Alexander Grothendieck.docx

Alexander Grothendieck.docx

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AlexanderGrothendieck

AlexanderGrothendieck; French:

;28March1928–13November2014）wasaFrenchmathematician,borninGermany,raisedandlivingprimarilyin France,andwhospentmuchofhisworkinglife stateless,whoisthecentralfigurebehindthecreationofthemoderntheoryof algebraicgeometry.Hisresearchprogramvastlyextendedthescopeofthefield,incorporatingmajorelementsof commutativealgebra,homologicalalgebra, sheaftheory,and categorytheory intoitsfoundations.Thisnew perspective ledtorevolutionaryadvancesacrossmanyareasofpuremathematics.Heconsistentlyspelthisfirstname"Alexander"ratherthantheFrench"Alexandre"; hisfamilyname,"Grothendieck"（fromhismother）is LowGerman,whichissimilarto Dutch,henceheissometimesmistakenlybelievedtobeofDutchorigin.

Afteraveryproductivepublicmathematicalcareerlasting1949–1970,particularly1958–1970（whenhewasatIHES）,Grothendiecklargelyceasedmathematicalactivityafter1970（age42）,thoughwithsomeprivatework1970–1988.Drivenbydeeppersonalandpoliticalconvictions,Grothendieckleftthe InstitutdesHautesÉtudesScientifiques,wherehehadbeenappointedprofessorandaccomplishedhisgreatestwork,afteradisputeovermilitaryfundingin1970.Hismathematicalactivityessentiallyceasedafterthis,andhedevotedhisenergiestopoliticalcauses,thoughhedidproducesomemathematicalworkprivately.Heformallyretiredin1988andwithinafewyearsmovedtothe Pyrenees,wherehelivedinisolationfromhumansocietyuntilhisdeathin2014.

Influence

Within algebraicgeometry itself,histheoryof schemes isusedintechnicalwork.His generalization oftheclassical Riemann-Rochtheorem startedthestudyof algebraic and topologicalK-theory.Hisconstructionofnew cohomology theorieshasleftconsequencesfor algebraicnumbertheory, algebraictopology,and representationtheory.Hiscreationof topostheory hasappearedin settheory and logic.

Oneofhisresultsisthediscoveryofthefirstarithmetic Weilcohomology theory:

the $\ell$-adicétalecohomology.Thisresultopenedthewayforaproofofthe Weilconjectures,ultimatelycompletedbyhisstudent PierreDeligne.Tothisday, $\ell$-adiccohomology remainsafundamentaltoolfornumbertheorists,withapplicationstothe Langlandsprogram.

Grothendieckinfluencedgenerationsofmathematiciansafterhisdeparturefrommathematics.Hisemphasisontheroleof universalproperties brought categorytheory intothemainstreamasanorganizingprinciple.Hisnotionof abeliancategory isnowthebasicobjectofstudyin homologicalalgebra.Hisconjecturaltheoryof motives hasbeenbehindmoderndevelopmentsin algebraicK-theory, motivichomotopytheory,and motivicintegration.

Life

Familyandchildhood

AlexanderGrothendieckwasbornin Berlin to anarchist parents:

afatherfromanoriginally Hassidic family, Alexander"Sascha"Schapiro akaTanaroff,whohadbeenimprisonedinRussiaandmovedtoGermanyin1922,andamotherfroma Protestant familyin Hamburg,Johanna"Hanka"Grothendieck,whoworkedasajournalist;bothofhisparentshadbrokenawayfromtheirearlybackgroundsintheirteens. AtthetimeofhisbirthGrothendieck'smotherwasmarriedtothejournalistJohannesRaddatz,andhisbirthnamewasinitiallyrecordedas AlexanderRaddatz.Themarriagewasdissolvedin1929andSchapiro/Tanaroffacknowledgedhispaternity,butnevermarriedHankaGrothendieck.

Grothendiecklivedwithhisparentsuntil1933inBerlin.Attheendofthatyear,Schapiromovedto Paris toevadethe Nazis,andHankafollowedhimthenextyear.TheyleftGrothendieckinthecareofWilhelmHeydorn,a Lutheran Pastor andteacher in Hamburg wherehewenttoschool.Duringthistime,hisparentstookpartinthe SpanishCivilWar insupportingratherthanfightingroles. Grothendieckcouldspeak French, English and German.

WorldWarII

In1939GrothendieckwenttoFranceandlivedinvariouscampsfordisplacedpersonswithhismother.Theylivedfirstatthe CampdeRieucros,andlater,fortheremainderof WorldWarII,inthevillageof LeChambon-sur-Lignon.Therehewasshelteredandhiddeninlocalboarding-housesorpensions.Hisfatherwasarrestedandsentvia Drancy tothe Auschwitzconcentrationcamp,wherehediedin1942.InChambon,GrothendieckattendedtheCollègeCévenol（nowknownasthe LeCollège-LycéeCévenolInternational）,auniquesecondaryschoolfoundedin1938bylocalProtestantpacifistsandanti-waractivists.ManyoftherefugeechildrenhiddeninChambonattendedCévenol,anditwasatthisschoolthatGrothendieckapparentlyfirstbecamefascinatedwithmathematics.

