Abstract：ForaboundedconvexdomainonaRiemannianmanifold,thefundamentalgapisthedifferenceofthefirsttwonon-trivialDirichleteigenvalues.Intheircelebratedwork,B.AndrewsandJ.ClutterbuckprovedthefundamentalgapconjectureforconvexdomainsintheEuclideanspace,showingthatthegapisatleastaslargeastheoneforaone-dimensionalmodel.Theyalsoconjecturedthatsimilarresultsholdforspaceswithconstantsectionalcurvature.Veryrecently,ontheunitsphere,Seto-Wang-Weiprovedthatthefundamentalgapisgreaterthanthegapoftheonedimensionalspheremodel,inparticular,≥3π^2/D^2(n≥3),providedthediameterofthedomainD≤π/2.InajointworkwithGuofangWeiatUCSB,weextendSeto-Wang-Wei’slowerboundestimatetoallconvexdomainsinthehemisphere.

研究生短期课程【戴先哲、韦国芳】课程名：OneigenvaluescomparisoninRiemanniangeometry授课人：美国加州大学圣巴巴拉分校戴先哲教授、韦国芳教授授课地点：东区管理科研楼数学科学学院1518室授课时间：7月4日、5日、6日上午10:00--11:30

Abstract:InthistalkIwilldiscusssomenonlinearproblemsinelectromagneticfields.Thefocuswillbeontheproblemsrelatedtothemicrowaveheatingandinductiveheatinginindustries.ThemathematicalmodelsconsistofMaxwell’sequationscoupledwithanonlinearheatequation.Somerecentresultsonexistenceandregularityofweaksolutionwillbediscussedinmytalk.

Mini-Workshop:Liegroupactionsonsymplecticmaniolds会议时间：2017年7月1日-2日会议地点：数学学院1518会议日程：7月1日:9:00-9:45张斌（四川大学）环簇的Todd类的重整化研究10:15-11:00贺琛（NortheasternUniversity）GKMdescriptionsoftheequivariantcohomologyringsofhomogeneousspaces11:05-11:50陈小杨（同济大学）Symplecticaspectsofpolaractions2:00-2:45李慧（苏州大学）Thefundamentalgroupofsymplecticandcontacttoricmanifolds2:50-3:35周久儒（扬州大学）BasiccohomologygroupdecompositionofK-contactmanifolds4:05-4:50杨向东（重庆大学）EquivariantformalityofHamiltoniantransverselysymplecticfoliations7月2日:9:00-9:45洪伟（武汉大学）PoissoncohomologyofholomorphictoricPoissonmanifolds10:15-11:00林毅（GeorgiaSouthernUniversity）Convexitypropertiesofmomentmapsintransverselysymplecticgeometry

摘要：Givenasmoothcompactcomplexmanifold,weconsidertheproblemonexistence,inafixedconformalclass,ofHermitianmetricswhosescalarcurvatureinducedbytheChernconnectionisconstant.Inparticular,wedescribetheinterestingroleinthepictureplayedbytheGauduchonmetrics(jointworkwithDanieleAngellaandCristianoSpotti).

摘要:TheKobayashipseudometriconacomplexmanifoldMisthemaximalpseudometricsuchthatanyholomorphicmapfromthePoincaredisktoMisdistance-decreasing.KobayashiconjecturedthatthispseudometricvanishesonCalabi-Yaumanifolds,andinparticular,Calabi-YaumanifoldsareKobayashinon-hyperbolic.Usingergodicityofcomplexstructures,togetherwithS.LuandM.VerbitskyweprovethisconjectureforallK3surfacesandformostclassesofhyperkaehlermanifolds.InthetalkIwillalsogivethealgebraicversionofhyperbolicity.TogetherwithM.VerbitskyweprovethatprojectivehyperkaehlermanifoldswithPicardrankatleasttwoarealgebraicallynon-hyperbolic.

