Symplectic field theory and its applications
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Eliashberg, Yakov
Symplectic field theory (SFT) attempts to approach the theory of holomorphic curves in symplectic manifolds (also called Gromov-Witten theory) in the spirit of a topological field theory. This naturally leads to new algebraic structures which seems to have interesting applications and connections not only in symplectic geometry but also in other areas of mathematics, e.g. topology and integrable PDE. In this talk we sketch out the formal algebraic structure of SFT and discuss some current work towards its applications.
 Ladies and gentlemen, it is my honor and my great pleasure to introduce Jakov Ilyashberg, a very generous mathematician. Yershey-Ilyashberg got his PhD in 1972 in Leningrad, in our Simpetersburg again, under the supervision of Vladimir Hochlin. The title of his dissertation was Singularities of Mappings. Yershey-Ilyashberg then took a professor position at Sik-Tikva in the Komi Republic. After a difficult period from 1979 to 1987, he was allowed to emigrate from the USSR to the United States of America in 1987. He visited the Mathematical Sciences Research Institute MSRI in 1988. He then joined the mathematics department of Stanford University in 1989, where he has been ever since. He now chairs this department. The present area of research of Yershey-Ilyashberg is Simplectic Topology, a discipline that grew from the fantastic developments Simplectic geometry underwent in the last 30 years under the influence of many people, in particular Mikhail Gromov, but also because of the push from theoretical physics. Magnificent geometry really appeared as a geometry of a new kind in the memoir of Joseph Frouillagrange of 1808 on la variation des éléments des planets, then was fully established by Sir William Hamilton with fundamental contributions by Darbou and Jacobine, the second part of the 19th century, and then the local content of the theory was completely uncovered. The beauty of the theory is praised by Felix Klein in his introduction of the history of mathematics of the 19th century, but it doubts its usefulness. The emergence of quantum mechanics in 1920s gave a major impetus for further developments of the theory and showed its relevance for a wide area of applications. The shift towards a global Simplectic geometry took some more time. A first major step was taken by Eugenio Calabi in the 1960s, but it is Mikhail Gromov who turned things around somewhat later. By exhibiting completely new rigidity properties of automorphism Simplectic manifolds. Conjectured by Vladimir Arnolv and a number of fixed points of Simplectic automorphisms played also an important role. Jascha Eliasberg started to be interested in these problems in 1979. At the time, Connley and Zander proved solved Arnolv's conjecture for surfaces. Further important developments are due to the regretted Andreas' flaw. Jascha Eliasberg started a very fruitful collaboration with Helmut Hofer in 1995. He presented the preliminary version of the now monumental theory he developed fully since 1999-2000 in collaboration with Hofer and Giventown. In his lecture, Jascha Eliasberg will tell us where Simplectic field theory and its applications stands today. Jascha Eliasberg. So of course it doesn't work. Okay. It doesn't work. Oh, okay. So now. So Simplectic field theory, as Jean Pierre mentioned, is some project which we initiated few years ago with Alexander Giventown and Helmut Hofer. And it's not yet far from being completed. A lot of people contributed to just building foundation, application of the theory. And this is of course not complete list. And there are a lot of people who not formally working in Simplectic field theory, but on the other hand they really, they work very relevant. And I will not give their names because I will definitely forget someone. So what I will start is some kind of crash, three-minute crash course of Simplectic geometry. So Simplectic geometry is just geometry of a non-degenerate skew-symmetric bilinear form, like Euclidean geometry, geometry of symmetric form. So symmetric form have at least one invariance, a signature, but skew-symmetric form is unique if it's non-degenerate. And viewed as a differential format always can be written in this form. And you also kind of important to think about Simplectic form in the context of complex structure. So if you think about even dimensional space and Simplectic geometry only exist in even dimensional space, and think about the space as a space of complex coordinates, then Simplectic form is just a minus-imaginary part of the standard termission product. So Simplectic manifold is a manifold which is locally modeled on this R2N with this special form. And equivalently according to Darbuth theorem, this is a manifold with a non-degenerate closed differential to form. And very important, again, like an Euclidean space to consider Simplectic form together with this complex structure, and in this case we want to consider almost complex structure, which means this complex structure on tangent bundle to the surface. And we say that it's compatible with Simplectic form if they relate to each other the same way as this standard Simplectic form and standard complex structure relate to each other on Euclidean space. So it's important, and this will be really crucial for the holder's development, that the almost complex structure, given omega, you can always find compatible in this sense J, and more or less this is a kind of convex choice of such J, so it's a contractible choice. But on the other hand, if you have a J, it's not always possible to find a compatible omega. So omega is much more important. So Simplectic manifold, unlike, on the first glance it looked like parallel to Riemannian manifold, okay, we have a manifold with symmetric form or manifold with skew-symmetric form, but unlike Riemannian case, Simplectic manifold has a huge group of different morphisms called Simplectomorphism. Well, this is probably due to the fact that you see that the symmetric form has a n times n plus 1 over 2 n trials and skew-symmetric only n times n minus 1 over 2, so it's less equation and this gives such a huge difference. So the group in the two-dimensional case, an n equal 1 for surfaces, Simplectic form is just area form and the Simplectic geometry is just geometry of area preserving transformations. But in high dimensions, this is a drastically different. So the group of Simplectomorphism form really proper and amazingly close to C0 topology. The C0 closure of this group is a proper subgroup of the group of volume preserving transformation. So this is a kind of really fundamental theorem which opens the whole area of Simplectic geometry. So the algebra of this group consists of Simplectic vector field and this is just equivalent that this is a vector field which has this property that if you contract them in Simplectic form, you have a closed form and especially we would like to have that consider the case when the dual form is exact and then this function is called Hamiltonian and the corresponding vector field called Hamiltonian vector field and I will denote by this Simplectic gradient of H. And H is called Hamiltonian function. So in the Darbouk coordinates, of course, everybody knows this canonical form for this Hamiltonian vector field. So this kind of Hamiltonian equation differential for the flow of this form p dot minus dh of dq, q dot is equal to dh of dp. So for any compatible omega j, you always have the Simplectic gradient just gradient rotated by j. So very important object in Simplectic geometry are Lagrangian submanifolds. The submanifold on which omega vanishes and maximum such dimension can be just half n. And in terms of complex structure, they characterized by the properties that when you multiply by j, the tangent place become its own orthogonal complement. So there is a kind of two most important examples of Lagrangian manifold. One when you take a cotangent bundle and cotangent bundle of manifold has a canonical Simplectic structure, which is differential of this famous form pdq. And then section is Lagrangian if and only if it's a closed form, the section. In particular, if you have a, say, differential of function and you have exact form, then its graph of derivative of function is Lagrangian manifold. On the other hand, if you consider Simplectic manifold and you have a Simplectomorphism, then graph in the product, if you take a Simplectic form omega on one factor, Mimolus omega on the other, then graph will be Lagrangian. And it's always kind of important kind of play in Simplectic geometry. Then when you have a kind of one split of coordinates, then Lagrangian manifold looks like graph of Simplectic transformation and another system of coordinates, they look like graph of derivative of functions. So there is because of this kind of huge group of Simplectomorphism, it's very difficult to find Simplectic invariance beyond some obelisk one. So of course, we have a total volume, it's invariant. Then there is this homotopic class of a compatible almost complex structure invariant. And in the closed case, the homology class of the Simplectic form is also invariant. But it was for till the beginning of 80s, nothing else was known essentially, more or less some minor things. And this was a Gromov's great insight who introduced the homomorphic curve as a tool for finding more subtle, specifically Simplectic invariance. And we will discuss this today. So let me talk a little bit about this kind of homomorphic maps between almost complex manifold. On the first glance, you can easily talk about homomorphic map between almost complex manifold as in this situation for integrable case, just say, okay, differential is complex linear map. But just unfortunately, if you, if you, if the dimension of the source manifold is real dimension greater than 2, then this is a highly over-determined system and you never have any solution generically even local. So it's a really kind of miracle that the Karschelian equation in integrable case and high dimension have solutions. But the dimension s equal 2 when s is the Riemann surface. Then, and by the way, in this case, almost complex structure and complex structure on the Riemann surface and two dimensions is exactly the same thing. So it's just a conformal structure. And in this case, you have a locally, the system is determined, you have exactly the same degree of freedom of constructing homomorphic map as in integrable case and they call homomorphic curve in this case also J, also J-holographic and or pseudo-holographic. I don't like the word pseudo-holographic because it doesn't carry any information. J-holographic sometimes useful because you want to say for which J. So I will fight with this kind of terminology. So if moreover, if s is closed and with appropriate boundary conditions, this is an elliptic system and its principle symbol the same as for standard debar operator. So you can apply standard thread home theory and under certain transversality assumption, you can define modulate spaces of homomorphic curves which are finite dimensional manifold or at least orbitals or algebraic geometry, they say stacks but okay, so here they just orbitals. So, but this is a kind of, of course I'm saying this thing but it is an extremely serious statement. So you see the whole point of, well one of the points of introduction, almost complex structure rather than integrable structure that like you have a much more freedom of constructing such object and in algebraic geometry, algebraic geometry you really restricted with your tools how you can achieve transversality but here you kind of hope that you can have almost as degree of freedom as an anthropology with achieving