 And so when you speak about general, about the Scaly Covici, there are two kind of perspectives you may have. And one depends kind of results to prove all the methods you use. And they just lead you in some way in different directions. So let me remind you what we're talking about this kind of topological. So where topology and geometry kind of become diverged. So topological, and I will call them negative, yeah, negative topological results. And this concerns non-existent theorems, so I'll give many a fault on certain topology of that, that it cannot have metric with poetry Scaly Covici. And this was actually up to some point of a starting topics in this metric from point of view of geometers. But on the other hand, there were these kind of coming from general relativity, there were positive energy or mass. It was conjecture and then there was a theorem and then there were various variations of that which were geometric. But they were quite, but they were quite, quite special. So let me remind what was happening here. And so what are the status of that? So one related to minimal surfaces, which I come to later, and this was partly motivated by and inspired by physics, I believe, yeah, which I couldn't extract exactly. This was what was happening in physical literature, though I think many things were already there. But let me describe this topological approach coming from Lucic and I described how we can prove that on the n-dimensional torus, why it admits no metric with point of Scaly Covici, and the logic is as follows. So first, you start DRock operator. So again, let me remind, what is DRock operator? There is certain bundle, which is spin a bundle over there, where it acts. What is essential, this bundle splits like that, so it is sum of two terms, for even dimension. And observe again, it's an artifact, I hate saying that. If you prove for even dimension, you automatically prove for odd dimension because, right? So we have torus of dimension to n plus one, you multiply it by one dimensional torus. If this has point of curvature, this has point of curvature, you reduce odd dimension, in case of even dimension, but this got more wrong, of course, yeah? You shouldn't do that, right? And in fact, if you do it correctly, you don't have to distinguish dimensions. Well, but anyway, so it's even dimensional, so the splitting. So this spin a bundle splits into plus and minus, and the operator exchanges paths. Because at the stands, the full operator itself are joined, so its index is zero. But because it consists of these two paths, the index is mutually canceled, but each of them has no zero index. And usually, the money is taken to here, we call it D plus. And that's essential because the rest of the manipulation is dero-cooperator, which you produce new elliptic operator, which is self-adjoined, but doesn't split. Which index zero, but still can be used for certain purposes. As was, I think, first observed by Minu for the rigidity of complex, for real hyperbolic spaces, for complex that's still unknown, I guess. Okay, so that's the point. So if this operator, whatever it is, you know, by the index theorem, which is kind of give you a simple, okay, a verifiable criterion of vanishing, or unvanishing this index. Then, you know, there are these harmonics, spinous, winding, kernel of that. But the way when people speak about the history of the, of the index theorem, they, of course, was proven by, I think, in, in, in, in, in 63. And it is announced, and then the proof we're appearing in various generality. Prior to that, several years before, equation was read by Gelfand, who, observing that index of elliptic operators, for the home operators in particular, in general, invariant under deformation, under homotopes. And therefore, it is a question of what it is, this index. And nowadays, there's just one line kind of argument. They say, you have denoted that. However, interestingly enough, there is a paper by Aleksandrov, when he proved his famous, give one of the two proofs, of this Aleksandrov-Fengelen equality, or basic inequality in convexity. And it's about 50 pages paper. In 49 pages, he proved the certain index invariant. Some particular elliptic operator, he proved the index invariant under homotopes. Right, so it was a, and I don't know if it was kind of, the first paper when I've seen it was certainly in, in 30s or something, 35. And so it is a really kind of simple, but fundamental fact. And then, making formulae, you can say, well, just trivial. You just look at sufficiently many examples, when you can compute it, and then deform it. And so, in, in this way, it was done by some people in Russia, just maybe, maybe 30 before we're at the sinker, but they made mistake, they, they lost some factorial coefficients. So the formula was wrong. So because there was some tricky formula, which involved algebraic formula, involved kind of funny numbers involved, which are not so easy to justify. Kind of, I, I probably, but anyway, you know, maybe zero, not zero. But for torus, okay, everything is zero. It's parallelizable, there is no kind of, so index should be zero. On the other hand, you do know that torus do have harmonic spinus, namely a parallel one, right? But so what, but they certainly have zero index. But if you deform the torus, this parallel spinus disappear. But the new spinus appear, but now we are not parallel for the flat structure. But you consider twisting it with a linear bundle over the torus. You just, you have a coefficient of a spinus with linear bundle, which is flat, but not have zero connection, but non-trivial. Which corresponds to non-trivial characters of the fundamental group. And so, yes, when you start deforming it, it does not disappear, it just shifts. And there is a more general theorem for family saying, yeah, it's there. So in some generalized sense, index is non-zero, but you have to understand the family. And in the family, it takes values, not in numbers, but you have the parameter space of all these line bundles. So these interlay bundles, it happens to be, again, a torus. Kind of dual torus, torus of characters of the fundamental group. So now, for every moment, for every position, you have the Syripti co-operator. It has kernel and co-kernel, take their difference. And so you have difference of two bundles. So if you think about the virtual bundle of a torus, which you have to make little effort to make sense of this, it's called variable. It's element of K theory. It's characteristic places define your index. And for this example, you can compute again. Also for families, it's kind of exercise and analysis to show index is invariant. This K-theatric element, when you deform this metric on the manifold, this vector bundle kind of changes, but only by homotopy. And therefore, it's representative of K theory doesn't change. And so here you can compute explicitly. You don't have to know index theorem for that, yeah? And then it follows that torus doesn't have metric or Poisson-Killicovich, right? And this is a kind of kind of elementary argument. But if you a little bit go the next step, then you see more. If you instead of this torus, you take manifold dimension and map here. And the degree of the map is not zero. Then by index theorem, still you have here everything provided this manifold spin, and this is a very annoying condition. Because we don't, you can't exclude it in this context. And so in this way, you prove that even if it's sort of torus, but had a rather complicated manifold, but spin. Then it has no metric or Poisson-Killicovich, but if you make, for example, in dimension four, if you just take connected sum of this, this Cp2, then you don't know because it's not spin, you cannot, Cp2 is, keep forgetting, Cp2 is non-spin here, just one by one, yeah? And so you cannot tell, cannot tell by this method if it has metric or not of Poisson-Killicovich. However, a posteriori, actually a priori it was proven by this dimension, but it's okay, it doesn't have this metric. And it's extremely unclear what is the role of the spin in here. And then from that, the next level of generalization, you can go in two directions. I don't know, maybe I give you another, there is another argument. Now I want to give you another argument where you carry a little bit geometry, because this is absolutely non-geometric because it depends on this very special property of the fundamental group that has this representations, there are flat bundles, so it's a very special property. Imagine you have many fault which the fundamental group has no, find dimensional representations at all, yeah, which things abundant, but then you can proceed in a different way, in a more geometric way, in which leaves you in a somewhere else. As follows, you go to the Euclidean space which covers, and we temporarily, I don't do that here, just to make it simpler, I take the high order covering of the stores, so that these stores become very, very big. We're getting bigger and bigger and bigger, kind of on the limit you have kind of Euclidean space, but so everywhere locally it looks like this Euclidean space on the certain scale, but now on the Euclidean space what you do, you can see the bundle which is flat in infinity, it's supported some way here, and again assuming dimension, dimension is even because it's zero over there, you can think it's coming from the sphere, and here on the sphere it has non-trivial churn class, top dimensional churn class, even dimensional. For example, in dimension two, if it were in dimension two, which is a quite easy case, you just take, pull back with some bundle, it's like a whole bundle, it's optimal in this situation. Now, and then you just do this on the torus, so to have closed manifold, again pure technicality, but essential, when you see it, it's kind of not pure technicality when you look further, but it's at this stage a minor issue, so you produced a bundle over the torus, but now because of so Euclidean space you can stretch it, Euclidean, you can scale it, and so this bundle become, this picture becomes like that, so the bundle become, when you sketch it, sketch it, sketch it, it has the same churn class, but its geometry become asymptotically flat, when you spread it it becomes indistinguishable from flat, of course to implement on the torus you have to go to high and high, high covering, and then become eventually flat, and if you twist your zero cooperative with this bundle, it will not notice it was non-flat, locally, because, because it's almost flat, just obituary small error, and this error becomes smaller than Scaly Kovachev there if you assume it was positive, now assume Scaly Kovachev was positive, you assume this error much smaller than Scaly Kovachev, Scaly Kovachev wasn't remained, because I just took covering of the torus, I didn't change Scaly Kovachev, it was normalized to be everywhere at least plus one, on the other hand the bundle become absolutely flat everywhere, and then in this basic formula the G rock squared equals this positive operator squared plus one quarter of Scaly Kovachev, plus error term coming from the bundle called the bundle L, it's Kovachev, and this term become epsilon, this goes to zero, you don't have to know what it is, bundle just disappear becomes zero, so you can forget it, and still you apply this formula, it's still positive, there is no spinners, but of course topologically nothing changed, just my perspective has changed, so spinners must be there, so observe this two, this is another proof saying there is no thing on the torus, I observe it's a different proof, because you use completely different spinners in the universal covering, in the first case of the spinners if you leave them from the flat torus, they were just trivial bundle, right, when you take this line bundle which was flat, when you look at the upstairs it becomes trivial bundle, so you prove and this kind of spinners were kind of spreading like that, and here they're quite different, yeah, these are different spinners, you know if you see what's happening upstairs