 Thank you very much. On behalf of the, well, on behalf of myself, I would like to thank the organizing committee, which includes myself but did not include myself at the time when I was invited to give this talk. Anyway, it's a wonderful occasion to be here. And happy non-birthday maximum, since I understand it's not right now. OK. So the title of my talk is similar symmetry for affine varieties. And that could include lots of very complicated things and very subtle issues, none of which I will address because I don't understand them. And instead what I have in mind is something simpler. And so among affine varieties, I mostly want to focus on something very simple, namely hyper surfaces in C star to the N. And when you need a concrete example, I will in fact think of a very simple kind of hyper surface, namely the pair of pants, which is after all a hyper surface in C star squared defined by the equation 1 plus x1 plus x2 equals 0, for example. And then when I need slightly less simple examples, I will think of, for example, other human surfaces in C star squared or higher dimensional pairs of pants. So this is the N minus one-dimensional pair of pants. And it looks like this but bigger. OK. OK. So what do I mean by mirror symmetry for these things? We have to decide first what the mirror space should be like. And conveniently for that, there are candidate mirrors, which I will explain now from the point of view of joint work with Mohamed Abouzaid and Ludmil Katzarkov from 2012. But you can also see other competing proposals by Gross, Katzarkov, Rudat. And earlier works by Katzarkov, Capustin, Orlov, Yotov, Patrick Clark. And it probably goes back way before. It all goes back. Different proposals by Katzarkov. Yes, yes. Well, but they're compatible. Different possible explanations for the same thing. So they're all consistent. OK. So anyway, so what's the mirror of such a thing? So let's say that we have a hyper surface defined by some equation. So some Laurent polynomial in N variables in C to the N. So we'll have a sum with monomials and inside C star to the N. Yes, HT, for example. And OK. So quickly speaking, A is a finite subset of integer weights, the exponents that appear in my favorite Laurent polynomial. Rho is weights associated to each of these integers, a way of basically thinking of a degeneration of these hyper surfaces to some tropical limit. So there's some convexity property. And T, depending on your taste, is either, well, you could think of it as a small or formal parameter. And in fact, you can think of this as either a degenerating family of hyper surfaces in C star to the N or as one non-archimedean hyper surface defined over the Novikov field. OK. And so what we've done in there and was done differently in other places is construct what I would call a generalized mirror in the sense of a Stromeger-Jouza slow conjecture to this thing. And so this thing will be a Toric-Landoginsburg model. So a Landoginsburg model for us will be just a non-compact scalar manifold equipped with a holomorphic function on it. And depending on what we want to do, we'll either study the symplectic geometry or the algebraic geometry of its singularities, of its critical values. So this Landoginsburg model here concretely will be a pair yw, where y is a Toric-Kalabiow N plus one-dimensional manifold, which is easiest to describe in terms of its moment polytope. So how do I describe the moment polytope of this thing? Well, first thing I will do is tropicalize this equation. So let's define, sorry, I will call that delta y, but I can't define it yet. So let's define the tropicalization of f to be a function, a piecewise linear function of n real variables, to see n. It will be a piecewise linear function defined by the max over all of these things of the linear function with slope alpha minus the constant rho of alpha. So if you know tropical geometry, you know how to come to this from there and why they're essentially similar. And then the moment polytope will be the set of all tuples c1 to cn eta in our n plus one, where eta is bigger than phi of c, and that defines for me a scalar-Toric manifold. And moreover, on that there will be a function which will be just a Toric monomial up to sine, the minus sine, which I don't want to explain, and it's the monomial with weight 0001. You can check that will be a regular function on this thing. Okay, so what does that look like concretely? So if I take the pair of pants in any dimension, so this function doesn't need powers of T to be tropicalized, it's already there. What the pair of pants look like is this. The tropicalization phi is going to be max of c1, cn, 0. Its domains of linearity look like that. I'm drawing the picture for n equals 2. And so here it's 0, here it's c1, here it's c2. Now imagine the graph of this thing in R3 and take everything that's above and that defines the moment polytope for you. What is that polytope? Well, it's nothing but just an octant in space, in this case. So in fact, y will be c