 So I have to add, if you commend what I was saying, last time concerning the issue of the C-naught closure of the space of a matrix, of a matrix G, there is a given sign of the curvature. Right? So there are two spaces when they were something greater than or less than. And the point I make, the statement which I made, that this space is closed and this is dense. And there are, of course, two different ways how we can think about that. You can say, well, that's very good and this is very bad, because this tells you that looking from very far away, you still can detect the positivity of the curvature. Go very far away, all kind of differential thing disappears. You look from a far and this corresponds to that. On the other hand, you can, of course, change perspective, yeah, and say on the contrary, once you know that, you know everything. All matrix, they are dense, so subject completely used just to matrix in general, so you forget about that. And this says, well, you don't know what to do. And so this depends what you want. So this is the end of the story. You know they're dense and probably have nothing else to add, and you can say the question is finished. But this is opening. You can say, huh, now something is there, and it depends what you like. For the second question, incident, is it a homotopic-less type of the space? Yeah, here is everything. Yeah, the homotopic is just contractable, yeah, just contractable. It says that the theorem of the camp in this kind of, in the philosophy of general phenomena that usually differential inequalities, when you impose the inequality of the type, usually it tells you nothing except what involved in the topology of this inequality. And this key topology is simple. And those of you who have different perspective, and originally this question was raised a long time ago for symplectic geometry, when it was unclear if the same thing is true about the group of symplectic differmophisms, and also it was not hard to show that there are two alternatives. It's either it might be C not closed, or C not dense in this space of all volume pre-genomic transformations, and therefore they were exactly depending how you are oriented to perspectives. And then eventually, after many oscillations back and forth, shown they're closed, which indicated symplectic geometry exists. It doesn't tell you what it is, it says there is something in there. And then some things are developing in kind of geometric sense. Of course, the word geometry has different meanings. For example, I think the papers on symplectic geometry, already I think by Siegel or whatever, understood in a very algebraic sense, but completely kind of oblivious or completely perpendicular to this perspective when you look from very, very far away and see what you see. See in classical calculus you do opposite. You look microscopically, and so you look at essential perturbation of the linear systems, infinitesimal perturbation. And calculus goes this way. But what is remarkable, of course, is that correct algebra, if algebra is so nice, it usually indicates when you go to the weak limit and still it survives. But that's exactly, exactly the point. So here in the case of scalar curvature, it is closed, and as I said, there are two proofs and none of them is kind of straightforward. The same as in the symplectic geometry. There is no kind of straightforward proof, because it exactly tells you something happened there, and the something must be detected. Okay, now I want to describe what are main topics in there. And when I just prepare this something I will put on the web later after the lecture in the new version of that, and I thought it would be 5, 6, 7, whatever. But when I wrote it now, I think it's 18 or 19. So it's a very kind of quite developed field, and some of them are very developed, and some are less developed, and some are only beginning, but they are amazing the many directions. And so the two major, which you may find in the literary direction, one is about positive mass theorem, which I explain again here. Oh, it used to be a conjecture in the Manian geometry, which is kind of geometric and motivated by general relativity, and in fact, in general relativity, you can find the germs of very many things which he has done afterwards, particularly because it's a relation between minimal surfaces in the scalar curvature. And another one related to c-stallgebras index, theorems, and knowledge of conjecture. These are the most kind of developed fields, which in there you have a huge amount of papers, and which I know only tangentially. I must admit, I know tangentially cannot say too much about that. And so let me explain what's in there. And just because they kind of correspond to two different masses involved. And so there is a following kind of. So one, of course, I say that knowledge of conjecture, maybe I say but it is because it's underlies many, many ideas. And knowledge of conjecture in its kind of simplest form, so I have to say too, these have great similarity between the two subjects. Though we don't quite understand what they are, there are some formulations saying they're kind of, in certain reformulations they become equivalent or one implies another, but in truth it's not so, right? But I just say, so you have to say two words about that, because I will be referring to this pretty often. So you have to distinguish when it's concerned many faults, x compact many faults of dimension n and large n. For lower dimensions it's kind of, we might be careful what it tells you. Now one knows, and this is kind of fundamental theorem of Browder-Novikov, that homotopy type of x doesn't tell you anything about the tangent bundle of x, so from homotopy type. And tangent bound x are essentially independent except for one relation, namely the signature of many faults, which is some signature of quadratic form for four dimensional manifold intersection of the cycles of product homology. So it's quadratic form, it has signature. This signature, which is homotopy invariant, equal to some pantheon class, which is invariant of the bundle completely regardless of anything else. Any bundle has pantheon class as some invariant of this. And that's the only invariant. So this is this relation, yes, follow some kabordian theory of Tom. This is Tom, here's the root kind of formula. But it is the only relation if you allow torsion, yeah, if you add here some constant, multiplied by some constant. I don't know, it's like n factorial. And then you know there is nothing else. If it's smaller than n factorial, actually it's the power of torsion. And then, and that's what you can think about that kind of the end of the story, you understand, simply connect manifold. And then for non-simply connected, there is some kind of formalism and wall kind of developed it and as if you have parallel theory. But in fact, it's completely different because when there is fundamental group, the world changes. You have kind of more structure in the simplest instance of this conjecture which is unsolved and it's kind of sufficiently representative is that if you have a manifold X, where there is nothing but fundamental group, namely such that this is contractable and this is universal covering of that. So this is, you can think about like Euclidean space divided by gamma. So all, all homotopy information is in gamma. Then all pentagon classes are homotopy invariant up to scaling. So it means that if you take this tangent bundle X and take it some with itself, say n times n factorial times or something big number of times, this bundle will depends only on the group gamma. So once you know a gamma, you know all characteristic classes. And this is still open. And on the other hand, this is kind of tricky situation because it's huge literature but we shall see Scaly curvature, the parallel question for Scaly curvature more or less like that, which says if we have this kind of manifold, it carries no metric with positive Scaly curvature. But this of course very weak statement compared to that. In fact, the, which is a little bit, maybe deceptive. And there are kind of reformulation version of this Novikov conjecture which I come to later on where kind of it become formally more general than what I said about Scaly curvature. But that's, you have to have in mind that. And so there's something here in the group. Seaching can something about the group. And actually the theorem can be formulated entirely in terms of groups. You can forget completely about topology and geometry. Some kind of algebraic property of group. You can formulate it and then you can prove nothing. That's amazing. So you can say, aha, topology reduced to algebra. Something about groups. When you have to prove something even about groups, you have to go back to geometry, even more than topology. So all theorems of this kind, whenever it's proven, proven by geometric means. Sometimes they're hidden geometry, sometimes hidden. And it's some algebraic dressing, very kind of deceptive. And not only it's unproven, we don't know if it's proven or not. There are many, many theorems of a different condition when it's true or not. We don't know this condition equivalent. We don't know if one implies another, except for some cases. And maybe an old case already covered, we don't know. So it's a completely chaotic situation. And there is more and more papers in the act of the scales, for all I understand. And the basic results, so there was original papers, long papers by logistical mission case, we don't know if anything was done afterwards. I mean, lots of things were done. But the class of groups to which it applies unclear. And we don't know what the problem is. The method is the property of the group. And the same is Kelly Covich. Well, Kelly Covich has extra points to it, which has been ineligible here. But this, to understand this, you have to look at the methods. And what is the reason? And what are the proof? And so there are two parallel lines of confrisoning in one related to minimum varieties in general terms. I'm showing Kelly Covich. And here is Diera cooperator. So you're talking about the proof of what now? I will be discussing what you do with Kelly Covich and what you do simultaneously for Novikov conjecture. So there are two things. So these are methods. And here is for Kelly Covich. Here is for Kelly Covich. And the parallel thing for the Novikov conjecture, this will be splitting theorem, coming back to Novikov, how we prove something. And here it will be signature operators. But the question is, where these operators act and what kind of index theorem you need for them? This is more geometric. But it's amazingly how closely can we can go from one to another? But every situation here, there is parallel situation here. And there is no formalization, despite certain claims. Some people say, oh, well, we have general theorem. Only because they narrowed the field. So this is rather easy to explain. And this is also easy if you know what Diera cooperator is. And of course, we are not supposed to understand this. Now, so maybe I shall formulate, again, in a form, the positive mass theorem because I simulated that. And say, these minimal surfaces in physics, we can see it all the time. And the various names, which I'm very confused there. I'm just a very scared way of doing physical terminology, which I don't understand. But the simplest form of the so-called positive mass theorem, and as we know, in a way, we're covering all possibilities. Modular, some relatively simple