 So I changed the title. It was a little bit vague because it was a little bit hard. And so I started with observing that although it's a very wide spectrum of the subject covered by this meeting, it doesn't cover all the work by Tomahand. I think the most cited his paper is on symplectic geometry, which is actually in all textbooks now on symplectic geometry, the classification of toric or symplectic manifolds. It's just quite a nice theorem. I say, but we don't touch it here, right? And my point is exactly, indicate relation between two aspects of this missing negative curve between Kelly manifolds. And the problem, here it is a problem. So we give in a group and it's maybe not any group. You always can find count example. If you do it too broadly, but group is some niceness to that, or at least without apparent pathologist, and you want to construct geometric object. Or the group where essentially the group is fundamental group, but something like that, right? You know, it's a classical number theory. It was probably still a famous problem to realize given fine group is a Galois group of extension of the curve. This is not like that. So it's a realization of a group in a nice way. And nice here is what they indicated. What the hell happened? How it just moved? With the arrows? You want to do that? It divides, you see? It's supposed to divide, yeah? Can you see it like that? Okay. And so it might be kind of canonical construction, preferably in terms of the group, some extra structure in the group, still described group theoretically. And it might be universal at leastifying that if you take the object in a similar way related, all of them will be induced from that. Similar how classifying space of a group in general classifies all spaces with this fundamental group and maps it into there, right? Like K by one spaces. And so there are two examples which kind of somewhat inspiring. And one of the classical theorem of Abel, I don't know whom, is A and it says it works very well in the diabetes case, that the fundamental theorem, if you have an algebraic manifold or a K-manifold, compact K-manifold, and you have an isomorphism, is homology more torsion to the homology of the complex torus, the fourth module of the right dimension, then it comes from a holomorphic map, but this holomorphic map is very kind of rare animal. This homomorphic map is a multiple kind of easy thing, but having it covered from complex geometry is quite amazing, and it is kind of a theorem. And so, except that this complex structure on the torus depends on your manifold and it's essentially unique, it's a unique structure in the map essentially in Capra translations. So it's completely canonical, right? So you can out of the group, you make this family of K-manifolds, and this is just modular spaces of all this, you have to polarize them just, the technicalities of this complex, complex tori, and then they represent all manifolds with this fundamental group in a very kind of nice way, and so it's a kind of, I must say, okay, it's one of the most famous thermo-mathematics. Is it due to lunas? Well, it's called Abel Jacobi or Vanessa, I don't know, I don't know who actually proved it, as we usually don't know. It's classical of the 19th century, but it has one, if you kind of know this, but you know how to ask nice questions, say, are there any examples? That's an interesting point, because of course, there are curves in the algebraic curve, for every curve there is Jacobian, and this is the classical construction, there are many specific constructions, how you make Jacobian complex torus out of the curve, right? Analytically, pure algebraic or any field, there are those algebraic manifolds. For Kepler, it doesn't, but this is not a kind of Kepler stuff, pure algebraic. But then if you look at the higher dimensions and ask, what are the examples? Can you give me manifold? Kepler manifold is a given kind of non-trivial H1, which is not like product of curves, and I've been in a variety, so I can say, well, if you know a little bit, you can say, take this complex torus algebraic torus, and take hyperplane section or genetic section, but then to be there from the statutes, and the fellows don't solve this. Actual algebraic geometry is a kind of cheetah, so they solve algebraic manifold defined by equation throughout the time and complex projection space. You say these words, are they examples? No. It's extremely hard to compute how to produce, they don't come easily, right? So I would say this, maybe kind of overstatement, but nobody has any idea what algebraic manifolds are. What are they? Because there are this naive complex intersection, there are modular spaces, and there are some, as I mentioned, arithmetic varieties coming from these key groups, and then maybe one, two, and then you, very few sources of them. So we don't know. Actually, they are conjectures, classical conjectures. I'm not certain how much of them are solved, that up to some extent there is nothing else in certain cases. And so, however, it's still a great term. And the core of this theorem is the case of the curves and my kind of quick skepticism about high dimensions not completely unjustified, because general case, trivially, more or less, follows from the case of curves, at least for algebraic varieties, but complex, projectile, algebraic manifolds. In the following way, because, so, given this homomorphism of groups and this typology, we know there is a continuous map implementing this homomorphism, but continuous in a defined up-to-homotopic. And then, on the other hand, your algebraic variety is built out of curves. They have a surface, they have pencil, right? They take some point and rotate plane around it, and have this family of curves, which meet at one point. For each of these curves, you know there's a complex structure on the torus and homomorphic map was there by this theorem. But at this point, they might disagree. Of course, it doesn't give you. However, if they don't disagree, it gives you map from this point, you have a rational map for rational curve to this modular space. But this modular space, you know, is simply