to the n plus one. And what is this function w? Well, it's the one that corresponds to the weight vector that goes straight out in this direction, which means it vanishes to order one on all of these facets. So in this case, w equals minus the product of the coordinates. So that's a good example to have in mind. Okay, so in what sense is this a mirror to the pair of pants or more generally to hypersurfaces? Well, that's what we're still in the process of finding out. Okay, so up there I've generalized SYZ mirror. SYZ mirror symmetry is about torus vibrations. And so you might ask, how does this work? This guy doesn't even have the right dimension. I started from something of dimension n minus one and ended up with something of dimension n plus one. And that's where the generalized thing comes about. So as far as I know, there's not any reasonable torus vibration in the sense of SYZ to be put on this hypersurface H. However, there's larger spaces closely related to H that do carry such vibrations. So in fact, the simplest thing to look at, I will call x0, is going to be... So the space defined by the equation uv equals f of x1 xn inside c2 times c star to the n. That's a conic bundle over c star to the n with discriminant locus given exactly by H. So the typical fibers of this thing look like cylinders, but then above my pair of pants on whatever hypersurface I have, these cylinders degenerate to unions of lines. So this is conveniently a nice Calabria manifold and it carries nice Lagrangian torus vibrations. Roughly speaking, all you have to do is pick... So there's a circle action. So what you do is you pick level sets of a moment map, which means you choose heights for circles on these conics, and then you pick Lagrangian tori in the base and you just lift them to obtain Lagrangian tori in the total space. There's a slight subtlety, but that's pretty close to the truth. What's the slight subtlety? Actually, you pick Lagrangian tori on the reduced spaces, which are all carrying slightly different symplectic forms from the usual one of c star to the n. In particular, the reduced symplectic forms are not toric anymore, but they're deformation equivalent to toric 1, so you still know how to find toric, I mean Lagrangian torus vibrations on the reduced spaces. The side effect of that is that this vibration is only piecewise smooth. OK. So this thing carries a Lagrangian Tn plus 1 now, vibration, with some singularities, of course. Singularity is basically when your tori passes through this point, which is singular along h. And you can use that to build a dual in a certain sense, torus vibration on a space which, after you stare at it for long enough, you find that it's most of this mirror space, y minus some hypersurface, which I will not elaborate on for now. Now, x0 is not the same as h, but what happens is there's a slightly larger space, x, which is the blow-up of c times c star to the n at 0 times h. And what this one looks like is almost the same, except, well, except slightly different in what way. So above a typical point, I just have, so if I just project again to the c star to the n factor, the typical fiber will just be a copy of c, which for your convenience I will draw like this. And above a point of h, the fiber will be a copy of c obtained by lifting to the proper transform union, a cp1, from the exceptional divisor. And now you should see that this is exactly the same as that one just adding one point in each fiber. So this is a nice setup for mirror symmetry in the s-wise sense, in that this one is no longer calabiow, but we know how to think of its mirror as being essentially the same as the mirror to x0, but with a superpotential added. And superpotential, roughly speaking, records, say if you're a symplectic geometry and doing symplectic geometry here, tells you how adding this partial compactification will deform flow theory for Lagrangians in this space. Sorry, so I should have said x0 will be mirror to something called y0, which is y minus some hypersurface, not telling you what it is. The mirror of this one will be again y0, but now equipped with a superpotential, which is exactly this w I told you about. Now, this is not equivalent to h either. It looks like I'm just playing games and making this space more and more complicated. Next step is I'm going to equip x with its own superpotential. Wx equals the coordinate from this complex variable. I would call that y. When I do that, I'm deforming again the geometry here, and the effect of that is to compactify the mirror to all of y. Sorry, not compactify, because it's still non-compact, but partially compactify to all of y. Okay, so now the claim is, sorry, the superpotential was still negative w. Next thing I need to do is also twist by some class in h2 with z mod 2 coefficients that count for basically discrepancies in the ways that signs are counted for holomorphic curves. That's the bane of symplectic geometers, that holomorphic curves always get counted with a wrong sign. And the effect