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1、BULLETIN(NewSeries)OFTHEAMERICANMATHEMATICALSOCIETYVolume33,Number2,April1996Lecturesondi erentialGaloistheory,byAndyR.Magid,UniversityLectureSeries,vol.7,Amer.Math.Soc.,Providence,RI,1994,xiii+105pp.,$35.00,ISBN0-8218-7004-11.IntroductionDi erentialGaloistheoryhasknownanoutburstofacti
2、vityinthelastdecade.Topinpointwhattriggeredthisrenewalisprobablyamatterofpersonaltaste;allthesame,letmestartthepresentreviewbyatentativelist,restrictedonpurposeto10on-obviouslydi erential"occurrencesofthetheory(andalso,asinthebookunderreview,totheGaloistheoryoflineardi erentialequation
3、sincharacteristic0):
4、In1984,J.-P.Ramis[16]discoveredthattheclassicalStokesphenomenononeencountersintheresummationofdivergentserieshasaGaloisiannatureandcanthereforebeviewedasthee ectofageneralizedmonodromyoperator.
5、In1986,F.Beukers,D.Brownawell,andG.Hekman[3]realizedthatthetechnicalhyp
6、othesisuponwhichSiegelhadbasedhisclassicalgeneralizationoftheLindemann-Weierstasstheoremamountstoasimpleconditiononadi erentialGaloisgroup.ThisrejuvenatedhisapproachtothetheoryofE-andG-functions.
7、In1988,N.Katz(cf.[10])startedanewwayofinvestigatingSato-Tatecon-jecturesonexponentialsumsb
8、yrelatingthemeasureinvolvedintheassociatedlawtoadi erentialGaloisgroup.
9、In1990,P.Deligne[5]rewrotethefundamentalsoftannakiancategories.Inthistheory,GaloisgroupspreexistGaloisextensions.ThisenabledhimtogiveanewconstructionofPicard-Vessiotextensions.
10、Algebraicextensionsoffunction eldsare
11、particularcasesofdi erentialex-tensions.Inthisway,di erentialGaloistheorycancontributetoourknowledgeof nitesubgroupsofclassicalgroups(see,e.g.,M.SingerandF.Ulmer[18]).
12、Di erenceequationstoohavetheirownGaloistheory.Anumberofauthorshaverecentlyapplieditwithsuccesstothestudyofrecurrencere
13、lationsandoftheirq-analogues.Thus,interestindi erentialGaloistheoryisnolongerrestrictedtospecialists.But(strangelyenoughforatheorywithsomuchhistoricalappeal),textbookin-troductionsarerare.Kolchin'sexhaustivebook[12]coversamuchbroaderarea(including,forinstance,theGaloistheoryofnonline
