Okay. Thank you very much. So, yeah. So, before I say, so my role in the sort of, you know,

asteroid belt of minor bodies that is orbiting Koncevich is the following one that I take a small slice of his things, mainly from the late 90s, and I combine them with a small slice of Fukaya's and a small slice of Donaldson's things. Okay. So, the rules

for this game have been known essentially for a long time. There is a geometric ingredient which is the classical, which is the geometry of Lefshitz vibrations, which is a very rich thing, and I think we don't really understand how complex it is, and the main goal of the theory is to try to capture some of that complexity as much as we can. Then, so that's

Donaldson's contribution. Then, there's Fukaya's contribution, which is, you know, the associated theory of pseudo holomorphic curves, which is also very complex, but, you know, in some sense, in a way that we understand, right? So, if you have a problem that you

want to solve using pseudo holomorphic curve theory, you have to decide which family of Riemann surfaces you're going to use, which auxiliary geometric data, but it's a finite amount of games, and then there is, you know, well, one of the things that Zebesha has contributed is a sort of particular flavor of homological algebra, which you will

use to sort of formulate your results and, you know, capture the information that pseudo holomorphic curves give you. So, there are these three kinds of building blocks, and then there is, you know, homological mirror symmetry, which is, you know, like the building plan that will help you, tell you where things should go, okay? So, it's a little

bit like putting together IKEA furniture, okay? And so, my success so far is really equal to my success in putting together IKEA furniture, which means that at each step, I'm convinced that this is exactly the way that things are supposed to go, and then when I look back, when I'm finished, it turns out that, you know, not only was this kind of clumsy what I did, but it's actually an obstruction to proceeding to the next

step. So, this has iterated several times. This is my latest iteration. I'm not particularly sanguine about it, but hey, you just, that's what I can do, okay? There won't be a particular amount of theorems in this thing, but I'm trying to make some kind of picture. So,

the basic geometry here in the case of, classical case of, you know, say a smooth projective variety, we have a line bundle, and we have two sections, and just, you know, it's classical. Well, I don't want any of these sections to be zero. I also don't want them to be linearly dependent, and in fact, I don't want the zero sections to have,

zero sets of common components. So, let's say, if you take both zero sets, you intersect them, it has at least dimension to it. You can assume that it's a generic pencil, if you want. So, obviously, there's a hypersurface X0, which is where S0 vanishes. There's a hypersurface X infinity, where S infinity vanishes, and then there's a whole pencil of hypersurfaces XZ,

which are defined by vanishing of some linear combination. And those hypersurfaces intersect all at the same thing B. Okay, so if you don't mind that they intersect, you can think of the whole thing as being given by some not quite well-defined map to the projective line, whose fibers are these XZ. Or if you mind the fact that the fibers intersect, then one thing you

can do is you can remove one fiber, let's say the one at infinity, and then the quotient of these two things becomes a well-defined function, which lands you in the affine line. The fibers of that function W are not the original, not this XZ, but the open parts where you

remove the base locus. Okay, so there are two different ways that I'm interested in to look at these things. One is to look at their topology and to look at their symplectic topology. So there we look at the singularities, we look at vanishing cycles and monodromy representations. And since the vanishing cycles are Lagrangian submanifolds,

it's natural to use symplectic topology to it. And look at pseudo-holomorphic curves, so pseudo-holomorphic maps into X or into the fibers with various boundary conditions. I think about this, it's still essentially topological. So you don't really use the

algebraic geometry very much. In particular, you don't really care about the single numbers. Like, for instance, where exactly the critical values are located on the complex plane isn't really important if you sort of deform things slightly, that will be sort of indistinguishable from this topological point of view, which is green, okay? So the other point of view,

which is sort of algebraic, which is purple, is to really look at these things here as defining a family of algebraic varieties, X, Z, and you can look at them from the point of view of deformation theory, from the point of view of cycles, from the point of view

of Hodge theory, the Asmanian connection. And to a large, I mean, not the Hodge theory, but everything else I said, you could do it over a, you don't need the fact that the brown field was the complex numbers. You could do it over a variety of other fields.

Just to mark your t-shirt of the green size, the green color. That's right. I'm definitely there, yeah. I mean, yes. My t-shirt is green, so. So here, actually, the numbers really matter, right? So they will be, you know, in your

family, there will be certain parameter values, Z, where special things happen, maybe a cycle becomes algebraic, and those things are relevant, okay? So okay, so mirror, it's a well-known thing that sort of these two points of view are exchanged under mirror symmetry, so I will do an example which is very close to

the most classical one, rather than trying to formulate things sort of abstractly, okay? So the example is this thing here, that, you know, on the subplastic topology side, I look at a pencil of cubics on CP2, so the line bundle is O of 3, which is the

anti-canonical bundle, which is always the case in mirror symmetry. It is not quite a generic pencil, so I could have done it for a generic pencil, but this is slightly simpler situation. So one section, I choose X0, X1, X2 on P2,

so the zero set of that is a chain of three, whoa, of three plus one curves. I wish I could edit this thing here, but it's too late, so this will stand as a mark of my shame. And S0 now will choose generically, and it will, so then it doesn't matter

very much which one you choose, okay? So the base locus of these things are nine points, and the generic fiber is an elliptic curve, so since I'm thinking topologically, let me say it's a torus, and there are nine critical points. In fact, if you

remove the fiber at infinity, you just get the open torus, C star cross C star, okay? So it's a little bit confusing, just a special feature of this example, that the mirror looks vaguely similar, but you shouldn't attribute any importance to these similarities.

