 Я не думаю, что я никогда не получил в моей жизни такие спекулятивные слова. Так что... В принципе, я хотел, когда я... Я announced this title. Я надеюсь, что у меня больше мечты. Но как-то как-то я не был достаточно... Слышен, чтобы... Так... Что? Ок. Да, но... Но тоже по-русски на нитмей. На нитмей. Ок, в принципе... Двадцать лет назад у меня была конференция. И я был говорим о том, что есть какая-то разработка, и это был флэксический сайт о симплектикной стополижеции. И я, как-то, выгадывал в следующем оптимистическом принципе, что в симплектикной стополижеции просто... Без холоморфа есть ничего, но есть х-пинцип. Так как... Как-то что-то не как-то... не конкурсировано в стополижеции. Это, в принципе, может быть конкурсировано и... и существует. И я как-то... Мой talk today is in a little bit in the similar spirit, so when I was talking then, then Foucaille was in the audience and he asked me questions, kind of, but can you just make some precise statement? What does that mean? Can you say a few more words in this direction? It's last year. So I have some kind of justification that I didn't have too many nights to dream of. Some... So again, kind of like my... general kind of direction is the following, that there is a... Apology, there is this methods coming from the theory of holomorphic curve or in its different incarnations. And there are some kind of results and kind of many wonderful results were proven in this direction, especially this holomorphic curve series powerful in dimension 4. For instance, in dimension 4 Gromov proved that there is a unique symplectic structure on R4 and kind of essentially if you have a symplectic manifold which is a standard at infinity look like R4 to infinity then with addition of to the MacDuff the result is that it's in fact either R4 up to maybe blow-ups. So in high dimensions kind of this holomorphic curve the same ideas they give only topological result kind of so there is also result by Floor MacDuff myself that we said in high dimension if you have a... using something just differential topology you can prove that if the manifold if standard like R2 and at infinity then it is standard provided topologically provided for instance there is no holomorphic curve in this manifold but then this is a kind of completely different type of result so also like in dimension in dimension 4 in the similar way there is a result we proved with Polterovich that if you have a Lagrangian some manifold in R4 which is at infinity looks like R2 then this Lagrangian some manifold is R2 up to Hamiltonian so in high dimension using all again powerful holomorphic curve methods we have a lot of results about constrainum for instance it's known that this manifold have to be R2N if it has this property but we don't know whether there is absolutely no I think hope using this holomorphic curve to prove that it's in fact Hamiltonian as a topic to this some kind of version of this so called nearby Lagrangian conjecture and my point is that probably because this is difficult because we kind of looking this problem in the long wrong domain so this is already part already belongs to H principle and then you need to prove it with H principle type statement and I want to kind of be talking something in this direction about so let me first review just few kind of recent development in the flexible size of synthetic topology so kind of two main flexible object which we know in synthetic topology and they are kind of start with contact world they are over twisted contact manifold and there is a notion of loosely rendering nodes due to MMRF so what is a over twisted contact manifold again it's maybe it's not so important but but for example one of this definition is the following so well maybe let me use the following definition so in dimension it's a mostly known kind of in dimension 3 and then in dimension 3 it just says that there is a certain disc which is a tangent in the context 3 manifold which is a tangent along the boundary to contact structure and in high dimension you can say that kind of one of the equivalent definition which is in fact due to presses Murphy and Casals which says that manifold is over twisted if it contains product of such disc with arbitrary large contact ball for instance but it doesn't matter the important thing that it's the contact manifold if there is a certain particular contains certain particular virus so there is a some kind of embedding of something and the something can be very small and somehow this kind of implantation of this virus makes the whole thing very flexible so in the same sense there is a Legendre knot and this is a notion of loose Legendre knot is technically it's a notion so first we introduced this notion of a twisted contact manifold meaning there is a Legendre knot whose complement is over twisted but then turned down that there is a very similar property exist in high dimensional contact manifold and in any contact manifold of dimension greater than 3 and again doesn't matter what is a particular definition for instance there is a certain local local construction called stabilization which you can make with Legendre knot and you can define this Legendre knot as loose if it's either topic of stabilization of some other knots so again any Legendre knot can be made loose via some local local