of that is to change this, too, what I wanted. Okay, and now the claim is, so what's the geometry of this function on x? You know, by now I've probably lost most of you, but I'm going to try to un-lose you. So the function which is just given by the complex coordinate from c, what is it, zero set? Well, it's the origin in each copy of c, so it's these new points I've added to each fiber. Union, well, the whole exceptional divisor, because after all, on the exceptional divisor, the complexity is also zero. So you see that it has two smooth pieces intersecting transversely along a copy of h in here. And in fact, this function is Morsebot, with Morsebot singularities along a copy of h, and then its singularity theory just reduces to the geometry of h. So there's good evidence for some of it, well, most of it is not written up in full detail, that there should be a factor from the Foucaille category of h to the Foucaille category of this thing. And that that factor should be almost unequivalent. At least we are hoping it's unequivalent. We don't know how to prove that yet. Sorry, it's unequivalent up to a shift. Sorry, I have a question about this s. Is this a question about, you know, being more spulted with the normal bundles? Exactly. It's Morsebot, but the normal bundle to it is the direction of two line bundles which have non-trivial second state for Whitney class. And it's the W2 of those normal bundles. And similarly, if you're an algebraic geometry instead, you probably know similarly when you have a vibration with Morsebot singularities, I mean, it's a case of basically Knower periodicity, as proved by Orloff, that D.B. Singh of this thing, the category that algebraic geometry would associate to this Landau-Ginzburg model, is the same as the category of coherence sheets of h. So for all purposes, this is a good replacement of h, and yes, yes, okay, sorry. I'm not that sophisticated. Okay, oh, I still have a whole blackboard here. I'm not used to having three. Okay, so after a long introduction, now we can talk a bit about homological mirror symmetry because, you know, you can construct mirrors by this sort of S-Y-Z argument, but that doesn't prove actually that there are mirrors in the sense of homological mirror symmetry, not yet. Someday, probably, the vast program that Kenji Fukaya and Mohamed Abouzaid are pursuing using family flow homology will tell us, once you have this, this is good enough, you're done. But we're not yet there, so I'm still in business. Okay, so one direction we can look at is check whether the wrapped Fukaya category of h is indeed equivalent to the derived category of singularities or matrix factorizations of the historic superpotential w. It should be somewhere up there. Yes, sorry, over there. And that's been studied in various cases. So that's known, for example, for the pair of pants and some other rim and surfaces in joint work with Abouzaid, F.M.O.F., Ketzarkov, and Orlov. So what's the game there? I'm not going to get into detail because I want to focus on the other side. But so wrapped flow homology is about you take Lagrangian submanifolds and you perturb them at infinity by a Hamiltonian that grows quadratically. So that couples flow intersection theory with contact dynamics at infinity. So concretely here, you might be looking at things like properly embedded arcs on the pair of pants and the perturbation that you do when you do flow theory is to wrap them around the cylindrical ends of your pants. And then you look at intersections and you get something big and infinite-dimensional and interesting. And then the game is to compare that with matrix factorizations of, in this case, C cubed minus Z0, Z1, Z2. So the kind of things you could look at are, for example, the two periodic complexes, sorry, not complexes, deformed complexes given by, for example, minus Z0 and Z1, Z2. And the claim is that this guy and its two friends will actually turn out to be mirror to these three arcs on the pair of pants. And you can check by brute force calculation that these things match. And then there's work on other remand surfaces done by Buckland on one side and also the thesis of my student heavily in progress. And there's work on higher-dimensional pairs of pants. For example, Nick Sheridan's first thesis result was the compact analog of that, looking at compact Lagrangians in the higher-dimensional pair of pants and matching that with singularities, I mean, basically the skyscraper sheaf of the origin in here. Okay, but the other direction, which is more what I would like to think about today, is slightly more mysterious. This one is far from being completely done, but it feels like it's almost under control in some ways. And the one where I think more work remains to be done is the opposite direction. No, how do I? Okay. So the other direction is about comparing the derived category of coherent sheaves of H versus some sort of Foucaille category of this Landau-Ginsburg model. And what the issue is is that this should be some