丘成桐大學生數(shù)學競賽數(shù)學專業(yè)大綱

/S．T．YAUCollegeStudentAlgebra,NumberTheoryandCombinatorics(seconddraft)LinearAlgebraAbstractvectorspaces;subspaces;dimension;matricesandlineartransformations;matrixalgebrasandgroups;determinantsandtraces;eigenvectorsandeigenvalues,characteristicandminimalpolynomials;diagonalizationandtriangularizationofoperators;invariantsubspacesandcanonicalforms;innerproductsandorthogonalbases;reductionofquadraticforms;hermitianandunitaryoperators,bilinearforms;dualspaces;adjoints.tensorproductsandtensoralgebras;IntegersandpolynomialsIntegers,Euclideanalgorithm,uniquedecomposition;congruenceandtheChineseRemaindertheorem;Quadraticreciprocity;IndeterminateEquations.Polynomials,Euclideanalgorithm,uniquenessdecomposition,zeros;Thefundamentaltheoremofalgebra;Polynomialsofintegercoefficients,theGausslemmaandtheEisensteincriterion;Polynomialsofseveralvariables,homogenousandsymmetricpolynomials,thefundamentaltheoremofsymmetricpolynomials.GroupGroupsandhomomorphisms,Sylowtheorem,finitelygeneratedabeliangroups.Examples:permutationgroups,cyclicgroups,dihedralgroups,matrixgroups,simplegroups,Jordan-Holdertheorem,lineargroups(GL(n,F)anditssubgroups),p-groups,solvableandnilpotentgroups,groupextensions,semi-directproducts,freegroups,amalgamatedproductsandgrouppresentations.RingBasicpropertiesofrings,units,ideals,homomorphisms,quotientrings,primeandmaximalideals,fieldsoffractions,Euclideandomains,principalidealdomainsanduniquefactorizationdomains,polynomialandpowerseriesrings,ChineseRemainderTheorem,localringsandlocalization,Nakayama'slemma,chainconditionsandNoetherianrings,Hilbertbasistheorem,Artinrings,integralringextensions,Nullstellensatz,Dedekinddomains,algebraicsets,Spec(A).ModuleModulesandalgebraFreeandprojective;tensorproducts;irreduciblemodulesandSchur’slemma;semisimple,simpleandprimitiverings;densityandWederburntheorems;thestructureoffinitelygeneratedmodulesoverprincipalidealdomains,withapplicationtoabeliangroupsandcanonicalforms;categoriesandfunctors;complexes,injectivemodues,cohomology;TorandExt.FieldFieldextensions,algebraicextensions,transcendencebases;cyclicandcyclotomicextensions;solvabilityofpolynomialequations;finitefields;separableandinseparableextensions;Galoistheory,normsandtraces,cyclicextensions,Galoistheoryofnumberfields,transcendencedegree,functionfields.GrouprepresentationIrreduciblerepresentations,Schur'slemma,characters,Schurorthogonality,charactertables,semisimplegrouprings,inducedrepresentations,Frobeniusreciprocity,tensorproducts,symmetricandexteriorpowers,complex,real,andrationalrepresentations.LieAlgebraBasicconcepts,semisimpleLiealgebras,rootsystems,isomorphismandconjugacytheorems,representationtheory.Combinatorics(TBA)References:Strang,Linearalgebra,AcademicPress.I.M.Gelfand,LinearAlgebra《整數(shù)與多項式》馮克勤余紅兵著高等教育出版社Jacobson,Nathan

Basicalgebra.I.

Secondedition.

W.H.FreemanandCompany,NewYork,

1985.xviii+499pp.Jacobson,Nathan

Basicalgebra.II.

Secondedition.