So the mirror is, you know, roughly speaking, it's an orbifold, so you take CP2, divide by Z3 in a way that gives you kreppen singularities, A2 singularities, which admit kreppen resolutions, resolve them, or you can just think of working on the orbifold, and you look at essentially

the same thing, except the fiber at infinity will now look differently, because you did this resolution, so it's a chain of nine rational curves, and because of this Z3 quotient, you now have three base points, and you have three critical points, and in fact,

you know, if you write down the function W in explicit coordinates, it is this one here, you know, which is well known, play a role in mirror symmetry for P2, okay? So this is this geometry here, so this, maybe naively speaking, I would say, it appears that this X check is slightly bigger than X, right, on account of, you know, its total betty number

or something, but in the way that the complexity is accounted for, they actually become equal, so this is the way that the complexity is accounted for, and I have to warn you that, you know, now there will be sort of, you know, when this subject started, there was like,

you know, a symplectic and an algebra-geometric side, and each side there was a single category, and as the subject developed, there's sort of various versions you can do, which give rise to all sorts of categories, so later, when we get to a couple of slides forwards, you will see that there are a lot of boxes, now I just make a list of stuff, and later we'll try to put the stuff into boxes, okay?

So the, you know, on the algebraic geometry side, maybe the simplest thing you could do is that you could take your variety X check, it's smooth, and you consider its derived category, by which I mean the bounded derived category of coherent sheaves, okay? So this thing here, it's what, it's K theory has rank nine, okay?

And so the corresponding thing on the symplectic side is, is that you take, you take this W, which is this Lefschet's vibration over the affine line,

and there is a Foucault category associated to it whose objects are essentially Lefschet's symbols, since we have nine critical points, we have a, you know, have a basis of nine Lefschet's symbols, which generate everything, and so this Foucault category will be the mirror of this derived category here, okay?

You know, obviously another thing, which is maybe more obvious on the symplectic geometry side, is instead of considering this vibration by itself, why don't we just consider a single fiber, so the fibers of W are this XZ minus the base locus, so it's a, in this case here, it's a punctured elliptic curve, T2 minus nine points,

and this, roughly speaking, this corresponds to the fiber at infinity of the mirror pencil. Now this X check infinity is singular, so if I write D of X check infinity, it's not particularly well behaved, so I replace it by something which is slightly better behaved,

which is, you know, the subcategory of perfect complexes, okay? But maybe I shouldn't have made a distinction, and I should just call everything D, and just say if it's singular, then D is perfect complexes, because that's what I like, and it's well behaved. It's a question about, you know, which finiteness properties you're going to impose, what you think of something being bounded and compact, in a sense.

Okay, so these do correspond, and then the next version is something which is, you know, where the subject actually, so now we're sort of going away, and we're converging, going backwards in time, towards the version that the subject actually started, so this is this XZ minus B, this is a punctured torus,

so I can put the punctures back, so the way that you do it, put the punctures back, is in the original thing, you know, you considered curves on this punctured torus, and you considered, you know, polygons, holomorphic maps on the punctured torus. Now let's consider holomorphic maps on the torus, and see how often do they actually go over the punctures, okay?

If they don't go over the punctures at all, now we're working in the complement of the punctures. If they go over the punctures one, you count them with a formal parameter Q. If they go twice, you count them with Q squared, so that gives rise to what's called the relative category of the torus relative to these punctures. It's a category which has a formal parameter Q, and if you set Q equals zero,

you can recover the thing that you had before, right? And so it's a formal one-parameter deformation of the previous thing, and so unsurprisingly, you know, if you have the, you know, here you had the mirror pencil, you have the fiber at infinity, the fiber at infinity is the fiber at infinity in this family of fibers,

so you can, you know, instead of checking just the point at infinity, you can take a formal disk near infinity, parameterized by some parameter Q, and so that gives rise to, you know, a scheme over this spec C double bracket Q, which is, you know, this family of elliptic curves,

which are Q equals zero, it's singular, and otherwise it's non-singular. So these correspond to here, but I haven't told you, here there's a natural choice of parameter Q, I haven't told you how I'm choosing the parameter Q on the other side, that is much trickier, okay? And so then the next step is relatively straightforward, which is to, we have this formal parameter Q, let's invert Q,

so we introduce Q inverse on the algebra geometric side, that means that I pass to the generic fiber, which is now a smooth elliptic curve, so it's over this ground field, so this is where, this is Kontsevich's original version of homological mirror symmetry,

which says that in this case here, it just means that, you know, we work over the compact torus and we forget the things, you know, we allow inverse, allowing inverses of the parameter Q basically allows your objects which are curved to sort of pass over the punctures, so it's effectively as if you actually worked on the actual torus, okay?

Now, so this is the classical version, now another thing which you can do, certainly would be on this side here, plop, you could just remove the fiber at infinity, okay? And then, so you get an open variety and you consider its derived categories,

this is smooth, so I don't have to worry about what I'm doing, and you know, at some point it was realized that there is a corresponding thing on the symplectic geometry side which looks quite similar, which is where I take my CP2, I remove this fiber here and

I, now I do something which is called, this is script W, so it's another version of the Fukaya category which I also script F, it's called the Vrap Fukaya category, so it has non-compact Lagrangian submanifolds, just like the previous Lefschet symbols which were also non-compact, but the Lefschet symbols are only allowed to go to infinity in a particular specified way

that's dictated by the Lefschet's vibration, here they can go to infinity in a rather freer way, and the price that you pay between them to make sense of this is that the morphism spaces are infinite dimensional, which is pretty good because the morphism spaces here are also infinite dimensional. Okay, and the last thing I'm missing from my collection is,

you know, how about mirror symmetry for CP2 itself? Of course, that's something that you might want to do instead of, you know, passing to fibers or removing parts of it. So there is a Fukaya category of CP2,

there's also actually a one parameter family of these things here that depends on a complex number Z, some kind of a bulk parameter, and so what happens here is that, there should be a W check here, typo, so you remove the, so here you have this dual path, you remove the fiber at infinity, then you have this function, W check to A1,

and there's a way to associate to it a category, the category Landau-Ginzburg brains, so locally speaking, what's happening here is that we have the total space, which is smooth algebraic variety, we consider Z2 graded,

Z2 periodic complexes of, you know, vector bundles on it, and with a differential, where the differential does not square to zero, but it squares to W, or if you can put in a parameter, which is a constant Z, they square to W minus Z. Paul, do you really mean bulk? Don't you mean like central charge?

