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1、ClayMathematicsProceedings5Volume5Mathematicalgaugetheorystudiesconnectionsonprincipalbundles,andLow-DimensionalTopologyFloerHomology,GaugeTheory,or,moreprecisely,thesolutionspacesofcertainpartialdifferentialequationsforsuchconnections.Historically,theseequationshaveco
2、mefrommathematicalphysics,andplayanimportantroleinthedescriptionoftheelectro-weakandstrongnuclearforces.Theuseofgaugetheoryasatoolforstudyingtopologicalpropertiesoffour-manifoldswaspioneeredbythefundamentalworkofSimonDonaldsonintheearly1980s,andwasrevolutionizedbythein
3、troductionoftheSeiberg–Wittenequationsinthemid-1990s.Sincethebirthofthesubject,ithasretaineditscloseconnectionwithsymplectictopology.TheanalogybetweenthesetwofieldsofstudywasfurtherunderscoredbyAndreasFloer’sconstructionofaninfinite-dimensionalvariantofMorsetheorythata
4、ppliesintwoaprioridifferentcontexts:eithertodefinesymplecticinvariantsforpairsofLagrangiansubmanifoldsofasymplecticmanifold,ortodefinetopologicalinvariantsforthree-manifolds,whichfitintoaframeworkforcalculatinginvariantsforsmoothfour-manifolds.“HeegaardFloerhomology”,P
5、roceedingsofthetherecently-discoveredinvariantforthree-andfour-manifolds,comesfromanClayMathematicsInstituteapplicationofLagrangianFloerhomologytospacesassociatedtoHeegaarddiagrams.AlthoughthistheoryisconjecturallyisomorphictoSeiberg–2004SummerSchoolWittentheory,itismo
6、retopologicalandcombinatorialinflavorandthuseasiertoworkwithincertaincontexts.TheinteractionbetweengaugeEllwood,Ozsváth,Stipsicz,andSzabóAlfrédRényiInstituteofMathematicstheory,low-dimensionaltopology,andsymplecticgeometryhasledBudapest,Hungarytoanumberofstrikingnewdev
7、elopmentsinthesefields.TheaimofthisvolumeistointroducegraduatestudentsandresearchersinJune5–26,2004otherfieldstosomeoftheseexcitingdevelopments,withaspecialemphasisontheveryfruitfulinterplaybetweendisciplines.DavidA.EllwoodThisvolumeisbasedonlecturecoursesandadvancedse
8、minarsgivenatthe2004ClayMathematicsInstituteSummerSchoolatPeterS.OzsváththeAlfrédRényiInstituteofMathematicsinBudapes
