 Thank you. I have known Masin for more than 20 years and congratulations for his birthday. I fly all the way from Taiwan for this conference, which is actually unusual for me. In any case, Masin certainly is one of the most original mathematician that I know of. He still takes time for me to understand his work. Since the first time I learned his work on homological media symmetry, which I'm still struggling to understand. Although I always like to learn things more in a geometric manner, I'm not an algebraic person. But still, that is probably a great deal about the geometry that I've been doing, including the SYC conjecture. So today's talk is somewhat related to trying to understand media symmetry in a geometrical manner to the extent that I believe what geometry tells me. It may not be accurate, but still a lot of interest in geometry come out from that. So this is the work that I have been doing with Zhang, who was my post-doc at one point. Now he is in Erwan. Five years ago we have been working on this. So Carabia would be defined to be those with first-train class EGO-0. And as we know, in the last 30 years, we have been talking about media symmetry, which is one of the most important symmetry for Carabia manifolds. It began with an important observation by several people in 1989, which is just a flip of the side of the representation of super conformal algebra. So a simple consequence is that the Carabia manifold will appear in pairs where the H11 and the H21 will be flipped, will be in the change. That's, of course, the simplest part of the media change. But still, there was an important observation and has inspired many people to do computer search for it at the very beginning. And then, of course, the very fundamental work due to Candidas and others is to compute the counting function of the rational curve on a queen take. That really makes mathematicians change their mind about what physicists are doing. And that was a very good, important consequence of understanding how to identify topological A and B model. And this has inspired many mathematical work. Some of them are actually contributed by conservation on understanding localization. And many, many people are contributed to the work. So I'm glad that some of them was proved rigorously, but still depending on the formula constructed by Candidas and all. So it's not a deep understanding yet, even though some formula were proved. Well, as we know, historically, in 1994, in ICM lecture, Concierge gave the first important talk on the homological metasymmetry conjecture. That was before the plane was invoked, the B plane was invoked by physicists. So that was fantastic. And he proposed that the derived Foucaille category on sympathetic manifold as a meta-equivalent to the derived category of coherence sieves, which we have learned in the last few days, a great deal about activities in the last few days that we saw, which I'm not competent to talk about actually, I'm not familiar with category theory. So we will be very much more low key, no category here. And we will talk about geometry that we learned. Most of the questions remain nonetheless because most of the work has been dealing with total geometry, which we know how to compute. But still the great deal of interest in question has not been answered for more general carburetor manifolds. So today I would like to consider those which are not Kepler. The reason I'm interested in not Kepler carburetor manifolds, not only because in diameter and thickness equal to 3, there are many more non-kepler manifolds than we can imagine, but also I want to single out the sympathetic structure of the manifold in distinguish from complex structure. And non-kepler complex manifolds give rise to a sympathetic structure by itself, and we can understand it better. So a large class of non-kepler carburetor free-force were known and due to a construction by Cremans and Freeman, Robert Freeman. So these are well known to many people, but I will talk about it first anyway. So we start out from a smooth carburetor free-force. Why? Let's call it. And there is a collection of mutually disjoint rational curves sitting in Y. These curves are supposed rational curves and has normal bundles O minus 1 plus O minus 1. So it can be blow down. And so you blow it down following Cremans, and you obtain singular carburetor manifold with orderly double pond. So there's a well-known construction. Then Freeman proposed a condition to deform X0 into a smooth complex manifold. So the condition was well-known. Freeman did the infinitesimal deformation, and 10 liters make it to be more than infinitesimal. Anyway, after you obtain a smooth complex manifold, the canonical light bundle of the resulting manifold is still trivial. So it can be deformed to a moisture or manifold, and it's carburetor in the sense that the canonical light bundle is trivial, so first trend class equals zero. But now in general, it's non-kepler. The reason is that each time when you do this, you are killing the second-bedded number. And eventually you can kill them all. So there's no second-bedded number whatsoever. And therefore, it cannot be kepler for sure. So in this