they're different, so I think there are three, four, five proofs like that, so by index theorem you prove that there are spinners of certain kind, and then you show they cannot be there if Scaly Kovachev is positive, and now you can go in two directions, and so what you can prove with that in general, and so one as I was saying, related to C star algebras, and so what you do, and this was became dominant in because related to Novyk of Conjecture, and so you can see the infinite dimensional flat bundles, you can see the infinite dimensional representation of your fundamental group doing like regular representation, you're already inside of this, it may harbor a lot of kind of non-trivial, non-trivial spinners, so if you consider this kind of bundles of your manifold and takes again, and the point is if you do it correctly, these concepts of a family of bundles being substituted by family of C star algebra, we should think as a family is kind of this fixed space, non-community space corresponding to this non-community algebra, sometimes you can show index again non-zero, and so there is a big conjecture saying that for all groups properly defined index is non-zero, therefore you can have medical positive character. However, when you start to specifically verify it, it boils down usually always to the same condition, at least there is no county example for R&O, and so it's possible, which I try to justify, that all this activity boils in for positive character in kind of fixtures, if you look whatever the proof is more or less obvious from different perspective, right, maybe, but just again it depends on conjecture, but in some cases it is solved, because, because of the folding thing, so let me give kind of specific argument, understand how it works, because the knowledge of conjecture, so next case after Lustig who proved it, so again, so what Lustig proved is that if you have this n-dimensional torus and take any manifold and map it here, and then they take pullback of this point, assuming this was n plus 4k, so I have this generic point, take a pullback, it's sitting here, and this is a 4k dimensional manifold, it's here the signature, and the signature is homotopy invariant of this manifold, if you take another small structure and have the same homotopy class of maps, this doesn't change, and this is kind of was novice of conjecture, it provides all high signatures, and if this were simply connected manifold, this you can do the same, and then it takes pullback, you know it's never true, right, if it is simply connected manifold, take pullback of the point, you can change structure, you take essentially any values, just divisible by something, right, there's no, the full flexibility of these invariants, this is the corollary of the Neukel-Brouwer theory, which is quite, quite simple, but the way it's really very simple, because you're also giving lectures on many faults in some Poincare symposium, improving where you think, you prove almost from zero everything in one hour, including this, but two things you cannot prove, two things are non-trivial, it's first one Poincare duality, and certainly third theorem of finite entomotopy groups, granted that all the rest of topology is topological, you just don't have to think, it's just some little, using some idea of Poincare, which is, you know, because if you use a homological algebra, homological theory, it takes several pages to do, but if you use the language adopted by Poincare, it becomes topology, but then at some moment people, at some time, I believe, I'm not studying, because carefully people thought it's rigorous, but it was just because language was not, proper language was not there at that moment, and the right concept, okay, this is not the issue, so once it's very elementary, Brouwer, Neukel theory is just kind of trivial stuff, and once you know it, of course, it's trivial, and here, because we still don't know the answer, it's non-trivial, and so the conjecture, Neukel conjecture, particular instance of which is unknown, if it is true, if here we have arbitrarily a spherical manifold, which is, meaning universal covering is contractible, and but this conjecture can still be kind of fundamental class, and it's, and the same can formulate for any cohomology class inside, and then the Neukel conjecture, and then if you look carefully, it has nothing to do with, it can be reformulated, it's some property of the fundamental group, which makes sense for any fundamental group, and particularly in this particular instance, it says something of the following kind, so we just formulate another shape of this conjecture, so given a group, it's corollary of the conjecture, but it is essentially as kind of difficult, as an approachable, as a general case, so given a group, you can form the, I think say a real ring of this group, maybe complex better, and then there's also conjugation, right, you can may conjugate and replace gamma element by its reverse, so it's a group ring, it's conjugation, and then you think about this ring, and you can see the width group over this ring, see it's considered of quadratic forms, you have to, might be Hermitian forms, on many variables, and there is usual rules, so if a, so we read the sum here, we can sum them, and if the group can be diagonalized without middle terms, you can see the trivial, right, it's kind of definition of width group, and so this is the kind of group group, and the point is it's never trivial, so if they say, if the group has homology in dimension 4k, non-trivial rational homology in dimension 4k, this group is non-zero, and that's the point, it has nothing to do with manifold, and it's quite difficult in a specific example, and if there is a group, and the group has non-trivial homology in dimension 4k, then if you take the width group over this group ring, right, you can see it's group ring, but you must like careful, there must be some involution there, some extra structure, so you can see the Hermitian, you can speak about Hermitian form, not just quadratic, but Hermitian, and you can see the width group, so it is a, you can see the quadratic form over that, and there's summation, you just add one to another, and there is a conception over trivial, or trivial one, and trivial one is just some subject like a squared minus b squared, but the tricky point is another identification, when you change coordinates, and then because you write it, it's in a ring, like always in some base like that, in another base it's another, you have to bring it to this shape in some change of coordinates, and this change of coordinates for non-commutative cases is a whole mess, that's kind of easy part, but who knows, I write you this quadratic form, how you can check that there is no some change of coordinates become trivial, and then essentially all arguments use the index theorem, implicitly, explicitly, you reduce it to something which you can get visualized, but if you look closely from certain perspective I'll show you, it becomes geometric property, and this geometric property can be proven, I guess in this case, by just, by very elementary means, essentially kind of simplified version of the original proof of Novikov topological invariance, right, and for Schiele-Kovich it's most transparent, so this I shall want to explain to you, these things are parallel, they diverge at some point, but so far I think the two subject matters very much close to another, so what you do in Schiele-Kovich, so again, so there was this history, and interestingly, of course, other people, first there was Novikov, then there was Leustich, and then there was Pepe by Mischenke, and then Mischenke proven that if you have many fault of non-positive curvature, then its fundamental group satisfies Novikov conjecture, so in particular, if it will close manifold, and you map another manifold, take pullback of the point, et cetera, signature will be invariant, but if he proved for all homology, and there is some little difference between fundamental class and non-fundamental class, so in how you prove the corresponding statement in the Schiele-Kovich, which is Schiele-Kovich, some of the statements are actually easier to prove, but if you look closer, they, of course, more or less become virtually the same, so if you have many, so we only just now show, we have many fault, which has metric of non-positive curvature, the simplest version of that, and I want to show it's impossible to construct metric with Poisson's character, so they are mutually incompatible. Now the argument we had with Leustich doesn't work, because you know nothing about this fundamental group, you don't know there are any non-trivial representation of this at all, though, I posteriori, you know there are this infinite dimensional representation, a regular representation carries enough structure, but let's do it more kind of geometrically, but what you do know that if you go to the universal covering, and you still can put this kind of a bundle, and because you know that this hyperbolic space universal covering of this manifold admits a map to the equilibrium space, dimension n, dimension n, which can, which goes kind of by homeomorphism, but essentially by, it's proper map of degree one, and it contracts as much as you want. Therefore if you take this kind of a bundle of Euclidean space, non-trivial, again assume dimension even, take a bundle localized infinity, non-trivial chain class, and make this map very much contracting, right? It's the most exponential map, of course, for negative curvature, but any map of this type we'll do, and pull back this bundle. So we create, now in this universal covering, you create a bundle which if you twist your operator with this bundle, you would make non-zero, it would have non-zero index, you would have harmonic spinus, but this harmonic spinus a priori will leave on this space, not on this space, on the universal covering, which is minor point. Again, it's some technical point which we can resolve in C-steel index theorem works in this context, though many fall non-compact, you might be careful, but if the group where you see the finite, if you could construct finite coverings here, which unravel this manifold, of course it can transplant it back here, and use usual index theorem, but you can also do that, and so the point is here you use bundle which nearly flat, and unlike flat, and the proof works pretty well, and again it seems to me you have index theorem, but then how we can de-huptch it with minimal surfaces. So the point I'm saying that in fact with minimal surfaces you can prove better statements as far as telecoach is concerned, and I think if you properly transform minimal surfaces to this language, for example the correspondence here would be, this was what the Schoen-Yau proof, they exactly followed the logic of the Novikov proof, they just make geometric, there is some analysis there, but the logic of the proof essentially parallel to the Novikov, and then if you kind of go backwards it's just what I'm going to say, you have translation again back to the poetry, and the point is that by using minimal surfaces in a second to explain how it works, what you can prove that you have any manifold with boundary, and you know it's scaley curvature is greater than say, then one, the manifold cannot be too large, cannot be too large means you cannot map it, there is no map to the sphere of, and the most, such as boundary goes to one point, and the map have non-zero degree, and simultaneously contractions, strongly contractions, so it cannot spread in all directions, too much, and then of course, you know because this exactly, if it has this negative curvature, it does spread a lot, if you change metric you only change by a fine amount, so this implied that, and so I'm using point, I'm using point because we have this Pepe Blaine Lawson about, you know in 83, so it's quite a while ago. Can you say again, the assumptions on the map to SN, you want them to have? The map to the sense of the boundary, so this manifold called also X, boundary goes