connected. I mean, it's universal covering. But you don't have to think about complex structure. It's just simply connected. It cannot contain any rational curves. So that's... At this point, I want to elaborate later on. So the key of all that and all generalization is apparently what happens to curves. And this is very similar to how you can think about hyperbolicity in group theory, right? Maybe you may think in terms of surfaces or in terms of curves, in terms of parallel language. But this is on the homomorphic side. But what's remarkable, and this is less known, though it's more elementary and equally beautiful theorem, due in this form, John Frank's, actually proven quite a few other theorems. And this is for dynamical systems. Now, instead of a complex scalar manifold, you take a manifold, any reasonable space, compact and locally contractible space with an automorphism, self-homomorphism. So this self-homomorphism replaces a complex structure. So a complex structure was in the morphism of the tension bundle, multiplication by square root of one to one. And here, just manifold itself transforms to itself. And I think there is a kind of much closer relation. But on the surface of things, you see, it's completely identical statement, right? And then there is homomorphism from H1 of your manifold to H1 of the torus. And then on the torus, there we needed scalar. Here instead of scalar, we say this automorphism induces hyperbolic automorphism on H1. Which means it's got generic. There is no eigenvalue by absolute value equal one. So it either stretches some vector contracted. No vector kind of goes unchanged. Of course, there might be an important element, which we don't want to. And then the same theorem is true. On this torus, there is unique, also an automorphism, but it will be the automorphism of the torus. And again, essentially unique, you can unique our translation. Here, you might as well be careful. Yes, sometimes it might be more constrained. Continuous map which preserves the structure. Meaning commuting with this in the morphism. So this kind of simple theorem, but what I want to say about that. Historically, it has kind of history of how people arrived at this. John Franks who finally founded the, but the first precursor of this was Smale, who I think about 62 announced stability of hyperbolic automorphism of the two torus. And then he announced it, I think somewhere, it's a meeting in Moscow, and when he left, soon thereafter, Arnold and Sinai published the paper giving the proof. And then Smale was kind of referring to that at some point and said that there is proof here and there, but the proof by Arnold Sinai has an advantage of being published. And this is quoted in the paper by Anosov, who said it adds one disadvantage. And this advantage is being completely nonsense. Not just wrong, it's nonsensical. The proof was... Absolutely, no proof. Yes, it was wrong. So, this theorem of Franks, it's proof implies that much recorded by all the magnitude of the powerful theorem, if you say it correctly, it takes five lines. And all these proofs are quite long. And then Anosov worked out some proof of stability. It was first correct published proof, but then it was realized it's not stability. It's much stronger theorem. It's not stability when you perturb the thermomorphism, whatever. You take any transformation in a different space, only remember about H1, and then reconstruct it. And it's a unique reconstruction. And when in dynamics, people apply it to the torus itself and use the word type semi-stability or something. Not because you see, but in certain quarters, still people think, oh, there are structures and there are isomorphisms. But from certain, but it was from here, and kind of... Think of growth and dick, you know. It's kind of minor, isomorphism minor, kind of small potato game. There are morphisms in the whole of this category. And there are things happen. The morphisms are just little, little guys there. Important, but little. Still not leading. And this is where theorem is. It is really categorical theorem. And so, I say a couple of words about that. So, the point is how far this goes. How far the analogy goes. How closely things are related. At the very beginning, I said, look at the one that already is complex. Polarized. Complex story. They already give fantastic this manifold in vector bundle over it and group action, semi-direct product. And the same automorphism, which acts here on the torus, appears there in the definitional module space. Because on the module space, on the story, you have all these monodrome elements acting in some of them. Hyperbolic and some, of course. Non-hyperbolic majority are hyperbolic. So, this thing closely related. But on one hand. On the other hand, there is parallel story. So, the logic here is as follows. You look at something in one field. And you know some little theorems or conjectures. And then you look what happens in the parallel one. And then you usually have conjectures. So, there is some way to generate conjectures, either about K-many faults in the fundamental groups or about dynamical systems. Or hyperbolic structure. You have a hyperbolic dynamical system where groups in the second will be in this box together. With hyperbolic and the automorphism of tori. So, what comes next? Excuse me. So, how do you do it in this Frank's theorem? So, what is the key concept? And this concept was coming from the work of Smale. Anyway, you see, what I failed to say. Why this picture? So, why this picture came from? So, who knows what it is? Niagara. No, why Niagara? No. What has to do with hyperbolic dynamics? It's HOS. HOS-SHU, absolutely. It's HOS-SHU shaped waterfall. That's where I have our list of dynamicals. First kind of pinpointed, I think, by Steve Smale. So, it's HOS-SHU. Because see just what I'm telling, so maybe I can go to the very end. Yeah, so there are references of how it didn't make them big. I wrote three papers and I wrote myself some. I put huge parentheses too early. And exactly about the matters I'm talking now. And so, nobody read the article. Now, I just read it and tried to explain it. I