argument, is that if I have a complete manifold of positive-scaling curvature, such that outside of some compact set, it's just Euclidean space. Then x is isometric to Euclidean space. And thesis, of course, formulated it differently. They have idea of a mass of a space. So there is a space containing some amount of mass somewhere. And you want to look at infinity, what this felt from the point of view of general relativity. This is, of course, a space slice. So it's not the whole space type. And so they defined mass in a way I must admit I never unable to understand. I haven't thought about that. But here, at least, this kind of mass is 0, for less space than infinity, mass supposed to be 0. But the point is that you cannot make this deformation. Of course, for dimension 2, it's more or less obvious. You have convex surface, flat infinity must be everywhere flat. But for higher dimension, it's quite tricky. And there is no quite simple proof. And as I mentioned before, there are probably seven or eight somewhat different proofs. They're closely related by different, the best and different phenomena. So is it flat? You know, this is just a flat space. So you don't have to be supposed to know what mass is. The mass here is just the terminology called positive mass theorem. Origin and physics. Why so? I don't know because I don't understand what mass is. And from a certain point of view, the typical it was written, it was not satisfactory. Because it was written in particular coordinate system. It was not invariant. I have some guesses I will explain in a second. So what it means on some examples. And the method to prove that is to use some kind of minimal surface around them and it reduces, in this case, to the Gauss-Bannet theorem. So this can be eventually explained today how to reduce the Gauss-Bannet theorem for surfaces. Modulate certain manipulations. And I'm just certain. I think this was a problem. If Gauss-Bannet was at the disposal of physics when they formulated that, I don't know the history. Because physical papers, I don't understand the verse. But that's mathematical statement. And so, but alternatively, you can prove this, again, in many ways by using Dirac operator. So minimal surface we understand. So let me explain how the proof goes here with this particular theorem, same dimension three. And so first, what you do, you reduce it to compact case. Which is, in this moment, you use very strongly the geometric infinity is kind of euclidean therefore periodic. Because you can say, well, you don't lose anything if you just take huge torus. And this perturbation was inside of the metric of the torus. So I can say, I have torus now, flat torus in this perturbation. And secondly, I say, why it should be like that perturbation? So I have just any torus, same dimension n, now dimension 3. And it has a metric g. And g has a point of scary curvature. And I say it's impossible. In fact, it might be flat. So let's prove that. And this, I think, was more or less formulated by Gerrach. And more or less, I think, the proof was already implicitly some steps of the proof were in the physics literature. T-Boy is not here because he said that there are certain formulas which are systematically used in physics and they were not used by divergent geometers. But here, this formula is essential. And so the formula you have to know, besides the Gauss-Bonne. So the Gauss-Bonne says that if you have a surface, so all I have to know that if I have a surface of a genus bigger than 1, and I have its scary curvature or Gauss curvature, which have to be factored to the same, this integral for this one is negative. And if the genus bigger than 1, it's strictly negative. And you don't even have to know the specific value of that. One thing you have to know, some technical statement which will go inside, which is kind of obvious. On the other hand, that was always, for mathematicians, it was up to point it was really a problem. Because they were earlier and are formulated. And then it says, for example, we have a torus or dimension 3. And take a homologous class surface there, which is not homologous to 0. Then there is another surface inside, which is minimizes area in this homologous class. So I push it, push, push it. It cannot shrink, so it must stop. It might be minimal. It's kind of obvious. It was kind of unknown for a while. And there were papers and geometry when people proving something kind of trying to bypass this difficulty. And then at some moment it was proven that always in every homologous class there is minimal surface, minimal really minimizing the area. And the moment it was proven, just everything else became just playing with formulas. It should already be ready. But somehow bringing this together was a big point made by Sean Yao that brought these points together. But I'm curious how much it was already done by thesis. Except, and so this is the fact. So what do you do? So imagine you have a torus with this point of scalar curvature. So take some homologous class, minimal surface. So what you know about this surface is homology. This surface must have genus at least 1. It cannot be sphere, right? Because universal covering is contractable. You lift up sphere, become contractable. It might be non-zero homologous class. So the theorem, kind of the technical theorem, is we have to prove is that if you have a manifold of positive scalar curvature, I don't care what happens outside. I have a surface inside, which is minimal. Then this surface is sphere. It cannot be anything else. And this is kind of obvious reformulation. And the paper of Sean Yao, they're proving. They gave their own proof of existence with minimal surface. So it was long paper. These people, they always write very difficult hard papers. And so they even prove that by that time, it was not really needed. Already it was known. But they prove in a certain way. Because at that time, people were very, very about that. I've thought the whole difficulty of mathematics is there. Once you know it there, you just take it for granted. But in fact, when you go to high dimensions, it still remains a problem. So what exactly happened to high dimensions from this perspective is not so clear. And now what you have to show, that if you have positive scalar curvature on the ambient manifold, and there is minimal surface, then this integral, so. So imagine you have this manifold scalar of this three-dimensional manifold, say positive. And this is the surface. What you have to prove? You have to prove it's spherical by showing that when it integrates scalar of this surface, called y, it will be positive. Now if you just take a minimal surface per se, it is not true. Minimal surface meaning, meaning, again, there is a point in terminology. When you take minimal, it says it is locally minimizing. If you take risk-ticket or this little ball, it's minimal. It's minimizing, and which is equivalent to the vanishing of the mean curvature. But it doesn't tell by no means it has the minimal area. For example, if you take equator in the three-sphere, and this is S2, it's, of course, minimal. It is locally calculated, minimizing, but globally can push it become smaller. And this is the point. And the key point, again, it was used again before, but never kind of to this extent before, by Badan Shonyal. You use the fact it is absolutely minimizing. So when you push it, it may become only bigger. And so you write down what happens there. And this is if you take a surface, you move it by normal direction and see how area changes. And this is called second variation formula. And, of course, I'm pretty certain, Fisk literally, she was done. But in mathematics, it was not written, so I just write it down, because I always have tendons to forget it, and so what it is. So the point, of course, eventually, you bring the formula to the shape that, so if you have scalar curvature of the end in space, is positive, then this quantity, the second derivative area, is smaller than the integral of the scalar curvature of x of y. This is why you surface. And because, which makes this integral positive, because this must increase, this increases. So this might be positive, and therefore, it might be sphere. And the general formula, so just we write it down, is that it is actually integral. So what's involved in this second derivative area, it is integral over y. And here is 1 half scalar curvature of y minus scalar curvature of x, and minus something else that is integral of the second fundamental formula, I don't know, square. So some extra term, which is by its nature, is here square, it's integral of somebody's square. And so if you know to have this inequality, you know this positive, this disappears, this point disappears, so it has an equality. And so just we explain how this works. So this is very simple meaning to that. So this formula, you said this is the exact formula of the second derivative of the area? Yeah, it is the formula of the second derivative. So you have 1 half. So it's 1 half different between two scalar curvatures, minus also square with the second fundamental form. So you need the 1 half to 0, too. Yeah, it may be also 1 half. But because it disappears, yeah, I think it's 1 half over here. No, it's a material, it's a negative here, it disappears. But they have a coefficient that will disappear at this stage. So again, maybe I'm missing something, what you're saying. Just said, yeah, there is 1 half on this inequality here. Here, 1 half. Ah, yeah. You're right. So this is essential formula you have in mind. What happens when you deform surfaces? And the general formula maybe is following, because it will be useful for. Again, I put them in this formula. And what I put on the net, and this is the following thing. If you have, again, any sub-manifold, and you start taking this normal field when you deform it. So it's y of dimension n minus 1, the side of x of dimension n. And you move it first by unit vector field normally, and then multiply it by any function. So it may go up and down, but this is your infinitesimal variation. Of course, you might be careful, because this is not extremal, it's not minimal. It actually depends how you move. And this way, you move a long, normal geodesics. If you change, take another vector field. Another kind of extension of this field you can get slightly different answer. Which is, of course, this term will be 0 if it was actually minimal surface. And the formula is as follows. It is will be, one term is will be quite clear. It will be integral of psi, or differential psi squared over i, before it was constant. So it was 0. It was not present. That's kind of a novelty. And the rest is what I said. It is plus integral of here. And here is 1 half, the scale of y minus the scale of x, minus the second fundamental form squared. Everything here 1 half, yeah? Squared. And there is an extra term, this one. And this is a mean curvature of the manifold. And the mean curvature, see, it's signed. So if you change the vector field, it will change the sign. I think it's muted, not muted. And again, if it is minimal variety, this thing disappears. Maybe psi squared? Maybe psi squared, because it's quadratic, we'll say? Yeah, type psi squared, yeah, right. Yeah, yeah. Time psi squared, absolute. There's a quadratic expression under this deformation. And this is a formula. And this is exactly what this format is used. And this is just keeping in mind that makes things very, very clear. And I guess it's what people are saying, that