transversality. But unfortunately it's not quite the case, it's almost the case but not quite and in some case indeed you can achieve transversality and just kind of forget about it but on other cases you in fact have to unreach the class of your object in talking about some what more general equations and homomorphic curve equation but I will ignore all this issue about it. But there is a huge subject and for instance one of my collaborators Helmut Hofer and this is Kosser devoted a lot of efforts of building appropriate foundation for this situation. Okay, so local theory as I said in the non-degenerate cases fine for homomorphic curve but if you want to get global result you need to have a compactness and here is this kind of one of the major result which was proven by Gromov he proved that for homomorphic curve you can prove similar result as a kind of this famous Knudsen-Dilling-Mampford compactification of just the space of Riemann surfaces. But there is a big caveat only if the area of homomorphic curve is uniformly bounded and it's extremely difficult to control this area so the role of symplectic form as we will see come precisely to in this point. So let me kind of recall few things about the Riemann for compactification so according to the uniformization theorem you can talk about this consider Riemann surfaces with mark point is the same as well for this condition which it's the same as hyperbolic matrix on the punctured surfaces so this condition precisely kind of called stability condition precisely just ensures that you have a hyperbolic geometry on this thing. And so this compactification Knudsen-Dilling-Mampford obtained by adding nodal surfaces and they look like union of two casps in the hyperbolic interpretation and the duration means shrinking a closed geodesic to a point. So the stability condition as I say have to be satisfied for every component. So Gromov compacting theorem is exactly the same but it's different the stability condition it needs to be satisfied only for constant components of nodal curves which are sometimes called good. So it is a kind of slight conserved modification of Gromov compacting theorem. So this is that you may have this phenomenon of bubbling off of holomorphic spheres so this is kind of doesn't exist in just case when we talk about Riemann surfaces. And as I said already the role of symplectic form is to cure area bounds and the point is that with compatible J omega area is the same as symplectic area meaning this is just integral of symplectic form and hence because form is closed it depends only on the homology class of this map of the surface and so when we vary our considered holomorphic curve in given homology class we don't have to be worried about control of area. The same true in the boundary value problem with boundary Lagrangian manifold. So Gromov scheme of defining symplectic invariant where holomorphic curve theory was the following. Pick a generic almost complex structure compatible with omega. Then the compactified modulate space of holomorphic curve of genus G and the fifth homology class can be viewed as a cycle as a kind of similarly unreached modulate space of just smooth surfaces which is kind of much more simple object and its topology can be understood. The compactness theorem ensures that the homology class of this cycle remain unchanged when one varies J while keeping it as a compagny of the omicron. This is an invariant therefore because all J kind of homotopic and hence we get an invariant that's what I call Gromov invariant which just invariant of symplectic structure. So let me give you example of kind of application and this is a Gromov theorem with kind of somewhat in this form it appeared in our joint paper with Gromov. So let's take a so-called symplectic camel problem it's probably called this way by Arnold. So let's consider half space in R4 of course you can go into high dimension but the first non-trivial first when this phenomenon appeared in dimension 4. There is a ball and half place with a small hole and the radius of this hole is smaller than the radius of the ball. And what we are trying we are trying to kind of to put this by symplectic isotope put this ball through this hole and the claim that it's not possible. So why this is not possible? So okay so suppose there exists such isotope. Then what we do we at every moment we consider almost complex structure which has the following property on the ball is just push forward of the standard complex structure at infinity it's standard and also near this our near the boundary of remaining part of hyperplanet standard. But and because this is a power of almost complex structure because it's just homotopic simple objects it's kind of easily to prescribe almost complex structure as you wish on some places and just continue to extend elsewhere with the conditions that it's compatible with symplectic form. So now we take a Lagrangian cylinder in this plane so just radius 1 Lagrangian cylinder sitting in this plane. And consider the modular space of homomorphic disk with one marked point this boundary in RL. So you see before we deform of course this flat Lagrangian cylinder was filled by flat Lagrangian disk. And we had this kind of our Gromov invariant this cycle this kind of one dimensional family of this Lagrangian manifold. And this cycle survive when we deform almost complex structure for any J we have such family of homomorphic disk. And this family together they form some kind of hyper surface spanning this Lagrangian manifold. And when the ball passing through the hole at some moment he has to intersect this kind of moving wall. So therefore in some moment we will have our homomorphic disk this green one kind of and it's centered precisely on one of this Lagrangian on this homomorphic disk. And then in this moment we can look at the area. On one hand we can compute the area of the intersection of the piece of homomorphic curve inside the disk. Because this homomorphic disk has a boundary Lagrangian manifold and therefore it's area is fixed. It just as I said it just depends it's a simplex invariant just depends on the radius of our Lagrangian manifold. It's just equal to one or pi. And on the other hand I can consider this area looking at the ball itself. And at the ball it's a ball with standard complex structure. And at the side you have a curve passing to the center and according to standard the perimeter inequality for minimal surfaces area should be bigger than pi r square and we get a contradiction. So gromofiton potential is a certain motivating motivated by physics way to package algebraic information contained in gromofinvariant. So let me kind of talk a little bit about this. So let's consider this compacted simplex manifold with a compacted complex structure and take a cohomology class. So let's think about this cohomology class as some differential form on our manifold. And then this is a kind of physically motivated notion that defines the acrylator just pulling back this to the small space of the homomorphic curve, evaluation map at the first mark, point second, etc. And integrating over our manifold. So, okay, so I just said what was written here. Okay, so this is the acrylator and then we would like to write some kind of generating function for all the acrylator. So we fix the basis in the cohomology and also just to talk about homology class we fix the basis in two dimension homology and identify any, so any homology class can be expanded in this basis with coefficient d1, dk and because this d degree and just will not distinguish between this d and this homology class. So we also can introduce this variable for this, all this d1 for this basis in cohomology and use this notation. So Gromov-Witton potential is a generating function for all, for all acrylators. So you just take this formal linear convention, it's important to take the t as a graded variable of the same grading or parity of grading as the theta, then we get this thing is a kind of each component even graded and this is a symmetric expression in this t. And if you write expansion in the t variable you get all acrylators. So the meaning of acrylator is kind of very simple, it just, if you consider cycles, duals to this cohomology classes then as it was explained in the talk of Andrea Kouinkov yesterday, what we're counting, we're counting homomorphic curve, some subject to this incidental condition of restricting some marked point to certain cycles. So also you can upgrade it further including so-called descendant but variable so if you don't know this you probably will not be able to learn from my talk so just what I just want to say is that you can further kind of elaborate this incidental condition for the homomorphic curve imposing some restriction on jets of homomorphic curve on this derivative and it's traditionally done in certain way but important thing that you get this homomorphic potential become dependent of infinitely many variables instead of finitely many which was dimension of our cohomology. So genus zero part of this expansion is called rational homomorphic potential and for instance for the case of CP2 let's take the standard basis of CP2, this one, the two-dimensional cohomology class and four-dimensional cohomology class, choose CP1 as a just projective line as a standard generator of two-dimensional cohomology and consider this function. We put two first variable corresponding to this theta 1, theta 0, to 0 and only take this one which correspond to point constraint and so this coefficient has this very simple and numerative meaning. This is as it was in yesterday Akonikov talk is a number of rational curve for given degree D in the complex in the projective plane which pass through k points in general position. So you have this standard number for instance you have through two points you can pass one line etc and well in general of course we have this relation between number of points and degree in order to have finitely many. And what I want to do I want to describe some kind of geometric algorithm generated by symplectic field theory which I didn't explain yet for computing this function. So this is a today there are many kind of recurrent procedure first started from Kansai for computing this number but I will explain some another one and this one is also actually in fact close to what Akonikov and Ponderpand the co-operator for my reason. Okay, so let's do the following thing. Consider this infinite dimensional space of this C2 very formal Fourier series just kind of think about this as a formal loops in C2 space of loops in C2 and so in coordinates I choose the coordinates in the space just for the efficient of this expansion. So notice there is no constant term kind of loop center than zero. So then we can see the some symplectic form in this space. So this is a symplectic form which also correspond to Poisson bracket versus PM0 bracket with QM1 is equal to K2M. So because of so it's also can be written kind of in this form. So if you think about this Poisson bracket as a tensor on just form defined on co-vectors and co-vectors themselves can be viewed as the same type of loop because it's vector space then is just given by this formula. Don't want to see my hand everywhere. Okay, anyway, so this is a pretty famous symplectic structure which appeared in the theory of integrable systems. So consider Hamiltonian function on this space. Hamiltonian function is functional on this U. So let's take this one and it defines some Hamiltonian flow and if you write the Hamiltonian equation you get this equation. And this is in fact turns out to be dispersionless Todor equation or continuum linearity of Todor lattice and written as one equation is just the second kind of order differential equation and I was told this by Dubrovian. I