 So, yes, now with the internet, when you start thinking about something, you start surfacing the net and see what is known about that. And every now and then I was hitting the back by Jean-Pierre, and in particular this one instance of that. So I took the subject, because if I have time, I arrive at a problem related to his work on spinners. And about spinners, again, the lecture, yes, Michael, he gave me personally about 20 or 30 years ago a meeting with some hotel during the breakfast. You don't remember that, probably, but exactly one of the things I remember that you're not supposed to understand spinners to back with them. And I felt very good, because I never understood spinners and still don't understand them. But if Michael didn't understand them all, what about the rest of us, I mean? And so just to encourage you, just make some kind of progress in this direction, and I try to convince you that you don't understand many falls as well, so. And the geometry is supposed to take a square root of. And this may look absurd, but strangely enough in science, sometimes non-understanding is more essential than understanding, just progress in non-understanding. And this joke made by physicists, I repeated it without truly understanding it, it's a problem in fundamental physics that Newton discovered the law of gravitation. He solved the problem of two bodies, but he couldn't solve the problem of three bodies. And then relativity came, and now even the problem of two bodies became unsolvable. And with quantum field theory, we don't know even what vacuum is, so. And that's as good as we. So where it goes. Now, but a geometry, I started something super, super, super elementary. And so here, it will be something about scalar curvature. And this will be about convex polyhedron. And I want to bring them together in this billiard. So what is scalar curvature? First, you have to know what is the Riemannian metric. And I take very kind of formalistic point of view. So the Riemannian metric is just positive, definite, quadratic, formal, and manifold. And everything will be local, this opposite to, as I say lecture, but Michael, I want to be really very local and the question will be globalized kind of rather unexpectedly. So the metric is just at every, it's a field of quadratic functions on a space. So to each x, we assign a quadratic function f on the tangent space, which is in the case of Euclidean space, the same space. So just in the case of quadratic function. Quadratic, meaning it's quadratic on every line, and smooth. It's not just margins, but quadratic. And of course, you can say how it's just family of functions. You can say if it's Euclidean space. The main Euclidean space, you just have so many functions. Yes, real, varied functions. But the point of course, it transforms differently, right? When you change the coordinate, it transforms differently. Now, what is a scalar curvature? And then there is a fact of algebra that there is unique up to scaling differential operator of second order from the space of Riemannian metric to real numbers, or to functions. Yeah, so this Riemannian metric to functions from the manifold, say manifold will be real. From functions here, it's a second order differential operator, which is linear in the second derivatives, invariant and the action of the film offices. And there is unique such operator up to scaling. So there is a question how I normalize it. And the normalization is that if you take unit, so normalization for two-sphere, is that for two-sphere, it equals two? Because two-sphere, for unit two-sphere, for that reason it's two. And then it's additive. When you multiply to many forces, surfaces, then it is this additive, it adds up. And essentially, all you have to know. It's uniquely determined by that. So you don't have to understand, of course, nothing of that. You just say this verse and that's it. Of course, it's also defined in a metric that has a positive definite, but then it's not so interesting. So you have that. Now, yes, to make some sense of that, again, reminding many foreign curvatures a tricky point. But if it's a surface, it's just twice Gaussian curvature. And if you have a general hypersurface, so what you do in high-dimensional space, you have hypersurface. And so what you do, you intersect it with three-dimensional planes kind of normal to this. You have a surface now, piece of a surface everywhere. In this surface, there is Gaussian curvature and average of all this over all the sections. So at every point, it's now defined dramatically. That's enough for purposes I want to discuss. It's this concept of scary curvature that I just sophisticated enough. Now I want to formulate a theorem in a kind of a way. We all want to understand the meaning of that. We want to understand if it has some geometric meaning and for geometry, meaning means you have to throw even derivatives. You have to express in the language that there will be no derivative. It's too complicated. It might be something more elementary. And one way