to one point, X goes to SN, map has non-zero degree, and so for example, its lipschitz constant is less than one van, you know, actually for the moment you know almost shop constant, right, up to one half, you might be careful, so it's better to say the scalar curvature greater than n, n minus one, so it's the same scalar curvature here, and then actually you can make it to the other half, you cannot have such map, and then of course everything follows, but what's amusing I just realized is actually last night in our Pepe Blaine we have different methods and two different pages, yes two pages apart, we give version of this argument and this argument and we get not just one implies another, kind of funny, because you can use different scaling, somewhere it was big number here, small number here, so pictures were quite different mental pictures, and but in interesting enough this, I realized only last night when I was exactly trying to explain similarity between the two results and they realized one implies another, so my, and so let me explain slightly more general kind of statement, which is more interesting and which is more general and which is more relevant to non-nuclear conjecture, and this will be as follows, when actually need slightly kind of more subtle version of all that, and it is as follows, so now today I want to make a break, last time I didn't make a break and it was kind of too tiring to everybody, so what I want to now I want to prove, so yes I just consider rather simple case, so here there will be manifold of negative curvature compact of these dimensions, background manifold, and this will be n-dimensional manifold close and there is a map, and here is picture this is bigger, so and we assume the map is not homologous to zero, so the image fundamental, so it's orientable manifold, this you don't have to say it, and so when I take the image of the fundamental class in here, so it will be hn of this manifold, it is non-zero, okay, then this manifold has no metric of positive scalar curvature, so I want to put it in this setting, so what I want to show from this data again we use it to geometric situation, and this is by the way, this was context where it was proven by by by by by by by mission kind of conjecture exactly in this setting, what I'm construction is, kind of geometric extract, what was done there, because there it was again on the language of bundles, Fred Holden. So you have no geometric assumption on the on the target? No, no, this is a manifold, I'm sorry you don't say it's again section curvature non-positive, I will make in the in the course of the argument, I make some simplifying assumptions to just make it clearer, so what happens, but this is essential, it's again, but the point is now it's not going to the top dimension, it goes inside, and there is no some manifold of negative curvature inside, however the same would work, so what you want to derive from that, so what you want to conclude is that there is sequence of, now make some assumption here, and the assumption just to simplify because otherwise again you have to go to non-compact space and you have to develop terminologies, not that you have to produce very something radically new, I assume that this group is easily finite, so I assume that this manifold has coverings, which kind of such that the inside they have bigger and bigger balls, which is more or less than the balls, and then using that I take a corresponding cover and I assume that this mapping being just embedded, it's just embedded there, and so I have this manifold and here is this embedding, and it's non-homologous to zero, so there is complementary kind of, where on credibility another manifold has non-zero intersection index with that, so we have this manifold, lift of my manifold, and then this, the one which intersected, and I take this product, so I look at discovering and they also have my, lift of my manifold, and I lift of another manifold, and what I am claiming is that I take high and high order covering, and I want to map it to the sphere, and now I want to use the following kind of normalization, so it's convenient to do this way, so it will be sphere of dimension this. So XI-Octagonal is what? This is just manifold which has non-zero intersection index with that, of complementary dimension kind of, for pipe on credibility, and their product is mapped to this sphere, and this is of this manifold, which has non-zero degree at point number one, and such that along this fibers its distance is contracting, and no assumption along this direction, but on each of them that it's remaining in metric with respect to this remaining in metric induced from, from here, it's contracting, and so it's kind of, kind of again, it's at least a little technical point how you do that, so you have this manifold downstairs, and you go into discovering, they lift upstairs, and so you, you, you restrict it to every ball, and every ball you just, you collapse all this bolt on the sphere, and then you move second parameter, and because they have non-zero intersection index, this degree will be exactly the intersection index, it's kind of elementary topology. There's some little cheating I'm making here, because I, I, I, what I secretly assume, that restriction of the tension bundle of ambient big manifold of this one, when it is restricted, this perp is trivial, so all this spheres, all tension space can identify them, it is not true, but it's minor error, it's again, just identification, they use error, negligible error, it disappears when the ball become bigger and bigger, and so this is the picture, and this is a kind of essential property of manifold, here I used universal covering, I used finite covering, but if, if they're not reasonably finite, I, here it will be universal, it will be, not universal, it will be covered, induced by universal covering of this, right, and then I have this map with these properties, and then the point is, when I'm concerned with Scaly Kovache, I always can assume that this main metric here is bigger than I want, I just scale it hugely, because I multiply Scaly Kovache as, when I scale it, this whatever it was with this, Scaly Kovache goes to zero, so I have again many thoughts, the same Scaly Kovache I did, map to the sphere by contracting a map, if n is even, I take here a bundle, with non-zero churn class, pull it back and apply Indic theorem, and if it was non-compact, you need some non-compact version of the Indic theorem, which is again, technicality, which is not quite, as I say, XP, it's technicality, but we look closer, you might be careful, sometimes it's true, sometimes it's not, but here's fine, and then with the Indic theorem, you do it, and now how you can do it with minimal surfaces, so again, what is the fundamental feature which is involved here? It is, I have a manifold X, such that if I take it to universal covering and multiply it by something else, now I'll call it another manifold, which is whatever dimension, this was n, this will be m, this new one admits a map to the sphere of dimension n plus m, such that infinity, this is universal covering, goes to one point, maybe, and it's, here's sphere of radius r, the map is, along this fibers, it's contracting, and r goes to infinity, so no matter how big r, I can construct such a map, and then this has numerical points of Skelly-Kovic by either using, taking pullback or some bundle in the Indic theorem, or using minimal surfaces, which I start now to, I want now to explain. Okay, so I have a manifold, which is, this is a covering of a compact manifold, but this, in fact, yeah, it will be not, very little of this will be used, essentially, of a compact manifold, and then I know that for some magnetic, and then down, so down, depend on this method, I can multiply it by some parameter space, another manifold, map actually doesn't have to be manifold, one can actually, can be just pseudo-manifold, right, it's just a parameter space. It goes to this sphere with non-zero degree, such that r can be making as large as you want, and along these fibers, for any fixed metric, it becomes distant decreasing. Of course, if you fix the metric, and then you choose r, but this doesn't depend which metric you choose, yeah, because it's decreasing as much as you want, so all, any two metrics differ by a constant, and if you apply it, if it happened to have positive scalar curvature, you run to contradiction, because this, you imagine, manifolds make it huge, so on these manifolds you construct this, assuming this number is, even, you always can make it even, because there's M under your control, you can add line, a circle, whatever, take, pull back on this product of this non-trivial bundle, which is as flat as you want, but still having non-trivial, non-trivial, churn class because spread a lot, because this manifold kind of, this may be just contraction, yeah, because this r goes to infinity, you don't have to say this contraction, yeah. If r is fixed, you have contraction, so because very big, this bundle over the sphere is almost flat, right, it takes, you standard any vector bundle on the sphere, makes phase bigger, the same bundle now from position with sphere become essentially flat, so it's almost flat here, and so locally Dirac operator look as if it was a twist, twist, this bundle look as if it's untwisted, so on one hand, by Indic theorem, you must have harmonic experience, on the other hand, the Neurovich-Bochner formula or whatever, says they're not there, and so this is all my understanding, it is, true in the following sense, I can give you instances of theorems, when you can prove it with using vector bundles, I give you in a second, in this Indic theorem, when this thing is invisible, it's not there, however, if you look at any concrete example, it is there, so, but that reason I'm saying that people have abstract theorems, they're given in the, of, and such, such condition, have numerical conjecture, and this formula is very general, however, if you look at any example, much stronger condition is fine, so you cannot find any example when all this big theorem wouldn't follow from very, very easy one, right, maybe they do, maybe they don't, let me give an example, where I don't know kind of, if I can make specific example, when this will not be covered by this scheme, but this is a, you see, this is kind of very, I'm using property, so if this X tilde is universal covering of a manifold X, with some fundamental group gamma, then this is essentially property of the group, you can formulate some property of the group, and it's very hard to kind of relate it to other properties, and the example which I want to say is like that, if you have a manifold such that it has a two-dimensional cohomology class, it's a manifold of dimension 2m, such that this class h to the power m equals the fundamental class, be multiple fundamental class, yeah, so it's a kind of homologically symplectic manifold, dramatically it means that you have a hypersorphy, hypersorphy of four-dimension 2m hypersorphy, m sub-manifold of four-dimension 2 is non-zero intersection index, and such that condition number one, when you go to the universal covering of this, this element becomes zero, so if you can make any manifold where it has a two-dimensional class, which power of each give you a fundamental class, but in the universal covering it disappears, then you can say there is no metric of poise of scalar curvature, but everywhere, whenever you use this irrecuperator, manifold must be spin, by the way, so they always must be spin, yes, manifold downstairs doesn't have to be spin here, but universal covering must be spin, and if it's not, it's not, yes, absolutely, there is no, no, for the moment understanding what happened there, on the other hand when we speak about minimal surfaces you never notice such spin condition doesn't exist, of course with spin you prove more subtler results, you can kind of, you can, even the spin you can prove more about some, there are more subtle invariants which you can detect, and