTrialsWingoftheS.A.A.A.wasatLydstepaboutamiletotheeastofManorbier.ThefunctionoftheTrialsWingwastocarryoutresearchontheperformanceofvari-ouspiecesofanti-aircraftequipment,bothexistingequipmentandequipmentproposedforfutureuse,andtoreportthereontothewaroceandMinistryofSupply.AttheTrialsWingtherewerethreeI.G.sandtwoI.F.C.s;andin1942IbecameoneoftheI.F.C.s,remainingthereuntiltheendofthewar.BeforethathoweverIwascalledupformilitaryserviceinthelatesummerof1940, rstasagunnerandnextasalance-bombardieratatrainingcampatArbor elduntilbeingsenttoanocertrainingcadetunitatShrivenham.Iwascommissionedasasecondlieutenantinthespringof1941andpostedtoananti-aircraftgunsitedefendinganarmamentfactorynearWorsham.AtShrivenhamIhadbeentoldabouttheexistenceofradar;andtheWorshamgunsitehadanearlypieceofradarequipmentwhichoperatedwithawavelengthofafewmetres.Itsperformanceinmeasuringthedistancetoatargetwasreasonable;butitsaccuracyinmeasuringthedirectiontothetargetwasprettyindi erent,relyingoninterferencee ectsbetweenvariousdipoleaerialsreceivingsignalsbothdirectlyandalsore\rectedfromalargehorizontalmatofwiremesh.Atanyrateitrepresentedthecurrentstateoftheartatthattime;anditinterestedmeconsiderably.Wantingtolearnmoreaboutthepotentialitiesofradar,ItooktheratherunusualstepoftelephoningdivisionalheadquartersandasaresultwasselectedtotraintobecomeanI.F.C.ThistrainingbeganwithasixweekscourseonbasicwirelesstechnologyattheRegentStreetPolytechnic,followedbyalongerandmorespecializedcourseonradaratWatchetinSomerset.AtWatchettheyhadaradarwithatencentimeterwavelength,3Of33candidatesfortheMathematicsPreliminaryExaminationsin1940,11wereplacedintheFirstClass,15intheSecond,and7intheThird. JohnMichaelHammersley(1920{2004)6whichatthattimehadnotcomeintogeneralserviceforanti-aircraftgunnery.ThereIlearntaboutthepropertiesofmagnetronsandwaveguides.OnpassingoutofWatchetasaquali edI.F.C.,whichcarriedtheautomaticrankofcaptain,Iwasposted rsttoanestablishmentatOswestrywhichtrainedoperatorsofradarequipment,andnexttoanti-aircraftbrigadeheadquartersintheOrkneyswhereIwasresponsiblefortheradarinstallationsofthegunsitesdefendingScapaFlow.Finallyin1942IwastransferredtotheTrialsWingatLydstep.AmongstthepersonnelatTrialsWingtherewasateamofabout40girlswhocarriedoutthecomputationsnecessaryforanalyzingtheperformanceoftheanti-aircraftequipment,andIwasresponsiblefordirectingtheircalculations.Oneoftheirjobsconsistedinoperatingthekinetheodolitesfortrackingatarget.Thekinetheodoliteswereapairofsynchronizedtelescopiccamerasateachendofabaselineaboutacoupleofmileslong,whichcouldgivesimultaneousreadingsoftherespectiveanglestoatarget(eitheranaircraftoraradarsleevetowedbehindanaircraft).Fromtheresultingdataitwaspossibletocomputefairlyaccuratepositionsofthetargetandhowthesepositionsdependedupontimeasthetargetmovedalongits\rightpath.Inpracticeitwasjustanuglypieceofthree-dimensionaltrigonometry;andwhenI rstarrivedatLydstepitwasdonewithpencilandpaperwiththeaidofa7- guretablesoftrigonometricfunctions,inaccordancewithtraditionsofmilitarysurveyors.Butwhilesurveyorsmayconceivablybeinterestedindeterminingapositiontothenearestfractionofaninch,itwasnonsensetodosoforanaircrafttargetinviewofthemoredominanterrorsinherentingunnery.Oneofmy rstreformswassimplytointroduce4- guretrigonometrictables,andtoequipthecomputingroomwithdeskcalculatingmachinesinplaceoflonghandpencilandpapersums.ThecalculatingmachineswerewinkledoutoftheTreasury,whowerekeepingthemmassedinabigcupboardincasetheymightbeoffutureservicefor nancialpurposes.Therewerecertainbitsofmathematics,ofwhichIhadnopreviousknowledge;inparticularIneededtolearnaboutnumericalmethodsandstatistics.ItaughtmyselffromWhittakerandRobinson'sbookaboutsubjectssuchas nitedi erencesandin-terpolation.Todescribethetrajectoryofashell,giventheangleofelevationofthegun ringit,rangetablesofthesumwereavailableintermsoftheCartesiancoor-dinatesoftheshellatsuccessivewidelyspacedintervalsalongitstrajectory.Ithadnotoccurredtothecompilersoftherangetablesthatitwouldbemorenaturaltorepresentthisdataintermsofpolarcoordinates;and,evenwhenthiswasdonethereremainedthenon-trivialtaskoftwo-dimensionalinterpolationofthisdata.Thereisaresult,duetoKolmogorov,thatacontinuousfunctionofdindependentvariablescanbeexpressedintermsofapolynomialin2d+1functionseachofasinglevariable;butIdidnotknowofthisresultuntilwellafterthewarwasover.NeverthelessIdiscoveredformyselfshortlyafterarrivingatLydstepthatthisresultwasexplicitlytrueintheparticularcased=2atleastforthepolarcoordinateversionsof3:7"and4:5"anti-aircraftguns.Accordinglywerecalculatedtherangetablesofthesegunsintermsofquadrantelevationsandtangentelevations;andwerethenabletocompletethepredictedtrajectoryusing1-dimensionalinterpolations.Acquaintancewithstatisticaltechniqueswastheothermaingapinmypreviousmathematicaleducation;andtocoverthisIobtainedleaveofabsencetoreturnto JohnMichaelHammersley(1920{2004)7Cambridgeforafewweeks.The rstvolumeofM.G.Kendall'sbookonmathematicalstatisticshadjustbeenpublished.IalsoreadR.A.Fisher'sbookonstatisticalmethodsforresearchworkers.StatisticaltechniquesplayedanimportantroleatLydstepinassuringtheperformanceofanti-aircraftradarsandpredictors,andinliaisingwithradardevelopmentsfromtheRadarResearchEstablishmentatMalvern.BytheendofthewarIhadbeenpromotedtotherankofmajor,andappointedaconsultanttotheOrdnanceBoardinLondon.Anti-aircraftgun re,whichhadbeenprettyinaccurateatthebeginningofthewar,hadgraduallyimprovedbytheendofthewar;inparticulartheV1bombwascomparativelyeasytoshootdownbecauseoftheintroductionoftheproximityfuseinshells.Againstthis,theV2bombwasaballisticmissileandsounassailable.InthenearfuturehostilitieswithnuclearweaponswouldrenderdiscussionswiththeOrdnanceBoardabouttheairdefenceofLondonnugatory.E ectively,thechapteronanti-aircraftgunnerywasclosed.POSTWARACTIVITIESIn1946IreturnedtoCambridgeasanundergraduateatEmmanuelCollege.FromtimetotimetherewereoccasionaltripsuptoLondontoful llmydutiesattheOrd-nanceBoard,butthesehadlittlerelevancetothefutureofanti-aircraftgunnery.BeforethewarIhaddoneacertainamountofskiing;andIhopedforahalfblueforskiing.OneofthedicultieswasthatforeigncurrencywasrationedbytheTreasury;andsoIneededtoearnsomeSwissfrancsbygivingsomelecturesonstatisticsatanySwissuniversitythatcouldbepersuadedtoemployme.ThankstoreferencesprovidedbyHaroldJe reys,theFederalInstituteofTechnology(E.T.H.)inZurichwaskindenoughtoprovidethenecessaryfunds.Howeverinthosedaystheuniversityskiingteamconsistedoffourmembers,andIwasranked fthinthetrials;soInevergotahalfblue,althoughIdidtakepartinajointOxford{CambridgematchagainstthecombinedSwissuniversitieswhichwasatwelve-a-sidematch.Needlesstosay,thecombinedSwissuniversitiesbeatthejointOxford{Cambridgeteam.AsaCambridgeundergraduateinthetwoyearsafterthewarIwasmuchmoremotivatedthanIhadbeenin1939/40;andIalsohadthegoodfortunetobetutoredbybettertutors,inparticularA.J.WardandJ.A.ToddforpuremathematicsandR.A.Lyttletonforappliedmathematics.In1948Igota rstclass(Wrangler)inPart2oftheMathematicalTripos.In1948IthoughtIwouldliketotrymyhandatanacademicjobinmathemat-icsormathematicalstatistics.TherewasnoopeningformeatCambridgethen.IappliedforvacantlectureshipsatReadingUniversityandatStAndrewsUniversity,butmyapplicationswerenotsuccessful.HoweverIdidgetanappointmentasagrad-uateassistantatOxfordintheLectureshipintheDesignandAnalysisofScienti cExperiment.ThisLectureshipwasasmalldepartmentheadedbythelecturer(D.J.Finney)andhavingtwograduateassistants(M.Sampfordandmyself)togetherwithasecretaryandacoupleofgirlswithdeskcalculators.AtthattimeitwastheonlyestablishedproviderofstatisticalservicesatOxford,anditsremitwasspreadquitegenerallyoveranyandallqueriesthatmightbethrownupinvariousbranchesofservice.Italsohadtoo erlecturesandinstructionsonstatistics;forexample,itfelltometogive JohnMichaelHammersley(1920{2004)8thelecturecourseintheDepartmentofForestryforoverseasforestocersonthecollectionandanalysisofdataontreesandtheirgrowth.OXFORDHammersleyheldthepositionofGraduateAssistant,DesignandAnalysisofExper-iments,atOxfordUniversity,untilhemovedin1955toAEREHarwellasPrincipalScienti