W.H.FreemanandCompany,NewYork,

1989.xviii+686pp.S.Lang,Algebra,Addison-Wesley馮克勤，李尚志，查建國，章璞，《近世代數(shù)引論》劉紹學，《近世代數(shù)基礎(chǔ)》J.P.Serre,LinearrepresentationsoffinitegroupsJ.P.Serre:ComplexsemisimpleLiealgebraandtheirrepresentationsJ.Humphreys:IntroductiontoLiealgebraandrepresentationtheory,GTM009.W.Fulton,Representationtheory,aFirstCourse,GTM129.Analysisanddifferentialequations(seconddraft)CalculusandmathematicalanalysisDerivatives,chainrule;maximaandminima,Lagrangemultipliers;lineandsurfaceintegralsofscalarandvectorfunctions;Gauss’,Green’sandStokes’theorems.Sequencesandseries,Cauchysequences,uniformconvergenceanditsrelationtoderivativesandintegrals;powerseries,radiusofconvergence,convergenceofimproperintegrals.Inverseandimplicitfunctiontheoremsandapplications;thederivativeasalinearmap;existenceanduniquenesstheoremsforsolutionsofordinarydifferentialequations,explicitsolutionsofsimpleequations.;elementaryFourierseries.ComplexanalysisAnalyticfunction,Cauchy'sIntegralFormulaandResidues,PowerSeriesExpansions,EntireFunction,NormalFamilies,TheRiemannMappingTheorem,HarmonicFunction,TheDirichletProblemSimplyPeriodicFunctionandEllipticFunctions,TheWeierstrassTheoryAnalyticContinuation,AlgebraicFunctions,Picard'sTheoremPointsettopologyofRnCountableanduncountablesets,theaxiomofchoice,Zorn'slemma.Metricspaces.Completeness;separability;compactness;Bairecategory;uniformcontinuity;connectedness;continuousmappingsofcompactspaces.Functionsontopologicalspaces.EquicontinuityandAscoli'stheorem;theStone-Weierstrasstheorem;topologiesonfunctionspaces;compactnessinfunctionspaces.MeasureandintegrationMeasures;Borelsetsandcontorsets;Lebesguemeasures;distributions;productmeasures.Measurablefunctions.approximationbysimplefunctions;convergenceinmeasure;Constructionandpropertiesoftheintegral;convergencetheorems;Radon-Nykodymtheorem;Fubini'stheorem;meanconvergence.Monotonefunctions;functionsofboundedvariationandBorelmeasures;Absolutecontinuity,convexfunctions;semicontinuity.BanachandHilbertspacesLpspaces;C(X);completenessandtheRiesz-Fischertheorem;orthonormalbases;linearfunctionals;Rieszrepresentationtheorem;lineartransformationsanddualspaces;interpolationoflinearoperators;Hahn-Banachtheorem;openmappingtheorem;uniformboundedness(orBanach-Steinhaus)theorem;closedgraphtheorem.Basicpropertiesofcompactoperators,Riesz-Fredholmtheory,spectrumofcompactoperators.BasicpropertiesofFourierseriesandtheFouriertransform;Poissionsummationformula;convolution.BasicpartialdifferentialequationsFirstorderpartialdifferentialequations,linearandquasi-linearPDE,Waveequations:initialconditionandboundarycondition,well-poseness,Sturn-Liouvilleeigen-valueproblem,energyfunctionalmethod,uniquenessandstabilityofsolutionsHeatequations:initialconditions,maximalprincipleanduniquenessandstabilityPotentialequations:GreenfunctionsandexistenceofsolutionsofDirichletproblem,harmonicfunctions,Hopf’smaximalprincipleandexistenceofsolutionsofNeumann’sproblem,weaksolutions,eigen-valueproblemoftheLaplaceoperatorGeneralizedfunctionsandfundamentalsolutionsofPDEReferences:Rudin,Principlesofmathematicalanalysis,McGraw-Hill.Courant,Richard;

John,Fritz

Introductiontocalculusandanalysis.Vol.I.

Reprintofthe1989edition.ClassicsinMathematics.

Springer-Verlag,Berlin,

1999.Courant,Richard;

John,Fritz

Introductiontocalculusandanalysis.Vol.II.

WiththeassistanceofAlbertA.BlankandAlanSolomon.Reprintofthe1974edition.