case, in general, well, if the homology is right, you prove that the manifold is actually a defumo to connect some of S3 cross S3 according to the freedom of war. So there's a bunch of very nice interesting manifolds, which is just like donuts, generalized Riemann surface in high dimensions. So they are handles given by S3 cross S3. So beautiful manifolds. And this is a very nice manifolds, although it's non-kepler. And so this construction sometimes is called conifold transition. It's a global version of conifold transitions. In 1987, Miles Rhee made an interesting proposal, although he refused to call it a conjecture. He called it fantasy. And so people sometimes call it Rhee's fantasy. He said this vast collection of Carabia manifolds in general with completely different topology can be connected if they are deformable to moisture manifolds. This manifold can fit into a single universal modular space in which each of them can be connected to another one precisely through the Clemens Freeman conifold transition. So that was nice if any Carabia manifolds can be tunneling to another one through such a construction. And then we can hopefully understand each other by this tunneling. But of course this very non-standard way of moving from one algebraic manifold to another one, not what algebraic geometry would normally do. But here it's clear that we have to go through non-kepler manifolds because the in-between are connect some or S3 cross S3. So if we take Rhee's proposal seriously, we got to understand this non-kepler Carabia manifolds. So we have to understand the tunneling, how they move from one to another one, and what kind of geometry we can give it to them. So that's one of the motivations I'm interested in studying non-kepler manifolds. At the same time, when Richard Thomas came to Harvard in 1999, we were interested in the middle of such a construction. So we looked at the sympathetic middle of the Clemens construction, calling for transition. So you remember when we go in the Clemens Freeman construction, we blow down the S2, the rational curves, which are S2, and create a S3, Lagrangian cycle. So we said why not go to reverse? Instead of shrinking the S2 to a point to move it to S3, we go backwards. We move S3 to S2 instead. And in this way, we can construct new sympathetic manifolds. So just similar to what Freeman do, there's a global condition for shrinking the Lagrangian S3, this Lagrangian S3 together, in order to form a smooth globally defined sympathetic manifold. So in this form, we kill S3, H3, the vertical homology in order to create second homology. So we shrink the Lagrangian free cycle, replace them by a sympathetic two sphere, and this was proposed by the paper, appeared in 2002, by Smith, Thomas, and myself. So that we were able to construct many sympathetic, new sympathetic manifolds in this way, but fortunately we could not go all the way in the most general situation, but nonetheless we have found a lot of sympathetic manifolds in this way. Eventually, we would hope to get manifolds, which is charnesum of Cp3. So we expect charnesum of Cp3 to play a role like charnesum of S3 cross S3. So charnesum of S3 cross S3 behave like non-calar complex manifolds, and charnesum of Cp3 will behave like sympathetic manifolds, which means not compatible complex structure. So we expect these two constructions are mirror to each other. One is charnesum of S3 cross S3, and the other one is charnesum of Cp3, which emits sympathetic structure, but not complex structure. So we call this sympathetic carabiaus, although they have no complex structure that we know of. And there are many other people who have contributed to this kind of construction, notably D.U.F.I. and P.N.O.F., who construct simply connect sympathetic carabiaus manifolds with a fair betting number equal to zero. So I like to study the geometry of such manifolds and hope to understand in what sense they are mirror to each other, although geometrically they are very tempting to see that they are mirror to each other. Well, for different perspective, another reason for considering non-calameter pair of carabiaus come from the SYC, description of mirror symmetry, where we study a T3 vibration. So T3 is a Lagrangian toroid, special Lagrangian toroid, and we apply T divaity on the fiber in order to obtain the mirror geometry. So this, of course, has been clarified a great deal in the wealth of gross cyber seabed. And extending the SYC idea, the T divaity, we also like to consider applying T divaity on non-calameter carabiaus manifolds that are toroid's vibration. We hope to apply this consideration to the pair that we discussed earlier. Now, the important question is what kind of geometric structure we should put on a non-calameter carabiaus manifolds. So, well, we have to put in constraints like supersymmetry and what kind of geometric structure we can put on. It's kind of tough to look at it from the point of view of geometry because we don't know much geometry of non-calameter complex manifolds or sympathetic manifolds for that matter unless there is some almost complex structure to help. Anyway, so we look for some motivation from physics. Well, already