 It's a great honor for me to open this journey, Gretchen and Barry Mazer. I should say that after the breakup of the Soviet Union I spent several years in Boston and was lucky to talk to Barry about some pieces of mathematics I was interested in at that time. And Barry is a universal mathematician, so he blends geometry and arithmetic and algebra in a kind of unique artistic way and very much admire how Barry does mathematics, how he thinks about mathematics, but also how he writes mathematics and how he speaks about mathematics. When I was in Cambridge I even went sometimes to the low grade courses where I knew the subject but I just wanted to see how Barry speaks about the subject. But more than that for me Barry is a man of renaissance, so it's not only mathematics, it's all many other human intellectual activities. So it's hard to name all of them, philosophy, poetry, and I just was lucky once to be at a Kashtan-Mazer seminar on philosophy and this actually was an unforgettable impression for me, so Barry spoke about Kant and I never thought that it could be so interesting. So I'm from the former Soviet Union, so I had some kind of attitude to philosophy and this dramatically changed my attitude to that, so just one Barry's lecture on the subject. Okay, let me go to mathematics, so I actually changed subject, topic a little bit, so what was announced will be hopefully in the end. And my main goal is to give some kind of update on the topic we discussed with Barry about a quarter-century goal. So today's look of this is like that, so it's mixed motives and geometry of modular manifolds. And of course format title was just an example of this relationship and so let me start from very kind of general remarks. So we're talking about mixed motives, but if you consider pure motives, then Langlands and then Closel make this precise. So they taught us that pure, let's say rank M, pure motives of rank M, let's say over Q. So they reflected in the world of modular forms by optomorphic representations of GLM or DELS, which are algebraic at infinity. So mixed motives, so these are pure motives. So mixed motives, they're built from pure motives, they're kind of successive extensions of pure motives and so it's very natural to wonder how the optomorphic world reflects this phenomena. So one can, for example, ask a question, can we see something or just all of these mixed motives? We are modular manifolds, meaning optomorphic world. And today I want to give some examples and just want to say again that the simplest of these examples is the one we discussed with Barry. And so if you want to look at mixed motives, so first of all, you want to mix the simplest of them, but you want to get a great supply of them. And so like a source of them is a motivic fundamental group, it's a great source for a curve. So higher, of course, this is a great source of mixed motives. And so what do you want it to do today? You want to look at two examples, two if I have time. But the first, you want to take the fundamental group of multiplicative group minus n through the unity with tangential base point at zero. And I'm going to show that this relates in a very strange way to geometry of this modular manifolds. And this is subgroup gamma 1 nn, which is the matrices, which is like that mod n. And I'm going to talk about this situation when m is 1, of course, but then 2, 3 and 4. The relation sometimes exists for large m as well, but this is where it's really precise. And what we discussed with Barry in his office was the first situation when it's 2. And then the second kind of update is that the story doesn't end here. So basically what's going on here is when we consider the field q. But we also can consider some other fields. For example, we can take q of i. And in this case, this relationship look as follows. So we would take elliptic curve. So this elliptic curve is Cm multiplication. And then consider, again, material fundamental group of this elliptic curve minus p-torsion points, or n-torsion points, let me take p, where p is ideal, prime ideal in the ring which acts by endomorphisms of the elliptic curve. And this similarly, in a very similar fashion, is related to these Bianca's rifles. So who they are, it's a very similar thing. Now it's three-dimensional hyperbolic space divided by a very similar subgroup now related to the prime ideal p. And it turns out that this relationship, a piece of it I can just, I will explain this later on, but this is nothing just but variation on Meiser's model of symbols. Here I've had to bear model of symbols, which actually, there is no paper space, I can understand. They appear in the work of Ashen Rudolf with reference to bearing in Vincenzo's in 79. Is this correct? But at least that's what's written there. Okay, so I'm talking motivically, Bianca, GLN. Okay, I'll define them. The same question. On the right-hand side, you seem to be limited to m equal to 2. The results are limited for m equals to 2. You expect the same. It's a good question, so I mean, please ask me this question at the end. I cannot explain the story before I explain what's about. Okay, so I started talking about motivically, but as we know, motifs, they can be seen in realizations like eladic or hodge. And I choose the language of eladic because it's kind of more aesthetic today. And so when we sit at the eladic side, the Galois side, so we see the following. First of all, we start with motivic fundamental group, but we have its eladic realization, which is given by the pro-air completion of the usual topological fundamental group of punctured C-star. And we can linearize the problem. So via motivic completion, we can replace the group, which is a huge pro-eladic group, by a Lie algebra, which is a pronulpotent Lie algebra