this was kind of standard formula in relativity. And of course, in geometry, it kind of was used, but not as kind of having not sufficient prominence it deserves. It's kind of a basic formula, one of the basic formulas of geometry. And actually, you can prove it by pure thought, without computation, right? Yeah, from symmetry, some form of symmetry. Knowing the scalar curvature kind of has this linearity. And this metric is flat up to some term, et cetera, et cetera. And then you have to normalize the coefficient in some example, like circle on the plane. If you check it for circle on the plane, it is correct. And therefore, this one have come because you differentiate r squared becomes 2r in this y2 come here, because it's squared in derivation. What do you have as information on mu? What do you know about mu? No, mu is a mean curvature of your manifold. If it is minimal 0, for minimal sorry, this term will be 0. But you shall meet situation when sometime it will be non-zero. Mu is some function x. But this is a deformation for anybody here. And this give you kind of a very good relation between these two scalar curvatures under motion. And this give you really some non-treatment in my view. Can be viewed as definition of scalar curvature is inductive definition. If you know what a scalar curvature of y, you know a scalar curvature of x, because all other terms, you see, they are simpler. They don't involve kind of curvature. The second fundamental form, it depends only on the first derivative of the metric. It doesn't in second derivatives. And also, these terms don't depend on the second derivative of the metric, which is not surprising. All integrals involving scalar curvature with any weight don't involve curvature anymore. They depends only on the connection itself. This kind of fundamental factor, because it's linear. Because scalar curvature is linear in the second derivatives. So by integration by part, you can shift or eliminate one derivative. But only one derivative, not the second one, you see? So it's easier from that to conclude that scalar curvature is sign, or its value, is stable under limit C1 topology, regardless of the sign. But it's positive or negative, right? C1 limits of metric, dover, a priori curvature-dependent second derivatives preserves scalar curvature. But it can go one level below. So that's the formula. And how this formula can be used to prove something about many photoscalic curvature in high dimensions. So what you have to know next are, so this is kind of clear. So for the three-dimensional case, we've done it. And so we've proved this version of the positive mass theorem of space flattened infinity. And then you may ask, aha, what is it in general, this mass, and how it is? So there is a theorem. There's some definition of mass in physical literature, whatever it is. But then there is a theorem of a log hump which is saying that if I have a space and mass is negative, and many of us have a positive scalar curvature, and if it's a mass negative, then I can deform the metric, keeping just outside of compact set, keeping scalar curvature positive and making flattened infinity. And so we don't have to know what mass is, if and only if. So sine of the mass tells you if you can flatten or not flatten it. So everything comes to this point. But now let me give justification of that. So I can try to consider more of the examples. And we also understand the meaning. And so my wife used to speak about this much. What is log hump theorem? What does it say, log hump theorem? It says that if you have x is Riemannian manifold complete, scalar curvature is non-negative. Say I would be more slightly more comfortable to say positive. Yeah, if it's non-negative, you might be more slightly more careful. You can reduce it in scale, but you might be slightly more positive. And it's mass defined. You don't know what it is at infinity. It's some sort of asymptotic property over this at infinity. It's negative. Then you can deform the metric outside of compact set, keeping its scalar curvature positive. And such that it becomes just Euclidean at infinity. And even on leave. So sine of the mass has this interpretation. And this is done by kind of elementary linear analysis. It's kind of a little bit of a mess. But this is a routine, yeah, it was. And there was some preliminary work done by Shonyauka. Partial interpretation of that. But this is just linear analysis, actually, close to that. But just to understand it, let's try to look at other kind of model cases. So you don't have any assumption of topology at infinity? No, absolutely not. No, no, it's Euclidean. It's Euclidean space. At somatic to Euclidean space. So it opposed to automatically standard definition. Inside maybe anything. But at infinity, it's just Euclidean space. And I'm saying that enough, the case in many respects, enough to understand flat, really, at infinity. And this is just Locke Hampton, 1999 thing. Prove this is called Cariffani. Something about Scholarly curvature hammocks. So actually, I didn't know this word. So what a hammock, sir. And so in this paper, you know, Locke Hampton, he's really, really, really very strong. He proved other theorem, kind of more difficult than that. And this was just a little kind of remark. And it was really in people in this relationship, or quite a while, they were not aware of that. And this is, well, so this is quite, quite significant kind of observation. You see, it killed lots of people. Because we really people like Witten and Scholarly, they spend a lot of energy handling this infinity. And this is all just in material. He just reduces something much easier, or I think we already know. And for that reason, maybe, he's not excited enough. But what is the kind of meaning of this positive energy and what can be justification of that? Look at much easier expression of a conical geometry. It looks like a sphere. And it may have corn like that. It may have flat corn. Or it may have corn, which kind of a more than 2 pi angle. It grows faster. So these cones will have positive curvature. They are convex. And so in particular, they have positive-scaly curvature. This is flat. And this will not have positive curvature altogether. They have negative curvature. However, I'm saying, and so what about mass? Mass properly scale. From physics point of view, your scale, this metric by constant, mass multiplied by the corresponding constant. And these are scale invariant, cone. So this mass will be plus infinity. And this mass will be minus infinity. So you go immediately out of physical range. So what happens, and then you can observe that this is one. You cannot flatten it. You cannot turn it around, make it flat. But this you can, right? Because you turn it inward. When you turn it inward, it becomes more convex. When you move something in this direction, you create convexity because it was opening more than nuclear space. Come back to nuclear space. You have to close up and close. And so this you can make flat infinity. If it had positive curvature only as to this, actually. You can make it positive curvature infinity, if you wish. But this you cannot. So this is a kind of baby, kind of a version. It would be weak version of positive mass. Because it says, if mass is plus infinity, then you cannot make it flat inside. And to understand the physical, meaning of that, you have to consider the basic example. So again, I don't know what mass is, but feces also were motivated by the basic example in this Schwarzman metric. So first, of course, it's very hard to write this name. It's again, the subject more than that. This actually I just read. I mean, here, the poor guy died during the First World War. And he wrote this metric. So this would be, of course, not what I'm going to write, but geometers write, not what feces write. Because it is, it must be Lorentz and I mean, Einsteinian metric, it must be in definite space. But you look at the Riemannian part. And this Riemannian part is the metric given by the following formula. It's symmetric on R3 minus 0, right? So this is, and this is conformally conformal to the Euclidean metric. So this Schwarzman is Euclidean metric with a factor. And the factor is 1 plus, I write something, it's Euclidean. It certainly looks a little bit horrible. This M is mass. It's called mass by definition. And so it's, you see, I have to understand a little bit of this metric. So it is actually the first reading of certain articles I misunderstood and thought that this has positive-scary curvature. In fact, it's supposed to be rich if it's flat. If it were not flat, just coming here, I found a counter example to a statement if you assume it's positive curvature. But it's flat. And so what I'm going to say will be OK. So this will be a major instance of something having positive mass when it must be here for positive. This conformal doesn't have singularity in 0. No, no, of course it's, aha. I'm sorry. You're right, you're right. I'm sorry, I'm sorry. It might be flat infinity, so it will go to infinity. This must go to 1. And formula says yes. But you have to, the picture of the space is like that. It's symmetric. Read the sphere with respect to its symmetric. You can verify easily that if you take transformation, if I'm not mistaken, r goes m squared over 4, 1 over r. The symmetric is invariant. So inside, there is a sphere of radius, I think it was this radius. And area of the sphere is 16 pi m over m, I guess. Sometimes you call this. You keep forgetting. But this is specific number. And this is what you can think about what is a mass. So you see, it goes to Euclidean geometry. Again, it's conical from this side, but much slower. So it is a cone that goes kind of with a linear rate. But this goes only with a flat and then slightly, slightly slower than the Euclidean. Only slightly. This is this mass. So you can see the spaces, which are not like conical. But we approach Euclidean geometry much slower, much closer. I mean, I'm sorry. They're closer to Euclidean than cones, but may be slightly above or slightly below. And that corresponds to the conceptual mass. So these are the type of metric. But this was kind of essential. This has zero scalar curvature. And so there is some kind of theorem here, which I will state in a little while. But this is what mass is. And then, of course, you have to make different attempts how to correctly define it. And what you stand the definition, I just cannot swallow. Now, this is about minimal surfaces and how they use. And then I said, the rest is high dimensional situation. So what you can do is high dimensional. And this was, again, in this idea, construction in the Utah-Schonielle. What about high dimensional torus? Gerrach actually formulated conjecture in dimension 3. Actually, I'm not certain where and how he formulated it. He said it on some meeting, because in the article, I read with him, he said, other things and this may be implicit in that. But of course, there is no reason to distinguish number 3. And of course, there is no more Gauss-Berner formula. But still, you know that in every homology class of the torus, of n dimensional torus, there is this minimal object. But this minimal object is not always smooth. It's known to be smooth only if n less than or equal to 7, even if it is absolutely minimizing. And the first example appears for n equals 8. And this very simple example, you take S3 times S3n in S7 in Euclidean space, so it's dimension 8. And you take this cone over it. This obviously is