in the mid-80s, 1986, people have been considering non-calameter carabiaus when we consider flux and planes wrapping, wrapping cycles of the internal six-dimensional space. So, when there are flux, such as similar to electric or magnetic fields, and the planes are introduced, they represent additional energy and that needs to have a bad reaction to deform the space itself. So, although the background could be scalar, but after you react to the flux or the planes, the manifold need not be scalar in general. And the supersymmetry condition of the string tells us that the internal space in general is not a scalar manifold when flux and the planes are introduced. So, this, for example, appears in the paper of N-distrometer in 1986, where he wrote down some equation in relation to heterotic string. So, N-distrometer's paper was on heterotic string, but here we are more interesting at this moment to 2A and 2B string freely for the general supersymmetric condition that we are interested in. And we kind of motivate by the generalized complex formulation introduced by Heaton. And there are equations which we have been studying with Zhang. They are motivated by several authors, especially Tomasillo, who was visiting Harvard at the time. So, we work with him on understanding these equations. We found some interesting comologies for generalized complex geometry, which I will describe here. So, these are generalized complex structural formulation inspired by supersymmetry. So, first of all, let us recall that the Kepler carburetor condition on a six-dimension manifold can be described as the requirement of a SU-free structure with a Kepler form, Hermitian form, small omega and capital omega, which satisfies the closed condition. One is the complex integrability for complex structure and the symphatic structure that d omega equals zero. And also the Ricci-Fleck condition, which is a complex monsoon pair equation. On the last one, the Ricci-Fleck condition. So, these are the structure related to carburetor structure, which give rise to SU-free holonomic. Now, in the case of type 2B, we flux. So, we studied the process of capacitance oriented for O5 planes, wrap over holomorphic curves. It becomes the following conditions, turns out. One is the capital omega is closed. So, complex structure integrable. And the balance condition, this omega, not necessary, is Kepler now, is omega square in the complex free manifold to be closed. And also, there's a distribution function f, satisfy this equation d d bar exponential f times small omega. d d bar v is equal to a distribution, low b. So, b is the sum of currents or holomorphic curves. And low b is a Poincare deal. So, it's a four form in the sense of distribution equal to the left hand side. And f is defined by the last equation here, replacing the carburet equation. Then the holomorphic free form, omega wedge, omega bar, was originally small omega to the power q without f. Now, with f, it becomes like that. So, these are the equation, the supersymmetric symmetry equation for the carburet manifold with complex structure, but no Kepler structure. So, the supersymmetry is replaced by the second condition, balance condition. Now, so the Keplerian condition is now related to the balance condition which is written down in here, plus a source term equation which is the Fred equation I wrote down earlier. Now, this is an interesting equation, the balance condition. It was introduced in physics literature without knowing some literature in mathematics, which was studied quite a bit earlier in 1982 by Mary Louise Michelson in Stony Brook. Already, she was studying manifold which is balance. So, this condition is simply says, given a Hermitian 1-1 form to the power a minus 1 is close, just in general, that's the balance condition. So, this is what an interesting condition actually, mathematically speaking, because it's preserved under proper holomorphic submersions. Most remarkably, it is preserved under birectional transformations. And we know that Keplerian condition is not preserved under birectional transformations. But on the other hand, it's known that balance condition is preserved under birectional transformations. So, it's natural for birectional geometry. And there are many such manifolds. In particular, the shifter space are all balanced, shifter space over four-dimensional anti-shifter manifolds. So, I construct some of the Carabiaw manifolds by taking a branch cover over the shifter space. So, in particular, you will take the connessum of CP2, which is anti-shifter, and the shifter space is good. It's a balance condition. And you can do a double cover to obtain non-keler Carabiaw space. So, there are many moistened manifolds, although they are not keler, but actually the shifter space could be moistened off. There are quite a lot of them, including the shifter space for the connessum of CP2. So, there are quite a lot of such manifolds. Now, so, in about five years ago, Foo, Jun Li, and I proved that in the deformation given by proposed by Clemens and Freeman, when you do the connessum of transition, the manifold, although it is non-keler, it does emit a smooth balance metric. So, although