over ql. And it's free with n plus 1 generators as it's obvious from this description. Now, what do we see when we look at this object? So first of all, this Lie algebra, this eladic Lie algebra, it carries canonical Galois invariant weight filtration. It's called W. And in this particular case, it's happened to coincide with lower central series for this Lie algebra. So it's filtered by commutators. Now, let's look how it actually looks like on the picture. So we start with the biggest quotient, which is first dimensional homology of gm minus mu n. I tend to omit the index, the upper script l, because eventually the story is eladic after all. And so this is kind of the whole thing, less than 1. Then the next quotient is double commutators, h2 of h1. So this is less than minus 2, and so on. Then goes triple commutators and so on so far. And so now what we want to do, so I wanted to explain how Galois groups enter through this business. So what we do, we restrict, so forget to say that of course the Gallo group, the absolute Gallo group, x by symmetries of this object, that's the main point. And we wanted to actually understand how the Gallo group acts here. This is the main goal. And we want to understand how it acts here and how it sees these modular manifolds. And so we restrict to representation phi nl of the Gallo group of q bar over fieldish, obtained by adding all l's through subunit nn. And so now it acts, so it maps to automorphisms of this huge eladically algebra. And so what we wanted to do, we wanted to consider the image and linearize it. This means that we introduce some graded Lie algebra which depends on n, which is the following thing. So you take image of this Gallo group and you take the Lie algebra of this. And don't stop, so you consider the associated graded quotient for the weight filtration. This makes this object kind of more canonical. Now what we get and what we're after is a Lie algebra, so it's g dot. So it's just direct sum of g minus one plus g minus two plus g minus three and so on. And we wanted to know what the spaces are and how the commutator works. It's a graded Lie algebra. Okay. Now let me explain in a kind of slightly different ways. This is just kind of the definition which is a little bit enigmatic. So what we're actually doing for any integer zero, one, two and so on, we look at Gallo group acting on the following things. So we have this filtration, so we take some number k and consider a quotient of amplitude m. So for each k, which is minus one and so on, and for given n, we consider this quotient of our Lie algebra. And then Gallo group acts on this already finite dimension quotient, but there are infinitely many of them for all k. We just kind of take some something of amplitude m and then push it to the end and look what we get. And then we see that we get some maximal Gallo subgroup, which x trivial is there. So therefore it produced us some tower fields. So we have a tower fields like we start with q. Then we have the field F1, which is nothing but q of zeta l infinity n. Then we have the next field F2 and so on. And we can record the Gallo group. So this is clear. This one is a new one. So we call it G1. Sorry, it's numbering is wrong. It's F1, F0, F1, F2 and so on. So this is G2 and so on. So we get these groups and they are billion, a billion eladic groups. So we can consider the Lie algebras. And the point is that, let me write general definitions, that GM is the Gallo group of this FM of F0. And the Gallo group of this extension, which is cyclotomic, with a little bit more quotient acts on this guy. And we wanted to consider the Lie algebra of this. You consider the Gallo group acting on W minus k divided by this. So what I wanted to say is that so far I'm talking about this a billion quotient, like for example, first of them. And so what I wanted to say is that I wanted to consider this Lie algebra and this is the same thing which we introduced before. What is the definition of Fn? I write it but didn't say. So you consider this quotient. So for example, you consider the zero quotient and then you're just talking about Gallo group acting on the homology, eladic homology. So you get what you get. So then you consider the first quotient. Now you're talking about X1s. So you have your filtration. This is associated graded. And you consider just extensions, all possible extensions. And so you ask the question what kind of, basically, what kind of extensions you get. And so you consider, I mean, you take the image of the Gallo group and you basically ask the question what the image is. And so F is the corresponding field. And so for example, you can ask the question, what is this G minus 1? And the first thing you see is that you better go to the dual because the dual has a very nice description. This is, you know, O star of the, let's call it scheme SP, which is just a spectrum of the cyclotomic field with 1p added, tensor QL of 1. So that's how the first guy looks like. And so what we learned here is that first of all, we better look to the duals than to the spaces because they have nice description. And so this is just a group of cyclotomic units. So the group of cyclotomic units tells you what this guy is. Okay, so now you can ask the next question. So how about, oh, where is this thing? Yeah, yeah, yeah. Okay. So you can ask the next question. How about the thing which has depth 2 from this point of view when M equals 2. And so when M equals 2, you're talking about the algebra structure on this G minus 1, which you already know, and G minus 2. It's a graded Lie algebra over QL. And so what you want to know, you want to know the commutator map, which goes from H2 of G minus 1 to G2, G minus 2 because there is no other possible commutators here. And it turns