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1、LecturesonSymplecticGeometryAnaCannasdaSilva1August17,20001E-mail:acannas@math.ist.utl.ptoracannas@math.berkeley.eduForewordThegoalofthesenotesistoprovideafastintroductiontosymplecticgeometry–thegeometryofmanifoldsequippedwithaclosednondegenerate2-form.Twocenturiesago,symplecti
2、cgeometry1providedalanguageforclassicalmechanics.Throughitsrecenthugedevelopment,itconqueredanindependentandrichterritory.Symplecticgeometryissignificantlystimulatedbyimpor-tantinteractionswithglobalanalysis,mathematicalphysics,low-dimensionaltopology,dynamicalsystems,algebraicg
3、eometry,integrablesystems,microlo-calanalysis,partialdifferentialequations,representationtheory,quantization,equivariantcohomology,geometriccombinatorics,etc.PartsI-IIIexplainclassicaltopics,includingcotangentbundles,symplecto-morphisms,lagrangiansubmanifoldsandlocalforms.PartsI
4、V-VIconcentrateonimportantrelatedareas,suchascontactgeometryandK¨ahlergeometry.ClassicalhamiltoniantheoryentersonlyinPartsVII-VIII,startingthesecondhalfofthisbook,whichisdevotedtoaselectionfromhamiltoniandynamicalsystemsandsymmetry.PartsIX-XIdiscussthemomentmapwhosepreponder-an
5、cehasbeengrowingsteadilyforthepasttwentyyears.Therearescatteredshortexercisesthroughoutthetext.Attheendofmostlectures,somelongerguidedproblems,calledhomework,weredesignedtocomplementtheexpositionorextendthereader’sunderstanding.Thesenotesapproximatelytranscribea15-weekcourseons
6、ymplecticgeom-etryItaughtatUCBerkeleyintheFallof1997,with2hour-and-a-halflecturesperweek.Thecoursetargetedsecond-yeargraduatestudentsinmathemat-ics,thoughtheaudiencewasmorediverse,includingadvancedundergraduates,post-docsandgraduatestudentsfromotherdepartments.Thepresenttextsho
7、uldhencestillbeappropriateforasecond-yeargraduatecourseorforanin-dependentstudyproject.GeometryofmanifoldswasthebasicprerequisitefortheguidingFall97course,sothesameholdsnowforthenotes.Inparticular,somefamiliaritywithdeRhamtheoryandclassicalLiegroupsisexpected.Therearebynowexcel
8、lentreferencesonsymplecticgeometry,asubsetofwhichisint