even more so in dimension four, which, but, but well in general we don't know what it is. In your example again, so in order to ensure that you have no poise of scalar curvature, what are the assumptions? So here, again, here we have a chemical homology class, which is here, in such that it's powered equal fundamental class, like some black man, and then, but when you go to the universal covering, this class might become zero, for example, manifold is spherical, it's contractable covering, and everything goes to zero, for example you have four-dimensional manifold, which is a spherical universal covering is, universal covering is, universal covering is contractable, and then, and it has no trivial H2, rational H2, then, of course, by duality square of some class will be non-zero, and it has no medical poise of scalar curvature. Indicate how you deduce in this instance. Yeah, but I, in this case, in this instance, so here the proof is, is based on the fact that you can, is that the, your line, the bundles you use, actually the line bundle, because when you have such a class there is, there is a line bundle, which has no trivial connection. When you go upstairs, they become trivial, the ones become trivial, you can take root of this bundle, you can take N's root, you want to take N's root, its curvature goes down, so take high, high, high root down, and become as flat as you want, and you twist with this bundle, and then you have, but you have to, to see that it's, it's, you need some index theorem, but this kind of rather elementary, the moment you say it, it becomes rather elementary, but there is no such picture, there is no contracting map, it's purely area-wise argument. So geometry behind this approach with index theorem is different from the one with minimal surfaces, even with the proof seemingly identical theorem on the bottom of something else. However, when you specifically come to compact manifold, and you, and you have this property I described, it's possible that still, is covering, still admit this map to a sphere which is contracting, which there is no example, the example which I know, which you can analyze like that, and some example you don't know, but so there is no convincing example saying, aha, this condition is satisfied, and another is not, right? And usually in my impression, people who work in this domain, there was no conjecture, they never try to do that, they hate to do that, yeah? And sometime it happened, they publish a paper, long paper, new condition, and somebody will say, well, it's just trivial, you follow it from another condition, having nothing to do with all this index theories. One instance, however, where it seems to me, at the moment that we're very far from relating this with minimal surfaces, which I shall explain probably after the correction, but I formulate this as the following, which is related to the Ellen Kohn's, what's he called? Longitudinal, I guess, longitudinal or something, index theorem for fallations, and this theorem says in particular, if you, or rather the proof of the theorem as I understood when I read a long time ago, now I say it by memory because they haven't, you know, and mathematics, it doesn't stick to my mind too well, but what it follows, the following kind of a geometric statement, that if I have a manifold X and such that it admits, it's non-compact manifold now, again to Rn, which has a positive degree, proper end distance, it's elliptic, it's a distance decreasing, then it admits no fallations, such that induced metric has positive scalar coverage, and this is a, yes, absolutely kind of, it's certainly geometric property with, it seems to me this is, it's related, it's much stronger than saying that manifold itself has no positive scalar coverage, in positive meaning, scalar, greater, something here you must be uniformly positive here, it's again, it's, there is big difference you're having uniformly positive and non-positive, right? You can make this kind of expanding thing like paraboloid, this curvature is positive, but goes to zero, but you cannot keep it uniformly positive for this shape, and this is a rather significant distinction, and for that, I mean, this kind of geometric picture doesn't quite fit, it's close to come to that, but you just, so far I couldn't really make it using minimal surfaces, it may or may not, but this would be extremely interesting, it's absolutely unclear why it should be true, if you think about, even for fallations when you understand very well, even if you assume kind of leaves compact or something, it's, I mean compact losses, if all leaves have compact losses, it's very unclear what happens, there's some basic issues with understanding geometry of fallations which is missing, kind of by these techniques you get it, but you don't know what it signifies, at least from some perspective, this is a, this is how it goes, okay, so let's now make a break and then after the break, we speak about minimal surfaces, so just, let me remind first how originally this minimal varieties were used here by Sean Yau, and in some sense there is also kind of some parallelism, we take this Sean Yau approach and this untwisted Dirac operator, this gives you topological information but doesn't give you geometric information, but if you slightly modify this, what we're doing, and then you can get also extract some geometry, right, so when you have flat bundles, it's dependent on the fundamental group, you don't see geometry, when you have this almost flatness, almost flatness of the bundle, bundles tells you some existence or non-existence of that, now become a geometric invariant, and though sometimes you can relate the two, and same situation is here, so the logic of the Sean Yau was quite simple, but it was based on previous work by Kajdan Vorna, it's again quite simple, of course once you do that, so what Kajdan Vorna prune is as follows, if you have a compact manifold, this is kind of Romanian metric zero, and it's supposed that the following operator is positive, minus Laplace operator with respect to this metric, plus in some, something like one quarter, it's slightly less than one quarter, yeah, but say one quarter, yeah, it will be slightly different number, depending on the dimension, yeah, but it's approximate or scaling curvature, and then this conformal change with the metric which makes scaling curvature positive, so this already, this is, so what means this operator is positive, it means if I take any function and take this integral, I'm sorry, deep psi squared plus this one quarter n scalar times psi squared, this integral always positive, right, so this term kind of, so it means if there is a little bit of negative curvature, it's okay, but in what sense a little bit, it's not so clear, right, and this by the way we shall turn, if we have time I explain what happens for this with Dirac operator, this little, so how much you allow negative, what kind of negative, it's kind of a very kind of delicate point, and one of them is kind of, appears basically, like in Pindarov's conjecture, when you can precisely say how much negativity you can, you can allow something like negativity, but anyway if, and so what is the proof, so once you have this operator positive, you take the first eigenfunction of that, and the first eigenfunction positive operator is positive, and then you multiply your metric by some f to the power, and I keep forgetting what power you have to take, and if you take the right power and just exactly the one we should get actually when you construct, when you construct, I don't remember if I explained this, Schwarz's metric, so in dimension, I think in dimension four it will be f squared, I guess, yeah, and then you just compute what happens to the metric and a conformal change, and exactly this term jumps up, and when it's positive, there will be some multiple of these functions, some power function, and this thing applied there, and this again is just one line computation, which I just don't know how to make it without computing. Of course, you know, if you look at the examples, it's kind of obvious, and then kind of these principles right, but the point it is right, and there is this coefficient, and what is essential, this coefficient is less than one half, okay, and that's what is essential, and that's if not nothing would work, right, this coefficient appears in this operator, for all dimension, and you know in Dirac operator it's one quarter, here's even smaller than that, but the correct number is one half for some reason, and so that's a little bit annoying, but then on the other hand this kind of plane of numbers is crucial here, and then so what you do, so you have a remaining manifold, and take some minimal sub-manifold, and minimal in some homology class, so it's locally minimizing, co-dimension one, and assume it's smooth since locally minimizing, because locally minimizing when you deform it, you lock back the field with any way, psi is, volume goes up, therefore certain operator, because it's, because it's linearized this equation you see, become, become positive, so I have to compute what the second variation is, and if ambient scalar curvature is positive, this operator is exactly minus laplace, plus one quarter of the scalar curvature of y in the induced metric, and again we just you write this formula, and kind of the point which was missing in earlier papers, my understanding is, of course I have not know what already was in physics literature, but in mathematics literature, this kind of computation in dimension three at least, has been done, but what was missing is the formula that if you like, if you, that, so a Ricci curvature in this direction, how it moves exactly, control the second variation of volume, but the Ricci curvature in this direction, plus all curvatures together in this direction give scalar curvature of the ambient manifold, so if you subtract the two, you come to scalar curvature of this manifold, so you can express it as scalar curvature of some manifold, rather than Ricci curvature in the ambient manifold, and with Ricci curvature of ambient manifold, I just, I posted it, there was an old paper by Buraga, Buraga, and the Panagov, when they do something about three-dimensional manifolds, and it was for certain bound on Ricci curvatures that they're showing, they cannot have too short geodesics, it's much more subtle geometric argument, but this algebra was missing there, not, they didn't need it, and they, but if they realized that, that would have more stronger consequences. Another interesting point, historically, when people doing these minimal surfaces, they're very much preoccupied with even for dimension two, what how do you think Goliari's were there? Which is strange, because already by the time, there was a work by Federer framing in, for, for, for surfaces at least, it was quite well understood. However, people kind of in dimension two, and originally, I think, also in this shown yellow paper, they proved themselves existence of minimal surfaces, instead of referring to the work by Fred Federer, which was done 10 yellow, in 1970, so 10 years prior to that. There was some kind of, kind of, much concern with that, yeah, which is justified only partly. But then, by the time already it was known, by 1979, when they wrote their paper shown yellow, that up to dimension seven, co-dimension one sub-manifers, when they're really absolutely minimizing their smooth, and this was proven by Federer in 1970, based on 60, 68 paper by Jim Simons. And this Jim Simons is, of course, the most kind of essential ingredient there, right, because all that is a general compactness that I, it's kind of clear. But again, in Simon, there is a trigger computation about minimal sub-varieties in spheres, with some particular condition, which come from the fact that the base of the cone, and this cone is minimizing, just stable. And the stability condition transformed to this surface, and it's kind of, you know, beyond my understanding, though, funnily enough, I was formally translating