摘要：Wewilltalkaboutverybasicnotionofpositivityoffunctionsandpositivedefinitenessofmatrices.Inparticular,wewilltrytoexplaintheclassicalHerglotz-BochnerTheoremanditsrecentvariantsarisingfromthestudyofsomeparticularstochasticprocessesontrees.

摘要：Inthistalk,werecallsomebasicpropertiesof$tau$-tiltingmodules,especiallythehomologicalpropertiesofself-orthogonal$tau$-tiltingmodules.Moreover,wegiveasufficientandnecessaryconditionforaself-orthogonal$tau$-tiltingmoduletobeaclassicaltiltingmodule.Weshowthata$tau$-tiltingmoduleoffiniteprojectivedimensionisaclassicaltiltingmoduleifandonlyifitisself-orthogonal.

摘要：Thewell-knownKPhierarchyisaninfinitesystemofnonlinearpartialdifferentialequations,whichdescribes,amongotherthings,theisospectraldeformationsoftheringsofcommutingordinarydifferentialoperators.Ingeometricterms,thesedeformationsaredescribedasflowsontheJacobiansofthespectralcurvesofsuchrings,whichcanalsoberegardedasrestrictionoftheflowsdefinedbythehierarchyontheSatoGrassmannian.Iwilltalkabouttheanalogueofthistheoryinthetwo-dimensionalcase.Inthiscase,ringsofcommutingdifferentialoperatorsoftwovariables,ormoregeneralringsofdifferential-differenceorpseudodifferentialoperators,andtheirisospectraldeformationsareconsidered.DeformationsaredescribedbyanaloguesoftheKPhierarchy—themodifiedParshinhierarchies,whichdefineflowsonthemodulispaceoftorsion-freesheavesofaspectralsurface.

摘要：Weconsiderarandombistochasticmatrixofsize$n$oftheform$MQ$where$M$isauniformlydistributedpermutationmatrixand$Q$isagivenbistochasticmatrix.Undermildsparsityandregularityassumptionson$Q$,weprovethatthesecondlargesteigenvalueof$MQ$isessentiallyboundedbythenormalizedHilbert-Schmidtnormof$Q$when$n$growslarge.Weapplythisresulttorandomwalksonrandomregulardigraphsandtoshuffle-and-foldmapsoftheunitintervalpopularizedinfluidmixingprotocols.

报告摘要：Inthistalk,Iamgoingtointroduceourrecenteffortsonunderstanding,designingandrigorouslyanalyzingpositivity-preservingnumericalmethodsforidealmagnetohydrodynamicequations.Importantrelationsbetweenthepositivity-preservingpropertyanddivergence-freemagneticfieldwillberevealedfromseveraldifferentviewpoints.

摘要:WeinvestigateabstractversionsofthemonomorphismcategoryasstudiedbyRingelandSchmidmeier.Weprovethatanaloguesofthekernelandcokernelfunctorsendalmostsplitsequencesoverthepathalgebraandthepreprojectivealgebratosplitoralmostsplitsequencesinthemonomorphismcategory.ThisisbasedonthejointworkwithJulianKuelshammer,ChrysostomosPsaroudakisandSondreKvamme.

Abstract：Wewilltalkaboutsomestochasticpartialdifferentialequations(SPDEs),whicharisenaturallyinthecontextofLiouvillequantumgravityandareproposedtopreservetheLiouvillemeasurewhichhasbeenconstructedrecentlyintheworkbyDavid-Kupiainen-Rhodes-Vargas.WeconstructglobalsolutionstotheseequationsandthenshowtheinvarianceoftheLiouvillemeasureundertheresultingdynamics.Asabyproduct,wealsoansweranopenproblemproposedbySun-Tzvetkov.

摘要：Letf_c(z)=z^2+cathefamilyofquadraticpolynomialsparameterizedbycomplexnumbersc.Itwasknownthatforanytwodistinctparametersc_1andc_2overthecomplexnumbers,thereareonlyfinitelymanypreperiodicpointsoff_{c_1}andf_{c_2}incommon.Inthistalk,wearegoingtoshowthatthereisauniformboundforthenumberofcommonpreperiodicpointsasc_1andc_2vary.Moreover,Iwillprovidesomemotivationsfortheresultofuniformbound.ThisisajointworkwithLauraDeMarcoandHollyKrieger.

Abstract：InthistalkIshallpresentrecentspectralboundsonthek-independencenumberofagraph,whichisthemaximumsizeofasetofverticesatpairwisedistancegreaterthank.Thepreviousboundsknownintheliteraturefollowasacorollaryofourmainresults.Weshowthatformostcasesourboundsoutperformthepreviousknownbounds.Someinfinitefamiliesofgraphswheretheboundsaretightarealsopresented.ThisisjointworkwithG.CoutinhoandM.A.Fiol.