I never... I'm saying, can't you use a bulk is not this? Well, this is bulk, yes, it's a constant bulk, it's a multiple of the identity that I'm using, no? Maybe it's not, okay, but probably wrong word, okay? Okay, anyway, so you get this thing here, which is called the

Landau-Ginzburg category here, and this is mirrored to this thing here. Now, when I say mirror, you know, in most cases, there are certain closures and the formal operations which you're supposed to do to make things match up,

but I'm just omitting all of them. And there's one more thing which we have no idea how to make an actual geometric counterpart, which is, you know, here, this is an actual, you get an actual family of elliptic curves. You know, we have a good mirror to the fiber at infinity, which was right here. We have a mirror to the fiber near infinity,

which is right there, so it's a natural question to ask, oops, what is the mirrors, are there mirrors to the other fibers, which are actual nice, smooth elliptic curves of a C? And we have no idea. I have a doubt in the previous slide. Well, you can see the relative fukai, how the intersection is B.

You should get kind of a curve, a formal family of Calabi-Yau, but in flat coordinates, and it should be not an algebraic curve, not your original family. I mean, I'm sorry, I didn't say that this parameter Q is equal to, I mean, well. No, no, but even after your optimization, it makes no, I don't believe it,

because if you consider, if you count here, intersection with B, yes, you get a curve, which is a straight curve in the flat coordinates, yeah, and let's suppose you have not, you know, five or something, you curve it, and this straight curve is not a general algebraic curve in the model of Calabi-Yau,

because here, consider, on the mirror side, you consider algebraic curve in the model of Calabi-Yau, yeah, it's a dunt free fit, yeah. I don't, I mean, there's only one parameter anyway, so. No, no, yeah, you have something, you have, you get one part of the family here,

totally different, you might have something, but in more complicated examples, the sample of gray curve in the dimension of the space, and it's not a straight line of flat coordinates. Oh, I'm not claiming that this is a description of what happens in general. Okay, this is just this case, because if I do general things, I have to put in caveats and assumptions and everything all over the place, okay,

but it will be a description in some cases, and if you don't think that's true, then you should tell me, okay, so I think I'll remind you of your objection later, and then you can kill me, so okay, so right, so okay, so who invented all these things, and who came up all these things, and you know, the answer is obviously not, it wasn't all me,

so the same thing we know with names, okay, so let's see, so the, you know, the Fukaya categories that I'm using just in their plain flavor, say for a puncture torus, obviously due to Fukaya, the version for Lefschet's vibrations is with these Lefschet's symbols, is as far as I know,

due to Kontsevich, and I sort of wrote it down, big deal, the full relative version, I also wrote down, but it's also no big deal, on the algebraic geometry side, well, this is Verdie's thesis, and I don't know who invented perfect complexes, but you know,

anyway, I don't think anybody's gonna ask me to assign precise credit for this thing here, so the, you know, the version of mirror symmetry in this case here, well actually, you know, there's a computation in Kontsevich's original paper which essentially proves it,

and then there was a lot more computations which sort of, I don't know, done by Polishuk and Zaslow, and the limiting version where you have sort of a puncture torus,

I mean, nobody really knows who should take credit for it, the one here is in a paper by Uruk, Katarkov, and Orlov. No, but if it's, if, now, that's a fully elliptic curve, it's a good example, not general variety, it's in view of its, it's, it's, it's, it's not related to the find and derive category of elliptic curves.

It's not Vergiero defined and derived category of, of, well, in this case here, it's of CP2 of the CP2 orbifold, I mean, you know, I know, okay, well, so, so, over here, well, you know, this is now, this is the Fakaya category of the, the actual elliptic curve with this parameter here, I don't know, I just wrote Fakaya order,

and obviously, their theory is much more general than this. So, in this case here, it was already in, in Kontsevich and in Polishuk-Zaslow, who also proved this thing here again. Well, these Raptochaya categories were, were introduced by, by a bunch of people, among them Abouzeid and myself, and actually, this particular special case here could,

you could do by hand, but, but, you know, these kind of arguments that actually compute the rap category probably also should be credited to the two of us. This thing here, this, this, this FZ done by, by Jung and Oh and, and Fakaya and,

and, and Cholly and Cho and a number of other people, you know, to prove this thing here. Essentially, the, these, in this particular case here, these categories are rather simple, so you can sort of prove that, sort of match objects one to one. The missing thing is to show that, you know, you have accounted for all the objects for CP2

and, you know, that, that's where this long list of names here comes in. Oh, and so matrix factorizations were invented by Eisenberg, and the categories, the triangulated categories, sort of, there's a long paper by Buchwald, which is somehow relevant, and then there is, you know, Olof, who sort of gave a different

definition and so on, developed the theory to a very large extent, especially beyond the case of affine varieties, which was the focus of the original theory. Okay, I hope I didn't, does anybody want to correct me on the names? I tried my best, okay, but, okay, so, so we, on both sides,

we have, like, all sorts of different categorical structures. How are they all related to each other, okay? Okay, fun. Let's look at the algebraic geometry side, okay? So we have the total space of the pencil, and we have one of the hypersurfaces, let's say the one at infinity, that's what appeared.