to think before starting formulating a theorem, I just say how we can kind of before throw even derivatives. And you can say, imagine your magic is only continuous. It's not smooth. Then can you say something if this curvature makes sense? In particular, if some condition on curvature you studied and the condition may be, say, scale of curvature, people either study it, say, positive or say negative, or it may be, of course, any constant on the sideward. It has geometric meaning. Before you start thinking about this meaning, you want to say if it is conditions stable on the limits which are only uniform, not respecting derivatives. And motivation for that, at least for me, comes from how it was developing in the simplex geometry. And there is some kind of way of similarity when it was a question raised by Eliasberg. If symplectic, if your morphisms invariant being symplectic on the uniform limits, and this was extending the work of Poincare, he eventually proved they are stable. And that was the starting point of the big development in symplectic geometry in the modern time after Poincare. There are two kind of big steps. One hand by Eliasberg and then by Rabbi Novitz, who actually proved an equivalent result in different languages. And then there was all this development followed with that. But in the scale of the curvature case, it seems that my feeling is there is a similar kind of theory, but it's never went as far as it should. And so I want to, and the reason is because we have to modify, I think, the concept of manifold. That's the problem with this. So this is a question you want to ask. And then you know that this is out of questions. Because for dimension bigger than 2, there is so-called log-camp, I call this confused about spelling, in which H and K. I think it's right, right? Correct, yeah? Log-camp H principle, who proven that uniform limits completely destroy these conditions. So any metric can be approximated by metric of negative curvature, scale of curvature, or even negative rich curvature. So there is no geometric message in negativity of curvature. But for positive curvature is different, and this what I rather recently approved, that it is stable under the limit. So the space of metric of positive curvature, or bounded by any constant, is stable under uniform limits, which means semi-continuity, of course, with that. Which means that there might be a geometric message. Of course, it was known before. There were some results, but they were of slightly different kind. Now, and just the easiest case of that, which I wrote a while ago, was base, base, and I want to formulate some very, very nice result by La Roule, I think I'm just talking about spelling. That was his thesis on the Blaine Lawson. And the kind of, hmm? I think it's too early on the beginning. It's too early on the beginning, I might be. Who was Blaine's Lawson student at that time, because this is who's proven a very simple looking theorem. And this is follows. So I have a sphere, and you have a standard metric, and you have some other metric G. And such that this metric is strictly greater than that. And for the manual metric, because the square root has not been taken, inequality makes sense. You know, even when we take square root, and you go to complex geometry, I mean, once starts when again there is positive curvature on a high level, right? Complex numbers have some positivity, right? Built into them. And so, positivity never disappears, yeah? You try to kill it, and it comes to a new kind of guide. After all, intersection is always positive between complex manifolds, yeah? And there are many kind of generations. So I imagine I have that. And then the theorem says that, scalar curvature, there is a point, S in the sphere, such that, this was standard metric, and this another metric, there will be points in the sphere such as the scalar curvature of G at this point, X, will be less or equal than scalar curvature of G naught at this point, X. So when you enlarge the metric, at least at some point, scalar curvature must go down. And this is a kind of simple-looking result, even when this metric G comes from this example of hypersurface, the hypersurface around that. There is no simple proof. It's very simple, but not elementary. And it depends heavily on dirac operator. You have to produce particular vector bundles, take twisted dirac operator, compute some curvature, tie in the proof takes one line, if you do it right. But, and there is no kind of any other way to prove it. And just, and this, which I want to indicate, it has to do something with global result about spheres. There is no topology involved. There is extra topological point, which one can do. And of course, it's better to formulate maybe in this more stable way, which is needed. Namely, it's better to say this way. I have a sphere cross torus, flat topological torus. And this maps to the sphere. And here there is a metric G. And here is metric G naught. And if this may map its distance decreasing, then this metric must have