there is no applied order geometry, in fact, we can put a balance condition over those transitions. So, in particular, therefore, we have a balance metric on connessum of S3 cross S3. Now, actually, we like to do more. There's some equation which we take into account of the back reaction to the supersymmetric condition here. And this will make the geometry to be more rigid. So, in other words, there's a, besides the equation of existing a balance metric, there's an equation similar to Carabia condition, which will make the modular space with finite dimensional. This need to be solved. And this is still not known for this case. But I expect that to be true. So, now, I want to look back in the meter situation of this whole thing now. So, this is very similar to the Maxwell equation. You think about it. So, the balance condition is d omega square equal 0. And there's a source equation, which I wrote down as a third equation, 2I dd bar EF omega equal low b, similar to Maxwell equation. So, if we emphasize omega to be right star of omega square, then the two equation can actually be written in this form. So, d omega square is equal to 0. And d d bar of c tau star omega square equal to low b. So, it's an analog of the Maxwell equation. So, in that way, we want to look at the meter of these two equations. So, while the Maxwell equation is this one. So, the curvature two form is closed, the Bianchi identity. And then d star F2 equal low e, there's a point grade deal of the electric charge configuration. So, this could be looked at as an analog of the previous equation sitting here. d omega square equal to 0. And d d bar star omega square equal to low b. So, if we do that, as I said, you will linearize the Maxwell equation, this exact analog of the equation I wrote down, the linearized part. Okay. So, here the important point is that there's a cohomology or the harmonic condition related to the Maxwell equation, namely closed and also co-closed. And that's the analog, the previous one equation is the analog is d of the four form, omega square is now looked at say unknown. And so, d of the linearized of the four form is equal to 0. And d d bar star of that equal to 0 is the second equation, the source term equation. So, this will be looked at as a harmonic condition for the 2, 2 element. And the natural cohomology here turns out is a Botchian cohomology. The Botchian cohomology is simply kernel of d as a PP form divided out by the image of d d bar in the PP form. While PQ in the making more general. So, this cohomology was introduced by Botchian and also Epile. So, this exactly fits the core for the condition that I imposed earlier. So, therefore, the Botchian cohomology is a natural one to use for studying complex balance manifold because that's how the balance condition come in. Well, of course, naturally, if the manifolds cater the d d bar lemma holes and the Botchian and the usual double core homology are in fact isomorphic. So, there's nothing really new. But if we are interested in non-cater manifold, the d d bar lemma no more holes and then this will be more important to use the d d bar core homology. So, now, we are interested in the symplatic analog of this condition that we just looked up. So, that's what the point of this lecture, like looking at the analog conditions for symplatic manifolds. So, first of all, the mere deal of the manifold is 2A freely and then we, in there, in the complex 2B freely is actually the exponential i omega is an important one. You look at the real part of exponential i omega, then the condition of omega square to be closed is equivalent to d of the real part of the exponential i omega equal to 0. And now, exponential i omega is supposed to be in the change with capital omega, the free form. So, therefore, the meter of the condition, or balanced condition, d omega square equal to 0, we interpret it as d of the real part of capital omega equal to 0. So, the free seal form has the property that d real part of omega equal to 0. So, this is also suggest by strength freely for the symplatic quantum transition that Smith, Thomas, and I did. So, this condition will be part of the type 2A supersymmetric condition in the presence of Oriental force, six planes that wrap over special Lagrangian sub-manifolds. So, now, putting the equations together, the mirror of the equation that I wrote down from the complex side would be this free equation, which is now in the symplatic side. So, we have a symplatic equation, d omega equal to 0, the symplatic form, and then we have a free seal form, which says that d real part of omega equal to 0, and also a n-arrow of d d bar, which I will define, d plus and d minus star of the exponential minus f mu capital omega equal to rho a, where a would be special Lagrangian sub-manifolds, which is meter to the holomorphic curves that we discussed earlier. So, d plus and d minus will be linear symplatic differential operator that can be fought as the symplatic n-arrow of the complex double operators, d and d bar. So, we shall now go into those definitions and the way that it evolves. Anyway, so, the source term equation that we were talking would be these two equations. After