 Спасибо большое. Мой話, наверное, будет идти к каким-то другим направлением от многих сюжетов, потому что, вместо объяснения, как разработать эту большую структуру, я думаю, в симпляке топологии, я хочу поставить какие-то лимитации. Так что симпляктика структура есть симпляктика манифолта, она манифолта, locally modelled на флэд симпляктике, и контакт структура это деменжерный аналог, это, можно сказать, прожигательный аналог с симпляктикой структура. Это гиперплеин, который locally modelled на флэд симпляктике, деменжерный аналог с симпляктикой структура. Так что, в симпляктике структура есть манифолта, или контакт структура, из локации. Но это так же, как в симпляке структура, это манифолта. Так что, у меня есть деменжерный аналог, в максимуме, в форме, и similarly, контакт-манифолта, это деменжерный аналог с полностью нонинтегробно-плэнфилдом, в котором полностью нонинтегробно-плэнфилдом, это означает, что в локации, в фрабениусном состоянии, это defined by one form, globally this is one form with twisted coefficient, such that the alpha restricted can be symmetric. So, it's at first glance, they kind of look completely different structure, but in fact, it's almost kind of the same thing, it's just right odd dimensional analog of symplectic structure, and just you can see it from the following construction that you have a symplectic manifold vector field such that the derivative of omega is equal to omega, or equivalently, you can say that if you take this interior product of z and omega, you get form alpha, which is a primitive for symplectic form. So, then in this case, if you have a trajectory of this vector field z, then you have a one-dimensional relation with transverse context structure, because this plane field alpha equal to zero is a plane field, which is supposed to say alpha, yes, so the z is a non-vanishing vector field, so alpha is non-vanishing, so this plane field is tangent to this trajectory of z and on any transverse of slice you get context structure and it's invariant under this monodromy and in case when space of leaves of z is some, say, household space, then on the quotient is a symplectic manifold and conversely starting from every y in this context structure, you can get this corresponding symplectic manifold which is called a simpletization, which, say, in the case when the context structure is corrientable, this symplectic manifold is just r cross r cross y and this vector field z is just vector field d o d s, where is this coordinate in this factor. So, in the main object in symplectic manifold is a Lagrangian submanifold and Lagrangian submanifold are kind of n dimensional submanifold which are isotropic for symplectic form and there is a, I think in my views the main importance of Lagrangian manifold come from the fact that they have this two incarnations, so one incarnation of Lagrangian submanifold can be provided by this example when you have a manifold which is a cotangent bundle of some manifold m and you take a, if you take a graph of differential of some function so you take some standard standard coordinates, say pq in cotangent bundle and you take a standard this pdq form and symplectic structure t of pdq then the graph of differential of function is automatically Lagrangian submanifold or more generally graph of any closed form is Lagrangian submanifold and conversely if you have a Lagrangian submanifold you can think about this well, in the so-called exact case meaning when you have pdq restricted to L is equal pdq restricted to L is exact then this can be viewed as a graph of so-called front or multivalued function so this is just one interpretation so from this point of view there's Lagrangian intersection theory becoming generalization of more theory it's something about critical point particular point of function and the second major example when you take a product of two symplectic manifold and you take a form omega plus minus omega then for every graph of some diffeomorphism so if you take a graph graph of the diffeomorphism then this graph is Lagrangian even only if f is a symplectomorphism so there's a big part of symplectic topologies coming from interplay between these two characterization so in contact case so as I said that contact manifold you can view as this symplectic manifold with this Louisville field and contact geometry can be viewed as a symplectic geometry or invariant with respect to this action by this Louisville flow which expands symplectic form and hence kind of analog like if you take a Lagrangian manifold which is a tangent to this Louisville vector field then it project to contact manifold as a sum n minus 1 dimensional sub manifold which is called Ligandrian and Ligandrian Ligandrian sub manifold play in contact geometry similar role as Lagrangian in symplectic geometry and their maximal integral sub manifold of this non-integral playing field Louisville vector field as long as you expand Louisville vector field it satisfies as long as you expand so it satisfies this equation and so therefore if you take a z t star of omega e to the t omega but contact form this form alpha also expanded primitive by this field and therefore kernel of alpha is preserved so what is if you kind of approach symplectic topology with kind of topological mind so you immediately see just a lot of similarity of like all symplectic problem with topological problem so because and also in contact geometry because this symplectic form has extremely huge group of automorphisms you have a unlike Riemannian case so in some sense you can say ok so this is just generalization