 Right, so the title is on the blackboard. It's going to be three lectures. And I'm very grateful to the organizers for setting this up in the first place and then for persevering and doing this in this format. Better than nothing. I mean, there are non-trial issues now, which don't appear in normal conferences. So I have to apologize. Today I have to run to finish five minutes earlier and run away to another seminar in Moscow. It compares to Moscow in five minutes. And so today I will not be able to take questions right after the talk, but I will also give a lecture on Thursday and then on Monday next week. And so you are more than welcome to ask questions around ceremony. Yeah. All right. So what I want to give you is some kind of overview of really, I mean, there will not be that much proofs and maybe not that many theorems even. Rather viewpoint on motives from something which eventually will be non-commutative, but it will come along kind of naturally. And so I have to again apologize. So it's a long story which developed over the years in kind of strange way. My own background is from algebraic geometry. This is where I come from. And this is where the story comes from originally. This is some of it, but a large part of it was developed by an algebraic topologist, and so we saw some of it in Tina's great talk yesterday, and we'll see more of it today in third lecture. And since I'm not an algebraic topologist, I mean, I might be not very, might not be very good to say references of who proved what. So I'll give you some statements which I know are correct, but maybe I will miss some attributions and I apologize in advance for that. It's very good that we actually have a course by Tina on this book that Sian Matz so on to work, which forms a very important part of the story. So I feel kind of covered on that part. So anyway, but let me start from algebraic geometry and even not algebraic, but kind of usual geometry. So if we have some kind of x, which is a same thing to manifold, and we want to consider this homology. And as we know, there are several ways to define it, which I'll give you in the end the same answer. I mean, there are many ways, but kind of the most standard ones are as follows. So first of all, you can think about homology and you can represent it by cycles, some actual geometric objects and x. Or if you want you can do a singular homology simplices. Your x, this gives you homology and homology with say integral coefficients or any other coefficients which you want. This is a nice theory which also works in the logical spaces. One approach. Another approach is instead of this, you can think more homological, you can think in terms of sheaves context and sheave homology. But more classically, maybe you might consider check. So instead of considering cycles, you consider open covers, nice enough open covers, you write the check complex or that kind of stuff. Again, this gives you homology with z and apostrophe or any other. And then there is a third thing which is specific for many folks now. And this is the RAMCOM. And it has no coefficients, meaning that coefficients are real, coefficients are your base field. It's represented by forms and it gives you the same result. So all, I mean gives you the same result when you compare it. So the RAMCOM is the same as usual, the same singular homology of coefficients and r. So all give the same result. And they share some properties which are kind of natural when you want to think about something as a homology here. So this is first of all the thing is homotopy invariant. If you have your x, you multiply it by an interval, say if you want the RAM maybe an open interval, then you get the same thing as you had before. And then a version of it, which in the same thing to setting is basically the same statement. But let me state it separately, nonetheless. So if you have something which you can think of as a manifold, for example, submersion, and you consider the homology of the fibers, then they form a local system downstairs. So they're locally constant. So fiber of some point. Now, this is locally constant. Again, this has several gauges. You can consider relative homology, which will give you a locally constant shift downstairs. When the RAMCOM homology, you can have some vector bundles downstairs with a flat connection. The point is that if you move along your base of your family, the homology doesn't change. Okay, now let's go to algebraic setting. And the point is that in algebraic setting, kind of all three approaches survive to some extent, but they give you different things now. Let me stick to smooth for now, algebraic variety. Let's see, let's say we're a field key. Well, the first approach, geometric approach, based on cycles, eventually have to think long and hard and invoke both of our theorems, gives you what is known as the chow groups, chow groups, or algebraic k-field. So the two things are intimately related. Maybe k-theory is more important for me because it's somehow more fundamental. But the point is that this is a very good theory in the sense that this is really intrinsic. You don't impose anything on your variety. You just work with whatever is there already. Either cycles for chow groups or vector bundles, so we can use for k-theory. But so it's intrinsic, very interesting, but hard to compute. It's notoriously hard to get handle. Okay, so this is the story. This is kind of the best replacement for a singular homologon, in the geometric sense. Now, the story about check coverings. Well, the reasons worked in the usual topology is that you can always cover your infinite manifolds by something, by some fine sets, which are small enough to be contractable. Algebraic is not possible anymore, because if you take an algebraic variety and you take, I mean, the one topology, you