you linearize, the fed equation is d plus d minus star of that equal to 0. So, this equation would be looked at as a harmonic condition for cohomology. Just like in the previous case, we used the delarm cohomology for Maxwell equations. So, we introduced some cohomology. So, delarm cohomology is kernel d divided by image d that's corresponding to the Maxwell equation. And the broad term condition for the complex deformation. And so, type 2b there. So, 2a here is the one that we introduced that kernel d divided by image of d plus d minus, which I will talk immediately. So, this is second order differential operators, d plus d minus. So, we search for a sympathetic cohomology. We are surprised to find that actually there are more than one finite dimensional sympathetic cohomology of different forms. And so, we found that it's related to the leftist maps and also some cohomology ring and a infinity structure behind those ring, which I will discuss now. So, we will talk about sympathetic form and then we will talk about primitive form just like in the complex case when we have a sympathetic manifold. So, primitive form are those that vanishes under contrast with omega inverse. Omega is not degenerate. So, you can talk about omega inverse. So, these are the conditions just similar to what we learn in complex geometry. So, we take the interior product of the form. So, b is called a primitive differential form. We satisfy this contrasting connection. And so, this primitive form will be due, pointedly due to Lagrangian or coisotropic condition on sub-spaces. So, coisotropic sub-manifold are those with dimension n plus k. So, manifold is 2n dimension. So, those n plus k manifold where omega to the power k plus 1 restrict to it is equal to 0. So, when k equals 0, these are Lagrangian condition. Omega restrict to the cycle is equal to 0. So, for us, coisotropic sub-manifold are these kind of manifold that we introduce. Well, that's well known. So, we understand that in fact, coisotropic sub-spaces are important for meta-symmetry. Also, especially when we want to expand fukar category in order for conservative homological meta-symmetry to hold. Anyway, so, we are interesting in the primitive co-mology because, partially because they are pointedly due to sub-manifold, which is coisotropic. Now, let's come into understanding of the Lepsys decomposition in general. So, given any different form k form, we can decompose it in the following way. So, the bk's are the primitive forms and decompose uniquely in that way, bk plus omega w-bk minus 2 plus omega square w-bk minus 4, et cetera. So, this primitive form in the decomposition, bk, bk minus 2, are uniquely determined by the form a k and the way that the sympathetic form omega there. These, of course, are the Lepsys decomposition from the SL2 algebra acting on the space of all different forms. So, we now have this decomposition. And we can write the, so, in the case of six-dimension, you can see how it looks like. These are the simple diagram that you can see. So, this is symmetric under the central x at the middle dimension. So, they flip from one side to another side. And we can define a reflection operator mapping from left to the right, which we call star r. So, this operator mapping left to right, because it's symmetric in the vertical x. So, we shall use this operator star r later. So, for example, star r b2 is omega w-b2 for that freedom as a metaphor. Well, since omega is sympathetic, d of a k form naturally decompose in a simple form. The important thing is, in case it acts on primitive components, when it acts on primitive form, exterior differentiation only sits inside p-s plus 1 plus omega w-p-s minus 1. This gives rise to two operators. So, d plus maps the primitive s form to primitive s plus 1 form. d minus maps s form to s minus 1 form. So, one increase, one degree, and the other one decrease one degree. So, it's similar to the d bar operators. And we, in fact, interpret in that way. So, we can see that, so these are natural forms associated to the sympathetic geometry. So, we can write exterior differentiation to be d plus plus omega w-d minus. So, this gives you a sympathetic decomposition of exterior derivatives. And the fact that d square equals zero implies similar condition for d plus and d minus. So, d plus square equals d minus square equals zero, and they are anti-commune with each other also. So, we are similar to d bar operators. And it turns out, associated to these operators, there's a elliptic complex associated to it. And it makes primitive forms horizontally. And there's one important connection is drawing the pn to pn in the middle dimension by d plus d minus. That one does not change a degree. One go up, one go down. So, we serve the degree. And this turns out to be elliptic complex. We found out this first, and then found out actually, Smith, R.T. Smith did this one for four dimensions in 1976. And also later, we also found Eastwood has done this around the same time independent with us. So, anyway, so we get the elliptic complex associated to the sympathetic geometry. Assocated with this elliptic complex, we found four different finite dimension of homology