Riemannian geometry from maybe not generalization some kind of cousin of Riemannian geometry where you replace symmetric form by skewsymetric form and hence you would think that this symplectomorphism or symplectic defiomorphism which preserves symplectic form should be something like isometries and indeed you see if you write this equation for preserving preserving preserving Riemannian metric you have to write n times n plus 1 over 2 differential equation and preserving symplectic form you write n minus 2 over 2 equation with respect to n function so it's fantastically both of them are determined but still this like n difference make drastic drastic difference so in this case the group of automorphism is extremely rigid and in this case it's becoming infinite dimension but only because the form is closed the form is not closed yes but even if you have in Riemannian case you have a flat metric still doesn't help because closed 2 forms it seems one form yeah okay in any case it has this huge group of difiomorphism okay let me not go into this most people here know this and hence it's kind of the same in the counter case so for instance group of difiomorphism externally on any finite set of point on much more things so really extremely huge group so therefore like handling symplectic manifold looks very much as some kind of topological problem so for instance let's consider in this context problem of say existence or extensions of symplectic in context structure so suppose say you have a sum manifold and you want on this manifold construct a symplectic structure so you want to construct closed closed non degenerate 2 form and of course like immediately necessary condition you said have to have at least non degenerate form which not necessarily closed so take a non degenerate form and existence of non degenerate form is the same as just reduction of tangent bundle group of tangent bundle to symplectic group which is a homotopy equivalent to unitary group and therefore this is the same as a construction from homotopical point of view construction of almost complex structure on the manifold so it's a kind of topological condition and now you're asking ok so you get it and if this sufficient or not so when you say well it's not quite sufficient because say well at least if it's sufficient or not and then Gromov proved that indeed if X is open then Gromov theorem says that this eta is homotopic to really symplectic form in the class of non degenerate form so there is no more abstraction for constructing of this symplectic structure on open manifold because all what is called almost symplectic structure I will refer to this and similarly that holds in the contact case so if you take a contact manifold and you want to on contact manifold consider almost contact structure what is almost contact structure almost contact structure means that I decouple so as I said what is the contact condition well again let's restrict to the currentable case you have a one form and you have a d alpha and d alpha restricted to alpha equal to 0 is non degenerate so it's differential condition but now you decouple this condition so you just instead of d alpha you just take eta which is any to form any to form and you say well let's call almost contact structure this pair alpha eta restricted to alpha equal to 0 is symplectic non degenerate which is again just mean and this is given up to conformal factor and hence this is just equivalent to having hyper plane field together with structure of symplectic bundle or the same as a hyper plane field from a topical point of view with complex structure on this bundle and this is called in topology stable almost complex structure almost complex structure on the manifold cross R so so Gromov theorem says that in this case indeed also you have almost contact structure and manifold to open you can deform it to context structure so let's just continue and let's analyze moreover Gromov proved here the same thing holds in parametric case so suppose you have two forms symplectic and they homotopic just almost symplectic forms and they are homotopic as symplectic as well unopened manifold so we can ask if the same thing holds in in the closed case and in order to get into closed case you need to solve extension problem right so for instance you take a closed manifold you remove one point and then you can apply Gromov theorem the complement of one point and what is left just you have this ball and you have this structure near the boundary of this ball contact or symplectic and this structure so we can kind of try to do that and kind of it turns out to be extremely difficult but in some turn in the counter case is not as difficult as in symplectic one so I'll come to this question a little bit later so we have this question so whether kind of given a germ of contact structure contact structure near sorry, context well let's say symplectic structure given germ of symplectic structure near boundary of two-dimensional ball and which is extended to be as an almost symplectic structure so is it possible is there is there a symplectic extension no for complex structure for complex so this would be wonderful but for open many fold situation there is a drastic gap because this kind of theorem holds up to co-dimension n plus 1 so we know that up to dimension n plus 1 and to n so this is a Grom of Land Weber theorem which says that like if you try to extend that up to n plus 1 skeleton you can do that in complex case and it's known that but in this symplectic symplectic contact you still have almost just one last