摘要:TheNorthAmericanPutnamCompetitionisconsideredtobeoneofthemostprestigiousuniversity-levelmathcompetitionintheworld.Inthislecturewe\'llsolvesomechallengingPutnam-styleproblemsinnumbertheory,geometry,differentialequations,probability,etc.报告人简介：LjudmilaKamenovareceivedherPh.D.atMITin2006,afterwhichshewasapostdoctoralmemberoftheIAS,Princeton.NowsheisaResearchAssistantProfessoratStonyBrookUniversity.Dr.KamenovatrainstheStonyBrookPutnammathteam.

摘要：Asalternativestopartialdifferentialequations(PDEs),nonlocalcontinuummodelsgiveninintegralformsavoidtheexplicituseofderivativesandallowsolutionstoexhibitdesiredsingularbehavior.Wepresentinthistalknonlocalmodelsofmechanicsanddiffusionprocessescharacterizedbyahorizonparameterwhichmeasurestherangeofnonlocalinteractions.ConsideringtheircloseconnectionstoclassicallocalPDEmodelsinthelimitwhenthehorizonparametershrinkstozeroandtoglobalfractionalPDEsinthelimitwhenthehorizonparametertendstoinfinity,wepresentnumericalschemesthatarerobustunderthechangesofthehorizonparameter.Wealsodiscussthecouplingofmodelscharacterizedbydifferentscalesofhorizon.

摘要：WrightStateUniversity,DaytonOhioUSAandHuaqiaoUniversity,QuanzhouFujianChinaAmatroidMisapair(E,I)whereEisafiniteset,calledthegroundsetofM,andIisanon-emptycollectionofsubsetsofE,calledindependentsetsofM,suchthat(1)asubsetofanindependentsetisindependent;and(2)ifIandJareindependentsetswith|I|<|J|,thenexistsx∈JIsuchthatI∪{x}isindependent.AgraphGgivesrisetoamatroidM(G)wherethegroundsetisE(G)andasubsetofE(G)isindependentifitspansaforest.AnotherexampleisamatroidthatcomesfromamatrixoverafieldF:thegroundsetEisthesetofallcolumnsandasubsetofEisindependentifitislinearlyindependentoverF.Tutte’sWheelandWhirlTheoremandSeymour’sSplitterTheoremaretwowell-knowninductivetoolsforprovingresultsfor3-connectedgraphsandmatroids.Inthistalk,wewillgiveasurveyoninductiontheoremsforvariousversionsofmatroid4-connectivity.

摘要:Einsteinmetricsonacompactmanifoldarecriticalpointsofthenormalizedtotalscalarcurvaturefunctional.SoitisnaturaltostudythebehaviorofthesecondvariationofthenormalizedtotalscalarcurvaturefunctionalatanEinsteinmetric.ThisisknownasthelinearstabilityproblemofEinsteinmetrics.Riemannianmanifoldswithnon-zeroKillingspinorsareEinstein.Weprovethatcompletemanifoldswithnon-zeroimaginaryKillingspinorsarestablebyusingaBochnertypeformula,whichwasprovedbyMcKenzieWangandthenwasrediscoveredbyXianzheDai,XiaodongWang,andGuofangWei.ThisstabilityresulthasalreadybeenprovedbyKlausKronckeinadifferentway.Moreover,existenceofrealKillingspinorsiscloselyrelatedtotheSasaki-Einsteinstructure.AregularSasaki-EinsteinmanifoldisessentiallythetotalspaceofacertainprincipalcirclebundleoveraKahler-Einsteinmanifold.WeprovethatifthebasespaceisaproductoftwoKahler-EinsteinmanifoldsthentheregularSasaki-Einsteinmanifoldisunstable.ThisprovidesusmanynewexamplesofunstablemanifoldswithrealKillingspinors.Moregenerally,weprovethatEinsteinmetricsonprincipaltorusbundlesconstructedbyMcKenzieWangandWolfgangZillerareunstableifthebaseisaproductofKahlermanifolds.