didn't know anything about Todor before he told me. So let's do this now the following construction. It's take zero section in this space. So zero section in this space is the kind of space where the Q-coordinate is equal to zero. So you can think about this if you think about terms of Fourier expansion then I just take Fourier expansion it's only if you have a positive coefficient and no negative coefficient. And then move this zero section under our Hamiltonian flow which I described. So that will be some moving Lagrangian manifold and I denoted by LT. And Lagrangian manifold I told you already they are graph of derivative of function. So they generated by some function if they are graphical. But they are in the world of formal power series and here everything is graphical. And so we can find this function GT and finally evaluate this function and this particular point. And this is precisely our function f of tz expansion of this function is this enumerative number of algebraic curve. Okay. So it's important that Gromov written potential is a symplectic invariant. It's Gromov written invariant. So it depends only on omega. So while omega explicitly does not participate in the definition, nevertheless it's really it just keeps this background controlling role. So how to compute Gromov written invariant? So it's easy to say okay take j take this modless space but nobody ever solved any kind of non-trivial d bar equation explicitly and kind of so it's just not possible. So the only way to do it either you kind of try to deform it to some kind of so you deform it to j or degenerate the j to some situation where you can understand think and then you say because it's invariant it was the same as before. And one way to do it is just kind of to deform it to algebraic situation and another way of doing this is to degenerate. And symplectic field theory is one way to degenerate it. And tropical geometry is the other one. So what we do is split manifold into pieces. So if you have a real hyper surface in almost complex manifold you can kind of start to stretch the neck as they say engage theory. So just insert bigger and bigger kind of piece where they solve almost complex structure independent on this vertical coordinates. And then in the limit you have a manifold with cylindrical ends. So this is impossible to do in integrable case. It's extremely restrictive but in non-integrable this is advantage you can always do that. And in fact in integral case sometimes this is also possible but it's much more restrictive. I'll say about this later. So what happened with holomorphic curve when we do this? They also split. So in fact what may happen that homomorphic model is based on homomorphic curve kind of split so you observe the curve which looks like this. So in upper part you have some piece of curve and lower part you have some piece of curve and also you can have a bunch of cylindrical parts. And there is a piece of curve stuck there and all together they match data and they fit in your original curve which you have in your original manifold. So this kind of brings us to studying these two objects. This cylindrical almost complex manifold, this manifold which are kind of cylinder where almost complex structure is invariant with respect to translation and this is the kind of definition. So I'll talk about this more detail later and also manifold with cylindrical end, this non-compact almost complex manifold which are cylindrical outside of the compact set. It's precisely those which appear as a result of such splitting. So let's kind of important that every cylindrical structure as a very cylindrical structure can assuage this vector field tangent to our manifold. Why? Because J times D or DT, where T is this vertical coordinate. So this is again the same picture. So I say that there is a splitting procedure which brings out of the manifold this manifold with cylindrical end. So any cylindrical structure is determined by what is called in complex analysis the CR structure. That's you have on our manifold you can in tangents why the odd dimensional manifold but it contains maximal complex subspace of real co-dimension 1, this intersection JT, YT, YT. So it's a, you have a distribution of complex tangency to Y and then you have this vector field R which is transferred to it and given R and given C you can always of course define some one form which satisfies this normalized by conditions of equal 1 on this vector field and equal 0 on this plane C. So generically all periodic orbit of the vector field R and periodic orbit will play important role in the story are non-degenerate and thus there are finitely many of them of bounded period. We will call this case generic and we will have this notation for the space of periodic orbit. So however it's kind of useful also to consider so-called Morse-Bott case when space of periodic orbit are organized as some kind of sub manifold for instance like all orbits are periodic. I told you that we cannot do anything in arbitrary almost complex manifold unless we have a symplectic structure which kind of control this manifold. So we also need something like this here. So we will assume that there exists close to form of maximal rank on Y. Notice that Y is odd dimensional so the maximal rank of close to form is 2N-2 if dimension is 2N-1 is one less. And so we want this form such that R is in the curve so R is a Hamiltonian vector field of this form. And we want this R form lambda be preserved by the flow of R. So this is the same as just to say that R also in the kernel of the form d lambda. So the pair we call this pair stable Hamiltonian structure and cylindrical almost come crutch is called compatible with H. So there are three important cases of Hamiltonian structure. In fact, I don't know any other kind which would not be like small modification of this one. So first is a floor case. So in this case our manifold just mapping torus we started with symplectic manifold M and then consider mapping torus of the symplectomorphism. And let's suppose for instance that the symplectomorphism is a topic to identity so it generated by time one dependent Hamiltonian. In this case R just equal do dt plus symplectic gradients of HT and periodic orbit are the same as in 1-1 correspond to this periodic point of f. So you already noticed that my picture of course pretty bad but I got as a present from a TNGist one of his pictures. Okay. So this is a TNGist picture of the mapping torus. So there is also contact case and in the contact case why is this contact manifold that means that C is maximum nonintegral plane field and lambda is a contact form and omega d lambda. In this case R is called red vector field of lambda. And another case circle bundle case when we have our manifold is circle bundle or symplectic manifold and R just generates this one action and so the space of periodic orbit is just our original manifold. So now we want to discuss homomorphic curve in this manifold. They themselves have to have the cylindrical end. First talk about cylindrical. So just cylindrical manifold. Notice that if you just take a straight cylinder over this orbit then you get a homomorphic curve because do dt direction is precisely j times this is equal R. So you have homomorphic curve and therefore you would like to consider just homomorphic curve which asymptotically look like this cylinder. So now I can say what just SFT is. It's the first approximation because in fact first of all the SFT is some kind of functor from the category which I call geometric category SFT where object are Hamiltonian structure with compatible J and morphism or symplectic abordism and algebraic category of the FFT the object are differential by algebra which I will discuss and certain Fourier integral operator kind of interwining them. So in fact it's just only zero approximation because the structure of the SFT is much richer and first of all it's not one category but at least two categories and second there's much more kind of structure in this object. So there is a kind of rational version of SFT when we're talking only about general zero curve and then instead of this while algebra we get a Poisson algebra or symplectic manifold like was in this example and instead of Fourier operator we get Lagrangian correspondences. So by the way symplectic abordism and just symplectic manifold bounded by this out to Hamiltonian structure and symplectic form restrict to the boundary as this omega plus and omega minus. I just want to say that you know that there's an interpolation there is this issue how you compose symplectic abordism there is not canonical operation we have to smooth but in this kind of world. In fact when we take abordism we immediately attach to it the cylindrical end and when we need to compose abordism we just take them one on the top of the other just without any glue. So let me now kind of describe this object. So first of all I need to say about periodic orbit. So periodic orbit themselves in non-degenerate case and restrict only to non-degenerate case can be odd or even depending on this return Poincaré map it's a kind of hyperbolic or elliptic orbit roughly speaking. So let us associate with every simple orbit of our R just sequence two infinite sequences of graded variable and I grade them say Z2 grading is precisely the same as I described here even or odd but there is a in principle there is an integer grading the mass of conical bits and rindices but I am not discussing this so then organize them in the kind of Fourier series. So in fact you can think that we have this big space where this like dimension is equal number of such gammas. So and you can think that this collection of those you gamma just coordinate so it's kind of loop collection of all of them is a formal loop in this big space. So in the first approximation and this is a even not first probably minus first approximation SFT associated with h and compatible J the following algebraic objects. So I will describe separately rational SFT and full SFT. So object in rational SFT is a graded Poisson algebra generated by this variable with the commutation relations that this Poisson bracket is equal to K and all other zero and this is the same as that you can consider corresponding symplectic form which is just one over K times this. And you have a special element which I call Hamiltonian which satisfies the properties that Poisson brackets with the self is equal to zero. So let me note that this h is in fact always outgraded but so this Poisson bracket is in fact not necessarily it's not zero automatically that we are in the even case. So in the full SFT you have instead Y algebra in the Poisson algebra so you just have a associative algebra with this commutation relation instead this Poisson bracket so you have one extra variable h bar and you have another element in that one over h bar of W which also satisfies this h commutator with itself is equal which is exactly the same as to say that composition of h is with itself is zero because it's odd object. So it's important of course all people who work in this area know that this well algebra can be represented as an algebra differential operator as this while representation which acts on this so-called Fox space is a space of function of this Q variable and kind of formal power series in this h bar variable and you oh I forgot here sorry I forgot most important thing. So here so should be k times h bar coefficient not just t or dQk but k h bar. So okay so I will not describe morphism in general I just don't have time so I just only consider rational case so in rational case the morphism between this two Poisson algebra is some function which have this properties that Hamiltonian vanishes on Lagrangian manifold defined by this function. So geometric meaning of this form