modification and again this is a similar similar situation that as soon as you have smallest kind of this loose virus then it makes the object completely flexible so so there is a on the first glance these are kind of two different object but they are tightly related so this loose Legendre knot and over twisted contact manifold are tightly related object for instance I don't think there is a such theorem can at least the theorem exist and kind of proven probably half of the theorem is proven but I'm sure the second half is also true which says the following thing so given given any manifold contact manifold then with this contact manifold you can make it's kind of like given symplectic manifold there is this kind of standard construction where taking product of the symplectic manifold with this manifold with opposite symplectic structure and the same you do with contact manifold you take contact manifold you simpletize it that mean you multiply by r so you get this symplectic manifold and then you take this symplectic manifold with kind of like inverted symplectic structure here and in this manifold you have this product and you have this diagonal and there is a diagonal action here you can quotient quotient by this r action here and then you get a contact manifold and the diagonal now after this quotient so you get new contact manifold and you get this kind of again like grandian diagonal remaining with Legendre boundary so you get you get our diagonal as a Legendre and some manifold in this product construction and the kind of theorem which should be true that this manifold is over twisted if and only if this diagonal is loose so there is a kind so I think in one direction it's pretty easy and this Roger I think proved it that it is if it's over twisted but in opposite direction I'm sure it's also should be true so also there are many other kind of connections between this looseness and over twisting for instance suppose you have a contact manifold and you have a Legendre and some manifold and then you can always have a construction have a given Legendre and some manifold you can attach attach Lagrangian handle to this so attach symplectic handle thinking about this our contact manifold is a boundary of this symplectic manifold of kind of part of its simpletization you can attach a handle to this and this handle by itself is just cotangent bundle disk of n-dimensional disk and so you can just attach and the boundary of the disk itself is Legendre and some manifold along the boundary and you attach it but also of course you can you have this disk and you have that disk and you can also kind of subtract a handle kind of like attach a handle to this manifold so you have also this construction and so so you get a kabordism between our original contact manifold and another contact manifold on negative side of this one and if the Legendre is not as loose then the result is over twisted so this is again kind of a simple pretty fact so as I said these two notions are tightly related and what I am kind of in fact in my talk I will be trying to explore this notion kind of parallel between kind of different version statement about Legendre not necessarily loose and contact manifold further so of course another kind of let me list few more kind of analogies between this loose Legendre situation and over twisted contact so so one common feature that homomorphic curve theory completely fails in both cases so if you if you have an over twisted contact manifold of dimension greater than 3 dimension 3 there is again special situation because of this embedded kind of another stuff kind of related to gauge theory but in high dimension if you try to define for instance this or any kind of invariant based on homomorphic curve you get nothing this algebra is identically equal to zero similarly there is a relative version of all this SFT type invariant for Legendre and some manifold for instance you can define this Legendre and homology algebra if it's loose it's identically zero there is absolutely no way of extracting any invariant out of homomorphic curve and hence this was a motivation for me and for Murphy in fact to prove flexibility that in fact there is nothing and you can prove the result so here you can prove in an over twisted case that any almost contact structure homotopic to genuine contact over twisted contact structure and to over twisted contact structure if they kind of formally homotopic they are homotopic and exactly the same result held in the Legendre category so another kind of result which we have in Legendre world is then when we are looking for a Legendre manifold with Legendre boundaries then this and Legendre boundary is on negative end so we consider the Simplactic manifold with say negative contact boundary so that means that there is a Simplactic manifold and there is some hyper surface d minus w you have a Lagrangian Lagrangian sub manifold with boundary on this single and this is negative boundary in the sense that our Simplactic form is exact near this boundary and it is dual corresponding Louisville vector field if you consider corresponding Louisville vector field and it enters our manifold it's a negative boundary then if this Legendre boundary is loose then it's very easy to construct Lagrangian manifold with Legendre boundary any formal