Abstract：Inthistalk,wefirsttalkabouttheSobolevspacetheoryandharmonicanalysistoolsfortheLaplacianopeartorassociatedwithHardypotential.Andthenweconsidertheenergy-criticalnonlinearwaveequationinthepresenceofanHardypotential.Inthedefocusingcase,weprovethatarbitraryinitialdataintheenergyspaceleadtoglobalsolutionsthatscatter.Inthefocusingcase,weprovescatteringbelowthegroundstatethreshold.ThispartisjointedwithChangxingMiaoandJasonMurphy.Finally,westudythelowregularityproblemfor3dcubicwaveequationwiththeHardypotential.ThispartisjointedwithChangxingMiaoandJunyongZhang.

Abstract：Inthistalk,wewilldiscussaSchrodingeroperatorinaconicsingularspaceandtheStrichartzestimateforthedispersiveequationassociatedwiththisoperator.Themaintechniquesarethemicrolocalanalysismethodandspectralanalysisargument.ThisisajointworkwithJiqiangZheng.

摘要：Inthistalk,IwillpresentrecentprogressinthestudyofheatkernelsandparabolicHarnackinequalitiesforsymmetricMarkovprocessesthathavebothdiffusiveandjumpingpartsongeneralmetricmeasurespaces.Undergeneralvolumedoublingconditionandsomemildassumptionsonthescalingfunctions,weestablishstabilityresultsfortwo-sidedestimatesforheatkernelsintermsofthejumpingkernels,thegeneralizedcapacityinequalities,andPoincareinequalities.StablecharacterizationsoftheassociatedparabolicHarnackinequalitieswillalsobegiven.Ourresultsholdonspacesevenwhentheunderlyingspaceshavewalkdimensionsarelargerthan2.JointworkwithTakashiKumagaiandJianWang.

2019流形上的整体分析研讨会会议时间：2019年7月28日（报到）－8月3日（离会）会议地点：管理科研楼数学科学学院7月30日（周二）下午、8月1日（周四）下午1318教室，其余时间1308教室会议报告题目：刘冰萧马普所(波恩)Anintroductiontosymmetricspaces申述巴黎第六大学IntroductiontothetraceformualforGL(2)伊泽霖南开大学Thetangentgroupoidandrescaledbundle张鑫武汉理工大学Hopfcycliccohomologyfornon-compact$G$-manifolds戴先哲加州大学（圣芭芭拉）、华东师范大学TBA王相生北京大学Weylgroupactionsongeometricobjects卢文华中科技大学BergmankernelforsequenceoflinebundlesonK"{a}hlermanifolds张野平日本京都大学BCOVinvariantandramifiedcover史鹏帅北京大学TheAPSindexproblemforCallias-typeoperatorsandtherelativeetainvariant余世霖厦门大学Deformationquantizationofcoadjointorbits宋言理华盛顿大学（圣路易斯）Ageometricformulafor$K$-typeoftemperedrepresentations刘宇航北京大学Onpositivelycurved6-manifolds苏广想南开大学Positivescalarcurvatureonnoncompactfoliations

