 So at the time when Poincare came to this mathematical world, mechanics was already a kind of mathematical method of mechanics were well formulated. So the work of these great people, Newton, Lagrange, Hamilton, Leibniz, there's a very kind of perfectly formulated mathematical foundation of mechanics. And essentially, the efforts of mathematician and physicist in this era were directly to solving some concrete system, some finding for concrete cases of integrability of classical mechanical system. And equation of mechanics, when you write this in either system of first order equation as a vector field, have this special form which was written by Hamilton called Hamilton canonical equation. And just written in this form, they have some special qualitative property. And of course, these people like Jacobi somehow knew this, but probably Poincare was the first one who kind of explicitly realized that, for instance, if you have a system with two degree of freedom and a conservative system and you have a level set, which is three dimensional, and then you have this flow on this level set of constant energy, then area of the transfer slice preserved by this flow. So also, I'm sure that he knew that the same property holds in high dimensions that this Hamiltonian flow preserve what is called, at that time, Poincare integral invariant, but then when this theory of differential form were developed, now known as symplectic form, and so he knew that these maps are preserving the symplectic structure. And I think that this kind of the Poincare had this dream that the Hamiltonian system should have this mechanical differential equation coming from mechanics should have special qualitative property just because of their form. Of course, some special maybe boundary condition, but essentially just these properties that have this form, for instance, the property of preservation of area of symplectic form, have an extremely strong qualitative property, for instance, for the problem of existence of periodic orbit. So Poincare, of course, was applied mathematicians. So at that time, just thinking of periodic orbit, though some classical mathematicians also that before him also did it, but it probably was something which one needed all the time kind of prove that it's important. Otherwise, he wouldn't get this funding from some agency, I don't know, whatever. But in every case, he just, for instance, this quote which already Arnold gave it, he just needed to somehow approve, have some kind of explanation why he want to find periodic orbit because they from his point of view there was this window in this otherwise chaotic world of dynamical system. So we already had this in this conference to talk concerning last geometric theory of Poincare. So unfortunately, I cannot avoid it, and I have to say something about this because that's really where the whole symplectic topology starts. So this theorem, which as was already explained early, kind of came to Poincare when he needed the theorem because he really wanted to get some results about periodic orbit in the restricted three-body problem. But nevertheless, he realized that there is just purely, because in the spirit of what I said before, that there is a purely geometric question about area-preserving map and about the fixed point, which he needed to prove and to find out. And so as it was already said, it concerns the area-preserving transformation of an annulus when the boundary component are rotated in opposite direction. But I should say that when you say that boundary component rotate in opposite direction, this is a little bit ambiguous thing because what does it mean? It's rotated, for instance, by pi. It's rotated counterclockwise or clockwise. So just an appropriate way of thinking about this, that instead of transformation of annulus, you look at the periodic transformation of the strip. So you just lift it to the cover of this annulus. And then there is this already unambiguous means what it means. So you can choose a covering map just on the one boundary. It's a shift in one direction, another in another direction. And say, if you take a kind of some vertical slice, then its image will be something like this. So Poincaré paper actually starts with some kind of sketch of the proof of the theorem, which I think also could be given by Jacobi or before that because he also knew such type of argument. And what I will explain today, this proof with some epsilon addition, in fact, really could be made to the proof, not only of this one, but also of high dimensional result. So this was a really right start. But then at some point, he just a little bit turned in the wrong direction. And that's why he couldn't prove it. And then it was things became even worse after Bergdorf proved the theorem. So let me explain this argument. So let's just assume that our transformation is small. So if our transformation is small and we look, so first of all, it's a transformation of the strip. But let's extend it as a transformation of the whole plane just by defining it as just translation by some constant c upstairs and minus c downstairs. So this is a transformation of the plane. And let's look at the graph of the transformation, which sit in the four-dimensional space. And like always used to kind of mechanics. And actually, you denote coordinates p, small pq, and the image p capital and q capital. And then what I want to do, if it's small, then it's also graphical with respect to coordinates q p capital. So you can consider this kind of splitting. Yeah, sorry, you should stop there. Oh, sorry. OK. That's a great idea. OK. So you have a splitting of these coordinates q p capital and pq capital, not like pq and p capital, q capital, but in these coordinates. And it's still graphical in this coordinate. OK, so suppose