Springer-Verlag,NewYork,

1989.V.I.Arnold,OrdinaryDifferentialEquations,Springer-Verlag,BerlinValerianAhlfors,AnIntroductiontotheTheoryofAnalyticFunctionsofOneComplexVariableK.Kodaira,ComplexAnalysisRudin,Realandcomplexanalysis龔升，簡明復分析Royden,RealAnalysis,exceptchapters8,13,15.E.M.SteinandR.Shakarchi;RealAnalysis:MeasureTheory,Integration,andHilbertSpaces,Princeton周民強,實變函數(shù)論,北京大學出版社,2001夏道行等，《實變函數(shù)論與泛函分析》，人民教育出版社.PeterD.Lax,FunctionalAnalysis,Wiley-Interscience,2002.《BasicPartialDifferentialEquations》,D.Bleecker,G.Csordas著,李俊杰譯，高等教育出版社，2008.《數(shù)學物理方法》，柯朗、希爾伯特著。ComputationalMathematics,AppliedMathematics,ProbabilityandStatistics(theseconddraft)ComputationalMathematicsInterpolationandapproximationPolynomialinterpolationandleastsquareapproximation;trigonometricinterpolationandapproximation,fastFouriertransform;approximationsbyrationalfunctions;splines.NonlinearequationsolversConvergenceofiterativemethods(bisection,secantmethod,Newtonmethod,otheriterativemethods)forbothscalarequationsandsystems;findingrootsofpolynomials.LinearsystemsandeigenvalueproblemsDirectsolvers(Gausselimination,LUdecomposition,pivoting,operationcount,bandedmatrices,round-offerroraccumulation);iterativesolvers(Jacobi,Gauss-Seidel,successiveover-relaxation,conjugategradientmethod,multi-gridmethod,Krylovmethods);numericalsolutionsforeigenvaluesandeigenvectorsNumericalsolutionsofordinarydifferentialequationsOnestepmethods(TaylorseriesmethodandRunge-Kuttamethod);stability,accuracyandconvergence;absolutestability,longtimebehavior;multi-stepmethodsNumericalsolutionsofpartialdifferentialequationsFinitedifferencemethod;stability,accuracyandconvergence,Laxequivalencetheorem;finiteelementmethod,boundaryvalueproblemsReferences:[1]C.deBoorandS.D.Conte,ElementaryNumericalAnalysis,analgorithmicapproach,McGraw-Hill,2000.[2]G.H.GolubandC.F.vanLoan,MatrixComputations,thirdedition,JohnsHopkinsUniversityPress,1996.[3]E.Hairer,P.SyvertandG.Wanner,SolvingOrdinaryDifferentialEquations,Springer,1993.[4]B.Gustafsson,H.-O.KreissandJ.Oliger,TimeDependentProblemsandDifferenceMethods,JohnWileySons,1995.[5]G.StrangandG.Fix,AnAnalysisoftheFiniteElementMethod,secondedition,Wellesley-CambridgePress,2008.AppliedMathematicsODEwithconstantcoefficients;NonlinearODE:criticalpoints,phasespace&stabilityanalysis;Hamiltonian,gradient,conservativeODE’s.CalculusofVariations:Euler-LagrangeEquations;BoundaryConditions,parametricformulation;optimalcontrolandHamiltonian,Pontryaginmaximumprinciple.Firstorderpartialdifferentialequations(PDE)andmethodofcharacteristics;Heat,wave,andLaplace’sequation;Separationofvariablesandeigen-functionexpansions;Stationaryphasemethod;HomogenizationmethodforellipticandlinearhyperbolicPDEs;HomogenizationandfrontpropagationofHamilton-Jacobiequations;Geometricopticsfordispersivewaveequations.References:W.D.BoyceandR.C.DiPrima,ElementaryDifferentialEquations,Wiley,2009F.Y.M.Wan,IntroductiontoCalculusofVariationsandItsApplications,Chapman&Hall,1995G.Whitham,“LinearandNonlinearWaves”,John-WileyandSons,1974.J.Keener,