课程摘要：TheincompressibleNavier-StokesEquation(NSE)isthefundamentalequationtodescribethemotionoftheincompressible,viscousfluidfromphysics.Itisusedtomodeltheweather,oceancurrents,waterflowinapipeandairflowaroundawing.CoupledwithMaxwell\'sequationandFokker-Planckequation,itcanbeusedtostudymagnetohydrodynamics(MHD)andcomplexfluidflowsincludingviscoelasticfluidsrespectively.NSEsarealsoofgreatinterestsfrommathematicalviewpoints,inwhichmanyfundamentalquestionsareyettobeunderstood.Theoutlineofthecoursemayconsistof:ThelinearStokesequation:Basictheorybehindbothsteadyanddynamiccases.ThesteadyNavier-Stokesequation:Existence,regularity,andstability.ThedynamicNavier-Stokesequation:Ladyzhenskya\'stheoryindimensiontwo,LerayandHopf\'stheoryindimensionthree,Serrin-Prodi\'sregularityanduniquenesstheorem,andCaffarelli-Kohn-NirenbergandEscauriaza-Seregin-Sverak\'sregularitytheories.

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Complexgeometryhasbeenaveryactiveareainrecentyears.Ithasdeepinteractionswithotherareas,e.g.algebraicgeometry,Riemanniangeometryandmathphysics.Despitetherecentenormousprogress,therearestillmanyinterestingdirectionstoexplore.Thiscoursewillserveasanintroductiontocomplexgeometry.Wewillcoverthefollowingtopics:holomorphicfunctionsofseveralvariablesondomainsof$C^n$,complexmanifolds,holomorphicvectorbundlesandChernclasses,Kahlermetrics,sheafsandtheircohomology,Hodgetheory,theRiemann-Rochtheorem(proofforRiemannsurfaces),theL^2estimatesandapplications.

Complexgeometryhasbeenaveryactiveareainrecentyears.Ithasdeepinteractionswithotherareas,e.g.algebraicgeometry,Riemanniangeometryandmathphysics.Despitetherecentenormousprogress,therearestillmanyinterestingdirectionstoexplore.Thiscoursewillserveasanintroductiontocomplexgeometry.Wewillcoverthefollowingtopics:holomorphicfunctionsofseveralvariablesondomainsof$C^n$,complexmanifolds,holomorphicvectorbundlesandChernclasses,Kahlermetrics,sheafsandtheircohomology,Hodgetheory,theRiemann-Rochtheorem(proofforRiemannsurfaces),theL^2estimatesandapplications.

课程摘要：Theaimofthecourseistoexploreanumberofdifferentconstructionsofnon-compacthyperkahlermanifolds,inparticularindimension4.Hyperk?hlergeometry,inJoyce’swords,“form(s)abeautifulandrichbranchofmathematics”andthecourseisanattempttodescribesomeaspectsofthistheory.Wewillstartwiththefirstexamplesofhyperk?hlermetricsgivenbyCalabi,wewillthendiscussthehyperkahlerquotientconstructionofKronheimer’sALEspaces,theTaub–NUTmetricandtheconstructionofcompletehyperk?hlermanifoldsasmodulispacesofsolutionstotheYang–Millsanti-self-dualityequations.Onemotivationforstudyingcompletenon-compactRicci-flatmanifoldsistheunderstandingofthemodulispaceofcompactEinsteinmetricsandtheirdegenerations.Asanillustrationofthispoint,wewilldescribetheconstructionoftheCalabi-YaumetricontheKummersurfacebygluingmethods.Introductiontohyperk?hlergeometry.ThegroupSp(n)asaRiemannianholonomygroup,thecurvatureofRiemannian4–manifolds,compactexamples,Calabi’smetriconT*CPnHyperkahlerquotientconstructionandALEspaces.SymplecticandK?hlerquotients,hyperk?hlerquotients,Kronheimer’sALEspaces.Gibbons–Hawkingansatz,twistorspaces.Gibbons–Hawkingansatz,theTaub–NUTmetric,gravitationalinstantons,thetwistorspaceofahyperkahlermanifold.Infinitedimensionalhyperkahlerquotients.TheYang–Millsanti-self-dualityequationsasahyperk?hlermomentmap,Kronheimer’shyperk?hlermetricsoncoadjointorbits.TheKummerconstruction.Donaldson’sgluingconstructionoftheCalabi-YaumetricontheKummersurface.References[1]Hitchin,N.,Hyper-K?hlermanifolds,SéminaireBourbaki,Vol.1991/92,Astérisque206,1992,Exp.No.748,3,137–166.[2]Joyce,D.,Riemannianholonomygroupsandcalibratedgeometry,Chapter10,OxfordGraduateTextsinMathematics,12,OxfordUniversityPress,2007.[3]Hitchin,N.,andKarlhede,A.andLindstr?m,U.andRo?ek,M.,Hyper-K?hlermetricsandsupersymmetry,Comm.Math.Phys.,CommunicationsinMathematicalPhysics,108,1987,4,535–589.[4]Donaldson,S.,Calabi-YaumetricsonKummersurfacesasamodelgluingproblem,Advancesingeometricanalysis,Adv.Lect.Math.(ALM),21,109–118,Int.Press,Somerville,MA,2012.

课程摘要：IwillgiveintroductiontothetheoryofminimalsurfacesinRiemaniangeometry.Wewillstartwiththefirstandsecondvariationformulaoftheareafunctional.ThenwediscussminimalsurfacesinEuclideanspaceandthesphere.AfterwardwewillfocusonstableminimalhypersurfacesinRiemannianmanifolds.Iftimepermits,wewillalsotalkaboutthePlateauproblem,geometricapplicationsofminimalsurfacesandsomeotherrelatedtopics.AsaprerequisitestudentsshouldhavesomebasicknowledgeofRiemanniangeometry.

《催化前沿讲座》暑期课程日程表地点：2210教室日期报告人报告题目7月10日路军岭教授中国科学技术大学9:00-11:15原子层沉积技术在催化剂设计中的应用7月11日江海龙教授中国科学技术大学9:00-11:15多孔金属有机框架材料在催化中的应用7月12日Prof.RolfJürgenBehmUlmUniversity9:00–10:30OxidationreactionsonoxidesupportedAucatalysts–Thecrucialroleofreactionconditionsandcatalystsupport10:40-11:40韩一帆教授华东理工大学OperandoSpectroscopy-ACutting-edgeTechniqueforR&DIndustrialCatalysts(动态现场原位光谱-工业催化剂研发前沿技术)7月13日陈昶乐教授中国科学技术大学9:00-11:15金属有机催化聚合反应7月14日曾杰教授中国科学技术大学9:00-11:15石油化工和金属纳米催化7月15日杜平武教授中国科学技术大学9:00-11:15光催化及稳态瞬态光谱表征7月16日邵翔教授中国科学技术大学9:00-11:15SPM在催化研究中的应用

课程摘要：Iwilldiscussbasichomotopytheory,fundamentalgroup,classificationofsurfacesandcoveringspaces.