sitting, naturally sitting product of sphere. Take the cone, and this cone is minimizing. And it was a big kind of discovery at some time. But if you think from general principles, it reduces to some very simple computation. Because if somebody is not absolutely minimizing, you can rotate it around. And then all this intersection together do something smaller. So absolutely minimizing here, because boundary symmetric must be symmetric. And once it's symmetric, it becomes ODE. And so you have to look at this ordinary differential equation and see it necessarily develops singularity. And I don't know whether it's true or not. This was considered one of the count of the example to the famous Hilbert problem, who said that if you have analytic elliptic equations, the solution must be also analytic. And of course, the question how interpreted. So this cone is not strictly speaking analytic variety. It has a singularity. However, it's kind of an analytic object. Yeah, it's not something horrible. And it's still unknown if this is kind of in general the case. And for different kind of equations that were done similarly, there are singularities. And you know by symmetry. And then you use the ODE. And then you solve the ODE. And then you see that for some moment, they develop singularities, which is kind of exercise. And actually, some of them were the thesis of somebody in Leningrad, who was solving another Hilbert problem. Of course, somebody put his professor talking. They just look at this ODE. And he just solved it. But this dimension, 8, yeah. But this doesn't happen in lower dimensions. And this is a real non-trivial. And again, there is a big theory for that, developed by Federer in Flaming. I think about that partly, I'm grand when someone further. When they developed general theory of minimal surfaces, but they certainly had no means to handle why there is no singularities. And this was done by really, by a very elaborate, in my view, incomprehensible computation. Computational algorithm by Jim Simmons, who has then extended this computation to the stock market also quite efficiently. And this is the main result of the geometric measure theory that there is no singularities below dimension 7. And there is no rational explanation, just computation. And then some numbers come out. And then another interesting point about this singularity, it's unstable. It's known to be unstable. You have this singularity in this dimension 8. I move it a little bit. I move boundary condition while slightly changing metric. Singularity disappears. And this was proven by son of Smale, not Smale. It's, again, I will show later on explain. It's very transparent why it happens. It just requires no knowledge of minimal rights as general principles, which kind of, again, completely transparent. But these are known for high dimensions. There's a big issue, even as what happens in big dimensions. And the reason by passing of that due to Locke-Hump, who recently published this paper, when he says, well, it's not exactly. That, but you do it, kind of, you have almost minimal rights, which are approximately minimal, and singularity disappears, which is for purposes of the present discussion sufficient. But his paper, this is rather short paper about 20 pages, but it based on another 200 pages of other papers which are hard to read. So nobody, of course, checked it. And, but this must be true. I'm quite confused. There's all indication of that, because when you try to unregoriously think about that kind of obvious singularity must be unstable. But then there are some very pathological phenomena which may come up. So if you believe in this Hilbert philosophy, of course, it must be so. But except that this singularity, when it's high dimensional, is so elaborate, kind of, can't accept and being stable at the same time, which is hard to believe. But anyway, accepting that, that you know this minimal hyposophist there, what you can do with that. Now you cannot apply Gauss-Bernet. You don't have Gauss-Bernet. But what you do know, still, when you push it down, some expression is positive. And the expression which I wrote down was integral was psi squared. I'm sorry, d psi squared. Plus, I keep forgetting already, minus 1 half. And here is integral. For minimal varieties, other term disappears. And so your integral will be the second variation. It will be the scalar of y. What's important, this quantity is positive, d f squared. f y phi squared. And so what it means, yeah? So it says on this manifold that this term, when you integrate square differential, outweighs negativity of the scalar for all functions psi. If you don't have too much negative curvature, right? If it were totally negative, you wouldn't do that, right? But there is a little bit positivity, a little bit negativity, right? So if you have not this term, not this psi, and just look at not the scalar curvature, it just says that the plus operator is positive, right? So integral of d d psi squared, the only ways to be positive to be zero is constant functions. And here, it's kind of more positive than the plus operator. So at this point. And then therefore, if you take for this operator, so corresponding operator will be minus delta plus psi, right? Look at this operator, apply to functions. Plus scalar curvature, right? Plus it's one-half scalar curvature, absolutely right. One-half, it's not one-quarter, no half. It's either one-half is essential, right? It's very important in many situations that one-half is greater than one-quarter, right? In some moment it's one-quarter, but this is kind of perfect balance. One-quarter is unclear why one-quarter enters. And there are some other numbers enters in different dimensions. But this is the kind of. And therefore, there is an eigenfunction. First eigenfunction will be positive. For this operator, it's positive eigenfunction. Well, we satisfy this equation with positive eigenvalue. And then it's another computation, another formula, which I haven't written down. It's going to be written in what I put on the web. But here, it's again, we can prove it by pure thought, is that if you take now the following method, you have manifold y, you multiply it by r. And you take the metric here, given by, it will be dy squared plus this function phi, I call it, yeah? Phi squared dr squared. Then, because it was coming from this operator, this one metric will have positive scalar equation, right? You just make some computation, tell it I, and get it. But again, this is a fact of life. And this exactly kind of, exactly this one half enter there. It's exactly perfect. So it's if and only if, right? You can do that. Therefore, I don't have metric of positive scalar equation on y. But I get this metric on this manifold and observe this metric invariant under transition. Now, I can repeat this process. Now, instead of y, I take next y prime and do it over there. But now, because this was an invariant under r, after the second round, become invariant under r2. And so I keep doing, doing, doing, doing. In the vanishing, I come in fully invariant thing. So it will be Euclidean space, invariant metric of positive scalar equation, which is impossible. That's the logic of the proof. It was somewhat differently done originally. Because again, because if you use ready up, too, that's usually what happens. You use ready there. If you plug in there, you have complicated proof. But if you go directly, you see it like that. All you have to know what I said. But the key technical point is this. And it's highly unpleasant. That's why there are the singularities. Because if you think a little bit about this formula, the singularities are only kind of always in this problem in your favor. The more singularities or inequalities become a little stronger. However, you can formally apply them. But I say more about that, what you can do about that. And so this proves that the torus scalar nomadic of positive scalar curvature, but it doesn't tell you still much about the geometry. We don't know how to fully use that. And so how to enlarge it, how to use this to extract geometric information. So if I had this topological information at the beginning, everybody was very happy proving this topological non-existent theorem, but from a certain perspective, who cares? So it doesn't exist. Forget it. So what do you do next? If you prove something, it doesn't exist. Not interesting. But still, it was kind of like with re-GH theorem. People enjoying re-GH theorem. Something is rigid. And then what? So the formation doesn't exist. It doesn't exist. Of course, it's only interesting if we can make the next step. And here, of course, you can make the next step, but you have to change perspective. But for that, it's useful, again, to look at this parallel with diarhap operator. So this was in a slightly different form, done by in 79, I think, or maybe 8, I think in 99, by Shon Yal. But 10 years, even more than that, in 63, as I said, Lykhnyrovic proved that there are the abstractions for Poisson scale curvature. And where the simplest example, we have hypersurfaces in CP3. So this is the Kuhn surface. Or you could put here any 2K. Any surface or even degree, is that they don't have Poisson scale curvature. Actually, now we know also for odd degree, they don't have it. But by more sophisticated means, but this is only limited dimension 4. But this was 4. And just to be sure, it can send all dimensions. If you take this surface, this surface to any power, and it's still multiplied many, many, many, many such surfaces, they all have, all admit, normatic of Poisson scale curvature. And the reason behind it is that there is a contradiction between two theorems, which I already mentioned. One is Schrodinger Lykhnyrovic's identity, which said that Dirac operator, whatever it is, is just operator you don't know what it is acting on some vector bundles, taken square equals something positive, some positive operator, plus 1 quarter of the scale curvature. Therefore, if many for the Poisson scale curvature, this has no kernel. In particular, index of the operator itself must be 0. Of course, you must be careful. That means square means, because when you say index, you map one space to another space. So this d squared, a little bit of abbreviation of d times that, but because for some reason, you can do it this way. Because organizing such a way, it makes sense, depends how you define Dirac operator. So more precisely, you have to do it this way. So the Dirac operator has to pass, and then being in the change. And then this is also, it applies to some class of possibilities, and tells you even less about the geometry of the manifold. So how we can bring in geometry? And for that, this part is much more transparent. And because. Excuse me. So you said that the non-adjustance comes from the, you have two results. So one fact that. Yeah, one fact that this operator has this formula. And that has. What is the other one? I'm sorry. Yeah, I didn't say it. Because it has non-zero index by HCI index theorem. You can compute index in this example, it's non-zero. And it's actually, what it's called, if you know, this is a roof genus. It's some, in dimension four, it's just a signature up to a constant here. A signature of the manifold. But in high dimension, it's something else. Some care is possessive of the first pan-triagin number in dimension four up to a constant. When you multiply them, this thing multiplies. It's multiplicative. In high dimension, it's a tricky expression. So I didn't