 Summary, which was happening, and then I try to prove something. I have a list of papers, formulas. And so our object at this stage is just many faults, with Riemannian metric. And Sajda Skelykovic of this metric is greater than something. Typically it is zero but not necessarily. Many features are common for other numbers. Maybe, yeah, positive or negative. And one thing to keep in mind is constrained. So we are concerned not so much with topology of that, though topology enters. But with geometry, constrained, it imposes. And it is very different from other conditions associated to curvature. Because other conditions or basic conditions use positivity of rich or sexual curvature. They roughly restrict your shape like you see for convex sets. They like convex and there are very few variations they might meet. And then you try to see more precisely what happens. But here the geometric picture in a very adequate geometric model will be hypersurfaces which have mean curvature positive or greater than constant. And these are very kind of a wild shape. And what's the example I was giving? Because you might have this kind of a, this sort of a has binominal positive curvature. But the symmetric thing you can arrange with positive and mean curvature. Which is quite counterintuitive. So it may have and more, in principle very simple that in order to have positive scale of curvature it's enough to have big positive curvature in one direction. Which dominates everybody and may spread anywhere else. On one hand and for scale of curvature you need two directions of positive curvature. And they dominate everything else. So as a simple example we have a subset, kind of polyhedral subset in the equilibrium space. So if this k, if its dimension is 11 and minus 2. Then if it takes small neighborhood and the boundary of this neighborhood it will have positive mean curvature. Like you have graph in the 3 space and around it you have positive mean curvature. And if it's n minus 3 then you have spheres around and this has positive scale of curvature. And you can arrange it in extreme kind of wide range of possibilities. That for instance, where is the reason for this. If you have flat torus and you can take my tiny little hole and attach huge bubble like spherical bubble, n dimensional torus. And it has positive scale of curvature. But here in this tiny thing you have a little bit of negative curvature. And it's unavoidable. And this is very tricky. I just cannot see it by any kind of elementary means. Because the more narrow it is, the curvature converges to zero. And in the limit you have this torus and this bubble and they become separated. So all naive stability theorem don't work. And kind of prototypical stability which exists. The stability of that in Schwartz's metric encoded by Penrose inequality which says that there is some particular flat scale of, it's a kind of Euclidean model of course, not Minkowski. There is flat scale of flat metric. And if you attach to it kind of little bubbles with positive curvature this section will be small. This area of this section will be small. And in this precise inequality saying what it is, it's exactly the response to the size of this neck in this metric. In Schwartz's metric, if you write down this formula. There is a unique spherical symmetric one which has spherical which is zero etc. Depending how gross infinity is. Gross infinity is a mathematical. So it's one thing to keep in mind. On the other hand, there are constraints. There are specific constraints. We saw that on one hand it's extremely flexible. On the other hand there are constraints. And the basic kind of prototypical result, the simplest one. The first sharp result will be the following suggestion by Blake Lawson. Which says that if I have a sphere, and this by the way has, let me formulate for three-dimensional sphere because there are, it's already quite, quite, quite, quite significant. Is that if you have three-dimensional sphere, on one hand and some other, reminding many falls on that hand, dimension n say close, I'm sorry, dimension three. You map it here by map f. Then, that was the best. So here the scale of curvature is six. Remember it's for n-dimensional sphere. It's curvature n, n minus one. So it's curvature is six. And then the following inequality. If you take differential and external power of the differential and this norm, you multiply it by six. This will be greater or equal than the scale of curvature of x. At every point x. We'll put six here. And this of x. So it means that if, in particular, if the map is decent decreasing, then this scale of curvature cannot be greater than six. So the metric was once we enlarge manifold beyond some reason, you cannot enlarge the sphere, even change the topology. Of course, what is essential? I'm sorry, degrees no zero. The topology, top-dimensional class, fundamental class enters the picture. And this is proven with some effort, because it is odd-dimensional situation, but still because you write what you do, you take some bundle here. It is a kind of poise of spin bundle pulled here. Then here you take dirac operator with this bundle. Of course, and then you take odd-dimensional index theorem, or you can reduce it to even-dimensional index theorem, and then there are some formulas as I explained, so that's impossible. It's one proof. It was not before. It was not sharp constant, and you don't have to make much computation as clear from general principles. But the rough of this, there are two other proofs. And one of them I will come later, which is give limited information, but has some advantage, because when you go to high-dimensions, immediately you need this manifold to be speed. In dimension three, of course, this problem does exist, everybody's speed. In high-dimensions, there must be speed. If you want to do it without spin, you have kind of different methods, but they never give this result. Except for dimension three, there is alternative result, which is even better, which is... So here, you see these areas. This means how much it expands at areas. So it's differential. This number measures area expansion. You see, it has the same scaling as Schuyler-Kovich, so it's got a perfect formula. But there is still better than that in dimension three, and it says that it will have only four spheres, unfortunately. It's theorem by Marcus and Neves that it will have three-dimensional spheres. It's metric of Schuyler-Kovich greater than these six. Then you can cover it kind of by two-dimensional spheres. You can slice it in a homologically phase-full way by two-dimensional spheres, area of which will be less than four pi. So for the usual sphere, of course, you have the sliding, and this you can do. Despite the fact, of course, this manifold will be not round. You may have the bubbles, you may have all the strange things. However, you can go and cover it. And this is quite remarkable, a simple theorem, and there is none of that in high-dimensions. It's a big question if there is counterpart in big high-dimensions. So you do have this La Rue theorem still holds. The same inequality, step instead of six, you have a scalar curvature of the sphere, so n, n minus one, and they have. And that was the theorem, and then it was generalized. Again, we did the Dirac cooperator, and in this extremely kind of useful generalization, which was not until recently fully exploited by Gert Heer. The assemblyman, where it says the same, proves essentially, but computationally somewhat heavy. And the result, this result still holds, because here, in fact, we have better inequality, which I don't want to write, because this better inequality doesn't generalize. And so what you need of the sphere? There are two things you need of the sphere. One is topological, index of certain operator must be on zero. So it is here to pull back positive speed, and it is here, and you know, this will be early characteristic of that. Exactly, which doesn't work for odd dimension spheres, so you need some effort to handle it. But on that hand, you need some kind of, the fact that when you pull back this bundle, this bundle will be sufficiently small here, and this curvature, this pull back bundle doesn't contribute too much to the formula expressing, you know, this basic formula for Dirac operator, which is equals positive operator plus one quarter of scalar curvature, plus curvature term of this twisted bundle. So positivity of this must overwrite this. And then this theorem says, and this is true even if it is positive curvature operator. You don't have to have metric of positive, usually metric on the sphere, but it's positive curvature operator, except then there are some trouble depending on topology. For spheres, everything fine topology is right, for other spaces with positive curvature operator. And what's so remarkable about this, this has its implication, it gives some elementary statement about Euclidean spaces, which you can prove otherwise, and it says that we have now surface here, I think even for a three-dimensional case, probably it's easier. So you have n minus one-dimensional hypersurface in Euclidean space, and its mean curvature of this hypersurface is everywhere greater than one of the sphere, and then this y, you cannot map it to the non-zero degree, to the sphere, such that this map will be distance decreasing, with respect to induce metric. I mentioned two, it probably followed from Gauss-Bernet or something, but in high dimensions it certainly