Obviously, there's restriction and inclusion factors, okay? Well, so then we have the, the, the fiber at infinity and the fiber in a formal disk near infinity, and obviously there's some kind of deformation theory there, and then when you want to pass to the punctured formal disk, you just invert the parameter, that's just like a, you know, a simple process, you just invert Q, there's nothing going on here.

When you, what happens when you try to remove X infinity, right, and pass to the complement, you can actually do it by a sort of, well, this is a localization you pass to an open subset, there's a correspondence of categorical notion of localization,

where you take the, the image of perf inside here and you quotient out, and then you get the, the drive category of the complement. And then the passage from this thing here to the Landau-Ginzburg category, you can think of it in a way as a, as a deformation, and then passing to the generic

fiber, you have to be somewhat careful because the, the deformation parameter has degree two, so the, the theory doesn't quite look like ordinary deformation theory, but I'm just going to allow myself to just sort of say this, okay? And exactly the same thing holds on this side here, you know, you have left shield symbols, they have boundaries which are vanishing cycles in a fiber,

so there's some restriction functor, there's an adjoint functor, a little bit more complicated to describe. Once you are in a fiber, there's a formal deformation that gives you to the relative category and so on, and maybe most substantially for the point of view of symplectic geometry,

you can obtain the wrapped category as a sort of categorical localization of this, this category of the vibration. So it would seem that, you know, we're all set, right? There's a vibe of relationships, there's a corresponding web of relationships. Now, this is really not good, okay?

And why is it not good? Well, for two reasons, one is, you know, I can make funny arrows here, but the funny arrows don't mean that things actually determine each other, so this arrow here with the dots is like a, you know, a little, you know, reproduction of the picture here on the right. This funny thing here with the dot and this thing here is a dot are deformations.

Well, you know, just because I say it's a deformation, it doesn't mean that you know what the deformation actually is going to be, there has been some success in, you know, using abstract deformation theory as a shortcut for computations, but still, you know, that just doesn't determine this one here, okay? And also the question is, you know, where's the center of the picture actually, right?

So I mean, you know, if you look back, it seems like with the arrow going, it seems that this pair here is the center of the picture, okay? But, you know, really, if you look at it from a point of view of geometry, the center of the picture is given by the pencil, right, itself,

and there's nothing here which is just pencil to pencil, because the objects that are here in the middle actually depend only on what the topology is after you throw away the fibers at infinities, and I'm talking about the green part, right? So they, you know, if I focus on this square here, I actually forgot that this thing here extends over CP1.

And clearly, I cannot hope to recover all the data if I forgot this crucial geometric fact. Okay, so actually, let me skip this thing here in the interest of clarity. I have one. I'll show it to all the specialists later, secretly, okay?

So then you have to also forget this box here, okay? So but let me say it as much as this thing here. So there are cases where, on the symplectic geometry side, it looks essentially the same. You have a pencil, but on the algebraic geometry side, you get something which isn't quite classical algebraic geometry.

It could be some kind of non-commutative deformations, or it could be something where, instead of the variety, which was supposed to be, you know, the X check, is itself not a variety by the Landau-Ginzburg model. And then you have some kind of vibrations of that, and so, formally speaking, things sort of look the same,

but on the algebraic geometry side, it's not really a pencil in the sense of classical algebraic geometry. And this is, it's actually a good thing about this game here, which is because it gives you a window into, you know, what non-commutative algebraic geometry and various exotic versions should actually look like, it corresponding to still symplectic geometry.

So what we're gonna think about is, you know, what is actually a pencil, okay? So let's think, let me be slightly less ambitious and think what is actually a divisor, an effective divisor, hypersurface. So what is it? So I have a variety. Now the notation is gonna switch temporarily,

because, you know, I'm gonna write X, but I'm gonna do algebraic geometry in it, because I'm going to forget about mirror symmetry for a while. So you have a variety, you have your line bundle, you have a non-zero section that gives rise to divisor. But instead of looking at the actual hypersurface, that it defines, I prefer to look

at a sort of DG resolution of this. So I take the structure sheaf, and I take the direct sum with the invertible sheaf L, placed in degree minus one. So this is, I think of this as the exterior algebra on L. So this is a Z-graded sheaf over X. I equip it with a differential that's given by S.

So this is a sheaf of DG algebras, which I think of as some kind of differential gradient scheme. And this is actually quasi-isomorphic to the hypersurface, which means this thing here is acyclic everywhere, it's outside the hypersurface, and on the hypersurface, it just, it's cohomology is O, okay? And it's actually somehow quasi-isomorphic

in some DG algebra sense, okay? So the motto is that when I think of this hypersurface here, I don't wanna think of the hypersurface, I wanna think of this hypersurface here as an extension of X by this line bundle, okay? And the same thing is if I have a pencil,

I wanna think about it as a family of extensions of X by line bundle that's parametrized by the projective line, so it's parametrized by two homogeneous coordinates. Happiness? Excellent. So assuming hypersurface being smooth, no? No, I didn't say, did I say that? I didn't say that, okay?

Okay, I mean, for instance, the fiber, not all fibers of the pencil can be smooth, right? Sorry, you said family. Excuse me? So you said family, family parametrized by? P1. By P1. So it's parametrized by two, it has two homogeneous variables. The notion of DG scheme over P1, is that what you're thinking?

Yeah, yeah, yeah, pretty much, yeah. I mean, over A2 with some homogeneity property. That's pretty simple, okay? So let's see what this, so now I'm gonna take this stuff here and I'm going to translate it all into abstract non-commutative geometry, okay?

