 Well, I'm very happy to participate at this conference in honor of my good friend, Luke Vizier. But in fact, I can tell that Luke is one of my teachers. Probably he is not aware of that. I learned a lot of mathematics from his work. And in particular, I just realized that most of things outside of the intermediate geometry I use, and which are the basis of this work. I learned from Luke's papers and I knew to him these are vanishing cycles, celebrating geometry, limit code structures, construction of casual complexes and toposes. So all this I learned from his work. And so in fact, I hope very much that these ugly masks are thrown away and we can meet again and again at Weissmann Institute in Israel and in Paris, of course. Okay, so now I start and most of talks at this conference are over the added field. And so it is about something. It's not about the added field. It's about a field which was discovered even earlier than the added field. And so let me introduce it. And so we work with a field we fixed, which is complete with respect to a noncommissioned variation, discrete variation. And first of all, I recall the standard mutations. K-node is the wink of integers. K with two nodes is the maximum ideal. It's the principal ideal. And K with time over it is the residue field. These are standard mutations. And so in fact, our field, the field we consider is non-communically isomorphic to the field of formal around power series. And so, but I do not want to fix this isomorphism. And so abstractly, I assume that this field contains the ring of integers contains the field of complex numbers and this field of complex numbers maps onto the residue field. And so what is an example of such a field, which is of course a isomorphic to it? It's the completion of a field denoted in such a way, calligraphic, where K is the fraction field of the ring of convergent power series at zero. And so we also fix evaluation of this field such that the value at a generator of the maximum ideal of both valuations of both generators are equal. And so, but the second field, calligraphic field, so in it we fix a coordinate function z, which is noted by z on the complex plane. And so this K-head is canonically isomorphic to the K-head, I'm sorry, K-head node isomorphic to the ring of formal power series. Okay. And, but in fact, the set of isomorphism between K-head and K, our field K corresponds to generators of the maximal ideas. And so many things depend on the choice of a generator of the maximal ideal. And so, okay, what I'll tell about it's how one can place it in some good picture. Okay. And so I start from the very end, I formulate the main result. And of course, after that, I start explaining what are these objects, because it's not clear. And so one can construct for every proper K-analytic space A, it's a non-archimetic and analytic space, and every N, a distinguished mixed structure over some funny K, and so which is a log-formal C-analytic space, I'll give the definition. And so it's just, I formulate, give this formulation because it appeals to something familiar. And so it consists of something which more or less integrates et alchymology of the space, then a rational et alchymology of the space provided with filtration, weight one of the filtration. And then the Ramka homology of the space provided with the Hodge filtration and something alpha. But only at the moment, I'll say at least only one property, that it's factorial in X. It lives in a non-archimedean. And now I'll give all necessary definitions. And so I start already, I do not use prepared nodes. I do not like follow prepared presentations because my brain is not scanning device. And so even I switch off, even when I myself use prepared nodes. Okay. And so first of all, I explain what is a mixed structure over K-node. It's something, I recall the definition of a mixed structure, but in a more general setting. And so definition. And so let us fix a generator of the maximal ideal. Then one can associate with it a conjugation on the, so we just called one pi conjugation on K-node. What is it? And so if we fix, because we fixed our generator, then K-node is canonically adjust isomorphic to the ring of formal power series over C in one pi. And so what is the conjugation? It's, it acts as a complex conjugation on C on constants. So A goes to A bar, and it acts trivially to on bar pi. And so it's a bar pi conjugation. Second, if V is a finitely generated R vector space, then this bar pi conjugation gives rise to a bar pi conjugation on the tensor product of V over R with K-node. It's an evident way. Now, suppose we are given a finite decreasing filtration F on this tensor product. Then one can define a bar pi conjugate of F. What is this? It's a filtration denoted in such a way such that its FQ is equal to FQ bar with respect to bar pi conjugation, which is defined on this space. So we say that F is N opposed to F, to its, to its bar pi conjugate if for any pair of integers such that P plus Q equals to N plus 1. The following is true that the canonical homomorphism from this direct sum to V K-node isn't isomorphism. It's a usual definition. Now, the next definition a bar pi mixed-watch structure, watch structure over K-node is a triple, which now I explained. And so where HZ is a finitely generated Iberian group. And so where W is a finite increasing filtration on HQ, which is a tensor product. Basically, it's Q. It's filtration by Q vector spaces. Now, F is a finite decreasing