say it. So for four dimensional manifold, just say if pan-triagin number doesn't vanish, then rational pan-triagin number doesn't vanish, then index is non-zero. When you multiply, it's also index is multiplicative. It's rather tricky, by the way. I must say, it is not immediately clear the formula for high dimension eaten by Hertz-Bruhitz kind of not immediately clear in coming from Riemann-Rochtherm, first proven by Hertz-Bruhitz here. But then if you think about what is the index of an operator, in particular, operator D, and then it is not only for given manifold. So one point which should be emphasized, it only makes sense if your manifold is spin manifold. Otherwise, it's defined always locally, but only up to plus minus sine, your operator. And to be sure it has constant definite sine, you need something like orientation, which is slightly different. It's kind of orientation of the one dimension up, being spin. And sufficient condition, just not to boss me about exactly too much about that, of manifold being spin, is vanishing of this group. If it is zero, it's always spin. By the way, these surfaces and this condition are not satisfied for surfaces in a complex projective space. But for even dimension, they're spin. For odd dimension, they are not spin. And so once you know it, you know the theorem, but your understanding of geometry is very limited. And then, but for here, you know it very well, that index is not a number in truth, but index give you a function on vector bundles on the manifold. So it's what is called k-homology class. So each vector bundle manifold will give you a number, because you can twist Dirac operator with every vector bundle, the complex vector bundle of a manifold. So you have differential operator. Like you have just repeaters, if your bundle is flat, it just takes a dimension of rank k, you just take d plus d plus d k times, kind of trivial thing. But bundle is not flat, but it has flat connection. Flat, you enter a connection like any bundle. So it's flat up to high order terms, Dirac first order operator. So it can make this twist. And it doesn't depend on choice connection. Index doesn't depend on choice connection. And so you assign now number to each vector bundle. And therefore, you have to always think about all these twists. And this tremendously changed perspective. And as far as topology is concerned, and this was got away, essentially, implicit contribution to the theory, this was the following argument to Lüstich related to the knowledge of conjecture. Where he proved the version of knowledge of conjecture, I don't let me give you. For example, for many faults of negative curve, I should say, for surfaces or something like that, yeah. Actually, it was originally done for the torus. But it's more transparent to mention you have like product of Riemann surfaces. You don't have much pan-traging numbers. We take such manifold product of Riemann surfaces. And the theorem says, if you change smooth structure, still all pan-traging numbers are 0, all the same for the torus. Right. This already is significant. You take a torus and take any smooth structure, still all pan-traging numbers might be 0. You cannot change pan-traging numbers of these manifolds by changing smooth structure. Because there are spherical manifolds. So everything you can remember from the mental group. For example, the 0 might be always 0. For rational classes, for integer, I'm not certain. More 2, there may be some problems. And so what was his argument? So the point was to express this somehow in terms of fundamental group. Now pan-traging numbers or classes enter the index theorem. So index theorem has two terms to it. When you have the Dirac operator or another signature operator which appears in second, you twist it with L. The index of this has two terms. This is one A-roof. Something depending on structure of manifold. Attached to this, and this is tricky term. Another simple term, plus chair of L. It's some combination of chair and classes of L. And these are completely of different kind of significance. So this is attached to manifold and this kind of external thing. And this is much easier to handle. To understand what happens when you change this bundle and prove the difference. You have one bundle and another bundle. And take the difference, this will cancel off. And then the formula becomes much easier. And the proof of it is much easier. And originally, in the proof Herzlbruch of the Riemann-Roch theorem, he first proved that in this kind of elementary, more or less, algebraic geometry. And then there is a tricky point to extract this. And the same as we shall see for both for signature and for that, they play a different role. This kind of weak term. And this is a strong term. And this is more delicate. On the other hand, this tells you nothing about geometry. Amazingly, this tells a lot, if you apply it to that. And what happens to, what parallel logic you can observe for minimal surfaces? What corresponds to this kind of variation of that? And that, instead of considering minimal surfaces, which minimize, minimize just the area, for simplicity, area of y, you add here a weak term. So you think the surface bounds something, some region. And so your y is boundary of some domain v inside of x. And you have some function on the domain. Actually, it might be measured. That's going to be a function, but say it be a function. I don't actually prefer to say it measure. But applications with some type of function or some type of function cannot be too singular measure. And your quantity, which you minimize, we minimize area of y minus or plus, but depending on convention, measure of this v. And remember, this is power. So it's a weak term. This kind of isopereometric problem. But sometimes you formulate it and say, you take area bounding minimal volume. And it's a bad way to say it. You just completely kind of don't give you right perspective. Of course, sometimes it's the same. When this function is constant, it borders the same. But this is the right way to think about that. And features call something brains or something. Yeah, there's very fine also physical terminology, which came later in this case. I don't think it appears in later literature. And solution to that, when you have a solution to that, this extremal object will have mean curvature exactly this one. And this give you much more flexibility, because you can choose this mu the way you want. It's something now applied to all functions mu. So all these functions play the role of all these vector bundles. And so this variation problem, it has got the right hand side. It's certainly pleasant to have equation delta f equals to 0. But also it's even better to have it equal to some function phi. So you have the non-homogeneous equation. And this is the one you do that both with vector bundles and with this vector function. You can accommodate them to the geometry of your manifold. So both vector bundles reflect property of the bundle, reflect geometry of the manifolds, and exist and so on exist so function with certain properties also easily related to geometry of the manifold. And thus, you get these results. So let instance of that, just mimicking some argument of Lucy, it's very simple. How you prove alternatively that on the torus there is no medical poiseus helicovic. If you take d recuperator itself, well, it's flat. So d recuperator is essentially the same as the square root of Laplace operator. So only solution of constants. So there is no nothing. Index theorem is back use. There is no index is 0. However, if you twist it with flat bundle, then locally it will be the same. So twist it with flat bundle. So it's actually, you go slightly beyond what I said before, by the way, how you understand index theorem. But I twist it with a flat bundle. But I make this, of course, the local formulas remain the same. It still will have the same positivity or negativity. So if this would have poiseus helicovic, this also would have no kernel or kernel. So but if your fleet is an individual flat bundle, of course, nothing happens. It still has zero index. Excuse me, so the flat bundle is it related to the metric? Wait, wait, wait, wait. OK, I haven't finished. So far it's not related to the metric. But existence of flat bundles depends on the present fundamental group. So at this level, I only show how the fundamental group interferes, the other picture. So far the fundamental group was not there. And this, by the way, what I'm saying is the beginning of all this huge development, the sister algebras, you know, in the Norvigov conjecture, et cetera, generalizes this pattern. But now it is fundamental group enters. In a second, they say how to make it more geometric. So far it's not very geometric. But it's in the right direction. So when you have this one flat bundle, it is no good. However, if you have n-dimensional torus, but all flat bundles, and namely, complexly, unitary bundles or S1 bundles, they make a dual torus. Just the dual torus is corresponding to all the representation of this fundamental group in this circle. So I have many bundles. And now they have the whole family of these bundles. Therefore, when I consider this indices, so I have a family of bundles, and index theorem doesn't tell you anything about what happened in the middle fiber. But it tells you if you take indices everywhere, but somewhere it must appear. Because index now takes value, not in numbers at all, but in the vector bundles in this dual space. We have a family of these bundles of this kind of, what's the lipstick operator? It's mapped between two spaces, two here, space AB. And this operator is Fred Holm. Fred Holm meaning it has final dimensional kernel and co-kernel. So although they are infinite dimensional, they are different well-defined vector space. So when you have family of them, you have a vector bundle. Of course, you have to say it precisely. There is a little bit of a way to think how to say it precisely. You have to observe that all this concept, admit some relaxation, it still makes sense. But what you get is a vector bundle here, and is, and then index expressed in terms of the characteristic of churned classes of this bundle. This is actually a complex vector bundle, some churned classes, and even dimensional torus. Again, the index theorem for families. Utah actually says, aha, and this was used for lipstick for a different purpose, it has non-zero index, meaning that at some moment, for some values of parameters, no matter which metric is there, there must be non-trivial kernel. And so, and therefore, it is obstruction to that. And, but this is still, doesn't tell you anything about geometry, right? It's still only about fundamental group. But now, let's give it up, but this was, yes, with Dirac operator one proof. Now let me give another proof, without using families. So what I was using here, that even if I have a flat, then still this squared will have the same formula, something positive, something positive, plus one quarter of a square root of two. But now imagine it's flat, it's slightly perturbed. It's not flat, but small perturbation of flat. Then you have this term, according to some term plus epsilon, my epsilon prime. Now, how we can make it? How we can create now, single, no family at all, but individual operator, individual vector bundle, which will be almost flat. And this, it's now geometry enters. So you have to be careful what you understand in this flatness, how you measure this flatness. And