丘成桐中国大学生数学竞赛大纲

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1、2010年中国大学生数学竞赛（丘成桐教授发起）竞赛大纲一．SyllabusesforGeometryandTopology      Geometry:Curvesandsurfaces1)     Planecurvesandspacecurves2)     Thefundamentaltheoremofcurves3)     Conceptandexamplesofsurfaces4)     Thefirstandsecondfundamentalforms5)     Normalcurvature,principalcurvatureandtheGauss

2、curvature6)     Orthogonalmovingframesandstructureequationsofsurfaces7)     Existenceanduniquenessofsurfaces8)     Isometrictransformationofsurfaces9)     Covariantderivativesonsurfaces10)  Geodesiccurvaturesandgeodesics,Geodesiccoordinates11)  TheGauss-Bonnetformula12)  Laplacianoperato

3、ronsurfacesGeometryonmanifolds1) Manifolds2) Vectorfieldsanddifferentials3) Tensorsanddifferentialforms4) Stokesformula5) DeRhamtheorem6) Liederivatives7) Liealgebras8) Maurer-Cartanequations9) Vectorbundles10)Connectionandcurvatures11)Structureequations12)Riemannianmetrics13)TheHodgesta

4、roperatorandLaplacianoperator14)TheHodgetheoremReferences:M.DoCarmo,Differentialgeometryofcurvesandsurfaces.SSChernandChenWeihuan,LecturesondifferentialgeometryQ.ChenandCKPeng,DifferentialgeometryT.Frenkel:GeometryfromphysicsJ.Milnor,Morsetheory TopologyPointSetTopology1)     Opensetandc

5、losedset2)     Continuousmaps3)     Haudorffspace,seperabilityandcountableaxioms4)     CompactnessandHeine-Boreltheorem5)     Connectivityandpathconnectivity6)     Quotientspaceandquotienttopology Fundamentalgroups 1) Definitionoffundamentalgroups,homotopicmaps2) Computationoffundamental

6、groups:VanKampentheorem3) Coveringmapsandcoveringspaces4) Applications:Brouwerfixedpointtheorem,Lefschetzfixedpointtheorem Complexesandhomologygroups 1) Simplex,complexesandpolyhedron2) Barycentricsubdivisionandsimplexapproximation3) Computationoffundamentalgroupsofcomplexes4) Classifica

7、tionofsurfaces5) Simplexhomologygroups6) Application:Lefschetzfixedpointtheorem Differentialtopology 1) Smoothmanifoldsandsmoothmaps2) Sard’stheorem3) Transversalityandintersection4) VectorfiledsandPoincare-Hopftheorem5) DifferentialformsanddeRhamcomplexes6) Orientationan