have a priority risk topology. And when you take kind of the smallest possible open cover, it will not be contractable at all. So if you're overseas, say you have a curve overseeing, you remove points. And that's the only thing you can do, remove points. You end up with an open curve without many points. And actually it's not contractable at all. The more points you remove, the more non-contractable it becomes. However, it is always a topological type KP1, so it's defined by its fundamental group. And then there was this great idea of growth and dig that you could generalize your notion of a cover. Instead of just open subsets, you should consider coverings of open subsets. And if you develop that idea far enough, and if you're growth and dig, you end up with a topology. Now, this still has some limitations because in algebraic geometry, we can only consider finite covers. We don't have infinite colors. So instead of the whole fundamental group of something, you only get to provided completion. And as a result, so you get some theory, which is nice, which behaves nicely, but the coefficients have to be, well, it can take fine coefficients, but in practice, for practical applications, people prefer L-addict homologous, so the coefficients are QL, where this is some L prime. And in characteristic zero, it's okay, but if you're in positive characteristic, then this should be different from the characteristic of your base fit. And the story I'm going to present is mostly interesting situation when you're in positive characteristic. So this will be some kind of P, and this has to be different from P. And now there's the third story, the round story. And this again works. Homology. So it again works nicely by theory. You can just do the very naive thing. You can define differential forms as exterior powers of the attention bundle. And you think about this algebraic thing. And then if you're overseas, a non-trivial but true statement is that this computes the same homology. So any homology class can be represented by differential forms with polynomial coefficients. So this works. This also works in characteristic P, but of course, here we have a problem, which is that the coefficients were homology theory. By definition, when you're working with Durham homology, the same as the base and originally lots of this was developed in the course of proving the way conjectures. And there are people wanted to have something positive characteristic, but homology theory with coefficient characteristic zero. So there are some kinds of extension of this, which comes up, which is called crystalline homology. So this has coefficients. K, but then there's crystalline homology, which is sort of better. So first of all, it can now be defined for a singular variety just as well. So X can be singular, singular. It doesn't have to be smooth anymore. And moreover, it can have coefficients. So if we start with something over K, there is a version of homology, which is defined coefficients in the ring of width vectors. So for example, if your K is just Fp, the prime field, then this will be a periodic number. I will speak more about this later. And not only they give you different things, it's not even clear how to compare them. So for example, if you want to compare Durham homology and et alka homology, then you see right away that if you're in positive characteristic, you cannot do it because the coefficients are different. One is a logic where L is not P. Whereas the other one, the Durham crystalline story has coefficients either in Fp or maybe in Zp or Qp, it's still the same P. So there is a very interesting involved and long kind of long story about what comparison you can do, which is called P. eddy-coach theory. And I mean, it's used to many people from 10 faultings and then in the end balance on and then in the end, I think it was finally completely resolved not very long ago by Peter Schultz and Matthew Morrow. But even there, the statement is not obvious because it's not hard to find. It's not that obvious. You can even find the context where I mean comparison theorem can be made. And that's for K theory. Normally, there is a map from K theory to either of the other two, but it's very far from being an isomorphism. We don't expect it to be an isomorphism. And what we expect to be able to do in the best of possible worlds is to be able to correct our Camology theory somehow that in the end, they compute something which is closer to K. So that's the general outcome. But let me now discuss the behavior of all these three theories with respect to those standard operations, which I mentioned in the beginning. So sort of homotopy invariance properties. So invariance. Here the story is again slightly different because we don't have an interval anymore. An interval is not an algebraic thing. You can either take the whole line, which is a fine line, or you can think really infinitesimally. You can take, I mean, we can't do it in the usual topology, but we can do it in an algebraic geometry. So we can consider infinitesimal neighborhood of a very small neighborhood of a point. And here the et al Camology is behaves exactly as you expect a Camology theory to behave. So first of all, it's what is known. So I think the Trimology is due to Weyvotsky. It was not originally, I mean, originally it was not thought it was, it would be very useful. So it's kind of a great discovery of Weyvotsky that this leads to very interesting theories. So the property is called AY homotopy invariance. And this means that the Camology of some x, it means what you think it means. If you multiply x by a fine line, you get the same thing. And this kind of a global state. And another important thing here is that the banner is that K theory behaves similarly the same way. If you localize it at some l, or