you can express this graph as explicitly in this coordinate. And then because dp, dq, it's area preserving, dp, dq is equal to dp capital, dq capital. And this means that form pdq plus qdp, this form is exact, is closed. But in the collision space, it means that it's exact. So it's a d of some h. And six points are points where p small equal pp capital q and q small equal q capital are therefore zeros of this differential form. You make a differential form with the coefficient of differences. So you get this form p minus pdq plus q minus qdp capital. And this is the differential of this function. So what is good that this function is periodic with respect to q variable? Because it's clear that the coefficients are themselves periodic, it's periodic transformation. And therefore, when you integrate periodic functions, you get kind of a covariant function after a period it's shaped by some constant. But what you observe that at infinity, this function is equal to cp because of our condition, it's just equal constant on plus infinity equal constant plus constant times p and other minus constant times p. And there we see that it's periodic in q. And therefore, it's everywhere periodic. This constant has to be 0. So therefore, this function is periodic with respect to q. And at p infinity, it looks like this. It's kind of the case this direction and the case this direction. And so clearly, you have a function periodic in q variable and kind of behaving this way. This function must have some critical point. And you can prove it just taking a minimax argument, this thing. So this will be, and the critical point are exactly 0, the differential. And hence, they are fixed point of our transformation. This is precisely what is written in Poincare on the first page of his paper. So, but then kind of what Poincare decided. Well, but this is, of course, applicable. It has very restrictive, restricted applicability because, well, when it's graphical, it's essentially that mean that our transformation, just every vertical slice, can look like this. But if you think, then it can be something like this. So it could be extremely complicated. And so therefore, this somehow forced Poincare in this way of thinking that now you have to kind of unwind this extremely complicated picture. So he started to consider that one special case after another. He considered many, many of them. But still they kind of kept growing and he couldn't consider all of them. So this was a kind of, this was the wrong term in his argument. So, but as I said, I'll just a little bit later like explain how he could done it correctly and kind of, and then the progress would be like 60 years or 70 years faster. So this theorem was proven, as was already told by George Birgolf, and it's of course great that he proved it. But he gave a very clever proof. And clever proof is never good because if it's a clever proof, it's kind of to use some trick and somehow it didn't go anywhere. So although I should say that Birgolf, in fact, understood at least the kind of I read in his book on dynamical system, he said there is some paragraph where he understood that this fixed point theorem is, in fact, has a high dimensional generalization as a properties of a symplectic map. So I would say that he formulated Arnold's conjecture but not really explicitly. So I can suspect that Bonkare also knew this because he kind of said, OK. But then came Arnold in 1965. And this is, I think, Arnold in 1977, so a little bit later. And he formulated this series of conjecture, which essentially just stated what was written on the first, in the proper form, what was written on the first page of Bonkare theorem that where he explicitly related this question about fixed point with critical point of some function. And then Arnold said that should be true in general. Somehow this question about periodic orbit of Hamiltonian system, it's some question about question kind of governed by property of critical functions, a critical point of function. For instance, more theory or as we, in fact, heard in the talk of David Gabbay yesterday that the more theory was invented by Bonkare, so it should be called Bonkare theorem. OK, so in particular kind of one particular question which Arnold formulated, it was a question kind of the first generalization of Bonkare theorem to the other surfaces, for the case of torus. So he said, OK, so if you have a transformation of two torus, then it should have an area preserving. Then it should have, well, in non-degenerate case, at least four points, or in general case, at least three points. And of course, this cannot be true just for anything. The Bonkare had this twist condition kind of ensuring this. And Arnold kind of formulated it as a condition of preservation of center of mass. So I'll say in a second what does it mean. So this condition of preservation of mass kind of known under many, many names. So it just means that if you take any curve on the torus and look at its image, then the algebraic area between this curve and its image equal to 0. So why it's the center of mass? Because if you go to covering, you can take a domain and take its image in the center of mass of this domain remain in the same place. And this is a kind of what's proven to be equivalent to many other quotations. So this is actually, this is also called flux. This homomorphism associated with every loop of this area is called flux. So you can say the flux 0. And Baniaga proved that it's exactly the same. The differmorphism is in the commutator of the group of area preserving differmorphism as a topic to identity and equivalently that you can just connect f with identity by Hamiltonian flow. So one correct theorem follows from Arnold's conjecture about torus because you see if you have a transformation of annulus