step left so whether you have the symplectic extension and immediately you say well of course it's not true because in this case if you have this form symplectic form near the boundary then the symplectic form near the boundary remember volume of the symplectic form if you manage to extend it because by stock theorem that omega near the boundary you can write it is equal to d lambda and then integral of lambda which omega n minus 1 over the boundary is equal to integral of omega n over the ball and so therefore if there is a symplectic extension it has to be positive so you have some one extra volume abstraction here which in the case of closed manifold translate to condition that this manifold have to have some two form whose power is non-trivial co-homology class fundamental class of the manifold so anyway you have this condition you modify this question by this extra condition that you have this and you ask is it possible to extend or not well and we know that it's not possible and I'll explain it just in few minutes so and you can ask similar question in contact case you have almost contact structure and you have a contact structure which is extended and you ask is it possible to extend it is it extended as almost contact structure is it possible to extend it as a contact structure and then you can ask the same question parametrically you have a two contact structure relative boundary homotopic as almost contact structure are they homotopic or not and what is the kind of additional price here in this case would be if you could prove such theorem that because of this grey mother argument this homotopy fix from the boundary is in fact an isotope so you would prove uniqueness of structure up to isotope so there is this so like in 60 70 there were no symplectic topology yet and kind of this questions very much were open and was for instance I I thought up some point up to about 72 I thought that this is actually all the answer to all this question for instance should be positive and that would mean that there is no symplectic topology so by the way it's very easy to deduce that if both the statement zero parametric and one parametric for a symplectic or contact is true then you can prove that the group of the symplectic is it's C0 closure is a group of all volume preserving diffeomorphism C0 closure of the group of contact diffeomorphism is C0 closure a group of all diffeomorphism and this would immediately show that there is nothing like Arnold conjecture could be true because this fixed point C0 property and then like the all symplectic topology would disappear and so therefore as I say C2 property cannot hold simultaneously zero dimensional and one dimensional H principle so I'll come to this in a second so then in 80's this symplectic topology was discovered and kind of there was a work of many people but kind of culminated in the work of Gromov when he introduced method of holomorphic curves so Gromov's idea was that well so as I said there exist this topology and there exist this symplectic topology inside and as I said like all this thing saying that the symplectic topology is a topology really not much different between this thing but on the other hand inside the symplectic topology there is also algebraic geometry because like there is an important and very prominent class of the symplectic manifold which is provided by Keller manifold and in Keller structure it's kind of much more rigid and we can prove a lot of result in Keller geometry and kind of you would like to say can we do something here there was an important tool in Keller manifold understanding model spaces of holomorphic curve and that was the idea of Gromov so at that time was already Donaldson discovered this kind of how fantastically you can apply elliptic PDE in topology and Gromov wanted to do the same thing in the symplectic case and so he was looking at the proper equation and he just thought well you cannot have symplectic manifold is not Keller manifold but it's almost Keller so if you have a symplectic manifold you can always introduce almost complex structure on this manifold such that this thing becoming not integrable but Keller we have a Keller form in its endings equation metric invariant with respect to J and its imaginary part is our symplectic form and moreover there is a homotopically unique choice the contractable space of such J and so now if you can for in this situation we can write some good equations and out of this you can by the Donaldson scheme you can extract invariants and put it in this program and the point is that locally almost complex manifold with respect to theory of homomorphic curve behave the same way as integrable homomorphic curve in integrable complex structure and this existence of this so-called taming symplectic form which allows you to talk about the modular spaces of homomorphic curve in symplectic manifold and that started the whole this fantastic development last 25 years with Gromov-Witton invariant and Foucaille category and kind of connection with mirror symmetry and kind of many-many Hofer metric and kind of a lot of great development in this thing so but my talk is not about this but about kind of limitations of this so all this time there were kind of some question which still despite all this progress remain like basic question for instance like the one which remain unsolved and also for instance the following question many people try to answer how kind of was clear that theory of homomorphic curve doesn't quite work there so what one of this great result with Gromov-Proof application of