consider the symplectic space corresponding to this Poisson algebra is coordinates pq and this is a as I already told this is a symplectic form which corresponds to this Poisson bracket and then in this symplectic space you consider Lagrangian manifold which is just given by this function and you have this Hamiltonian vector field kind of everywhere given by this function h plus h minus. So the condition that h plus plus h minus constant on Lagrangian manifold just means that this Hamiltonian vector field is tangent to this Lagrangian manifold. So in the case when one of this when you have a keyboard with an empty set so you have just a manifold bounded by one just keyboard spinning one odd dimensional manifold then we associate with this just Lagrangian subspace in this symplectic space. And composition of morphism is a composition of Lagrangian correspondences well I don't have time to explain what is this is well for instance for those Lagrangian manifold which are graphs of symplectic morphism this is just correspond to composition of this map. So where is all the structure come from? This is the above algebra describes the structure of the boundary of the modular space of holomorphic curve with cylindrical ends. So I need to describe this thing so this the appropriate compactification consists of so-called holomorphic buildings. So this is a kind of typical holomorphic building so you have several stories so you have kind of one holomorphic curve another holomorphic curve and you have a matching data on each between each floor of this building. So the co-dimension important thing the co-dimension of boundary stratum in the cylindrical case is equal to the number of non-trivial mean not different from this trivial cylinder component here for instance here 1, 2, 3, 4, 5, 6 you have this 6 components as a co-dimension of the stratum and hence you always have only two story building when you looking as a co-dimension one stratum of the boundary. So in the case by the way in the case of the manifold this cylindrical ends this in the count of co-dimension you only count this non-trivial stratum in cylindrical parts. So why this while algebra? Because this while algebra is precisely the algebra which describes the gluing of Riemann surfaces. So look at this two surfaces and suppose I would like to consider all possible way of gluing there for instance I glue this to end and just extend this trivially here or I don't glue anything or I in this one I glue this to this end this to this end and don't glue anything to this one etc. And this is exactly the same as I just associate with each of this so end of this curve marked by I'm sorry this is not possible to see in this picture but it's marked by some of this p's and q's I marked by p's upper end and by q's lower end and I associate with every such curve I associate monomial we just describe its end and this is h bar to the power of minus error characteristic of this. So you get a so this one you get p1, p2, p3 not error characteristic g minus 1 the g minus minus minus 1 half of error yeah half of error characteristic okay so you have a p1, p2, p3 and this is this one then you have q1, q2, p1 this is this one and then you multiply them in our while algebra just means that transpose transpose all p to the right all q to the left or think about p the h bar d over dq and take a composition of this operator and you precisely get this number of terms. So appearance of this k efficient k in this because of this k h bar because there are k way of gluing of multiple orbits so you see if you have a kind of you can glue this way or you can glue this way this way or this way. So then you can kind of also do upgrade the whole theory in the same way it was done in Gromov-Witton theory so you define correlators and this is the same correlators we kind of count a whole more significant curve with cylindrical end and subject to constrain and the only difference we have to remember what are cylindrical ends I mark them by gamma minus gamma plus here and then organize them in the correlator which now become element of this while algebra and finally consider this generating function which I call this h h bar and I will call this h and then you have the following kind of master equation. So notice I didn't tell that theta consists of the closed form I could take any forms and then turns out the equation of the following if we consider h associated with this set of theta form and d theta and then you get a kind of the composition of them is equal this operator applied to h where this is a s variable correspond to d theta form t variable to theta form. So in particular if d theta is equal to 0 you have that h composed with h is equal to 0 so this is precisely the Stokes formula as I said this is our composition of differential operator describes the boundary of the modular space so the left hand side it's a so it's a in Stokes formula I integrate some form over the boundary and get this one and I integrate form over the manifold and I get this one. So in many many interesting cases you have this in fact h of 0 is 0 and in this case differentiating different sorry differentiating this identity in t variable you get this commutator of h i and m j equal to 0 where the h i j derivative and so you get sequence of commuting differential operator moreover you can add this descent variable so you can in fact this h depends on infinitely many variable and hence by this procedure you get infinitely many commuting differential operator so it's kind of sign of some kind of quantum integral system and indeed for instance if you do it for the case of S1 you get infinite sequence of commuting integral for the so-called quantized burger hierarchy this is non quantized thing and you need to take a kind of quantum version of this replacing this ODE by PDE and then this will be commuting integral for this hierarchy and for instance if you do the same thing for three dimensional sphere and then then you get integral for quantized so called this dispassion of Toder hierarchy which already appeared before in our county hall Morphicor in CP2. So let us talk a little bit about invariant so thanks to the master equation h h is equal to 0 we can define on this while algebra the differential just commutation with h and it still satisfies this property d square is equal to 0 and therefore you get defined homology which provide us with many cases with powerful geometric invariant for instance in floor case you get some far-going generalization of homology floor homology theory which is not really explored at all so there is a kind of wealth of new algebraic structure and kind of cosmological operation on the on the floor homology which which were not studied so in the context case you it leads to so-called quantum homology theory which really extremely successful story in terms of finding invariant of many a fault and in high dimension in dimensions there is a kind of competing construction of invariant via OJVSABO theory but in high dimension there is nothing or cyberquizan theory but in high dimension nothing else is known so alternatively one way use h to define differential on the space of function of q on the so-called folk space and then leads to what is called BV infinity algebra kind of structure on the space and this was recently studied by chili bark and lachev so so let's let's consider the case when you have a x which bounded x is simply a co-ordinate and bounded by y just co-ordinate them with one end so I already explained that you can associate with this just counting homomorphic curve in this combordian with cylindrical end you get some Lagrangian manifold but also you can get this Lagrangian manifold you can do the same thing with this parameter t including this form and including precisely as was done in the case of h and if dceta is equal to zero then similar to this naive case h vanishes on this corresponding Lagrangian manifold but if dceta is not zero but only zero on the boundary then you get the following following equation with a similar d and if you apply to this case of four ball bounded by three and kind of we get then you in fact get so you take this equation and notice here notice here that you have this operator t d or dc so if you differentiate with respect to t you get something d or ds is equal something like this and so and this is exactly Hamilton Jacobi equation for the evolution of this f is that how this so if you do this for the case of of a three then you can explicitly easily compute this Hamiltonian and then you get this dispersion is sort of floor which was used for immigration of rational course and sepital okay so let me do final remarks so just this will be to do least well so there are plenty of things one need to do with this kind of symphactic field theory which currently some people working on on this and on some think people not working here but so for instance there are huge possibilities of application Hamiltonian dynamics so homomorphic for curve successfully were applied to homiltonian dynamics but nobody ever seen any serious application of homomorphic curve different from of general greater than zero to dynamics and and using this I'm sure you can do this for instance you can generalize Poltorovich end of quasimorphism but which it end is top and etc so there is this kind of mysterious relation the theory of quantum integrable system so as I already hinted you have a this odd dimensional manifold Hamiltonian structure and you always get some kind of infinite sequence of commuting differential operators and in rational case it's a good infinite sequence of commuting Hamiltonian and moreover this kind of evolution property allows us to write equation solve them and left hand side of this and the solution of this equation will become left hand side of new integrable hierarchy so it's kind of self generating system of integrable hierarchy if one can really solve them explicitly so one need to study algebra of SFT what is actually invariance okay so there is this homology but of course there is much more in this kind homotopy type of this algebra and there is much more structure and how to extract through this actual computable invariant that's a problem so there is a lot of work now of getting topological invariance where this idea of SFT there was a for instance so the idea is the following you take a topological manifold and do some kind of canonical synthetic construction take a cotangent bundle of union cotangent bundle or and then you study symphonic invariant of this one and in many cases it turns out to be powerful topological and very powerful for instance there's a I just mentioned Lenny and work on knots invariance of nodes but there are a lot of work to do here so there is this kind of current work or attempt of constructing relative SFT which is half constructed which work with holomorphic curve with not only this kind of puncture but also with boundary and it's related with string topology and cluster homology and and there is a work here of Foucaille and of Chillybacke and Lachov and many others and Carnell along to cluster homology he'll speak about this today and so there is a certain procedure which is called symplectic surgery which also tightly related to serious symplectic left spin cell and in order to be able to compute invariance one need to kind of to develop this and this also will shed light all this mirror symmetry theory and there is this connection between topological boundary field theory of SFT there is a kind of some work of Tereman and Voronov here and there is a relation is tropical geometry so this generation of tropical geometry is a kind of another very much related generation to this SFT generation and there are a lot of other things to do thank you.
