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.

Abstract：Inthisintroductoryreport,wewillconcerntheMcShaneidentityanditsvariousgeneralizationsandapplications.WewillpayspecialattentiontoMirzakhani’scelebratedrecursiveformulafortheWeil-Peterssonvolumeofthemodulespaceofcurveswithmarkedpoints.

摘要：Let$kge1$beanintegerand$mu$bethehalfofa{itnonzero}integer.Thefollowingstatementsholdfortheellipticco-adjointorbitoftherealLiealgebra$mathfrak{so}(2,2k+2)$thatcorrespondstothedominantweight$(underbrace{|mu|,ldots,|mu|}_{k+1},mu)$.1.Thisorbitisdiffeomorphicto$mathrm{SO}_0(2,2k+2)/mathrm{U}(1,k+1)$.Asaresult,itispre-quantizable.2.Thisorbitisthetotalspaceofafiberbundlewithbasespacebeingthetotalcotangentspaceofthepuncturedeuclideanspaceofdimension$2k+1$andthefiberbeingdiffeomorphicto$mathrm{SO}(2n)/mathrm{U}(n)$.Asaresult,itadmitsacanonicalpolarization.3.ThegeometricquantizationofthisorbitwithitscanonicalpolarizationyieldstheHilbertofsquareintegrablesectionsofaHermitianvectorbundleoverthepuncturedEuclideanspaceindimension$2k+1$;moreover,thisHilbertspaceprovidesageometricrealizationfortheunitaryhighestweight$frak{so}(2,2k+2)$-modulewithhighestweight[(-k-|mu|,underbrace{|mu|,ldots,|mu|}_k,mu).]TheaboveresultsinLietheoryisobtainedfromthestudyofmagnetizedKeplermodelsindimension$2k+1$.

美国伊利诺伊大学厄巴纳-香槟分校李一川博士学术报告会题目:ADistributedSecondOrderMethodforConsensusOptimization时间:2020年1月6日15:30地点:西区电三楼632学术报告厅

摘要：Thispaperpresentsageneralframeworkforhigh-dimensionalnonlinearvariableselectionusingdeepneuralnetworksundertheframeworkofsupervisedlearning.Thenetworkarchitectureincludesbothaselectionlayerandapproximationlayers.Theproblemcanbecastasasparsity-constrainedoptimizationwithasparseparameterintheselectionlayerandotherparametersintheapproximationlayers.Thisproblemischallengingduetothesparseconstraintandthenonconvexoptimization.Weproposeanovelalgorithm,calledDeepFeatureSelection,toestimateboththesparseparameterandtheotherparameters.Theoretically,weestablishthealgorithmconvergenceandtheselectionconsistencywhentheobjectivefunctionhasaGeneralizedStableRestrictedHessian.Thisresultprovidestheoreticaljustificationsofourmethodandgeneralizesknownresultsforhigh-dimensionallinearvariableselection.Simulationsandrealdataanalysisareconductedtodemonstratethesuperiorperformanceofourmethod.

课程简介：K-stabilityisaconceptfromdifferentialgeometry,whichisoriginallydefinedbyTiantodeterminetheexistenceofKE-metriconagivenFanomanifold.Recently,algebraicmethodshavebeenintroducedintothisfieldtoadvancesitfastly.Inthisseriesoftalks,IwillindetailpresentthepictureofthesocalledalgebraicK-stability.

Abstract：WewilltalkaboutourrecentworkontheproofofthegluingformulaofBismut-Lottanalytictorsionforms.ThegluingformulawillplayanimportantroleinestablishingahigherversionofCheeger-Mllertheorem.ThisisajointworkwithMartinPucholandYepingZhang.

摘要：Intheyear1970,Kobayashiintroducedanintrinsiccanonicalmetricforeverycomplexmanifold$X$,andwhenthisobtainedmetriciseverywherenon-degenerated,$X$issaidtobecomplexhyperbolic.Heconjecturedthat,inalldimensions,mostmanifoldsshouldbecomplexhyperbolic.Inthistalkweshallpresentthehistoryandsomerecentdevelopmentsinthissubject,includingtheGreen-Griffiths-Langconjecture.

摘要:Iwillsurveysomerecentprogressondifferentialcalculusofquiverswithpotentialsandgivesomeapplicationsofittoalgebraicgeometryandrepresentationtheory.

摘要:OnebasicphilosophyinDiophantineGeometryislocal-globalprinciple(Hasseprinciple)forrationalpointsandintegralpoints.Inthistalk,Iwillfocusonhowtouselinearalgebraicgrouptostudythelocal-globalprincipleforintegralpointsandtalkabout:(1)thedescentobstructionanditerateddescenteobstruction;(2)invariantBrauersubgroupandstrongapproximationofhomogeneousspaces;(3)arithmeticpurityofstrongapproximationforhomogeneousspaces.