his method of homomorphic curve is his famous non-squeezing theorem and this non-squeezing theorem in particular said that if you take a poly disk and let me denote this poly disk some kind of so it's a poly disk in CN and this is just idea of the corresponding disk and I say let me order them in increasing order and suppose you have two such poly disk and you are asking when one poly disk one poly disk can be simplically embedded to another poly disk and of course there exist volume obstruction which we just said that product of this one Rn is less than product of this radii and Gromov-Proof that in addition you should have the smallest radius of this one is less or equal than this one Gromov theorem and of course there was a lot of effort to try to understand is there any other obstruction so like in dimension 2 not much beyond this but let's suppose is there any kind of intermediate intermediate obstruction for this embedding and kind of like nothing worked essentially and people kind of trying to find some maybe intermediate equation some for not in for curve but for something high dimensional but kind of nothing worked and until Larry Gooth Larry Gooth proved that in fact there is no other obstruction at least modular some constant so there is some slightly room left for extra invariant but what he proved he proved that there exist such some constant some constant C such that if the say R1 is less or equal than C say R1 R1 less or equal than C, C R1 and the product R1 Rn is less or equal than C R1 Rn then there is a simplectic embedding in this case so well there is some subtle issue about the size of this constant and there are some bounds for this but this already shows that you cannot go too far so this is a like for me this is the first first example showing this limitation limitation and saying that there is a kind of like where holomorphic curve is not applicable and you couldn't really nobody invented any way of using holomorphic curve for kind of getting high invariant and that maybe they don't exist and so therefore like I from that point started to take seriously this principle so let's just say this kind of general principle that there is no symplectic life beyond holomorphic curve so so that essentially kind of everything which is not in symplectic topology not prohibited by the theory of holomorphic curve is in fact possible and this is of course kind of very vague principle holomorphic curve can be applied and were applied in very kind of many ingenious ways and so it's not clear what it means but at least there is in some cases strong feeling that like there is no nothing like holomorphic curve exist and can be applied and therefore it should be should be true so let me start kind of my next example discussing this problem distance of an homotopy of structure which I mentioned so let's start with this problem of extension of symplectic structure so I said that this problem of extension of symplectic structure is there any extra kind of invariance not coming directly from topology it was I think it's not published anywhere but it's a follows immediately from Gromov non squeezing theorem that indeed should be some additional abstraction namely to extension of symplectic structure so let me kind of explain you how it goes so let's consider the following example so let's take 2 poly disks 1 thin poly disk of big volume and another kind of thick poly disk but of small volume so the volume of this is less than this but this is kind of thick and Gromov theorem does not allow us to embed this one into this one symplectic so you can kind of position then in the R2N everything is here in R2N and I can position them in such a way that they share a piece of common boundary and then I take this piece of common boundary and I remove it and then I get a piecewise smooth emerged sphere kind of difference of this 2 poly disk so you can extend this to immersion or maybe piecewise smooth but you can smooth it, it doesn't matter and you get some spherical annulus and you can take induced symplectic structure so asking is it possible to take this symplectic structure and is it extendable to the ball so again there is a volume is precisely formal volume which I wrote this abstraction somewhere this one is precisely difference of volume of this poly disk so it's positive and also it is a regular homotopic to the standard immersion so there is no problem with kind of extension formally so question can you extend it I claim no, why because suppose you can extend it and then now you can return back this our poly disk which you removed so that means how it looks like so this is our symmetric disk with induced structure and then I added on outside this poly disk which I removed and now I get some symplectic manifold which near the boundary is exactly coincide with our thin poly disk so you have a thin poly disk and inside you have a thick poly disk and of course this thin poly disk does not have a standard symplectic structure but it does not matter because Gromov methods perfectly work for this you can sweep this by family of this homomorphic disk and you can find homomorphic disk through the center of this of this ball and which says that this ball have to have radius smaller than the smallest radius of this one so in symplectic case there is no such this kind of each principle type claim for extension is not true remember I said that the symplectic rigidity kind of result proven in 80's in particular implied that the group of symplectic difomorphism is C0 closed in the group of all difomorphism also group of contact