So instead of my original variety X, I'm going to have a non-commutative space here, which is, you know, and for me, I'm kind of, I got addicted to this language of A infinity structures. So I would like this to be an A infinity algebra. And what if you don't like or don't want to like

A infinity structures? Well, you can assume that it's a DG algebra, okay? But you'll have to pay for it in the next step. There's no loss of generality, but you'll have to pay for it in the next step. So the next step is to say, okay, this guy here, now I need something which replaces the line bundle itself.

So I'm going to replace it by bi-module, which is invertible with respect to tensor product. So you treat this as another bi-module, which you tensor them together, you get the diagonal bi-module. Let's say tensoring with P is an invertible operation. And so, you know, what you will have to pay

is that, you know, in the A infinity world, any A infinity bi-module is, you know, homotopically flat and homotopically projective and homotopically anything. Whereas if you work in the DG world, you have to put in a bunch of assumptions on P. So you have to find a suitable, choose a suitable resolution. But there is no actual substantial difference, okay? So what do I want to do is now I want to make an extension.

So I take A, I add P, shift it down in degrees by one, and I want to make this into an A infinity algebra, okay? And this I will call non-commutative divisor. There's some conditions that I want to have. So if you know an A infinity algebra, it's given by multi-linear operations on B, okay? So my first condition is obviously I have to somehow,

they have to have something to do with the original A. So the condition is if you take your multi-linear operations and all your inputs are in A, then your output is in A and it recovers the original structure of A, okay? So A sits inside there. The next simplest case would be to say I take all the inputs in A except one, okay?

And oops, I didn't say it. And the one lies in P. And here I didn't write it right. So when you have the one input in P, you could land in A or you could land in P. If you just consider the component that lands in P, it recovers the bimodule structure of P, okay?

So you could do the same thing here in a more classical context here. So then, you know, it's a kind of extension of this DGA by this DG bimodule. But it's not a square zero extension, okay? So you can have operations which takes several Ps and outputs something non-trivial, okay? So this thing here, I call it,

I decided now to call it a non-commutative divisor, but it's not entirely new. There's a version of this where P is the dual diagonal bimodule, which, for instance, appears in Kontsevich-Blasopoulos. Excuse me? Will the second summand be an ideal in P or not? Will the second summand be an ideal? No, absolutely not.

In fact, that's, here I wrote it in a confusing way, but you'll see in a second. So A could be on the right hand side. Yeah, there you go. So let me consider, so there is, in fact, there is a part where you take all inputs in A except one in P, and you land in A, okay? So let's look at this part here.

And the way to say that part here is let's consider B itself, not as an algebra, as a bimodule over the subalgebra A, okay? So here's B. So this means that we consider only those operations where all inputs except possibly one lie in A, okay? So then there's, you know, the assumptions say that, you know, there's A sits inside of it

as a bimodule over itself. The quotient bimodule is P, but the extension is not necessarily split, okay? And in fact, the boundary map of this extension is, because of the way that I shifted the grading, is a bimodule homomorphism from P to A. Since A is invertible, it's also a morphism from A to P inverse, which you want to think of as a section of P inverse, okay? And this thing here I call the first order part

of the non-commutative divisor. And so the corresponding thing is, I actually want to use geometric language. So when I say A, I just call it the ambient space where the divisor lives. P, I call the line bundle. So this sigma here, I call the section of the inverse. That's what defines the divisor. And B, I just call the divisor by itself. That's what it is. It's this extension.

So if we, you know, forget about A, just consider B as an infinity algebra. That's the divisor by itself. So there's a slight surprise here that, you know, sigma in general is not really everything, right? So there are higher order terms. And those higher order terms, you can analyze them into obstruction theory. There are obstruction groups.

The pieces of those obstruction groups are homes between higher tensor powers of P and A. They lie in certain degrees, okay? So you can ask them, you know, why didn't I see those pieces in classical algebraic geometry? You know, classical algebraic geometry, you give the line bundle, you give the section, and you're done, right? There's no extra data needed.

It's because in the classical algebra, these all live in negative degrees, the higher order obstruction. The classical algebraic world, there are no negative degree homes, and you will not see these things. These are homes of bimodules. These are homes of bimodules, yes. Okay, yes. Well, these aren't effective divisors, right? This is always effective divisor. When I say divisor. Do you think about, you know, metamorphic sections of these guys?

Not yet. Okay, when I say divisor, I always mean effective divisor, yeah. So one thing that has occurred before in the story and the algebra geometric side is where I removed the divisor and considered its complement, right? And I said that on the level of categories, this can be interpreted as a categorical localization

of quotient construction, and the same thing is true here. You basically, you take a bimodule, sorry, you take a modules, and b is an a module, and you kill b, okay? And so this gives you a localization construction in the sense of, you know, Keller and Drinfeld, and I just write it geometrically as saying,

you take a and we remove b, okay? But you could write, that's obviously a highly non-traditional way of writing it. Okay, so now we know what a non-commutative divisor is. What is a non-commutative pencil, okay? Well, it shouldn't be very hard, right?

It's a family of non-commutative divisors that's parametrized by two homogeneous variables. So we just have to get the homogeneity right. Okay, so let me fix a, you know, the two parameter space. If you want, you can set it equal to c squared, but you know, as usual, maybe you don't want to. There could be a space and a dual space.