found out it's very hard to read. I had difficulty reading it. But a little understood from this I'm telling you. Well, but definitely, just I pinpointed quite a few. Point is the interesting observation. And one of them was motivated by the book by Toledo whose contribution will come in a short while. So, coming back. So, orbits and quasi-orbits. So, when you have an action of a group somewhere and it doesn't have to be a group, just set of transformation. It will be generating set of transformation. An orbit meaning you have a map from your group, a semi-group to the space, such that it's compatible with action on these elements. So, I have finitely many elements which will be generating such finite. And you have a group itself, from one side, from the left or from the right. I cannot, when I say this word, I don't know what it means, yeah. You know, what mean left or right? No, no, it's not a joke. You better communicate with somebody along the way, say along what, left or right. So, what is, how you can explain it? You know, there's only one explanation. I know only one, when you know a second. You know what? Well, left, right, right, from right. It's a weak, in the weak interaction, yeah, this particular, you know, nuclear decay when left and right slightly different. And so, you have to know details of that, if you don't know that, don't say these words. And I don't quite know the details of the experiment, neither really young logic, so I don't say left and right, because you have to be responsible for what you say. Mathematicians are mostly responsible for scientists, because they don't refer to the real world. They speak about their intuition, right. Which is certainly least reliable of all kind of, of all measurable instruments we have. So, it acts somewhere, somewhere, right, from some side, whichever it is. On the other hand, it acts in the space, and these two actions must commute. But if they commute up to by an error, so if they commute here and here in the finite error, it's quite the orbit, if it's finite. If it's epsilon small, it's epsilon small, very small. And so, these are orbit, quite the orbit. And how do you construct your, in the case of a space, it acts by homomorphism, and you want to build up the stores with an action in an economical way. What you do, you go to universal covering, and then action of the finite generating set, in this case, there's one element, and it's universal if you want. Act on the space, they act on that, right. And so, there are orbits and co- and action time approximated by corresponding sequence in the group, this will be quite the orbit. And then the remarkable thing, every quite the orbit is followed by a unique orbit, right. The scheduling principle. Why so? And this is exactly as in geometry because for orbits, for actual orbits, distance between two orbits is concave function, yeah. In some way they're closed, and then they diverge in a very sharp way. So it is a stable concave or convex function, I don't know how it's called, convex. And just once it's being observed for this action, okay, you have this theorem, it's always consequences, and it's quite simple. However, there is another way to approach that. And this was in classical dynamical system of a north of Smael real, you know, who defined them in this final way, which can be done in a more formal method without appeal to a group, or to universal covering, but it has kind of only partial power of what I'm going to say. So instead of saying this universality for which you need have concept of a covering or something like covering, and so just bound that error without specifying by how much, it will be just epsilon bounded and epsilon will be small depending on the station. So how to describe in dynamical system in a kind of, what is simply dynamical systems on compact spaces. So we have, we're going to describe. And you see the simplest system will be the one which I'm going to describe, this is what I call Markov hyperbolic. They are more the same as hyperbolic, but not quite. There is slightly more general, but the full extent I am not quite certain by how much. So number one, dynamical system for any group. It is a Bernoulli shift. You can see the function from your group to your final set. And this is a compact counter space. And then in a group, quite simple. And then you say, you can take subset there. So you take some, in the group, you take some finite region and say I only allow particular function there. When I restrict to this set on all the shift, I want to be in a certain subset of all functions. So I impose finite to many conditions. And then I have what is called Markov shift. In combinatorial terms, it means you have a directed graph and it is easily reduced to that. And the system consists of all maps of infinite graph and infinite path going along the line and then the transformation, the shift. Because you don't take the full graph, you don't allow all propositions. But this is Markov systems and they also counter. But now we have something like torus. And the torus you have is endomorphism. There is apparently no counter set. And then you can reduce to that. And this is reminiscent of what I'm saying now, unfortunately. Unfortunately, there is nobody who knows that. There was this recent wave of ideas called going under the name of condensed spaces. Implemented by Schultz and his collaborator, I've forgotten his name, but when they describe continuous objects in terms of category of essentially counter spaces over this object. And then, in particular, they were describing in a fairly, fairly algebraic way. Because counter states and their projective limits of finite sets. They're still not waved by using Markov partitions. No, no, no, no. Markov partition, I agree with you. Markov partition gives you slightly more. The point is, what I'm saying, listen, what I'm saying makes sense for any group where there is no Markov partitions. Indeed, traditionally, actually, before Markov partition it was used by North of using the discovering by smell. Then I'll come back. Markov partition is something else which is related to that. Absolutely. But what