twist one. So you can just by this doubling trick, you can make a transformation of the torus and you can kind of insert inside some band where you can compensate area just inserting that it would be area is equal to 0. So it's just generally generalization of Poincare theorem. So I kind of I spent many, many years in 70s trying to prove this thing. And finally, I proved it in 79. And but in fact, kind of now going back to what I actually proved, I proved some kind of version of Broward translation theorem, which I didn't know at the time. And it essentially just said that if you have any differmorphism of torus without fixed points, then on this torus, you can always find some embedded homotopically non-trivial curve such that it's kind of disjoint with this image. And then, of course, this contradicts to this preservation of mass condition for area preserving thing. And this is Broward. It's not me. So let me kind of I need to a little bit more terminology. Let me talk slightly a few minutes about Lagrangian sub-manifold. So we already spoke about symplectic form. So symplectic form in two-dimensional space is just a form which associated with every pair of vectors, some of its area of projection to all coordinate plane Q1, P1. And this is a skew symmetric form. And it has a some isotropic sub-manifold. Maximum dimensional, which has a half dimensional, is called Lagrangian. And if you restrict symplectic form to Lagrangian, then it's automatically because d of this, then it's a 0. And therefore, form PDQ, which is a preimage of symplectic form, when you restrict it to Lagrangian, is closed. And if it's exact, then this Lagrangian manifold is called exact Lagrangian sub-manifold. So for instance, if you take a circle in the plane, this circle in the plane is not exactly Lagrangian sub-manifold because this form is a restricted form and just d-heater form on this one. So what you need is that when you integrate form PDQ, which give you enclosed area, then you get 0. So therefore, on the plane to get exactly Lagrangian manifold, you're kind of forced to have self-intersection form. And you'll see that this is, in fact, general property. So we already encountered in proof of Foncare, we encountered two main examples of Lagrangian manifold. Namely, if you have an area preserving, so this graph of this area preserving map in this kind of space where I consider this symplectic form is this precisely dp d2 plus dq dp. Turns out to be Lagrangian, and this was the main thing which we used. And another thing which we used is that with respect to different splitting in the two Lagrangian coordinate plane, if it's graphical, then it's automatically graph of differential of some function. So in this time, in the 60s and 70s, when Arnold formulated his conjectures, it was completely unclear whether this should be true or not. So in particular, though it was, for instance, Foncare theorem was proven, so you could say something was proven. And well, I later proved that are not theorem for torus and other surfaces. But this very much, both Berghof proof and my proof, were extremely two-dimensional. And so it could be just some kind of phenomena from two-dimensional topology rather than something symplectic. And Gromov, at that time, formulated a proof of the following symplectic alternative. So you see, when you have a symplectic transformation, then symplectic transformation also preserves volume. So this, in fact, was known like Liouville proved that Hamiltonian flow preserved volume. The phase flow of Hamiltonian is preserved volume. And Foncare understood it preserved more than that. It preserved the symplectic form. And so the group of transformations which preserve symplectic form is, of course, inside the group of those preserved volume. And, of course, preservation of volume is C0 property because it can formulate as a major preserving property. And Gromov just asked, what is the C0 closure of the group of symplectic defiomorphism in the group of all volume preserving transformation? And he proved that could be only two answers. Either it's a whole group or it's just itself. There's nothing intermediate. And so this is just kind of dramatic kind of alternative. Either symplectic topology exists or it does not exist at all. So that means that then kind of nothing true because for volume preserving transformation in high dimension, you can prove there is no fixed point properties. And because fixed point is something C0 kind of question, that means that all our null conjecture, all similar questions are wrong. So at the time, in fact, most of evidence was pointing that it should be wrong. So Gromov himself proved a lot of kind of result, very much close to this one. For instance, I will not go explain you all, but just mention just one small thing. So as we've seen, there's this kind of essence of all this symplectic rigidity is some question about Lagrangian embedding. Because for instance, fixed point theorem is again something about intersection of this Lagrangian diagonal with Lagrangian graph is diagonal of two Lagrangian manifold. And so kind of related question about Lagrangian embeddings of possibility of make Lagrangian embeddings of some manifold into symplectic manifold. And for instance, what Gromov proved that if you look at epsilon Lagrangian embedding, so that means that if you just consider maps which Lagrangian up to some angle epsilon, deviate more and more, then you have a kind of complete freedom you can extremely easy to construct such thing. And so you just pass to the limit with epsilon goes to zero, that's again proof of everything. So was there even kind of more motivating? So I said that in the beginning of 70s, I was like one week I was proving symplectic rigidity and next I proved that it's actually just opposite. So