out that as before, it's much better to dualize the picture and look to G minus 2 dual to commutator map of H2 of G minus 1 dual. So this is the guy which we wanted to see. And we wanted to describe this guy. Okay, I can actually start describing this guy, but I want to make story kind of more elementary, at least for exposition. So what I wanted to do now, I wanted to go to some quotient. You can call cyclotomically algebra. And so this guy is G minus 1 plus G minus 2 plus and so on. And I want to consider subjective map to some sequential introduced in a second, C minus 1, C minus 2, and so on. So this actually will be, it's actually not even isomorphism here. So I'm going to get some quotiently algebra in a natural way and then talk the whole lecture about this. So how we define this quotient. All right, you notice that geometrically Gm minus Ns of unity embeds into Gm. And therefore, if you can consider the corresponding fundamentally algebra, the group of the fundamental group here, which is just Ql of 1. And you can map the fundamental group to say the corresponding the algebra to this one, subjective, of course. And then you get the kernel. It's the codimension one kernel of this map. I forgot to say kernel. And what I wanted to do, I wanted to consider some depth filtration. Its role will appear later, so far just a definition. So it's by definition just lower central series of this codimension one ideal I. And then I have two filtration weight, which is completely canonical and works for any variety. And depth, which is very specific, it just uses this specific case. And so we introduce the following quotient. So it's G minus W minus M. So I define this as a quotient with respect to depth filtration and with respect to, I can just say, what I already introduced, G minus W of mu M. And this is minus M. Once again, so what I'm doing, I'm introducing one more filtration on the topological object, on the fundamentally algebra. Then it filters everything, like derivations, the image of the Galois group. It induces some filtration on the image of the Galois group. Miley algebra by introducing the C minus M of mu M to be just diagonal part. This is just G minus M minus M. So here one can easily see that W is always not less than M, but here we just took the diagonal part. And as you will see, this makes just presentation simpler. Okay, so what do we want to do? So we wanted to understand this Lie algebra. And so the same question is about this Lie algebra. How the commutator looks like and how the... Actually, I just realized that I told you a lie. I beg your pardon. So I mean this isomorphism with cyclotomic units is C minus 1. That guy is big, I beg your pardon. This is... You need some beds, but that's... Actually... Actually, it's okay, yes, it's okay. It's okay, it's okay, it's okay, it's okay. It's okay, it's okay, it's okay. So everything else in a bigger weight. It's okay, it's correct. Okay, so there's no lie. So I want to understand this Lie algebra and what does it mean to understand some Lie algebra? So we wanted to consider the co-chain complex of this Lie algebra and in particular it tells us about the commutator. So we consider the standard co-chain complex. So we take... I introduce notation. This Carly C w of mu n is going to be dual Cw and this is supposed to be minus m and this is supposed to be m. It's supposed to be dual vector space and now this vector space is formed, of course, the Lie co-algebra. So this means that I have... I will omit n's now. I have a co-bracket which is going to weight squared of C dot and then I can continue the story. I can go to weight cube in a natural way, weight 4 and so on. So we got this complex, which in particular contains the information we were looking for, the co-commutator. So if you know this complex, we, of course, know more than we wanted to know. We know the Lie algebra and its co-homology. But this complex is graded by this degree m. And so what we can do, we can take the m's part of this complex and consider its co-homology. And so this is the guy which is denoted co-homology of degree m of the C dot. All right. Now I can state the theorem. This is the main theorem at this moment which tells you who this co-homology are assuming that my level is prime. Okay. So let me give you somehow a picture of this co-homology computation. So on one hand side, you wanted to put the co-homology of this Lie algebra, cyclotomically algebra, of mu p, level p, of degree m. So m is going to be 1, 2, 3, and 4. And so if you want to consider the corresponding co-homology, we are talking about complex. And this complex is very simple at the beginning. This is just C1 of mu p. And here I'm going to put the answers. And we know what it is. It's just O star of this sp. And it's just one vector space. So this is this guy, tensorql. Then if you go to the level 2, you get C2 going to h2 of C1. And the co-homology here turns out to be h1 of gamma 1, 2p with the coefficients in determinant representation. Everything turns out ql. I'm not going to keep writing this. And here we get the same thing but h0. And this is exactly what we discussed in the office. Now we want to continue. So we take C3. Then we go to C2 tensor C1. This is just the standard complex in degree 3. And we get, so we get here h1 of gamma 1, now 3p and standard coefficients. And then we get, let me think actually, I think, yes. So here we get h2 and here we get h3 of the same group. Now we go to the weight 4. Then we have C4. It goes to weight 2 of C2 plus C3 tensor C1. Then it goes to C2 tensor lambda 2 of C1 and goes to lambda 4 of C1. And here of course we get gamma 1, 4p but the numerology is that we get here h3, h4, h5 and h6. And little caveats. So here we get some quotient which actually I don't know, I