filtration, finite decreasing filtration on, now here, H K-node, which is a tensor product is K-node. It's finite finitely generated Iberian group. But such by K-node modules, sub modules such that for any N, the filtration on N's graduated quotient of H K-node induced by F is N opposed to its bar pi conjugate. It is a definition. And so it's a usual definition, but instead of C, I consider K-node. Now, first of all, one can show easily that in this situation, all F pi have direct compliments. They have direct or it is equivalent that quotient H K-node have no adoption. Now, it follows that if we can assume it should be such an object, following object H-tile, which is H Z bar pi and F bar pi, which is F-tile, where F-tile is a induced filtration on H C. And so if you consider H C as a quotient by the maximal ideal, by some model generated by maximal ideal, we get the usual classical mixed-coach structure. Okay, but also let me notice if on the opposite direction, if H is a classical mixed-coach structure, one can associate to it a bar pi close structure, but essentially, which doesn't depend on bar pi. Namely, if this means a classical mixed-coach structure, this means that F comes from H C. And so F is a tensor product of some filtration on H C with K-node. And so in that case, bar pi conjugation doesn't change X as usual. So bar pi conjugate doesn't depend on bar pi. So we get a constant mixed-coach structure. Okay, now the main object, it's a log formal C analytic space, C analytic space, which is denoted in such a way. So it is associated with our field K. In fact, it's a very simple object, which is a formal punch to define line, but which depends on K. And so we just, we do not fix such an isomorphism. So first of all, general perception, if X is a formal scheme of finite type over K-node, then one can associate with it the following log C analytic space. Here is the function. And so for R integer R, we define, we consider the following scheme. As a topological space is the formal scheme itself, but the structural shift is quotient of the structural shift of the formal scheme by R's power of maximal ideal. And so it is a scheme of finite type over C. And so we can associate with it its analytification. And our object, so this log C analytic space, it just conducts a limit of this complex analytic space. And so it's a formal, log formal analytic space. So now, in fact, I consider two examples, which the first example is very simple. So yes. The log structure comes from the base? Just a moment. So far, in my examples, I have introduced log structure. It's not in general constructions. There is no log structure. It just a log, a log, I'm sorry, I should hear, yes, your question is legitimate. And so it's like a formal C analytic space, not log. Now we consider, first example, if it is a formal spectrum of K naught, then we get the point. So it's a very simple as a topological space. It's one point. And now it is a log space, log structure. It's an evident one. It is just the monoid, corresponding monoid, I think. And so it is embedded to K naught. It's a formal C analytic, log formal C analytic space. Now, if our, in fact, we start with this calligraphic K, then we get singular log point denoted transition this way. And it is also log formal C analytic space. But what is the difference? There is a fixed element Z in this monoid and this monoid. And by the way, the set of isomorphisms from one point is another one. There is a canonical bijection between the set of isomorphism with a set of generators of the maximal ideas. So just if we have such a isomorphism, we consider, we take the image of Z and we get a generator of the maximal ideal of K. Okay, now the main example, and as it's a very simple example. So it will be, I start our formal scheme, I start with, is a function, is a completion of function defined line, isomorphic, something isomorphic first. So consider the module of all K naught calligraphism from the maximal ideal to K naught. And so it's a free module of rank one of K naught. And so if bar pi, a generator of K naught of the maximal ideal, one can associate an element here. In A, this element as a comomorphism, it takes bar pi to one. And so it is a generator of such, each such element is a generator of our model. Okay. Now notice if we have another generator, also generator, then bar pi is A times bar pi, where A is an invertible element, the link of integers. And in this case, bar pi equal to A times bar pi prime. And now consider the algebra, the following algebra, symmetric algebra of L, of this model L, and consider the localization. And so this localization doesn't depend on bar pi because of the previous equality. And so we, after that, consider formal completion, completion of this symmetric. And so this algebra is isomorphic to the algebra of algebraic functions on a function defined line. So consider its completion. And so if we now take formal scheme x, we apply the previous construction to this completion, and we get a space. And so it's a formal c-analytic space. It's for this formal scheme. Now, so it's a formal c-analytic space. And so there is in fact, it's the inductive limit of c-analytic spaces. And so in fact, only the first space is reduced and all other are need emotions. But the first space, the set of points, it's easy to see, it's canonical. There is a canonical projection with the set of non-zero elements of the one-dimensional c-vector space, which is quotient by the, of