Chairs: Torsten Ekedahl & Jan-Erik Roos 12:30 Registration 13:00-13:05 Opening address Prof Staffan Normark, Permanent Secretary of the Royal Swedish Academy of Sciences 13:05-13:50 The general background and history of the classification of finite simple groups Prof John Griggs Thompson, University of Cambridge, UK 13:55-14:40 Applying the classification in other areas of mathematics The Rolf Schock Prize Laureate in Mathe- matics Prof Michael Aschbacher, California Institute of Technology, CA, USA 14:40-15:15 Refreshments 15:15-16:00 Michael Aschbacher’s work and the Classification of Finite Simple Groups (CFSG) Prof Stephen Smith, University of Illinois at Chicago, IL, USA 16:05-16:50 A glimpse into the future Prof Ronald Solomon, The Ohio State Uni- versity, OH, USA 16:50 End of the symposium 3 Nov Mathematics The Classification of finite simple groups Symposium in the honour of the Rolf Schock Prize Laureate in Mathematics Prof Michael Aschbacher. The Beijer Hall, the Royal Swedish Academy of Sciences, Lilla Frescativägen 4A, Stockholm. Open to the public. Registration is required and must be made before 25 October 2011 at http://kva.se/events. THE ROLF SCHOCK PRIZE IS AWARDED BYTHE ROYAL SWEDISH ACADEMIES OF FINE ARTS, MUSIC AND SCIENCES COMPLETE PROGRAMME FOR THE ROLF SCHOCK PRIZE 2011 EVENTS, AVAILABLE AT HTTP://ROLFSCHOCKPRIZES.SE/ AND HTTP://KVA.SE ROLF SCHOCK PRIZE 2011 Registered participants are invited to participate in an informal mingle after the symposium
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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.
THEROLFSCHOCKPRIZEISAWARDEDBYTHEROYALSWEDISHACADEMIESOFFINEARTS,MUSICANDSCIENCES
COMPLETEPROGRAMMEFORTHEROLFSCHOCKPRIZE2011EVENTS,AVAILABLEATHTTP://ROLFSCHOCKPRIZES.SE/ANDHTTP://KVA.SE
ROLFSCHOCKPRIZE2011
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.
THEROLFSCHOCKPRIZEISAWARDEDBYTHEROYALSWEDISHACADEMIESOFFINEARTS,MUSICANDSCIENCES
COMPLETEPROGRAMMEFORTHEROLFSCHOCKPRIZE2011EVENTS,AVAILABLEATHTTP://ROLFSCHOCKPRIZES.SE/ANDHTTP://KVA.SE
ROLFSCHOCKPRIZE2011
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.)
THEROLFSCHOCKPRIZEISAWARDEDBYTHEROYALSWEDISHACADEMIESOFFINEARTS,MUSICANDSCIENCES
ROLFSCHOCKPRIZE2011
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