so what measures bundles being flat or being non-flat? What makes kind of structure in the bundle? So this is a connection. So I have a vector bundle, and there is a horizontal field, and there is parallel transport, right? And so there are operators when going around the holonomy. Flat, meaning if it takes a small circle, this operator is identity. Almost flat, but be close to identity. So what we mean close? And this is tricky point, in what topology? And this is operator topology. So each vector moves by small amount. And the constant involved here, which is essentially epsilon, doesn't depend on the rank of the bundle. No matter how big it is, if each vector moves the model, by epsilon, it's OK for this formula. And this, of course, you have to think a little bit. But again, follows from the Schellen principles. And that's kind of crucial for all the c-star algebras. Because the c-star algebras, you exactly make completion in this norm. Why you can kind of go to infinite dimensional thing? But the point is there. But how you can create these almost flat bundles? And this is, for example, for the torus. So I want to say that now that for any vector bundle, at least a multiple of the bundle, can be realized, may have connection as flat as you want. The moment I know that, I take the bundle which I give you in a trivial Schoen class in this index formula, and then apply that and come to contradiction. But now geometry enters. On which manifold you can do that? So that will be the issue. And this will be geometric. I have a manifold X. And I'm asking, if for given epsilon, there is a vector bundle, must be a complex vector bundle, such that it will be epsilon flat on one hand. On the other hand, some churn number of L will be non-zero. It's churn number. It's not first churn number. It's some top dimensional churn, elementary churn character must be non-zero. So it must enter non-trivial, it requires little computation why this is sufficient. Because in the index formula, what enters is a particular characteristic class. But if you take the whole kind of algebra, the tensor algebra with atoms separation, it's enough. But this is my algebraic point. So when you can do that? So I'm saying it is now a geometric property, a priori of your space. Which a posteriori will be topological. Because I have a specific manifold, I take some charts. In each chart, I take the small loops. And it all depends on geometry and the length of these loops. And with respect to this set of loops, I want to choose this bundle. But then you can show the posteriori, because epsilon may be arbitrarily small. You don't fix this epsilon. Then depends on the topology. Therefore, for every manifold, we have this invariant, which I call, which is, by the way, in a second, I will modify it. It is kind of, well, the best fecalate terminology called length. It's not area, it is. Area we enter in a second later. So how we can produce such bundles? So this is geometric invariant, this minimal epsilon. So we can create a vector bundle with these properties. And this is how kind of hard, maybe a little bit kind of, general, you may have to understand how both bundles are, et cetera, et cetera. But this very simple criterion, which is now truly geometry, and this is follows. Say for compact manifold x of dimension n, I can do the following thing. I can map them to the sphere by some map, such that this map has non-zero degree. And assume n is even. As odd, you usually have to be more careful and use odd index theorem. And you ask, how big differential this map can be? And what I am saying that if this less than epsilon, then again, slightly different epsilon, I have this small epsilon. So if I can construct a map here, and degree is non-zero, but still map rather contracting, then it will have such a bundle. And here, it's quite, quite easy. So what I do, I just take, stand any vector bundle here. It's even dimensional sphere. It's non-zero, churned class. And take pull back of this. And whatever size of the non-flagness was here, it will be multiplied by this epsilon. In fact, there is perfect choice here. There is a perfect choice. In a second, I explain what is the perfect choice for this bundle, l. Point it out by, actually, by Blaine. And it's worked out by his student, by Blaine Lawson. But first, being rather careful. But now, how to reduce it opposed? Look, for example, for the torus. If I take this N torus, it goes to SN with some geometry, but you have no control about this JF. Yeah, it's something. But what you can do, for the torus, you can take phi order, take covering of very large degree. And then torus become huge. These huge torus, I can shrink back to this size, become more of a unit size, map here. Now epsilon is as small as I want. But of course, now my bundle lives here, not here. But there is a push forward map. And push forward map, I just rotate. So according to the monodromia of this transformation, add up all these bundles. And because my norm was operator norm, it doesn't add up. And that's crucial. You don't add the norm, become soup. And that's a very essential point. As I said, this makes a possibility to go to infinite dimensional situation and still kind of theory works. So that's the point. So you can take a manifold, take sufficiently fine large covering. And if this covering go to the sphere with no zero degree in the contractable map, it's OK. Of course, this depends on special feature of the fundamental group, which is reasonably finite. But in general, you don't want to make this assumption. But always in this kind of theory, it's OK. Imagine you just have infinite covering. You have your x. And here is infinite covering. And this infinite covering admits a map to the sphere with non-zero degree and very contrasting. For example, imagine this manifold is roughly like Euclidean space. Euclidean space, you can compress as much as you want. And where I want, and where I say non-zero degree, I mean infinity goes to one point. So a degree makes sense. Then, again, I have vector bundle here as flat as I want. But when I go down here, when I said aging the map, it's infinite sum. And so I have little trouble. And so what you do, again, you relativize. You cannot define it for one bundle. So we say, ah, you must have, again, something like map between two bundles. So you define, instead of vector bundle, you define them by short exact sequences. And so this makes sense. And so the object you have, it will become, again, thread home operator and index defined by that. It's technicality. But the principle is, if something, because, again, the norm was a operator norm. It's very unsensitive to the number of terms. So a fully unsensitive. How many terms you add? Therefore, you can go to infinity and make sense of that. And that is what these stars are doing. They make sense of this kind of rather simple thing. And then, immediately, you have a score of many faults. And you have geometric criteria. You have geometric criteria about non-existent scalar curvature, when you can go from there. And here, now, as just, so they still were very much related to the fundamental group. How you can go away from that? And so there are two paths. So first, maybe, at this point, we'll already, it's worthwhile, compare advantages and limitations of these approaches. Here, by the way, when I map to this sphere, if you look slightly more carefully at the formulas, when you write this formula, d squared tensor with l equal poise squared plus 1 to water scalar plus something else, this something else depends on the curvature of this bundle. Some expression, depending on the curvature of the bundle. And again, it's a norm of the curvature. So what advantage of this expression? Because it says it's not important so much what was differential of this map. But it was essential what was the norm of the exterior power, first exterior power of that. How much it was contracting or compressing surfaces. And this is a different geometric feature. So it's not so much lengths involved, but area which is involved. So that's one property. So many fault doesn't have to be large. So again, the idea behind the positivity of curvature that many fault is positive curvature tend to be small. And this says that this cannot be large area-wise. It may be kind of long like that. But cannot spread in two directions. What may spread in three dimensions, it doesn't go beyond that. Curvature is two-dimensional object. So this is kind of characteristic feature of that, of this approach. It tells you about many fault with poise of scary curvature. Here they compact many fault. Or this applies to complete many faults. So for closed manifold, for complete manifold, and also for many faults looking like that, which I explained later on. This is the most interesting contribution due to a general who just said died recently. You may have small part of the boundary. You don't have to complete in one direction. It must have infinite part, but it may dominate somewhere. So this gives you some information. But the basic limitation, it doesn't apply, for example, to the ball or anything that compacts the boundary. There is no meaningful, I'm sure there is, but it's for the moment, theory of your operator for such many faults. It might be some degenerate object, which is not quite your operator. Something can be done, but so many questions remain open. So this is one limitation. And the second, you need spin. Everywhere manifold must be spin. Or more precisely, the universal coverage must be spin or something. Somebody else must be spin, but for very simple cases, you cannot handle the theorem. For example, if you have a torus, four-dimensional torus, they connected some, you see, and you cannot rule metric of Poisson-Scary curvature here. And in a way, you can construct such a metric. Everywhere you have Poisson-Scary curvature. And tiny little bit is negative here. So it's not kind of a joke. They connected some of two things, this kind of Poisson-Scary curvature. Everywhere you have a lot of Poisson-Scary curvature. But this will be little, little, very small neck. And this neck is not to be small. And I may not today explain this penrose inequality. But in one case, it says it's very tiny little place where it's there. But it's not everywhere. But on the other hand, the minimum surface is absolutely oblivious of the spin. They have probably a little bit singularities. And as we see, they apply to many of these boundaries. And which is more or less the same as you throw in with extra mu. But they tells you nothing about area. It's impossible to say something about area. Now what are the ultimate results which I want to present now? So one of them, the first one was just, as I said, one done by some following suggestion by Blaine Lawson by his student. The following theorem, which I think is quite impressive, which says, if I have a manifold of Poisson-Scary curvature, I'm sorry, scary curvature greater than that of the sphere. It's Poisson, but it's greater than of the sphere. And it goes to the sphere F. It goes to the sphere F. By map, it's known as 0 degree. Then, in fact, scary curvature is equal to this number. Oh, no, I'm sorry. And such that the exterior power of the F less than mu, which means that all surfaces may become only shorter. Area only goes down under this map. Then this map actually is, well, let's say dimension 2 is something exceptional. It's a symmetry. So you cannot enlarge the metric in the sphere. Yeah, but I'm sorry. This might be