 Ladies and gentlemen, it is my honor and my great pleasure to introduce Jakov Ilyashberg,  a very generous mathematician.  Yershey-Ilyashberg got his PhD in 1972 in Leningrad, in our Simpetersburg again, under  the supervision of Vladimir Hochlin.  The title of his dissertation was Singularities of Mappings.  Yershey-Ilyashberg then took a professor position at Sik-Tikva in the Komi Republic.  After a difficult period from 1979 to 1987, he was allowed to emigrate from the USSR to  the United States of America in 1987.  He visited the Mathematical Sciences Research Institute MSRI in 1988.  He then joined the mathematics department of Stanford University in 1989, where he has  been ever since.  He now chairs this department.  The present area of research of Yershey-Ilyashberg is Simplectic Topology, a discipline that  grew from the fantastic developments Simplectic geometry underwent in the last 30 years under  the influence of many people, in particular Mikhail Gromov, but also because of the push  from theoretical physics.  Magnificent geometry really appeared as a geometry of a new kind in the memoir of Joseph  Frouillagrange of 1808 on la variation des éléments des planets, then was fully established  by Sir William Hamilton with fundamental contributions by Darbou and Jacobine, the second part of  the 19th century, and then the local content of the theory was completely uncovered.  The beauty of the theory is praised by Felix Klein in his introduction of the history of  mathematics of the 19th century, but it doubts its usefulness.  The emergence of quantum mechanics in 1920s gave a major impetus for further developments  of the theory and showed its relevance for a wide area of applications.  The shift towards a global Simplectic geometry took some more time.  A first major step was taken by Eugenio Calabi in the 1960s, but it is Mikhail Gromov who  turned things around somewhat later.  By exhibiting completely new rigidity properties of automorphism Simplectic manifolds.  Conjectured by Vladimir Arnolv and a number of fixed points of Simplectic automorphisms  played also an important role.  Jascha Eliasberg started to be interested in these problems in 1979.  At the time, Connley and Zander proved solved Arnolv's conjecture for surfaces.  Further important developments are due to the regretted Andreas' flaw.  Jascha Eliasberg started a very fruitful collaboration with Helmut Hofer in 1995.  He presented the preliminary version of the now monumental theory he developed fully since  1999-2000 in collaboration with Hofer and Giventown.  In his lecture, Jascha Eliasberg will tell us where Simplectic field theory and its applications  stands today.  Jascha Eliasberg.  So of course it doesn't work.  Okay.  It doesn't work.  Oh, okay.  So now.  So Simplectic field theory, as Jean Pierre mentioned, is some project which we initiated  few years ago with Alexander Giventown and Helmut Hofer.  And it's not yet far from being completed.  A lot of people contributed to just building foundation, application of the theory.  And this is of course not complete list.  And there are a lot of people who not formally working in Simplectic field theory, but on  the other hand they really, they work very relevant.  And I will not give their names because I will definitely forget someone.  So what I will start is some kind of crash, three-minute crash course of Simplectic geometry.  So Simplectic geometry is just geometry of a non-degenerate skew-symmetric bilinear form,  like Euclidean geometry, geometry of symmetric form.  So symmetric form have at least one invariance, a signature, but skew-symmetric form is unique  if it's non-degenerate.  And viewed as a differential format always can be written in this form.  And you also kind of important to think about Simplectic form in the context of complex  structure.  So if you think about even dimensional space and Simplectic geometry only exist in even  dimensional space, and think about the space as a space of complex coordinates, then Simplectic  form is just a minus-imaginary part of the standard termission product.  So Simplectic manifold is a manifold which is locally modeled on this R2N with this special  form.  And equivalently according to Darbuth theorem, this is a manifold with a non-degenerate closed  differential to form.  And very important, again, like an Euclidean space to consider Simplectic form together  with this complex structure, and in this case we want to consider almost complex structure,  which means this complex structure on tangent bundle to the surface.  And we say that it's compatible with Simplectic form if they relate to each other the same  way as this standard Simplectic form and standard complex structure relate to each other on  Euclidean space.  So it's important, and this will be really crucial for the holder's development, that  the almost complex structure, given omega, you can always find compatible in this sense  J, and more or less this is a kind of convex choice of such J, so it's a contractible choice.  But on the other hand, if you have a J, it's not always possible to find a compatible omega.  So omega is much more important.  So Simplectic manifold, unlike, on the first glance it looked like parallel to Riemannian  manifold, okay, we have a manifold with symmetric form or manifold with skew-symmetric form,  but unlike Riemannian case, Simplectic manifold has a huge group of different morphisms called  Simplectomorphism.  Well, this is probably due to the fact that you see that the symmetric form has a n times  n plus 1 over 2 n trials and skew-symmetric only n times n minus 1 over 2, so it's less  equation and this gives such a huge difference.  So the group in the two-dimensional case, an n equal 1 for surfaces, Simplectic form  is just area form and the Simplectic geometry is just geometry of area preserving transformations.  But in high dimensions, this is a drastically different.  So the group of Simplectomorphism form really proper and amazingly close to C0 topology.  The C0 closure of this group is a proper subgroup of the group of volume preserving transformation.  So this is a kind of really fundamental theorem which opens the whole area of Simplectic geometry.  So the algebra of this group consists of Simplectic vector field and this is just equivalent that  this is a vector field which has this property that if you contract them in Simplectic form,  you have a closed form and especially we would like to have that consider the case when the  dual form is exact and then this function is called Hamiltonian and the corresponding  vector field called Hamiltonian vector field and I will denote by this Simplectic gradient  of H. And H is called Hamiltonian function.  So in the Darbouk coordinates, of course, everybody knows this canonical form for this  Hamiltonian vector field.  So this kind of Hamiltonian equation differential for the flow of this form p dot minus dh of  dq, q dot is equal to dh of dp.  So for any compatible omega j, you always have the Simplectic gradient just gradient  rotated by j.  So very important object in Simplectic geometry are Lagrangian submanifolds.  The submanifold on which omega vanishes and maximum such dimension can be just half n.  And in terms of complex structure, they characterized by the properties that when you multiply by  j, the tangent place become its own orthogonal complement.  So there is a kind of two most important examples of Lagrangian manifold.  One when you take a cotangent bundle and cotangent bundle of manifold has a canonical Simplectic  structure, which is differential of this famous form pdq.  And then section is Lagrangian if and only if it's a closed form, the section.  In particular, if you have a, say, differential of function and you have exact form, then  its graph of derivative of function is Lagrangian manifold.  On the other hand, if you consider Simplectic manifold and you have a Simplectomorphism,  then graph in the product, if you take a Simplectic form omega on one factor, Mimolus omega on  the other, then graph will be Lagrangian.  And it's always kind of important kind of play in Simplectic geometry.  Then when you have a kind of one split of coordinates, then Lagrangian manifold looks  like graph of Simplectic transformation and another system of coordinates, they look like  graph of derivative of functions.  So there is because of this kind of huge group of Simplectomorphism, it's very difficult  to find Simplectic invariance beyond some obelisk one.  So of course, we have a total volume, it's invariant.  Then there is this homotopic class of a compatible almost complex structure invariant.  And in the closed case, the homology class of the Simplectic form is also invariant.  But it was for till the beginning of 80s, nothing else was known essentially, more or  less some minor things.  And this was a Gromov's great insight who introduced the homomorphic curve as a tool  for finding more subtle, specifically Simplectic invariance.  And we will discuss this today.  So let me talk a little bit about this kind of homomorphic maps between almost complex  manifold.  On the first glance, you can easily talk about homomorphic map between almost complex manifold  as in this situation for integrable case, just say, okay, differential is complex linear  map.  But just unfortunately, if you, if you, if the dimension of the source manifold is real  dimension greater than 2, then this is a highly over-determined system and you never  have any solution generically even local.  So it's a really kind of miracle that the Karschelian equation in integrable case and  high dimension have solutions.  But the dimension s equal 2 when s is the Riemann surface.  Then, and by the way, in this case, almost complex structure and complex structure on  the Riemann surface and two dimensions is exactly the same thing.  So it's just a conformal structure.  And in this case, you have a locally, the system is determined, you have exactly the  same degree of freedom of constructing homomorphic map as in integrable case and they call homomorphic  curve in this case also J, also J-holographic and or pseudo-holographic.  I don't like the word pseudo-holographic because it doesn't carry any information.  J-holographic sometimes useful because you want to say for which J.  So I will fight with this kind of terminology.  So if moreover, if s is closed and with appropriate boundary conditions, this is an elliptic system  and its principle symbol the same as for standard debar operator.  So you can apply standard thread home theory and under certain transversality assumption,  you can define modulate spaces of homomorphic curves which are finite dimensional manifold  or at least orbitals or algebraic geometry, they say stacks but okay, so here they just  orbitals.  So, but this is a kind of, of course I'm saying this thing but it is an extremely serious  statement.  So you see the whole point of, well one of the points of introduction, almost complex  structure rather than integrable structure that like you have a much more freedom of  constructing such object and in algebraic geometry, algebraic geometry you really restricted  with your tools how you can achieve transversality but here you kind of hope that you can have  almost as degree of freedom as an anthropology with achieving transversality.  But unfortunately it's not quite the case, it's almost the case but not quite and in  some case indeed you can achieve transversality and just kind of forget about it but on other  cases you in fact have to unreach the class of your object in talking about some what  more general equations and homomorphic curve equation but I will ignore all this issue  about it.  But there is a huge subject and for instance one of my collaborators Helmut Hofer and this  is Kosser devoted a lot of efforts of building appropriate foundation for this situation.  