have a conjecture what the subspace is. But I can, yes. Yes, yes, thank you Maxim. The alternates. Thank you very much. So what you see here is that this is what I promised that you look, you start with a Galois theory of gm minus mu n and you write down the complexes and complexes delivers you when you look at the different grades co-homology of these groups. And now you can ask whether it's maybe coincidence because if you look at this group, all right, so I claim they're the same but dimensions of this group can be easily calculated. Just some number which depends on p you can find formula in any textbook and maybe it's a combinatorial accident and this will be zero. So maybe it's not there. But if you look at the other groups here if you look at this whole region then this is precisely the hospital range. It's quite amazing that you catch precisely the hospital range of these groups here and for the hospital co-homology we know of course that's sporadic if m is bigger than 2. So you cannot write any formulas for this dimension. We know that these vector spaces can be non-zero but we have no formula, no prediction, nothing. So we cannot say basically anything about the spaces as far as I know except running program and tell us computer what the number is. But nevertheless that's what we get so now we can say unmistakably that yes we see this model or manifolds sitting somewhere in the fundamental group and of course this was not an accident. Okay, so now the next question was behind this. So I said that we have the isomorphism between co-homology of this guy and this group co-homology but when one proves this of course we get much more so we get isomorphism of complexes and so I'm going to tell you that I mean we are going to see in a few minutes that actually not only co-homology but these complexes are realized inside of the symmetric spaces and in order to do this I need to remind you a little bit of geometry so how we approach co-homology of discrete groups so we consider the corresponding symmetric spaces and so let me remind you that we have this symmetric space H m which is defined as the quotient of G L M R modulo R star plus O M and for us it's going to be the collection of all positive definite quadratic forms on a vector space of dimension M over real numbers divided by the positive real numbers and it's also important that it sits inside of its natural compactification not borderless but just some compactification and this is just the same definition when you consider non-negative quadratic forms so do the same thing now we can introduce the corresponding modular manifolds officially so to speak so Y 1 M p is going to be the quotient of the symmetric space by gamma 1 M p and so we are after topology of this guys now how we approach this so first of all so when you consider this compactification using semi-definite form you also divide by R plus star and what you do with the 0 you include it I delete 0 write it not equal to 0 but what do you think about this how do you think about this we take a lattice so you want to say what is G L M Z of rank M in the dual vector space and then we do the following construction so if you have any element of this lattice it produces a quadratic form p sub f which is defined as follows so we have a functional f cc vm star so we can take this functional and evaluate this on any vector so we get a linear function and so we take it square so it's a quadratic form and it generates this rank 1 quadratic form so it leaves in the compactification now if you have a basis in this lattice so if you take a basis f1 and so on fm of L M then we can do the following thing we can take a convex hull let me just do it like that convex hull of these degenerate forms and so this is a very simple thing it's just a linear combination of those just remind you what's convex hull phi i when i goes from 1 to m and we assume that lambda i is non-negative so that's the definition of the convex hull okay so what do we get so now if you take this thing and project to the modular manifold so what we get I want to have some notation for this let's denote this like that f1 star and so on phi pm which belongs to homology I'll tell in the second y of gamma 1 mp Borel-Moll homology and this is the definition of these are measures modular symbols so look at the paper by Ash and Rudolf and in 79 and so on you'll find the same definitions there so what there we notice so first of all you want to see an example first so the example is when m equals 2 then you have an upper half plane and so on the upper half plane you have this famous modular decomposition and so what you get you get at this moment you get these lines for example line going from 0 to infinity and so when you project them down let me run it this way so when you project them down to modular curves you get some strange objects so you get triangulated surface topologically triangulated surface because there are casps at infinity and so what Barry did so he generalized this picture for GLM now for GLM you have a similar guys and a very nice thing about them that they are cycles because like here the boundary of this guy belongs to the boundary of the space it's at infinity here again if you take a very triangle then its sides really belongs to infinity because this is linear combination of m minus 1 rank 1 form it's a rank at most m minus 1 in the space of dimension m it's on the boundary this means that we got a cycle it's a very very simple observation it's a crucial so we got a cycle we got a homology class they generate the homology group and so these are modular cycles ok now the next thing to do we wanted to relate this to the original problem and so we wanted to use major modular cycles to relate them to the following thing so