difomorphism C0 closed so therefore corollary of this international age principle and one dimensionless principle cannot hold simultaneously because if they were true then you can disprove this approximation and hence here you don't have so now we have this negative result but this negative result is in fact a source of some hope of positive hope because now one parametric theorem still holds so that means in particular what it says it says suppose you take a ball and on the ball you have a two symplectic structure for instance you have on the ball you have a symplectic structure standard near the boundary and the question is it either topic to the standard one or not so like age principle type result says well one parametric that formally in the same class should be should be isotopic and in fact this is still open in all dimension Gromov proved that in dimension 4 this is true up to symplectomorphism but in dimension 4 unfortunately we don't know anything about pi0 of the group of difomorphism also up to isotopic this question is still open and high dimension is open and so this is a still completely open problem so now let's move to let's move to contact case in the contact case contact case the one parametric one parametric age principle one parametric age principle fail so this was already demonstrated first in three-dimensional case by Benekin who constructed on three-dimensional ball context structure which is in the standard almost contact class but not as a topic not even as a morphic to the standard one and then example of this type were constructed in all dimension of non-standard context structure on the ball so we know that one parametric age principle fails in this case yes so therefore again this is a source of some hope that zero parametric age principle could be true and in fact turns out that it is true and this is our recent okay it's our recent theorem with Strom-Borman Amy Murphy and myself that so let's formulate it anyway so let let why why say I write something okay eta be and almost be an almost contact manifold and suppose suppose that before you almost this alpha okay let's take alpha eta and call it beta so suppose you have this almost contact structure almost contact structure such that this beta on the neighborhood of some closed subset on the neighborhood of some closed subset is a contact structure is a kind of genuine genuine contact structure then then beta is homotopic is homotopic to a contact structure to a contact structure we are almost contact structure fixed fixed on the neighborhood of A so in particular if you have a near the boundary you have almost contact structure which is formally extended then it extends non-formally so in fact it turns out that even more true so there is a so some kind of one parametric and even two parametric also holds if you kind of restrict from all contact structure to certain number of component in the space of contact structure so so like similarly to similarly to the similarly to the three-dimensional case similarly to three-dimensional case turns out that there is some kind of dichotomy dichotomy of contact structure on any manifold which is called tight and over twisted so what is the definition of over twisted contact structure so you can define its follow so let's let's consider the following map so let's take a r2n plus 1 and consider the map r2n plus 1 kind of standard map which I call wrinkle so it's a map with some kind of standard singularity which have a fold of singularity and some kind of casp equator so it's in like if you are doing in two-dimensional case it would look like this so you can say this is a map which is a so let's say let's fix so now I would think about this as a contact space and I have this standard Darbou coordinate with this form de minus yi dxi and let me pick one coordinate I want to pick yn so this map w I split all variable I split all variable except this yn variable and in all other variable this map is identity and with respect to last thing this is just standard kind of cubic thing so this is a so you write yn cube minus some function alpha u times yn and you want this function alpha to be positive on some ball so it's a map non-singular here create to critical point and then so you get this picture in high dimension so you get this thing so as I said this map has some sphere of singularity sphere of locals and let denoted by p the ball bounded by this sphere so now I take I take r2 r2n plus 1 r2n plus 1 and subtract subtract this ball closed ball and then I say subtract this ball so I have this open manifold neighborhood of this thing and context structure is called over twisted if it admits embedding of some open neighborhood of this sphere db into your manifold so if this neighborhood some small neighborhood some spherical annulus around this singular sphere can be embedded then it's called over twisted I take w sorry I didn't say the main thing you take this w and now on r2n there is a standard symplectic form so let's call this form lambda standard contact form lambda and now you take a w star of lambda so w star of lambda is degenerate along the singularity it's kind of negative contact form inside and positive contact form outside so I removed this interior ball and on the complement on this rn minus b w star lambda is an honest genuine contact form and so now this is a source of over twisting neighborhood of this sphere so if some small neighborhood of this sphere lambda can be constantly embedded in your manifold then manifold is called over twisted so in fact if you consider this concentric