maybe complete some l, so complete that l with the same provisor as before. So l is not equal to characteristic. By the way, if there are any questions along the way, I don't think I can see it very well. So if some, I don't know, it could be great. All right. Dmitry, sorry to have interrupted you. There is a question for crystalline K or DVF in the question chart. For crystalline, let me go back to that slide, where W of K, F or DVF? So K will be the residue field of and coefficients is WK or fraction field of W. Either, I mean, there are both. If there is WK, you can take the fraction field there. Yeah. I was about to answer, but okay. And then, and for K theory, this actually hailing on trivial statement in the end, with this is, I mean, just what I wrote is actually easier, but some generalization are not. So it is basically a discovery of Susan, which is called Susan rigidity. And then another thing which I mentioned, you can work with large-breed geometry is infinitesimal rigidity. And this is again just true. So telecomology of some X, which is maybe non-reduced, maybe some kind of, you know, infinitesimal neighborhood of X and some ambient variety is exactly the same as telecomology with reduction. It doesn't feel nearly potent. So this is the story for the talk. Now, K theory. So as I said, if you consider K theory, say, find coefficients or completed at some L, then it's fine. But then generally, it's important to realize that K theory is not a one or more ambient variety. Well, it is if X is smooth. But as I said, all the theories actually make sense for singular X and it's useful to consider them with full generality. So for smooth X, it's still true. But if you just multiply by a fine line, the K theory doesn't change. It's one of the basic theorems of Quillen when he kind of developed K theory. But in general for singular X, it's just not true in a very bad way. And it's not infinitesimal like what I put like this. So it's not true anymore that K theory of X is the same as K theory with reduction. So if we consider some kind of infinitesimal neighborhood, then K theory would change very much. In fact, we already saw this yesterday in Tina's lecture, where you consider, for example, well, basic example, Z mod p and Z mod p square. And K theory of Z mod p, we know K theory of Z mod p square, we don't even know. So I mean, not completely different, but they change drastically. There's a whole story about how K theory changes when you're doing infinitesimal deformations. So in this sense, K theory is not, does not behave in the way you would expect homologies needed to do. So here it's maybe useful to mention that there is this complete, especially if you're in positive characteristics, there is this complete dichotomy. So you can sort of take your K theory and localize it at some prime different from the character or at the prime equal to the characteristic. And the answers are totally different. Already for the point, so if you just consider the prime field, both have been computed by Quillen, but the answers are different. So if you computed lk efficiency, l not equal to p, then I get some kind of both periodist, you get some polynomial algebra in one generator in degree two. Well, if you do Fp bar maybe. So it's kind of similar to topological k theory. And if you do it at p, then K theory of a point only, there is only K zero, there is nothing else. So already here the behavior is different. But when you start, say, taking some kind of intasimal neighborhood of a point. So for example, lifting it to z mod p to the power n, or maybe considering truncated polynomials, then there is this relative k theory which we saw. And that relative k theory is entirely p local. So if you localize at l, you get rigidity, it doesn't change. And if you look at p, then it changes very much. So the two kind of stories have been very different. And the kind of p local story is not at first similar to topology. And now the crystalline topology. Dmitry, there is a question, does there any kind of a1 invariant k theory? You can force k theory to be a1 invariant. But for my purposes, this is exactly what you don't want to do, because it completely kills off what you want to study. This is some kind of dichotomy. So just like the whole theory of motives of wayward is kind of based on the ideas that things should be a1 invariant. And it really leads you very far when you do a l-addict stuff. But for p-addict stuff, I don't think it's basically kills off the interesting part. So I would prefer not to do that. But even before you go to k theory, which is very difficult, let's discuss what happens with crystalline topology. And here it's like this. So it's kind of an infinitesimal invariant. I mean, not completely an invariant. But at least if you have a family, it's kind of locally constant. So if you have a family and you look at the fibers, then downstairs you get a bundle of flat connection. Of course, the problem is that, say, if you don't even do crystalline, but say do just the RAMc homology in positive characteristics, it's true that you get a flat connection. It's the same story over any base field. The problem is, of course, that in characteristic p-flat connections give you less than you expect. So it doesn't give you like a trivialization to all orders. But at least it gives you a trivialization of some kind of what they call Frobenius neighborhoods, up to the power of p or k. And in general, I think the full statement is that crystalline homology doesn't change if you do some kind of things in a visible neighborhood, which has an additional structure called divided powers. So there is some story you can do there, kind of locally constant. But certainly not a one homotopy environment, not a one homotopy environment. And actually, there is a very easy