some way, curvature greater than that of the sphere. Or smaller than that of the sphere, I'm sorry. The bigger the space, the smaller the curvature. And so you can add the torus. And so topology is not as any manifold. But it might be topological torus. So topology is kind of hidden, but it's very elementary topology. No characteristics, classes, nothing like that. So this index theorem appears in a very kind of hidden way. And it's very unclear why it is there at all. But now what it has to do is this limit question. And then this theorem can be combined with another kind of idea we used in Scaly Kovic, which was brought, yes. So Gira Cooperative was brought there by Norwez. And in fact. But in fact, Gira never had the idea of this operator. It was your operator, right? Gira has only one on flat space, yeah? In the indefinite metric. The people with relativity theory knew how to extend it to curve space at that time. They knew it at that time, yeah? Because, so another idea was about the idea of minimal surface and Scaly Kovic. Let me explain this in two words. So in this Lichnerovic idea, Lichnerovic approach, is censored as a bochner formula. Because there are all these things, the periodic elliptic, tata, tata, tata, but they would be completely kind of vacuous, if not for some particular kind of formula and some identity and positivity of some expression, right? And so the same for minimal surface. It's simple. The first instance made by Sean Yao. And I don't know, I guess it was a similar thing we've done by Phyllis, but unregoriously, that if you have a surface, a minimal surface inside, inside of a three-dimensional Riemannian manifold, it's a minimal surface. And you can see it's pushed by a normal field. And see what will be the second derivative of the area. Because minimal by definition, the first definition of area is zero. And the second derivative is, would be just integral of the Gaussian curvature plus or minus integral of the ambient Scaly Kovic. And this is this formula. This is similar to the bochner formula. I want to say that in fact that, and once you know that, it follows that if many fold, ambient many fold have positive Scaly Kovic, this surface must be sphere. It cannot be anything else because it allows beneath theorem prevention from doing that. And then it's immediate to show that, for example, if you have torus, three-dimensional torus, it cannot have metric of positive Scaly Kovic, because you will have minimal surface, which will be tori or hygienist, and you have contradiction, right? So this is another point. And now the two things, but then there are kind of some elaboration of this idea. Now I want to say the first thing which I knew about that was that if you have metric of constant curvature, so I have many fold and I have G naught of constant curvature, then Scaly Kovic was semi-continuous at this point. So if it wasn't constant curvature, you can't approximate it uniformly by metrics which have more positive curvature. All near biometrics may have only smaller curvature. And the point was, consider not so much minimal surfaces at around this point, constant curvature, but consider these spheres around them. And if you perturb a little bit the metric, they are not minimal, but as minimal bubbles, they remain stable. They kind of got to disappear, like for the same reason, essentially, as minimal surface in the torus, this equator is stable. And therefore you have this kind of minimal bubble and similar argument applies. And then you can show that not these spheres themselves around them, but this time circle would have Scaly Kovic greater than the sphere. And therefore it cannot, so if you gain some positive curvature here, it will be gained here and you have contradictions a little. So you can derive this local result which is just limited near, obviously near small point, but from this side of the global thing, we shoot the whole thing of a category. Index theorem and full kind of power index theorem, right? And because here there is a circle, the torus, you need not just usual index theorem, but index theorem for families. So it's really the whole thing amazingly goes to this local stuff. And then, yes, for many years, I just couldn't generalize for general thinking, eventually I kind of managed to do that. And then came from completely different kind of way of thinking, which I actually was not thinking about that and now let's look at the second. Part of this, that's about Bill Wilson convex polyhedron. So we want to understand convex polyhedron. And so I just said, understanding means we just to show we don't understand that. Right, so we want to show something we don't understand. So one question is somewhat amusing because I remember when I thought about that, I found something on the net and I thought it was obvious and so I didn't register it. But then later on I realized my obvious wasn't correct. And the statement is that if you can see the old polyhedral, where all the hydral angles bounded from above, so they all sufficiently