“PrinciplesofAppliedMathematics”,Addison-Wesley,1988.A.Benssousan,P-LLions,G.Papanicolaou,“AsymptoticAnalysisforPeriodicStructures”,North-HollandPublishingCo,1978.V.Jikov,S.Kozlov,O.Oleinik,“Homogenizationofdifferentialoperatorsandintegralfunctions”,Springer,1994.J.Xin,“AnIntroductiontoFrontsinRandomMedia”,SurveysandTutorialsinAppliedMathSciences,No.5,Springer,2009.ProbabilityRandomVariables;ConditionalProbabilityandConditionalExpectation;MarkovChains;TheExponentialDistributionandthePoissonProcess;Continuous-TimeMarkovChains;RenewalTheoryandItsApplications;QueueingTheory;ReliabilityTheory;BrownianMotionandStationaryProcesses;Simulation.Reference:SheldonM.Ross,IntroductiontoProbabilityModelsStatisticsDistributionTheoryandBasicStatisticsFamiliesofcontinuousdistributions:Chi-sq,t,F,gamma,beta;Familiesofdiscretedistributions:Multinomial,Poisson,negativebinomial;Basicstatistics:Mean,median,quantiles,orderstatisticsLikelihoodprincipleLikelihoodfunction,parametricinference,sufficiency,factorizationtheorem,ancillarystatistic,conditionallikelihood,marginallikelihood.TestingNeyman-Pearsonparadigm,nullandalternativehypotheses,simpleandcompositehypotheses,typeIandtypeIIerrors,power,mostpowerfultest,likelihoodratiotest,Neyman-PearsonTheorem,monotonelikelihoodratio,uniformlymostpowerfultest,generalizedlikelihoodratiotest.EstimationParameterestimation,methodofmoments,maximumlikelihoodestimation,unbiasedness,quadraticandothercriterionfunctions,Rao-BlackwellTheorem,Fisherinformation,Cramer-Raobound,confidenceinterval,dualitybetweenconfidenceintervalandhypothesistesting.BayesianStatisticsPrior,posterior,conjugatepriors,BayesianlossNonparametricstatisticsPermutationtest,permutationdistribution,rankstatistics,Wilcoxon-Mann-Whitneytest,log-ranktest,Kolmogorov-Smirnov-typetests.RegressionLinearregression,leastsquaresmethod,Gauss-MarkovTheorem,logisticregression,maximumlikelihoodLargesampletheoryConsistency,asymptoticnormality,chi-sqapproximationtolikelihoodratiostatistic,large-samplebasedconfidenceinterval,asymptoticpropertiesofempiricaldistribution.ReferencesCasella,G.andBerger,R.L.(2002).StatisticalInference(2ndEd.)DuxburyPress.茆詩松，程依明，濮曉龍，概率論與數(shù)理統(tǒng)計教程（第二版），高等教育出版社，2008.陳家鼎，孫山澤，李東風，劉力平，數(shù)理統(tǒng)計學講義，高等教育出版社，2006.鄭明，陳子毅，汪嘉岡，數(shù)理統(tǒng)計講義，復旦大學出版社，2006.陳希孺，倪國熙，數(shù)理統(tǒng)計學教程，中國科學技術(shù)大學出版社，2009.GeometryandTopology(theseconddraft)SpacecurvesandsurfacesCurvesandParametrization,RegularSurfaces;InverseImagesofRegularValues.

GaussMapandFundamentalProperties;Isometries;ConformalMaps;RigidityoftheSphere.TopologicalspaceSpace,maps,compactnessandconnectedness,quotients;PathsandHomotopy.TheFundamentalGroupoftheCircle.InducedHomomorphisms.FreeProductsofGroups.ThevanKampenTheorem.CoveringSpacesandLiftingProperties;Simplexandcomplexes.Triangulations.Surfacesanditsclassification.DifferentialManifoldsDifferentiableManifoldsandSubmanifolds,DifferentiableFunctionsandMappings;TheTangentSpace,VectorFieldandCovectorFields.TensorsandTensorFieldsanddifferentialforms.TheRiemannianMetricsasex