sphere sufficiently close their image look like this so image of the sphere so like you have a family of the sphere which kind of like as we approaching this singularity they start to look like this and in fact there is some for every dimension there is some critical smallness of this loop so in fact you don't need to embed the whole thing up to the boundary but sufficiently small but how small I cannot say okay and so turns out that over twisted contact structure satisfy complete age principle so that means that for over twisted contact structure you have classification for instance like if you if you take a sphere S2 I'm sorry if you take a sphere S2-1 and you want to know how many this over twisted contact structure up to S2-1 and you want to know how many this over twisted contact structure up to isotope exist on the sphere so this is a precisely question of computation of the homotopy group of so this homotopy group is a metastable group and it was computed in sixties well this particular case I think by Bruno Harris and so so you know precise answer and for instance we know that if you take a 5-dimensional sphere there is a unique one so okay so let me kind of move on so I'll stop about this contact structure and let's move to some other example so I said that like core of symplectic topology is a Lagrangian manifold and that's like the main rigidity property of Lagrangian manifold they want to intersect in more point than the kind of topology required so that's all Arnold conjecture about this so Lagrangian self-intersection and etc so we can ask the following question kind of that yeah so this is Lagrangian manifold and in content geometry the corresponding projects are Lagrangian manifold and the question about Lagrangian manifold it's not about the embedding but about the isotopic so asking if you have a two-Lagrangian manifold are there isotopic or not and again there's a whole symplectic topology development and contingent case provided using holomorphic curves a lot of invariance to distinguish two-Lagrangian manifold up to the Lagrangian isotopy that in fact in dimension in contact manifold in contact manifold of dimension greater than 3 there is a class there is a class class of of Lagrangian node called loose such that for this loose Lagrangian nodes you have a complete H-principles that means that like formal Lagrangian isotopy like forgetting like in the same sense I don't have really time I want to say a few more things so I don't want to go into definition but formal Lagrangian isotopy is implied genuinely and it's a I will not give you precise definition but it's sufficient to say you can get loose Lagrangian node by very standard modification in the neighborhood of one point so you can take any Lagrangian node and make some infection kind of put this virus and then after that this this Lagrangian node become viral they become completely flexible so this was a kind of important step and then what we did with with Emmy is we kind of extended the thing to Lagrangian Lagrangian Lagrangian manifold with concave concave Legendrian Legendrian N so let me explain you what we are talking about so for me the starting problem was the following suppose you have a ball suppose you have a ball in R2M the question is it possible to find Lagrangian disc in the complement of this ball with Legendrian boundary on the ball so is there is there a Lagrangian Lagrangian Lagrangian ball with db with d Lagrangian sorry Lagrangian disc Lagrangian disc let's call it delta Lagrangian disc delta in the complement of the ball delta inside db Legendrian so one can prove using beniken inequality that in dimension 4 this is impossible and there are no known examples on high dimension and what turns out that in fact there are plenty of them what we proved and all of them kind of there is a complete flexibility provided that boundary you make the loose Legendrian knot and why I kind my motivation for this problem that I was interested in problem of topology of so called polynomial and rationally convex domain so I don't want to go into the definition but the point is that if you have a say ball and you have this Lagrangian manifold attached with Legendrian boundary then the neighborhood of this union is you can get a neighborhood so called rationally in particular it means that any homomorphic function can be approximated by rational function there and so and this kind of provided complete answer to what possible topology of rationally and polynomial convex domain in high dimension in dimension greater than 2 complex so what we proved we proved kind of more that the Lagrangian manifold with this what I call concave Legendrian boundary if boundary is loose you also have H principle so what it implied that the following so I don't want to waste time for handling board so what it implied is the following thing so Gromov proved that there are no exactly Lagrangian submanifold so exact meaning that say primitive of symplectic form restricted to the submanifold is exact so in R2N the Gromov Gromov say in R2N there are there are no exactly Lagrangian submanifold what this H principle implied that there are plenty of Lagrangian submanifold which has one singular conical point so if you can make on Lagrangian manifold one conical point which kind of looks locally it's automatically cone over Legendrian submanifold then you have a complete kind of freedom of constructing such thing and this also allowed to do the following thing take this point and when you if you study the structure or near the point it's very possible for some of