geometric way to see why this happens. So now I have to, it's a geometric reason. I don't give you the definition of crystalline homology nor of k-theory. Well, because there is no time, of course, but also because it's not that important, what's important is the properties. And the basic property of crystalline homology, which you need to know, is the following. So if you have some x over this k, sorry. And for now, let me fix k to be perfect. Okay, perfect fill of characteristic p. Say finite field of characteristic p would be good. And it can happen that this thing can be actually lifted to something different over the sweet vectors. W of k, there are some kind of lift. Then it always happens. If it happens, then the crystalline homology of x just computes with the drum homology of this lift. And then if you have a lifting, crystalline homology of x is the same as drum homology of this lift. And if you look at something like a projective line, for example, then it has, of course, a lifting. So crystalline homology of p1 is what you expected to be. So there's this base, which is W of k is one dimensional. There's a special point in the base, and the fiber of a special point is p1. And then you want to lift. Okay, now if you want to consider a1, you just take some point, the special fiber and remove it. There is no point. Does this have a lifting? Well, it does have a lifting, but it's not what you expect. So here we have a1, 1 minus infinity. But this lifting is not this minus any kind of lifting with infinity. Formally, by definition, this x twiddle has to be pedically complete. So you have to complete. This means that you not only remove one point, removing one point would correspond to removing a single section of your family. But basically, you remove all the sections which pass through this point infinity at zero. So you remove this, this, this, this, all these have to be removed. And in the end, you end up with lots of holes in the general fiber of your story. Lots of holes in the general fiber of your lifting. So it's not just p1 minus infinity. It's basically p1 minus anything which reduces infinity to modulo p. And this has lots of converges, as you expect, because you take a curve and remove lots of points, each point creates your homology class. So h1 crystalline on a fine line is huge. And so some people think it's a bug and try to correct for this. And there are ways to correct for this using rigid analytic spaces and maybe putting log structure. I mean, there are lots of interesting stories about how to correct this and make this telling homology behave more like you would expect homology theory to behave. But on the other hand, you can think about it as not a bug of the future as something which is intrinsic to the nature of crystalline homology and work. And this is the viewpoint I want to adopt. Dmitri, sorry to have interrupted you. Really, Kartinas asks when a lifting exists, that is telling homology with p-addict integer k-efficient. Lift to the drum with same k-efficient or you need to pass to p-addict rationals? No, no, no, no, no. It's through integral. So a lifting has to be smooth in larger break sense. But then it's through w of k-efficient. Of course you can, after that, invert p, but you don't have to. But what you have to do, you have to complete your lifting periodically. So it really has to be some kind of formal scheme because the story is really about infinitesimal, a series of infinitesimal lifting that are any geometric lifting with a genuine geometry. So it's through integrally, but it has to be adequately completed. So this for A1 what you get will be actually torsion, but there will be lots of it. So it will annihilate by p, but there will be lots of it. Okay, now the kind of my general goal in these lectures would be the following, or after this picture. So I want to leave completely aside the tale story, which I thought I wanted to want to present. But I want to think about k-theory and crystalline homology. And I want to show that the two phenomena are actually related. Show that the two phenomena meaning that neither k-theory nor crystalline homology really behave in a way to expect from what are related. And actually, if you start thinking about k-theory, it's just inevitably leads to crystalline homology. If you don't know about crystalline homology at all, but you think about k-theory long enough, you discover crystalline homology automatically. And that's kind of the message which I want to explain. And then as an additional bonus, it will turn out that this whole story is naturally defined in a much bigger generality. Namely, you don't start with a ring which is commutative. You don't have to start with a great variety. You can actually do it for any associative unit or not necessarily commutative ring, or even digital. So this whole story turns out to be non-commutative, which is kind of an added bonus, which I think was originally unexpected, but now it's accepted. But I mean, let's not rush. We'll just come up by itself. Okay, so let me start the story. I told you what the story should be, and now let me actually start doing it. So I need to tell you at least something about crystalline homology, right? I told you one thing is that when you have a lifting, it gives you a lifting. But one other thing I want to know about crystalline homology is the following. I see that X is smooth now. Again, over K, which is perfectly characteristic. Then although it was not the original definition, there is a way to compute h-crease of X in a way which is similar to the rank homology. I mean, it's canonically as a morphology. So the risk typology of coefficients and certain replacement of the derangue complex, W omega X. So this W omega X is something which behaves as a derangue complex. So it's canonical, it's