acute, then there are only finally many combinatorial types. For convex polygons clear because the sum of angles, the more you add, the sum grows. And also it's easier to see that you have really acute angles. So there's a pi over two. There might be simplices. If you allow pi over two, only have products of simplices. But then it's not so clear. At least I couldn't find, I asked all people around and nobody knew that, even people who wrote textbooks on that. But I remember it was mentioned in some survey and I couldn't find it anymore. But this is not exactly, this is a kind of relevant issue, but another property which you want to know anyway, what are possible angles, the hydral angles of this polyhedral. And then in this respect, there is one result, very simple, and just immediately raises the question, is for simplices, just for triangles, of course there are obvious but already for two simplices, simple but non-trivial, is that if you have two simplices and you, and corresponding the hydral angle in one, greater than another, in fact the two are congruent. Oh yes, hematheic. You cannot enlarge the angles. And this is just, as we shall see, the same, exactly the same phenomenon. Right there, the side of the same phenomenon that you cannot enlarge the scalar curvature of the sphere. And so this is what I want to explain in this bill. And this is a very old result, it's due to Kirchbaum theorem, it's called Kirchbaum theorem, where it's lemma kind of, it's kind of, it's kind of algebraic. Yeah, it's just positivity of the scalar product. It's not just seen geometrically, because here there's some of angles like that and then you see it. And here it's, well, one line per rule but it's kind of, you have to write some simple formula to do it. But the way I get this count is this is some kind of, well, maybe a much development associated in different direction. Okay, so, and for example in particular I don't know for what are the polyhedron when the same remains true. I take a polyhedron and imagine I can deform it such that the combinatorial type doesn't change but all the radial angles go up. Is it, although if it's possible, except for moving some faces and parallel one to another. Say for cubes, yeah, it's also true for product of simplices. The same result, for example. So I make modified square like that but that's all in the same high dimensions. But then my next question was, okay, this is a convex theory, it's two kind of two regions and you want to say something more dramatic and allow curved linear polyhedron. In curved linear, the point of course, naively you think that, so here say I cannot make angles bigger or angle smaller, this is how they say. And so it's emphasized when I can make angles smaller or when I cannot make them smaller. And it's clear that if I allow just convex bending, yeah. So I have something polyhedral but face a convex. Then angles may only become bigger. How much of this is true in high dimensions? And convexity is again, is two rigid condition. What I want to say is mean curvature convexity. Which means my faces must have positive mean curvature and positive mean curvature in this direction means when I push it inside, it gets in area goes down and when I push outside, it goes up. In smooth category, this is the same. But in the more inspiration I want to go, it may be not a priority in the same, yeah. The fact that it goes down here doesn't mean up abstractly it goes up there. But for a smooth surface it's like that. And so the question is, now, is as follows. So I have polyhedral in three spatial, high dimensional space, three spaces, we know slightly better. If you can deform it, so keeping this mean curvature of the faces, so the angles become smaller. And conjecturally, for the most polyhedral movement, never it's possible. You just, if you do that, I'm just the only thing is parallel translation. And well, sometimes you can prove it, sometimes not. And then by actually this theorem about parallel spinus may enter, or as of here. So the case when I can handle is kind of almost, yeah. Is just like taking a cube. And this will be relevant to our problem. If it is a cube, what I can prove that you cannot deform it in such a way. Namely, whenever you make them faces, with a positive curvature, some of the angles will go up. But that cannot prove that the extremal configuration necessarily flat. That's the usual thing. And this is exactly where you need this thing about parallel. Think and just you have to modify the concept of Dirac operator, I guess, and spinner. So, how it goes. And again, this even, this kind of result, the proof depends on the use of the Dirac operator. Maybe in this case you can use because there is actually recently there was some paper block hump when it does something with minimal surfaces. But I think it's questionable if you prove and know what you think about his work. It was solid or not? I think it's, as far as I understand, it's solid. It's solid enough by now, yeah? Well, I guess he's fighting with some