Lagrangian class is realized by such Lagrangian manifold and again you have a complete H principle if the two such things formally they are Lagrangian Lagrangian isotopic Hamiltonian isotopic even you require boundary need to be allowed to move so I am slightly lying because this is true provided this is a like I think this is an infinite end and during this isotopic this Lagrangian may be movable maybe changing Lagrangian at some compact part but not at infinity so in other word when you take a Lagrangian and then you kind of spoiled this Lagrangian with one singular point again you kind of have this virus in this Lagrangian manifold which look like cone over this Legendre Legendre node which is loose then such object is completely flexible which is again no symplectic topology and again kind of like another feature of this if you try to define any homomorphic curve invariance for such thing you will fail because somehow they all disappear in this black hole so now let's look at this symplectic another part for this sorry just on this side so we have now kind of natural to assume that when we consider now symplectic manifold so we consider now symplectic manifold with contact boundary with negative contact boundary and if the boundary if this boundary is over twisted then this object should become completely flexible so like it should be easy to construct symplectic structure on manifold with this kind of one single or conical point and again I am talking about manifold of dimension at least 6 dimension 4 is very special so with a single or conical point and a single or conical point and such that near this point this our symplectic form look like simpletization look like simpletization of some over twisted contact structure on the boundary so indeed unfortunately so we proved with some result in this direction but unfortunately not quite this and I very much hope that this all also true in this form but let me formulate what did we prove so we prove the following results so suppose you have a we started with caborgi why isn't this diagonal trick just work again for this I mean embed this thing in the diagonal then here's how in this picture you just discussed what embed the diagonal so I mean same trick you used before then you should have this thing sitting in the diagonal but how how I get it back so from symplectic manifold I can get this Lagrangian but how from the Lagrangian I can get symplectic manifold I don't know so you are going to do something but maybe not the same shape as this W if you have a symplectic manifold then you get Lagrangian diagonal but if you have a kind of I construct so I don't know what anyway so we have a so suppose we started with some smooth caborgi so we started with smooth caborgi between two manifold D minus and D plus and well we want to make this caborgism into symplectic manifold and this contact boundary so we prescribe here some contact structure and prescribe here some contact structure and we want this to be negative contact boundary and this is positive contact boundary so we want to construct symplectic manifold which is near the boundary which looks like simplification kind of negative and positive simplification of corresponding contact boundary and so so we want to construct it but of course you need to to have almost complex structure in order to do that and the the theorem is a following that so this is a theorem of Murphy himself so in this situation again dimension dimension let's assume so dimension is at least at least 6 so then you can always find some Louisville form lambda so it's a Louisville form Louisville form on w such that so lambda on d-minus w is defined defines c-minus lambda d-minus defines c-minus on the boundary this again negative and positive boundary provided provided so this is Louisville form that means that d-lambda is symplectic compatible compatible with j provided that the following conditions that first of all this c-minus is over twisted this contact structure negative end is over twisted and unfortunately I need the condition that this boundary upper boundary is not empty so in this situation this thing with boundary which is non-empty always constructs symplectic caborism on such thing if this is over twisted so there is a using this result which I told you before that the surgery kind of inverse surgery on loosely gendered node is on loosely gendered node is over twisted you can easily to prove the following result that you have always in dimension in dimension greater than 4 you can always find concordance between any contact structure and over twisted contact structure so there is a if you take manifold and multiply manifold by interval then you can find on this manifold symplectic structure always in dimension beginning not 4 where this thing is any contact structure and this over twisted you can just kind of use this fact and this was observation of Kazar's Murphy and Presses and so therefore this statement you can replace it by just assuming that both boundary are over twisted because positive boundary you can replace any contact boundary instead using this trick so essentially the statement that you have a two over twisted contact structure and you have any topological cabardism between them almost complex you can make it symplectic and in high dimension this is