doesn't, and there is no elementary proof, and the fact you can induce it, and see it's very tricky because it depends, it's not really embedded hypersurface. If it's immersed, you may have everything, you see, you may have a immersed circle, with any curvature you want, any long you want, in the same high dimension, you take a kind of torus, two torus in embedded, in a standard way, in R3, no, it's a thickened circle, it has very poisonary curvature, you take it covering, it becomes huge, but it's immersed, you can't make it embedded. So you have to, you cannot do it by looking at the surface itself, you have to know it feels something with poisonary-scaled curvature, and the way you prove it, you apply the theorem for very special degenerate metrics, that is, degenerate metrics like that, you have say, ball, and it's sitting in Rn, and this is seeing Rn plus one, and now in this bigger space, I take an episode neighborhood, so I think in this direction, it'll become like that, and then this mean curvature, degenerate-scaled curvature, in this absolute neighborhood, and this is positive thing, so it's still positive curvature operator, so you do it, and you get this result, but what's so certainly bizarre, that when you work with a direct operator, the only term which matters here, but in the limit, it's unclear what happens, I don't understand what is the object in the limit, it's super, of course, but you see, it cannot work totally forgetting the embedded space, it's very important, it bounds subject of positive curvature, and so this is one of the results, so I want to emphasize because elementary is that, but again, it requires many falls to be spin, it's at some moment, of course, when we're in the Euclidean space, everybody is spin, so it says if I have a hypersurface in the Euclidean space, why we encourage, and I map it to the sphere Sn-1, it's non-zero degree, this is my map, then mean curvature of this y at a point y times norm of the differential of my map must be less or equal, then mean curvature of the sphere, or in fact, maybe not a sphere, but anybody, it's true for any convex, but maybe actually any convex hypersurface in the sphere, at the point f of y, and it's true if you replace all these pictures, many of them, except in its spin, it's a moment in its spin, however, it's unknown, it's unclear if spin is necessary, and there are weaker statements of this non-sharpening qualities, and the proof, which is actually quite painful, and you didn't check all the detail in them if you don't want to be spin. So it's one. But you see, they're nice, they're sharp, and because they're sharp, you shall see it raises, sphere is extremal for this purpose, you can prove that equality holds only if it is actually sphere. And dimension two, I'm not certain, it's obvious or not, dimension two, it's probably, it might follow from the Gauss-Bernet. No, it cannot, because no, no, it depends on the interior, even dimension two, I think it's not obvious. Okay, so it's one type of thing, the painting on the spin. On the other hand, there are other things where you can prove by different means. No, it's not this, I want to give another corollary. And maybe more about that, there is some, I deprise it, how coming he may give lecture, next project related to that. And another thing is concerning the following, I'll say it in the following question. So let me give you an example. Or dimension there, if we have a simple, and dimensional simple, and in Ukrainian space, and it has some, the hidden angle, so there are faces, and there are the hidden angles between the faces. And now imagine, it's curved linear simple, curved linear simple. And it has other angles. Now these angles, of course, variable, yeah, because curved linear, it variables, but this means the supremum of these angles. So it's angles, so what I say what is impossible under certain conditions, this you cannot have. Provided all faces have positive mean coefficients. So if you said deforming your simple, complex, and so making these faces slightly more convex. Naturally you expect, so if you do it in here, and you enlarge it, of course this angle become bigger. And you say, all this happens like that. And so what are the status of that? And how this can be done. What is, definitely I think, using this theorem, you can prove it in any dimension, in n dimension is simple. And yeah, the first one is flat. Yeah, let's come to the simplest case. Imagine simple is flat. And this deformation also flat. Can be large angle, you cannot be large angle. And this is kind of elementary geometry, but not quite trivial. You cannot do it. So given convex simple, move the faces, some of the angles goes up and some goes down. You cannot make all smaller, all bigger. And that's a kind of nice exercise to do, which is not trivial. I mean, simple, just one line. Can we replace simple as a polyethylene? Yeah, that's a big question. That's exactly unclear. It isn't clear. So it is, for some, probably you can know, for simple ones, you know it's impossible. But in general, it's unclear here, the guy I reprised that told me that if you have a high dimensional polyhedron and it is simple. So at every point, you have only kind of the minimal number of faces in general position. Then you cannot do it infinitesimally. And this follows from some kind of Hoch theory. It's not kind of elementary, infinitesimal definition. And for global, it's unclear, profiled definition. And he will, he'll come here, he's exactly invited here for this purpose to discuss. But here, you see, it's about curved light. In dimension three, there is some guy who claims to prove it, but I'm slightly worried about when angles, it is significant difference in geometrically, where all angles will be bigger or smaller than pi over two. Because this very special spherical geometry for angles which are smaller than pi over two. Of course, polyhedra where all angles are less than pi over two are product of synthesis. Like cube, for example, or simply, but they are regularly simple. You multiply them and for those, this argument proves that. If you have such a polyhedron in a creative space, which is product of synthesis. And the synthesis have all the, all the little angles, less than pi over two, you can don't deform it, keeping faces with positive mean curvature without enlarging some of the angles. You cannot make angles small. And this follows again from this kind of, from this argument. Argument as as follows. So when I have this simplex, it is kind of, you can smooth it like that. Taking kind of small neighborhood. And when you smooth it, kind of scaling curve, mean curvature here depends on this amount of mean curvature depends on this angle. Mean curvature depends only on how, the epsilon, but the amount depends. And then you compare to the standard model to the flat one. So this is, has positive, this is, after the second procedure we have positive curvature operator and the theorem applies, except you must be careful what happens at the vertices. And characteristic feature of these triangles, what happens, this kind of piece of spherical geometry, very simple, very simple, but it doesn't show. It just follows that in Euclidean space you have a simplex triangle we can enlarge it, enlarge it kind of like that, but it doesn't become bigger. It becomes smaller and smaller. However, if you have spherical simplices and their all sides are greater than pi over 2, it will, simplices dual to our polyhedron. Then this cannot happen. If we have it, if we have it in spherical simplices, all faces, all A is greater than pi over 2 and you enlarge them, it becomes only bigger in very precise sense. And this is what you can use to make this argument kind of rigorous. But again, when you look at the bottom of that, something happens to some twisted deroc operator and all action takes only on co-dimension to skeleton. What the hell the object is there? What are the correct object involved? Yeah, it's some kind of limit of this deroc operator will go to the limit there is no limit in the usual sense. It's certainly absolutely kind of begs to find correct definition and concept what is this deroc operator. And this problem I think is crucial here understanding, generalizing concept of a deroc operator for this purpose. You're not an operator, you'll be not quite defined but you'll have enough structure to prove all that. I see. Isn't that related to you now? No, it's not. It's just, it's a kind of elementary proof plus this, and then there's some computation of which operator is kind of a simple, simple kind of linear algorithm can mess with one page computation in this paper. But the point it has is kind of you see I posterior it may even prove that for flat polyhedral, right? If you can do this little bit if you don't careful it immediately works for all angles and prove it for all this polyhedral and of course even for this product polyhedral it's not totally obvious. It's easy, you see. When angles this elementary lem reduces it to something simple. But in principle it can give you something which you cannot get by elementary argument. And because you see there are many terms in there this were a rough argument and they sometimes cancels some of your favor and it's rather tricky. It's exactly about the stability. What stability means for this problem? And so this is how you use how you use a Dirac operator in one essential point which is I was explaining in a different lecture there's