this paper to Russian, in Russian. But it was not, I was not translating, by the way, yeah. And this how I learned the subject, because at some point there was a mathematician, Fett, who was actually a professor of topology, who had some problem with authorities, because he was, I think, I think it was about the time, he was not very happy about the invasion of Czechoslovakia, and he lost his job. And so to survive, he was making translations for some day, but there he couldn't sign it. And so I was supposed to sign this paper. And so I read a little bit, I understood nothing, but I just remember that, yeah. And this how I can equate it with the subject matter. And I didn't appreciate at that time that it was a really great paper, except I couldn't understand the paper by myself. But this really kind of day, and this domain is kind of one of the really brilliant papers. And which is, well, significantly, I don't think anybody followed deeply enough along this line, and then what happened, the white dimension seven. Of course, from a certain point of view, it's clear why it fails. This Bambieri's Justi, which people will say are very happy about that, that's not, breaks the high dimension, but it's kind of obvious. I mean, just from general principles, routine computation, you do it. It doesn't require, I mean, any, any efforts from, from modern perspective, intellectual effort, just routine. But this Simon's paper is kind of mysterious, the white dimension seven, and seven has nothing to do with Kelly Kovach's, Kelly Kovach's, all phenomena, dimension one, two, three, and then this stabilizes. But anyway, it's going on. But now, so, so that's the proof. Once you have it, like in the torus, with some method, you take this cycle, you construct another medical point of scale, you go down, and just to the surfaces, and of course, there is no surfaces. And the point is, of course, that the condition which is, do it's not being a torus, but admitting a map of degree not equal to zero to the torus. This property inherited by hypersurfaces, which are non-homologous to zero, because you can project them to one of the coordinates, non-homologous to zero, some projection also non-homologous to zero. Had poised with degree, and so the induction works. But I don't remember if I said it, but in fact, their theorem proves something stronger than that. The topological condition is stronger because, and so the class of Manifoss, when they rule out the point of scale, which is not the one which cannot be covered, and there is a real theorem due to Schick saying, cannot be covered by indexeretic techniques. All invariants coming from, the example when all invariants vanish, coming from deregoperate all those indices generalized in this vanish, but this argument still works. So it's quite, quite simple, logically simple, and this is technicality, what to do in high dimensions. So one point you don't need his spin, which is kind of pleasant. It's kind of maybe not so essential, but still. But then, but still you want to carry the argument and to go to high dimensions, there are singularities. And this absurd that you have to bother about them, if you look at all formulas, how you prove positive view of this operator, the more, kind of, if it's smooth, but the closer to singularity, the better formula it becomes. It's only add, kind of, make operator only more positive. But you cannot use it. You have to find the singularity a couple of years ago, Shona Yau wrote a paper when they explain how to go around singularities. But you don't know, but you don't know actually singularity is truly there, because in dimension, this way I can explain now, it's very simple. But in dimension eight, when you have, you know, a singularity is isolated as follows again from Federer, you know they're unstable. You slightly move your data, add the boundary value or something, and then it becomes smooth. And this very, I'll explain in a second argument, it's a very simple argument, and this looks must work in all dimensions. But so how it doesn't? It's negligible difference, yeah. So, and if you prove, we could prove that in a conceivably, there may be very simple proof, because in this case you don't have to know nothing. Oh, that's all. We can explain it, you know nothing. Just compactness or something. Just it's from hand waving. But then this hand waving breaks down in higher dimensions for some stupid reason. And then still you can bypass it, there are two kind of, rule one by Schoen-Yau, which still doesn't, it's not as good as on in other dimensions. And another is due to log-camp, log-camp, which is, both prove very complicated. So I haven't read either of them in detail. And I'm more inclined to trust Schoen-Yau, because well, at least I understand intermediate statements. And log-camp is just, you have to understand, not only prove a statement of about 10 other theorems, quite sophisticated about geometry of minimal varieties. And it's very difficult papers. But so let me explain how you eliminate singularity in dimension eight. And this related to the following problem I remember the undergraduate. So if you have this kind of figures in the plane, you put them everywhere in the plane. The number of them may be the most countable. You cannot put uncountably many of them. Like intervals you can put uncountably many intervals, right? Segment, but you cannot put uncountably many of them wise on the plane. Corollari, in dimension eight singularity unstable. Is it clear for you why? Nobody see how you can put. If you put one inside, you always have space. You always kind of, they don't want to come close here. They just don't want to come close. But that's exactly kind of geometry behind it. And the reason, of course, why should be, in general, should be so. Why singularity should be unstable? Because if you move within a continuous family, figure why? This singularity must disappear. You have this family of curves, one inside with another, right? They cannot be all singular, because you only have countably many of them. But in a continuous family, they're uncountably many. So, and that is the geometry, and actually was proven by nuts and the smell. I haven't read his proof. His proof, I look, I just said. He uses too many theorems, you don't know. But if you, now let me explain this from this position. So what is the idea? You have the singularity. You have this, what makes a singular is singular cone, right? So if you take minimum variety at some point and scale it, become bigger, bigger, bigger, bigger, bigger. And then, from general principle, for those converges, and moreover, it sub-converges to a cone, right? And so it ends up with a cone. And if this was flat cone, the point was smooth, if it's singular, it's singular. What is, by the way, still unknown if this limit is unique? It's in some cases known to be unique, but a priori only sub-limit. And so what you use, why there is sub-limit? Because you know that the volume is monotone function. It has tendency to increase faster than a flat case, right? And on that hand, it's compactness. And therefore, if you have monotone function, you know, it has tendency to converge. But it's here, you don't know which shape it takes. It may be approximately on here shape and here, it's like a different shape. It may kind of oscillate. But by the way, one of the problems, I think, if you knew uniqueness of the cone, it would be very helpful. And actually, a law can prove something in this direction. But he doesn't prove this instability, but it comes very, very close to that. He says if we have, if I understand correctly, because you have to be vague in his statement, that if you have this minimum of variety, you can little bit more modified, become non-singular, but you'll be approximately minimal. It will be very, very close, but not totally minimal, not truly minimal. So, so, so what is this cone when it is, say, isolated point? And they are not flat, so there is a sphere. Inside of sphere, you have some minimal variety and this cone over the sphere, right? And this cone. But now imagine you move such a family of this minimal variety, start moving them and take these cones. But then you have one cone inside of another cone. But this is exactly what I was saying. It's important we have some single thing. It's really going in all directions, yeah? Because this minimal thing is not switching to one hemisphere. Either minimal thing can be one hemisphere. So it's spread over the sphere and it's not being sphere. So you cannot push one cone inside of another because if you take this minimal variety, move it a little bit, it intersects itself, right? Because it's kind of, you cannot move it a little bit without intersecting itself. And that's a rational behind it. But what's the difficulty? Now, if you try to make it rigorous, so you go to this limit. So imagine I have a family of this minimal thing, one inside of another. And so the theorem is, if you have this family of minimal varieties, one inside of another, then that the number of points where they will be singular will be close no way then. It's closed, obviously you have to prove it's no way then. So it cannot be, you cannot have the whole interval of them. So you take these cones and then say, huh, I have moving family of cones. So cones go inside. That's impossible. And say, huh, I prove it in all directions, in all dimensions, right? Because about cones is true in all dimensions. You cannot put cone inside of a cone except in obvious way. However, this is a subtle point. When you blow it up, so what may happen, you have the singularity, the cone shifts, the cone moves here, the singular point moves along. And you don't have family of cones. But when it happens, it's the only way to happen, this singularity is pressed over. And then in some limit situation you have one-dimensional singularity. But you know in the dimension they cannot do one dimension. But this exactly, the sliding which only kind of makes singularity bigger does not, does not allow you to make high dimensions. But it's kind of minor thing. I mean, I just, I'm pretty certain that, well, that they may have one-page argument be, if you know a little bit more about minimal, minimal varieties. Because the argument by, by, by Borslach Khamyshev is too heavy. And besides, there should be argument without which you only welcome singularity. And they only make results better. Whenever, whenever you know, whenever you can, but you will arrive them, they only improve all inequalities. But now, whatever it is, it doesn't give you again much information about your manifold. So all you have is a manifold, you know, it's, again, this poison of poison. But what do you say about the geometry? And how you can extend it to more general geometry? It depends on topology. You need this exactly, this nested family of one-dimensional objects. So you need, you need a strong assumption on topology. But what, interestingly enough, if you're slightly, there are two modifications which you should main introduce to this method. And one is that you allow boundaries. So you have to consider minimal surfaces with boundary. And boundaries, there are two kinds of situations. When you fix the boundary and when you have free boundary. So you just, boundary is somewhere. Number one. And number two, you allow and introduce a weak term. This weak term kind of similar as what you twist your, your spin bundle with another vector bundle. So in this Dirac theorem, there are two. The main term is just this A, A, A genus, which is come from manifold, from smooth structure of manifold, kind of subtle power from the variable. And then there's something kind of more, more elementary in this Schoen class. And we relate to geometry. And here it's similar to that. What you, the function you consider is this. You consider, so your manifold X. You consider domain, there's an omega. It has boundary Y, the omega. And what you do, it is you take this, it's n-dimensional manifold. So you take volume of your Y minus some measure of omega. And mu