it was extremely difficult to decide. So another interpretation of this question of Gromov alternative is that in symplectic topology, all non-trivial question are tightly related. Like if you have any kind of progress in one of them, kind of non-trivial, then it's immediately give you some progress in other. So let me give you some example. So here is some basic question. So like you question about extension of symplectic structure. So you have a boundary of the ball and near the boundary of the ball, you have a symplectic structure given and you're asking is it possible to extend it to the ball? There are some obvious obstruction. For instance, the symplectic form give you some almost complex structure you want to be extendable. And then you also, symplectic form on the boundary via stock theorem remember what should be volume of symplectic form inside, so therefore it should be positive. So this is just only two non-obstructions and you ask is this obstruction sufficient or not? So a similar very related question. You have a two symplectic structure. For instance, you have on the ball, you have a symplectic structure which is on the boundary standard. And you're asking and supposing again formally it's by the almost complex structure or as a non-cloth non-degenerate form you can deform one to another. Some kind of homotopical conditionment. Is it true that they are in fact the same? They are isotopic? Or so as I already say this Lagrangian intersection principle is it true or not? That like intersection of Lagrangian is governed by Morse theory rather than just homological intersection theory. Or there's kind of partial question of this are there exact embedded Lagrangian in some manifold in symplectic space? As we see here in R2, they do not exist. Can we find exact Lagrangian torus in R4? So, well, so then kind of some revolution happened and kind of many great results were proven. So Daniel Benneken proved, well, he proved a lot of great results but one what I want to quote if he particularly shows that not every contact is different morphism in dimension three and why contact has to do with the symplectic I'll say in a second later. It can be approximated by any, not every different morphism can be approximated by contact once, so it's kind of analog of the Gromov alternative in contact case. So, and then Conley-Sander proved our null conduction for torus of any dimension. And so these both two result imply rigid resolution Gromov alternative because if they prove something non-trivial and they say any non-trivial result would imply this. But probably at least one of these people in this audience probably they didn't know about Gromov alternative so therefore they didn't prove it. Okay, so what I just want to do now as I promised I will prove our null conjecture for the torus and well I will do it only in two dimensional case but in fact my proof will be perfectly kind of good in general case and this proof as you'll see is just essentially the same proof which is given by Poincare on the first page of his paper but it just required one kind of epsilon addition and this epsilon addition required at least 70 years to kind of to be realized. And so this is some kind of the argument we'll explain it some kind of adaptation by Gromov and myself of Mark Chaperon finite dimensional version of Conley-Sander's proof but you don't need to know neither of them to understand what I said. That I said just Poincare argument just I'll just repeat it with few more words and that's it. So okay, so I will be proving just Poincare fixed point theorem for the torus for the two dimensional torus. So again as before I lift everything to R2. So you have a like Z2 periodic, Z plus Z periodic transformation on the plane and it's preserved this center of gravity or center of mass and you'll see how it play around. So again let's first think that it's small. If it's small then I repeat word by word Poincare argument. So I just again I take a graph and there's this graph is can be expressed in this with respect to the splitting PQ capital QP capital kind of graphically and then again you have this property which imply that this form is closed therefore exact and again like before when you take this function H minus PQ capital by the same argument by Poincare argument critical point of this function are fixed point of F and what we can easily check that this function is periodic and this is exactly periodicity of this function is where I use the conservation of mass property because again other was it would be kind of not periodic but just a Q variant and to prove that this constant are equal to zero you need this condition. Okay so now let's consider general case and now this additional fantastic idea that we have a transformation which is not close to identity. It's still a composition and it's an answer either topic to one which is close to identity. Then it is composition of those which are close to identity. So this is a I think well I think Poincare could realize this probably. Okay so then we have a N map. So you presented F as this product and let me just I'll give you this argument when we have two of them. If you have more it's just exactly the same you have to just write this thing. Okay so now you have two graphs of F1 and F2 and now in this eight dimensional space I consider product of these two graphs. So notice then you take a product of two graphs of small diffeomorphism. It's again some kind of surface in this space which is close to the diagonal there in this eight dimensional space. So this product of graph is given by product of this equation just given by this four equation. And exactly for each of them you have a Poincare argument which say that you have this PI DQI plus QI capital DPI capital R differential and if you oh sorry this