fundamental difference of using just usual Dirac operator or Dirac operator to be some vector bundles. It's like we can see the differential equation without right hand side and with right hand side. On one hand it's a weak term if you like Laplace equation plus equal something this something is a weak term. However this exactly which control geometry of the object. And for that reason when you prove something with Dirac operator per se you get topological result just do this index nonzero something nonzero but tell you nothing about geometry. The moment you introduce that this you can build to your pleasure is remembering something of the geometry and the same will happen to the second method which was used by Sean Yao I guess following thesis which I never exactly read they had many of this construction but in different language sometimes hard to decipher that using minimal services and they again gained a lot if instead of minimizing the functional so we have now manifold X we have hyper surface Y and instead of minimizing just call area just for just to skip just for simplicity for keeping meditation instead of minimizing area Y you subtract measure of what they bound so it bounds something on this side because what is bound not essentially crucial actually will be not a functional it will be closed one form right so but exactly when you cut off a material but you have this some measure here and measure is not and a priori if it's constant just usual so bubble and this and solution of that we have this function mu mu on the space these surfaces have exactly been curvature mu extremely one and using these you can do something else and some but then you run into problem of regularity of this and so it was from the from the very beginning was a big problem when the logical that once you have this now I have to look at my paper when you have this y and y next and it minimizes this functional so I call it area symbolically it means volume or dimension one y minus measure what it bounds right so the first variation of that will be integral of this it is in variation of this area in variation of that so the first variation is integral of this mu over y so extremely one so the extremely one will have mean curvature exactly equal to this mu you see this mu is directed function it's not quite function important what's called intention and this kind of crucial to remember how you from which side you look and if you do it in the wrong you get kind of crucial because positive or negative essentially but the second variation is another matter in the second variation it is as follows so and so what's important about second variation because the moment is really minimizing you know second variation must be positive and this second variation is certain operator on here so some operator become positive and the quantity here is as follows it is so if you take the second variation you take normal direction and wait with some function psi of this of this function I know how called a something it is equal to integral one thing is that like for a plus operator that d y and here you have one half and here you have scary curvature of y minus some term which I described in a second psi squared d y and here is important term and this important term is n times here is a mean curvature which on the hypersurface if it is extremely the same as my function mu squared n minus one and here is quite essential here is it will be normal derivative of my function mu in the direction in the direction of mu is a dependent co-orientation so it's sign dependent and you see they have different scaling but it's strangely now but it's okay and plus scaling curvature of x so there is this kind of term and you see what happens in your favor so this is good because it eventually makes you what you want of course this some manifold have positive scaling curvature so you can use induction or something and this is against you this derivative so and scaling curvature of the ambient space of course in your favor and when you in the end of the day so what you do once you have that you know this positive operator is positive you take first eigenfunction and construct a new metric over this y is it the middle term you know sectional curvature of y sectional curvature never enters here no this term last term you know it is normal derivative of my mean curvature and his scaling curvature of the ambient manifold it is coming from where is it a term of this last equation or the no no I'm describing what this is right it's not to repeat it this is expression every way here it's separate from x x everything internal everything external and therefore when you change properly so I have my y in x it has some induced in magnetic g sub y and what I do I multiply it by real line or by circle whichever I want and and I take first so this operator so this is second variation so I like this take operator corresponding to this energy take the first Eigen function and because it's positive it's positive Eigen function that the model is that the metric which I obtain here will be invariant under this action and this is right and its scale its its its scaling curvature will be a plus plus something positive plus yeah so when this this quantity is sufficiently positive again for example if it's a minimal surface this term disappear so if this was positive then that will be positive and otherwise we have here and just we have idea so what are basic examples you see this is kind of a it's almost the same as if y itself here this metric it's not quite the only after stabilization you stabilize and this again principle it's hard to work with a space itself you always multiplied by high dimensional euclidean space invariant under the action of the space and inequalities now related there but that's almost as good as a digital manifold but not quite it's almost as good but not quite but we can pretend up to some point that it's a digital manifold here at the property and the basic example which is look at this concentric spheres they're all exactly extremal and kind of minimal you know strictly what are the functional the mean curvature itself right so if you take now in the euclidean space mean curvature of those and solve this problem of course you come back to this and because they make a family they actually minimizing not only extremal but minimizing right and at any time you have this picture you can take more general manifold feed this in so you imagine you have so so just a typical example now I'm saying take the following manifold remind manifold here it will be exactly what it was positive positive mean curvature at least as positive as here and this has as positive as here and inside its scalar curvature is positive and distance between these two as big as it was and then if I construct this very function you can construct with this called small gradient it means that inside there will be the hypersurface which has mean curvature at every point as I'm sorry scalar curvature as this sphere and now if you again combine this with the previous thing with scalar curvature operation when mapped to the sphere you can say you use this kind of cannot happen again using the dirac operator argument and one of the basic corollaries as follows if you take a punctured sphere these two punctures exactly into opposite points you cannot enlarge its scalar curvature and enlarge the metric at the same time if you take three points it's probably not true it's very highly inclined see original theorem of La Roule was without punctures and this was about closed manifold and now it's about two punctures and this applies to give some non-trivial information for all manifold with boundary right because many manifold can be not only for this manifold but anybody who maps maps here but again for spin manifold the moment you change topology you have to bother about spin and this is about two points and actually even for dimension even for dimension for surfaces I'm not certain in dimension about three points or if the points are not opposite I take two sphere usual metric puncture out of two points one point to the K because I can take this and opposite take two points which are not opposite can they enlarge the metric and simultaneously enlarge the section of curvature for two points it is okay and so it is a rather delicate argument using both diraco-operated minimo surfaces so it's another kind of argument and the last one the last kind of argument where the diraco-operated doesn't end it's purely it's purely a variational method so just maybe just for summary what we had won so one can think about the spheres mean curvature greater than n-1 so this might be large so here you spin and we don't know why the second closely related when you have this polyhedral again we need spin we don't know why we need angles to be less in pi over 2 we don't know why whether it's irrelevant or not so we have this these two results maybe because it's picking spin I was speaking about this much on here there's another result of this kind and this is as follows so if I have I think it's already true for you imagine you have euclidean space on some other manifold exactly to have this emphasized spin which is is admits admits a map here of let me say it I think I was saying correctly which admits in this direction distance decreasing map of positive degree and proper map of non-positive degree distance decreasing so it's bigger than euclidean space but again somebody have to say it must be