どうもありがとうございました。実際にここで話をしています。オーガナイザーのためにとても良い話をしています。とても高い話をしています。私は quite sure that I can give a talk with such a high standard.So, I'm hesitated but finally I decided to talk here.Anyway, I would like to thank the organizers to make this conferenceand to speakers that give so good talks.Ok, so today I'm going to speak with my favorite correspondence,Andrea Plastransforms and the artist part with joint work with Andrea Daniello.And so I think many of them are rather now oldbut I think I will try to explain here someit reminds me of correspondence in irregular caseand perhaps I change a little bit the presentation.Ok, so we consider a complex mind-fordand we consider the category of regular holonomic demodules.So that is the derived category of DX-modules with regular holonomic comodules.And so it is known that I've heard a long time agothat that is equivalent to the deconstructible shifts.So it means that is a category of vector spices on Xwith constructive comodules.Cconstructible means say there is a filtration of closed complex analyticssub-sets such that the chief is locally constant of finite rank on the differences.And of course that is contained.So there is another notion, all constructibles.Persuant all constructible themes are less familiarbut more or less that is the same thing.Here that is we take the C-analytic space, complex analytics spacebut here we take the so-called sub-analytic space, sub-analytic set.But I think so it's something like semi-analytic setor semi-analytic set.So I think I will not go into the details.And of course that is context.And there is a function called drum functionand it gives the equivalence.So drum function is m to omega x tensor m.So that is a drum complex.So that is the C4 with the highest weight form.So that is a right model.So that is more or less the regular case.So what happens in regular case?So this one is contained.So that is a DX module with the holonomic comodules.So that is it contains regular holonomic onesas a full subcategory.So now we don't take this onebut we take X times R.So we add one variable.And so here so it containsR constructives on X cross R.And so we don't take that.We change a little bit.So how we change it?So this is in fact monoidal category.So DB or in fact you don't need constructives.But that is a monoidal category.Commutative monoidal category with respect to the tensor productthat I denote plus.So that is by definition P1 inverse L to inverse GSo here X cross you consider R and P1 P2.So that is a projection to the first componentand the second component.And mu is the addition map T1 T2 goes to T1 T2.So that is a kind of a combinationwith respect to the second variable.So with this this is a commutative monoidal category.And unit object is just C X cross 0.So that is the unit object.So and now we considerI think there are many interpretationbut I will not go to the details.We consider this one.So the positive line.So this is a kind of item importantwith respect to this tensor product.So that is C X cross 0.Ok.So that is a setting.And so we don't consider thisbut we consider some kind of quotient.It's not exactly quotient.Q and I denote X.C X.So what is this?So that is object isF R such thatF tensor C T positiveis isomxtable.Ok.So that is the object.Andisom is a little bit different.In fact it appears in the talk ofGiomo in fact.But so C X.Is the isomorphism part of the data?You just really just pick an object now.So there is a canonical mapC T equals 0.And that is a unit object.So that is a canonical isomorphism.And the statement is that is isomorphism.Ok.So that is a property,not a structure.Ok.SoHome FF primeis home limit,inductive limit.She goes to infinity.Home F.So that is a in factbecause of this onethat is a new C orStar F primewhere new C istranslation by C.Anyway,so this is aninteractive systemand you take the whole.So that is a homomorphism.And so that is a category.And of course Q of Fis the one in this herewhich is just C T plusthen of course it satisfies this conditionbecause that is very important.So it's a kind of quotient.And so the theorem isit isthere is some canonical embedding.So that we denotethe enhanced drum.So we call that is enhanced.Is it fully faithful?Yes.So that is fully faithful.Ok.So what is this?I can't explain details.That is more or less.So you add one variableand you consider Mand you consider DI think minus T.That is drum and solutions are different.I think that iscorrect.And so we don't take thisbut we take some so calledtempered systems.Tempered functions.But I think that I'm not going tothat details orbut so this functiongives a function from here to here.So that is the one.Ok.So for exampleF isCR.So when X is a pointand FCR.So this oneif we consider thatthen any objectis a direct sum.So the structure is