first of all so yeah there are some lines here so this modular cycle major modular cycles they produce you a way to catch those guys precisely those guys so you consider this line and so you can call it major's line because that's exactly where the homology which we constructed leave and it shows also that if you consider anything to the right you get 0 so this is the top-dimensional homology which you can possibly have and they leave here and of course what we wanted to do we wanted to relate them to this line directly and so I would call it millner line because this reminds very much to me this kind of idea of millner-casir in the resolution by my TV pohomology but the cohomology of this line close related to millner-casir okay so you wanted to send that line to this one so let's do this and this is just one word construction after little explanations so we noticed and this works for any M so we notice first of all very elementary observation that the group GLMZ acts on the basis of our lattice and if you mod it down by the subgroup millner-casir after and take linear combinations of the points in this finite set then actually even before it takes it just when it takes a set I can describe it because this is just finite field P to power M minus 0 point so very elementary observation and so now we have the key point so we consider a map as I said this measure goes to millner so we consider a map from what? so we take this cycles which we get by which are span of measure's modular cycles they live in Borel-Mull homology of dimension N minus 1 of this modular manifold notice I didn't take the homology group they generate although they generate homology I just took cycles, measure's modular cycles I map them directly to wedge M of O star of the cyclotomic P cyclotomic scheme and actually as you will see I will do a little adjustment I will add one dimension of subspace later on which corresponds to Euler's Gamma constant but so far don't worry about this so how it works so first of all you take any collection which is this mod P and now you know that it represents some element here and now when your group GLMZ acts on measure's modular cycles so the guys you get are parametrized by this set but they satisfy some relations like symmetry relations and when you put plus minus they do not change so there is little relations where you satisfy but it's basically elements of this set so we take the corresponding cycle here this is again this is measure's cycle and we map it to the following unit 1 minus zeta P to alpha 1 wedge 1 minus zeta P to alpha 2 wedge and so on wedge 1 minus zeta P to alpha M now if you I think offer is going to ask me a question at this moment you play the role of the offer so your question is what you do if alpha is 0 exactly so you just say that we use a symbol I call it a motivic gamma this is just some extra element which I just add to the game which do not talk to anybody else but it makes the answer much simpler I can avoid doing this but the answer will be very explicit but a little less nice so it's just doing this for convenience but I believe it actually makes sense so I believe it's really you really add to your tape motives one more okay so we consider this map this map is surjective and so after that you can ask questions what happens with this map is there any kernel of this map and you can guess that we're going to say that we want to construct a resolution because this measure's model cycle they live on the right and I wanted to introduce now kind of high analogs of major model cycles and they will provide me a map from that complex to this complex so now I wanted to do kind of high what? so alpha 1 alpha Fp to the M minus 0 and what is CH oh CH is it just chains it's a Borel Moore chains so there are cycles I just use this notation because my C is overloaded so you have in this manifold you have chains of dimension M minus 1 this is measures model cycles they actually cycles but I just set chains at the moment and they infinite so they produce your cycles but I just use word chain but the map is defined only on those yes yes yes only on those chains so I take these chains and for each I basically take the span of measures model cycles just take the linear combinations for example if you get 0 it's supposed to go to 0 and so on so is that ok? ok so this is as for under GLM ok this is yeah ok so now I wanted to do the resolution of this ok my blackboards are the nice ones are limited so let's go here no beautiful so what kind of resolution so remember this model picture so I'm going to erase it but what I wanted to do I wanted to take this group which generated by measures guys and I wanted to write a resolution M I skip Borel Moore here but they all kind of in this setup and so in the end of the day I'll go up to the chains of the grid 2M minus 2 and this is on this model manifold Y1 of M so I wanted to build this so we don't yet have it so is it a group of chains or model or model? let's wait till the definition I think it would be better when it says definition so far just some a billing group and so they are not cycles any longer so there is a differential inherited by the differential on chains and we wanted to map this exactly where we wanted to map them we want to map them to which M of all star of SP this guy we wanted to map to the next guy which is C2 tensor lambda M minus 2 of C C M minus 2 no no C2 M minus 2 of C1 and so on and the most interesting one we wanted to map to the Cm of my new P so why emphasize this? because we really after this guy we really want to know what this guy is and so what we did so far was from the point of view of this Gallo-Ali algebra okay it was okay but we didn't catch the main guy this is where our interest is so how we do this is a