true in all dimension including 4 but in high dimension this is in addition implies that contact structure on the positive boundary could be standard so in particular notice that what we get we get a symplectic structure you can take any manifold so take a suppose now say m to n is any closed manifold and then any closed manifold and say let's suppose it almost come it's almost complex almost complex in a complement of one point so then almost complex structure in the complement of one point and then you can first delete some small ball of this and make this into some kind of cabardism which is a look like standard r to n standard contact structure in infinity and you're asking is it possible to construct symplectic structure on this manifold so you have this manifold minus point from the manifold minus point constructs symplectic structure which is standard at infinity so the theory in which I mentioned just at the beginning of of Lord Magdalf myself that this manifold topology if it's really symplectic structure topology is fantastically restricted essentially it just have to be r to n but if I allow one singular point and there's a singular point will be like cone over cone over twisted contact structure on a sphere then you can do it on any symplectic manifold so in particular it answer this question which I said in following ways you take any manifold which you want to make a symplectic and then if you connected some with already another manifold which is symplectic then this manifold always admit a symplectic structure with one conical point okay but also as I hope that in this situation much stronger facility result holds first of all all this completely should be unnecessary and moreover you should have a kind of even parametric version of this just symplectic topology of this manifold kind of non existent so let me kind of now move to kind of so this was a still not dream it was result except this was like small piece of dream so so let me kind of now move to more to dream path again let's kind of try to push this analogy between this Lagrangian relative and closed situation further so in so I was talking about this kind of symplectic caps and Lagrangian caps think this negative boundary but of course we much more interested we think this positive boundary so suppose you have a like Weinstein manifold so Weinstein manifold so you thinking about this as a kind of well we can think about it as some symplectic handled body so we started with a ball and we started to attach handle along this first isotropic manifold and we get some kind of manifold which is what is called subcritical and then you start to attach handle along Lagrangian submanifold and you get this Weinstein handled body so you get this manifold so you get a manifold with contact boundary so this is kind of important class of this manifold of course all Stein manifold like this of course all final algebraic manifold so it's important class class of symplectic manifold and we would like to understand their symplectic topology so because known since long time ago that this subcritical Weinstein manifold are not so interesting in the sense that if there is no handle of middle index then this manifold is a symplectic topology is determined by its pure topology and again it's manifested in some kind of homomorphic statement that the symplectic homology vanish so this was generalized by in our book with some class which we called flexible Weinstein manifold so flexible Weinstein manifold means it's object again of dimension greater than 4 it's always starting from 6 and flexible Weinstein manifold this is the handle body where all handles attached along loose-ligendron knot or kind of more precisely to say if you kind of attach them simultaneously they have to be attached along what is called loose-ligendron link that means that every attaching ligendron sphere have to be loose as a complement of the rest so so this loose Weinstein manifold Weinstein handle body again have vanishing symplectic homology and again we prove that there satisfy kind of complete H principle in a sense that any any manifold which potentially can be Weinstein manifold admit this loose flexible Weinstein structure and any to such structure are homotopic provided that they are formally in the same class so the question so there is a so on first glance this is a kind of again non-interesting object flexible Weinstein manifold but not quite because boundary of this flexible Weinstein manifold still have a very non-trivial context structure for instance the theorem of Alec Lazariff says that if you have such manifold that context structure if you know that you have a contact manifold and it's contact manifold bound this flexible Weinstein manifold then this boundary then the contact boundary contact structure remember for instance homology rank of homology of this manifold and in fact we have some kind of work in progress with Lazariff and Ganatra kind of saying that in fact much more subtle topology remembered by by contact structure so it's not completely obvious kind of in the first glance of course it's straightforward because a simplect homology trivial then what is called linearized contact homology have to be isomorphic to to or what is called sh plus simplect homology which determined by the boundary have to