摘要：Thistalkfocusesonexploringtheglobalstabilityofnonlinearstochasticfeedbacksystemsonthenonnegativeorthantdrivenbymultiplicativewhitenoiseandpresentingacoupleofsmall-gainresults.Weinvestigatethedynamicalbehaviorofpull-backtrajectoriesforstochasticcontrolsystemsandprovethatthereexistsauniquegloballyattractingpositiverandomequilibriumforthosesystemswhoseoutputfunctionseitherpossessboundedderivativesorareuniformlyboundedawayfromzero.Ourresultscanbeappliedtowell-knownstochasticGoodwinnegativefeedbacksystem,Othmer-TysonpositivefeedbacksystemandGriffithpositivefeedbacksystemaswellasotherstochasticcooperative,competitiveandpredator-preysystems.ThisisajointworkwithDr.LvXiang.

摘要：通过两类具体方程，浅谈有关随机微分方程和随机偏微分方程渐进稳定性的研究感想。

Abstract：Inthistalk,weuse2-Dharmonicmapasanexampletoillustratethebasictheoryofneckanalysis.Theproofswillbealmostself-contained.WealsocomparewiththeYang-Millstheoryandoutlinesomegeneralizationoftheargument.

摘要:Idiscusssomerecentresultsonboundednessandcompactnessof(generalized)ToeplitzoperatorsactingonBergmanspacesoftheunitdiskandonFockspacesofthecomplexplane,andstatesomerelatedopenproblems;inparticular,thereisstillnocompletecharacterizationofboundedToeplitzoperatorsonBergmanspaces.WhilespectraltheoryofToeplitzoperatorsonHardyspacesiswellunderstood,muchlessisknownabouttheminBergmanandFockspaces.Inparticular,thereareveryfewresultsonFredholmnessofToeplitzoperatorsonFockspaces.IdiscusssomerecentgeneralizationsoftheresultsintheHilbert-Fockspacetotheother(generalized)Fockspaces,andlistrelatedopenproblems.

摘要：Inthenumericalsolutionofpartialdifferentialequations,itisfrequentlynecessarytoensurethatcertainvariablesremainpositive;otherwiseunphysicalsolutionswillbeobtainedthatmightresultinthefailureofthenumeralalgorithm.Positivityofcertainvariablesisgenerallyensuredusingpositivitypreservinglimiters,whichlocallymodifythesolutiontoensurethattheconstraintsaresatisfied.Thecombinationof(positivitypreserving)limitersandimplicittimeintegrationmethodsresults,however,inseriousproblems.Manylimitershaveacomplicated,non-smoothformulationthatisdifficulttolinearize,whichseriouslyhamperstheuseofstandardNewtonmethodstosolvethenonlinearalgebraicequationsoftheimplicittimediscretization.Inthispresentation,wewilldiscussadifferentapproachtoensurethatthenumericalsolutionsatisfiesthepositivityconstraints.Insteadofusingalimiter,weimposethepositivityconstraintsdirectlyonthealgebraicequationsresultingfromadiscontinuousGalerkinmethodbyreformulatingtheDGequationswithconstraintsusingtechniquesfrommathematicaloptimizationtheory.Theresultingalgebraicequationsarethensolvedusingasemi-smoothNewtonmethodthatiswellsuitedtodealwiththeresultingnonlinearcomplementarityproblem.ThisapproachallowsthedirectimpositionofconstraintsinimplicitdiscontinuousGalerkindiscretizations,withouttheconstructionofcomplicatedlimiters,andresultsinmoreefficientsolversfortheimplicitdiscretization.

摘要:Oneofthecentraltopicsinhypercomplexfunctiontheoryisinterpolation.InterpolationbyinnersphericalmonogenicsprovidestheadequatecounterparttotheclassicfiniteFouriertransformandiswidelyusedinapplications.Inmoregeneralterms,interpolationinhypercomplexfunctiontheoryiscloselylinkedtointerpolationinreproducingkernelHilbertspaces.Butinterestinglyenough,indifferencetothecomplexcasetherearemanyapplicationsofhypercomplexfunctiontheoryinwhichHilbertspacesarenotthenaturalsetting.Forinstance,studyingnull-solutionsofultra-hyperbolicDiracoperators,polefiguresofthecrystallographicRadontransformwhichbelongtothekernelofanultra-hyperbolicLaplacian,orevenmoregeneralproblemsinMinkowskispace-timeandstringtheoryrequireinnerproductspaceswithanindefiniteinnerproduct.AmongotherthingsthisisduetothepossibilityoftheunderlyingCliffordalgebratohaveasignature$(p,q)$and,therefore,tobelinkedtoPontryaginmodulesinsteadofHilbertmodules.AsweshallseethenaturalsettingforsuchproblemsistheClifford-Kreinmodule.InthistalkwegiveageneraloverviewoverClifford-Kreinmoduleswithreproducingkernelsandwearegoingtodiscussinterpolationinthesemodules.

摘要：Generalisedpolynomialsareexpressionsformedfromordinarypolynomialsusingtheintegerpartoperation,additionandmultiplication.Theirdistributionhasbeenextensivelystudied,andthankstotheworkofBergelsonandLeibmanisknowntobeintimatelyconnectedtodynamicsonnilmanifolds.Automaticsequencesarethosewhosen-thtermcanbecomputedbyafinitemachinegiventhedigitsofnoninput.ThestartingpointfortheprojectwhoseresultsIwilldiscussisthefollowingsimple-lookingquestion:Arethereanynon-trivialexamplesofautomaticsequencesgivenbygeneralisedpolynomialformulas?Iwillpresentapartialsolutiontothis,aswellasexamplesofsequenceswhichsomewhatsurprisinglyturnouttobegeneralisedpolynomials.

课程简介：Inthelectures,IwillintroducevarioustopicsrelatedtothePoincare-Einsteinmanifolds,includingexistenceanduniquenesstheorems,topologyandrigiditytheorems,spectrumandresolventfortheLaplacian,scatteringtheoryandGJMSoperators,aswellasfractionalYamabeproblems.

摘要：Afewpreconditionedsteepestdescent(PSD)solversarepresentedforthecertainoptimizationproblems,inwhichthesolutioncorrespondstoaconvexenergyfunctional.Thehighestandlowestordertermsareconstant-coefficient,positivelinearoperators.Byusingtheenergydissipationproperty,wederiveadiscreteboundforthesolution,aswellasanupper-boundforthesecondderivativeoftheenergy.Theseboundsallowustoinvestigatetheconvergencepropertiesofourmethod.Inparticular,ageometricconvergencerateisshownforthenonlinearPSDiterationappliedtotheregularizedequation,whichprovidesamuchsharpertheoreticalresultovertheexistingworks.Somenumericalsimulationresultsarealsopresentedinthetalk.