is some measure function, some measure on your manifold, which may pose you for negative. And in, in, in simple example, it's given by a continuous density function, but not necessarily. Today. Good example when it is not, when it's supported on, when it's supported on hypersurface itself. And I shall explain some of that today. And so this is a function. And if this measure is just constant, right? So what, what is the solution of that? There will be hypersurface, mean coverage of each, of each point equal to that, to this measure. And they kind of vary, actually thesis called them kind of brains or something, yeah? I, I saw some literature, they say brains, brains, something. There is no chunk loss here. It's volume minus measure. Which chunk loss? It's a pure job. Here, no chunk loss here. Everything will be geometric. Chunk loss disappear, right, in, in, in the story. The, the equation would be transcribed mean coverage or? Stationary point with prescribed mean coverage, but if they are minimizing, they will be stable. And it will be stability condition. And typical example, the model example, they have in mind. So you see, the point is, usually people say, oh, take something, can, surface of minimal area containing given volume. But that's inconvenient because you, but what you say, huh, I, I, I, I not minimal given volume, but given integral. For example, can see the kind of concentric spheres. They're extremal for that. But what's the function mu? Exactly mean coverage of this. You see? In the same hyperbolic space, whatever. And they're extremal. It means that if you now have the same kind of geometry, but it changes, it makes scaling coverage more positive. And you look what happened there. You see that this operator, again, will have some positivity such that if you take now the, I, I'm sorry, I'm running a little bit ahead of myself. So, so again, you can use this for positivity or some operator and to reduce the, to kind of reduce dimension, but in a more, in a more subtle way than we were doing before, which I now have to explain. So there are these two major extension of the minimal surface. Case and what you can get with them. So let me give instances. So eventually what you can obtain kind of results, which follow are, yeah, no, I'm not, I'm not ready to say because I haven't said thing before. Now I explain kind of some other scheme, how we can use this second variation equation. So the second variation equation, we have so many for X and has some positive, say positive scaling coverage, but this applies to any scaling coverage. You have to kind of change the formula. And you take this minimal sub-variety, then the operator which is positive is this minus Laplace operator plus one-half of the scaling coverage, right? So if you write down all this variation, some term disappears, this positive term, there are some other term in this, this norm of curvature of this manifold. And this quality is exactly kind of like that. If this is a minimal surface, it's totally, it's kind of like totally judicious. It was actually quite positive operator. Lots of positivity. And the more singular surface, the more positive this operator, but this point. And what you know about this operator if you now, so it's one-half is bigger than this one-quarter, right? So if you have to use a full power of that, what you do, you take this manifold times real line. The metric here was dx squared plus you have this first eigenfunction, I keep forgetting maybe, of this maybe squared times dt squared. So you take some power of this. And then this will have positive scaling coverage. I might be wrong about whatever, whether it's squared or something, there might be some other power, I think squared. So the point is that once this operator, we want to have a stable minimum sub-variety, then you can construct here a metric which, when you project back, you have your old metric. And in this direction, it's invariant under action of r. So here you have ambient space where God knows what, and there was minimum sub-variety. And now the ambient space becomes just product. Not exactly product, because this arm maybe scale this or that way, right, where it's invariant under. Now we can repeat this process, like we did in Shengyao. But the advantage is we now have a pretty good track of geometry. It was not some conformal change, there you lose completely what's happening to geometry of this sub-manifold. Maybe I better call it y. This was minimum sub-variety y, y embedded in x. x was complicated manifold and side minimum surface. Now this y inside becomes actually totally genetic, and you have the same picture, but now it's symmetric under r. Keep doing that, and when you keep doing that, you arrive just at full symmetry. If you have dimension n, what you have, metric of point of scale, and invariant under translations. And that's kind of ultimate thing, certainly impossible. And an old geometry of this y is still there. If y was big, and this would be big. You see, when you take conformal change, you lose control of the sign. And here this manifold, of course, bigger than y, because in this case, you change it in one direction, but you don't change the y direction. And when you do that, for manifold with boundary, you immediately conclude, and this was done sometime ago, although it was a coarse argument, that if you have manifold with boundary, it cannot be too large. What you do with argument was like that, take minimum, so boundary here, construct minimal surface, which may go somewhere still the same picture applies. And you have symmetrization, but you shrink by a half, do it again, and then it has a very bad exponential factor. So you just, it was very, very an imprecise argument. However, this can be done sharp. Now, certainly explain how this can be done without losing kind of any constants in a very simple and transparent way, using the same kind of computation we had before. There, but now, so what I want to prove is this kind of statement, that if I have manifold topologically, torus cross interval, such that scalar curvature, and this is symmetric here, so it's symmetric homomorphic to it, and scalar curvature of x is greater or equal than scalar curvature of the sphere, which is n, n minus 1. And here boundary consists of two parts, so there is boundary called g plus and d minus, corresponding, I better start with minus and then plus, distance between two ends. Then this distance in this geometry x is less than or equal, I hope I remember it correctly, to pi divided by n, and this sharp inequality. In dimension two, in dimension two you have just sphere, two-dimensional sphere, right, in distance here to pi, when you said to pi over n, here is pi, so it's pi over n. It must be, it must be, ah, no, it's correct, correct. N is two, yeah, yeah. N is two, it cancels pi, yeah, distance between two points is pi. So this is an extreme object. So in high dimension it will be no subject of constant curvature, it will be of variable curvature, but there is an extreme object, which is not kind of the obvious one, right, but this sharp inequality in all dimensions, and when here there is a torus, and so you don't use implicitly that there is no metric here, but, and then there is another kind of more precise statement, which is combine our other things, but this more general one in the high dimensions, so here again there is a problem with singularities, and here you only need the Schoen-Jauh theorem, which is a kind of that, more, more inclined to trust, but the same statement is true, we have instead of torus, I take any manifold, this kind of is two ends, and with the property that if I take any hypersurface here, it admits the metric of positive scalar curvature. For example, it may be like this Kuhl-Mesoffice times interval, which we know admit, and then the same is true. So if scalar curvature of the ambient space is what I said, this distance will be like that, but here I need this log hump, in dimension for Kuhl I don't need it, because dimension is five, when dimension goes up, like if you have exotic sphere of this Hitchens sphere, which have numerical positive scalar curvature, I need this theorem by law hump, which is not exactly his theorem, I need generalization of his theorem, I'm sure it follows if his argument is right. No, if his logic is argument, it might be stable in some way, but because I don't know the proof, it's just conjecture, but I'm fairly certain if what he's written in his paper is okay, then it is okay. But for Schoen-Jauh, at least I can use the lembo from their paper, which is, if correct, it applies literally. So how this proof goes, let me again explain. And this again goes by symmetrization, so for this case, instead of taking minimal surfaces here, homology class has a cylinder, and I take this condimensional one surface, again based on the low dimensional cylinder, and use the same logic of symmetrization, right? Every time I have the minimal one, so I can multiply by a circle, and eventually I reduce the situation when this metric is invariant on detection of the torus. It becomes one-dimensional problem, and you solve it explicitly, and this was actually done already in our pay-by-slowson, and you solve this metric with the tricky, it will be metric of the time dt squared plus some function phi squared d times ds squared, so also flat, but this is the tricky function, some integral, some tangents or something. But this kind of elementary, very kind this type of equation you solve it explicitly, and you get this number. This is, and then if you look slightly more carefully at this argument, you don't have to be actually torus if you, what is actually true, yeah, well. And so what's amusing about that, so let's discuss this point. Already that is quite, quite kind of, has strange consequences. Namely, if I have n-dimensional sphere, and I have torus there of k dimension k, k either n minus one or n minus two, then it cannot have thick, say, k-dimensional one just two, right? It cannot have big neighborhood around him, because if you put it aside, I cannot put them aside more than this distance, because sphere has poise of scalar curvature. I don't use the fact that it has poise of sectional curvature on your scalar, so you cannot have kind of wide, vertical bend inside of the sphere. And because this property is in variant of the Glybchus maps, the same is true for the ball. Inside of the ball, I cannot have this torus with a wide bend. In particular, it must have big curvature. If you have small curvature, you can have wide bend. And I don't see elementary proof of that. So a corollary, if I have a torus inside of here, its curvature must grow roughly like n. n-dimensional torus, k-dimensional one, see, you can do it in a ball. It's one of the principal curvatures, must be of other n. For small dimension, at least some constant, of course, yeah. But for large dimension, become the asymptotics. And the same, up as the word true, if I have this immersed, actually, or embedded, don't exist, immersed exotic sphere of this kind. Even very special sphere. And nothing you can say about other spheres. And it looks like absurd. That's elementary situation, Euclidean space, and you use all this kind of, all this strange stuff. And you cannot kind of go through usual, make this argument inside immediately, only scalar curvature being kept track of. Everything else just disappears, all this geometry become invisible. And so that's very bizarre, and I don't know what happens to the example. If instead of torus you take product of the spheres to the power, and I have no idea what happens. Of course, absolutely zero. No estimate at all. It may happen, you can do it with bounded curvature. So I don't know if you take this S2 to the power of my hand, and you want to put it into the space of dimension I know 2n plus 1. If you, in the ball here, in the unit ball, if you can do it with bounded curvature, regardless of dimension. Of course, the obvious, the obvious kind of thing, kind of, it's, it's, it's, curvature grows