be equivalent to homology of the manifold but this thing depends definition depends on limitation which depends on the feeling so the fact that it's independent of feeling is not trivial fact so anyway so as I saying that still boundary is kind of interesting of this object or contactly so also things here turns out to be much more subtle than kind of what we thought with chili bug at the beginning so there is a recent work of Kyler Siegel and Amy Murphy which says that answer the following question which we ask where the flexibility is invariant under Weinstein homotopy so suppose you take a Weinstein structure and deform this Weinstein structure in the class of Weinstein structure and the question is it remain flexible and critical question is the following so suppose you have some Weinstein structure on flexible manifold and you have some Weinstein sub level set is it flexible for instance it also have to have trivial homology but turns out there is example when it's not flexible so the sub level set of Weinstein function on flexible manifold for kind of wrong Weinstein function can be not flexible so things are pretty kind of subtle here ok but let's let's move to let's move to to these dreams so what is the analog of flexible Weinstein manifold in the Lagrangian-Ligendrian category so suppose you have now this Weinstein manifold and you have a Ligendrian sub manifold on the boundary and now we have a Lagrangian Lagrangian sub manifold we have a Lagrangian we have a Lagrangian we have a Lagrangian sub manifold with this Ligendrian boundary so I'm talking that Weinstein manifold with contact boundary and we have a Lagrangian sub manifold so let me just take two words for you that there is a in the class of Weinstein manifold there exist this class what is called flexible Weinstein manifold whereas you have a kind of handle body construction where top dimensional index and handle are touch along loosely Ligendrian nodes and this object have this kind of like symplectic topology disappear on this one and so my question kind of like all this theme of my talk is kind of to try to have this analog of of this result analogous result in relative situation and the closed situation so suppose we have a Lagrangian now you have a contact boundary Lagrangian sub manifold so I would like to what does it mean that L is flexible so there is a of course some obvious example when we want to be flexible so this is a by the way what I'm talking is again part of kind of something which we again discussing currently with Chilganatra and Oleg Lazarov so so one obvious example so you started with this W and you take take this product of this W this kind of inverse symplectic structure and in this product you have this Lagrangian diagonal so you have this Lagrangian diagonal and this Lagrangian diagonal has this Lagrangian boundary so I want this to be flexible if my W is flexible so this is kind of one one important example so the second example which I want with the following so suppose I have any Weinstein manifold and now suppose I am attaching this handle along this loose Lagrangian knot so loose Lagrangian knot by the way they cannot bound say exact I'm actually interested in exact Lagrangian some manifold I can say this so they cannot bound exact Lagrangian manifold because if they do what kind of homomorphic curve theory prohibits this but now this Lagrangian this handle which I attach has this homomorphic co-co Lagrangian disc so I attached this is a core disc and this is a co-core disc and I want this co-core disc in this case to be flexible Weinstein manifold flexible Lagrangian manifold so so what is a kind of natural definition so here is a kind of conjectural definition what loose Lagrangian manifold so this is a full following object so notice that this when we consider when we consider this Weinstein handle kind of critical Weinstein handle this is a cotangent bundle of n-dimensional disc and this this this handle of course contain two obvious Lagrangian disc this is a zero section and and the fiber but it also you can take a conormal bundle you can take a conormal bundle of full cotangent bundle I take a disc bundle cotangent cotangent disc bundle of this one so you have this just handle and now we can take now we can take a conormal bundle of some sub-disc of smaller dimension so if you take if this case and then this is our core if case equals zero this is co-core but there are all intermediate things so there are a lot of this Lagrangian disc and what I want I want when I do this my handle body if I if I do my handle body then then I attach handle simultaneously simultaneously to the manifold and some manifold so in other way if you're talking in terms of so you have a this pair WL and you have this Morse function on this pair and I want this critical point I want this critical critical points critical points of phi restricted to L are also critical points critical points of phi itself so when I so each time I have a say contact manifold and Lagrangian come some Lagrangian sub manifold and now I have attaching sphere for the handle which is again Lagrangian and this attaching sphere need to intersect my boundary of my Lagrangian manifold what is called cleanly along the non-transversely but along some sphere some equatorial sphere of smaller dimension and