I'm saying is there is much more kind of high-brow thinking along possibly similar line that very many continuous objects describable as categories of particular categories of profiled sets. And here, from this point of view, you replace your dynamical system by a general dynamical system by all Markov system over it. So those provisions are quotient with all kind of diagrams which commute to this quotient map. And this is very much how it goes. And so this is now the definition. So what's the definition? So you want to represent your dynamical system as a quotient of a Markov system which is zero-dimensional. You know any topological space quotient, oh, compact. Because door space is quotient of a canter set. But you can do it together with dynamics. You can say that, but that's not a big deal. It's not so interesting. So what's interesting is that the relation which defines it. So you have a quotient space. So it says you have binary relation on your original space. So you have a Markov space. Forget about this Markov, that Bernoulli shift. And then you identify certain orbits. Now you do that. You identify them again by a Markov rule. You can see the system by itself, times itself. Now it's an X on a product or space by itself. And again, you identify pairs if this identification given on pairs or on final sets. So on each final set, you take this binary relation, generate corresponding relation on infinite sequences. And they say, I wanted to be equivalent relation. Extremely strong condition. And miraculous is the right example. So for some reasons, for one parameter dynamic system, when you have groups? It makes sense for any group. Any semi-group. For any semi-group. Now already for dynamical system, it gives you the following way of representing them. Very much in the spirit of small cancellation theory. On one hand you have a graph. This is directed graph. And then you identify finite pieces of the orbits. So there are this segment and this segment. And you attach square to this. And to this segment and to this segment you attach square. This will give you binary relation and say two sequences, two strings equivalent. If any time they pass here, they satisfy this relation. What you get, you'll be not in general equivalent solution. You can spend equivalent solution. But then you have non-house door space. You have very ugly space. But miraculously there are examples. Conjecture. There is nothing else but this miracle. All these conjectures kind of have a fancy fancy name of the genitive theorem. You have a miracle and then you say, ah, it doesn't repeat itself. And it's called the genitive theorem. First it was very exciting. Most of the genitive were good arithmetic. At some moment I think it's a ball. Of course we have miraculously kind of tight object. It's very unlikely it appears twice. But here, however, it's unknown. Conjecture, he is going in a way going back to Smale in a special form saying that if you have such a system and resizing space is a manifold then it will be one of the classical examples. It will be endomorphism when it's being manifold or something of the kind. On the other hand, I guess genetically a huge amount of such system where the phase space will be solenoid. It will be cantoset times line or something. And then there might be kind of parallel theory like small cancellation theory where in Smale cancellation theory we describe a whole group where boundary one dimensional and here it will be a similar thing. It will be a generic dynamical system. Generic such combinatorial scheme will give you such. This is none of this is known. And there are a lot of high score of problems which is unknown here and I mentioned only one of them. But this concerns that and this kind of more or less rethinking of the classical conjecture of 50 or 60 years ago development by Smale and also Boeing and other people on hyperbolic dynamics. But now what about other groups? So the example when the first definition appeared there was only one example and these were boundary of hyperbolic groups. Boundary of hyperbolic groups also like that. And they can be described exactly in a similar way. You can see on the hyperbolic group kind of the map to the space of self mapping of the balls or something. It is described in a way that is the mark of presentation of the boundary. By the way, I was referring to the question about mark of partition. So mark of partition in this context mean that this question map is finite to one. And the proof kind of extremely unpleasant because you artificially order some elements in the graph and using this to prove it and in exactly what is wrong you don't work in the right category. So this categorical thinking which I mentioned in the categorical description is just a dream. It was not implemented as a tool to prove theorems. Because eventually when I think if you develop it you get by audible profound theory of hyperbolicity. And now we are coming to the general case. So one example of that. But then when I was writing this article I just realized there is another nice class of examples. Namely if you take any semi-simple lig group which satisfies Marguerite's rigidity theorem. So you have to throw away throw away the real complex hyperbolic spaces as a factor. And this lig group semi-simple lig group divided by compact by co-compact lattice then the action of this lig group has this universality property. Why? You take semi-simple lig group divided by by Gamma and then Gamma acts there from another side. But this action has this universality property. Which means if you take this group this anywhere else. And this anywhere else you have a homomorphism from the universal fundamental group here. Then there is a unique continuous map implementing that. And the proof. Quite simple modular no known result by by client with consequence of quasi-asymmetrical rigidity of this corresponding spaces. And then there are other examples you can make. But still it's unknown. If you have just torus and there is a subgroup of homomorphism of the torus. We show them have this property. Some of them do, some of them don't. And this is one of the question