people there, but I guess we shouldn't go into those details here. Yeah, no, but it's a, no, but myself, I didn't read the paper, I don't quite understand them. So I can't watch them, and then kind of unpublish. But those are, there may be alternative just way to handle that with minimal surfaces, but Dirac operator is applied here and that. So what we know is actually this falls from a hump theorem that was known before, that if you have a torus and torus, then this was conjectured by Geroch, and I wonder exactly what was his motivation, and then solved in dimension below seven by minimal surface, but yeah, I'm sure. And apparently the same method now by Loham, but in between with Blaine Lawson proven for all dimensions using twisted Dirac operator. So it admits nomadic or point of scalar curvature, nomadic G with scalar curvature which is positive. And actually if it is zero, then it is flat and this actually rigidity is tricky in what I'm going to say. So if you have such a creature with positive mean curvature on faces, you may start reflecting it. And when you reflect it, it kind of develops and you can close it up and become a torus. And then it's not hard to show and this again was shown with Blaine Lawson, we proved it, that you can smooth the angles and then we strip little point with the corners, it will have positive scalar curvature. If this mean curvature was at some point strictly positive, it will be strictly positive mean curvature. But if it was minimal surfaces, we did quite clear. And then we will have some kind of singular objects with abstractly positive scalar curvature or non-negular scalar curvature to which usual techniques don't apply. At least, and then it says I come to that later how we can handle that. So this kind of, this kind of a thing, after reflection, become like a torus and you can, and if you can smooth it in this category, you can apply either of the technique either scalar, either minimal surfaces if they work or dero-cooperator, twisted dero-cooperator. And then this is a point about what you can say billiard. Because if you think what happens, how it looks inside of this, so you reflect it and close it up and you kind of think about this as a torus and then being fundamental domain in the torus. And look at minimal surface in the torus. So how do you look here? And this will be tremendous mess, yeah, it will look a little bit like orbits of a billiard and in dimension one it will be billiard. And it's even more tricky for dero-cooperator because the boundary condition here is not the one which can be handled by dero-cooperator, right? So you up-say on the torus, if you use either, you use family of twisted with a family of flat bundled dero-cooperator or with some non-flat bundle. And so it's in a big space, something happens. And so what you see here is extremely tricky. It becomes extremely condensed picture and so what is this and this can be billiard there. And so I will explain later how probably you can understand this, yeah? And then again I say you have, you'll have to modify concepts, concept of manifold. Now, how this can be used? What it has to do with our original question. So now let me come back to that. So we want to understand the matrix. So this is the theorem which I want to prove. On this side, e that I have a remaining manifold and I have family of metric such that they scale the curvatures are greater equal some function on v. Then and they, this metric uniformly converge to g. Then the same is true for this metric. So scale the curvature at every point v with the same condition, right? So if scale the curvature were greater than something on all these metrics, the same inequality will be true for the limit. So it's semi continuous. I keep forgetting from up or below. It's a principle thing impossible to remember. And then, so what has to do with this, yeah? So you want to characterize, you want to characterize sign of the curvature in terms of something inside. And this must be kind of on the middle size. And just for that, it reminds you how you do it for usual curvature. And for usual curvature, one of the way to characterize it and was kind of implicit in the book by Robert that it is the sum of angles of the triangles, yeah? And so in particular positive curvature dimension to characterize, but if we take any such triangle, the angles will have tenders to be bigger than the angles in the flat case. And for negative, negative, smaller, but negative it doesn't have extension. And here, so what you want to do is we combine this Euclidean picture with Riemannian one and you say, huh, positive scalar curvature is characterized by the fact that whenever we take inside this kind of domain, and in fact, what is, we can prove it from a larger class of domain, it's cubical domain, it might be cubical only in combinatorial sense. Downs have to be topologically cubical. There's formally cubical domain. And if all faces have positive mean curvature, then you cannot have all angles less than pi over two. And this would be