then when you attach smaller handle which is one of the conormal disk to this Lagrangian sub manifold so I always always my critical points always critical points on Weinstein manifold of index N but on Lagrangian that can be of arbitrary index so this is that so this is this picture but now what is the flexibility condition so flexibility condition is the following first this critical so critical points of phi of index N I always want them of index N this is a definition this is a definition this is the condition of the definition and I don't know whether it exists or not it's a question question so suppose such thing exists suppose such thing exists and in addition I want the following condition so first of all all critical points of phi over L are of index less than N so on my Lagrangian sub manifold I am not allowing critical points of top index for instance this Lagrangian cannot be closed it must have a boundary so second I want so each time when I have this attaching sphere which intersect my boundary of Lagrangian along so this is some attaching sphere Ligandrian and intersect this boundary along some equatorial sphere I want this sphere to be loose in the complement complement of this intersection so this kind of virus is somewhere have to be outside of this intersection so this is a so such Lagrangian is if you have we call it flexible so if you can find on our manifold Weinstein function which have this property you should notice this line in that sphere what's this line in your picture the sphere I have a contact manifold in contact manifold we have already we started with I am building handle body so I have already this my manifold and already piece of Lagrangian sub manifold and now I am attaching handle and I am attaching handle along some Ligandrian sphere and this I denoted this S and this is DL so this is a boundary of Lagrangian and this is a Ligandrian along which I attach which require to intersect this along some equator sphere and kind of simultaneously attach handle to both of them of course this also have a very transparent interpretation in terms of this left shits handle body picture so if you have this left shits so now with all this work of beginning from Donaldson and kind of latest of Jirupardin so we know we can take a Weinstein manifold and you can kind of present it there exist left shits vibration over the disc with some bunch of this critical left shits fiber and and the fiber itself is this Weinstein sub manifold of smaller dimension so we have this left shits presentation and now nobody kind of proved this yet I think but certainly this is true that there is this kind of it can be proven in the same spirit as all this Jirupardin result that if you have a Lagrangian sub manifold in this Weinstein manifold then you can find this left shits picture in such a way that this Lagrangian sub manifold will project to certain ray going to the boundary so this such result exist in closed case but kind of this should be true also in this relative case and then in this case so this our Lagrangian would pass through the bunch bunch of this critical point of this left shits vibration and like it will be precisely like I am looking at this complex analytic function and on our Lagrangian manifold it's take a real value so it's kind of real analytic complex analytic extension of real analytic function this is the picture so then this this Weinstein handle here are exactly will be this symbols of this critical points core of the Weinstein handle so anyway Yasha, can you see the flexibility in that picture? I wish I could see the flexibility in absolute case in this picture I don't know so this is I think it kind of like we have a world expert here Kyler Siegel in this audience but I think he was thinking a lot about this but I don't think it's understood what flexibility means in terms of left shits picture but anyway so the answering to I think or it's your question or somebody asked question that I think for every Lagrangian manifold such presentation exist but when it's flexible not flexible of course it's subtle question so what was the condition that made it flexible? conditions flexible that all these handles are LEGENDARY so definition of flexibility for closed case that all these LEGENDARY are loose attaching and here just one additional thing that they are loose in the complement of this Lagrangian so now I want to formulate kind of three conjecture kind of one wilder than the other the first I'm kind of pretty sure that this is true and I'm almost see how one could prove this but I don't know the proof okay so the first conjecture so first so I have a three statement kind of I I have the statement in the closed case and I want to formulate them kind of they are analog in the generative case so one statement in the closed case is that when we have that any almost complex manifold if this handle body have homotopy type of half dimensional thing then it admit this Weinstein structure more or less flexible Weinstein structure so therefore I claim the same thing true here so suppose you take any Weinstein manifold and you take some any sub manifold so you take any any embedded sub manifold not just take any any smooth sub manifold which is formally Lagrangian which means that I can take this Lagrangian embedding and I can kind