which is unclear in general for this group and so in terms of the group this orbit and quasi-orbit what acts as the group? In terms of the group it's an in-automorphism. So the group of in-automorphism may have this property that quasi-orbit uniquely followed by orbits. But this orbit now represents elements in the really group enveloping the group. However this is not the final story because what you get topological structure how you can get geometric structure of the group. And this is not quite clear. I think I know what it is but I can indicate it is definitely so. It's canonically constructable. We know it's episterior. But there is the following way to think about it which you can see strangely enough most clearly in the case where there is no this rigidity, namely in the real hyperbolic case. Now in real hyperbolic space so you divide by co-compact lattice or even co-volume lattice. How we can reconstruct out of the group how we can reconstruct the space. This is geometry. Every way except of course for dimension 2. In dimension 2 there are modularity it's not that you have to throw it away but it requires more thinking. Starting with dimension 3. So what you do, you construct the boundary you know the boundary. Topologically. But then what you have extra. In your space if it's co-compact if it's not compact you have distinguished homology class. You have fundamental homology class of your manifold. So you have fundamental homology class. And this class is bounded in some way represented by measure in the space of simplices representing this class that is actually bounded. And this is an extremal one. An extremal one, some plotted on regular simplices at infinity. And once you have them you can reconstruct the full structure. So you have sphere at infinity you have topological sphere and you have distinguished configuration of points corresponding to regular simplices and this reconstructs all geometry. And the same nowadays we know that the same truth for all local symmetric spaces the fundamental class bounded there is a distinguished measure and this measure at infinity I think reconstructs all geometry but they never got out. It's hardly questionable. However, we still have very special examples. So the question is how we can go further. And so what will be the next steps over time? Not very much time. Because they wanted to say something to turn to complex geometry. What goes along? So I say it just works as an indication. An instance of a theorem. So there are two basic theorems there. One of the grower's theorem it says its solution on Levy on Levy problem and it's properly interpreted. It tells the subject about Kelley von Demenz group. Then there is theorem by Schuch. Together they imply as it gives a case corollary. I indicate some corollary. What does it say? Schuch's theorem? Schuch's theorem says in this special case we have a Kelley manifold. A compact Kelley manifold. A map to say compact a manifold is a complex geometry. The complex is divided by a compact lattice and you have a map which is rank at least real rank map more than two. Then the map is hematopic to halomorphic or anti-halomorphic map. A grower's theorem says that if you have a polyconvex, the manifold with a polyconvex boundary then all cycles above middle dimension are realizable by complex manifolds. This is a grower to the great theorem because a very soft condition makes the complexity of the boundary and it gives you cycles. Cycles as you know in the boundary area they function in origin. As I said to produce interesting you have to secretly solve some differential problem which may or may not be solvable. They shouldn't be there. But this gives you a soft mechanism in generating them. Not that they truly understand the implication I never thought to but this one corollary combining the two saying they're related. And then maybe say just one word all this construction which I know associates with some distinguished cycles. For example, manifold we have a group where there is top dimensional homology class and this class organizes the geometry of the class. And then there is an issue what are these high dimensional homologies? And this is very tricky because the fact that there are groups where there are ample kind of high dimensional rational homologies I forget about torsion of course there are lots of these final groups but in general it's very hard to produce specific examples of groups with interesting homologies. Again you start making examples pretty fast around other examples. And here is a conjecture because they were speaking about problems which related to that and it's one of the rather actively studied problems these days which is a version of the Novikov conjecture about high signatures. So Novikov conjecture itself is purely algebraic and says that homology of the group of any group which is conjected probably wrong it's too general to be true is expressible in terms of roughly speaking in terms of quadratic forms of a group ring of the group. Well I don't want to formulate more precisely but sensitively what it says. And geometric counterpart of that which is again there is no count example for the moment but probably it's not true says the following which I formulate in a very special case we have a spherical Riemannian manifold we have compact probably with a boundary a spherical Riemannian manifold so universal covering contractable so all I know. And then conjecture says in a rather tricky way which takes time to explain to what I'm saying before is as follows that if I take any complex vector bundle of this if I add it to itself sufficiently many times then it admits given an epsilon then I can take some of the bundle with itself so in K theory I multiply it by big integer then the resulting bundle admits unitary connection where curvature will be everywhere point-wise less than my epsilon so every element of rational K theory is a bundle this is a it's not as general as a Noigo conjecture but much stronger would relate to that but it takes some time to explain what it has to do with what I said before about structure of this structure of this cycle and again it's