pretty, pretty fast. But, but for, for this embedding, yeah. But it's very unclear in, but in good dimension one. Yeah, this, it's actually there is no obvious simple embedding. It seems obvious embedding have in the middle dimension, and then it grows like square root. In examples, but we, but you, and then you cannot do better than square root in any dimension. So it's completely, completely strange situation. So, so again, so the question is, I have this S2 to the power n, I embed it to the ball of, of unit ball. What kind of the minimal section of curvature can achieve? And the best embedding I have on n goes to infinity. It's like square root of n. And, and, and then you follow co-dimension, you can construct some rather artificial embedding slightly. Also curved. And they, in co-dimension one will be curved more than n actually. They will like n to some power. But this very strange and only Scaly Kovac tells you what happens for, for, for all, for all non trivial kind of examples. Very, very, very strange situation. And I think it's quite interesting, I mean, exactly that you realize how poorly understand things. This is really makes you, makes you happy. The, of course at the beginning, then we keep an understanding you're not so happy. Right? So, that's, so, so what happens here. And, and then there is a, and how this goes along with, so at which moment you need this, this function of what I call mu bubbles. So, what they're good for. So, what you can prove with the, the, and this is exactly what I said that if I have again this kind of picture, many forms, two ends, and you know, no hyposurface here has metric with positive, positive Scaly Kovac, then, then there is, then the distance might be one what I said. So, what you do? You construct a function. So, we have this model example. I said there is extremometric and there is familial hyposurface. And they have certain mean curvature. Right? And so on the shrink, of course, this mean curvature from outside because very kind of, very, very negative. But then you just take this function mu and just solve this variational problem with this function mu. Right? And if the distance was a little bit bigger, you couldn't, the function go a little bit slower. And therefore, to hear these two things, this, you can solve this problem because by maximum principle the solution cannot hit either of the two boundaries. So, they, it's usually as a barrier for this problem. So, if distance was big enough and you, you, you slightly, you make, you could make this mu move slightly slower from this value to this value, right? Because mu kind of becomes first positive, then become negative. When you go in, in this direction, right, mean curvature depends on the sign. You go in this direction, first positive, positive, then become negative, negative, right? Like on, on, on the sphere. From this point of view, it was growing, growing. So, positive, then shrink, become negative. So, we can solve that. And you can show that if you look this, this thing and look what happened to a second variation equation, it tells you exactly that it is, that again, this operator, 1 plus, 1 half minus Laplace in scale of curvature is positive. And therefore, this conformal change, you don't have to do anything. It has positive scale of curvature. So, my assumption was the reason is such manifold. Therefore, this might be like that. But here you see, because I, I need to just not just kind of, in to be, step, I need exactly this manifold to, to, to be there. And I need lochamp, lochamp theorem for that purpose, which is, he says it slightly differently, but I think it's, it's almost formally followed from what he said, but he's saying it's so, and specifically it's, it's, it's, it's so. And, and then, so what follows from this, for example, no, this probably still wouldn't follow. There are some other things you should need more, more, more, more, more, more to say. Let me give another kind of corollary of the segment, which is, I think I'm using, because it's kind of topology, completely hidden there, right? And which is almost sharp, and it is as following. But imagine I have a cube with, or dimension n, and it doesn't have, of course, actual cube, but anything cubically shaped. And it also comes with a medical poison scale, give medical poison scale, which I want to give another way how to bound the, the size. All right. This, by the way, all can be used to bound the size, but they're not very sharp. And this is almost sharp. And scaling again of this greater than n, n minus one. And then there are the following functions here. It is, you have distances, you have opposite faces. On cube, there are kind of n pairs of opposite faces. And there are these distances. And then the statement is that, maximal distance of that is greater or equal than one over square root of n. I don't remember the constant, some numbers say, put me two. But I know this, it's not sharp. It's sharp up to fact of pi over two or something. So the quality which I have is not sharp for the, for dimension two. For dimension two, you know the extremal distance might be pi. We, by the way, maybe just exercise for you. You have a square in metric of poison bigger than for the sphere. Then at least two pairs must be within distance, no more than, no, I said maximum, I said minimum. I'm sorry. Of course, the one direction will become very big, but some must be small. These are actually relations between all of them, which I am afraid to write. But in dimension two, the sharpening quality is pi. But in general, and this square root is kind of close to optimal. You see, you cannot do it better. And the example is we take a ball in this sphere. And this sphere take a cube. And this become your cube. So divide it as a cube. And then, and here distance will be about square root, all sides of the cube. But this not optimal situation, you can write more slightly better. But even up to your concept, the universal concept, there's the same. And so what I'm using here, topology disappeared here. But it's secretly there, because tube, in a way, related to the torus. But then this gives you a kind of very transparent thing that why, why you cannot have big thing, yeah? You cannot have very big cube, and this are more sharpening quality. And probably, well, there is a reason to, for argument principle, maybe you can eliminate this constant. But that's a minor issue. And now, the last, I'll start here, but we will not finish. How we can use, kind of both of them, and how you can use in a more kind of amusing way, direct operator argument for kind of, for geometric purposes. So as far as topology is concerned, it seems to me that everything obtained is direct operator can be superseded by superseded. I see this written, but never spoken, by minimal surface arguments. And in the noise of conjecture by, by kind of elementary topological argument. But when it comes to geometry, then there is something. One of the instances that did the following statement, which concerns scale, mean curvature, and this is as follows. That if I have now a manifold, and again it's scalar curvature and dimensional manifold X, and the, the boundary Y, now this boundary. And again, scalar curvature of X, greater or equal to the length of the sphere. And, assume that mean curvature of this Y is greater than curvature of the sphere. Where what I say is quite, you'll be already, at least I don't know the independent proof, even if you, I don't know, I'm sorry. Here I want to say zero. I can say for the sphere, but prefer to zero. Just point your scalar curvature. Even when curvature is zero, for example, maybe just hypersurface in the Euclidean space. This point is of mean curvature, which mean curvature greater than that. Then this Y, if I map this Y to the sphere of dimension n minus 1, such that it is smooth, distant decreasing map, then either it's contractable or it's isometric. I say smooth. If it's not smooth, actually I cannot prove it. It's kind of funny. If I just say elliptic, this is decreasing. I don't have to say smooth. And then I don't know how to prove it. So it's either it's contractable or it's isometric. And of course, when it's isometric, there's usual sphere. Right? There's nothing can be there. And for that, you need, you need rather sophisticated level computationally. You use the same kind of mathematics, but you use a Dirac theorem, but apply to some twisted bundles where you have to compute curvature as a carefully. Right? So let me say something about that. And then, and this is a careful thing. It can be combined. It can be combined. And to prove, and this way I'll prove next time. So it's one thing. Another is that if you have a n-dimensional sphere with one or with two punctures, but the punctures might be opposite. Okay? It's isometric. Then you cannot enlarge the metric without making scalar curvature very smaller. And it's one or two points and two points opposite. If it's slightly changed, I don't know what happens. Actually, I even don't know what happens very well for dimension two. And this uses a combination of both Dirac operator and minimal surface argument. And this is kind of sharp results. In particular, it says that any manifold with a boundary, forget about a second point, yeah? Can't be big with scalar curvature, which is big because, you know, otherwise we'll be contained. It'll be covered the sphere easily. I'll just shrink. Right? Because you see that my bro opened up this point. If manifold is complete, it's rather, it's much easier to show, but that's not so interesting. And this is also sharp. So if equality only, if this is a situation, it's like that. And if you throw away any other subset, I have no idea, well, I don't know whether it's true or not. For three, four, if you take the two points, which are like that, I really don't know what happens. Even for two sphere, I was confused. Okay? If you can enlarge the metric. For two sphere, of course, there are some cases where if you think a little bit, there are obvious arguments. And if they work, it's work. If they don't work, they don't work. And then you don't know what to do. So what is the, what goes into that? I already said about this minimal surface. And the last ingredient when geometry comes is as follows. So I was talking about manifolds mapped to the sphere, and they take pullback. Somebody on the sphere, it comes here, you take the recuperative twisted bundle, and the flatter the bundle, the better geometric constraint you have of the map, right? So you want to show that map of non-zero degree, non-contractable map will be incompatible with large-scale curvature. And the better, the smaller curvature of the bundle you can find is non-zero kind of chunk, chunk less than better. What is the optimal bundle on the sphere? Of course, over S2, it's a hop bundle, right? It's square root of the, of the tension bundle, so it's better curvature by twice as much. What corresponds to the hop bundle to high-dimensional sphere of dimension, even dimensional? But odd dimension of the way might be so tricky. You can do it also odd dimensional, but then you have to, it's so artificial. So what is the optimal bundle over n-dimensional sphere, which corresponds to the, which corresponds to the hop bundle? Because, and I remember that because we're doing this with Blaine, and he just, this question arises, and of course he, obvious for him, obvious bundle. If you know a little bit, I have topology, you know with the obvious bundle. It's spin plus bundle. You take the bundle of positive spinners, and this bundle has a minimal possible curvature among all bundles. And if you take any manifold, spin manifold, and map it to the sphere, and take this spherical plus bundle and pull it back, then the index of twisted dirac operator is non-zero. If the map has non-zero degree. Because you see, for the sphere itself, if you take dirac operator twisted with this bundle, it's, what will be the index? It's one half, but it's every characteristic of the manifold. And stroke for any manifold. Take any