maximum idea by the square minus zero. And so just I introduce notation, there is a canonical map. And so we'll denote the image of Wi-Fi generator by small hat of a Wi-Fi. Now, I want to provide the space with a log structure. And so since our field, the ring of integers embedded to the ring of functions on this space, then there is a morphism from our space to the point. And we provide the space with the induced log structure. This means that it just, any generator of our maximal idea defines a chart of this log structure. But we notice immediately, which is important. And so if we consider element var pi times L var pi, then this element in the monoid of this space doesn't depend, depend on var pi. And so this follows from, now I show, from this relation. And so we denote this element by z. So this element doesn't depend on var pi. And so since we have a fixed element, canonical element in this monoid, this defines a morphism of log spaces from here to here, to this log space. And so what do we have? And also it is a steep morphism. It's easy to see steeped because var pi and z are related, are different by an invertible element, invertible function. So what do we have? We have our space to projection, canonical projection here and here. Now, in fact, I formulate several properties, very easy properties. First of all, the set of sections of pi k, pi k sections of these morphisms is canonically, there is a canonical bijection with the set of isomorphisms. And so if you have a section, if you have a section, then we can consider the composition. We have an isomorphism from here to here. And so we get an isomorphism from one point to this point. And it is a set of generators of the maximality. Okay, now, so first of all, let us consider the one differential one form on this point. And so it is, of course, it's free k-node model of rank one, which is generated by the logarithm of var pi for any var pi. And one can easily see that if you consider tensor product, this k-node with the structure and shape of this log form of analytic space, then we get the shape of relative differential form, one forms of k of this space or this space over this space. Okay, that's very easy to see. If e is a vector bundle, this connection on our space, so connection of this type, which is used as a previous, then the set of the subshift of horizontal sections is a local system of finite k-node model. Finite k-node models of rank equal to rank of e. Now, how can we construct such a vector bundle if e is a demodern over k-node? And so what is demodern? A finite free k-node model provided with the connection, then we can, if we consider define such a vector bundle on our space, which is a tensor product, we get a vector bundle with connection, plus connection of the previous type. Now, if you have such a demodern, then one can associate with it a residue of connection. Let me recall the definition, and so if you have such a demodern, then our connection gives rise to a connection the space on the space, which is e factorized by the maximal ideal, by the submodule generated by the maximal ideal. And so it is a connection, let me write here. There is a canonical generator. It also can be considered the quotient of c can be considered as a lock point, and it has a canonical generator of the chief of one form, and there is an associated connection on this quotient. And so this connection would sense any element to such an element, where delta is just endomorphism, delta is endomorphism of the space, and it is called a residue of the connection. By the way, also it's a residue, and it's an endomorphism of severe space. It has additive Jardin decomposition, additive Jardin. Okay, now I definition again. Unfortunately, there are many definitions. e is called distinguished demodern, distinguished if the eigenvalues of delta of the residue lie in the following set. They are a rational number between zero and one. And one can show that in this case, there is a spectrum of the composition. e is a direct sum over all these eigenvalues of d submodules. So d submodules. And here our residue, just a semi-simple part of the residue, just multiplication by delta. Okay. And just I'll use the following definition, k-node submodule of e. And so submodule p in e is called demarkt, if it has a direct complement. And p equals to direct sum of the intersections. Okay. Now what is the distinguished, the next definition? Distinguished the round structure. On this space is a triple, is a triple hz e alpha, where hz, a local system of a finitely generated this space. In fact, it is essentially only on the first level. So it's a topological space, this space. Now, e is a distinguished demodule over k-node. And alpha is an isomorphism of vector battles disconnection. Okay. Now it follows easily from the definition that this local system is quasi-unimportant. And in fact, if you have, moreover, if you have a generator, if you are a generator of the maximal ideal, then one can associate with, to it, a homomorphism from the stock of our structural shift and the corresponding point of our space to k-node. And so it is defined just that element goes to one. And so if you have alpha, then we can construct alpha whereby from the stock of our local system is k-node. And isomorphism to e, if you can see, apply this homomorphism to the previous top alpha. And also we can consider, like reduction, it's a homomorphism from the stock tensor product with c to the quotient of e. And so the