a character abstract characterization of positive scalar curvature. And this makes sense. It don't need much regularity of the space because you can define positive mean curvature as I said. You don't need any regularity of the metric. You don't have to say the word curvature. You just, how it behave on the inside to outside variation of area. So I have to notion of area, but here there's subtle point. So what you call kind of inside and outside variation are the same. And so it's one of the easiest, which was occupying me, what are singular spaces and this of say positive or whatever scalar curvature. And when you arrive at this question, I guess the answer is there are not spaces in the usual sense here. And this way you need to change concept of space. However, so what you can do here, you're stealing this kind of naive setting, the point is that if you have a point of negative curvature, scalar curvature, if there is such a point of negative scalar curvature, then around it always there is such a cube. You can surround it by a cube where all faces will have points of mean curvature and all angles will be smaller than pi over two. And this little algebra you can just construct it using a particular frame. And then if you could approximate by point of curvature, you arrive at the contradiction with the previous thing, right? So but, so what goes into that just, excuse me, my little remark, it's so inside of this, you need some geometric measure theory, right? Because constructing this construction require construction of surface of point of curvature, which is not completely as simple as you may think, and just like an instance of that of some different nature, where this problem enters in here in a slightly bigger form, measure and take euclidean space, and you just picture this metric a little bit in the semi-informatopology. And then the statement, you know, the usual, the first euclidean space can be exhausted by balls, with boundaries, with convex boundaries. And by geometric measure theory, it follows, you can now, if it was more perturbation, you can, with respect to pneumatic, you still can exhaust it by similar shapes of points of mean curvature. And there is no elementary proof of that. You have to go through whole, it's this problem, no geometric measure theory, constructively we use regularity, a theorem of Armgren and Allard, to show that we really smooth surfaces of points of mean curvature. And this is very strange, you cannot, this looks very soft question, to be exhausted by this point of convex, convex is looking surfaces. And there is no simple proof. And the same thing enters here, and the interesting class of problem appears that you have now this kind of polyhedral, which builds a, where boundaries are the minimal surfaces, or they kind of solve a bubble of constant curvature, and you have prescribed angles when they meet, for example, 90 degree angles, right? And this is obtained by consequently solving the plateau problem. So, first you have something like that, minimal surfaces, then you take minimal surface like that, then you have minimal surface here, but then there are singularities at the corners. And it is, you know, the singularity of the boundary where the problem in the corners may be enormously unpleasant. If you just, when the corner gets in deeper and deeper, it becomes horrible, kind of a horrible mess, and even for linear equation, for Laplace equation, it's not understood. Actually, I spoke to some expert on that, who wrote, I said, two volumes on that, and they're very complicated estimate, I'm asking, why this kind of estimate? Why not in this case? Well, it's unknown. All simple cases are unknown, and they're known to be missing, but the sharp result in simple terms are unknown. However, in this situation, you can bypass them for some reason. You can go around them, and this will go there. You know, a few words remaining about, why I'm saying it's something wrong because there are spaces in many forms. So, you want, it seems that all the singularities, which either appears to be minimal surfaces, or they suggest that you have to have spaces which have nothing regularly in them. They might be singular spaces and still have scalar coefficient. And when you start doing that, it becomes kind of very unpleasant to prove anything. And then, if you hope it will work, you will have to have a full measure, a geometric measure theory there, which is conceivable, but then you have to have derock operator which looks kind of very unlikely, to have it on two singular spaces. And the space may be extremely singular. For example, you can produce super, you know, fractal spaces which is in some sense of positive curvature. And derock operator should be there at least enough to say that it's under some conditions possible in some possible. And then there is the idea which I think might be elaborated in due to maximia. And I don't know what maximia I can comment on that, is that how you can think about many faults. And one of them is to say that many