And so I want to have the same thing. I start with a, which is my space, my line bundle, which is this bimodule p, and I do the direct sum, as usual, which gives this b. Now I want to have operations here, but the operations are parametrized by these two variables,

so they landed additional tensor product, which is the symmetric algebra. So you have to take care of the homogeneity, right? So you have two homogeneous parameters. If you set the two homogeneous parameters to zero, you're supposed to recover such as the trivial non-commutative divisor,

which is just given by taking a and p and putting them together with nothing, okay? So the way to do it is to say, well, b is a direct sum p. You introduce some kind of weight grading, where this thing here is weight one, and that has weight zero, and then v has to have weight one, and then you ask for this thing here to preserve weight.

And you also ask, I mean, before that, there were some conditions about that certain restrictions recover a or recover p, and let me just say that if you take any w and you pair with w, so you make this thing here into scalars by evaluating a w, then you recover a non-commutative divisor. But you can also formulate it in a general way.

Okay, so this is a generalization of an A infinity structure. You can think of it as an A infinity structure over this symmetric algebra, but that doesn't quite capture all the structure because of the homogeneity that's involved, which is sort of non-traditional, okay? So it doesn't quite reduce to something

that you've seen before, but it does produce a family. You can specialize to any w, and then it produces a non-commutative divisor, which we saw before, okay? And it actually, if you rescale w,

then you get quasi-isomorphic specialization, so it's really actually parametrized by the projective line, the non-trivial part of it, okay? And so obviously, this is non-commutative geometry. One test is that the ordinary commutative geometry fits into it, so if you have a smooth algebraic variety, and you have a pencil of hyper-surfaces

as defined before, it actually gives rise to a non-commutative pencil, oh my god. So where a corresponds to the derived category somehow, and p corresponds to l and everything like that. And did I have to work hard on this? No, you don't have to work hard because as I said, there's some obstruction theory, all the higher obstructions vanish,

so it's kind of straightforward. This is not a one-to-one correspondence. For instance, when I first defined divisors, I wanted sections to be non-zero and so on, but in principle, you can let the sections be zero, other degeneracy, and you will still land in something that's in non-commutative geometry. I don't know, I don't think of that as being a problem.

Okay. So just as before I said, the non-commutative divisor, you can think of it as this bimodule map, which is the first-order part and there's obstruction theory. So the same thing is true for the pencils here. So you have, you know, you look at a part of it, so you get a two-parameter family of bimodule maps,

or if you just choose two basis vector, you just get two different bimodule maps, which are called the first-order part. So these are, think of sections of p inverse. So those are your two sections that define them. And then there is a sort of deformation theory treatment. Actually, the way to understand this deformation theory in a reasonable way is to put it into the framework

of the Mara-Katan formalism, okay? So when you actually, there's a bi-graded DG-Lee algebra such that if you have a non-commutative divisor structure with fixed a and p, that corresponds to a solution of Mara-Katan in this DG-Lee algebra. Or what is the same thing is that a solution of Mara-Katan is the same thing

as an L-infinity map from a trivial one-dimensional Lee algebra into your target G, okay? And so what we're doing here is just simply, instead of having one, we have two, okay? And the reason why you see, from this point of view, why you see symmetric powers here, that's just part of the formalism of L-infinity maps. Yes? Sorry, a bit close to it,

because Lee algebra depends on line bundle, yeah? No, no, no. You fix the, yes, the Lee algebra depends on the line bundle, that's right. A bundle in pencil is, I see. No, no, it's a fixed line bundle. I fixed, yeah. Fixed line bundle, yeah. Okay, so now, I have defined a non-commutative pencil. What can I do with it, okay? Well, you know, originally I said if you have a divisor,

you know, you have the space, you have the infinity algebra corresponds to the divisor, and you have the complement, okay? So now we can have more fun. Obviously, we have the fiber. We can take the, you know, there's an A-infinity algebra which corresponds to the divisor associated to any point in CP1, but you can also do more general points.

So for instance, you know, instead of considering a single value of Z, I can consider a formal disk around some value, and then I get an A-infinity algebra which is associated to that, which is over a one parameter, you know, it's parameterized by this formal disk, and is a deformation of this BZ, okay? Obviously, what you can also do as before, you can take this quotient construction

which formally removes the fiber. You could do it to any fiber, of course, but, you know, the fiber infinity will be more interesting to us, and most importantly, actually, in parallel, so, you know, when I first, the slide number one, I said a pencil, I said you remove the fiber at infinity and then you have this function W, okay,

which is S0 divided by S-infinity, and there is actually an analog of this thing here which I call the non-commutative Landau-Ginzburg model, which is sort of a formal deformation of this guy here but with a deformation parameter of degree two. I don't understand it very, very well so far. I've wrote down the definition for it, and it seems to make sense in examples,

but so this is maybe the most, this is something where you really need, you know, for both this B-hat and this one here, you really need the pencil except here, you just need the pencil in a formal neighborhood of the point. This really uses the entire pencil, okay? So is this some kind of two periodic, can you, can you watch it?

Yes, yes, yes. All right, so two periodic infinity algebra or something, which is, and you, usually when I say the algebra, it secretly means some kind of category associated to it. Okay, so now we finally get back to my original thing of symplectic geometry, right?

How many parameters are in this definition? One. And in all examples, it'd be one? In all examples, it'd be one. I mean, there is no obstruction to introducing non-commutative higher dimensional linear systems. They follow the same thing, but I haven't found any fantastic use for them at the moment. Okay, so let's go back to symplectic geometry, okay?