operator, one can show that the action of the overgenerator of the fundamental group of this space and the action of residue of connection on e-tiles are related just generator of fundamental group is exponent minus the bar by i delta. And so, and the operator, if you consider the operator, this operator on the right hand side, it corresponds, one can show that one can construct an operator n from the local system h, I'm sorry, hq rational to hq minus one, homomorphism of the local system. Now, the next several definition and after that distinguished mixed-coach structure, mixed-coach structure over this space definition and distinguished, so mixed-coach structure over this space, a formal c-analytic space is a table, which is looks very familiar with theorem of high-formulated, where first of all, hz is a local, no, not on hz, I'm sorry, the triple hz, if you only consider hz, e and alpha is a distinguished theorem structure, which I defined previously. Now, the next, what is w? A finite increase in filtration on hq, by local subsystems of k q-vector spaces, but with the property that n of vk is contained in wk minus two. I think that this appeal to something familiar. Now, if a finite decreasing filtration by of e, by d-marked, you remember that d-marked k-node submodules with the following property, which is also looks very familiar. Nabla of fk is contained in this tensor product, it's a Griffith's transverse entity, such that now the property, such that for any bar pi, which is a generator of the maximal ideal, the following triple, now I will explain. So first of all, we consider the stock, our local system at the corresponding point of our space, the induced filtration of the 8-monotlometer filtration on the space, and now filtration, I will explain what it is, which is dependent on bar pi. This strip is a bar pi mixed co-structure over k-node. So I gave, it was my first definition what it is. Now, I'll explain what it is. And so these are stocks, the first two entries are stocks at this point, and what is f bar pi? And so we have a homomorphism, and so since it from, I already used it. And so we have such an isomorphism, if we fix bar pi, then this filtration is that just preimage of f, preimage of this filtration on the left hand side. Okay, so everything is defined. Now, I can turn back to k-nodality geometry. So it's like Ho Chi-Chi vector, k-nodality geometry. To recall the connection with the classical theory, like Steinberg and so on. So if you have a something of a punctured disc, projective, or let's say, which is semi-stable reduction, so you can consider the variation of whole structure on a punctured, this is a limiting one. So what you define is the analog of the variation of both. Yes, it's something like formal completion of this variation of mixed co-structure. Yes, yes. In fact, I formulate the theory of, I did not formulate properties. And so I will spell out that. Yes, yes, yes. There is a direct relation. But non-archemidian point of view brings something new. And so you know that all this exists already, lives in the non-archemidian world. And this more general notion comes from there. But even in the classical situation, we get something, in a sense, a new view to what we know in the classical situation. Okay. Okay. And so first, the Rammcammology. In the theorem, we had the Rammcammology. And so first of all, I recall that if X are, and so X is, first of all, let me, so let X be a compact k-analytic space, strictly k-analytic space. I just, I do not want to give also definition of bounded k-analytic space, which I mentioned in the abstract of this talk, because it's already too much. And so I restrict with compact k-analytic spaces. Then, first of all, if X is so-called Riggs smooth, and so in the notion, usually in rigid geometry, one tells that they are smooth. And so in this case, one can define the Ramm complex. And so the Ramm complex is a reasonable object to consider. And the Rammcammology for such a space are defined over k, are defined as the hyper-charmology of the complex of the Ramm complex of X over k, differential form, or X over k. And so facts, first of all, I can show facts. First of all, this space has finite dimension over k. It's not evident at all. It is provided with Gauss-Mannig connection data. So it is a demodern in the previous, not yet, it's over k, okay. Third property. One can define a lattice. And so it's a finite dimensional vector space, but one can define a lattice in this vector space, which is denoted in such a way, which is functorial in X. And such that the image of this connection, of this lattice lies in the, I'm sorry, here in the tensor product, I'm sorry, here. I should write only the one k over k. I'm sorry, Gauss-Mannig connection should be written in such a way here. But, and so really it is a demodern in the previous sense, I define the next property. One can extend this construction in a smart way. The construction, which takes any compact strictly k-analytic space, one that defines such a pair, and can, together this connection to arbitrary compact, compact k-analytic space. And moreover, such that all these are distinguished demoderns. Okay, these are Durand Camology. Now I want to say something about integral Camology. So in the formulation of the theorem, they were denoted by calligraphic age, fact, and so again, fact. One can construct for any compact strictly k-analytic space, space X, if for any one negative integer in a local system or