spin. Or if you look a little bit better, you are still covering it to be spin. I think it's slightly more technical. But essentially, you need spin condition. In dimensions, in dimensions 4 probably, you can prove it without spin condition. But for other dimensions, you need spin. And this is actually extremely annoying. They need a spin condition. It's such a nice general geometric theorem. It says you cannot enlarge, of course, if you don't change topology, no problem. You cannot enlarge metric in the sphere. Enlarge even not only lengthwise, but only areawise, such that scalar curvature necessarily somewhere goes down. What, by the way, is unknown if you don't say anything about enlargement, which may be true to some indication of that. If you take a small piece of a sphere without caring what happened there, then can you modify metric at all being localized in a small ball? So the scalar curvature goes up. So one knows if you allow this deformation in a hemisphere, it's possible. If you take a sphere, I forgot exactly, of particular radius, you cannot do it by small deformation. But in general, for example, that's unclear. And again, I explained again why it's quite delicate. Because if negative curvature enters, maybe tiny little place where it enters, completely invisible. But somehow, crucial for this phenomenon. So this is a theorem due to the rule. And kind of a big problem is to eliminate here spin. Even if you take maybe just without area, it is, from dimension 5 on, you need spin, strangely enough. And so the point is just, if you again start comparing, and then we ask what if you puncture me, or you take many for slightly smaller than sphere. Take something half-sphere. And just what I shall explain later on, if you take punctured sphere, it's still true. You can't enlarge metric in punctured sphere without scalar curvature going down. But then it also needs spin. But then you do it combining this and minimal surfaces, or rather, these surfaces with this new term. You cannot prove it. So it says eventually even if you take any manifold whatsoever, here any spin eventually is true without spin, you cannot enlarge metric even with boundary. If you start enlarging it, and it has poison scalar curvature, it eventually must become small. But exact bounds for that are not terribly clear. And so this will be kind of what I'm going to prove in my following collection varies instances of that. So that's the situation. But again, I want to emphasize the following point. So because in this index theorem, there are two terms. One is a roof of manifold, and another is churn of the bundle by which you twist. And this one, of course, can spin inevitable here. In all results, when the region did not know this term, only that one, we know theorems are not true without assumption of being spin. Easily there are count examples. On the other hand, if you subtract a relative case and this disappears, and only churn plays the true role with your proof not vanishing, actually by adapting some vector bundle with churn bundle, spin seems to be irrelevant. And so the theory becomes very close to what you have obtained by minimal surfaces. So the whole theory definitely bifurcated at some point. So here either you have this pi 1, non-zero, and spin probably irrelevant, sometimes questionable. And another concerning manifold, which are indeed spin and using kind of really full differential topology. And when you go in this direction, it seems differential topology become completely more or less irrelevant. You use kind of coarse geometry rather than all homotopy theory, and differential thing disappear. Of course, exceptional is dimension 4. And some more special is dimension 3. They're special, and they're not entering here. Now, so how this kind of geometry can be encoded, how we can see this Dirac operator even before they come into play? Because you see this as they look extremely artificial. They still look like artificial. The minimal surfaces, geometrically, don't look artificial, they're natural objects. The remarkable thing, of course, there is this formula. As for second variation, I repeat, it is just what enters is scalar x, scalar y minus scalar x. The scalar of the ambient manifold is not absolutely obvious why it enters. It's little computation. And I think it was this kind of formula which I was writing, more or less. It equal, this was the formula which is little algebra which interferes. And when you use this formula for the Dirac operator, it's kind of similar computation involved. Though here you spin this in Clifford algebras, but it's, again, some symmetrization argument involved. And the question is, I think the main issue in my view the most kind of challenging and annoying, is actually Dirac link between the two. So you see the algebra is similar. But again, there is no even algebraically kind of absolutely parallel treatment. And so from this formula, which is purely if you look at the bottom, it's algebraic formula. Analysis is not involved. Analysis was involved when you relate d and d squared. So why vanishing of this kernel of d squared and vanishing of d essentially the same? For that, you need manifold to be either compact or under some case complete. And so you have index theorem. But this formula was purely algebraic. Also here, this formula has nothing to do with analysis, with integration, with minimization. Just pure algebra, depending on pure linear algebra. And so anytime you have such a nice linear algebra, it's usually have manifestation. In geometry, probably it must have kind of extension here also, which kind of they might be linked by something algebraically more general. And then you can guess what can be analysis behind it. But anyway, it's kind of general. How can you, and say something about that for Dirac operator and other operators? And so what is this geometry which being involved? And how you can say it? And now I want to say a few words about the raw algebras. So again, I want to point out that there is a really kind of wide spectrum of things. So I was speaking about positive mass and relativity. And now we come to this amusing algebra. How we can encode a coarse geometry of space x, symmetric space, algebraically? And why differential or some elliptical operators enter this game? And so which in particular, when you have a group, when you have a space x, and it x is by a group gamma, gamma x by asymmetries, a discrete group is compact motion. Then x and gamma are closely related. They are kind of geometrically very similar. On the large scale, if you look from FA, all points merge together and you see your x. And so this space is kind of roughly the same. And now, but there is a nice way to linearize it and go to the framework of sister algebras. And this is what I want to explain now. And then differential operators, or elliptical operators will be formally linked to that. So as I mentioned before, this here, prior to that I say just one word again. So I was describing all these formulas, et cetera. But then there is formalization of what I say in terms of sister algebras. So you don't have to know what they are. There's some infinite dimensional objects, linear algebraic objects, period. But they serve some purpose of the following kind. The Hilbert space, for me, this algebra is a Hilbert space with some extra structure. It has some extra structure, which is make it look like finite dimensional space, though the infinite dimensional. And as far as I do know of conjecture, or our problem about scary curve, which is concerned, you can say, aha, I was Dirac operator twisting with finite dimensional bundles. Now there may be infinite dimensional. There must be of this kind. It is a Hilbert space acted by sister algebra. And then index still well-defined. There is an index that may be 0 and non-zero. But now it takes values, not numbers. Like in some example, what we had before, we have family of vector bundles over torus. Family of flat vector bundles was dual torus. And then index was a family of linear spaces over this base parameter space, which was, again, a vector bundle. So it was element of k theory of this modular space. Yeah, so it's class of these vector bundles. And this will be the same, but in the kind of philosophy of today, it will be like non-community of space. So they represent this object family as if it's family of bundles over some non-community of space. But again, it has non-trivial characteristic class, and it may be 0 and non-zero. But you don't know much about this. I described finite dimensional picture, and this is just formal generalization of that. There's a limit of that. And so the formalism is follows. So you can see that Dirac twisted with this bundle. And there is an index theorem saying the index theorem properly formulates through index maybe 0 and non-zero. But the point of being non-zero, it says necessarily there must exist spin satisfying Dirac equation. Now it's coefficient with a bundle, but with many, many Dirac. Therefore, if you can produce on such a manifold, such a sister algebra, such that the index of the twist operator non-zero, there is no scalar coverage, no positive scalar coverage. So you have to produce, and the old construction I described before, this bundle is almost flat. If your epsilon goes to 0, you go to the limit and have this infinite dimensional object. As far as example are concerned, I don't think it gives you any new examples. At least I've never seen one, right? But lots of different how I look at the version of that there. But the point is that it's very advantageous still to look at other sister algebras, which are not coming, but this limit process. So you don't just add infinitely many something, but something else. And this is one way to proceed. And conjecturally, there are enough of sister algebras that are always such a flat bundle over any space, which represents, which has sufficiently many to represent non-zero sections of everywhere. So it's lots of them, and they allow you, prevent you from scalar coverage, which you don't need too many. Or for no kind of conjecture when you need a wide spectrum of them to represent all k, 3, or all co-homologes or homologes of the space. But now come a row algebra, which I think is very good, very nice class of the sister algebras. So it generalizes what happens for the sister algebras of groups. So first, let's do it geometrically. So given metric space x, this kind of a semi-group, called a row semi-group, which encodes very nicely large-scale geometry of the space. And this is very kind of simple semi-group. You consider just self-mappings. Then you kind of have to a little bit limit them, but they don't have any continuous from x to x, such that distance of x to fx less or equal to c infinity. Constant depends on f, but doesn't depend on x. So you move everything by finite amount. If you compose such maps, then you get a map of this type. So motivation may be as follows. If you have a group, you have an infinite group, and you think about this group as a metric space, which act on a sub-biasometer. You have a group, and if you take a distance on a finite set, they find to generate, then translate it. So you have a distance defined there. Of course, this translation, except group, is a billion, turns things a lot. But here, there are two actions. This is a horrible thing to say. There is left action, and there is right action. And nobody knows what it is. And actually, it's not a definable notion. This, by the way, is one problem with