cOcer.HereturnedtoOxfordin1959asSeniorResearchOcerattheInstituteofEconomicsandStatistics.ThiswasapositionofroughlythesamelevelasaUniversityLecturerbutwithneitherformalteachingdutiesnoralinkedCollegeFellowship/Tutorship.ItwasduringthisperiodthathebegananassociationwithTrinityCollegewhichwastolastfortherestofhislife.WhenP.A.P.MoranleftOxfordfortheAustralianNationalUniversityattheendof1951,HammersleytookoverhistutorialdutiesatTrinityasLecturerinMathematics.ItwasnotuntilhiselectiontoaSeniorResearchFellowshipin1961thathebecameafellowofthecollege.In1969hewaspromotedto(University)ReaderinMathematicalStatistics,andwaselectedtoaProfessorialFellowshipatTrinity,twopositionsthatheretaineduntilhisretirementin1987.ItissometimessaidthatHammersleywasonlythesecondmathematicsfellowatTrinitysinceitsfoundationin1555,followinginthefootstepsofThomasAllen(electedin1564).Hewasinfactarguablythe rstsuchfellow.Inthelate16thcenturyallTrinityfellowswererequiredtotaketheoathofsupremacy,anobligationthatAllenavoidedbydepartingtheCollegein1571.ItwasduringthatperiodandlaterthatAllen'smathematicalactivitiesdeveloped,although,unlikeHammersley,heissaidtohavewritten\littleandpublishednothing"(see[A1]).DespitethefactthatHammersleyheldnoocialpositionattheUniversitybetween1955and1959,hetookonhis rstfourOxfordDPhilstudentsinOctober1956.HeretainedanoceinKebleRoad,andheseemstohavespentalotofhistimethere.From1959untilhisretirementin1987,heworkedinwhatappearedtobesplendidisolationinhisoceintheInstituteofEconomicsandStatisticsinStCrossRoad.Asfarasonecouldjudge,apartfromseeinggraduatestudentsandteachingafewTrinityundergraduates,hehadhistimefreeforresearch.ItwasoverSundaylunchinOxfordshortlyafterhisarrivalthathemetGwenBakewell,whobecamehiswifein1951.Their rsthomeinLongwallStreetwassoonreplacedbyWillowCottageontheEynshamRoad,wheretheirsonsJulianandHugowereborn.Althoughhisuniversitypositionwasnotinmathematics,hewasamemberofthesubfaculty,andhelecturedandexaminedunderitsauspices.Hegainedacertainnotorietyforhishighexpectationsofundergraduates.Forexample,oneyearheo eredanon-examinablelecturecourseon`SolvingProblems'inwhichfewstudentslastedverylong.AsaFinalsExaminerin1966,heset(orwasatleastblamedfor)whatwasthemostdicultsetofcompulsorypapersinlivingmemory.1966becameknownasthe`yearofthecarrot'inhonourofonequestionondi erentialequationsthatopenedwiththephrase:\Ifaslicedcarrotisimmersedattimet=0in -indolylaceticacid:::"BasicmathematicaltechniquesmatteredalotmoretoHammersleythanmanyan JohnMichaelHammersley(1920{2004)9advancedresult.Ononeoccasioninanexaminers'meeting,hewouldnotwithdrawfromthepositionthatarelativelylargenumberofmarks,inanadvancedprobabilityquestion,begivenforthecorrectuseofpartialfractions.ItwasnotalwayseasyforstudentsandcolleaguestorisetotheuncompromisinglyhighintellectualstandardssetbyJohnHammersley,butitwasalevelplaying eld,andheappliedhisstandardstohimselfjustastoothers.Totheknowledgeofthecurrentauthors,hetookononlyeightdoctoralstudentsduringhiscareer,andatleast veofthesecontinuedtosuccessfulscienti ccareers.Studentswererequiredtoshowtheirworth,asexplainedbyJohnHalton:AcousindrewmyattentiontoanadvertisementintheObserver:::,seekingappli-cantsforUKAtomicEnergyAuthorityResearchStudentships,tostudyMonteCarlomethodsforaDPhilatOxford.:::Inafewweeks,Iwasinvitedto\presentmyselfforexamination"attheUKAEAsiteatDidcot.Withverylittleideaofwhatthiswouldentail,Iwent.ThereIfounda[numberof]equallybemusedapplicants,whowereusheredintoalargehallfurnishedwithasuitablenumberofsmalldesksandsatdown.JohnHammersleystrodebreezilyuptothepodium,introducedhimself,andaskedustowriteafour-hourexamination,consistingofadozenorsotoughmathe-maticalquestions.Iattemptedtosolveeachprobleminturn,suggestedpossiblelinesofapproach,andtriedtoanswerthequestionsposed,withlittlesuccess.Attheendoffourhours,thepaperswerecollectedandwewaitedanxiouslyfortheoutcome.PeterMarcerhastakenupthestory:WhatasleeplessnightI(andIexpectothers)hadbeforetheinterviewsthenextday,wheneachofusaskedmembersofthepanel,whichincludedJohnandProfessorFlowersashewasthen,whattheanswerswereandhowonedidthequestions.OnlytobetoldthatJohnhaddonetheroundsofthetheoreticalphysicsdepartmentatHarwell,andcompiledtheexaminationoutofthequestionsthatthemembersofthatdepartmentwereinthecourseoftryingtoanswer!Thatis,therewerenoanswerstothesequestionsasyet,andthepaneljustwantedtoseehowwe,thecandidates,mightbegintotacklethem!IthinkthatepisodesumsupJohnforme,agreatmindsometimespuckishlyinclinedbutwithgreatpurpose,andaboveallagreatgentlemanoftheoldschool.Hewasadelighttoknow,andwillbesorelymissed,andIowehimagreatdeal.Asaresultofthisexercise,Halton,Marcer,DavidHandscomb,andJillianBeard-woodwereawardedstudentshipsunderHammersley'ssupervision.As`MonteCarlo'students,theywereprivilegedwithaccesstotheFerrantiMercurycomputersatbothOxfordandHarwell,aswellastotheIlliacIIwhilevisitingtheUniversityofIllinoisatUrbanain1958.HammersleywasforaperiodequallyathomeinCaliforniaandOxford.HewasaregularcontributortotheBerkeleySymposiaonMathematicalStatisticsandProb-ability,andwasaclosefriendofthedistinguishedstatisticianJerzyNeyman.HespenttheMichaelmastermsof1958and1961atUrbana,IllinoisandBellTelephoneLaboratories,MurrayHillrespectively.Onboththesetripshewasaccompaniedbyhisgraduatestudents.HeneverstudiedforaPhD,perhapsbecauseofhisagefollowingwarservice,buthewasawardedanScDbyCambridgeUniversityin1959,followedinthesameyearbyanOxfordDSc(byincorporation).HewasawardedtheVonNeumannMedal JohnMichaelHammersley(1920{2004)10forAppliedMathematicsbytheUniversityofBrussels(1966),theGoldMedaloftheInstituteofMathematicsanditsApplications(1984),andthePolyaPrizeoftheLondonMathematicalSociety(1997).HewaselectedtotheRoyalSocietyin1976.Hegavethe1980RouseBalllectureatCambridgeUniversity,andpublishedanaccountin[141].OnretiringfromhisOxfordReadershipin1987,hewaswelcomedattheOxfordCentreforIndustrialandAppliedMathematics(OCIAM).Hereciprocatedthisactofhospitalitybymakinghisextensivemathematicalexperienceavailabletoallwhoasked.ManyofHammersley'sfriendsandcolleaguesgatheredin1990attheOxfordMath-ematicalInstituteforaconferencetorecognisehis70thbirthday.Avolume[G5]en-titled`DisorderinPhysicalSystems'waspublishedinhishonour,withcontributionsfrommanywhoseworkhadbeentouchedbyhisideas.Hammersleygavetheclosinglectureofthemeetingunderthetitle`Isalgebrarubbish?',butheuncharacteristicallyrefrainedonthisoccasionfromansweringthequestion.InmorerecentyearshewastobefoundatWillowCottage,reading,doingthecrossword,andworkingonEdenclusters.Hediedon2May2004afteranillness.JOHNHAMMERSLEY,MATHEMATICIANJohnHammersleywasanexceptionallyinventivemathematicianandaremarkableandfearlessproblem-solver.Hehadtherareabilitytopinpointthebasicmathematicsunderlyingascienti cproblem,andtodevelopausefultheory.Hepreferredwhathecalled\implicated"mathematicsover\contemplative"mathematics;thatis,hefoundthesolutionofproblemstobesuperiortothe\high-risemathematics"ofwhichhecouldbesharplycritical(see[126,131]).Theconventionalmodernclassi cationofmathematicsintopure,applied,andsta-tisticscanaccentuategapsbetweentheseareas,gapsthatneedtobe lled.Hammer-sleyspurnedsuchanattitude;whenfacingapracticalproblem,heusedwhateverhecould ndinordertosolveit.This`barehands'approachdoesnotalwaysleadtotheneatestsolution,although,inHammersley'scase,muchoftheresultingmathematicshasstoodthetestoftime.Severaloftheproblemsthatheformulatedandpartlysolvedhaveemergedsinceaslandmarkproblemsofcombinatoricsandprobability.Forexample,hisworkonself-avoidingwalksandpercolationisfundamentaltothetheoryofstochasticLownerevolutionsthatisnowcausingare-thinkoftherelation-shipbetweenprobabilityandconformal eldtheory;hisresultsontheUlamproblemunderlytheproof[B1]thattherelevantweaklimitistheTracy{Widomdistribution.Thesetwogeneralareasareamongsttheliveliestofcontemporarymathematics,aswitnessedbytheawardofFieldsMedalsin2006toWernerandOkounkov.Paper[29],writtenjointlywithK.W.