very simple.CRATB.And shifted by some number.And A is a real numberand B is bigger than that.And B can be plus infinity.So that is more or less half line.So that's that.EPointCPoint orHoward University.CPoint.So that isin fact any elementisa direct sumofCN.So there is only one objectand the translation.So it is exactlysame asvector space.The derived category ofC vector space of finite dimension.So that is very small categoryin that sense.But in general it is not.So that is the solution.And sothe problem isso you have the functorfromfull face functorfromshould bethe long functorfrom here to here.Sothe problem iswhat is the imageof this one.And one answeris given in factTakuroand I thinkI willexplain it.And before I explain itI would like todo something.Sothere is a mapX,Y,F.Okay.And I thinkwe are asking for some suitable conditions.But I thinkI am a little bit teaching.For example,it isconstructible here.It isconstructibleso that isconstructiblelocal constantonsubalytic setX times alphabut it includingat infinity of R.Sothere is some problem of infinity.SoI will skip that partand so that issomewhat irrelevantin this talk.So thenyou haveusual operationto direct imageand toinverse image.Sothat is rather easy.So youconsider X cross Rto X cross Rand for exampleEFQFisQF tildeFSTARQFisQofRFFand soand similarlythere isD-module sidethere is awe need some conditionsbut I thinkto talk about thatthere is a corresponding operationsoI like DDFtopush forwardand toinverse imageand sothe proposition istheycommute each otherthey are compatible withenhancedDM FunctorSoyou can calculatesothe DM Functoris one.So that is one partand I'll make another remarkyeahsosolet us takeOX star Ywhere Y is a closed hypersurfaceandHolmec Functionswould pose in Yand then you can considerDEFysoin terms of connectionsthat corresponds toOX star Yto D plus DFand so you considerenhanced one.So that can be calculatedyeahin some sensesoQlet me seeDEFyandperhapsit's not exactly correctSo EJ star EJ minusanywaySo J isYXandsoD means QSoin this case you can calculatevery explicitlybutthis one is rather hard tocalculate but anywayso you can calculatesoand another remark isfor the curve caseso nowtypical caseyou take the discandyou can considerHolmecDEFyMand withsinality atonly at the originthenand you can considerenhanced drum ofMso that iswell knownconditionso it satisfiesfollowing propertyso that isI think it already appears several timessoyou take anydirectionthen you can findsmalltube orsectorialneighbourhood Usuch thatpi isX cross R to Xone deltasoyou restrict tothis onethen that isin a particular formand fjis aprice seriesor Uand that isprice seriesI don't knowhow to sayZ1 overMstate inverseokso you can write it in this wayso buthere of courseyour part fjis isomorphic tocefj plus gifg isboundedbecauseof the definition of the homebecause of thedefinition of the homeand so ifshe is bigthenit partkills this partso in factwe can consider fjbut it is modularbounded functionanywayyeahso we sayI thinkMaxim says somethingbut I can't remembersowe write mj ismultiply stateoverMwith respect to fjso that isverifiedokso nowI can statethefilmdue toTakuroso image ofdrum consists ofk such thatfor anyf delta to xyou consideref inversek and you considercommodigrouphas the formthis formthis formso in this casewe say that k isshe constructablethensoin that terminology that isshe constructableso it meansyou cut by a curveand if for any curveif that is a normal formthen that is an imageso that is a characterizationsocurve testso that is trueso nowthe problem hereso that is a rathergood characterizationbut I think it's notstill not very satisfactoryin the sense thatfor exampleso you considerthose categoriesso those categoriesI meanthe category of ksatisfying those conditionsand that can be definedforany base fieldso herewe takecomplex number fieldas the base fieldbut you can take another oneor z orthosebase ringthen still you candefine itand the problem isfor exampleso you candefine those categoriesand then the problem isif that ispush forwardpull backpush forwardI don't know how to prove itso that isone pointso that is notwellfor meI don't understand wellandsoanotherprogramor not problemanother thing that I don't knowisfor exampleif you consider the demodulesthen it'scharacteristic varietyis very importantand usuallymany propertiesismicrolocal propertiesso it meansyou can consider myou cananalyze itfor examplehonomic demodulesto be regularhonomicso that is a microlocal propertyso you cancharacterize it microlocallybutI don't know this partyeahI didn't tell what isso that issomealgebraic partthat is a topological partandwhat is a functionthat isf goes topiI seethat is a functionandso by thisso for examplethis oneis foriffsothose partI don't stillunderstand that is afuture problemoksoso