this if topology allow to resolve it just with one extra intersection point so this immediately provided kind of another counter example to hold this line of problems about the minimal number of Lagrangian self intersection so so this is how by the way this is about the self intersection our joint work with Murphy and also with the column and Smith so so along with this Arnold conjecture about minimal number of intersection of two manifold I don't know I don't know if Arnold ever wrote it in writing certainly he was talking about this about this problems about what is the minimal number of self intersection of exact Lagrangian submanifold R2N for instance there was a kind of conjecture that you have a torus and it's a you have exact Lagrangian immersion then of n dimensional torus to R2N it should have 2 to the n minus 1 so essentially half of the rank of homology kind of natural conjecture and this show you this is completely wrong because you can kind of essentially one extra point extra intersection point allow you to do everything for instance corollary I will not give you precise definition precise formulation but for instance corollary that if you have any three dimensional manifold you can always find Lagrangian immersion to R6 with exactly one self intersection point so let me kind of finish so this is a kind of essentially all what I know mostly about what I can prove but let me tell you some kind of conjecture which I cannot prove but I kind of in the same spirit I think they should be true so one is a question about construction of symplectic structures so this is already conjectured so kind of construction of symplectic structure so symplectic manifold this can cave can cave over twisted contact boundary satisfy H principle in particular this would imply that given symplectic manifold whether symplectic well maybe you cannot construct symplectic structure or maybe even you can but we don't know this but at least you can always construct symplectic structure with one singular point and near the singular point it would look like cone over some over twisted S4 so I'm pretty sure that this is true in four dimensions kind of I even have some sketch how to prove but I think this is true in all dimensions as well so the second question is the following okay I would also conjecture well this is not so original because many many people conjecture this but I would also conjecture that in fact even construction of symplectic structure so extension problem for symplectic structure have extra abstractions construction of symplectic structure and closed manifold is in fact I think have no abstraction in dimension greater than 4 and even in dimension 4 it also should not have abstraction beyond some kind of obvious example so now another kind of type of conjecture so construction of symplectic structure on closed manifold and then the last one which I want to mention are bounds bounds and okay let me recall anti-Arnold conjecture okay so Arnold conjecture about like classical Arnold conjecture about intersection of say deformation of zero section and cotention bundle it's what does the Arnold conjecture not clear you can say that Arnold conjecture is a statement bound in terms of frank homology you can say it's a statement in terms of bounds of the minimum number of critical point of function and certainly it was expressed in this extreme form many times so kind of what I want to say is that on the other hand if you look at the halomorphic curve method then it's completely kind of inconceivable that halomorphic curve method can prove anything beyond bound on stable morph number so like if you have a manifold there is a stable morph number there is a number of minimum number of critical point after quadratic stabilization of this function and this bound is proven for this particular problem and I think this is at least in sufficiently high dimension is exact bound so should be some kind of H principle that if you have some example when this stable number is less and such example for instance were constructed by Damian then we in this case should be possible to in fact remove extra intersection point from zero section ok thank you so you said that your principle is that if something cannot be proved by a halomorphic curve then it shouldn't exist or something or it should exist yeah halomorphic curve essentially proving you something ok first of all I'm trying to stay away from dimension 4 dimension 4 kind of that's a special story but maybe also it's subject to H principle subject to the same principle but with their halomorphic curve much more powerful for the ball with standard boundary you think that it's not possible to say anything for for particular reason or requests no no no but that I am extremely afraid to say I'm not conjecturing this so like I don't think there is enough evidence yet that this question whether there is a standard symplectic structure non-standard symplectic structure on ball standard with the boundary we explored all possibility of halomorphic curve it seems to be very difficult to construct such something which halomorphic curve would distinguish but maybe there is a way just not clever enough but still I will maintain if somehow we can fix those whole halomorphic methods then probably it exists there are no more questions so about the statement that in symplectic well anything possible do you think that at some point one can cook a rigorous conjecture which makes this rigorous rather than as a principle how can I say I cannot make it rigorous at the moment so maybe somebody will be able to