Okay, so local theory as I said in the non-degenerate cases fine for homomorphic curve but if you  want to get global result you need to have a compactness and here is this kind of one  of the major result which was proven by Gromov he proved that for homomorphic curve you can  prove similar result as a kind of this famous Knudsen-Dilling-Mampford compactification of  just the space of Riemann surfaces.  But there is a big caveat only if the area of homomorphic curve is uniformly bounded and  it's extremely difficult to control this area so the role of symplectic form as we will  see come precisely to in this point.  So let me kind of recall few things about the Riemann for compactification so according  to the uniformization theorem you can talk about this consider Riemann surfaces with  mark point is the same as well for this condition which it's the same as hyperbolic matrix on  the punctured surfaces so this condition precisely kind of called stability condition  precisely just ensures that you have a hyperbolic geometry on this thing.  And so this compactification Knudsen-Dilling-Mampford obtained by adding nodal surfaces and they look  like union of two casps in the hyperbolic interpretation and the duration means shrinking  a closed geodesic to a point.  So the stability condition as I say have to be satisfied for every component.  So Gromov compacting theorem is exactly the same but it's different the stability condition  it needs to be satisfied only for constant components of nodal curves which are sometimes  called good.  So it is a kind of slight conserved modification of Gromov compacting theorem.  So this is that you may have this phenomenon of bubbling off of holomorphic spheres so  this is kind of doesn't exist in just case when we talk about Riemann surfaces.  And as I said already the role of symplectic form is to cure area bounds and the point  is that with compatible J omega area is the same as symplectic area meaning this  is just integral of symplectic form and hence because form is closed it depends only on  the homology class of this map of the surface and so when we vary our considered holomorphic  curve in given homology class we don't have to be worried about control of area.  The same true in the boundary value problem with boundary Lagrangian manifold.  So Gromov scheme of defining symplectic invariant where holomorphic curve theory was the following.  Pick a generic almost complex structure compatible with omega.  Then the compactified modulate space of holomorphic curve of genus G and the fifth homology class  can be viewed as a cycle as a kind of similarly unreached modulate space of just smooth surfaces  which is kind of much more simple object and its topology can be understood.  The compactness theorem ensures that the homology class of this cycle remain unchanged when  one varies J while keeping it as a compagny of the omicron.  This is an invariant therefore because all J kind of homotopic and hence we get an invariant  that's what I call Gromov invariant which just invariant of symplectic structure.  So let me give you example of kind of application and this is a Gromov theorem with kind of  somewhat in this form it appeared in our joint paper with Gromov.  So let's take a so-called symplectic camel problem it's probably called this way by Arnold.  So let's consider half space in R4 of course you can go into high dimension but the first  non-trivial first when this phenomenon appeared in dimension 4.  There is a ball and half place with a small hole and the radius of this hole is smaller  than the radius of the ball.  And what we are trying we are trying to kind of to put this by symplectic isotope put this  ball through this hole and the claim that it's not possible.  So why this is not possible?  So okay so suppose there exists such isotope.  Then what we do we at every moment we consider almost complex structure which has the following  property on the ball is just push forward of the standard complex structure at infinity  it's standard and also near this our near the boundary of remaining part of hyperplanet  standard.  But and because this is a power of almost complex structure because it's just homotopic  simple objects it's kind of easily to prescribe almost complex structure as you wish on some  places and just continue to extend elsewhere with the conditions that it's compatible  with symplectic form.  So now we take a Lagrangian cylinder in this plane so just radius 1 Lagrangian cylinder  sitting in this plane.  And consider the modular space of homomorphic disk with one marked point this boundary in  RL.  So you see before we deform of course this flat Lagrangian cylinder was filled by flat  Lagrangian disk.  And we had this kind of our Gromov invariant this cycle this kind of one dimensional family  of this Lagrangian manifold.  And this cycle survive when we deform almost complex structure for any J we have such family  of homomorphic disk.  And this family together they form some kind of hyper surface spanning this Lagrangian  manifold.  And when the ball passing through the hole at some moment he has to intersect this kind  of moving wall.  So therefore in some moment we will have our homomorphic disk this green one kind of and  it's centered precisely on one of this Lagrangian on this homomorphic disk.  And then in this moment we can look at the area.  On one hand we can compute the area of the intersection of the piece of homomorphic curve  inside the disk.  Because this homomorphic disk has a boundary Lagrangian manifold and therefore it's area  is fixed.  It just as I said it just depends it's a simplex invariant just depends on the radius of our  Lagrangian manifold.  It's just equal to one or pi.  And on the other hand I can consider this area looking at the ball itself.  And at the ball it's a ball with standard complex structure.  And at the side you have a curve passing to the center and according to standard the  perimeter inequality for minimal surfaces area should be bigger than pi r square and  we get a contradiction.  So gromofiton potential is a certain motivating motivated by physics way to package algebraic  information contained in gromofinvariant.  So let me kind of talk a little bit about this.  So let's consider this compacted simplex manifold with a compacted complex structure  and take a cohomology class.  So let's think about this cohomology class as some differential form on our manifold.  And then this is a kind of physically motivated notion that defines the acrylator just pulling  back this to the small space of the homomorphic curve, evaluation map at the first mark, point  second, etc.  And integrating over our manifold.  So, okay, so I just said what was written here.  Okay, so this is the acrylator and then we would like to write some kind of generating  function for all the acrylator.  So we fix the basis in the cohomology and also just to talk about homology class we fix  the basis in two dimension homology and identify any, so any homology class can be expanded  in this basis with coefficient d1, dk and because this d degree and just will not distinguish  between this d and this homology class.  So we also can introduce this variable for this, all this d1 for this basis in cohomology  and use this notation.  So Gromov-Witton potential is a generating function for all, for all acrylators.  So you just take this formal linear convention, it's important to take the t as a graded variable  of the same grading or parity of grading as the theta, then we get this thing is a kind  of each component even graded and this is a symmetric expression in this t.  And if you write expansion in the t variable you get all acrylators.  So the meaning of acrylator is kind of very simple, it just, if you consider cycles, duals  to this cohomology classes then as it was explained in the talk of Andrea Kouinkov yesterday,  what we're counting, we're counting homomorphic curve, some subject to this incidental condition  of restricting some marked point to certain cycles.  So also you can upgrade it further including so-called descendant but variable so if you  don't know this you probably will not be able to learn from my talk so just what I just  want to say is that you can further kind of elaborate this incidental condition for the  homomorphic curve imposing some restriction on jets of homomorphic curve on this derivative  and it's traditionally done in certain way but important thing that you get this homomorphic  potential become dependent of infinitely many variables instead of finitely many which  was dimension of our cohomology.  So genus zero part of this expansion is called rational homomorphic potential and for instance  for the case of CP2 let's take the standard basis of CP2, this one, the two-dimensional  cohomology class and four-dimensional cohomology class, choose CP1 as a just projective line  as a standard generator of two-dimensional cohomology and consider this function.  We put two first variable corresponding to this theta 1, theta 0, to 0 and only take  this one which correspond to point constraint and so this coefficient has this very simple  and numerative meaning.  This is as it was in yesterday Akonikov talk is a number of rational curve for given degree  D in the complex in the projective plane which pass through k points in general position.  So you have this standard number for instance you have through two points you can pass one  line etc and well in general of course we have this relation between number of points  and degree in order to have finitely many.  And what I want to do I want to describe some kind of geometric algorithm generated by  symplectic field theory which I didn't explain yet for computing this function.  So this is a today there are many kind of recurrent procedure first started from Kansai  for computing this number but I will explain some another one and this one is also actually  in fact close to what Akonikov and Ponderpand the co-operator for my reason.  Okay, so let's do the following thing.  Consider this infinite dimensional space of this C2 very formal Fourier series just kind  of think about this as a formal loops in C2 space of loops in C2 and so in coordinates  I choose the coordinates in the space just for the efficient of this expansion.  So notice there is no constant term kind of loop center than zero.  So then we can see the some symplectic form in this space.  So this is a symplectic form which also correspond to Poisson bracket versus PM0 bracket with  QM1 is equal to K2M.  So because of so it's also can be written kind of in this form.  So if you think about this Poisson bracket as a tensor on just form defined on co-vectors  and co-vectors themselves can be viewed as the same type of loop because it's vector  space then is just given by this formula.  Don't want to see my hand everywhere.  Okay, anyway, so this is a pretty famous symplectic structure which appeared in the theory of  integrable systems.  So consider Hamiltonian function on this space.  Hamiltonian function is functional on this U.  So let's take this one and it defines some Hamiltonian flow and if you write the Hamiltonian  equation you get this equation.  And this is in fact turns out to be dispersionless Todor equation or continuum linearity of Todor  lattice and written as one equation is just the second kind of order differential equation  and I was told this by Dubrovian.  I didn't know anything about Todor before he told me.  So let's do this now the following construction.  It's take zero section in this space.  So zero section in this space is the kind of space where the Q-coordinate is equal to  zero.  So you can think about this if you think about terms of Fourier expansion then I just take  Fourier expansion it's only if you have a positive coefficient and no negative coefficient.  