of primitive form, which can be looked at as a sympathetic invariance for non-calor sympathetic metaphors. So, associated with this elliptic homology, they are all finite dimensional because this elliptic complex. So, we draw as a direct anorogue with double homology, apollic homology and botanical homology, exactly the same way we look at. So, all acting on primitive forms. So, the left hand side is analog on the right hand side, as you can see from this picture. And we are interested in this, the last one, kernel d in the center of p3 and image d plus d minus in the center of p3, the primitive free forms in the case of six dimensional space as relate to the sympathetic type IIa string equations. Well, a simple example was a codiola, first in four metaphors, which is a t25 base and over t2. They globally divide one forms, left in random one forms, e1, e2, e3, and e4. So, these are taller than the over taller. And they have a naturally sympathetic form given by e1, e2, plus e3, e4. And they are close, the omega is close, easily checked. And you can check that the primitive form has extra element, e2, e3, which is in trivial delamco homology, but non-trivial in this d plus d minus co homology that we are talking about. So, the primitive co homology is not the same as the delamco homology. And, well, we like to not understand what they mean and they can be thought as a point grade deal of a Lagrangian sub manifold L that wraps the tallest fiber over a point on the base taller. And this taller is trivial in standard co homology. So, we have a probability choice of an almost complex structure, left sent by the omega 2 CO form as written here, e1 plus i, e2, e3 plus i, e4. The Lagrangian's, this L becomes a special Lagrangian cycle, as you can see from this calculation. The real part over it equals zero, imaginary part is a volume. And this is a trivial in homology, and therefore it's not the same, while it cannot occur in scalar geometry. So, this is for the corollary first example. So, we have a non-trivial element in the primitive co homology, but not so in delamco homology. Anyway, this should be interesting, and so we like to talk about this more. So, we look at the filter forms besides the form that we discussed earlier. It turns out to be more natural to look at filter forms. The k forms, as we said, can be decomposed uniquely as Bk plus omega wedge Bk minus 2, etc. And we look at the filter according to the degree, pi CO is Bk, pi 1 is Bk plus omega wedge Bk minus 2, etc. So, we filter the k forms by this filtering property, and so we have space on k filter forms. So, we have written it in this form. So, starting from primitive forms to the space of all forms in that way. So, getting little bit step by step more complicated. So, there are elliptic complex and corollaries for these p-filter forms as well. And we found that the following complex is elliptic. So, we have D plus mapping this way and coming back and D plus D minus. Where D plus is the projection to this filter complex of the exterior differentiation, D minus is the joint star R D star R that the star R was defined earlier. So, we have this filter corollary and all that relate to this elliptic complex. Well, we have this relate to, in the co-module level is relate to the standard Lepsys map which is just obtained by taking wedge product with omega to the power R. And well, of course, this maps on the co-module level are purely co-module, co-mological. So, this maps as kernel and co-kernel. This is, turns out to be exellent encoded in the sympathetic co-module. And these are given by this long excess sequence involving primitive co-mology that we defined earlier. So, the L is a map from deram co-mology and then there's kernel and co-kernel which implies this last equation in the case of D equal to 6, co-kernel plus kernel or the Lepsys map. So, in fact, there is a triangle, a simple exellent triangle, the map, the Lepsys map from deram co-mology and in between there is this filter co-mology. The sympathetic co-mology we are discussing earlier. Of course, the kernel and co-kernel on the map can change according to the co-mology class of omega. So, the dimension of the sympathetic co-mology can change accordingly. So, they are not topological invariance, but sympathetic invariance. Now, interesting thing is actually define a product for the primitive forms. And more generally for pure, for P-filter forms. This gives the filter co-mology ring structure which is different from the deram co-mology ring. So, the question is to understand a product which will preserve the filter co-mology. And so, there is a natural algebraic structure on this filter form, this form turns out. That takes a while for us to come up with such a product. And the product, the ring structure that we define on this primitive co-mology ring turns out to be different for sure from the deram co-mology ring. And we shall give some examples. The product we define is interesting. It involves first-order derivative operators. The product itself involves differentiation. So, I will tell you what that is. So, first of all, this follows from understanding the grading that we have before. We just