of deform to Lagrangian tangent plane and also with Ligendrian boundary so formally it look like this thing then it is either topic to Lagrangian sub manifold with to flexible Lagrangian sub manifold so any any sub manifold you can realize as a flexible Lagrangian sub manifold if topology allows you any Weinstein manifold any Weinstein manifold ok, so this is kind of one thing the second thing that that this flexible thing have no the second thing that the flexible object have no symplectic topology namely, again if you have two such objects so you have this kind of L1 and L2 so you have now this is a already flexible Lagrangian with Ligendrian boundary and they are formally Lagrangian isotopic so that means there is an isotopic together with this kind of covering deformation of Lagrangian homomorphism then they are Lagrangian isotopic in the class of Lagrangian some kind of Hamiltonian isotopic talking about only exactly Lagrangian manifold Hamiltonian isotopic but of course boundary is moving it's not this fixed boundary boundary moving with some Ligendrian isotopic so for this thing I kind of have some idea how I would try to Can you say again that I put this because what you said it didn't look like it implies Lagrangian boundary so do you ask that boundary is moved no I'm not asking I'm not the boundary is not boundary cannot be loose because it's bound Lagrangian yeah boundary is not yeah but so this this is strong conjecture of course so it implies there are no examples of Lagendrian which are formally isotopic and bound formally formally isotopic Lagrangian then they cannot and then kind of like the wildest of all conjecture is the following that in a flexible in a flexible Weinstein manifold all Lagrangian are flexible so I just remind that I consider only the close does not kind of applicable to close Lagrangian not saying any statement close Lagrangian because I require this property that they this because I want them with my handle body picture I want them to be loose in the complement of complement of intersection it's a top dimension and nothing left so there's no place for looseness just understand the definition of flexible just understand the definition of flexible it's that there exists one more function there exist one more function there is one more function with this property again this is a completely kind of open question well in subtle question in closed case we know that this flexibility condition for this Weinstein manifold can hold for one function and not for the other function and kind of precisely this need to be understood similar is here but you still think they are exactly right yeah everything exact all Lagrangian closed one could be for instance you cannot take a tangent model of a sun in principle I'm saying the definition of flexibility prohibit closed but it's not because they are exact yeah flexible Weinstein manifold and so black models you should vanish by your exact sequence and so on I mean for the last one for the last one yes no close yes yes yes so so this what kind of in particular like take wonderful manifold which is a ball so this is a flexible Weinstein manifold and you consider this and take a say Lagrangian Lagrangian disk with the gendron boundary so this if this kind of all conjecture through that means that any two Lagrangian disk with the gendron boundary are Lagrangian is a topic and so this is a kind of again this is not quite this nearby Lagrangian conjecture because I allow boundary to move but kind of some step in this direction so that's kind of what I just wanted to say that that this kind of this nearby Lagrangian conjecture already after all work which were done using holomorphic curve theory maybe the remaining part is complete H principle so I think my time actually expire so no more dreams any questions I just wanted to make sure about your results about the existence of primordialism so if you take a closed manifold which you like to have which is always complex you certainly can't say this is a defective form on that but if you remove two points then you could make a symmetric form which is like infinity at one point and sort of an over twisted conjecture of the other but if you just remove one point you can say yes if I remove not yet you should be able to get by just removing one point you think it's kind of different I hope because I cannot think because I don't know how to prove it I tried to prove it but I couldn't so maybe it's wrong but it's okay if you remove two points then you're good you said something about an example of a flexible white steam domain with a non-flexible white steam sub-domain but not a sub-level set sub-level set so you have a Weinstein function and for some smaller phi less than or equal to c for some the sub-level set is non-flexible this is an example of Murphy and Ziegler but does that mean that this function you're looking at is not the one that tells you that the load is... no, not the one so like you start with this one then you do some kind of handle slide and at some moment you end up with this strange situation is that written down somewhere? it is... the text was shown to me but I don't know in your dream? no, in my computer so it's kind of yes, I dream in my computer maybe okay there are no other questions the author in the audience so it can be