mathematics. You cannot say rigorously what it means, left action and right action. If you have my left hand, the right hand, so how we say it. If you space to speak somebody in the other universe where they have a break of the symmetry, as you know it's possible. And so you cannot tell them, and just what's left or right. So what does it actually do? Somebody knows how to formalize it here. So it's indeed, it's kind of tricky. And actually, there are two actions. There are two, we know, from different sides. But which is left, which is right, is, of course, but again, in this geometric framework, one action turns things like that. And another action moves everything by a fine amount. And therefore, the semi-group is extension. In this example, just extension of the whole group. And this perfectly, I think, is one to one. So I guess, I haven't thought about this carefully. But I think it's easy to say that if you have corresponding between two spaces, such that points which were within certain distance remain within bounded distance, remain within bounded distance, if and only if these two, if these two semi-groups, well, in some sense equivalent. I wouldn't say some of them here. Might be careful, but kind of equivalent in certain sense. And but this is still, this is still kind of groups. And how we can go to algebra. But already, from some moment on, we can say, you can forget, you want to describe geometry, your space for this purpose in terms of the semi-group. So you have this manifold, you have a semi-group, you forgot about manifold. You only have the semi-group. In a second, you have to add something to that. And what you do, you make out of this sister algebra. But the problem is, first, you have to make it to act on some Hilbert space. And here is this little problem, which I must admit I don't fully understand. But if the space is discrete, then there is an underlying measure, discrete measure on your space. And then there is Hilbert space, and then there is act on this space. So we have semi-group of operators on the space, and then we can complete it by sister algebra. So what is completion? You can take linear span with that, but completion will take operator norm. So two things are close, if on every vector they act closely. And this is very weak norm. And then remarkable, so we have huge algebra, but still it carries enough structure to remember about your space. So how much, now, after this completion, how much of the geometry remains? I think there are many conjectures, which I don't know too well. But the point is that you can now consider spaces which out have infinite spaces, and without any action or any group of them, so really open manifolds, and formulate existence or nonexistence of medical poetry in terms of Dirac operator to visit with this algebra on this space. Of course, they are not compact, so I might be careful. However, it works, and so let me give an example and give some motivation so what you can see by elementary means and whatnot. So again, as usual, so when you go to the bottom, something rather simple being concorded, but sometime in a way you cannot see in the more elementary ways. So it would be, I was saying this kind of, imagine you have this kind of space which is infinite in one direction, it has this thing, and imagine it has scalar curvature greater than a constant. You have your right sigma. So I want to rule it out by some condition and principle in my case, easier space. But some pose just for simplicity, that is very topological in a simple space, like torus, time, poise de frais. And let me just give some partial reasoning why it should be true. Actually, I must admit, I never fully understood this argument. I understand in some basic cases, and there is alternative proof of that, but this is quite instructive. So what you do? So the theorem is that if you take this, you take the algebra of this space of this space, this row algebra, and twist it with c star algebra of that, then it has no zero index. So you have to make some computation, you have this kind of, and this c star algebra convenient, but the way of this geometric construction, they're a little bit harder to manipulate, yeah? Like this construction, you cannot take the tensor product immediately, you have to say many words. In c star algebra, everything goes into the one board. And I said before, in groups you have to keep track, you cannot keep, everything's flat, you cannot say thing approximate. And this algebra partly can recapture that, yeah? So many folks doesn't have to be, doesn't have to be symmetric in a sense. No groups should act, but the semi-group always act there, right on this space. And this semi-group, what matters? This kind of translation, semi-group. And so, but how we can see it from a different perspective? So you have this DRock operator here, imagine a really kind of this tube-like shape. And you have this DRock operator, and you can imagine that you take, if you have a story, you take finite covering, so it's kind of big and thick tube, yeah? It's kind of huge tube, and it goes, goes, goes, goes, goes. How DRock operator can prevent you from happening that? Because DRock operator feel very badly at this corner, it's not defined because of this boundary. So just stupid thing, you just double it. Now DRock operator is defined, but of course there is no control of the geometry here. Scaly curvature will be negative here, right? So you only can use vanishing property, this formula set infinity, but there will be some error coming in there. So what you can do there, before what I was saying, I was having manifold X map to the sphere, with degree non-zero, which is enough to have some contribution on non-zero index. But now, because I have such a huge big thing, I can have degree greater than anything I want. So it may be huge. So I map it to the degree many, many times. And when it happens, of course contribution to the index, index become bigger, bigger proportionally. So it means on this space, if it have such a big and thick space, after twisting with this bundle, which is kind of minor, minor point, you have huge number of sections. So here, and everywhere here, your formula says, ah, ah, they impossible. Therefore, all this section might be concentrated to this region, where we have absolutely no control, but there are many of them. And here still we have this formula. We have d recuperator squared equal. Poiseve operator plus one quarter of scale. Here scale is negative, so it's minus one. And so you compare with this operator, but this operator also has only so many eigenvalues below this level. So there's a limit for that. So say, oh, it's impossible. However, there is some weakness to this argument, which says, ah, that's good if you have this kind of picture through tube. But theorem is true without that. Your manifold may become more and more narrow to infinity. My exponentially becomes small. And when you apply this kind of trick and do the same, you have to take high and high covering and constant become bigger, bigger, bigger, and you know where. However, and this is the point of this, just alibis, you can say, ah, no, the whole thing, you can go to this ideal limit, so to speak. You cannot go it on the level of operators, but on the level of formalism or algebra can pretend everything happens over there. And this becomes so far, this contribution still will be small. But then again, this is a kind of nice point. However, I mean, I must admit, I haven't followed this argument to the end, but there is a, and still we need to see a spin condition. You can prove it without spin, but using minimal hypothesis. So strangely enough, you can recapture all these kind of things from a raw algebra for minimal hypothesis. However, directly, it's very unclear what they tell in this language. But I think it's extremely nice, still very nice point of view, which allows you to speak about spaces in purely algebraic terms on the large scale by using this first semi-group and then this algebra. And there are, of course, the question of how you completed and technically what happens, you don't work with Dirac operator, but work with some kind of localized version of Dirac operator. So there is kind of a little bit of analysis involved. And I hope maybe too, but at some moment I will understand it exactly, because if you feel very uncomfortable with this argument exactly, because kind of this contradiction, but then the posteriori, you know, it's true anyway from a different perspective. So this argument must work somehow, but what, and that's what's so interesting, of course, the spectral issues enter here very kind of essentially, it's not just index of the operator, but the whole spectrum enters in this geometry. And so positivity of the scalar curvature is just something on the bottom of the spectrum. And the rest of the curvature talks to the other part of the spectrum of various operators or geometric object like these minimal surfaces where again in a small dimension spectral, there is a kind of indices of Morse indices, which correspond to the spectral enter. So this will be for the next time. But again, if we have questions, just ask me, because I know it's maybe not always understandable. Yeah, I'll put on the web something or I said already, I put something and I keep kind of refreshing it. And many of course issues you're not able to describe discuss here, but because I put in there, you can ask me about what is written there, even not what was not said here. And I think the fun of that is that this really goes in different directions and they must converge and they diverge psychologically because some people want relativity and they never heard of system of the index operator and other people do other ones. It's amazing how servers are written. They have servers here in curvature and each text, I never saw anything covering what I did. For example, the theorem of La Roule and there is your generalization, I think it's absolutely fantastic theorem. The proof is very simple except some calculation but it's completely remarkable theorem in simplicity. And if you don't know nobody mentioned to you deregoperator just say on a sphere you can enlarge the metric without changing the scale which you even just having graph of a function already there. Absolutely hopeless. No geometric means otherwise we have to prove it. You say deregoperator comes in spontaneously and it's geometric theorem and I bring for some other instances of that. Do something very elementary geometric theorem and you never guess if you don't know a priori that you have to use these techniques. I mean, in some of them even don't manage to scare the curvature. You say I was giving some examples last time but it needs to scare the curvature involved. So, but again the point is there is two ways to think about that. Two opposite and both very valid kind of point of view. One is global, you look from infinity and so you see on the cost properties and want to in core and kind of relate them to curvature. In the opposite, the infinitesimal calculus they're just nice formulas and one there they have some meaning but you have to know them and organize them in a certain way and it's like they're very simple formulas like this. Bohm's formula, some, you see this. Clifford algebra's or just Gauss formulas and relate. They're all related to symmetry and representation of certain simple groups and they give you this very nice formulas but understanding their kind of development is not apparent.