functorial with respect to at least say local azimorphisms and gluings and so on. It's called the RAM complex. I think it was the original idea of Bloch. And it was later developed and the story was fully done by Eleusie, but with lots of input by Delinia, apparently. I mean, the paper is by Eleusie, but if you read it, he says very often that this and this and this was done by Delinie. And the story from like mid to late seventies. So I don't really want to know, don't need to tell you what the whole complex is, but it's first term kind of what replaces functions. It's something which has been around for quite some time already then. So W omega zero of X. Omega zero of X is just functions. And then here we have something which is this thing called ring of weight vectors. Now this already appeared, of course, but there I did it for a field. And if you didn't know what this was, you could just think that this is, you know, like ZP. But now all X is certainly not a field. It's a polynomial ring of many variables of some relations. So I need to explain you really what weight vectors are. So recollection, kind of this was recollection on h3. So now kind of sub recollections. What is, what is this? So it's a very least it's a function from community rings to community rings. But actually at this point, the fact that it's a ring will not be that important. Let me first describe it as an a billion group. So it's a function from rings to a billion groups. And there are many ways to present this story, but one is the following. You first define something bigger. So as an a billion group, you can see that something which is denoted blackboard. So you have started some A now, which is commutative ring. That's the only requirement. And then you find some a billion group, which is called big universal weight vectors and denoted by blackboard W of A. And by definition, it's the following. So big. It's basically the most naive thing you can do. So you have something which is possibly in characteristic P. And the goal is to have something which is not not not not not inflated by P have a commutative ring. You have its additive group. And that's of course the same characteristic as a, but you also have the multiplicative group. So let's look at this. Of course, multiplicative group of A may be very difficult, but you can add one form of variable of denoted by T. So you can see the form of power series in A, one coefficient in A, one form of variability, and you can see the invertible powers. So there is the map, which the power series, of course, invertible, if it's leading term is invertible, if and only if. So there is a map, which sends something to the leading term, and this is split. Because you can always take any invertible element in A, you can take it to the constant power series. So this splits as, so this is a star plus something else. And this something else. We do not W. Sometimes you can always write, you can see it in literature, it's written as one plus T, A. So these are form of power series of leading term one. And the group operation is multiplicative. But that's W. Now, there is a way to look at it, which is slightly, which looks a little bit too complicated. So instead of considering the multiplicative group, I can do the form. So instead of A star, we can actually consider algebraic k-theria, but not like higher k-theria, but just k1. Now the definition of k1, we actually saw in Kinostok yesterday, fortunately, it's actually very close to A star. So this is A star. Occasionally plus something else. But there's something else. It can be non-trivial, but it will be the same for A and for power series. So when you do kind of the difference, it will not change. I mean, formal definition is you take JL infinity and you take its first homologial habilization. And you can take determinants. So this gives you a map to A star. It's not always an isomorphism, but it's not that far. And it certainly will be kind of relative isomorphism if you look at power series. So in fact, it's also true that k1 of A is k1 of A canonically splits into this plus Now, why is it useful? So I gave you, I mean, we saw the definition of k1 in terms of JLn, but there is also a more invariant way to define it. And the point of more invariant definition is that it's actually not an invariant of the ring, but an invariant of the category of projectively generated modules. So actually k1, in fact, all k theory depends on the, let me denote this by p of A, which for me will be the category of projectively generated A modules. And particularly it's a functional aspect. So if you have some function between this And there is one obvious function. If you have some vectors, let me now stick with the situation A is k algebra, the simplicity over some k. Then if you have some k prime dimensional vector space, flip to the endomorphism, then you consider a functor, which just, you know, just turns the product to v. And it's a functor from A of t from projective modules for t to itself. So take your module, your module turns it to A and you twist the action of t by this small a here. And so this is a functor. So we need to use the map on k1. So for any v A, to get an endomorphism of example, take some integer, consider the cyclic group, take the corresponding, well, group algebra and dimensional vector space. And A, let A just act by the generator. So it's just, you know, shift, just a free action of the mod n on itself, kind of termitation of cycle of order n permutation. So this is a vector space in the endomorphism. So it gives you a well defined, defined map. So get let me denote by epsilon n of this v. But then if you take a product of v with itself, you see that as a vector space of the endomorphism is just a sum of n copies of it. This means that this epsilon n is actually almost an