isn't should be H not F. And if you just correct them by this quadratic function that they are the two periodic. Now let's take a G is equal sum of these two functions. So now you use the following set theory. So you say okay so you have a product of two maps. You have a map, one map F1 and you have another map. So this is a map F1 and you have another map F2 of something to something. So what, so now we're looking at composition. You have a graph of this map and a graph of that map and we're looking at composition of this map. What are fixed point of the composition? So fixed point, how to find fixed point? You should say okay you start with some point then go to some point here. Then you need to take the same point here take its image and then you say that this point is the same as this point, right? So let's just write this property. Write this property. So fixed point of F in one to one correspondence with this intersection of the graph with this condition P2 equal P1. So this is precisely two conditions which express that this point is equal to this point and this point is equal to this point. That's what I wrote. And again, this just mean that this fixed point are in one to one correspondence with zeros of this one form. And this one form is a differential of this function G which is periodic with respect to all variable and plus some quadratic form. So let me kind of rewrite it. Just I change variable to write P1 minus P2 is equal to U1 and let us make this change of such variable and then you get this formula that this differential form is just differential of G which is again periodic function minus quadratic form. Okay, so again we have exactly the same situation. We have a, so this periodic function so therefore this periodic function descend to the torus with respect to the, you can quotient by this Q translation in Q1 and P capital one of the there. So it's a descend to this torus. So on torus cross R2, you have a function which is a periodic function plus some quadratic form, non-degenerate quadratic form. And this is a subject of what is called stable Morse theory. If you have a periodic, if you take a quadratic function has a sum zero, that's a critical point. You can find it by minimax. And if you perturb the quadratic function by something with kind of bounded, then this thing is still have a critical point which is we can find by minimax. Anyway, it's a kind of elementary Morse theory which called stable Morse theory and it's imply existence of this critical point. So this is a whole proof of Poincare of Arnold conjection. And so it's, so really, really it's a proof which just can be written in one page and it's in addition to Poincare argument it just require just only these properties that any map can be the composition of small one. Okay, so actually I jumped ahead because this proof, this my proof of Poincare theorem required four dimensional symplectic geometry. I kind of passed to this graph and I was talking about Lagrangian submanifold in four dimensional space. But in fact, kind of like logically first case to consider B, so like it's very kind of, you know, when you study, of course when you study complex geometries and you're jumping from complex dimension N to N plus one and this is kind of you jumping in real sense by two but this is a kind of real mathematics and so therefore we would like, we do not want jump by two. So we want to study symplectic geometry also in odd dimension and analog of symplectic geometry is on dimension it's called contact geometry and let me just talk a little bit about contact geometry. So what contact geometry is about? So if you have a, let's take a, let's talk about geometry of function of one variable. So if you have a function of one variable and you can somehow I would like to see simultaneously this function and its derivative and I want to draw the kind of graph of the function and its derivative. So if you take this curve which is a graph of function of derivative then it satisfies this equation of course GZ is equal PDQ and hence in three spaces the curve which is a tangent of the plane field given by this path in equation so and this is a precisely called contact structure and curve with this condition which is tangent to plane field called Legendre and curve. So this is a kind of picture, picture of contact structure you have this non-integrable plane field so they kind of rotate being tangent to some line field and this is a picture of Richard Montgomery some graphic artist Gregoire Lyon of Legendre and curve and tangent to this one that's kind of roughly what they look like and generic like a generic Legendre and curve of course need not to be graphical so it can, so if you projected to ZQ plane it would look like some kind of graph of multivariate function with this cast point but when you projected to PQ plane you still always get a smooth curve possibly with self-intersection so you can think that you can just looking at the functions with curve like this emerged and you integrate the area and you're getting picture like this. So this curve is not something abstract it's again it's applied mathematics because this front you see when you're skating you're kind of on the skating you see the fronts of Legendre and curve and projection of fronts you see when you're looking at this beautiful picture which is called caustics. So let's discuss the following kind of question which I, okay so there is this in elementary calculus there's a called roller theory, right? So we have a, if you have a function which take equal value say at the end then it has to be, you have a point where derivative is equal to zero. So in other way to say that Legendre and graph of this curve cannot be contained if it contained in upper half space P greater than zero then the right end of