spin some moment somebody will be spin condition so in euclidean space there will be no spin condition if it was just euclidean space itself and I take a foliation of any direction k then induced metric on this foliation cannot have scaling curvature greater than plus 1 so if it's euclidean how many more euclidean spaces through resistance and it's extremely unclear so because so what it signifies so it particularly implies you can't have compact simple leaves quite a few topological corollaries but still it looks very mysterious there's something about structure of these foliations I couldn't quite figure out even that you cannot have foliation when all leaves have uniformly bound closures or something Michel is that related to what we are discussing? Yeah it follows from your theory if you it's little effort here but you have to take non-compact version add some L2 index to the table essentially follows this is what we were talking about Right exactly this is so so so bizarre it's just induced metric more or less euclidean geometry and then it tells the leaves cannot have particular shape they cannot kind of spread in all directions and there is no way to understand without it and even formulate it properly But when you are saying that it was conceptually what did you have in mind? No but just understand what is the geometry of the leaf dimension of the leaves as an example which we understand then we know the three-dimensional leaves if they because they have point of scalar curvature each of them as a uniform each leaf admits a map to one to the graph so as you pull back or fall points have diameter less than 10 say so imagine now I'm asking this question imagine an experience can you make this property forgetting scalar curvature intuitively no this would mean that the whole euclidean space probably admits a map to this graph and so each of these leaves simultaneously contracts is it true or not so the theorem says it's kind of in some sense it is true but in what precisely so this is what you have this is what I want to say there are intermediate statements and it's unclear what you mean you want to find intermediate statement there may exist intermediate statement which you can I see but this is what you are and then I think for fallation may have many many consequences if you slightly modify it so this is again this is more sophisticated index theorem than this one but of course by the way I'm pretty certain that your index theorem and this must have common kind of descendants and there might be results in cooperation both fallation and all assets here because all these have counterpart for non-compact index theorem and they have become generalized significantly even if you say aha I don't want spin it's enough to have universal covering spin and already you have to go to universal covering and use more 3k index theorems but it's okay but with your index theorem I don't know if we have to do that because see your index theorem in a way in a way it says morally that you have this many fall and there is fallation and poise telekirvich and many fall itself and the little modification carries medical poise telekirvich and it's true but it doesn't really say that it says that no no after some fibrations after stabilization after stabilization it's kind of kind of heavy but this is so the object where it exists is not properly defined that's the problem and now so when it comes to pure variational methods so emphasize kind of geometric theorem you see they have kind of geometric infallations and this geometry is somewhat hidden here is how you can still really take it extract it and now two theorems were pure variational one of them is this follows you have a many fall with two ends like for example cylinder and its scalar curvature is greater than nn minus one which is scalar curvature of sphere then the conclusion is that if I take any hypersurface here y which separated the two ends then this hypersurface y admits metric with positive scalar curvature if this is I forgot to say something I assume otherwise I assume it's not true so no hypersurface admits such metric and there is this and then concludes because the proof will be using that by contradiction then distance between these two this is two parts of the boundary is less or equal than to pi with n see it's n it goes to zero you see even for constant negative constant it takes sphere so the corollary of that that if I have sphere and I have inside of a hypersurface or actually I mentioned some manifold which admits the metric of positive scalar curvature like torus then it cannot have big bands around it particularly its curvature must go to infinity with rate n and it's absolutely unclear for any other reason because we have to use the facts of torus we have a torus inside and we know if it takes slightly different topologies unclear and then this can be the same form as for the ball so the conclusion we have a ball and we have torus in this unit ball then curvature of the torus at some point must be at least n up to a constant probably bigger actually I think in example we in Texas make example it looks bigger and there is absolutely no hint of elementary proof of that and that's kind of very bizarre and because in the course of the proof I use scalar curvature no matter how much I know about this curvature after that it disappears it's a material right so this is the quality and it can combine not only the results of existence non-existent from medical poise cake like for torus it doesn't exist or there are exotic spheres it doesn't exist of course exotic spheres never embed but here it is true even for immersions here it may be even immersed and it doesn't have to be embedded and of course the spheres do immerse into the ball but if you take another exotic sphere which does have medical poise cake we don't know if it is any non-trivial constraints on this curvature like local constraint by the way it's easy to see square root of f by local argument like knowing of course they don't have symmetric but global is n and it may be even n to the alpha maybe one class episode I guess this is conjectural but the the argument we have which is sharp for scalar curvature is n 2 pi over n we see this is because the bend around that must be quite narrow for these there are two slightly different proofs one of them just I'll describe in the spirit of what I was saying and this is follows the both depends on the fact that there is a standard metric I prefer separate to to say one to be half periodic it looks like that this kind of this manifold where in this section maybe torus or maybe space but something flat metrics just you multiply it by constant you go from here to here this each of this segment is pi over n and the and the metric of this thing everywhere is exactly n n minus 1 n by n minus 1 so as for the sphere it shrinks to 0 here shrinks to 0 here and it's this kind of weight functions I keep forgetting it's something like you take integral of something octangular there is some kind of particular function you can solve certainly ordinary differential equation where it's easy because the equation you can solve it and then you compare to this kind of and so it's and the theorem says this picture is extremal you cannot stretch them further keeping this curvature and there are two arguments to that so one is I look at this kind of tori flat and they have they have some mean curvature and so on my if the distance is bigger you can construct here similar functions so a function goes like that it is from minus infinity from plus infinity to minus infinity so what is essential about this function is log and cave so it's its first derivative goes like that it's monotone and because and then so you can construct it here and then you and you can construct then this extremal object exactly as described use it as a weight of the function and you see there is boundary a periodic and hit the boundary but this because it goes to infinity it creates a barrier and so you can solve it that's very essential this function blows up at infinity and exactly at these two points and it blows up and of course and the first approximation it goes like 1 over x is the derivative 1 a x squared and remember they must match exactly in this formula x and x squared and there is a gap in this gap because it's not exactly 1 over x but it's a little term but this number and my sources very well balanced kind of situation you do it and then you make this situation and then you know that your manifold time circle has an equivalent metric but then formula follows the manifold itself has a metric except there is a problem with singularities and so when dimension n greater than 7 there are problems with singularities I'll come in a second to that alternative proof because here at some moment when we to apply it we may or may not use in the theorem but there is slightly different argument instead of this at least for special topologies when this was like torus cross interval what you can do instead of taking this section with these barriers you can take minimum surfaces in this direction and you can do the same process and reduce it the situation is equivalent of the action of the torus it's inductive argument explicitly using conduction and this has advantages in a second to explain what kind it gives you weaker result but it has advantages as far as singularities are concerned so what about singularities so it is kind of kind of remarkable theorem that up to