(Bill)Morton,isalandmarkofhisearlierworkintworegards.Firstly,itmarksabeginningofHammersley'sextensivestudyofdiscreteproblemsinprobabilityandstatisticalmechanics.Secondly,thepapercontainstwoproblemsandatechniquethathaveattractedagreatdealofattentioninthe50yearssince.Despitethetitleofthepaper,`Poorman'sMonteCarlo',thelastingcontributionsaretheclearstatementoftheproblemofcountingself-avoidingwalks,theuseofsubadditivitytoprovetheexistenceoftheconnectiveconstant,and JohnMichaelHammersley(1920{2004)11thediscussionofrandommediathatculminatedinSimonBroadbent'sformulationofthepercolationmodel.Theseandothertopicsarediscussedfurtherinthefollowingparagraphs,comple-mentedbysummariesofhowJohn'sworkhasstimulatedtherelevant eldssince.Computing/Calculating/EstimatingHammersley'searlyscienti cworkwasbasedonthemathematicshehadbeendoingduringthewar.His rstpublication[1]arosefromindependentcontributionsbyMajorsBayleyandHammersleytothediscussionfollowingthereadingofapaperonrandomprocessesbyMauriceBartlett[B2]atasymposiumonAutocorrelationinTimeSeriesheldin1946attheRoyalStatisticalSociety.TheproblemconfrontingBayleyandHammersleyaroseintrialsofanti-aircraftequipment.Thedetailswereembodiedin\reportsnotgenerallyavailable"but[1]containsincondensedformsomeoftheresultsobtained.Therefollowedasequenceofpapersonessentiallyunrelatedproblems,manycon-cernedwithhardcalculationsorestimation.Probablyhis rstsigni cantworkwashispaper[11]ontheestimationofparameterswhentheparameterspaceisadiscretesetofpoints.Heshowed,forexample,that,iftheunknownmeanofanormalpop-ulationwithgivenvarianceisassumedinteger-valued,thenitsmaximumlikelihoodestimatoristheintegernearesttothesamplemean.Hisinterestinissuesofthiskindarosefromaproblemofestimatingthemolecularweightofinsulin,andthismayhavecometohisattentionduringhisworkasaconsultantonstatisticalproblemstomembersoftheuniversityinthenaturalsciences.Itwasamathematicalproblemarisingin[11]thatledtohispaper[14]onasymp-toticformulaeforthesumsofproductsofthenaturalnumbers.Paper[14],readinisolation,mayappeartobescantilymotivated.However,itdoesdisplayHammers-ley'sformidableanalyticskills,anditattractedtheattentionofPaulErd}oswho,in[E2],settledoneoftheopenproblemsposed.Itisnowclearthat,in[14],hewasinfactcalculatingwhatCramer[C2]describedrecentlyas\remarkableexpressions"forthemodeofStirlingnumbersofthe rstkind.Throughouttherestofhisscienti ccareer,JohnHammersleycontinuedthisin-terestincomputingmethodsandcomputerscience|principallythroughhisworkonlarge-scalesimulations(seebelow).AppliedProbabilityIntheperiodbetweenleavingthemilitaryandstartinghiscollaborationwithMor-ton,Hammersleyappearstohavetriedhisluckatarangeofproblemsinappliedprobability,hardanalysisandlarge-scalecomputations.Forexample,in[9],hecon-sidersaproblemarisinginthedesignofexperimentsthatmaybeexpressedasfollows:givenacollectionofkcounterfeitandn kgenuinecoins,howmaywedetectthecounterfeitcoins?Hisinterestinstochasticgeometrywasdevelopedin[10],wherehestudiedthedistributionofthedistancebetweentwopointsindependentlyanduniformlydistributedoverthesolidn-sphere.In[15],heprovedaspecialcaseofaconjectureofFejesTothaboutthesumoftheside-lengthsofaconvexpolyhedron JohnMichaelHammersley(1920{2004)12containingasphereofunitdiameter.Hispaper[27]onMarkovianwalksoncrys-tallatticesoriginatedfromastudyofdi usionofelectronsincrystalssuchasthehexagonalclose-packedlattice.Around1953,heconsideredaproblemoncountingbloodcellsthathadarisenattheClinicalPathologyDepartmentoftheRadcli eIn rmaryatOxford.Themathe-maticalproblemhereturnsouttobeequivalentto ndingtheprobabilitydistributionofthenumberofgapsbetweenintervalsofrandomlengthplacedrandomlyonacircle.Hammersleyshowed(bytypicallyhardanalysis)thatitwasasymptoticallynormal.CyrilDombhasgivenanaccountin[D]ofthehistoryandrami cationsofthispar-ticularproblem,andthisworkillustratesHammersley'sgiftforpickingouthard,genuinelyinterestingproblemsfromtheappliedsciencesandtranslatingthemintovalidmathematics.In[40],heextendedaclassicalresultofMarkKac[K1]concerningthenumberofzerosofapolynomialwithrandomcoecients.Kac'sresultswereforthemeannumberofrealzeroswhenthecoecientsareindependent,identically,andnormallydistributed,andHammersleygaveasubstantial,albeitcomplicated,generalisation.Forrecentactivityinthis eld,see[F2,F4,R1].Hammersley'smostin\ruentialworkinappliedprobabilityisthatonpercolationandonthelarge-scalegeometricalpropertiesofnpointsdroppedatrandomintoaboundedregionofEuclideanspace.Wereturntothesetwoareasinlaterparagraphs.HavingsketchedHammersley'searlywork,wemovetohisworkpost-retirement,almostallofwhichwasconcernedwiththegrowthofcrystals.HeworkedwithMazzarinoonathird-orderdi erentialequationarisingasamodelforthegrowthofacrystalinasupercooledliquid,see[116,117].This`classical'workwasfollowedbyhis naltworesearchpapersdirectedatthestochasticmodelintroducedin1961byMurrayEden[E1]forgrowthinbiologicalcells.Despiteitsapparentsimplicity,theEdenmodelhasattractedagreatdealofinterestovertheyears.Inthesimplestversion,the`cells'aretakentobeclosedunitsquaresofthe2-dimensionalsquarelattice.Allcellsbutonearecolouredwhiteinitially,andsubse-quentlycellsareblackenedoneatatime.Themechanismofgrowthisasfollows.Anedgeofthelatticeiscalledactiveifitseparatesablackcellfromawhitecell.Atstagen,anactiveedgeispickedatrandom,andtheassociatedwhitecelliscolouredblack.Attimen,thereisaclusterCncontainingn+1blackcells.TheshapesoftheCnhavethesamedistributionasthoseofthe rst-passagepercolationmodeldiscussedbelow,whentheedge-passage-timesofthatmodelareexponentiallydistributed.Naturalquestionsofinterestaboutthisprocessare:(i)whatisthe`shape'ofCnforlargen,and(ii)howlargedothe`lakes'ofenclosedwhitecellsgrowbeforetheyareeventually lledinbyblackcellsanddisappear?In[120],Hammersleypresentednon-rigorousargumentssuggestingthatalllakesinthe`island'CnliewithhighprobabilitywithinadistanceO(logn)ofthecoastline.Inhispenultimateresearchpaper[119],Hammersley(withMazzarino)carriedoutalarge-scaleMonteCarlosimulationinwhichclustersofsizeoforder109aregrown,andvariousquantitiessuchasthemeancluster-radiusareestimated.Theauthorsevincedprideinbeingabletocarryoutthishugecomputationaltaskusingonly24megabytesofaConvex220machine,incontrasttocomparablesimulationsof JohnMichaelHammersley(1920{2004)13ZabolitskyandStau er[Z]usingaCray2with4parallelprocessorsandavast(fortheperiod)storeof2045megabytes.Asubjectofprimaryinterestinthesetwopapersisthe`surfaceroughness'ofatypicalcluster.Thetheoreticalanalysiscarriedoutin[120]makesuseofthetheoryofharnesses,asintroducedbyHammersleyin1967.Harnessesmaybedescribedlooselyasaspatialgeneralisationofamartingale;theyappeartohavereceivedverylittleattentionsince1967,althoughHammersley'soriginalpaper[81]wasoneofthe45articlesselectedandreprintedin[F1]asoneoftheseminalcontributionstothescalinglawsthatcharacteriseroughsurfacesgeneratedstochastically.MonteCarloMethodsFromtheverybeginningofhiscareer,JohnHammersleysoughtmethodstocarryoutlargecomputations.Theequipmentthenavailablewaslimitedandunreliableand,ratherasinhisarmydays,hebecameamasterofdeskcalculatorsandearlycomputers.Heconsidereditavirtuetousecomputingresourcesinaneconomicandecientmanner,andthisattituderemainedwithhimallhislife.Heonceboastedofholdingthe1961worldrecordforkeepingacomputer(atBellLabs)workingwithoutbreakdownfor39hours.Creditforthenameandthe rstsystematicdevelopmentofMonteCarlomethodsisusuallyaccordedtoFermi,Metropolis,vonNeumann,andUlam.ThisareafascinatedHammersley.Theideaisthatonemayestimateaquantitythroughcomputationsinvolvingrandomnumbers.Aprincipalobjectiveistoreducethedegreeofvariationintheestimate,therebyimprovingtheaccuracyoftheresult.Hammersley'sinterestinMonteCarlomethodsseemstohavebeensparkedbyhisattendanceatasymposiuminBerkeleyintheearly1950s,andhegaveaMaster'slevellecturecourseonthesubjectonhisreturntoOxford.IntheaudiencewasBillMortonwhohadjustgraduated(in1952)fromOxfordandheldanappointmentatAEREHarwell.ItwasaroundthistimethatHammersleyorganisedtheworkshoponMonteCarlomethodsatHarwellduringwhichhemetSimonBroadbent.ItwaswithMortonthatHammersleywrotehispaper[29]entitled`Poorman'sMonteCarlo',ofwhichthebasicthesiswasthatonedoesnotnecessarilyneedlargehigh-speedmachinestouseMonteCarloe ectively.Inordertoillustratethismainpoint,theauthorsdrawonarangeofexamplessuchasself-avoidingwalks.AmongthemoredivertingoftheexamplesisthetestingofaquantumhypothesisofAlexanderThom.Thomhadmeasuredthediametersof33DruidcirclesinWesternScotland,and,basedonthe(integer)data,heconjecturedthatthesediameterswereintendedtobemultiplesof11.1feet.Theevidenceforthiswasthat27ofthecircleshaddiameterslyingintherange11:1(n14)forintegraln.HammersleyandMortonusedsimpleMonteCarlomethodstotestthehypothesisand,asDavidKendallsuggestedin[K4],theirworkledtoastatisticalexaminationwhichwentalongwaytowardscon