And then move this zero section under our Hamiltonian flow which I described.  So that will be some moving Lagrangian manifold and I denoted by LT.  And Lagrangian manifold I told you already they are graph of derivative of function.  So they generated by some function if they are graphical.  But they are in the world of formal power series and here everything is graphical.  And so we can find this function GT and finally evaluate this function and this particular  point.  And this is precisely our function f of tz expansion of this function is this enumerative  number of algebraic curve.  Okay.  So it's important that Gromov written potential is a symplectic invariant.  It's Gromov written invariant.  So it depends only on omega.  So while omega explicitly does not participate in the definition, nevertheless it's really  it just keeps this background controlling role.  So how to compute Gromov written invariant?  So it's easy to say okay take j take this modless space but nobody ever solved any kind  of non-trivial d bar equation explicitly and kind of so it's just not possible.  So the only way to do it either you kind of try to deform it to some kind of so you deform  it to j or degenerate the j to some situation where you can understand think and then you  say because it's invariant it was the same as before.  And one way to do it is just kind of to deform it to algebraic situation and another way  of doing this is to degenerate.  And symplectic field theory is one way to degenerate it.  And tropical geometry is the other one.  So what we do is split manifold into pieces.  So if you have a real hyper surface in almost complex manifold you can kind of start to stretch  the neck as they say engage theory.  So just insert bigger and bigger kind of piece where they solve almost complex structure  independent on this vertical coordinates.  And then in the limit you have a manifold with cylindrical ends.  So this is impossible to do in integrable case.  It's extremely restrictive but in non-integrable this is advantage you can always do that.  And in fact in integral case sometimes this is also possible but it's much more restrictive.  I'll say about this later.  So what happened with holomorphic curve when we do this?  They also split.  So in fact what may happen that homomorphic model is based on homomorphic curve kind  of split so you observe the curve which looks like this.  So in upper part you have some piece of curve and lower part you have some piece of curve  and also you can have a bunch of cylindrical parts.  And there is a piece of curve stuck there and all together they match data and they  fit in your original curve which you have in your original manifold.  So this kind of brings us to studying these two objects.  This cylindrical almost complex manifold, this manifold which are kind of cylinder where  almost complex structure is invariant with respect to translation and this is the kind  of definition.  So I'll talk about this more detail later and also manifold with cylindrical end, this  non-compact almost complex manifold which are cylindrical outside of the compact set.  It's precisely those which appear as a result of such splitting.  So let's kind of important that every cylindrical structure as a very cylindrical structure  can assuage this vector field tangent to our manifold.  Why?  Because J times D or DT, where T is this vertical coordinate.  So this is again the same picture.  So I say that there is a splitting procedure which brings out of the manifold this manifold  with cylindrical end.  So any cylindrical structure is determined by what is called in complex analysis the  CR structure.  That's you have on our manifold you can in tangents why the odd dimensional manifold  but it contains maximal complex subspace of real co-dimension 1, this intersection  JT, YT, YT.  So it's a, you have a distribution of complex tangency to Y and then you have this vector  field R which is transferred to it and given R and given C you can always of course define  some one form which satisfies this normalized by conditions of equal 1 on this vector field  and equal 0 on this plane C.  So generically all periodic orbit of the vector field R and periodic orbit will play  important role in the story are non-degenerate and thus there are finitely many of them  of bounded period.  We will call this case generic and we will have this notation for the space of periodic  orbit.  So however it's kind of useful also to consider so-called Morse-Bott case when space of periodic  orbit are organized as some kind of sub manifold for instance like all orbits are periodic.  I told you that we cannot do anything in arbitrary almost complex manifold unless we have a symplectic  structure which kind of control this manifold.  So we also need something like this here.  So we will assume that there exists close to form of maximal rank on Y.  Notice that Y is odd dimensional so the maximal rank of close to form is 2N-2 if dimension  is 2N-1 is one less.  And so we want this form such that R is in the curve so R is a Hamiltonian vector field  of this form.  And we want this R form lambda be preserved by the flow of R.  So this is the same as just to say that R also in the kernel of the form d lambda.  So the pair we call this pair stable Hamiltonian structure and cylindrical almost come crutch  is called compatible with H.  So there are three important cases of Hamiltonian structure.  In fact, I don't know any other kind which would not be like small modification of this  one.  So first is a floor case.  So in this case our manifold just mapping torus we started with symplectic manifold M and  then consider mapping torus of the symplectomorphism.  And let's suppose for instance that the symplectomorphism is a topic to identity so it generated by  time one dependent Hamiltonian.  In this case R just equal do dt plus symplectic gradients of HT and periodic orbit are the  same as in 1-1 correspond to this periodic point of f.  So you already noticed that my picture of course pretty bad but I got as a present from  a TNGist one of his pictures.  Okay.  So this is a TNGist picture of the mapping torus.  So there is also contact case and in the contact case why is this contact manifold that means  that C is maximum nonintegral plane field and lambda is a contact form and omega d lambda.  In this case R is called red vector field of lambda.  And another case circle bundle case when we have our manifold is circle bundle or symplectic  manifold and R just generates this one action and so the space of periodic orbit is just  our original manifold.  So now we want to discuss homomorphic curve in this manifold.  They themselves have to have the cylindrical end.  First talk about cylindrical.  So just cylindrical manifold.  Notice that if you just take a straight cylinder over this orbit then you get a homomorphic  curve because do dt direction is precisely j times this is equal R. So you have homomorphic  curve and therefore you would like to consider just homomorphic curve which asymptotically  look like this cylinder.  So now I can say what just SFT is.  It's the first approximation because in fact first of all the SFT is some kind of functor  from the category which I call geometric category SFT where object are Hamiltonian structure  with compatible J and morphism or symplectic abordism and algebraic category of the FFT  the object are differential by algebra which I will discuss and certain Fourier integral  operator kind of interwining them.  So in fact it's just only zero approximation because the structure of the SFT is much richer  and first of all it's not one category but at least two categories and second there's  much more kind of structure in this object.  So there is a kind of rational version of SFT when we're talking only about general  zero curve and then instead of this while algebra we get a Poisson algebra or symplectic  manifold like was in this example and instead of Fourier operator we get Lagrangian correspondences.  So by the way symplectic abordism and just symplectic manifold bounded by this out to  Hamiltonian structure and symplectic form restrict to the boundary as this omega plus  and omega minus.  I just want to say that you know that there's an interpolation there is this issue how you  compose symplectic abordism there is not canonical operation we have to smooth but  in this kind of world.  In fact when we take abordism we immediately attach to it the cylindrical end and when  we need to compose abordism we just take them one on the top of the other just without  any glue.  So let me now kind of describe this object.  So first of all I need to say about periodic orbit.  So periodic orbit themselves in non-degenerate case and restrict only to non-degenerate case  can be odd or even depending on this return Poincaré map it's a kind of hyperbolic or  elliptic orbit roughly speaking.  So let us associate with every simple orbit of our R just sequence two infinite sequences  of graded variable and I grade them say Z2 grading is precisely the same as I described  here even or odd but there is a in principle there is an integer grading the mass of conical  bits and rindices but I am not discussing this so then organize them in the kind of Fourier  series.  So in fact you can think that we have this big space where this like dimension is equal  number of such gammas.  So and you can think that this collection of those you gamma just coordinate so it's  kind of loop collection of all of them is a formal loop in this big space.  So in the first approximation and this is a even not first probably minus first approximation  SFT associated with h and compatible J the following algebraic objects.  So I will describe separately rational SFT and full SFT.  So object in rational SFT is a graded Poisson algebra generated by this variable with the  commutation relations that this Poisson bracket is equal to K and all other zero and this  is the same as that you can consider corresponding symplectic form which is just one over K times  this.  And you have a special element which I call Hamiltonian which satisfies the properties  that Poisson brackets with the self is equal to zero.  So let me note that this h is in fact always outgraded but so this Poisson bracket is in  fact not necessarily it's not zero automatically that we are in the even case.  So in the full SFT you have instead Y algebra in the Poisson algebra so you just have a  associative algebra with this commutation relation instead this Poisson bracket so you  have one extra variable h bar and you have another element in that one over h bar of  W which also satisfies this h commutator with itself is equal which is exactly the same  as to say that composition of h is with itself is zero because it's odd object.  So it's important of course all people who work in this area know that this well algebra  can be represented as an algebra differential operator as this while representation which  acts on this so-called Fox space is a space of function of this Q variable and kind of  formal power series in this h bar variable and you oh I forgot here sorry I forgot most  important thing.  So here so should be k times h bar coefficient not just t or dQk but k h bar.  So okay so I will not describe morphism in general I just don't have time so I just only  consider rational case so in rational case the morphism between this two Poisson algebra  is some function which have this properties that Hamiltonian vanishes on Lagrangian manifold  defined by this function.  