摘要：Inthegyrokineticmodelandsimulations,whenthedouble-gyroaveragetermincorporatesthecombiningeffectcontributedbythefiniteLarmorradius,shortscalesoftheperturbation,andsteepgradientoftheequilibriumprofile,thelow-orderapproximationofthistermcouldgeneratenonnegligibleerror.Thispaperimplementsaninterpolationalgorithmtocomputethedouble-gyroaveragetermwithoutlow-orderapproximationtoavoidthiserror.Forasteepequilibriumdensity,theobviousdifferencebetweenthedensityonthegyrocentercoordinateframeandtheoneontheparticlecoordinateframeshouldbeaccountedforinthequasi-neutralityequation.AnEuler–Maclaurin-basedquadratureintegratingalgorithmisdevelopedtocomputethequadratureintegralforthedistributionofthemagneticmoment.Theapplicationoftheinterpolationalgorithmtocomputingthedouble-gyroaveragetermandtosolvingthequasi-neutralityequationisbenchmarkedbycomparingthenumericalresultswiththeknownanalyticalsolutions.Finally,tomakeadvantageoftheinterpolationsolvermoreclearly,thenumericalcomparisonbetweentheinterpolationsolverandaclassicalsecondordersolveriscarriedoutinaconstanttheta-pinchmagneticfieldconfigurationBasedonSELALIBcode.Whentheequilibriumprofileisnotsteepandtheperturbationonlyhasthenon-zeromodenumberalongtheparallelspatialdimension,theresultscomputedbythetwosolversmatcheachotherwell.Whenthegradientoftheequilibriumprofileissteep,theinterpolationsolverprovidesabiggerdrivingeffectfortheion-temperature-gradientmodes,whichpossesslargepolarmodenumbers.

Abstract：In1987,FukayaprovedthecontinuityofeigenvaluesofRiemannianmanifoldswithuniformlyboundedcurvatureanddiameterwithrespecttothemeasuredGromov-Hausdorfftopology.WiththemajorconcernbeingasequenceofcollapsingRiemannianmanifolds,hisresultwasbasedonaso-calledkeyestimate,assertingthattheeigenfunctionsarealmostconstantalongthosecollapsingdirections.Inthistalk,wediscussageneralizationofFukaya\'skeylemmatothesettingofcollapsingEinsteinmanifolds,withoutassumingauniformcurvaturebound.

Abstract：Theaimofthistalkistopresentsomenewideasontheasymptoticexpansionofpositiverealsequenceswhosegeneratingfunctionisgivenbyanexplicitparametricequation.Toprovidecontext,Iwillfirstintroduce/recallbrieflysomeclassicaltheoryofasymptoticanalysisinenumerativecombinatorics.ThispartfollowsthebookofFlajoletandSedgewick.Afterthat,Iwilltrytoexplainwhy"havingaparametricequation"definesaninterestingclassofgeneratingfunctions,andcompareitwiththeclassofalgebraicfunctions.Finally,Iwillpresentsomemethodsfortheanalysisofthesegeneratingfunctions,andgivetwoexamplesofapplication.

Abstract：Inthistalkweconsidermass-criticalnonlinearSchr?dingerequation(NLS)withaharmonicpotentialandanangularmomentum.WeshowthatthesharpthresholdforfinitetimeblowupisthemassofthegroundstateforthefreeNLS,andthe"log-log"blowuprateunderthespectralassumptionwithinitialdataslightlylargerthanthegroundstate.

摘要：Amatroidisapair(E,I)whereEisafinitesetandIisacollectionofsubsetsofE,calledindependentsets,suchthat1)?∈I;2)ifI∈I,theneverysubsetofIalsobelongstoI;and3)ifI,J∈Isuchthat|I|<|J|,thenthereexistsx∈JsuchthatI∪{x}∈I.Itisclearthatthesetofcolumnsinamatrixformamatroid,whereasubsetofcolumnsisindependentiffthecolumnsarelinearlyindependent.AmatroidisbinaryifitcomesfromamatrixoverGF(2).Inthistalk,wewillshowhowtousebipartitegraphs,calledfundamentalgraphs,torepresentbinarymatroidsandtostudytheclassofbinarymatroidswithoutcertainminors.

Abstract：Inthistalk,weshowthatanyasymptoticallyflatmanifoldwithcontrolledholonomyadmitsarefinedtorusfibrationoveranALEmanifold.Inaddition,weprovethatanycompleteasymptoticallyflatRicci-flatmetricon$mathbbR^4$mustbeisometrictotheEuclideanortheTaub-NUTmetric,providedthatthetangentconeatinfinityisnot$mathbbRtimesmathbbR_+$.ThisisjointworkwithXiuxiongChen.