rmingthisproposal.MonteCarlomethodsarebasedupontheuseofpseudo/quasi-randomnumbers,andthisraisescertainissuesofprinciple.Hammersley'simpatiencewithphilosophicaldiscussionsinvolvingtheethicsorcorrectnessofusingpseudo/quasi-randomnumbersinplaceoftrulyrandomonesiscapturedinhisreplytothediscussionsattheSympo- JohnMichaelHammersley(1920{2004)14siumonMonteCarloMethodsatwhich[29]waspresented:\Thediscussionhasraisedseveralquestionsaboutrandomnumbers:dotheyevenexist;cantheybeproducedtoorderandifsohow;cantheyberecognisedandcanwetestthattheyarenotim-posters?Thesearedivertingphilosophicspeculations;buttheappliedmathematicianmustregardthemasbesidethepoint."Indeed,hisintoleranceofphilosophyasanacademicsubjectseemedtostaywithhimthroughouthislife.TheOxfordjointschoolofMathematics&Philosophywasoneofhisb^etesnoires,andvariousamusingstorieshaveaccumulatedabouttheyearwhereheendedup(bydefault)asChairmanoftheExaminers.WhentheopportunitycameforhimtochairtheFinalsexaminingboard,hegraspeditenthusiastically,andtaped(withhiscolleagues'permission)apost-meetingdiscussiononthevalueofthedegree.Hisfurtherstrenuouse ortscouldnotintheendpersuadeeitherthemathematiciansorthephilosophersthatthedegreeshouldbeshelved.Hammersley'smostsigni cantcontributiontothetheory,asagainstpractice,ofMonteCarlomethodsisprobablyhisworkonantitheticvariates.Thisisatechniqueforyieldingestimateswithvariancesconsiderablylessthanthoseobtainablebyanaiveapproach.Thisistypicallyachievedbyrepresentingtheestimatorasasumofcorrelatedrandomvariables,anditisoneofthemostpopularvariance-reductiontechniques.Itsdrawbackisthatmanyantitheticsamplingplansaretoocompu-tationallycomplextobeofpracticaluseinsimulations.Despitethis,theworkofHammersleyandMorton[38]iscurrentlyregardedasamajorcontribution.(See,forexample,[R2].)Itisinteresting,therefore,thatin[74]HammersleyandHand-scombclaimonlythename,nottheoriginalideawhich,aspointedoutbyTukey[T3],canberegardedasanimportantspecialcaseofregression.Thistechniqueisnow,perhaps,oneofthemostimportantintheapplicationofMonteCarlomethodstohigh-dimensionalnumericalintegration,withapplicationsinmanyareasincludingmathematical nance.TheHammersley{Handscombmonograph[74],publishedin1964,isalandmarkinthestudyofMonteCarlomethodsandisstillmuchusedtoday.Hammersley'sinterestinthe eldseemstohavedeclinedfollowingitspublication.PercolationPercolationwasbornasamathematicalobjectoutofthemusingsonrandommediafoundin[29],andithasemergedasacornerstoneofstochasticgeometryandstatisti-calmechanics.Oneofthediscussantsof[29],SimonBroadbent,workedattheBritishCoalUtilizationAssociation,wherehewasinvolvedinthedesignofgasmasksforcoalminers(see[G3,110]).HammersleyrecognisedthepotentialofBroadbent'sproposalfor\rowthrougharandommedium,andtheycollaboratedontheseminalpaper[41],wherethecriticalpercolationprobabilitywasde ned.Thereareearlierreferencestoprocessesequivalenttopercolation,see[W]forexample,butitwasHammersleywhoinitiatedacoherentmathematicaltheory.Thebasicmodelisasfollows.Consideracrystallinelattice.Wedeclareeachedgeofthelattice(independently)tobeopen(tothepassageof\ruid)withprobabilityp,andotherwiseclosed.Fluidissuppliedattheoriginofthelatticeandallowedto\rowalongtheopenedgesonly.Thefundamentalquestionistodescribethesizeandgeometryof JohnMichaelHammersley(1920{2004)15thesetCofverticesreachedbythe\ruid.Thesigni canceofthismodelisfar-reachinginstochasticgeometryandstatisticalmechanics,andtheassociatedmathematicsandphysicsliteratureisnowveryextensiveindeed.Ofprimaryimportanceistheexistenceofaphasetransition:thereexistsacriticalvaluepcsuchthatCis nitewhenppc,andCisin nitewithastrictlypositiveprobabilitywhenp�pc.Thenon-trivialityofthephasetransitionwasprovedbyHammersley,asfollows.HammersleyandBroadbent[41]establishedalowerboundforpcintermsofcountsofself-avoidingwalksandtheconnectiveconstant.(Anaccountoftheconnectiveconstantmaybefoundinthenextsection.)Thisresultwasstrengthenedin[43],whereitwasshownthatjCjhasanexponentiallydecayingtailwheneverithas niteexpectation.Themethoddevelopedin[43]isaprecursorofanowstandardargumentattributedtoSimonandLieb[L1,S3]andusuallyexpressedas: nitesusceptibilityimpliesexponentiallydecayingcorrelations.In[46],heprovedanupperboundforpcintermsoftheboundarysizesofneighbourhoodsoftheorigin,andhededucedbygraphicaldualitythatpc1fororientedandunorientedpercolationonthesquaregrid;thisisthepercolationequivalentofthePeierlsargumentfortheIsingmodel,[P1].Thisgeneralroutetoshowingtheexistenceofaphasetransitionisnowstandardformanymodels.Inanalternativemodel,itistheverticesratherthantheedgesofthecrystallatticethataredeclaredopen/closed.Hammersley[54]provedtheusefulfactthatCtendstobesmallerforthe`site'modelthanforthe`bond'model,therebyextendingaresultofMichaelFisher.Thebestmodernresultofthistypeisbyoneofhisstudents,see[G4].Aninveteratecalculator,Hammersleywantedtocalculateorestimatethenumeri-calvalueofpcforthesquaregrid.TheodoreHarrisprovedinaremarkablepaper[H3]thatpc12,andHammersley'snumericalestimatesindicatedpc12;\whatbetterevidencecouldexistforpc=12",hewouldask.HewasthereforethrilledwhenHarryKesten,[K8],provedtheholygrail.Thiswashoweveronlytheendofthebeginningforpercolation.PercolationTheoryhasgonefromstrengthtostrengthinrecentyears.Themainquestionsarelargelysolved(see[G2]),andcurrentattentionisfocussedonthenatureofthephasetransitionintwodimensions.Schramm[S1]predictedthatthescalinglimitoftheperimetersoflargecriticalpercolationclustersconstituteastochasticLownerevolution(SLE)4withparameter6.Smirnov[S4]provedCardy'sformulaforcrossingprobabilitiesofcriticalsitepercolationonthetriangularlattice,andindicatedhowtoachievethefullscalinglimit.See[S2]forasurveyofSLEandassociatedproblemsandconjectures.Self-AvoidingWalksandtheMonomer{DimerProblemIntheparadigmofstatisticalmechanics,asystemismodelledbyasetofcon gu-rationstoeachofwhichisallocatedaweight.Thesumofallweightsiscalledthe`partitionfunction'andthestateofthesystemmaybedescribedviaananalysisofthisfunctionanditsderivatives.Inasystemofpolymers,the rstcalculationisto4OftentermedaSchramm{Lownerevolution. JohnMichaelHammersley(1920{2004)16JohnHammersleyandHarryKestenintheMathematicalIn-stitute,OxfordUniversity,November1993. ndthenumberofsuchpolymers.Whenthepolymersaresimplechainsrootedattheoriginofalattice,thisistheproblemofcountingself-avoidingwalks(SAWs).LetsnbethenumberofSAWsoflengthnonagivenlattice.The rstseriousprogresstowardsunderstandingtheasymptoticsofsnasn!1wasmadein[29].Thekeyisthe`subadditiveinequality'tm+ntm+tnsatis edbytn=logsn,fromwhichtheexistenceoftheso-calledconnectiveconstant=limn!1n 1logsnfollowsim-mediately.Thisobservation,regardednowasessentiallytrivialinthelightofthecomplicatedanalysisachievedsince,hashadaverysubstantialimpactonspatialcombinatoricsandprobability.Itmarkedtheintroductionofsubadditivityasastan-dardtool,anditinitiatedadetailedstudy,stillongoing,ofthegeometryoftypicalinstancesofgeometricalcon gurationssuchaspathsandlatticeanimals.Thesubadditiveinequalityimpliestheboundsnn.Hammersleyinvestedagreatdealofenergyintotryingto ndacomplementaryupperboundonsn,butwithonlypartialsuccess.WithhisstudentWelsh,heprovedin[65]thatsnnexp(n1=2)forsome1.ThiswasimprovedbyKesten[K7]ford3,andsuchboundswerethebestavailableforsometimebeforeitwasrealisedbyothersthatalaceexpansioncouldbeusedforsucientlyhighdimensions,see[M1].Asaresultofalargeamountofhardworkandsomesubstantialmathematicalmachinery,theproblemofcountingSAWswassolvedbyHaraandSlade[H1]in veandmoredimensions.Thecaseoftwodimensions,forwhichtheboundof[65]remainsthebestknown,hasattractedmuchinterestinrecentyearswiththeintroductionbySchrammofstochasticLownerevolutions(SLE),andtheconjecturethatarandomSAWintwodimensionsconvergesinanappropriatesenseasn!1toaSLEwithparameter JohnMichaelHammersley(1920{2004)1783,see[S2].Thisconjectureisoneofthemostimportantcurrentlyopenproblemsinprobability.Hammersleywashappyinlaterlifetolearnofprogresswithpercolationandself-avoidingwalks.Hefeltthathehad\helpedthemintoexistence"forotherstosolve.Thetwo-dimensionalpercolationandSAWproblemsaretwoofthehottestproblemsofcontemporaryprobability,intestamenttoHammersley'sexcellentscienti ctaste.ThereisasecondcountingproblemofstatisticalmechanicsthatattractedHam-mersley,namelythemonomer{dimerproblem.Thisclassicalprobleminsolid-statechemistrymaybeformulatedasfollows:Abrickisad-dimensional(d2)rectangu-larparallelepipedwithsidesofintegerlengthsandevenvolume.Aunitcubeiscalledamonomer,andabrickwithvolume2adimer.Thedimerproblemistodeterminethenumberf(a1;a2;:::;ad)ofdimertilingsofthebrickwithsidesoflengtha1;a2;:::;ad.Hammersleyprovedin[77]thatthesequence(a1a2ad) 