this curve the z-coordinate of the right end have to be above z-coordinate of the left end. Question, is it true for any Legendre and curve? Answer of course negative this is the front of Legendre and curve where slope at every point is positive but nevertheless at the end it's lower than the beginning. So how it turns out the answer is positive which you impose the following condition you consider all Legendre curves say between two planes q equal zero and q equal one and consider such curve which is a topic in the class of Legendre and curve to the graphical curve and turns out for any such curve you have the following theorem and this is a kind of, this is a, I've proved some version of the theorem in 81 and this is a kind of like follows another version from then you can see in 83 and it also follows from Chicano result in 87 and it's also followed from argument of Jean-Claude Cicara for in 87. So it says that if you take a front of Legendre and curve in this class then this front inside contain graph of continuous piecewise smooth function. So you can, on this front there are a lot of unnecessary pieces and if you just leave only this red part so you get graph of continuous piecewise smooth function and because derivative is positive even for such piecewise smooth function it has to have, and here is higher here. So this is a kind of essentially equivalent to Daniel Benekin's theorem and so it's kind of again it's implies rigidity, symplectic rigidity in content dimension three and in fact the proof and the kind of I had in mind different proof but the proof by Cicara is exactly in the spirit of my proof of one kind of theory. So the true new era kind of started in symplectic topology with Gromov paper to the holomorphic curve and what Gromov did, he just, he needed some tools to kind of to how he, well actually I should say the Gromov this father of all this H principle types mathematics but he always hated it from the very kind of beginning and kind of he always always wanted to do something more rigid and so kind of this and in particular, in particular very much wanted this symplectic kind of rigidity structure and he would kind of, symplectic structure is kind of too flexible like topology there is nothing to hold and so he wanted some structure which would rigidity and then would allow us to define some invariance and then hopefully this invariant would be independent of this extra structure and this scheme of course by the time was already realized by Donaldson and for dimensional topology using model space of solution of some different elliptic problem of this so-called anti-Seldulian mill situation and Gromov's idea of Gromov was just to realize similar thing and not only idea that what he did and he realized that it's using instead model space of all the morphic curve and probably I will kind of not talk about this in detail because it's required more than 51 seconds which I left but still I have a lot of things to say okay so just I want to say that a lot of kind of great development follows after Gromov and this guy first of all in his paper kind of fantastic number of new results for instance as I said he proved that there is no embedded Lagrangian sub-manifold in R2N and then he defined first kind of genuinely symplectic by genuinely symplectic I mean something which is dependent on symplectic for not volume form invariant as we called for instance Gromov with now and it's allowed him to prove the squeezing theory non-squeezing theorem that as a ball cannot be symplectically embedded in poly disk of large volume but kind of narrow in one direction and he proved uniqueness of standard that infinity symplectic structure in R4 and contractability of the group of compact supported symplectic diffimorphism of R4 and many, many morphic curve. So Gromov promised even more great result in his paper Grot 2. So unfortunately Grot 2 was never written but then kind of like despite this kind of the development continued with kind of fantastic speed and so first floor kind of modified Gromov approach making more like connoisseur proof to define his floor homology theory and then Hofer applied this instead of from defining invariant of domain he used similar scheme to apply to define invariance of diffimorphism and this kind of created all this area of Hofer geometry which is really in some sense brought closer to realization this Poincare dream of getting great result out of nothing just kind of like using only symplectic property of a map and not analyzing equation in detail and so this is also kind of independently was done by Claude Viterbo and then Cliff Taubes united this with cyberquit and severe in dimension four which again brought a lot of great progress in for four dimensional topology and then furthermore this was a kind of by physics development and suggested that you can define what is called quantum product it's some kind of deformation of a cosmological multiplication on symplectic manifold and it's also suggested some kind of nice way of packaging invariant coming from Gromov Gromov homomorphic curve to what is called Gromov written potential and this kind of found link with theory of integrable system with mathematical theory of mirror symmetry, et cetera so I also want to say that flexible site of geometry which is kind of another start amazingly still alive well so for instance, Larry Gooth proved in some sense almost converse to Gromov non-squeezing theorem so Gromov non-squeezing theorem says that when you embed say one polydisk to another then the smallest radius of the first one have to be smaller than smallest radius of the second and of course volume which is a product of radius should be less than the product of this one and there was a lot of effort of people try to prove something about intermediate radius