infinity of question comes from playing back and forth between these two domains I didn't say what was another essential part and this we can find in paper by Toleda in mine about Toleda hyperbolicity how you can reformulate and improve the few theorems in this kind of hyperbolic type of language very similar to how works in dynamics and in manifolds in geometry of manifolds of negative curvature but I have to stop at 45 minutes, right? I have 3 minutes I have 3 minutes you have more information I have 15 minutes I have 15 minutes great so let me say what is relation between this making of the structure this implicit volume and the Noigo conjecture so I want to give yet another question which is related to so when I speak about a simple volume of manifold or a homologous class but say fundamental class of manifold so what you do you look at all pseudo manifolds when I say pseudo pseudo manifolds meaning oriented pseudo manifold dimension equal to a manifold in map to my manifold and then I see how many simplices are there divided by D at the degree of the map right? so for any manifold you can see that all the pseudo manifolds map there and take a degree of the each map and divide the number of simplices by degree and then minimize this number and go to infinity when D goes to infinity so I have a number for most manifolds simply connect to get 0 but for interesting manifolds it's not 0 it's a number and it's an variant of your manifolds topological variant not obvious so there is a variant ever non-zero but apparently it's non-zero very very general class of cases on one hand but not covering just just say non-covering would be kind of p-p-carapinas exactly it's not covering any map but now let's do the following on one hand we say instead of pseudo manifold we say manifold this seems kind of little difference after all any pseudo manifold there is a manifold which goes there with about the degree by tom therm but this degree magnification is significant but now instead of number of simplices needed for triangulating it you take the minimal number of cell on a Morse function minimal number of critical points on Morse function there or the number of cells in the composition and also have an effect and now if you look at the first invariant then it's non-zero for all locally symmetric space of negative type compact space universal covering symmetric space with non-negative curvature without flat factors this invariant non-zero which was recently proven for generality I'm sorry I've forgotten the names they were proven after I wrote this paper I don't have it in my list of references but on that hand if you take a manifold on one hand you restrict class of object but now you relax it and say ah instead of number of simplices I just look at the number of cells in the cellular composition and then the picture will be different because again for all even dimensional manifold of constant negative curvature negative constant negative curvature all the formulas have symbolic spaces or generally for all those locally symmetric manifold with non-compact type which have non-zero early characteristics its invariant non-zero which is already non-trivial for product of two surfaces right and on the other hand if you take three dimensional manifold as we know of course it's false because this manifold sends you five over circle and the five over circle of cells doesn't grow and then that's all we know we don't know nothing more about this invariant and it's certainly related to the simplification volume and it's exactly a decor of this knowledge of conjecture it's just kind of quantification of the knowledge of conjecture because the proof of what I said goes through the index theorem applied to some twisted deregoperative there is no elementary proof of that even for product of surfaces the proof is elementary geometry is a simple system but there is the only proof is by index theorem it would be a very kind of amazing thing saying how much manifold differ from pseudo-manifolds and so the question of course about hyperbolic groups for example do all even dimensional manifold of negative curvature have this invariant non-zero or it may be zero yes even for very simple examples we just cannot cannot say because those which map to manifold of constant curvature that's fine but you know there are very simple examples of a different kind even when the boundary is topological sphere of negative curvature they are not simply related to manifold of constant curvature there are quite a variety of examples and we don't know so what happened there because the proof at some moment you is very special representation of the kind of dual Lie group to the semi-corresponding semi-simple Lie group due to Lucic so that's kind of question so what are the questions that I was mentioning so what are you going to mention I don't understand I don't understand that no, tori of course is zero tori of course is zero so for tori of course is zero it's a semi-simple group yes of course I'm asking even dimensional manifold of negative curvature celeron and celeron celeron and celeron no, it's unknown I mean it's only known being non-zero if there is a locally symmetric space of non-compact type we discreet it's being serious, it's not zero early characteristic it's obvious for covering because early characteristic takes care of that but it's true for all maps and the first example, non-trivial example it's product of surfaces so the proof kind of you don't need your representation just a very simple representation just at your hand but still you need index theorem or actually computation of a signature in coefficient in some flat bundle so it is a categorological computation but it's very bizarre question that when we come to manifolds say of negative curvature and we look at basic invariance we don't know if there is zero or non-zero another question of course which I mentioned for dynamical system of course appears for manifold or hyperbolic manifold where the boundary is spherical so again they are very limited for all we know and they all may have rigid geometric structure behind them but we don't know if it is true