1logf(a1;a2;:::;ad)ap-proachesa nitelimitdastheai!1,butwhatisthenumericalvalueofd?Thereisa`classical'resultofstatisticalphysicsofTemperley{Fisher[T1]andKaste-leyn[K3],whoshowedindependentlyin1961that2existsandisgivenby2=exp(2G=)=0:29156:::whereGisCatalan'sconstant.Hammersleydevotedmuchenergyontheoreticalandcomputationalapproachesto ndingacorrespondingresultford3but,asfarasweknow,theexactvalueisstillunknownevenwhend=3.Initsmoregeneralform,themonomer{dimerproblemamountstothepurelycombinatorialquestionofcountingthenumberfG(N1;N2)ofdistinctarrangementsofN1monomersandN2dimersontheedgesandverticesofagraphG,suchthateachdimerisplacedonanedge,eachmonomeronavertex,andeachvertexofGeitherisoccupiedbyexactlyonemonomeroristheendvertexofexactlyonedimer.Forthistobepossible,GmusthaveexactlyN1+2N2vertices,andthedensitypofthecon gurationisde nedastheratio2N2=N1.Hammersleyprovedin[77]thatthenumberofp-densitycon gurationsonthecubeofvolumeninddimensionsisoforder(d;p)nforsomefunction.Hespentmuche ortonobtainingboundsfor,but,eventodayin2-dimensions,ourknowledgeisverylimited.See,forexample,[F3].Thedimerproblemisverymuchalivetoday.Thetwo-dimensionalmodelturnsouttoberelatedtotheGaussianfree eldandtostochasticLownerevolutionswithparameters2,4,and8,see[K5,K6]forexample.First-PassagePercolation,andSubadditiveProcessesPercolationisastaticmodelinthesensethateachedgeiseitheropenorclosed,andwaterisconsideredto\rowinstantaneouslyalongopenedges.HammersleyandWelshformulatedatime-dependentversionofthismodelin[75],anddubbedthis` rst-passagepercolation'.Toeachedgeofthelatticeisassignedarandompassage-time,andthetimeax;yforwatertoreachagivenpointy,havingstartedatx,isthein mumoverallpathsfromxtoyoftheaggregatepassage-timeofedgesin.Thispioneeringpaper[75]isnowrecognisedasoneofthe rstworksofmathematicalsigni canceinthetheoryofthespreadofmaterial,whetheritbedisease,\ruid,orrumour,througharandommedium.Thebasicproblemwastoprovetheexistenceofaspeedfunctionx=limn!1n 1a0;nx,where0denotestheoriginofthelattice. JohnMichaelHammersley(1920{2004)18HammersleyandWelshrealisedthatthekeylayintheuseofsubadditivity,a0;x+ya0;x+ax;x+y,thedi erencewithpreviousapplicationsbeingthatthisinequalityinvolvesrandomvariablesratherthandeterministicquantities.Theyprovedaversionofthesubadditivelimittheoremforstationarystochasticprocessesindexedbyd-dimensionalspace,the rst`subadditiveergodictheorem'.Theyrealisedthatthisisbestdoneinthecontextofageneralsetofassumptions,ratherthanthespeci csituationoutlinedabove,andthustheirpapergavebirthtooneoftheprincipaltechniquesfortheanalysisofspatialrandomprocesses.Thesearchbeganforthe`right'combinationofde nition/theorem,andthiswasfoundbyJohnKingman[K9]inoneoftheclassicpapersof20thcenturyprobability.Despitelaterelaborations,itremainsfascinatingtoreadthisearlyliterature,andespeciallythedialogueof[K10,95].Kingman'sinvitedreviewarticle[K10](withpublisheddiscussion)appearedintheAnnalsofProbability.Hammersley'scontributiontothisdiscussionwastooextensivetobeacceptedassuchbythejournaleditor,anditappearedlateras[95].Itistherethattheconditionofpathwisesubadditivityisre-placedbytheweakerassumptionof`superconvolutivity'oftheassociatedprobabilitymeasures.InanearlierapplicationofsubadditivitytospatialsystemspursuedjointlywithhisstudentsJillianBeardwoodandJohnHalton,Hammersleymadeafundamentalcontributiontothestudyoftypicalinstancesofproblemsinoperationsresearch.DropnpointsatrandomintoaplaneregionRof nitearea.Whatisthelengthoftheminimalspanning(Steiner)treeandtheminimaltravellingsalesmanpathonthesepoints?Theyshowedintheirclassicpaper[47]thattheansweris(inessence)proportionaltocRpnforsomeconstantcR,andtheydevelopedalsoahigher-dimensionaltheory.Thekeywastoencodetheprobleminsuchawaythatthenaturallength-scaleispn,andthentouseatypeofspatialsubadditivity.ThistheoremwascentraltothelaterworkofKarponaprobabilisticanalysisoftherandomEuclideantravellingsalesmanproblem,[K2].FurtherdevelopmentsaredescribedintheFestschriftpaper[S5]bySteele.Thetitleof[S5]makesplayonHammersley'sownfamoustitle`Afewseedlingsofresearch',publishedin1972intheProceedingsoftheSixthBerkeleySymposium.Inthisinspiringaccountofhowtodomathematicalresearch,Hammersleyshowedinparticularhowtousesubadditivitytosolve(inpart)thenowfamousUlamproblem:inarandompermutationofthe rstnnaturalnumbers,whatisthelengthlnofthelongestincreasingsubsequence?Itturnedoutforgeometricalreasonsrelatedto[47]thattheanswerisasymptoticallycpn.Thiswasthestartingpointofamajorareaofprobabilitytheory.Hammersleyclaimedaback-of-the-envelopeargumenttoshowc=2,buttheformalproofeludedhimandwasfoundbyVershikandKerov[V]andLoganandShepp[L2]inthecontextofrandomYoungtableaux.Interestturnedthentothesizeofthedeviationln 2pn.ManypartialresultsprecededtheremarkableproofbyBaik,Deift,andJohansson[B1]that(ln 2pn)n 1=6convergesasn!1tothefamousTracy{Widomdistributionofrandommatrixtheory.RandomFieldsOneofthemostimportanttopicsinmodernstatisticsistheBayesiantheoryof JohnMichaelHammersley(1920{2004)19imageanalysis.Inthisstudyofspatialrandomsystems,itisusefultohaveaclassi -cationofthoseprobabilitymeasuresthatsatisfyacertain`spatialMarkovproperty',namely:thecon gurationinsideanyregionVdependsonthecon gurationoutsideVonlythroughthestatesoftheverticesonitsboundary@V.Somelimitedthe-oryofsuchmeasureswasdevelopedbyAverintsev,Dobrushin,Spitzer,andothersaround1970.ThiswasgeneralisedtoanarbitrarynetworkbyHammersleyin1971followingasuggestionofCli ord(see[C1,87]).Theensuingtheorem,commonlytermedtheHammersley{Cli ordtheorem,thoughneverformallypublished,ismuchusedinprobabilityandstatistics.ItstatesthatapositivemeasureisaMarkov eldifandonlyifithasaGibbsianrepresentationintermsofsomepotentialfunction.ThemethodsusedbyHammersleyweremuchclari edbylaterauthorsincludinganotherofhisstudents,Grimmett[G1],whoreducedtheprooftoanexerciseintheinclusion{exclusionprinciple.InMichaelmasTerm1971,Hammersleyo eredagraduatecourseonMarkov eldsattheMathematicalInstitute.Hepromisedasolutiontothecorrespondingprobleminwhichtheassumptionofpositivityisrelaxed.Itwastypicalofthemanthathehadnotyetprovedtheresult,andindeedthe`theorem'wasdisprovedthroughthediscoveryofacounterexamplebyaRhodesScholar,JohnMoussouris,intheaudience.See[M2,131].EducationalIssuesGreatchangesweremadeduringJohnHammersley'slifetimeintheteachingofmath-ematicsinschools,andhewasforaperiodattheforefrontofthedebate.Fromthe1950sonwards,heargued ercelythatschoolchildrenandundergraduatesshouldbetrainedtosolveproblems,andthatthecurriculumshouldbedesignedaccordingly.HelecturedonthistopicaroundtheUK,andhecontributedtothedevelopmentoftheSchoolMathematicsProject(SMP).Notbeingamanofequivocalviews,hisun-compromisingstancewasseenbysomeasaprovocation,buthehadmanysupportersandadmirers.However,theSMPprovednopanaceaforhim:whileit`modernised'aspectsofmathematicalteaching,itintroducedabstracttheorywithoutasucientproblemelement.HammersleyfrequentlypublishedhislecturesintheBulletinoftheInstituteofMathematicsanditsApplications(IMA).Hisprincipalarticle[126]onmathemati-caleducationappearedthusunderthetitle`Ontheenfeeblementofmathematicalskillsby`ModernMathematics'andbysimilarsoftintellectualtrashinschoolsanduniversities'.Thisserious,iftypicallyprolix,critiqueofschoolmathematicscom-pelledarebuttalfromBryanThwaites,[T2],temperedasfollows:\Ihave,however,aprofoundreluctanceto[replytoHammersley's\charges"].Thereasonisthatmyadmirationofthemanandmyopinionofhispaperareingreatcon\rict.Muchofmyadmirationstemsfromhismathematicalachievements;butitalsorests rmlyonmyjudgementthatitwashe,morethananyotherEnglishman,who nallysetgoingthelong-overduereformsinschoolmathematicalcurricula."Throughhis`popular'articles,Hammersleyexpressedhispowerfullyheldviewsonmanymatters,primarilyscienti candeducational.Thesewritingsareerudite,provocative,andskilfulwithlanguage,ifsometimesself-indulgent.Histhoughtson JohnMichaelHammersley(1920{2004)20mathematicalresearchwerepublishedalongsidethoseofMichaelAtiyahin[A2,131],andincludesomenotableexpressions:\:::perfusehisprofessorialpiddledom",\Puremathematicsissubjecttotwodiseases,resultingfromrigourandfromaxiomatisa-tion",\whateveralgebracanaccomplish,someotherbranchofmathematicsoughttobeabletoaccomplishmoreelegantly",\:::andtheproductionofneatersolutionsismerelyamatterfortheorybuilders".Helovedagoodphrase,even(perhaps,espe-cially)whenitriskedgoingabittoofar.Inreality,hewouldacceptanytheorythatproveditsworth.AsHammersleywrotetoAtiyahin[131]:\Idon'tquarrel,butIampreparedtoenterthelists.:::itisthejostlingandjoustingbetweendi erentsortsofmathemati-ciansandscientists,betweendi erenttemperamentsandunliketastes,thatadvancesknowledgeasawhole.Somuchthemorefun,varietyisthespice,andsoon!"ACKNOWLEDGEMENTSWethanktheHammersleyfamilyforpermissiontoquote(initalics,andwithminorcorrectionsandchangesofpresentation)fromJohn'saccountofhisearlylife,writtenapparentlyinresponsetoarequestfromtheRoyalSocietydated1994.Christo-pherPriorandClareHopkins(archivist,TrinityCollege)haveadvisedusonCollegematters;JohnHaltonandPeterMarcerhavereminiscedabouttheirexperiencesasPhDstudentsofJohnHammersley.WethankPeterCollins,DavidHandscomb,andBillMortonfortheirmemoriesofHammersleyinOxford,HarryKestenforkindlycommentingonthisbiographicalmemoir,andNicholasCoxforsomecorrections.ThephotographsweretakenbyGeo reyGrimmett.References[A1]Allen,Thomas(1540{1632),DictionaryofNationalBiography(2004),OxfordUniversityPress,Oxford.