dimension 7 there is no singularities and the first singularities appears you want to take you know s3 times in s6 I'm sorry in s7 in r8 right you take a cone over it and this cone minimizing and this is a trivial thing one line computation somehow it will be a big excitement because obviously the cone must be a querying because a querying becomes an ODE and it develops singularities but this was kind of big thing but non-trivial part why doesn't it happen before 7 and see the most trivial kind of in the whole theory and then from that from general principle it was derived by by Federer that all singularities of minimal minimizing varieties have co-dimensions 7 at least and then there is a case extra case dimension 8 and this is exactly where it is singularities but it's unstable because so I was explaining that may repeat argument it's very simple his argument actually I couldn't know he used too much he knows too much but it's obvious just from general principle this singularity must be unstable and the proof is as follows so what you do you imagine so one point is you have a cone like this kind of round cone then you cannot move it inside without dissecting it argument so here is the kind of lemma which I learned you know maybe one month but first year at the university if you have a figure Y topological you only can put countably many of them on the plane only countably many you can't put on countably many of them on the plane they cannot you can't move it there is no room yeah and this is the same because this cone on the base of the cone it's not a hemisphere it kind of doesn't sit in one equator whenever you move it it will intersect itself and you can't move it so for cones it's kind of obvious right cones cannot be moved for that reason but now if you have a general thing you know in the limit it's a cone so I take this my hypersurgery singularity take some family which move in this direction each serve barrier for the previous one you can construct this family and I'm claiming the singularity in some way it must disappear it might be nowhere then set it might be open open then set where there is no singularity because I blow it up so I have a cone and say I have moving cone I'm done however the trouble is that this when I blow it up a nearby thing slides away so there is no contradiction in the limit so what happens in the limit in the limit this point spreads up and you have one-dimensional singularity it's also impossible but in the high-dimension it doesn't work in the high-dimension I have aceratic singularity it spreads off and that's all and then we have a problem very very close but it's a it's still a known high-dimension and there were two paper two competing kind of things so the story was many years ago and I think about H2 it was Sean and Yaw announced they can do it and I remember we worked with Blaine okay they don't forget so we dropped it but then in 2017 they eventually published the paper and because in between Locke published his papers when he claimed proving that his papers nobody can read and he is a very very technical paper very long and then after that paper he published another paper in 1998 which is almost full solution it's almost proven stability up to a small error so it doesn't say you can illuminate singularity by minimal but there will be almost minimal and which for all application is as good however so it's it's short paper with five pages but it refers to other for 300 pages or something else nobody can read them some people read and say they are okay those who could manage and he writes in a pretty bad way so I don't know how and Sean Yaw published their proof they proved much weaker statement for me for some applications okay and for some of this it's not okay and it's still extremely heavy very very heavy and it's kind of absurd because in all these inequalities if you kind of forget singularity only you can help you they make all the inequalities stronger they add positive term to inequalities exactly equal to the weight of the singularity but you the trouble is the equation makes no sense they become only better all the equation better but they make no sense especially for minimal varieties they still make sense but you have to prove something but for these varieties which are have extra term they have they have no geometric sense by the way for minimal varieties there is very good explanation of the streak which I was using of symmetrization so I'm saying I have this minimal sub-variety when it's smooth and then there is a new metric on this manifold times the line which is invariant under this and which kind of gains in positivity so let me give geometric proof of that without using linearization I saw linear linearized variety use the fact positive operated has first eigenfunctions positive this eigenvalue multiply use analysis how to do it geometrically what makes sense like that so imagine I have something like this and then this is locally minimizing variety I assume it's really minimizing so it's in small neighborhoods minimal therefore if I add a little bit mu it makes slightly positive it still be minimizing and therefore I have this little band surrounding it but now this band will have positive mean coefficient very very narrow it's still singular but it has positive mean coefficient again by this theorem this will be actually non-singular you can make it non-singular in general it will be singular that's the problem however this singularity is very easy to correct but creating something far from minimizing because if you have this kind of singularity when I can do it I can push it a little bit inside in the moment I do it this singularity will disappear I'm sorry what I'm saying what happens now singularity was very tricky it will become like that become simple singularity like cross singularity and when it's cross singularity I can do backwards and make it smooth and so I create something of positive mean curvature and now it's smooth so this small variation exactly corresponds to the solution of this linearized equation and they go to the limit but the point is they make sense even when the singular now the moment I have this thing with positive mean curvature and this I was saying this basic relation between positive mean curvature and scary curvature if I have many for this positive mean curvature if I take a double it had a singularity but now this mean curvature can be regarded as singular scary curvature and there is this construction I described I can smooth it and have positive mean curvature so I can reflect it in this particular case what I do I would reflect it around this face and I have positive mean curvature and I keep reflecting and I have this band I go to zero and go to the limit and then you can show the limit it's not quite trivial but the limit will be many for this positive scary curvature and this and this will and this will be invariant so it can asymmetrize the problem except if I have singularity and do all that the limit may have wrong topology it will be not topological cylinder something blows up and this happens because the moment when I have the singularity a priori probably doesn't happen when I take this band this distance here forgetting will be either too small or too big they will not proportional distance if I knew it distance was more or less proportional then it would finish the story but maybe not I keep forgetting it's too small or too big what may happen here and I think it's too small I want maybe too big so when I shrink to the epsilon this may be having a bad way and if not however it will not work for this one this I don't know if you have like sphere which is minimal for this energy and indeed induced metric has pointed exactly by means of you in general by this linearized operator I don't see how do you geometrical and that's of course rather annoying and this causes other problem further because I don't understand another thing which I don't understand technical and I just was surprised I just couldn't find the literature it must be of course known if you consider so what happens when you have the following variational problem you have a domain or binary manifold and you have the following function it is dirichlet f squared plus some integral of the boundary and here is some function some weight so so it's a dirichlet functional inside plus extra term some measure on the boundary there is this y nice function on the boundary if it's inside so what kind of operator I couldn't find really it's kind of obvious I'm still I'm confused Michel what is the difference omega in the air it's some function on the boundary omega is a function on the boundary smooth function on the boundary it's my function dirichlet functional I'm sorry I'm sorry as usual I will forget this you always forget that right so it has extra extra weight you integrate on the boundary and it's okay you can show it's really well defined because of this term it makes sense but I don't know where what for example corresponding operator with respect to which human structure you have to make operator in linear dirichlet functional boundary conditions no they are normal but it's variable you see this is after here it's fixed we don't say it's normal we just say integral involved it's kind of close to this condition but not quite it's not quite right and there is little proof why it makes sense I need some sort of inequality it makes sense because function with this regularity makes sense when you stick to the