sort of fiber-wise-wrapped Foucaille category. And, well, such things are not defined in general, at least not yet. But in this case, fortunately, we can do it. Okay, so what I'm going to say from now on is actually joint work with Mohammed Abu Zaid very much in progress. It's more in progress than the last time that I gave a similar talk in that now we've actually started writing. But we are still, well, still in the geometric preliminaries about whether the Lagrangians we're going to look at in a moment are geometrically bounded and in a wuss sense exactly. Maybe I want to say gate first name. Oh, does he have something on that? No, he was the first person. Okay, and there's probably others too. Okay, all right. So, okay. So what's the main steps that I want to discuss? So one is can we actually define and construct and calculate in some way a fiber-wise-wrapped version of our theory in this case? And the answer is yes, we can in a very limited context and for very specific Lagrangians. So I'm going to do this only for a very restrictive kind of category. There's a more general approach. Well, so I'm not familiar with Gabe's work, actually. So I don't know how much he constructs. Okay. Okay, so. Okay, and my student, Zach Sylvan, is also working on a generalization of this which should work but might be too complicated to use. So who knows? Anyway, in this particular case, we can do it. And then we can construct an object. Yes, yes. So the problem is, okay, so I'll explain in a moment what the issue is, but we're going to deal with non-compact Lagrangians of a certain kind and we'll want to do something that's halfway in between. So when you have a function with, basically, a set of critical values is proper, you can look only at Lagrangians that are fiber-wise compact and escape to infinity in an orderly manner. And basically, this is the setting of Paul Seidel's framework. Well, after I guess the ideas of. Probably I should have, well, I should probably have started with saying that Maxime had the first idea that, anyway, we should have Fouquet categories for such pairs. Sorry, that's, you know, the generic concept originates with Maxime. Okay, so in Conceivage, then Seidel and so on, one looks specifically at things like fimbles for a left-hatch vibration. Lagrangians that are non-compact in the total space but fiber-wise compact. The Vrap-Fouquet category looks at non-compact Lagrangians and applies a Hamiltonian flow that perturbs things in all directions at infinity. And what we want to do is a hybrid that does wrapping within the fibers of this function and only slight perturbations in the base directions. That's what this is about. Okay. So having defined fiber-wise Vrap theory with very restrictive assumptions, the next step is to see if there's any Lagrangians of interest that actually satisfy those assumptions. And the claim is there is one. So I'm going to call, maybe, whatever. We'll call it W of Y and W. And I could conjecture that this object generates the category since right now it's the only object I know how to put in there. It's obvious that it generates the category but that would not be a very fair claim to make. Okay. Anyway, I will still write should generate because, I mean, so the goal, I mean, the purpose of this guy in life is to be the mirror to the structure sheaf of H. And on an defined variety, the structure sheaf generates the derived category. So this should be the only Lagrangian I should care about. Do you have also some single Lagrangian kind of scale it on description of the story? He probably... So I don't have a good one because I don't know how to do these things. There should be one. What you're going to see is that this L0 turns out to basically be skeletal in nature ultimately in that it looks like it's defined very smoothly but if I try to draw a picture of what it looks like at infinity you will not believe that this thing was meant to be smooth. So I'm sure that a skeleton description might be possible. I don't quite know how to do it at this point. So I'm not going to say anything more. And the third step, of course, is to calculate endomorphisms of L0 in this fiber-wise wrap sense and find that it does match with endomorphisms of a structure sheaf. And so this tells you that the direct category of H embeds into this wrapped category. And actually, well, there's nothing else right now because we don't have other objects in there. But again, that's not exactly... Well, we don't know other examples of objects in there. That doesn't tell you exactly what you want to know. OK. So let me say a little bit of content about what goes into there. OK, so what's the basic idea of a fiber-wise wrapped category of these things? So, OK, so first of all, I could try to define that in more generality but really I will be in the historic setting. So maybe the example you should have in mind is when I explain the mirror to the pair of pants, Cn plus 1 mapping to C by the product of the coordinate. So this is something where the