[A2]Atiyah,M.F.,Howresearchiscarriedout,BulletinoftheInstituteofMathematicsanditsApplications9(1973),276{280.[B1]Baik,J.,Deift,P.,Johansson,K.,Onthedistributionofthelengthofthelongestincreas-ingsubsequenceofrandompermutations,JournaloftheAmericanMathematicalSociety12(1999),1119{1178.[B2]Bartlett,M.S.,Onthetheoreticalspeci cationofsamplingpropertiesofauto-correlatedtimeseries,JournaloftheRoyalStatisticalSociety(Supplement)8(1946),27{41.[C1]Cli ord,P.,Markovrandom eldsinstatistics,in[G5],19{32.[C2]Cramer,E.,Asymptoticestimatorsofthesamplesizeinarecordmodel,StatisticalPapers41(2000),159{171.[D]Domb,C.,OnHammersley'smethodforone-dimensionalcoveringproblems,in[G5],33{53.[E1]Eden,M.,Atwodimensionalgrowthprocess,ProceedingsoftheFourthBerkeleySymposiumonMathematicalStatisticsandProbability(Neyman,J.,ed.),vol.IV,UniversityofCaliforniaPress,1961,pp.223{229.[E2]Erd}os,P.,OnaconjectureofHammersley,JournaloftheLondonMathematicalSociety28(1953),232{236.[F1]Family,F.,Vicsek,T.,DynamicsofFractalSurfaces,WorldScienti c,Singapore,1991.[F2]Farahmand,K.,TopicsinRandomPolynomials,Longman,Harlow,1998.[F3]Friedland,S.,Peled,U.N.,Theoryofcomputationofmultidimensionalentropywithanappli-cationtothemonomer{dimerproblem,AdvancesinAppliedMathematics34(2005),486{522. JohnMichaelHammersley(1920{2004)21[F4]Friedman,J.,RandompolynomialsandapproximatezerosofNewton'smethod,SIAMJournalofComputing19(1990),1068{1099.[G1]Grimmett,G.R.,Atheoremaboutrandom elds,BulletinoftheLondonMathematicalSociety5(1973),81{84.[G2]Grimmett,G.R.,Percolation,2ndedition,Springer,Berlin,1999.[G3]Grimmett,G.R.,Percolation,DevelopmentofMathematics1950{2000(Pier,J.P.,ed.),Birkhauser,2000,pp.547{576.[G4]Grimmett,G.R.,Stacey,A.M.,Criticalprobabilitiesforsiteandbondpercolationmodels,AnnalsofProbability26(1998),1788{1812.[G5]Grimmett,G.R.,Welsh,D.J.A.(eds.),DisorderinPhysicalSystems(AvolumeinhonourofJohnM.Hammersley),seehttp://www.statslab.cam.ac.uk/grg/books/jmh.html,OxfordUniversityPress,Oxford,1990.[H1]Hara,T.,Slade,G.,Self-avoidingwalkin veormoredimensions.I.Thecriticalbehaviour,CommunicationsinMathematicalPhysics147(1992),101{136.[H2]Hardy,G.H.,Mathematicsinwar-time,Eureka1(1940),5{8;seealsoAMathematician'sApology,Section28.[H3]Harris,T.E.,Alowerboundforthecriticalprobabilityinacertainpercolationprocess,ProceedingsoftheCambridgePhilosophicalSociety56(1960),13{20.[K1]Kac,M.,Ontheaveragenumberofrealrootsofarandomalgebraicequation,BulletinoftheAmericanMathematicalSociety49(1943),314{320.[K2]Karp,R.,ProbabilisticanalysisofpartitioningalgorithmsforTSPintheplane,MathematicsofOperationsResearch2(1977),209{224.[K3]Kasteleyn,P.W.,Thestatisticsofdimersonalattice,Physica27(1961),1209{1225.[K4]Kendall,D.G.,SpeechproposingthetoasttoJohnHammersley,1October1987,in[G5],1{3.[K5]Kenyon,R.W.,Anintroductiontothedimermodel,SchoolandConferenceonProbabilityTheory(Lawler,G.F.,ed.),ICTPLectureNotes,vol.XVII,AbdusSalamInternationalCentreforTheoreticalPhysics,Trieste,2004,pp.267{304.[K6]Kenyon,R.W.,Wilson,D.B.,Boundarypartitionsintreesanddimers(2006);arxiv:math/06088422.[K7]Kesten,H.,Onthenumberofself-avoidingwalks,II,JournalofMathematicalPhysics5(1964),1128{1137.[K8]Kesten,H.,Thecriticalprobabilityofbondpercolationonthesquarelatticeequals12,Com-municationsinMathematicalPhysics74(1980),41{59.[K9]Kingman,J.F.C.,Theergodictheoryofsubadditivestochasticprocesses,JournaloftheRoyalStatisticalSocietyB30(1968),499{510.[K10]Kingman,J.F.C.,Subaditiveergodictheory,AnnalsofProbability1(1973),883{909.[L1]Lieb,E.H.,Are nementofSimon'scorrelationinequality,CommunicationsinMathematicalPhysics77(1980),127{135.[L2]Logan,B.F.,Shepp,L.A.,AvariationalproblemforrandomYoungtableaux,AdvancesinMathematics26(1977),206{222.[M1]Madras,N.,Slade,G.,TheSelf-AvoidingWalk,Birkhauser,Boston,1993.[M2]Moussouris,J.,GibbsandMarkovrandom eldswithconstraints,JournalofStatisticalPhysics10(1974),11{33.[P1]Peierls,R.,OnIsing'smodelofferromagnetism,ProceedingsoftheCambridgePhilosophicalSociety36(1936),477{481.[P2]Pitcher,H.,TheSmithsofMoscow:aStudyofBritonsAbroad,SwallowHouseBooks,Cromer,1984.[R1]Ramponi,A.,Anoteonthecomplexrootsofcomplexrandompolynomials,StatisticsandProbabilityLetters44(1999),181{187.[R2]Roach,W.,Wright,R.,Optimalantitheticsamplingplans,JournalofStatisticalComputationandSimulation5(1976/77),99{114.[S1]Schramm,O.,Scalinglimitsofloop-erasedwalksanduniformspanningtrees,IsraelJournalofMathematics118(2000),221{288. JohnMichaelHammersley(1920{2004)22[S2]Schramm,O.,Conformallyinvariantscalinglimits:anoverviewandcollectionofopenprob-lems,ProceedingsoftheInternationalCongressofMathematicians,Madrid(Sanz-Sole,M.etal.,eds.),EuropeanMathematicalSociety,Zurich,2007,pp.513{544.[S3]Simon,B.,Correlationinequalitiesandthedecayofcorrelationsinferromagnets,Communi-cationsinMathematicalPhysics77(1980),111{126.[S4]Smirnov,S.,Criticalpercolationintheplane:conformalinvariance,Cardy'sformula,scalinglimits,ComptesRendusdesSeancesdel'AcademiedesSciences.SerieI.Mathematique333(2001),239{244.[S5]Steele,J.M.,Seedlingsinthetheoryofshortestpaths,in[G5],277{306.[T1]Temperley,H.N.V.,Fisher,M.E.,Dimerproblemsinstatisticalmechanics|anexactresult,PhilosophicalMagazine6(1961),1061{1063.[T2]Thwaites,B.,Waysaheadinsecondary-schoolmathematics,BulletinoftheInstituteofMath-ematicsanditsApplications5(1969),49{53.[T3]Tukey,J.W.,Antithesisorregression?,ProceedingsoftheCambridgePhilosophicalSociety53(1957),923{924.[V]Vershik,A.M.,Kerov,S.V.,AsymptoticbehaviorofthePlancherelmeasureofthesymmetricgroupandthelimitformofYoungtableaux,SovietMathematicsDoklady18(1977),527{531.[W]Wood,DeVolson,Problem5,TheAmericanMathematicalMonthly1(1894),99,211{212.[Z]Zabolitsky,J.G.,Stau er,D,SimulationoflargeEdenclusters,ThePhysicalReviewA34(1986),1523|1530.PublicationsofJohnM.Hammersley1.Bayley,G.V.,Hammersley,J.M.,Thee ectivenumberofindependentobservationsinanautocorrelatedtimeseries,JournaloftheRoyalStatisticalSociety(Supplement)8(1946),184{197.2.Hammersley,J.M.,Ageometricalillustrationofaprincipleofexperimentaldirectives,Philo-sophicalMagazine39(1948),460{466.3.Hammersley,J.M.,Anelementaryintroductiontosomeinspectionprocedures,Rev.SuisseOrg.Indust.17(1948),315{322.4.Hammersley,J.M.,Theunbiasedestimateandstandarderroroftheinterclassvariance,Metron15(1949),189{205.5.Hammersley,J.M.,Thenumericalreductionofnon-singularmatrixpencils,PhilosophicalMagazine40(1949),783{807.6.Hammersley,J.M.,Electroniccomputersandtheanalysisofstochasticprocesses,MathematicalComputing4(1950),56{57.7.Hammersley,J.M.,Calculatingmachines,Chambers'Encyclopedia(1950).8.Hammersley,J.M.,Harmonicanalysis,Chambers'Encyclopedia(1950).9.Hammersley,J.M.,Furtherresultsforthecounterfeitcoinproblems,ProceedingsoftheCam-bridgePhilosophicalSociety46(1950),226{230.10.Hammersley,J.M.,Thedistributionofdistanceinahypersphere,AnnalsofMathematicalStatistics21(1950),447{452.11.Hammersley,J.M.,Onestimatingrestrictedparameters,JournaloftheRoyalStatisticalSocietyB12(1950),192{229.12.Hammersley,J.M.,Atheoremonmultipleintegrals,ProceedingsoftheCambridgePhilosoph-icalSociety47(1951),274{278.13.Hammersley,J.M.,Onacertaintypeofintegralassociatedwithcircularcylinders,ProceedingsoftheRoyalSociety,SeriesA210(1951),98{110.14.Hammersley,J.M.,Thesumsofproductsofthenaturalnumbers,ProceedingsoftheLondonMathematicalSociety1(1951),435{452.15.Hammersley,J.M.,Thetotallengthoftheedgesofthepolyhedron,CompositioMathematica9(1951),239{240.16.Hammersley,J.M.,Theabsorptionofradioactiveradiationinrods,NationalBureauofStan-dardsW.P.1929(1951),1{11. JohnMichaelHammersley(1920{2004)2317.Hammersley,J.M.,Thecomputationofsumsofsquaresandproductsonadeskcalculator,Biometrics8(1952),156{168.18.Hammersley,J.M.,AnextensionoftheSlutzky{Frechettheorem,ActaMathematica87(1952),243{257.19.Hammersley,J.M.,Lagrangianintegrationcoecientsfordistancefunctionstakenoverrightcircularcylinders,JournalofMathematicalPhysics31(1952),139{150.20.Hammersley,J.M.,Tauberiantheoryfortheasymptoticformsofstatisticalfrequencyfunctions,ProceedingsoftheCambridgePhilosophicalSociety48(1952),592{599;Corrigenda:49(1953),735.21.Hammersley,J.M.,OnaconjectureofNelder,CompositioMathematica10(1952),241{244.22.Hammersley,J.M.,Capture{recaptureanalysis,Biometrika40(1953),265{278.23.Hammersley,J.M.,Tablesofcompleteellipticintegrals,JournalofResearchoftheNationalBureauofStandards50(1953),43.24.Hammersley,J.M.,Anon-harmonicFourierseries,ActaMathematica89(1953),243{260.25.Hammersley,J.M.,OncounterswithrandomdeadtimeI,ProceedingsoftheCambridgePhilosophicalSociety49(1953),623{637.26.Antosiewicz,H.A.,Hammersley,J.M.,Theconvergenceofnumericaliteration,TheAmericanMathematicalMonthly60(1953),604{607.27.Hammersley,J.M.,Markovianwalksoncrystals,CompositioMathematica11(1953),171{186.28.Hammersley,J.M.,Theconsistencyofstop-watchtime-studypractitioners,OccupationPsy-chology28(1954),61{76.29.Hammersley,J.M.,Morton,K.W.,Poorman'sMonteCarlo,JournaloftheRoyalStatisticalSocietyB16(1954),23{38.30.Hammersley,J.M.,Morton,K.W.,Transposedbranchingprocesses,JournaloftheRoyalStatisticalSocietyB16(1954),76{79.31.Hammersley,J.M.,Morton,K.W.,Theestimationoflocationandscaleparametersfromgroupeddata,Biometrika41(1954),296{301.32.Eyeions,D.A.,Hammersley,J.M.,Owen,B.G.,Price,B.T.,Wilson,J.G.,Morton,K.W.,Theionizationlossofrelativisticmu-mesonsinneon,ProceedingsofthePhysicalSociety(A)68(1955),793{800.33.Hammersley,J.M.,Storageproblems,MathematischeAnnalen128(1955),475{478.34.Hammersley,J.M.,Nelder,J.A.,SamplingfromanisotropicGaussianprocess,ProceedingsoftheCambridgePhilosophicalSociety51(1955),652{662.35.Hammersley,J.M.,TheareaenclosedbyPolya'swalk,ProceedingsoftheCambridgePhilo-sophicalSociety52(1956),78{87.36.Hammersley,J.M.,Percolationincrystals:gravitycrystals,UKAEATP13(1956).37.Hammersley,J.M.,ConditionalMonteCarlo,JournaloftheAssociationforComputingMa-chinery3(1956),73{76.38.Hammersley,J.M.,Morton,K.W.,AnewMonteCarlotechnique:antitheticvariates,Pro-ceedingsoftheCambridgePhilosophicalSociety52(1956),449{475.39.Hammersley,J.M.,Mauldon,J.G.,Generalprinciplesofantitheticvariates,ProceedingsoftheCambridgePhilosophicalSociety52(1956),476{481.40.Hammersley,J.M.,Thezerosofarandompolynomial,ProceedingsoftheThirdBerkeleySymposiumonMathematicalStatisticsandProbability(Neyman,J.,ed.),vol.II,UniversityofCaliforniaPress,1956,pp.89{111.41.Broadbent,S.R.,Hammersley,J.M.,Percolationprocesses.I.Crystalsandmazes,ProceedingsoftheCambridgePhilosophicalSociety53(1957),629{641.42.Hammersley,J.M.,Percolationprocesses.II.Theconnectiveconstant,ProceedingsoftheCambridgePhilosophicalSociety53(1957),642{645.43.Hammersley,J.M.,Percolationprocesses:Lowerboundsforthecriticalprobability,AnnalsofMathematicalStatistics28(1957),790{795.44.Hammersley,J.M.,Discussiononrenewaltheoryanditsrami cations,JournaloftheRoyalStatisticalSocietyB20(1958),287{291. JohnMichaelHammersley(1920{2004)2445.Egelsta ,P.A.,Hammersley,J.M.,Lane,A.M.,Fluctuationsinslowneutronaveragecross-sections,ProceedingsofthePhysicalSociety(A)71(1958),910{924.46.Hammersley,J.M.,Bornessuperieuresdelaprobabilitecritiquedansunprocessusde ltration,LeCalculdesProbabilitesetsesApplications,CNRS,Paris,1959,pp.17{37.47.Beardwood,J.,Halton,J.H.,Hammersley,J.M.,Theshortestpaththroughmanypoints,ProceedingsoftheCambridgePhilosophicalSociety55(1959),299{327.48.Hammersley,J.M.,MonteCarlomethodsforsolvingmultivariableproblems,AnnalsoftheNewYorkAcademyofSciences86(1960),844{874.49.Hammersley,J.M.,Limitingpropertiesofnumbersofself-avoidingwalks,ThePhysicalReview118(1960),656.50.Hammersley,J.M.,LettertotheEditor,TheMathematicalGazette44(1960),40{42.51.Hammersley,J.M.,Onnote2871,TheMathematicalGazette44(1960),287{288.52.Hammersley,J.M.,Thenumberofpolygonsonalattice,ProceedingsoftheCambridgePhilo-sophicalSociety57(1961),516{523.53.Hammersley,J.M.,AshortproofoftheFarahat{Mirskyre nementofBirkho 'stheoremondoubly-stochasticmatrices,ProceedingsoftheCambridgePhilosophicalSociety57(1961),681.54.Hammersley,J.M.,Comparisonofatomandbondpercolationprocesses,JournalofMathemat-icalPhysics2(1961),728{733.55.Vyssotsky,V.A.,Gordon,S.B.,Frisch,H.L.,Hammersley,J.M.,Criticalpercolationproba-bilities(bondproblem),ThePhysicalReview123(1961),1566{1567.56.Frisch,H.L.,Hammersley,J.M.,Sonnenblick,E.,Vyssotsky,V.A.,Criticalprobabilities:siteproblem,ThePhysicalReview124(1961),1021{1022.57.Hammersley,J.M.,OnSteiner'snetworkproblem,Mathematika8(1961),131{132.58.Hammersley,J.M.,Onthestatisticallossoflong-periodcometsfromthesolarsystem.II.,ProceedingsoftheFourthBerkeleySymposiumonMathematics,StatisticsandProbability(Neyman,J.,ed.),vol.III,1961,pp.17{78.59.Hammersley,J.M.,Onthedynamicaldisequilibriumofindividualparticles,ProceedingsoftheFourthBerkeleySymposiumonMathematics,StatisticsandProbability(Neyman,J.,ed.),vol.III,1961,pp.79{85.60.Hammersley,J.M.,Ontherateofconvergencetotheconnectiveconstantofthehypercubicallattice,TheQuarterlyJournalofMathematics.Oxford12(1961),250{256.61.Cranshaw,T.E.,Hammersley,J.M.,Countingstatistics,EncyclopaedicDictionaryofPhysics2(1962),89{108.62.Frisch,H.L.,Gordon,S.B.,Hammersley,J.M.,Vyssotsky,V.A.,MonteCarlosolutionofbondpercolationprocessesinvariouscrystallattices,BellSystemTechnicalJournal41(1962),909{920.63.Frisch,H.L.Hammersley,J.M.,Welsh,D.J.A.,MonteCarloestimatesofpercolationproba-bilitiesforvariouslattices,ThePhysicalReview126(1962),949{951.64.Hammersley,J.M.,Generalizationofthefundamentaltheoremonsub-additivefunctions,Pro-ceedingsoftheCambridgePhilosophicalSociety58(1962),235{238.65.Hammersley,J.M.,Welsh,D.J.A.,Furtherresultsontherateofconvergencetotheconnectiveconstantofthehypercubicallattice,TheQuarterlyJournalofMathematics.Oxford13(1962),108{110.66.Hammersley,J.M.,Themathematicalanalysisoftraccongestion,Bulletindel'InstitutIn-ternationaldeStatistique39(1962),89{108.67.Hammersley,J.M.,MonteCarlomethods,Proceedingsofthe7thConferenceontheDesignofExperimentsinArmyResearch,DevelopmentandTesting,U.S.ArmyResearchOce,1962,pp.17{26.68.Hammersley,J.M.,AMonteCarlosolutionofpercolationinthecubiclattice,Meth.Comput.Phys.1(1963),281{298.69.Frisch,H.L.,Hammersley,J.M.,Percolationprocessesandrelatedtopics,JournaloftheSocietyforIndustrialandAppliedMathematics11(1963),894{918.70.Hammersley,J.M.,Walters,R.S.,Percolationandfractionalbranchingprocesses,JournaloftheSocietyforIndustrialandAppliedMathematics11(1963),831{839. JohnMichaelHammersley(1920{2004)2571.Hammersley,J.M.,Long-chainpolymersandself-avoidingrandomwalks,Sankhya,SeriesA25(1963),29{38.72.Hammersley,J.M.,Long-chainpolymersandself-avoidingrandomwalks.II.,Sankhya,SeriesA25(1963),269{272.73.Lyttleton,R.A.,Hammersley,J.M.,Thelossoflong-periodcometsfromthesolarsystem,MonthlyNoticesoftheRoyalAstronomicalSociety127(1963),257{272.74.Hammersley,J.M.,Handscomb,D.C.,MonteCarloMethods,Methuen,London,1964.75.Hammersley,J.M.,Welsh,D.J.A.,First-passagepercolation,subadditiveprocesses,stochas-ticnetworks,andgeneralizedrenewaltheory,Bernoulli,Bayes,LaplaceAnniversaryVolume(Neyman,J.,LeCam,L.M.,eds.),Springer-Verlag,Berlin,1965,pp.61{110.76.Hammersley,J.M.,Subadditivefunctionalexpectations,TheoryofProbabilityanditsAppli-cations(Russian)11(1966),352{354;311{313(English).77.Hammersley,J.M.,ExistencetheoremsandMonteCarlomethodsforthemonomer{dimerproblem,ResearchPapersinStatistics(FestschriftforJ.Neyman)(David,F.N.,ed.),JohnWiley,London,1966,pp.125{146.78.Hammersley,J.M.,First-passagepercolation,JournaloftheRoyalStatisticalSocietyB28(1966),491{496.79.Hammersley,J.M.,Mallows,C.L.,Handscomb,D.C.,Recentpublicationsandpresentations:MonteCarlomethods,TheAmericanMathematicalMonthly73(1966),685.80.Bingham,N.H.,Hammersley,J.M.,OnaconjectureofRademacher,Dickson,andPlotkin,JournalofCombinatorialTheory3(1967),182{190.81.Hammersley,J.M.,Harnesses,ProceedingsoftheFifthBerkeleySymposiumonMathematicalStatisticsandProbability(LeCam,L.M.,Neyman,J.,eds.),vol.III,UniversityofCaliforniaPress,1967,pp.89{117.82.Hammersley,J.M.,Animprovedlowerboundforthemultidimensionaldimerproblem,Pro-ceedingsoftheCambridgePhilosophicalSociety64(1968),455{463.83.Feuerverger,A.,Hammersley,J.M.,Izenman,A.,Makani,K.,Negative