Chairs:TorstenEkedahl&Jan-ErikRoos

12:30 Registration

13:00-13:05 Openingaddress Prof Staffan Normark, PermanentSecretaryoftheRoyalSwedishAcademyofSciences

13:05-13:50 Thegeneralbackgroundandhistoryoftheclassificationoffinitesimplegroups

Prof John Griggs Thompson, UniversityofCambridge,UK

13:55-14:40 Applyingtheclassificationinotherareasofmathematics

TheRolfSchockPrizeLaureateinMathe-maticsProf Michael Aschbacher,CaliforniaInstituteofTechnology,CA,USA

14:40-15:15 Refreshments

15:15-16:00 MichaelAschbacher’sworkandtheClassificationofFiniteSimpleGroups(CFSG)

Prof Stephen Smith, UniversityofIllinoisatChicago,IL,USA

16:05-16:50 Aglimpseintothefuture Prof Ronald Solomon, TheOhioStateUni-versity,OH,USA

16:50 Endofthesymposium

3Nov Mathematics

TheClassificationoffinitesimplegroupsSymposiuminthehonouroftheRolfSchockPrizeLaureateinMathematicsProfMichaelAschbacher.TheBeijerHall,theRoyalSwedishAcademyofSciences,LillaFrescativägen4A,Stockholm.Opentothepublic.Registration is required and must be made before 25 October 2011 at http://kva.se/events.

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Registeredparticipantsareinvitedtoparticipateinaninformalmingleafterthesymposium

3Nov MathematicsTheClassificationoffinitesimplegroups;symposiuminthehonouroftheRolfSchockPrizeLaureateinMathematicsProfMichaelAschbacher.Registration is required and must be made before 25 October 2011 at http://kva.se/events.

The general background and history of the classification of finite simple groups ProfJohnGriggsThompsonUniversityofCambridge,UK

Applying the classification in other areas of mathematicsTheRolfSchockPrizeLaureateinMathematicsProfMichaelAschbacherCaliforniaInstituteofTechnology,CA,USATherearemanyapplicationsoftheclassificationofthefinitesimplegroupsinmanyareasofmathematics.I’llbrieflymentionafewexamples,butmostofthetimewillbespentgivingsomeideaofhowaproblemonfinitegroupscanbereducedtothesimplecase,andwhatinformationaboutsimplegroupsisthenneededtocompletethesolution.

Michael Aschbacher’s work and the Classification of Finite Simple Groups (CFSG)ProfStephenSmithUniversityofIllinoisatChicago,IL,USAAnattemptismadetosurveyhowtheclassificationwasfinished.Amoredetailedabstractisgivenontheenclosedsheetofpaper.

A glimpse into the future ProfRonSolomonTheOhioStateUniversity,OH,USATheclassificationofthefinitesimplegroupsleavesopenmanyfascinatingquestionsconcerningfinitegroups,andpointstowardnumerousdirectionsforfutureinvestiga-tion.Iwillhighlightsomequestionsinthemodularrepresentationtheoryofgroups,notablytheWeightConjectureofAlperin,whichinturnfocusinterestonthecategoryofsaturatedfusionsystems,establishinganewinterfacebetweengrouptheoryandtopology,whichisactivelybeingexploredbyAschbacher,Chermak,Oliver,andothers.

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3Nov MathematicsAbstractProfStephenSmithUniversityofIllinoisatChicago,IL,USA

Michael Aschbacher’s work and the Classification of Finite Simple Groups (CFSG)Since“most”finitesimplegroupsGareinfactmatrixgroupsoverfinitefields,anearlyresultdeterminingtheoverallshapeoftheCFSGwastheDichotomyTheorem---whichshowsthatanabstractsimpleG(awayfromcaseswith2-subgroupsofrankatmost2)iseither:ofCOMPO-NENTTYPE(resemblingagroupoverafieldofoddorder),orofCHARACTERISTIC2-TYPE(resem-blingagroupincharacteristic2).

Thetreatmentofthe“oddcase”,namelycomponenttype,wasbasedonAschbacher’snotionofaquasisimplecomponentLofSTANDARDFORM,inthecentralizerinGofanelementtoforder2.ThevariouspossibleLweretreatedbyAschbacherandvariousotherresearchers.

Thetreatmentoftheremaining“evencase”,namelycharacteristic2-type,wasobtainedviasuit-ableanalogiesoftheabovecasedivisions---butreplacingtbyanelementofODDprimeorderp.

Heretheinitial“small”casecorrespondstoQUASITHINgroupsG;namelywheretherankofsuit-ablep-subgroupsisatmost2.

Thissituationinvolvesmanycomplications;itwaseventuallytreatedinalengthyworkofAsch-bacherandSmith.

Fortheremainingcasesinvolvingp-subgroupsofrankatleast3,theaboveDichotomyisre-placedbyaTrichotomy---establishedbyGorensteinandLyons(withcontributionsfromAsch-bacher).

Thethreebrancheswhichemergeare:

a(p-component)branchcalledStandardType;

a(roughlycharacteristicp-type)branchleadingto“GF(2)type”;andafurther“disconnected”branchcalledtheUniquenessCase.

ThegroupsofstandardtypeweredeterminedbyGilmanandGriess.

ThegroupsofGF(2)typeweredeterminedbyvariousauthors,includingAschbacher,Timmesfeld,andSmith.

AndthefinalcontradictionintheCFSG(althoughquasithingroupswerechronologicallythelasttobetreated)wasestablishedbyAschbacher,whoshowedthatnogroupcanactuallysatisfytheUniquenessCase.

(ThisoutlineoftheClassificationofFiniteSimpleGroupsisfurtherdevelopedinarecentbookofAschbacher,Lyons,Smith,andSolomon---SurveysoftheAMSvol.172.)

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