only critical value is 0. Other 0 will have something which is just the union of all the coordinate hyperplanes. The singular locus is the union of the coordinate axi... Sorry, the union of all the strata of co-dimension 2 or bigger. And the other fibers will be smooth and look like C star to the n. The kinds of ligands we want to look at will want properly embedded Lagrangians of manifolds satisfying various conditions. So the first one is about in which directions they can escape in the base. And that condition is similar to what happens in the more familiar case of things that are fiber-wise compact, things like fimbles and so on. You want roughly speaking your Lagrangians over arcs, at least outside of a compact subset. So one way I could state it is that my Lagrangian L projects under something which, well, I don't care what happens in a compact part, but outside of a compact subset, this is going to look like a union of radial lines and there's a forbidden direction which for me will be the negative real axis. So the image of L under this projection map, W, outside of a compact subset is a union of radial straight lines and not real negative. And then there's going to be, of course, some other asymptotic conditions. Sorry, there's going to be some other reasonable conditions that we want to impose. So we want these things to be properly embedded and we want when they escape at infinity fiber-wise to have some bounded geometry property in a fairly weak sense, that they don't look too bad in small balls. Just enough to be able to do some flow theory. And we'll want some extra conditions, some of which are going to be, well, I think I need to wait a bit to explain some of them. I don't want my Lagrangians to bound any holomorphic disks. That helps with flow theory. Well, the next condition is just too technical, so I will not explain it yet. So that first I need to explain what kind of... I've lost a blackboard. So how do we want to perturb these Lagrangians when we are going to take their flow homology? Or rather, what kind of Hamiltonian perturbations do we want to introduce in flow theory? No, we shouldn't do that. What kind of perturbations should we do to the Lagrangians? That's how we actually do it in this case. It should be equivalent, but we don't know how to do it the other way. Okay, so two things I need to do are perturb in the base and perturb in the fiber. And perturb in the base, well, let's pick a flow on C, which is identity on a compact subset and maps... Well, let me just draw a picture of what it does. So it's identity inside, and then if you take a collection of radial straight lines, what it will map them to is it will bend them towards the real negative direction and then go straight again. That's supposed to be all the radial and straight. So this sort of thing. So all the radial directions remain radial at infinity, all being bent. Except, sorry, the real negative direction is fixed, and all the other bends eventually they accumulate towards the real negative direction. Okay? So now this flow, well, I could try to work harder to make it area-preserving, but I don't actually care because what I'm going to do to it is take a horizontal left of rho t to the symplectic orthogonal of the fibres. And the claim is I can do that. I will obtain a flow upstairs, which is autonomous, and it's not a symplectomorphism, but it maps fibred Lagrangians to fibred Lagrangians. So in particular it maps this kind of Lagrangians to Lagrangians of the same kind. Okay? It's still an admissible Lagrangian. And then there's something else I can do, which is I can define phi t to be the flow of a Hamiltonian on the total space, which is invariant under parallel transport and fibre-wise proper. So what this will do is it will preserve all the fibres, but within each fibre it's going to rotate things at infinity. And what I want to say is I want this guy to have linear growth in a certain sense. And for those of you who already know Wrapped-Fleur theory, this is the linear growth kind of Hamiltonians in Wrapped-Fleur theory. And then these two flows actually are both autonomous and they commute. This one is not symplectic, but actually it shouldn't bother us because it maps Lagrangians to Lagrangians. And so now we'll define L t to be the image of L under both of these in either order. It bothers me because it's so simple. We're going to make it isomorphic by definition. The point is the area on C is not relevant because the area on the total space is very different from the area on C. So you could change the symplectic form of the total space, but I don't know if that would be really legal. No, because then that will not preserve fibrousness. If you just take the pullback, you will have to rescale the flow by an amount which varies in the fiber. So I don't know a better solution, but I claim this should not bother us. Well, okay, sorry. Rho t of L is Hamiltonian isotopic to L just by a Hamiltonian which I do not want to make explicit because it's a Lagrangian