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1、BULLETINOFTHEAMERICANMATHEMATICALSOCIETYVolume82,Number1,January1976TOPOLOGYANDLOGICASASOURCEOFALGEBRARetiringPresidentialAddress1BYSAUNDERSMACLANETableofContentsIntroduction11:Separableextensions12:Homologicalalgebra53:Thecohomologyofgroups84:Thebarconstruction17
2、5:Coherenceandcanonicalmaps246:Naturalityandcategories337:Whatisalgebra?36Introduction.EachPresidentoftheAmericanMathematicalSocietyisrequiredtopresentaretiringpresidentialaddress.Bycustom,thisaddressisnotinanywaydirectedtotheexaminationoftheadministrativeorprofes
3、•sionalproblemsoftheSociety.Thisisnotbecausetheseproblemsareunimportant,butispresumablybecausethePresident'scontributiontothemshouldhavelongsincebeenmade,ornotmade,asthecasemaybe,wellbeforehisretirementfromoffice.InsteadthecustomisthatthePresident'saddressisconcer
4、nedwithcontributionstoourscience.Thisisagoodcustom,sinceitisthemathematicaltheoremsandthesolutionsofmathematicalproblemswhichpossessthequalityofpermanenceandthesymmetryofstructurewhichisourprimaryobjectiveinmathematics.Myownresearchworkhasbeenlargelyconcernedwitha
5、spectsofalgebra—avarietyofaspectsbynomeanscoveringallofalgebra,butchieflyinvolvedinsomeexplorationsoftherelationbetweenalgebraandtheneighboringfieldsoflogicandgeometry.Thesestudieshavegivenmethelivelyimpressionthatmanyoftheideasofalgebradoindeedarisefromtheseother
6、fields,andthatthisoriginhighlightsthesenseinwhichthescienceofmathematicsexemplifiestheinterdependenceofitsparts.Hence,thisaddresswillbedevotedtoanexaminationofcertainofthewaysinwhichtheproblemsfromgeometryandlogicarisinginmyownresearchworkhaveilluminatedalgebraorc
7、ontributednewconceptstothatfield.1.Separableextensions.Myfirstexampleisaprobleminthetheoryoffieldswhosesolutionrequiredtheuseofideasfromgeometry—moreexactly,ideasaboutanaxiomatictreatmentoflinearindependence.1Deliveredatthe81stAnnualMeetingoftheAmericanMathematica
8、lSocietyinWashing•ton,D.C.,onJanuary24,1975;receivedbytheeditorsJune16,1975.AMS(MOS)subjectclassifications(1970).Primary18-02,12-02,55-02,02-02,00-xx.1C