construction the construction goes as follows the generalization of major cycles but it starts to remind you'll see what by the way so if you do this construction for M equals 2 you will recover triangles in that picture so for M equals 2 you have this guys and this guys this is a complex now key geometric construction so let's take any F1 and so on Fm and let's complement it by one more vector F0 which makes this sum of them equal to 0 and then I wanted to construct some chain which depends on this vectors and this is going to be after projection this is going to be something I mean right now it's going to be something which leaves in 2M minus 2 chains space and later on we will project it down so we're just doing some kind of generalization of this modular picture so the construction goes as follows so we take a circle and we put this M plus 1 vectors on the circle put here F0 F1, F2, F3 and so on Fm and then we take a plane trivalent 3 which is which labeled by this vectors so what dimension is that just the graph this is just the graph so you take a graph and you make this plane trivalent 3 let's call it T now first of all you notice that if you happen to have an edge of this tree for example this one then what you can do you can cut this edge and then you have 2 trees so you have this tree and you have the other tree but each of these trees is labeled by the vector so what you can do you can introduce vector so you assign to E vector F sub E which is defined as some of several of these vectors so in this case it will be some from F1 F2, F3 and up to this point you can say okay why don't you take the other trees your construction is not canonical the answer is well we can take the other one but since the sum to zero we get negative of that so if you take this tree so here you get F sub E so you get negative F sub E so it's almost canonical so now we can use this as follows okay let me come here so I define the cycle phi of F0 and so on Fm to be sum over all plane three valent trees T labeled or decorated by your F0 and so on Fm so basically you take all pictures which I draw like that could be different pictures you consider all of them and for each of them you do the following thing so the most important thing you do is take the convex hull of the following guys so remember I got vector F sub E but if I have a vector F sub E it's give me quadratic form F sub E squared which I denoted by phi sub E and so I take the convex hull of this many vectors phi E1 and so on phi E2m minus one exactly 2m minus one of the edges of this tree if you label by this basis and therefore it produce your dimension 2m minus 2 as we wanted but that's not the end of the story because we have to take the sum of those guys and we don't label the edges of the tree so we have to pay little price for this so we put the sign of E1 and so on of E2m minus one and this is the sign which is related to the canonical orientation of the plane so there is one way to introduce a sign here I'm not going to talk about this so that's the main construction and for example if you take m equals 2 that you really get these triangles so then the cycle is just one because you have this three vectors on the circle don't mix this circle with that circle that circle is combinatorial and this is the boundary of the hyperbolic plane but then if you consider three measures actually classical model cycles there is unique 2D triangle here and this triangle is exactly the guy which I signed to it if you go for example for m equals 3 geometry gets a little bit more complicated I just give you a cartoon what you get but not very complicated so you basically get a pyramid because there will be in this case we can have only two different plane trival entries and so what you get you get two tetrahedra but these two tetrahedra are joined in the prism so this is the cycle which we constructed for m equals 3 and so on and so on alright so we constructed a map we didn't construct map we defined this group so far now before I go to the rest it's very easy to go to the rest of the complex from here but I wanted to tell you more important things so who goes to whom here I don't have unfortunately time to explain the construction I'm just saying that it's kind of key motivate construction it sounds that one can define for any if you have any rules of unity for example in mu n then you can define actually if you take any collection of points on gm you can still define some elements so I call it motivate correlator of this in this case units and I assume for technical reasons that their product is one to match the construction and so where this guy leaves it leaves precisely in this c minus m of mu p start so I claim that I can define for you absolutely canonical elements which leave in the motivically algebra they become so canonical because I took at the beginning graded for associate grade for the weight filtration but they are absolutely canonical elements there is no dual algebra very important dual so all kind of work in the dual you never work with the algebra you also work with the dual that's where you see this objects okay so now you have this objects but you also remember that there is one more game you can play with motives you can take the virtualization and so what happens because I didn't tell you anything about this elements and actually forgot to tell you the main thing that how this construction works so okay let me do it here so the main construction takes a cycle phi of alpha 0 and so on alpha m this is a cycle I can define it upstairs the generalization of measures model symbols so now it's a cycle of dimension 2m-2 not a cycle it's a chain it's boundary not 0 and so we map it just to this motivic correlator of zeta p you just take some root of unity raised to