isotope. And you can check easily that it will be exact. So that's probably the best answer. It is a Hamiltonian isotope of L, just the Hamiltonian is not the same for all L. What is the exact? The key one will be exact. It will be actually contractable topologically. There's others I want to look at which would not be exact, but they don't bound any disks. There's no disks in the page. That's a very good point. For the pair of pants, everything can be as fine. If we are not with a pair of pants and a few other examples like that, total space is not exact, so that doesn't happen. And, yes, sorry, so that means, I didn't say exactly in what sense we had things. So in case I don't get there, the last step is, you know, in the case of a pair of pants, it should be completely clean. In cases that are other than the pair of pants, this is modulo some correction terms and some Gromov-Witton theory corrections. The calculation of the endomorphisms of L0 being what you think. There's a calculation of the floor differential. Let me just try to get there. It's true that it's well-defined, and to calculate it, you need to use things that are well-accepted, but not as clean and elementary as the others. Okay. Okay. So if you set this up properly, the various things that will happen are that, okay, so I said the fibers, they look like C star to the n's, and what my flow will do, really, is some rotations inside the argument directions of the C star to the n's, the phi t, the fiber-wise flow. And I can arrange that this flow happens always at speed t in some direction, at unit speed in one of the directions. So that means points can come back to themselves in times other than multiples of 2 pi. So the claim is I can arrange that the distance between L t and L t prime at infinity is bounded below uniformly. One other t minus t prime is not a multiple of 2 pi, 2 pi. That's a specific feature of the Hamiltonians I know how to build in the historic case, and I will not care about non-toric examples. So that's, you know, a big restriction in generality. It means for a fixed t and t prime, it's bounded below infinity. Yes, yes, sorry, yes. Uniformly by quantity which depends on the distance of t minus t prime to 2 pi z. Okay, and there's other reasonable features which I'm going to just skip. Okay. Oh, sorry, but important. So the other thing I should discuss is what happens since I'm doing this linear perturbations. So what happens fiber-wise is that in a finite, in a small amount of time, I have a Lagrangian which might look like this in my c star to the n, and I will start perturbing it. And of course as I perturb more and more, it's going to intersect more and more because the ends are rotating at a constant pace. But for small times, there's only one intersection, a distinguished one. In the base, if I take something that fibers over an arc and I push that arc by rho t, I will just push the arc around. And for very small times, as long as this angle doesn't hit that angle, I'm going to still have a distinguished intersection. So the claim is for small t minus t prime, there's a canonical element intersection point of l t with l t prime. And we're going to call that the identity of this thing, which is kind of a strange thing to say because they're not the same Lagrangian anymore. The correct thing to say is that we will define flow theory by basically letting this flow operate and considering that we can replace freely l by l t for large values of t. So technically you set this up as a limit. I don't know whether it's direct or inverse. Actually, oh, direct. Yes, OK. So what we're going to do is really look at some direct limit of the flow complexes of l. So I'm going to take, well, actually since I say 2 pi, I could say probably 2 pi n plus pi with l prime is what I will define to be hub l l prime in my category. Where the connecting map is given by multiplication by this thing I called identity. So, OK. How do we find Lagrangians of this kind in here? And what's interesting? So if you've seen left-hatch vibrations, you know what's very tempting to do. When you have an isolated singularity, what you do is you take the vanishing cycle in the nearby fibers all over a path that starts at the critical value and continues forever and you get a left-hatch symbol. And that's a very nice thing to play with. So in this case, we have singularities along the union of various strata. In the C-cube case, that's just the union of the coordinate axes in C-cubed. And the first obvious thing you might want to do is say, hey, this is a Morsebot singularity everywhere except at the origin. So I can take a circle in one of my coordinate axes and push this by parallel transport along a straight line in the base to get something that will look like... So in this case, the vanishing cycles are S1s for each of the points. So you'll get a torus in the nearby fibers and this will produce for you a Lagrangian R squared times S1, a solid torus. And these are interesting things to look at, but