boundary so this integral makes sense right so your character should give you exactly yeah but we should with respect to which measure because here is actually measure right there is a measure on the boundary involved not only inside so which measure to use to diagonalize it and what you do I can function etc yeah well if the space probably is some of two bigger spaces well for the boundary yeah maybe yes maybe no I'm not certain when you diagonalize which one we give the right answer maybe the some maybe the one I don't know but I couldn't find the text books I mean on the web I don't know what are the there might be key words which I don't know right this exactly what happens when you linearize this problem with my main for this boundary there we go yeah which is a section in boundary so yeah but no there are so many you know with PD there are so many papers and never never can find what you want yeah but anyway this kind of simple we can figure it out but there are computational okay anyway so let's make little break and then we continue after the break so in the last geometric theorem I want to mention which is unfortunately not quite sharp the cube or actually maybe any kind of cube shape painful where topology is hidden and again the scary curvature will normalize it greater than then of the sphere then there are two opposite faces so we have this gi plus gi minus d minus i d plus i pairs of opposite faces and so minimum of these distances will be less or equal is one over the square root of n with some constant which is by pi versus expected optimal case actually it's not quite clear what is expected optimal case but there is one example which is closed optimal when you take just sphere and an inside this kind of cube and you can project it so this may be my cubicle picture of the geometry of this sphere and the distance between these two in this sphere is of this order but Misha I mean this constant depends on then I'm sorry it's constant it's so it comes kind of 4 it's it can prove it's pi over 2 but by pi differs from this one which you have in this picture and here's one with square root so it's amusing here's there is no topology as it stands and and so but what probably true is also for for hypothesis of positive mid-covid it's probably the same as true if you have this hypothesis divided into these pieces about this is an equality but then this exactly has some problem because of this variation linear problem which I don't know how to handle at this point but this is true in the sharpening quality is not clear what should be the it is because in dimension 2 it is pi so extremal configuration it's not but obviously the extremal configuration by the way for dimension 2 the extremal configuration you take a sphere take two tiny little holes and takes covering of very high order so it will be kind of very narrow it spreads very much here and this this will be about pi and this will be you can make this as big as you want and this is about pi and this when you make this argument it doesn't come to this picture it's by by fact of square root of 2 it's greater than that but this is not optimal picture either you can show you can make better than that but here so it's it's unclear if this is extremal picture and for with it's been curvature strangely enough at the moment I'm not certain I can prove so that's it's here and so what else one knows and what we don't know so now I want to make some general remarks so on one hand we have kind of questions and then we have some genetic results and genetic question and there are techniques as far as techniques are concerned so there are two methods and both have kind of some parallelism in them so one when you have Dirac operator and what's crucial when you twist it with a vector boundary and that's completely changes perspective and also then you have these minimal varieties so minimize of y and this corresponds kind of to Dirac itself and then there is extra term you can add measure given measure of the complement I'm sorry of the one of the half of the complement and this exactly also because like this knows the geometry there is property of this of this vector boundary's connection what it is and how it enter this formula depend on geometry unlike the whole operator which only remember scalar curvature and also this one when you construct this function you can do it accommodate the geometry and derive some geometric information and so but then kind of there is hidden relation between these two methods either there are certain formula and here is what is the why it's in the box if you know Schrodinger I know square of Dirac operator equals to positive operator but it also not as easy this operator you know this it is a plus operator so it just corresponds to energy when you integrate the square of the old covariant derivatives of the section then there is one quarter of scalar curvature and plus term depending on e some curvature this by the way also not as to understand this nature it's not only positive it's more positive in some sense than usual a plus operator and this difference in positivity the best there are two ways to do it one there is some local formula cut-off formula relating to operator but it doesn't give you a good intuition what is much better is the cast final formula so the cast final formula tells you why diffusion along this path in the vector bundle will be always faster than diffusion without connection and the reason is very simple because the cast final formula says how you what will be parallel what will become how heat flow in the vector bundle so you have to take orbit here then lift it here it's heard and still transparently defined go there and then average of the wind image so this rotation in the bundle cancels thing so it cancels faster than here here nothing cancels just like in this direction here it rotates so the vector conservation therefore the more curvature in the bundle the better this inequality however there is no simple way to write it down this error term right and this is a kind of essential essential for many purposes indicates some problem when immediately this kind of this comes up but in any way this and the formula which is here I think the key formula which is very simple but also this of course in one way you will say this pure algebra formula the moment you wrote it was like this pure algebra however when you relate d and d squared the kernel of one analysis is analysis you integrate by parts and this makes it unapplicable to many forms of boundary or to the strictly boundary term to that so d recuperator works extremely bad with many forms of boundary you have to do some very tricky points and do something and the best which is done which I must admit I never could believe I could convince myself works it's so called partitioned index theorem using coming back to John Raw and so they say in effect what they kind of theorem they prove that they have many fault with positive curvature just instance of that the boundary going like that imagine here is a torus in this section you cannot have it with positive curvature and here is the boundary in the spin case again you can you can imagine you can have any kind of handles but they might be spent by minimal surface you can prove it but it is here but you don't need even this infinite tube you just take any cut off in the size bigger than to pi over n it cannot go however from the point of view deracupirated is bizarre because there is a boundary and the tricky point is so when you have such a thing so you can produce lots of kind of section of your deracupirator but to do that you have to go to some covering of the space and then you go to the covering of that but so if it were tube rather thick and so and then you don't have to go to much covering so something positive and you have a lot of section so they must concentrate here and so they give you a lot of spectrum of this thing but when you go to the covering the spectrum will go up and if manifold doesn't if it exponentially shrinks it's unclear how it would be you know if a posteriority is true moreover it seems that it cannot shrink exponentially under this condition at least this is one of the questions you don't know however they come how they manage by some manipulation avoid this difficulty if all doesn't shrink right morally a kind of thick infinity in a very precise way it's okay this we can prove by argument I said because there are too many sections therefore they must concentrate here and give you a lot of section of this of this even of Dirac itself here I cannot have too many whatever operator localized here but still it's very strange you cannot do it so this is interesting point here right and then of course but here the key formula which is also kind of semi-algebraic which is local is Gauss formula and the Gauss formula is saying the curvature of manifold and curvature of submanifold related and so it is not quite trivial formula because there is some cancellation involved because you see curvature of this depends on the second derivative embedding the first derivative and this second derivative should be third derivatives but it cancels and this however simple that makes thing work for you everything dies you don't have that and so there are these formulas and then they can be they can be embodied here in Dirac operator in some analysis and here is minimal service and this reminds me analogy as people say what are we as human being so we kind of body for what