power alpha 0 and so on zeta p raised to power alpha m and so you land as I said in this cm of mu p and so in the end of the day that's how you get the map so this map now constructed model is the fact that I didn't explain to you who these guys are but I'm going to tell you not who they are but how they look in the virtualization and okay why don't I read this so notice the following that I have this li coalgebra cm and I was talking all the time about it's a latic realization now we do have mixed motives mixed state motives we do have motive fundamental group for example defined in our paper with the link and so we can consider now this as a motivic guy now we have a new game to play we can go to the hodge realization of this li algebra and then there is a canonical period map which is not at all obvious what's constructed in my paper on hodge correlators some absolute canonical map from here to real numbers so in the end of the day if you start with this guy which leaves here or here then you get a number and so you can ask the question what this number is actually this motivic correlators if you understand a little bit this motivic philosophy then all you need to know you need to know co-product in this li algebra and I can tell you it's explicitly maybe in a second and you need to know the hodge realization of this guy then it's uniquely determined of a number fields so all I need to tell you I need to tell you what is this hodge correlator and it's given by construction which is very similar to this one that's why I wanted to present it so what you do you again take a circle but now you label this circle not by vectors but by your roots of unity you can take any actual points on GM just for convenience I take roots of unity and then consider the same plane trivial entry so in this case like that it's exactly the same combinatorial setup but now they are decorated so they are decorated by those points outside and by some points inside like y1 y2, y3 so before we took the h and h e and we assign to this h something which was used as a building block now we do the same but this something is going to be a logarithm of absolute value in this case y1 minus y2 and in general this is just a green function of this points y1, y2 this answer we get because we normalized appropriately using the tangential base vector so we just assign this guy I just call g y1, y2 for short and then we do the following construction so we consider this green functions assigned to the edges g e1 which dc of g e2 which and so on which dc of g e2m minus 1 so notice that this is form of dimension 2m minus 2 and then we integrate this over our cp1 to all internal vertices this y's and then we take the sum of all plane trivial entries as before so this is kind of identical combinatorially to this construction you also put a sign of course so it's kind of identical but we put green functions instead of putting here this degenerate quadratic forms so you get a number and so there is a theorem that this number this is the canonical period of the culturalization so this means that it determines uniquely who this guy is and I actually don't have a couple of minutes because we start a little later but I unfortunately don't have time to tell you how we define the intermediate steps it's very easy and kind of mimic the construction of this co-chain complex I defined you c4 and c1 and the rest is kind of wedge products here I defined you this cho and the other wedge products of them in some sense so it's very easy but I skip this but other than doing this I wanted to tell you a little bit more on what happens next so now we come to the story about Bianchi's riffles finally so okay let me do it just take few minutes two minutes to sacrifice so as I said the story before we started with gm-muN and we played basically with roots of unity and this is kind of high cyclotomy game because we started with cyclotomic units they described the simplest piece of the Galov group and then we went up and got the rest of the cyclotomically algebra now we want to go to elliptic curves and here is what we get so first of all the game with motivic correlators can be played on a universal elliptic curve so take universal elliptic curve then here is something which I called Euler complex just the same game so we get some complex c2 of elliptic curve and n torsion points going to veg2 of c1 of elliptic curve so this is the same game with motivic correlators so this is motivic kind of motivic guy and this motivic guy related to pi1 of universal elliptic curve minus n torsion points in a way which can be specified very precisely so we start here now we can go 3 different roads so you can go to the cusp which is our cusp SP and then we get today's story for m equal to it turns out that this complex specialized exactly to the complex which describes this first non-trivial portion of cyclotomically algebra so it is an explanation in talking to Barry so in your office I was talking about this this is an explanation where does the modular curve come from from here this is really modular curve so geometry is there secondly why it is called Euler why I call it Euler so I define this guy but you can go here to this guy produce you two Ziegler units you can take their class in k2 and this is precisely balancing cut Euler system so that's why the name so this leaves in k2 and this leaves at the genetic point but you can also go to cyclotomic point to cm point like the one I choose today and then you get a new complex which relates to Gauss and integers and the answer is that for this complex for this specific complex so you can call it something like c2 of some prime ideal the same notation kind of h2 of c1 so it relates to Bianchi manifold so let me this is the last thing I'm telling