they're not what we want. They're actually going to be the mirrors and the skyscraper sheaves of points on the pair of pants. So this corresponds to the skyscraper sheaves of points. Actually, on H in general, this will be the same. I will have just taking turi inside the smooth strata of the critical locus and pushing them gives me the mirrors to point. So in fact, part of the conceptual problem was that for the longest time, we're kind of stuck with this issue that we thought what we should be doing for the structure sheave is instead take the real positive part of this critical locus. So in this case, the R plus inside each axis parallel transport that and then you get a piecewise smooth Lagrangian and we don't know how to study it for flow homology and we don't know how to smooth it. So that was the problem of smoothing this thing in unsecured was assigned to several of my graduate students who went on to instead write very long, complicated things about infinity structures and decided that algebra was easier. So I still don't know how to do that. Instead, the new idea is that we don't need to obsess about what you would call maybe generalized fimbles and if you really can't deal with a singularity then maybe it's best to just bypass it. So what do I mean by that? So we have a singular fiber which is the union of coordinate axes but we're not going to look at it. We're going to look at a smooth fiber. So remember, my potential is minus the product of the coordinates. The other fibers are c star to the n and specifically the fiber at a real negative value say minus 1 is v c star to the n where the product of the coordinates is 1. So that one has a well-defined positive real part. So let's take that for this kind of arbitrary but it's more elegant. Okay, so in this one, I will look at r plus to the n and then I will take a path in the base that does something like this, goes around the origin and escapes in two directions kind of to the right and then I will parallel transport this along that arc and what that looks like, well down here in the middle it looks like a nice smooth thing and then at infinity it looks like a crazy thing which is really... Well, the best I can do is draw... Okay, so the claim is I will get something that will be an arc... Sorry, isomorphic to r to the n plus 1 and if I slice it by setting one coordinate to be constant in the case of c cubed then I will get something that's like Lagrangian r2 and that Lagrangian r2, the way it looks, you'd think, okay, take a very nice straight Lagrangian parallel transport it along a smooth curve this is a very simple explicit thing, what could go wrong? Well, what it looks like when you slice it by setting one of the coordinates constant equal to some larger value is where moreover these two things come together as you go out to infinity. Anyway, it's messy, it looks very piecewise linear that leads credence to the idea that there's a skeleton hiding behind there and that in fact this is not so far from the one I had complained was bad over there but anyway, here it is, technically it's smooth even for infinity it looks less and less smooth the curvature is unbounded and the injectivity radius is going down to zero it still has barely bounded geometry fiberwise in the sense of C Corav I mean, I don't think he invented that condition but I mean in the sense of the technical condition he uses to study holomorphic curves with boundaries and Lagrangians and so the claim is this thing actually exists and is suitable for doing flow theory so this is what I will call at zero maybe if I were slightly less, well, it's okay so why is this not completely unreasonable? maybe I should point out so this construction of taking a Lagrangian far out at infinity and moving it around all the singular fibers and bringing it back to infinity has been known for a little bit to be mirror to the inclusion function from the mirror of the fiber to the mirror of the total space and in this case because of a strange way that we arranged for this generalized SYZ construction the inclusion of... well, the fiber is mirror to C star to the N and the inclusion of C star to the N into H is of course, well, secretly into this bigger space is exactly what we want to do and what we do actually includes C star to the N into this big blow up space I was considering and then we restrict to the exceptional divisor which gives us H as the image anyway, so there's some justification for why this should be the right thing but of course we need to calculate and I will not do the full calculation in front of you but I want to convince you at least that the answer might look like what you want so the question is what do... I never wrote number 2 but anyway what do the endomorphisms of this object look like? so to answer that question I need to take it and its image under this flow that I was talking about which means in the base I will see one copy L0 and the other copy L0 T for T very positive will look something like that both branches will