we just we go for our genes and genes tell you what to do and this body you can put genes somewhere and do the same therefore here is the same picture embodied in some analysis and analysis does what the genes tell equation what is the actual information there how to understand it we know what genes are actually we don't right because that's why we don't understand because information in genes which may change the variable still they determine our phenotype which means there is information there but this information is not readable unless you have phenotype so it's very tricky kind of chicken and egg problem and here it's something like that so there might be hidden information in this formula which exists independently on this analysis and if you understand we understand many other things Misha when one looks at the heat expansion for the main focus boundary then what happens is that you get the extrinsic culture of the boundary which is entering right but it's for but not for the rock operator not for the rock for the rock operator you need this topless projection that has kind of message you lose control of geometry right it's very unclear because they see the point but when you have this complete manifold the point is how you do that you use character of function is very small very small gradient ah but it depends on which boundary condition you take for the rock operator right of course but it's topless operator right what I mean is that so the type of boundary condition that John Rowe was taking was it a global no boundary condition not the whole point he was doing like a doubling and then using because index was located in infinity the point is because everything located in infinity apparatus are local and these don't few but the computation which I see in my mind if the exponentially compressive propagate too fast it still may come here in naive in naive logic and that's hidden in some kind of formalism I couldn't check if it's correct or not I'm just it's probably there's no reason to believe being incorrect but usually all this argument and see model example on the other hand episteriori I know it's true by different reason it's much stronger but completely different reason which is absurd I mean they have and all of this argument I described they have many variations in really different argument in secretly different features and then that's both encouraging and confusing it's encouraging and confusing so what are so it's kind of an issue from the technical we forget about what you prove you have these formulas and they develop and and these bodies kind of interact among themselves but this formula seems to be separate but I probably there not it's one thing another thing is stability question so let me indicate what stability problem is so I take this model I think for example you have a torus and you know it doesn't have metric with point of scalar curvature of course if you don't say anything you can scale it and become converges to zero it's not interesting imagine you still have kind of bounded shape let's try easiest thing imagine the overall geometry is more or less fixed namely imagine you have family of metric on the torus conversion to the limit which is just continuous metric and and all this metric scalar curvature of this greater or equal to minus epsilon so it converges to zero so in the limit you have but not potentially positive scalar curvature we know if it were smooth it would have positive non-negative scalar curvature you know it doesn't exist now what happens in this case and with dirac operator I don't quite understand at all so what to do with it with minimal surface on the other hand how you just understand what happens with me see how we can use how we can use these solar bubbles so you say remind me what you do with the torus how you prove something with the torus with minimal surfaces if it has positive scalar curvature delete this inside of this minimal sub-variety we cannot say this minimal sub-variety it has positive scalar curvature itself positive scalar curvature itself but if I multiply it by real line then on this product there would be a metric invariant under this transition with positive scalar curvature which is most as good I repeat it it's not as good like that if you repeat it time after time after time I will produce produce on the Euclidean space invariant metric with positive scalar curvature which is certainly impossible because it might be flat now imagine now I want to prove that what happens in the flat case imagine my manifold has zero scalar curvature then apparently there is no contradiction but I want to show that actually manifold is torus flat so what you do you do the following so the key point I'm showing that if I have this minimal thing it will slide around here it will not be never localized it cannot be actually minimal imagine it was actually minimal so you may slide it a little bit and then it becomes slightly bigger but if it becomes there slightly bigger I can again reduce this measure term a little bit bubble it up but the moment I do it I remember there was m squared term there positive and that will positive scalar curvature so they slide around and if you look slightly more carefully they might be totally geodesic as well so the same is true in the limit here the limit object can continuous metric all this minimal surface slide around and they all in a way might be totally geodesic but that tricky right in what sense right so what happens it's unclear so this for example yes it's kind of very plausible to show the limit might be flat but I don't see I don't see the proof and also this derock operator also unclear because so you what you would like to know that when you go to the limit of the metric the the spectrum of the derock operator may only go I keep forgetting up or down right so so if you had so it's twisted or no just derock operator picture so what happens to the spectrum but like scalar curvature scalar curvature may become only in the limit only more positive so derock operator also might become only more positive however if you look from this perspective if you write it like that this if I have some many fault like flat and approximated by some metric query closely or slightly variable this becomes more positive right when I go away from positive opposite state must become slightly more negative but it's not true so this term become more negative but this become more positive so it's very tricky so what happens you do know the derock operator its positivity has strong constraint on geometry of the manifold yeah this is a if you look at this it was in a general term it was absorbed by Waffa and Wheaton and it says if I have a big manifold then spectrum of the derock operator must be close to zero it cannot be right for usual derock operator with usual delta Laplace operator it's kind of paradoxical but you can enlarge manifold like three spheres starting bigger bigger but spectrum doesn't go to zero you know that right which is of course many times people ask me how to prove that and oh you know the expander the expander you can incorporate expander geometry into there but for derock operator it cannot happen because the derock operator a priori may have spectrum index and if it becomes bigger here the bundles with small curvature and carrying non-trivial topology twist them you have now you have non-spectrum but there's just perturbation adding that just small perturbation therefore original one may have non-zero modes but you'll have small logarithms therefore it has the same effect so but in the limit of this matrix it cannot be anything it cannot really become kind of outrageously positive or negative I keep forgetting when say semi negative same positive because you have to remember this or that and this depends of course on the gradient of the gravitation and it's very difficult to incorporate gravitation especially with this kind of planet into the mathematical language you know it's not a joke you cannot say it otherwise I think I was saying for inexpressible mathematical language left or right up and down and this conventional and for that reason very hard to remember what upper continues lower continues what goes up or down but in one mathematical because it's because how your brain can decide it so I have to remember gravitation all the time it's it's tough in algebra it's left and right we should nobody how no left we're right no no what is left and what is right right because absolutely if you forget about weak force and you speak to a person from other galaxy how you can explain it what is left and what is right there's no way absolutely right and it's and your brain kind of doesn't have to build in this machinery though your cells as you know they all have some polarity or biological cells they either left or right it was discovered by Sturff it's rather rather rather amazing but so but anyway so for direct cooperative only partial only partial and but on the other hand there is one in extremal case if I am not mistaken I keep forgetting that it is name of this people who done it that if you have many thought of constant negative curvature it extremely in the following sense because if you because you know in the universal covering you know exactly the bottom what is the bottom of this of the usual Laplace operator on the other hand if it's sufficient to large it has lots of this flatness and have lots of sections or many kind of zero most of the twisted gerak operator and