finitely generated on this space, such that, first of all, it is factorial in X. Second, it is quasi-important. So for any prime L, there is a canonical isomorphism. If you tensor this local system with DA, we get something we know. Excuse me, what is X bar again? Ah, it's an notation. X bar is a notation. All this is a notation. But the notation for what? Each for something related to X. Something related to X. Yes, X bar is something like X our space X, and X bar stands for X lifted to algebraic closure. But here, now, I hope that you understand. So for any prime L, so all this is a notation. And so a local system, it's a notation. Now, if we consider tensor product of this local system with DA, we get something which now I will explain what do I mean by the right-hand side, because we know what are et alka homology of non-archimedian analytic spaces, erratic homology. Not integral, but erratic. Now what it is, here I should, and so to define the right-hand side, there is a local system on this space, on the topological space. And so it takes a point to some field, which is denoted, which is an algebraic closure. Closure of that, I'm sorry, I'm sorry, by calligraphic, it's k. It's algebraic closure of k. So there is a local system on this space of algebraic closures of k, which are not complete, and et alka homology. And so local system on the right-hand side, now I define. And now you'll see what is X bar. So if you consider lift of X over k to the complete, again, to the completion of this algebraic closure. I'm sorry. So it's the H, the stock of our local system on the right-hand side at the point, bar pi hat, is just et alka homology of the lift of X to algebraic closure, which is related to k to bar pi, et alka homology. Okay, I hope that it's now clear. And now the last thing here, before formulating, the RAM theory, the first step, the RAM theory, for any n, the tuple. So I denote X, so it's X over k-node, which consists of this local system, X bar Z. And so X bar, I hope you understand what is it. It's a notation and the RAM homology of X over k-node. And there is a tuple, one can, the RAM theorem, one can define, so an isomorphism. This means that one can define an isomorphism alpha in the previous slide. And so wait me right. And so it is distinguished the RAM structure, the RAM structure. And so, in fact, in this triple, so this means that there is an isomorphism of vector bundles with connection. There is an isom alpha, such that it is a distinguished RAM structure. By the way, if we have bar pi, then this homomorphism induces an isomorphism of the stock of this local system at this point, tensor Z k-node, with the RAM homology. Now I can formulate, if we have one more minute, probably I'll need a couple more minutes, that's our theorem. So one can associate, construct for any proper, now here, and so all previous results are to hold for compact k-analytic spaces, but here we need proper k-analytic space, or in more general k-analytic space, x and any n, distinguished mixed-forge structure over k. So again, now such that I have list several properties. So first of all, I already listed it's factorial in x. Now, second property, I can, if x is smooth, then f is induced, the Hodge filtration on the RAM homology group is induced by the stupid filtration on the RAM complex. And so it's a usual RAM complex, but you remember that the RAM homology group over k-node are embedded lettuces in the RAM homology groups over k. And so it is just a pre-image of the filtration used by the stupid filtration on this complex, and the spectral sequence. So Hodge-deram spectral sequence degenerates, degenerates at e1. Now, the third property comparison is what we know in the classical situation by the work of Steinbrink. And so if x is the following project, so let me explain you. And so where x is a proper scheme of a calligraphic node, which is the ring of convergent power series at zero, and the complex plane at zero. And so proper scheme such that its closed fiber is a slick normal crossing device and produced, then in fact, Steinbrink constructed limit Hodge structure. So what is the relation to the previous object? And so let me, and so you remember when we have a distinct mixed Hodge structure over this funny space, we can consider it's like reduction only when we factorized by the maximal idea. And so it's something already looks like only some local system over the topological space, not on the formal scheme on the topological space, but at least I'm sorry. And it's not I should use calligraphic k. And so if I use, since I use calligraphic k, this means that there is a fixed generator of the maximal idea or a fixed point of this space. And so let us consider stock of this local system at this point, fixed point. And so it is the limit mixed Hodge structure of Steinbrink. In fact, he considered also different points at this space, like how can this structure transforms when we consider another generator of the maximal idea. So everything depends on the generator module, the square of the maximal idea. And so everything is consistent. Okay, I think that it's enough. I think I finished. Thank you. I want to ask just to clarify. Okay, so when you write the last theorem, so a proper catalytic space in your sense, is it is it again strict in your sense or not necessarily? No, but it is strict by definition, in fact, because it's in my theory, it's a good analytic space. And so every point has an affinate neighborhood and the