manifold, say with zero signature. Signature, we don't, signature also enter in the formula. And you take dirac operator twisted with this very bundle and this manifold. And then the, by index theorem, you get early characteristic. What is essential for us if early characteristic is non-zero, this is non-zero. I forgot maybe there is some coefficient in the characteristic. And this bundle has a minimal curvature, and it's optimal. So if you take any other manifold, and map it to the sphere, and the map is contracting, then this bundle has a smaller curvature than here. And therefore, dirac operator with coefficient with bundle will be positive. This requires some computation. There is no proof without kind of, morally obvious, but it requires some computation. And what is, and moreover, and this, for some situation, is essential. It's not necessarily distance decreasing map. It's enough area decreasing map. Because curvature lives in two dimensions. And so it may, it may expand in some direction, but then be compensated by contraction with other directions. And so that the theorem was proven by La Roule, so he actually, he made this computation for this bundle. And it says, if you have a spin manifold, and you map it to the sphere, and its scalar curvature is n minus 1 greater than that, and the map is non-zero degree, it's either contractable or symmetry. If it's area contract, you might be careful for g, it's still, so there are some tricky points. And then the last what I want to say today, and then I will be using it, I will be using it next time, but this is kind of crucial for geometric applications, that the same is true if it is not surface, but any convex surface in Euclidean space. So if you have a map, which is distance decreasing, and at every point, scalar curvature here, bigger than scalar curvature here, it map either contractable or symmetry. So in this spec, this was proven by to people, I think, got a similar amount of something about five, six years ago, seven years ago, no more than that, I think, but now, no, it's more probably 15 years ago, that this is true, they prove it to actually for any manifold, which have a positive curvature operator. Yes, of course, the dimension must have non-zero degree, and either if it has non-zero degree, pull back the same bundle, and they make the same computation, and then you, yes, it's enough to have positive curvature, which is kind of remarkable. For applications, it has quite interesting application, exactly for convex hypersurfaces, but for spheres. And even for very simple kind of convex, you would collect phosphorically symmetric hypersurfaces, you would think what's happening in there, because here's a flat pattern here, there, and it's some, for example, from this theorem, I think even from the, yeah, from this theorem, you can already have at least some form of the positive mass theorem, which kind of follows, rather trivially, because you have very, very beautiful possibility here, you can change this convex hypersurfaces, and you have different geometric results, when they would take. In particularly, the last corollary I want to say, of the theorem, which I will be discussing last time, is that if I have a convex polyhidron in the ectodian space, where all the hidden angles less or equal than 2 pi, I'm sorry, pi over 2. Of course, there are not too many of them, the simplices and cubes and product of simplices, and that's it, but still, then you cannot deform it in such a way, that you cannot simultaneously make curvature, curvatures of boundaries more positive, mean curvatures are more positive, scalar curvature inside more positive, and all the hidden angles going down. And again, even if I get, even if you forget about scalar curvature, it is still not, not obvious, even for flat deformation, it's not fully obvious. Actually, I don't, I haven't thought, no, for flat, the proof, which I have, you kind of, you prove it on the way. And this condition is, it's unclear how essential is condition, and the situation when you know it is unneeded, and probably it's never needed, and just, I communicate it with Karim, Prasat, who comes here, and he's kind of expert on these angles, and he said that for small deformation of flat polyhidron, you can prove it using ground sophisticated Hodge theory. It's also by different kind of mathematics, and here what you prove, you prove by Dirac operator, and so there may be quite interesting link here, they kind of come together, these two different mathematical elliptic theories. So, so there are definitely instances when it's proven for, for example, we have a kind of n-gone here, and this kind of prism, it's okay, and then it was proven by Chaouli recently for certain simplices, but I am not certain again, 100% certain, you know, sometimes he assumes this condition, sometimes not, but he's an argument by minimal surfaces, and this is fully by Dirac operator. And so that's how things can be, can be, can be combined. So here, you know, here you don't choose minimal, you can, they enter both, but in different setting, but in some, some way or sometimes you can combine them, right, because the basis of this combination, they said this property, that if you know that this section has no positive scalar curvature, then they have some distance here, and this, sometimes you can prove the only way to prove it is using, using Dirac operator. So this, of course, there is no chance of proving something like Lichnerov's theorem or Hitchin theorem by minimal surfaces, right, because they're completely oblivious of, of smooth structures, or maybe I'm wrong here, but, but for all you know, there is no, no hope to see it geometrically, if, whatever you call geometry. So, so what is, an objective, I just, whatever I stated, there are various inequalities and some of them sharp, and they want to prove them, and if there is time, I shall discuss stability. So how stable this inequality, and in relativity, again, this basic stability property is proven recently, is this Penderow's conjecture saying that this Schwarz's time, Schwarz's space, you know, time, spacial part of Schwarz's geometry, is, have some stability, slightly perturbed, you cannot change much shape if you slightly perturbed the, the metric. Because this, this, the subtle point, I have to keep in mind, we have some things like this point of scalar curvature, or curvature greater sigma, greater scalar curvature, greater any number of points of negative sigma, or you always can say, make this bubble. Big bubble, any kind of positive scalar curvature greater than one, and in only this moment, it becomes slightly better than the sigma. On this tiny little net, it will become sigma minus epsilon, right, and the, and picture will be exactly, and then become again positive. So, nothing can be stable in the naive sense. If you have something, you immediately have these extra bubbles. But the point is, they cut, you can cut them by narrow necks. And how narrow the necks, it's sticky. These sharp result only available for the, for this, by understanding the Penderow's situation, when there is sharp estimate, and the extremal, extremal object is the Schwarz's metric. But in general, this, it can prove that there are some kind of partial results in this direction. So, certainly, exactly saying, saying what, what I said, that there are these possibilities of instability is kind of very limited. But this kind of has interesting, of course, interpretation, like physical interpretation, you have kind of universe, my bubble, but little, very small, modification of geometry here, lack of positivity, and things like that. And again, also, the interesting idea you have if you multiply in general, your metric, by Schiele-Köwitzer, and by Schiele-Köwitzer, it's positive. This reminds me of metric, it's a nice metric. When it's negative, it's kind of negative, well, it's negative energy, and they can play one against another. And how they play, partly, it's encoded by this conformal Laplace operator, but not fully. And there are quite interesting relations between positive and negative part. They can balance, but not in a simple way. And sometimes, they are reflected in something simple, but when it comes to the boundary effects, like in positive mass conjecture, but this I must admit I don't quite, I'll tell what results are there, but which I have not quite understood. Okay, so that's for today. So, it's a question, now I keep this, and you ask questions, and it's going to be recorded. So, it might be a really good question, yeah, because it's recorded. So, Maxim, you must have a question. No more, it's a question, but no more. Everything was clear, fantastic. So, you see, there is an interaction of this method, and none of them is perfect. And eventually, of course, you have to either or both completely change concept with scaling curvature or invent different kind of mathematical formalism, and then scaling curvature becomes redundant, becomes kind of trivial matter. It will be just on the side of the issue. But you don't know what this method is, right? So, we have to bring together this variation of techniques with minimal surfaces, and basic formulas, of course. And there are some speculations, what kind of objective, how we can generalize, for example, minimal surface equation, right, and there is a kind of interesting, and you go very, very far from it. But, well, it's, and this spirit close to quantization, but not quite, which I don't know quantization, but it's kind of the same flavor, and the same, of course, for the deregoperator, which is more common, how you quantize it, or sline, and then it can meet somewhere. Okay, but if you don't have questions, or if you have questions, no? Pierre, no? Yeah, I'm sure. So, I'm going to prove, what's the master of this technique? So, what is making the master of this technique? No, no, there are two proofs. So, one point is that, at some moment, it was shown by Locke-Hamp, that positive mass theorem, this perturbation, which preserve energy of certain sign, equivalent, reducible by simple linear kind of analysis, to the case when perturbation are constant infinity, flat infinity, right. Prior to that, there were this partial reduction, and then it's used by Schoen-Jauer by minimal surfaces, and there is no reduction, direct argument, by Witton. And the Witton argument, you construct certain deregoperator, harmonic deregoperator, and if you look at this formula, Bohner, Lichnero's formula, Schoen-Jauer at infinity, is boundary term, exactly becomes what is considered to be mass. And so, it becomes quite, except you have to work out this analysis. And this was generalized by various people. But on the other hand, the moment you reduce it to this flat infinity, you can compactify it to the torus, and then any proof applies. And it's one argument. And there are many arguments for them. And in particular, using these sharp estimates of Goethe-Zimmermann, you can give yet another proof. And actually, what you prove with them, it's a better result anyway. It proves something you cannot prove by other methods. So this, for the moment, looks the most powerful kind of general method. But it depends on rather, well, for somebody, it's easier. It's a behalf of page computation. For me, it's horrible. You have a positive Goethe operator, you just play a little bit, diagonalize this operator, this operator. That is not nothing kind of outworldly, pure algebra. But it's very remarkable that it works. And it gives you, and here it's everywhere. It's some algebra, it's some moment interferes and gives you these results. And this algebra is no kind of systematic understanding of this. So I think that if you understand this algebra and develop it fully, and then you find some analytical geometric implementation of that. But there is, of course, no such theory for the moment. But this is what you would like from point of view of methods. But from point of view of results, you, of course, well, there are still cases when there are obvious conjectures you can prove.