idempot. It squares to n times itself. And this can be used. So if, so we're now in a situation where, so it's over k. So I assume that now that k is characteristic p as I, as I always assume, then it's easy, very easy to prove and easy to see actually that as a group, as an abelian group, this w of a is actually p local. So everything not divisible by p is inverted. This means that if n is not divisible by p, you have a well defined endomorphism of your w of a. You can divide this epsilon n by n. This is well defined. And this is an idempot. So it squares to n. In fact, you can show that the whole w of a actually splits into a copy of, so you have a family of commuting idempot and endomorphisms numbered by all positive integers not divisible by p. And in fact, what happens is that the whole w of a just splits into product of copies of a single thing which is denoted now by just w without blackboard. And this is called a group of, so this is just if you want the common kernel of all those idempot. And so this is known as a group of p typical width vectors. And this is what I want. Where am I? Yeah, p typical vectors. There's additional theorem that it has a product so it's actually a ring. And this is my width vector. But for me, it's not important that it's ringed now. But what is interesting is that it turns out that exactly the same construction now gives you the whole, the wrong width constant. And this was not realized at first. So there is one correction. So I was working with K1, but we have the whole algebraic k theory. So we have a n. And then of course, sorry, of course, formal power series, by definition, are just projective limits of truncated things, right? To take a of t model t and plus one, maybe I'm taking less than with respect to n. You can ask whether it commutes with k theory and it does commute with k1, but not with high k groups. So what I want to do, I want to find a completed k theory of A of t as this inverse limit. Well, it has to be a multiple limit. Some type of things there doesn't matter. Okay, groups of truncated polynomials. And again, there is always the documentation map to the leading term, which is split. So you can say that this thing splits as k theory of A plus something. What do you know about the value of x? For now, exactly the same. So again, all k theory depends on the k theory of model. So you have the same kind of relative. So this x is the second term. You can also think about it as the root k theory of this. This is p local. So you can always also look at the, on those side importance I have, look at those kernels. And the theorem is that this gives you exactly the terms of the Rambut complex. So A is now smooth commutative, maybe, maybe find it generated. So just on a smooth edge break, we're right. But there's this complex and then in degree i, this thing is naturally identified with this common kernel, seven i on relative k theory. So let me rewrite this. So I take this k complete relative degree i plus one. I take the common kernel of all those side importance epsilon and acting on this guy. And this gives you the term of the Rambut complex. So the point is, I mean, the gist of this is that if you just replace k one with high k theory, you automatically, without thinking, are led to crystalline commut. The Rambut complex and crystalline commut. Now, the theorem has a long history. So actually it was originally conceived by Bloch because the paper where he actually introduced the ideas for the Rambut complex, it was called on p typical curves, large break k theory. But it was maybe even before or at least right after Coolant's k theory appeared. So he worked with the previous version, which was Milner k theory. And he could only do it in like small degrees and so on. But then the story developed and I think eventually the theorem is due probably to large threshold. I think people on the Sorry, a quick question. Is it obvious that the chain complex structure on these relative k heads? It's a priori spectrum. Well, when I don't know what k i, I write, I mean, I mean the homotopy group posteriori, it is actually, but it's not obvious now. So it is an Elberck McLean spectrum. It is a chain complex. And then so this is one question. And then another question is so I get this W individual terms. Another question is what is the, where is the differential right? The drum differential. And this is kind of the next question, which I'm going to address on Thursday. And the kind of one line kind of punchline is that it comes from circle action on this relative case. But this will really have to do. I mean, there's no time for that today. I'll explain this on Thursday. Okay. Yeah, but the result itself is due to large threshold from, I don't know, like, he has a paper whose title is very similar to blocks. It's called on PTIP-Calcurz and Quillen's case. But even then, it uses lots of technology. So I'm not really sure what is the good reference for this. It's also direct to prove. And I'm not going to prove it in these lectures. But the one I'm going to present in these lectures is the way to see that this thing here is actually very compute. It's related to some pure environmental environment which you can compute. And then how to identify that to the drum with complex in a separate story, which is not that interesting. But kind of the main thrust will be this. And I will explain first, the first thing I'm going to explain on Thursday is how to get the drum differential on this thing. So it's really not just terms in the drum with the complex, but the whole complex. Okay. Now, I think I have to run now more or less. Okay. Let's thank the speaker for today's lecture. And yeah, please prepare your questions and comments for Thursday and Monday lectures, please. Yeah. So see you very much. Thank you. Thank you. Bye-bye.