So geometric meaning of this form consider the symplectic space corresponding to this  Poisson algebra is coordinates pq and this is a as I already told this is a symplectic  form which corresponds to this Poisson bracket and then in this symplectic space you consider  Lagrangian manifold which is just given by this function and you have this Hamiltonian  vector field kind of everywhere given by this function h plus h minus.  So the condition that h plus plus h minus constant on Lagrangian manifold just means  that this Hamiltonian vector field is tangent to this Lagrangian manifold.  So in the case when one of this when you have a keyboard with an empty set so you have just  a manifold bounded by one just keyboard spinning one odd dimensional manifold then we associate  with this just Lagrangian subspace in this symplectic space.  And composition of morphism is a composition of Lagrangian correspondences well I don't  have time to explain what is this is well for instance for those Lagrangian manifold  which are graphs of symplectic morphism this is just correspond to composition of this  map.  So where is all the structure come from?  This is the above algebra describes the structure of the boundary of the modular space of holomorphic  curve with cylindrical ends.  So I need to describe this thing so this the appropriate compactification consists of  so-called holomorphic buildings.  So this is a kind of typical holomorphic building so you have several stories so you have kind  of one holomorphic curve another holomorphic curve and you have a matching data on each  between each floor of this building.  So the co-dimension important thing the co-dimension of boundary stratum in the cylindrical case  is equal to the number of non-trivial mean not different from this trivial cylinder component  here for instance here 1, 2, 3, 4, 5, 6 you have this 6 components as a co-dimension  of the stratum and hence you always have only two story building when you looking as  a co-dimension one stratum of the boundary.  So in the case by the way in the case of the manifold this cylindrical ends this in the  count of co-dimension you only count this non-trivial stratum in cylindrical parts.  So why this while algebra?  Because this while algebra is precisely the algebra which describes the gluing of Riemann  surfaces.  So look at this two surfaces and suppose I would like to consider all possible way of  gluing there for instance I glue this to end and just extend this trivially here or I don't  glue anything or I in this one I glue this to this end this to this end and don't glue  anything to this one etc.  And this is exactly the same as I just associate with each of this so end of this curve marked  by I'm sorry this is not possible to see in this picture but it's marked by some of this  p's and q's I marked by p's upper end and by q's lower end and I associate with every  such curve I associate monomial we just describe its end and this is h bar to the power of  minus error characteristic of this.  So you get a so this one you get p1, p2, p3 not error characteristic g minus 1 the g  minus minus minus 1 half of error yeah half of error characteristic okay so you have a  p1, p2, p3 and this is this one then you have q1, q2, p1 this is this one and then you  multiply them in our while algebra just means that transpose transpose all p to the right  all q to the left or think about p the h bar d over dq and take a composition of this operator  and you precisely get this number of terms.  So appearance of this k efficient k in this because of this k h bar because there are  k way of gluing of multiple orbits so you see if you have a kind of you can glue this  way or you can glue this way this way or this way.  So then you can kind of also do upgrade the whole theory in the same way it was done in  Gromov-Witton theory so you define correlators and this is the same correlators we kind of  count a whole more significant curve with cylindrical end and subject to constrain and  the only difference we have to remember what are cylindrical ends I mark them by gamma  minus gamma plus here and then organize them in the correlator which now become element  of this while algebra and finally consider this generating function which I call this  h h bar and I will call this h and then you have the following kind of master equation.  So notice I didn't tell that theta consists of the closed form I could take any forms  and then turns out the equation of the following if we consider h associated with this set  of theta form and d theta and then you get a kind of the composition of them is equal  this operator applied to h where this is a s variable correspond to d theta form t variable  to theta form. So in particular if d theta is equal to 0 you have that h composed with  h is equal to 0 so this is precisely the Stokes formula as I said this is our composition  of differential operator describes the boundary of the modular space so the left hand side  it's a so it's a in Stokes formula I integrate some form over the boundary and get this one  and I integrate form over the manifold and I get this one. So in many many interesting  cases you have this in fact h of 0 is 0 and in this case differentiating different sorry  differentiating this identity in t variable you get this commutator of h i and m j equal  to 0 where the h i j derivative and so you get sequence of commuting differential operator  moreover you can add this descent variable so you can in fact this h depends on infinitely  many variable and hence by this procedure you get infinitely many commuting differential  operator so it's kind of sign of some kind of quantum integral system and indeed for  instance if you do it for the case of S1 you get infinite sequence of commuting integral  for the so-called quantized burger hierarchy this is non quantized thing and you need to  take a kind of quantum version of this replacing this ODE by PDE and then this will be commuting  integral for this hierarchy and for instance if you do the same thing for three dimensional  sphere and then then you get integral for quantized so called this dispassion of Toder  hierarchy which already appeared before in our county hall Morphicor in CP2.  So let us talk a little bit about invariant so thanks to the master equation h h is equal  to 0 we can define on this while algebra the differential just commutation with h and it  still satisfies this property d square is equal to 0 and therefore you get defined homology  which provide us with many cases with powerful geometric invariant for instance in floor  case you get some far-going generalization of homology floor homology theory which is  not really explored at all so there is a kind of wealth of new algebraic structure and kind  of cosmological operation on the on the floor homology which which were not studied so in  the context case you it leads to so-called quantum homology theory which really extremely  successful story in terms of finding invariant of many a fault and in high dimension in dimensions  there is a kind of competing construction of invariant via OJVSABO theory but in high dimension  there is nothing or cyberquizan theory but in high dimension nothing else is known so  alternatively one way use h to define differential on the space of function of q on the so-called  folk space and then leads to what is called BV infinity algebra kind of structure on  the space and this was recently studied by chili bark and lachev so so let's let's consider  the case when you have a x which bounded x is simply a co-ordinate and bounded by y  just co-ordinate them with one end so I already explained that you can associate with this  just counting homomorphic curve in this combordian with cylindrical end you get some Lagrangian  manifold but also you can get this Lagrangian manifold you can do the same thing with this  parameter t including this form and including precisely as was done in the case of h and  if dceta is equal to zero then similar to this naive case h vanishes on this corresponding  Lagrangian manifold but if dceta is not zero but only zero on the boundary then you get  the following following equation with a similar d and if you apply to this case of four ball  bounded by three and kind of we get then you in fact get so you take this equation and  notice here notice here that you have this operator t d or dc so if you differentiate  with respect to t you get something d or ds is equal something like this and so and this  is exactly Hamilton Jacobi equation for the evolution of this f is that how this so if  you do this for the case of of a three then you can explicitly easily compute this Hamiltonian  and then you get this dispersion is sort of floor which was used for immigration of rational  course and sepital okay so let me do final remarks so just this will be to do least well  so there are plenty of things one need to do with this kind of symphactic field theory  which currently some people working on on this and on some think people not working here  but so for instance there are huge possibilities of application Hamiltonian dynamics so homomorphic  for curve successfully were applied to homiltonian dynamics but nobody ever seen any serious application  of homomorphic curve different from of general greater than zero to dynamics and and using  this I'm sure you can do this for instance you can generalize Poltorovich end of quasimorphism  but which it end is top and etc so there is this kind of mysterious relation the theory  of quantum integrable system so as I already hinted you have a this odd dimensional manifold  Hamiltonian structure and you always get some kind of infinite sequence of commuting differential  operators and in rational case it's a good infinite sequence of commuting Hamiltonian  and moreover this kind of evolution property allows us to write equation solve them and  left hand side of this and the solution of this equation will become left hand side of  new integrable hierarchy so it's kind of self generating system of integrable hierarchy  if one can really solve them explicitly so one need to study algebra of SFT what is actually  invariance okay so there is this homology but of course there is much more in this kind  homotopy type of this algebra and there is much more structure and how to extract through  this actual computable invariant that's a problem so there is a lot of work now of getting  topological invariance where this idea of SFT there was a for instance so the idea is  the following you take a topological manifold and do some kind of canonical synthetic construction  take a cotangent bundle of union cotangent bundle or and then you study symphonic invariant  of this one and in many cases it turns out to be powerful topological and very powerful  for instance there's a I just mentioned Lenny and work on knots invariance of nodes but there  are a lot of work to do here so there is this kind of current work or attempt of constructing  relative SFT which is half constructed which work with holomorphic curve with not only this  kind of puncture but also with boundary and it's related with string topology and cluster  homology and and there is a work here of Foucaille and of Chillybacke and Lachov and many others  and Carnell along to cluster homology he'll speak about this today and so there is a certain  procedure which is called symplectic surgery which also tightly related to serious symplectic  left spin cell and in order to be able to compute invariance one need to kind of to develop  this and this also will shed light all this mirror symmetry theory and there is this connection  between topological boundary field theory of SFT there is a kind of some work of Tereman  and Voronov here and there is a relation is tropical geometry so this generation of tropical  geometry is a kind of another very much related generation to this SFT generation and there  are a lot of other things to do thank you.
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