or not so there are groups many groups seem to go along with some very rigid geometric structure however we know little about that and some structures of course related directly and directly to lead groups for example we may start with manifold of constant curvature and then they can be fight covering or fight in quotient and still have interesting group or make more complicated surgery out of them we can build very very many things but they all come eventually from these locally symmetric objects and the point is that we cannot produce directly interesting cycles of high dimensional group for high dimensional groups so many for the negative curvature is really very even it's not so easy to produce they are there but for them for example all these conjectures are known to be true the cohomology is kind of simple in a way they are presentable by most flat bundles so cohomology is kind of group fully control the cohomology which is in a way similar to what happens in algebraic geometry where you know if you think about homology of manifold in etal terms it actually come from group homology of open the risky open sets so if you know group homology of all the risky open sets and how one goes into another all these errors from that you build your etal homology or homology of the group according to Grothendijk and the same picture appears here so again I think that there is no way to show this problem without really rethinking basic concepts in making them more categorical which is of course goes we lag behind what happens when people doing algebraic geometry but for example another question is again I can formulate it very precisely one of the properties of this hyperbolic dynamical system Markov hyperbolic defined combinatorially is the data function is rational so the counting function the number of periodic points and because it defined combinatorially you think they might be combinatorial proof and moreover everything indicates there is not a function but some kind of much more general object function in the variant category of graph or something like that which have this rationality however the only proof which was found at some point by David Fried is actually repeating argument with Markov partitions so you reduce the system repeat the argument we should do partitions use topological argument and you cannot do it purely combinatorially however this purely combinatorial theorem and actually about this categorical combinatorics which we don't know how to do that there are infinite number of problems they just cannot of course go through all of them you can look in my paper where it's written but maybe I emphasize again one simple and elementary problem if I had some time I can say what I wanted to say about Talerda convexity so see your theorem based on Bohner type formula Bohner type formula for maps from many fault from Kali manifold to other manifolds and this Bohner type formula rules out certain maps this kind of local computation however there is another approach close to what they indicated for how you reduce the curves in the case of Albanian theorem how to use it to classical Jacobi you can look what happens to curves and you observe that under curvature conditions where SU theorem is true the following more general property is true if you can see the map from a curve it's remaining in manifold to another manifold and observe first energy of the map is conformal invariant It's invariant of a complex structure. So if you consider a demon's surface, we are dealing with the magnetic. You can see these maps to another manifold. And look at the Dirichlet energy of that. This energy depends on conformal structure. So harmonic maps and minimum of this energy is conformal invariant. Therefore, becomes a function on the modular space of curves. So every remaining manifold and homotopic class of maps from a given curve into there give you a number. It is a minimum of this energy on this modular space. This function is true for every complex structure, for any conformal structure. When you vary this, you have a function on the modular space of curves. So every remaining manifold as an invariant now, and given these curves and homotopic class of maps, give you a function on the modular space. And this function is what is called Plutius subharmonic. So it's kind of convex in a homotic sense. Its restriction to any curve is harmonic, subharmonic, kind of convex. And now we can forget about complex structure. And you think, this is the key property from which everything follows by rather simple arguments. And so in this very of the same spirit, as I said, what happens to orbits. So what characterizes orbit of this way under the stability is that function between orbits have this convexity, this weak convexity. And this is the convexity in the complex sense, which, unfortunately, serves only true for closed surfaces. There is no counterpart for surface with boundary. If it were with a boundary, it would give you much more power, much more control about what happens. But still this kind of interesting. And for particularly geometries, besides those described in where it was observed first, I wonder why it's true. For example, if you take natural geometry from two-dimensional polyhedra, small cancellation group, whatever, you may have this property. And this will tell you something more that you usually get by other arguments. But this is certainly quite, quite, I think, basic concept in this theorem of Toledo, emerging from this theorem of Toledo. And the final, because I mentioned in my reference, I have another paper by Toledo which I received. Yes, that's the final. The final paper by Toledo I received a week ago. They eventually produced examples of algebraic manifolds, compact algebraic manifolds with negative curvature, which are not coming from the complex ball. It was a long-standing question. And the obvious candidate, we are removed covering over totally the genetics of manifold. But to constructs removed covering, you need to find a group. You have to know the corresponding group as subgroup of finite index. And they proved that the group is easily finite. And therefore, you