Studiesandcontactwithresearchmathematics

Afterthewar,theyoungGrothendieckstudiedmathematicsinFrance,initiallyatthe UniversityofMontpellier wherehedidnotinitiallyperformwell,flunkingsuchclassesasastronomy.Workingonhisown,herediscoveredthe Lebesguemeasure.AfterthreeyearsofincreasinglyindependentstudiestherehewenttocontinuehisstudiesinParisin1948.

Initially,Grothendieckattended HenriCartan'sSeminarat ÉcoleNormaleSupérieure,butlackedthenecessarybackgroundtofollowthehigh-poweredseminar.OntheadviceofCartanand Weil,hemovedtothe UniversityofNancy wherehewrotehisdissertationunder LaurentSchwartz infunctionalanalysis,from1950to1953.Atthistimehewasaleadingexpertinthetheoryoftopologicalvectorspaces.By1957,hesetthissubjectasideinordertoworkinalgebraicgeometryand homologicalalgebra.

TheIHÉSyears

Installedatthe InstitutdesHautesÉtudesScientifiques （IHÉS）in1958,Grothendieckattractedattentionbyanintenseandhighlyproductiveactivityofseminars（defacto workinggroupsdraftingintofoundationalworksomeoftheablestFrenchandothermathematiciansoftheyoungergeneration）.Grothendieckhimselfpracticallyceasedpublicationofpapersthroughtheconventional, learnedjournal route.Hewas,however,abletoplayadominantroleinmathematicsforaroundadecade,gatheringastrongschool.

Duringthistimehehadofficiallyasstudents MichelDemazure （whoworkedonSGA3,on groupschemes）, LucIllusie （cotangentcomplex）, MichelRaynaud, Jean-LouisVerdier （cofounderofthederivedcategory theory）and PierreDeligne.CollaboratorsontheSGAprojectsalsoincluded MikeArtin （étalecohomology）and NickKatz （monodromytheory and Lefschetzpencils）. JeanGiraudworkedout torsor theoryextensionsof non-abeliancohomology.Manyotherswereinvolved.

The"GoldenAge"

AlexanderGrothendieck'sworkduringthe"GoldenAge"periodat IHÉS establishedseveralunifyingthemesin algebraicgeometry, numbertheory, topology, categorytheory and complexanalysis.Hisfirst（pre-IHÉS）discoveryinalgebraicgeometrywasthe Grothendieck–Hirzebruch–Riemann–Rochtheorem,ageneralisationofthe Hirzebruch–Riemann–Rochtheorem provedalgebraically;inthiscontexthealsointroduced K-theory.Then,followingtheprogrammeheoutlinedinhistalkatthe1958 InternationalCongressofMathematicians,heintroducedthetheoryof schemes,developingitindetailinhis Élémentsdegéométriealgébrique （EGA）andprovidingthenewmoreflexibleandgeneralfoundationsforalgebraicgeometrythathasbeenadoptedinthefieldsincethattime.Hewentontointroducethe étalecohomology theoryofschemes,providingthekeytoolsforprovingthe Weilconjectures,aswellas crystallinecohomology andalgebraic deRhamcohomology tocomplementit.Closelylinkedtothesecohomologytheories,heoriginated topos theoryasageneralisationoftopology（relevantalsoin categoricallogic）.Healsoprovidedanalgebraicdefinitionof fundamentalgroups ofschemesandmoregenerallythemainstructuresofacategorical Galoistheory.Asaframeworkforhis coherentduality theoryhealsointroducedderivedcategories,whichwerefurtherdevelopedbyVerdier.

TheresultsofworkontheseandothertopicswerepublishedintheEGAandinlesspolishedforminthenotesofthe Séminairedegéométriealgébrique （SGA）thathedirectedatIHES.

Politicsandretreatfromscientificcommunity[edit]

Grothendieck'spoliticalviewswere radical and pacifist.Thus,hestronglyopposedboth UnitedStates interventioninVietnam and Sovietmilitaryexpansionism.Hegavelectureson categorytheory intheforestssurrounding Hanoi whilethecitywasbeingbombed,toprotestagainstthe VietnamWar.[13] Heretiredfromscientificlifearound1970,afterhavingdiscoveredthepartlymilitaryfundingof IHÉS.[14] Hereturnedtoacademiaafewyearslaterasaprofessoratthe UniversityofMontpellier,wherehestayeduntilhisretirementin1988.Hiscriticismsofthescientificcommunity,andespeciallyofseveralmathematicscircles,arealsocontainedinaletter,writtenin1988,inwhichhestatesthereasonsforhisrefusalofthe CrafoordPrize.[15] Hedeclinedtheprizeonethicalgroundsinanopenlettertothemedia.[16]

WhiletheissueofmilitaryfundingwasperhapsthemostobviousexplanationforGrothendieck'sdeparturefromIHÉS,thosewhoknewhimsaythatthecausesoftherupturerandeeper. PierreCartier,a visiteurdelonguedurée （"long-termguest"）attheIHÉS,wroteapieceaboutGrothendieckforaspecialvolumepublishedontheoccasionofthe IHÉS'sfortiethanniversary.TheGrothendieckFestschrift wasathree-volumecollectionofresearchpaperstomarkhissixtiethbirthday（fallingin1988）,andpublishedin1990.[17]

InitCartiernotesthat,asthesonofanantimilitaryanarchistandonewhogrewupamongthedisenfranchised,Grothendieckalwayshadadeepcompassionforthepoorandthedowntrodden.AsCartierputsit,GrothendieckcametofindBures-sur-Yvette"unecagedorée