faults is a contra variant factor from the category of graphs to category of measure spaces, right? And this is a really nice way to think about many faults. Right, it looks good. It's a point about Ricci curvature. This stuff was about. Yeah, exactly. This was applicable, and now I want to indicate kind of Ricci curvature. It's related to Ricci curvature. But the point, again, if I want to say this for the audience, what's maximum? No, that, so what's so good to say it, yeah? When you say these words, right? So this being functor contains kind of a volume of complicated green type formulas, right? You write all the integral formulas and they own this word contra variant functor. It's an extremely concise way to have the whole book, yeah, you just read the book, 100 formulas, and you can say, huh, all they say, the certain thing is functorial. And so what is this functor? And this functor for many faults is quite simple. When you have a graph, and graph with length being assigned, and here is the remaining manifold, you assign the space of maps there between the measure. And that's the object. And this knows all information about Laplace operator there. All green function, all fundamental solution and then because we know measure is defined here perfectly in this situation is kind of, not conditional measure, but honest measure is also functorial. And now what about Dirac operator? So in this thing, I think only half of the Dirac operator is defined. You cannot quite say what Dirac operator is, but you can have kind of a bound, a lower upper bound on this operator. And the point is when you map, a graph, this is my speculation, this is my speculation, maybe not true. When you have a graph going there, you can pull back connection and define along this graph, even if discontinuous. And at this vertices, you know there's something, this is kind of bundle, tension bundle, but because at the vertices, you know this correspond to this edges. And so you can incorporate in this picture the Dirac operator, or rather connection and therefore spinus. And this requires less regularity. It's partly integrated. The point is that you already integrate connection, so you don't need this regularity. It's already kind of in there, but again, I don't know if, and then interestingly enough at this point, you also can incorporate information about minimal surfaces. And they come together because when you have hyperservices there, they control how graph intersects hypersurfaces. So hypersurfaces will be reflected by some kind of points in there and this family. And I guess, or guess I dream, you can completely forget about many faults, shingling and shingling develop this kind of theory and then everything become kind of very nice and there will be no many faults. So generally Mark, many faults is Michael mentioned, we are, I think the way we know it, we invented it by Herman Wail about 70 years ago. It's a long time. I mean. 100 years ago. 100 years ago. It's a long time. It's time to have something new. Move one. Herman Wail. Oh. For being foundational. So it's time to invent something new. I mean, and the basis I believe there, what is developed in both cases is kind of direct and bohno formulas. And for minimal surface also, there will be purely positive equation. You see, there's nothing square root there. It's completely hidden. And they must be part of a bigger body of some, if you have this algebra and somehow will be unraveled in this kind of factorial picture. But this kind of a dream and well, I just, I wrote nautical get writer to give more justification of them. Of course, I don't know. Any other question or questions? Misha, how do you make sense with the condition that the manifold is spin in this context? Well, you see, in many cases there, you don't need spin. Spin being suppressed. Yeah, of course, for the Lichnerowitz type theorem you need spin. But in this geometric theorem, spin is irrelevant. It's only just a problem. We know it, if it comes true, it's a need. So Dirac doesn't operate on the spin field? Yeah, it will be not a Dirac operator. It will be something. It will be not an operator. It's something that remains of Dirac in this picture. It will be less than Dirac was sufficient to encode some of the geometry of the space. But this my understanding, yeah. So about spin, of course, it's a big issue. And you see all the time this issue appears called dimension two kind of play a special role here. So I must say, I don't know the answer. It would be the most honest answer. They don't know it here. But partly maybe it is. But the way she started Dirac and the technical question is, what is the behavior of the spectrum of Dirac on this weak limits of mathematics? And this we have to understand. So what remains of Dirac under this weak limit? And when you spoke about the angle, somehow you jumped directly from sectional curvature characterization to the scalar curvature leaving out Ritchie curvature as an intermediate step. That's correct. But somehow one would maybe think