you so what Bianchi manifolds do first of all you start with the n-loc of the modular decomposition so you take rectangle with vertices at zero i1 and 1 plus i and do it like that now if you put this on a hyperbolic space you get a picture like that and you kind of make it grow to infinity and then you also add this guy octahedron and this octahedron is the n-loc of my triangles and now the game goes as follows this when you project down to the modular manifold this guy goes here this triangle goes here the cohomology proceed to be given by h1 of gamma1 to p and h2 gamma1 to p I want to notice here some portion which I don't completely control but here it's an isomorphism I notice that that's exactly the range where the casplocochomology are so you again get casplocochomology in the end starting from the Galov theory and even more interestingly you see that you get the octahedrons and it had some cycles here start to rush so it's this is 1 and 2 it's homology so there are some homology classes here which is sporadic casplocomology classes they tell you that the elements the motivic correlators which you see here satisfy some relations which we do not expect at all and we don't know how to describe them except going through this picture so we know that they exist but that's it so that's the end of the story so the very last thing that I have a student Kolya Malkin who in his thesis explains how the story goes when you go to the whole Galovly algebra don't cut it as we did and we get here co-homology of these guys with coefficients and symmetric powers of the two dimensional presentations the same story okay sorry questions actually I remember in the 90s sort of about graph homology as it was which never developed it's almost the same you can see the not punctured curve but closed curve and you can see the derivation of the combination of the algebra you get the co-homology of this algebra as the same as the co-homology of the kind of graph complex which is responsible for auto automatically of free group the version of graph complex yeah and this is very similar it's the same as punctures which is very close relative but it's broken linearly so I think it's really both graph complex yes yes I think so I didn't tell you how to define motivate correlators and I do use your work so for sure yeah so the cycles the chains that generalize modern symbols they're not cycles and their boundary what can you say about that boundary what can you say? thank you, you give me a chance to tell you the answer but it's quick okay maybe so you see better I have a follow up question is there a similar structure for HECA operators and they would be on the logic of symbols that's a good question so if you go to the kohomology sure the HECA operators so on the level of complexes but look the answers are falling so you start with this guy which basically circle this collection of vectors f0, f1, f2, f3, f4, f5 and so on and what you do you cut it so you take one of the vectors you like for example this one and it takes an arc here you cut it so after you cut it you see that you have two semicircles but when you cut it you inherited like here inherited three vectors so you put them here but they're supposed to sum to zero so you put the fourth vector here so you get one, two, three, four vectors but you get one more that's the answer so you have to take some of all possible cuts so the answer is very, very simple and of course the theory means that the motivic correlators can multiply in the motivic algebra exactly the same way that's why it's a map of complexes and you also see why it's easy to define kind of all cycles in between you kind of take which product in this case the convex hull of the construction you already know for example your construction the simplex is just convex hull of these three guys that's why we have the wedge product wedge M that's it that's very simple so first of all a remark so here the H6 is still in the capital range in the bottom line I think H4 and H5 in the capital range I think H4 and H5 if my memory is correct the middle degree is 4.5 so it's 9 divided by 2 so it's as a segment which is again it's 1 for SL2 2 for SL3 2 for SL4 and the center it's middle 4.5 but ok the next question two years ago we were listening you and Katush so do you understand the plausible relation with no first of all I wish I so this story is the beginning it's very old of the story so it's like quarter century old and at that time I saw about this you know numbers like 1, 2, 3, 3 and so on and noticed that they correspond to X1 but I have no idea X1, H1, X1 I thought I have no idea why it appears this interpretation of Zuckerman and Wogan and so what when Katush does he explains to give a huge evidence that actually indeed this is not the case and so this kind of capital range is related to X1 of material cosmology and I have no idea why so in particular you don't get the whole range of I get what I get up to 4 it's a difficult theorem to prove on the level 4 so like before I did this I did the case mx with 2 then couple of here is n equals 3 and then in a number of here is n equals 2 it's already like 20 pages and I don't claim anything about higher m and my last question is at the beginning when we are starting from the realization of the get one group it was erratic and then it seems that after a while we are after a while there was a confusion to the motivic world through the wall and we end the motivic does that mean that there is a queue structure of course of course of course the whole story is motivic I started erratic just because for perception it's kind of easier because when you start talking about motivic other groups you lost your audience so that's why but the story is 100% motivic and that's very important