And let's do some stuff, okay? So let's, let me look at a pen, Lefscher's pen as the pencil, now from a symplectic geometry point of view. I'm gonna now, because I wanna state an actual theorem, I will make all sorts of simplifying assumptions. So I'll take a symplectic manifold

and the symplectic class, it's like the symplectic counterpart of Fano, so the symplectic class is equal to the anti-canonical class. And now I wanna take a pencil, and now I don't care anymore whether it comes from algebraic geometry or not, as long as it qualifies as a pencil and a symplectic geometry, a pencil associated to this anti-canonical line bundle,

which is ample. And so it has fibers, which are symplectic, hyper-surfaces, it has a base locus. But now I'm gonna assume that it's really a Lefscher's pencil, in particular the fiber at infinity is smooth, and the other fibers have generic singularities. So my assumptions that I've done here,

this assumption here and the fact that I'm using the anti-canonical implies that the fibers have a zero first turn class, so they are symplectic Calabi-Yau's. It also implies that if I remove this fiber at infinity, the symplectic form becomes exact on the complement, and all of them help me to state stuff in a reasonable way.

So as we saw, so the basic category I'm gonna use is the category of the Lefscher's vibration, which is the one that uses Lefscher's symbols, and the one that's supposed to be mirrored to the total space if we had mirror symmetry. And the theorem is that this comes

with a canonical structure of a non-commutative pencil. So how does this pencil relate to the one that comes from Donaldson's theory? No, I mean, look, okay. So the word pencil appears on this slide in two different ways, right? So you start with a pencil. This could be what comes out from Donaldson's theory here, right, the green thing?

It's exactly that, would qualify, yeah? And except, well, you know, Donaldson's theory usually requires you to pass to a power of this thing here, but here for simplicity, I mean, you know, it wouldn't break my heart to pass to a power, but I have to modify the formalism a little bit, okay? And then you get a pencil, but not in the ordinary sense,

but in this kind of non-commutative geometry sense, a pencil living on this category here. So how is this pencil made up of, okay? So to make the pencil on this thing here, we need a bimodule. That bimodule is always the same. It's the dual diagonal bimodule, which corresponds to the, it's some sort of non-commutative geometry version of the, excuse me, of the canonical line bundle.

So this has no additional information. It's just constructed from, canonically associated to any a. And then I said that there were two, going to be two bimodule maps here. So there are two bimodule maps from a dual to a if you wanna think of it in categorical terms as two natural transformations

from the CERF factor to the identity. And they actually play wildly different roles. So the one that I call sigma infinity, exists actually for any Lefshitz vibration over c. And it has to do with the, it's hard to say, the CERF factor in this category here

has a geometric interpretation which involves rotating your direction in which the Lefshitz symbols go. And because it's part of a one parameter family, that family gives rise to this sigma infinity. Now the sigma zero is totally different. It actually only exists for pencils that actually can be extended over infinity. Maybe it can't be extended smoothly over infinity,

may not be necessary, but definitely there's some, it encodes the fact that if you have this guy here that maps to c, the monodromy around a large circle is essentially the identity. And you will use this to construct this thing. Here in more, maybe in more intrinsic ways, it sort of counts holomorphic sections of your pencil that go through the fiber at infinity.

Okay? Yes. So you said the Lefshitz pencil, the symplectic Lefshitz pencil gives rise to a non-commutative pencil. What if you just take a symplectic anti-canonical divisor, a single one? Will you get a non-commutative divisor? That's right. But that's not new.

I mean, well, I have to be a bit careful, sorry. Principle, yes. As usual, the literature is sort of sketchy.

Will you say that the pencil, so if you take the generators of the original pencils, take the two symplectic divisors? No, no, no, no, no, no. That's the whole point I'm trying to make here. This is a pencil of a P1, but the P1 is not homogeneous. The point infinity plays a special role in this story here

and so does the point zero, in fact, to a slightly lesser extent. But so this is really a parametrized P1 and the point at infinity is important. Okay. So now I have this symplectic geometry situation and I associate to it a non-commutative pencil.

On the other hand, the non-commutative pencil has all sorts of things associated to it. So let's see what they could mean geometrically, okay? So non-commutative pencil, one thing you can do, you can take a fiber, say a fiber at infinity, and that should correspond to the category of a fiber of W, so a fiber minus the base locus.

But on the other hand, we could take the fiber of the formal disk near infinity and that should correspond to the relative for chi category where you put in the base locus. After a suitable change of, I mean there's a question of, you know, what the deformation variable is,

and which is rather tricky. And so this at least is internally a consistent picture because this relative for chi category is a formal deformation of this one here and this one here is certainly a formal deformation of that one here. And it seems to work out well in example. And then obviously, if you take the fiber at the punctured formal disk of the non-commutative

pencil, then that would correspond to inverting the deformation parameters so you would land here. On the other hand, you can, you know, you can remove the fiber at infinity just by this quotient construction that should correspond to the wrap category of X minus this divisor. And then there's the most sophisticated thing, which is this, you have this non-commutative pencil,

you can do this non-commutative Landau-Ginzburg model. And that should actually correspond to the for chi category of the, of the original thing, X that you started with. Or maybe I should say it should be closely related to.

And you can also get, there's these versions of the chi category which have what I maybe erroneously called bulk. And there's just the same thing that the Landau-Ginzburg model will take, you subtract the constant, which is not a big deal. So the reason why, you know, all my statements tend to be a bit sketchy, but here I am particularly careful

for the following reason that, you know, the objects of the chi category of X, as we define it these days, are closed Lagrangian sub-manifolds of X. So if you made a statement, if I made a statement that says this is this, this actually amounts to a way of constructing closed Lagrangian sub-manifolds inside X, which actually nobody knows how to do in general.

There's no way to predict whether this will be zero or not, okay? Now, my point of view is that, you know, this, just our definition of F of X isn't right. But, you know, you have to be careful what you say here. Okay, so this is my wishful thinking thing. And here you can make your objection.

So I like this a lot better than 10 slides back. Why? Because we have one object in the middle and all these arrows that I've drawn, they are strictly, the object in the middle determines these things by, you know, strict construction. Okay, you have this guy here and you get everything.