that if you look at this for Ritchie curvature, it might be easier in particular in terms of the recent developments in terms of transportation inequalities and stochastic process which might also be high. Now they can make the general kind of remark why I emphasize scalar curvature. Yes, from perspective I take that kind of thing you want. You have a maximally soft environment like clouds and still to find some structure in it. If you're listening to something rigid like Ritchie curvature or sectional curvature, well, you know there is a lot of structure there. I mean, it's not so much surprise to add more and more. But here in scalar curvature, it's absolutely like topology, almost amorphous. There is nothing. All that symplectic structure is very, very soft. And then amazingly it hit the wall. And then it hit it something remarkable happens. And one instance, of course, for many faults. Topology is very softish. Actually originally this was kind of was upon correct. People were kind of saying, what do you do? It's a kind of flabby thing. Why do you do this mathematics? But then condense and become a weather rigid. Then it will happen with dimension four. And then another instance, symplectic topology. And now I think it's that. So the whole point to start from something extremely soft and then must condense to something rigid. And Ritchie curvature, not against it, but it's a different kind of mathematics. It's already very structural. And the right relation, of course, I think the Ritchie flow equation has something to do with that. In some cases you can do with that. Ritchie curvature does enter. But not very again in this setting. It's exactly as Maxim said. Ritchie curvature is seen in this picture, in this pictorial picture. And this transportation measure would have been implicit there. But not by itself. This use sectional curvature only as motivation, I guess. Because many statements from there can be transported to scale the curvature. Sometimes conjectually, sometimes it can prove them. And, but Ritchie curvature, I guess, again, I can discuss it, but this is actually first, I thought about Ritchie curvature in this context before I turned to scale the curvature. This is really a comment. I learned from a physicist recently, which I'm not very interested in. It has to do with combinatorial polydra and dirac operate. You try to set up, for example, triangulation of manifold and then imagine it getting smaller, finer and finer, and they get to the limit. Then, of course, you try to set up this up so that you get to the index of the dirac operation at the end. Only succeed if you put the index at the beginning because for finance systems, the index is always fixed. So it was always a mystery to me how we could get from a combinatorial picture to it. But the physicists, they answer. For physicists, if you take a combinatorial structure, you can write down and dirac operate formally and write down a spectrum, and then what they should look at, for instance, is not the zero eigenvalues, but what they call the low-lying eigenvalues. It's not a high eigenvalue. And these are, of course, a very vague notion. And the idea is that when you make a sort of finance, then the low-lying eigenvalues get closer and closer to zero, and the other eigenvalues stay in the sky. And this way, the answer is not to look at the zero eigenvalues from a combinatorial. You don't need to look at small eigenvalues, whatever this means. Now, of course, the scalar curvature enters in precisely in estimates to do with the zero eigenvalues. So that makes some kind of link with these ideas. Do you think it's very easy? Yeah, of course, it's this kind of thing. But my point is that to try to move to the direction of what Maxim suggested from concept to point space. It's another point space of, actually, even metric spaces. We also are thinking as functors from very simple categories, as a category of two point spaces, something. And that gives you a certain perspective, and also by passing singularities. Because Maxim has another occasion of how to bypass singularities and Poisson structures by a certain kind of algebraization. And the same happened in symplectic geometry. Relaxation, you can bypass singularity, but you have to keep algebra with you, but in a different world. The algebra is always there, the dentists are always there. But how you embody them should change. But how I don't know. I know, I just want to comment a little bit. Because those graphs appeared as a limit of surfaces, kind of string students. String student target space would be very similar. So it's kind of like in cold spaces. Possibly, possibly. And this I don't understand. This you might understand. I don't. I have a question. If we consider a counter-variance on a category of graphs, can we be able to give the conditions, to help us to give the conditions necessary and sufficient so that this filter is represented by a variety? No, I think you do understand what I'm saying. Another question? Brother Poisson? Good question.