have been pushed past by this flow rho T in the base and fiber wise, what do I get? well, so all the intersections will happen over these two points so I might as well focus on these so this one what I will see is I mean C star to the N and inside that L0 was just the real part and its image under wrapping so that was something that I was just going to twist around a lot so I think I throw it the wrong way but it's not worry and here essentially a similar picture so what are the intersection points that we see come up? well we get two copies of the wrap flow complex of the real part in C star to the N and that is already understood so see in the case of a cylinder for example this wrap flow complex has a Z-worth of generators and you can calculate explicitly the multiplicative structure and find that what you will get for your flow homology just for this picture of R plus to the N C star to the N will be Laurent polynomials in N variables for each cylinder factor you get things that I would call maybe one here in the middle X, X squared and so on X inverse X minus 2 and so on where one is actually the one that's in the interior you have a minimum of your Hamiltonian and same for all the others so that means here I will get a copy of Laurent polynomials as my flow complex and here I will get another copy times a generator I will call H okay now except the multiplicative structure of course I need to now work in the total space but fortunately I'm in a situation where the projection is holomorphic so holomorphic curves project to holomorphic things with boundaries on these things that means the only holomorphic curves I need to look at are either entirely within the fibres or they project to this bygone in this case and when I multiply I will need three copies and then we will project to triangles with corners at the intersection point between these arcs in the base so it's all going to be explicit very computable well almost computable anyway so the claim is the multiplicative structure is in fact we need to write Laurent polynomials in n variables which all have degree zero and with an extra generator I will call H which is degree minus one odd and H square equals zero as you would expect for an odd generator okay so the multiplicative structure on this floor complex is on the nose something reasonably pleasant and there's no higher operations mu three and beyond however there is a differential I didn't tell you about the differential yet so what's interesting and where really the meat of the argument ends up being is in the calculation of the floor differential so the main part of the calculation is actually going to be the floor differential applied to H so what this means in practice whoops yeah but it doesn't look the same so I should probably, it's the same but it doesn't look the same okay zero was here and I'm going to draw it similar so I want to count over this region which encloses the origin I need to count pseudo but actually in this case really I'm going to take sections over this thing with boundaries on well here L0 here L0T and with corners at well this corner needs to be at the guy I can say it was in the middle that I called H and this one can be wherever it wants I will get some Laurent series in principle but polynomial once you know that you have compactness in the variables XI okay and now how do we count sections of these things well this fits in the general framework of these ideas that Seidel and then James Pascal in his thesis and so on have developed for computing hopefully speaking what you do is you push the origin out of here and try to understand what happens when you do that and the answer is some disks bubble off sorry not in this case but I'm skipping a step I think I should anyway so the claim is this can be done completely for the case of Cn plus 1 and modulo some grammar written theory in the general case grammar written theory of a sort that can be found in the papers of Chenlau and Lung or other papers of that sort and so what you find is that D of H will end up being exactly there will be one term in this for each Torek divisor and the Torek divisors of your Torek variety do correspond exactly to the monomials in your defining equation and so labeling them in the same way you will find things that are of the form so 1 plus dot dot dot this dot dot dot is some grammar for written invariance which are going to be correction terms T to the row alpha X to the alpha all that times E and so now that tells you that if you take the flower cohomology so the cohomology of this thing H of course doesn't survive in fact the this piece maps injectively into here so this all dies and what dies in here is exactly everything that's divisible by H so the cohomology is exactly Laurent polynomials mod this defining equation but I called F I think initially and that completes the proof except I didn't give you any of course of the actual arguments and details okay this is all in degree 0 so it's automatically formal yes yes it's just a rank