write down the filter algebra in this way. This is the elliptic complex that I discussed earlier. D plus at the first line and the second line is D minus and drawn by D plus D minus. So, this complex has different maps from Fj to Fj plus 1, the filter co-mology. And for the first part it is just D plus and the middle part is D plus D minus as you see here. And the last part is D minus. So, there is a second-order operator appear certainly. And it has a property that is a complex in the sets Dj plus 1, Dj equals 0. So, every guy is first-order except the middle guy. So, this middle guy that caused some difference. And we would like the differential that we are operating on the filter complex satisfy the grading and satisfy the line as rule. So, that is what we want. So, this will make sure the product will be well defined in the primitive co-mology that we define. So, for example, in the case of the product in P1 cross P2 equal to F1 cross F2, it requires the falling property. This is the property that is different from the other first-order operators. We need to have this properties in order for our line as rule to be true for D3. So, there is a second-order operator on the left in this equation and first-order operator on the right only. And yet, we insist them to be equal. So, therefore, the only possibility comes out from the fact that the product itself should involve some differentiation. So, that is what we do. So, we define the product on a filter primitive form in a following way. So, Bj product Bk is defined on the right-hand side. So, it involves differentiation D and D minus and projection and star operator. The rest is simpler except the first one. So, this turns out they are great commutative and satisfy the line as rule that we expect. And L inverse is defined by this operator, which is the inverse of the left-hand map. So, this gives rise to the primitive product and this generates the filter forms, gives rise to ring structure for the filter commodity, FPH. This looks a little bit strange. On zero forms, is it the usual problem? Yeah. So, this is an interesting product. It is a kind of example of this A infinity structure which I never learned before when I was a student. Anyway, so, I remind myself, presumably, everybody here knows what they are. So, these are the maps, multilinear maps, and on a greater better space of degree 2 minus k satisfy the strong homotopy associative conditions. And this, so M1 increase the grading by one at X-ray differentiation and M satisfy the grading corresponding to the product structure. And therefore, it adjusts the line as product rule when M1 is the differential. M3 measures the associativity of the product map. So, these are the conditions that we like to exhibit. And in fact, in the product structure that we propose, M4 and M5 and on are all equal zero. So, I only care about M1, M2, and M3. So, for primitive forms, the M3 looks like that. So, what happens is that our product is not associative and it is measured by this obstruction. And yet, it is commutative and sets the line as rule. And therefore, it defies a nice product on the cohomology nonetheless. And it's still associative in the cohomology level. So, therefore, while all the problem as I said is caused by the fact we have a second order operator in our elliptic complex, D plus D minus. So, it seems to be natural because as I said, it comes out from the study of the attempt to understand metasymmetry in 2A theory. I think we are still developing this theory and understanding the harmonic theory behind it. If you give me an almost complex structure J, we have an elliptic operator associated with it. So, you can talk about the similar thing as what people have been talking. So, given the J structure, we can talk about the elliptic operator, but we have a elliptic complex, no matter what. So, we are still developing this whole theory and whether there's some quantum question or not, we will see. I think it should be there, but we have not developed into it. It is still doing the classical elliptic complex associated with this whole thing. So, these are naturally attached to sympathetic geometry, which I hope would be interesting and useful. Thank you. I was wondering if there's an equivariant version of this, if you have a Hamiltonian group action on the manifold. I'm sure that should be, but I haven't thought about it yet. I mean, it's still developing. Great. I first thought this thermo-interval, primitive connoisseur degree, responsible for Lagrangian manifolds up to Hamiltonian isotope, and in fact, we can have unicyclic samples in two-dimension making. And actually, the control is limited. The stability structure is very exotic for purpose on the surface. And the question is, if you consider conditions of volume and I have no idea how to relate this real three-formant as a galvanic condition. Yeah, because it's a real part of three-formant. And how it depends on generalized dimension of the mass of these people. There is a generalized complex structure. Well, there's a generalized complex structure formulation, more complicated, which I don't want to go into. Any more questions? Well then, thank you again.