ndingforthethree-dimensionaldimerproblem,JournalofMathematicalPhysics10(1969),443{446.84.Hammersley,J.M.,Sequencesofabsolutedi erences,SIAMReview11(1969),73{74.85.Hammersley,J.M.,Calculationoflatticestatistics,Proceedingsofthe2ndConferenceonComputationalPhysics,InstituteofPhysicsandPhysicalSociety,London,1970,pp.1{8.86.Hammersley,J.M.,Menon,V.V.,Alowerboundforthemonomer{dimerproblem,JournaloftheInstituteofMathematicsanditsApplications6(1970),341{364.87.Hammersley,J.M.,Cli ord,P.,Markov eldson nitegraphsandlattices,unpublished(1971).88.Hammersley,J.M.,Afewseedlingsofresearch,ProceedingsoftheSixthBerkeleySymposiumonMathematicalStatisticsandProbability(LeCam,L.M.,Neyman,J.,Scott,E.L.,eds.),vol.I,1972,pp.345{394.89.Hammersley,J.M.,Stochasticmodelsforthedistributionofparticlesinspace,AdvancesinAppliedProbability(Supplement)(1972),47{68.90.Hammersley,J.M.,Maximsformanipulators,BulletinoftheInstituteofMathematicsanditsApplications9(1973),276{280.91.Hammersley,J.M.,Contributiontodiscussiononsubadditiveergodictheory,AnnalsofProb-ability1(1973),905{909.92.Bell,G.M.,Churchhouse,R.F.,Goodwin,E.T.,Hammersley,J.M.,Taylor,R.S.,ProofofaconjectureofWorster,BulletinoftheInstituteofMathematicsanditsApplications10(1974),128{129.93.Hammersley,J.M.,Anisoperimetricproblem,BulletinoftheInstituteofMathematicsanditsApplications10(1974),439{441.94.Hammersley,J.M.,AratherdicultO-levelproblem,BulletinoftheInstituteofMathematicsanditsApplications10(1974),441{443.95.Hammersley,J.M.,Postulatesforsubadditiveprocesses,AnnalsofProbability2(1974),652{680.96.Hammersley,J.M.,Somespeculationsonasenseofnicelycalculatedchances,SIAMReview16(1974),237{255. JohnMichaelHammersley(1920{2004)2697.Hammersley,J.M.,Grimmett,G.R.,Maximalsolutionsofthegeneralizedsubadditiveinequal-ity,StochasticGeometry(AtributetothememoryofRolloDavidson)(Harding,E.F.,Kendall,D.G.,eds.),Wiley,London,1974,pp.270{284.98.Hammersley,J.M.,Somegeneralre\rectionsonstatisticalpractice,FestschriftforProfessorLinder.99.Hammersley,J.M.,Thewideopenspaces,TheStatistician24(1975),159{160.100.Hammersley,J.M.,Ruminationonin niteMarkovsystems,PerspectivesinProbabilityandStatistics(PapersinhonourofM.S.Bartlett),AppliedProbabilityTrust,Sheeld,1975,pp.195{200.101.Hammersley,J.M.,Lewis,J.W.E.,Rowlinson,J.S.,RelationshipsbetweenthemultinomialandPoissonmodelsofstochasticprocesses,andbetweenthecanonicalandgrandcanonicalen-semblesinstatisticalmechanics,withillustrationsandMonteCarlomethodsforthepenetrablespheremodelofliquid{vapourequilibrium,Sankhya,SeriesA37(1975),457{491.102.Hammersley,J.M.,Thedesignoffuturecomputingmachineryforfunctionalintegration,Func-tionalIntegrationanditsApplications,ClarendonPress,Oxford,1975,pp.83{86.103.Hammersley,J.M.,AgeneralizationofMcDiarmid'stheoremformixedBernoullipercolation,MathematicalProceedingsoftheCambridgePhilosophicalSociety88(1980),167{170.104.Hammersley,J.M.,Biologicalgrowthandspread,LectureNotesinBiomathematics,vol.38,Springer,Berlin,1980,pp.484{494.105.Hammersley,J.M.,Welsh,D.J.A.,Percolationtheoryanditsrami cations,ContemporaryPhysics21(1980),593{605.106.Hammersley,J.M.,Criticalphenomenainsemi-in nitesystems.Essaysinstatisticalscience,JournalofAppliedProbability19A(1982),327{331.107.Hammersley,J.M.,Torrie,G.M.,Whittington,S.G.,Self-avoidingwalksinteractingwithasurface,JournalofPhysics.A.MathematicalandGeneral15(1982),539{571.108.Hammersley,J.M.,Oxfordcommemorationball,Probability,StatisticsandAnalysis,LondonMathematicalSocietyLectureNoteSeries,vol.79,CambridgeUniversityPress,Cambridge,1983,pp.112{142.109.Hammersley,J.M.,Thefriendshiptheoremandtheloveproblem,SurveysinCombinatorics,LondonMathematicalSocietyLectureNoteSeries,vol.82,CambridgeUniversityPress,Cam-bridge,1983,pp.31{54.110.Hammersley,J.M.,Originsofpercolationtheory,Percolationstructuresandprocesses,AnnalsoftheIsraelPhysicalSociety,vol.5,Hilger,Bristol,1983,pp.47{57.111.Hammersley,J.M.,Mazzarino,G.,Markov elds,correlatedpercolation,andtheIsingmodel,TheMathematicsandPhysicsofDisorderedMedia,LectureNotesinMathematics,vol.1035,Springer,Berlin,1983,pp.210{245.112.Hammersley,J.M.,Functionalrootsandindicialsemigroups,BulletinoftheInstituteofMath-ematicsanditsApplications19(1983),194{196.113.Hammersley,J.M.,Whittington,S.G.,Self-avoidingwalksinwedges,JournalofPhysics.A.MathematicalandGeneral18(1985),101{111.114.Hammersley,J.M.,Threealgorithmicexercises:::,TheCollegeMathematicsJournal16(1985),12{14.115.Hammersley,J.M.,Mesoadditiveprocessesandthespeci cconductivityoflattices,ACelebra-tionofAppliedProbability,JournalofAppliedProbability,vol.25A,1988,pp.347{358.116.Hammersley,J.M.,Mazzarino,G.,Adi erentialequationconnectedwiththedendriticgrowthofcrystals,IMAJournalofAppliedMathematics42(1989),43{75.117.Hammersley,J.M.,Mazzarino,G.,Computationalaspectsofsomeautonomousdi erentialequations,ProceedingsoftheRoyalSociety,SeriesA424(1989),19{37.118.Hammersley,J.M.,Self-avoidingwalks,Currentproblemsinstatisticalmechanics,PhysicaA,vol.177,1991,pp.51{57;Corrigendumvol.183(1992),574{578.119.Hammersley,J.M.,Mazzarino,G.,PropertiesoflargeEdenclustersintheplane,Combina-torics,ProbabilityandComputing3(1994),471{505.120.Hammersley,J.M.,FractaldynamicsofEdenclusters,Probability,StatisticsandOptimisation,Wiley,Chichester,1994,pp.79{87. JohnMichaelHammersley(1920{2004)27Publicationsonothertopics121.Hammersley,J.M.(ed.),ProceedingsoftheOxfordMathematicalConferenceforSchoolteachersandIndustrialists,TimesPublishingCompany,London,1957.122.Hammersley,J.M.,Thevalueofmathematicsanditsteachers,ibid(1957).123.Coulson,C.A.,Hammersley,J.M.,ThebottleneckinBritishscienceandtechnology,NewScientist10(1961),499{500.124.Hammersley,J.M.,Levine,H.,Planningforthedistantfuture,TheTimesEducationalSup-plement(15September1961),293.125.Hammersley,J.M.,Industryandeducation:prospectsandresponsibilitiesinmathematicsinSouthAfrica,Kwart.Tyd.Wisk,.Wetenskap.5(1967),11{17.126.Hammersley,J.M.,Ontheenfeeblementofmathematicalskillsby`ModernMathematics'andbysimilarsoftintellectualtrashinschoolsanduniversities,BulletinoftheInstituteofMath-ematicsanditsApplications4(1968),66{85.127.Hammersley,J.M.,Nomatter,nevermind!,BulletinoftheInstituteofMathematicsanditsApplications7(1971),358{364.128.Hammersley,J.M.,Symposiumonteachingofmathematicsinschoolsinrelationtotheteach-ingofphysics(EtonCollege,2October1971):impressionofthemeeting,BulletinoftheInstituteofMathematicsanditsApplications8(1972),39{40.129.Hammersley,J.M.,Howisresearchdone?,BulletinoftheInstituteofMathematicsanditsApplications9(1973),214{215.130.Hammersley,J.M.,Modernmathematics,thegreatdebate:Motionproposingthatthishousedeplorestheenthusiasticteachingofmodernmathematics,particularlyinschools,BulletinoftheInstituteofMathematicsanditsApplications9(1973),238{241.131.Hammersley,J.M.,Pokingaboutforthevitaljuicesofmathematicalresearch,BulletinoftheInstituteofMathematicsanditsApplications10(1974),235{247.132.Hammersley,J.M.,Somethoughtsoccasionedbyanundergraduatemathematicssociety,Bul-letinoftheInstituteofMathematicsanditsApplications10(1974),306{311.133.Hammersley,J.M.,Statisticaltools,TheStatistician23(1974),89{106.134.Hammersley,J.M.,Thetechnologyofthought,TheHeritageofCopernicus(Neyman,J.,ed.),MITPress,1974.135.Hammersley,J.M.,LehrsatzeandLeersatzediPolentaeSegu,BulletinoftheInstituteofMathematicsanditsApplications11(1975),117{121.136.Hammersley,J.M.,Sweetnothing,BulletinoftheInstituteofMathematicsanditsApplications14(1978),146{147.137.Hammersley,J.M.,Obituary:J.Neyman,1894{1981,JournaloftheRoyalStatisticalSocietyA145(1982),523{524.138.Hammersley,J.M.,Theteachingofcombinatorialanalysis,BulletinoftheInstituteofMath-ematicsanditsApplications19(1983),50{52.139.Hammersley,J.M.,Probabilityandarithmeticinscience,BulletinoftheInstituteofMathe-maticsanditsApplications21(1985),114{120.140.Hammersley,J.M.,Threealgorithmicexercises,TheCollegeMathematicsJournal16(1985),12{14.141.Hammersley,J.M.,Roomtowriggle,BulletinoftheInstituteofMathematicsanditsApplica-tions24(1988),65{72.