like in dimension if you take four dimension there is nothing left but you have in high dimensions there are a lot of intermediate radius and what Larry proved that at least up to some constant there is no other constraint so also kind of recently with some collaborator I proved that in fact when you consider problem of Lagrangian self-intersection then in fact you don't have anything like what was Lagrangian intersection theory says it says that if you have two Lagrangian manifold that they have to intersect and kind of more in the number of points and then it's kind of required by topology and also I don't know if Arnold formulated it or not Arnold but at some point all questions in symplectic topology became known as Arnold conjecture so it was Arnold conjecture but maybe it was Arnold conjecture maybe not that if you have a say any if you take any exact Lagrangian immersion the number of self-intersection point also should be bounded below by kind of something like half of rank of homology and turns out that this is a completely wrong and for instance like this is one corollary that is a three dimensional case every manifold admit Lagrangian immersion to three dimensional into R6 is just only one transverse intersection point and kind of similar result exists in another dimension so I in fact wanted to say much more but probably I will skip this thing so what I just let me just say what I wanted to say but I will not say so I wanted to say that amazingly there is the symplectic geometry came kind of on the new round like this this Gromov written theory it's some kind of Gromov written theory you can interpret as a kind of enumerative algebraic geometry in some sense with the counting of holomorphic curve subject to certain conditions and so what turns out that this Gromov written theory can be formulated as a symplectic geometry on some other space some infinite dimensional space and just developing the symplectic geometry in infinite dimensional space you amazingly get result about your finite dimensional manifold for instance like this is a question of counting rational, rational curve in the complex projective plane and this is a kind of many now beginning from conservative many and recurrent relation for this is known but what I just wanted to say just kind of brief you so if you consider this generating function for all these numbers then you can do the following thing you consider some infinite dimensional space space of some formal Fourier series with this symplectic form in this one and now you consider this particular Hamiltonian in this space and you consider flow of this Hamiltonian and then this flow is in fact kind of if you write this is if you write as one equation is a so-called total equation dispersionless total equation and then you take a zero section and in this space and flow it with this total flow with this flow of this Hamiltonian and so this moving manifold satisfy what is called Hamiltonian Jacobi equation this is one written then and then you take it and then you take this Lagrangian manifold and this Lagrangian manifold has some generating function that it's a graph or differential of some function you take this function and evaluate this function at certain particular point namely at the point with coordinates z e to the ix zero so evaluate it and fantastically you get this generating function for this number okay so just well so this is a kind of great symplectic highway from Fonkare to Gromov and it's still very busy and going very far I'm squeezing so much in the 51st century something that confused me a little bit about the history of the Arnold conjecture did he formulate it for two dimensional surfaces only or did he? No, so Arnold certainly formulated in general case Arnold formulated in many, many times so there was a kind of, there was his 65 paper and then he repeated in many other publications so I'm not sure what exactly in 65 paper but by 69 definitely kind of all cases of known today Arnold conjecture formulated and Arnold conjecture as you know it's kind of not proven yet right because Arnold conjecture Arnold conjecture what is Arnold conjecture? It says Arnold formulated in the following way that there's a number of fixed point of symplectomorphism is bounded below by the number of critical point of function and of course people immediately replaced it okay it's bounded by some of beta numbers so sometimes it's correct bound but quite often it's incorrect bound and kind of in general case they're just completely open and there is absolutely no even close to be proven maybe it's wrong You mentioned two different things that's the fact that you come to a new flexibility result and also it exists is a homological theory What is the logical link between these two trends? Well you know that the whole symplectic topology I hope to convey this idea it was kind of was born out of this fantastic tension on flexible side and on rigid side right somehow that was a some result on flexible side some other rigid side but suddenly this fantastic explosion happened and so kind of what I just wanted to say is that this pressure continued Are the manifolds? Okay so great question Okay so it's certainly people I think generalized this line of idea to many manifold no not just torals for instance like you have like Arnold so it's essentially what I said this idea of generating function right so this is just a slightly packaging of this the question where generating function can be applied and so you know this is a in more no kind of modern language this is now kind of reformulated on this categorical language of shifts and so Tamar can claiming but I don't know that this can be somehow he can define Gromov with invariant using this so maybe it's possible so then it will be like development of this idea