And from a symplectic geometry point of view, you know, this is interesting because the difficulty of computing the things in the outside boxes actually varies enormously, okay? So, you know, for this one here,

strangely enough, for these breadth categories and for this one here, we have pretty good computational approaches. This one here is very hard. This one here is very hard in general. So, okay, so this was as far, was that? In the constructions of all these arrows, was it really essential that the original

symplectic pencil you took was Lapschitz? I mean, you could. So, for instance, if you have a toric funnel and one of the generators of the symplectic pencils is the toric divisor and then you just take any other.

Yeah, yeah, yeah, so what I said is, you know, the condition that the fiber at infinity is smooth is probably not essential. What happens is I used it improving this theorem here, so I have to think again. But if you think of this, the reason why I formulated it in this way here is to open up the idea that, you know, if I just consider a case of fiber at infinity is smooth,

I would have written here sigma zero depends on the fact that the monodromia at infinity is trivial. But I didn't quite wanna say this, so this is in case you want generalization. Okay, so this is how it looks like in my most optimistic moments. Where is the things now and where I have the problem?

So, at least I proved that in some cases this non-commutative pencil structure exists. There's this relation here with the fiber, with the higher category of the punctured fiber. I proved this one here to first order with respect to this weight thing, so ignoring the higher order parts of the pencil, which is, you know, a reasonable indication, I think.

The relation to the wrap categories is proved in large part by Mohammed and myself. But these two things that remain are by far the most interesting things, okay? So, this thing here, the relation between the structure of the non-commutative pencil

and the actual structure of the fakaya category of XZ, or if you want, the relative fakaya category, this is completely conjectural. I have some ideas how to do it. That doesn't mean that I know how to do it. And there are a couple of warning signs, okay? So, one thing that is, I mean,

so this thing here, this is a formal family of categories in this parameter Q, so you can look at, you know, the Hochschild term and, let's say, periodic cyclic homologies, and it carries a connection, which, you know, in many examples is equal to, you know, well, there's a map, and if that map is an isomorphism, then that connection is equal to the, sort of,

quantum connection, or A-model connection, okay? And obviously, if this is true, then that corresponds, that A-model connection is the non-commutative Gauss-Mannin connection that's associated to this family here in a form of this near infinity, which means that if you change variables back to the variables that are rational on the P1,

then this connection extends rationally, okay? And so, I don't know that this is always the case from a symplectic geometry point of view. If it's not always the case, this is clearly wrong. On the other hand, if you know how to prove that this is always the case, I would love to hear this,

because that would help me, you know, help my confidence as far as proving this is concerned. Okay, so there's a prediction here which says that the A-model connection, after suitable change of variables, extends rationally, okay? And in fact, mod some, you know, open work in non-commutative geometry,

this should extend to a regular connection away from infinity. So, and yeah, this one here is also conjectural, but at least here, because you have a formal deformation with a parameter of degree two, the amount of reparameterization that you could be wronged by is very small,

essentially none. So, you know, and in a sense, you know, there's a denier rule's kind of, you know, interpretation of mirror construction in terms of discounting going through the fiber at infinity should basically be a model for what I'm supposed to do here. Okay, and you know, now, so I,

now I talked about symplectic geometry for a while, so the, you know, there may be overly naive picture of homological mirror symmetry is that, you know, you have two pencils on both sides, and one, both pencil give rise to a non-commutative pencil structure. This one here in a more straightforward way,

you take the, you know, the derived category, and you know, you encode the commutative algebraic geometry into a non-commutative one, and this one here in this way that I've described, where you pass through for higher categories. And so the hope is that these are actually the same, you know, there's an isomorphism of non-commutative pencil structure, which then by this big thing here

would imply all the other equivalences of categories. So we sort of cleaned up this relationship. And, you know, so on this side here obviously, there are cases where you have a symplectic, especially if you drop this assumption here, that it's a pencil, anti-canonical pencil, you get cases where the mirror is in all sorts

of weird and wonderful versions of algebraic geometry. But, you know, here you can generalize things easily because, you know, it's just algebra, so there's a hope that they fit all in. But, you know, I'm not claiming this right. Okay, so this is the end.

Sure, yeah, but we see that on a normal set, it's, you don't have continuous parameters in sense, yeah? That's right. Yeah. So it means it should be kind of algebraic pencils kind of, both given the kind of integers and... Well, it is, I mean, this side here, yes,

is defined over the integers, or, you know, at least over Q. Yeah, no, because this is kind of in the same direction equation. Let's say you plan over right here, you can see the, it's like a, it's like a sense, you can see the quantum products which is connected to a class, you get some integer matrix.

Also, you get gradient matrix, you get two integer matrix, then you can build from them some connection on, like we have, it should be, it should follow from all the speech, but it will be really object over integers which are structured.

And these are objects, really canonical pencils to build the right here. Do you have? Yeah. I mean, yes. I mean, I reformulate the objection in case somebody didn't hear it, or rather the comment was that this thing here has a canonical parameter, so there should be, and it's defined over Z,

so here there should be a canonical pencil, and especially if you have, if you look at the modular space of Calabi-Yau's, it's not clear that there is where the line should go through. But however, this predicts that there should be some way to, you know, single out a canonical

line. Yeah. Also, some remark that we have really made for two years, and still preparation paper is Stony and Goodmill, maybe you know about Vanoch, how to predict hoge numbers, and there are plenty of conjectures about degeneration of hoge to derive for an exponential

6, almost all of them proven by Saba and Seito and so on, but something is still not proven yet, I can share it. And also a few tables, very similar tables, plenty of categories, yeah. Other questions? If not, let's stand for a game.