valuation is non trivial and so on. And it's, this space has no boundary. And so it follows Okay, okay, okay. So you just explained the point. And now I want, maybe I want to ask again about the mixed structure thing. So the weight filtration here, when X let us say is smooth, do you get that the filtration W is trivial or not? Is it like the limiting, is it the limiting weight filtration or is the weight filtration in? Yes, yes, yes. It's a limited weight filtration. And so everything, you see that in the last line here. And so what I'm writing, it's the whole tuple. It's the whole tuple. It's in fact tuple. It's a limit mixed structure of stembrings with weight filtration, everything, of course. What I'm asking is the classical theory, let us say you have something over the puncture disc analytic. So you have a, let us say the smooth, proper smooth case on the puncture disc. So you have a usual pure variation of old structure on the puncture disc. You can extend it and you have then, and then you can, you have also a limiting one on the kind of the special fiber of the ring extension. So in your context, do you have an analytic of the one, not the limiting one, but the one, the variation of pure structure? No, I have no pure structures outside zero. And so zero, what I constructed, it's only leaves over zero. You see, I have no something which leaves outside. If you are asking this question, if I understand your correct answer, you don't have an analog of the variation of old structure on the puncture disc whose limit is the one under the stem. That's true. That's true. I know what, of this, that's true. I only have without limit of what, yes, yes, yes. And so that's true. I just can tell you that something, what I can do. And so there is, in fact, this theory is extended to a more general analytic spaces I called bounded. And by the way, also probably in formulation, I should say that the classical theory of the line was only covered for proper schemes, not, it doesn't cover non-proper schemes. And so if our scheme X comes from proper scheme OEC, then we get a constant mixed construction. So like very constant, but I cannot do non-proper schemes. You cannot. May, may I ask a question? What is the last question about non-proper schemes? The last, and so there is also not necessarily proper schemes, but which I can probably, and so if you consider a formal scheme of finite type over K-node and closed-up scheme of the closed fiber, which is proper as a scheme. And so the whole formal scheme is not proper, but closed-up scheme. And we can see the formal completion of the X fiber. These are bounded analysis, which, to which the whole scheme is extended. Yes, we have all the people who want to ask questions. Yeah, there is, can you switch on your microphone and ask your question? Yes. Can you hear me? Yes. Can you hear me? Good morning. Hi Vladimir. So my question is the following. Do you have an invariant cycle, an idea, an invariant cycle theorem? So in the sense, what is the kernel of the monodome? I have in mind to think about it, but I didn't yet do. I still have some problems in like based foundations, I would say. I believe that invariant cycle theorem also should come here. So in general, what should be the, let's say, the the the homology of the invariant? So what should be, in general, you have the sequence, and after you have the homology of the special fiber in the classical stibling setting? Yes, I think something exists. It's also, it is extended, not only you see that here I use homology x-bar, x-bar, but in fact, one can withdraw bar, one can consider just integral homology of x itself. And also, there is a hodge structure, but which is constant. It's also parallel to a classical situation. And something it is described in terms of this invariant cycles. Okay. Thank you. But I didn't do it. I just, I have in mind, but I cannot claim that I can do something yet. Okay. Thank you. Thank you. Okay, I think that Professor Evesi has a question. Yes, sorry. At the beginning, you assume that x is a quarter of a k, but not necessarily smooth. Not necessarily smooth. So then you, when you write h on the rank of x over k naught, do you mean that you replace x by some hyper-covering, proper hyper-covering with the smooth? Yes, yes, yes. That's right. That's right. Do you use some kind of the, of the young iteration thing or similar things? No, no, no. I use, it is based on Tjomkin's result because he yes, yes, yes. For formal schemes, it's a very essential. In fact, I started all business after Tjomkin proved his results and which I applied also to formal schemes. And so, yes, everything is constructed. And so, the underlying things is a vanishing cycles here. Yes, yes, yes. That's true. Because even for a smooth case, I need it. Yes, because reduction is only so-called dissinguished, but I need semi-stable. And so, yes, yes, yes, yes, yes. It is. Also, second question, your, your, your filtration, your limiting filtration, W. So, do you expect some monotromic filtration here and in some cases, or? No, no, no, no, not yet, not yet. I understand. Not yet, but does it mean that you, you expect something to be true? I can tell you, I expect many things should be true and monotromy and also even one can formulate a Hodge conjecture here. Because this Hodge structure is not pure, but nevertheless one can formulate. And so, at least, I hope that, I hope it's also true. But I, okay, thank you. So, let's send the speaker again.