my titleis Laplace-Trump-Spoutand I think itare a little bitLaplace-TransformSo Laplace-Transform isyou have vector spaceand thedr of vector spacethendemodulesand saythat is algebraic demodulesso that is isomorphicso that is well knownandthat is comessaywso there is the correspondencethe well known correspondencez to wso by thisdremodulecorrespondenceorokso that can be interpretedin this wayso that is theso you considerdmoduleorsuggesterandso you considermandp1p2dp2oror so that isin terms ofdemodule sideand byenhanced drumyou can go to theenhanced drum sidethenisomorphic to star in a similarway so that isk goes toepep1inverse kisopanddremoduleby this correspondenceso that is an isomorphismso it corresponds tothis oneokthen we can askwhat happenswhat happens herefor exampleif you give demodulewhat you getand for example in the rightdimensional caselocal behavior is well knownthen how it changesby the transformand of course that isit already knownI think it already known bythe work of many peopleserverノースペンサーmalgrangebutI think I willit is not exactlyformulated in this wayI think I willgive a formulationit is somewhatit is microlocaldescriptionso I think it isworse to writeexplicitlyor I thinkit is already knownbut soso this kernelthis kernel leavessomewhata contacttransformof zctspace tosay wdata is aco-tangent orco-vector correspond to wsay t primeand let me seesow is minus zetaz equals cso it correspondto this correspondenceandt prime is t plus cso by this correspondencet t pluszeta dw isdt plusokso that is a well known correspondenceok sonow you take somepointin zeta spacelet me seelet me seez equals zetaw is minus cI am sorryI think it is correctanywayso you havesay those ucorrespond to u choose zetaand u thenthere is a piecefunction definedon uthat is given bythe piece seriesok sowe cross this dataand utransformby this correspondencesofirst you consider cfcf isthis oneatx3c is tfzandatx0w is f primeokand you saythis correspondencewe shall write lso you consider lcfthenin the general caseif f is boundedit is not butotherwise so that can be written cgwhere g isso there is somew0and some v and g isso uit has the same formand w0 is of courseI thinkminus or plusw is minus cso that is minuswe needsobyusing this oneandby thisphlogicalisomorphismwe can prove thefollowing thingfor exampleso fthere isso some constanttimesz-z0power same minus lambdaplus somehigher order termbecause that ispcc seriesdegree of fandso lambda must besolet me seeorder of fso sayif z is vand order of fis positivethen w0 isinfinityandz0 is infinityand lambda is positivethen w0 isinfinityandw0or equalsstill is notsotheycorrespondingso thenwe can prove thatso I definedmultiplistmultiplistof fso mismultiplistof gofthe transformI don't knowI don't knowthe same LL of mso I thinkso as I saidit is already knownI think bymany peopleI don't know who is the firstbut as I saidthatperhaps at the end ofsponsor block is the firstI don't knowyou know it sirclaw itmany peopleI don't knowmany peoplesothere are too many peopleinvolvedanyway so that isobtainedby thisconcelerationso herethe problem isso that isin fact Idelivered it microlocallymicrolocally is still notwell adaptedhere sobecausebecause f is notuniquely determinedso you canchoose another fthen the micro supportchanged in some senseand I don't know howto deal with itanyway soI hope thatthosethere are stillmany things thatwe don't understandon the topological partofreality modeland I hope we canI can continue toclarify those partsok thank you very muchthank you Masakithere are questionsyes aboutsoif you want to change the base fieldyesyou can change the base fieldat the level of eyeswhy isn't it sufficientto change the base fieldat this level to definethe q structurefor example q structureyesmaybeyou can go tocharacteristicsoI don't understandyour question correctsomething thatthe u theoremwas notenoughto defineto change the base fieldthe theorem of Takuroyeahsoif you changethen you can definethe category yeahso that part is easy butfor exampleI don't know if they arestable by those operationsand that wastensor productand that is important forthe applicationso I thinksothis partis a substitutefor theirregularwhat is the good wordirregular counterpartof the purpose shiftand so I thinkit isI think it is importantto knowwith otherbase fieldmy question isis it notenough to change the base fieldfor eof cxchange by e of some base fieldyeah yeah we can changeyou know thispart is oksofor examplethat is well definedso for any kyeah so the problemis this partso that one iswhat we wantand so saythat is ifwe do not seeconstructible k weeksso you change all kand thenso we can definethe category butwe don't knowthe invarianceby those operatorsI think forinverse image it is easymaybeare there other questionsno sothank you