have lots and lots and lots of manifolds of negative curvature, which are not coming from the ball, and they are cally and moreover, they're complex algebraic. And this is pretty nice. It is expected, but still it's a usual problem when you construct hyperbolic groups. By various constructions, you're getting stuck because you don't know the group is easily finite. And you, constructions don't go. Like there will be five covering, three-rate constructions. And here they overcame this difference, which, of course, it shouldn't be there. I mean, in a way, the construction must go anyway, but in a more general category, which is not at the moment well-defined. So maybe it doesn't be time yet. OK, thank you very much. Are there questions? No questions. How come? Yes. Yes. So in general, in dynamic system, what is the analogous, the modular space? In dynamic spaces, you have your torus and you have different anamorphism. And so it's a group of the torus, so a group of automorphism acting on this modular space. How to get modular space? Modular space of the torus is kind of a symmetric space, mod gamma. And this gamma is a modular space for the dynamic system. So they're really coupled. They're kind of two parts of a general picture. And on the case of the SU, there is no modular spaces. The map is just in the, because it's strictly negative curvature, it's just only one structure. Space is rigid, right? And therefore, and this, by the way, has a lot of questions related to that, which of the spaces is rigid or not? Immediate question appears when you have other manifolds. How you can get other manifolds in the same fundamental group? You just take a generic surface there. You take the plane of the right dimension. So it's in high dimensional, that locally have a modular space, you have a surface. So generic is completely, and then it can move. It's very movable. When you move it, of course, it has modular spaces. Conjectually, the only modular it has. Again, there is no geometry there, maybe somebody knows it. But this again, according to this logic, there shouldn't be exterior, extra, extra deformation. This extra deformation is like external automorphisms of the group, right? Like this, a group of high symmetries, and like surface groups, have no outer automorphisms. And here it corresponds, there is no outer deformations. So everything here is strangely enough, counterpart there. It just says there was slightly different linguistically. And you arrive at questions. And again, this logic, it's not fully justified. Example would say you must be careful. However, that's a very interesting question, algebraic geometry. What are modular spaces of this hyperplane section of this complex algebraic variety? And most likely, the modular space is exactly only the one which we get inside of there. There is no other deformations. Of course, I guess that it may be easily done by people who know the ropes, how to do that. But then, once you have it, it has against them good rigidity implication. Of course, it depends how you respect the rigidity theorem. My respect for them can weigh more and more and more than the general overview. But the question is to construct new objects, yeah? Which is more interesting here. Thank you. Question, question from line. Seeing any question on line, I don't know. OK. Yeah. So you mentioned several times there's small consolation phenomenon for this model. The dynamical system, right? Yeah. So what kind of applications do you? No, you can produce. The question is, using small consolation on random groups, you know the exact. You do produce groups with certain properties, right? There are groups obtained by small consolation. And these are groups in your hands about which you can prove something. There is no such effective description of this dynamical system, of this hyperbolic system, yeah? So, of course, it doesn't, it wouldn't describe all of them, only this low dimensional one. But you, but they look superficially, they organize exactly small consolation groups, right? They are not groups, they kind of collect semi-groups because their graph is directed. And this kind of your two cells are also directed. So it's a two dimensional complex with extra structure. It's much more rigid object, right? Because it can respond roughly not to hyperbolic group, but to solvable groups, so to speak, yeah? But the question is, well, I can say this was all definition, but I cannot work out specific examples, look at the group, compute the entropies, et cetera. And this is kind of obvious thing to do, compute the, effectively compute the entropies, the zeta function and it is a world of nobody looked into. There's definition and that's it. And this somehow was forgotten by people of dynamical system because they focused on smooth system. And they're definitely non-smooth, right? And the problem which I left behind, the topological dynamics beyond resources which exist at the moment, yeah? And in my view, we have to redo the field in this more categorical way and then have a new perspective on that. It's certainly very, very old fashioned, but the beauty of that, unlike kind of what is done in dynamical system in a classical, traditional way, is categorical theory. It's very functorial. You just do these and things come by themselves if you do it right. And this is exactly what happened with geometry. And this was, I think it's interesting. And it may illusion, of course, the similarity may be an illusion because we, in our mind, we make things similar. Maybe it is similar, I don't know. I made this formulation personally. Of course, that sounds very similar, right? Maybe it was my cheating, right? I don't know. They look very similar. I put it this way, right? Who knows, who knows, who knows. But again, anyway, regardless of what I say, these are two great theorem, and especially the theorem of Franks, I highly, highly advertise this remarkable theorem in his simplicity and universality and potential for generalization. I think it's a remarkable theorem. Okay. Yeah. Okay. Thank you very much. There are no more questions. So, thanks again. Thank you. Thank you. Thank you.