they and this apparently kind of extremal for this purpose but I I've forgotten exactly so again the idea is that if you have many thought who's in the universal covering spectrum of of Laplace is away from zero then spectrum of gerak also can be too too much you can have too many too many some small eigenvalues on the other hand if many thought is too large it has too many flat bundles and it can twist with them and you produce lots of this more so it shows that there are strong constraint on geometry even when here even for scaling curvature being here negative but it's greater than minus here and minus one but in some sense it's extremal so what is unknown however for negative curvature for example for such for hyperbolic manifold is that and this is that if you have this constraint on geometry and per given topology it volume it volume cannot be too small too large when you scale it down when you scale it like that its volume must be so volume must be smaller than for this one so this has maximum possible or minimal possible one look sorry the volume of the standard point so okay right so that's that's unknown however what's interesting it's almost known in dimension four if it is complex hyperbolic manifold and this by by the work of LeBron when you use cyberquitting equation so it has sharp so it proves slightly different thing it proves the bout on the integral to scale the curvature norm squared dx and if manifold has certain topology because you see this is scaled variant integral it's very nice integral scaled variant integral the trouble is paper use absolute value maybe you can use only negative part of this poison probably doesn't enter but even that it's quite remarkable quality because it's sharp for not only for not only for complex hyperbolic space which you expect for other reason but for all algebraic surface or general type algebraic surface codire dimension is whatever the maximum then you have sharp inequality of this kind this integral must be greater than something anthropological variant today is this rich constant curvature no, no, no, no we don't use it no, yes it writes it's a cyberquitting and scaling curvature and then constant no, but he's constant curvature have to check how it's related to the original one it's any it's any curvature it doesn't have to go of course you can relate it to constant but what happens to this integral it's another story he doesn't use existence of constant curvature matrix in high dimensions it's not true in high we still have constant curvature but this yes it's not true in high dimension you can see the trouble is different between dimension three and four and in high dimensions in high dimensions say if you have non-spin many faults it's simply connected dimension more than five it always has medical curvature all the integrals they can completely kind of control them big, small, whatever and if spin you know exactly this is one invariant which give you obstruction in some dimensions that's the policy but in dimension four it's in dimension three you can it follows from parallel argument that in dimension four this LeBron by Savick Wittner in general we don't know so the naive conjecture of course we have a hyperbolic manifold in scary curvature is small the volume must be large because it's it's true if you replace scary curvature by rich curvature but this we don't know what is the you more concordant dimension sorry so if this is in quality this is scary some topological invariant from the Schoen number whatever and this is true if this X was algebraic surface of general type which is con dire dimension equal to 1 again because there are enough in the in the canonical bundle in the in the canonical bundle and this because Savick Wittner invariant when you compute it there is Kelly Kovic entering in the formula did he recuperate entering the formula and you just write this properly it comes up some little argument but I'm not certain about the sign I think you only need negative part by logic but it's formal is it's what's written in his paper and so this is one one probably conjectually what you expect dramatically and we don't know that and another issue dimension stability what to expect when you slightly perturb your situation right for example let me give so you have many faults like like a sphere you slightly perturb the geometry curvature a little bit may go below on the sphere and then what can happen of course you may have this bubble immediately emerging but is it true there is really always a good core which will be unchanged and this thing will be really as narrow as example show an example show you can make this think localize a subject of co-dimension 2 but in co-dimension 3 yeah dimension 3 for example you have connected some so these sections you expect to have small area and this was proven in some cases in this pen rose inequality so when you go to the limit all thing disappears and this becomes separate and you come back you converge to the sphere but in what sense exactly to give precise definition you have to prove precise theorem it's not so clear right because lots of lots of stuff and this has kind of physical meaning right this now universe and other can go down but the bridges are small and this exactly motivate pen rose and this guy in the eye conjecture specific models so this what is what is unknown another general question goes again related to that saying that and accordingly what you may expect that if I say that if you have a manifold of dimension n scale equation normalize it like that everything beyond the size of the sphere so it may have a bubble like sphere but everything else will be concentrated of sum to call dimension n minus 2 not even n minus 1 but n minus 2 right so that manifold kind of one way to say it that means the map to be polyhedron of dimension n minus 2 and so all pullbacks are small and small you expect them to be both small area wise and size wise but exactly maybe saying carefully we don't know and in dimension 3 we know there is such a map which is small size wise pullbacks have small diameter on one hand and independently for the sphere dimension sphere perfect result area wise sharp dimension it also true but it's not a map in this example but you have to think about this as a this family of these two spheres represent this one dimensional cycle in the space of two cycles right and this is how you can think about that what does it mean you can slice manifold by small things slicing has different meanings one you take pullbacks of points or it may be homological slicing and we don't know what the different kind of examples suggest different possibilities and this absolute can clear on that hand if if if what I say would be true this would probably imply the proof of this would imply no bit of conjecture which is probably not true right which is very unlikely to be true and so this probably must not be true or at least in some correct formulation should be not true but of course we don't know that how this this go one from another but we also no conjecture can parallel story but it's at some point the two things diverge as I mentioned so one because there one is more more more more more geometrical no in a very naïve sense naïve formulation here we know not not not not being true so water hmm at the point maybe we are explaining we now we have a little bit time you ask me some questions and I mean if I can remember what else I have to tell you which I didn't tell you yet when you were talking about one dimensional cycle in the two dimensional cycles with coefficients where you know or what two dimensional when you were talking when I speaking about this map what is known we have three dimensional manifold the scale is greater than some say one normalize it then there is a system map to one dimensional space such a diameter of pullbacks it must be complete manifold otherwise be careful the F minus one of all point less or equal than 10 and we don't know if it's true if it is n dimensional this will be n minus two dimensional and this is unknown it's unknown this kind of basic idea about the shape but I'm saying if it were true this would be suspiciously close to prove the knowledge of conjecture but the F minus of points they have a non-trivial homology class no, here we don't know nothing we don't know no, no, no, no it's lacking the sphere it's non-trivial homology class in the space of cycles in the space of cycles but here it's not they are not even cycles they are not even cycles it's a deconstruction which we have it doesn't give you cycles a priori and here they are cycles even two spheres and this proof this is actually very nice and simple proof so how you prove that let me prove that yeah, because it's so simple so I have the S3 and here is familiar spheres and when you have this family sweeping around among them there will be a minimal one and there will be this pretty but there will be but not minimal in a sense minimizing area but minimal of index more one but then there is a following theorem that you have to show that this one have small area because it's exactly minimax area so this is exactly what they proved these guys didn't prove what they said they didn't formulate it they only say this sphere of index one and this little argument showing this one and play another you must have area less than standard sphere and what you use that you know that we have forgetting all the story this is the following fact if you have a sphere and look at Laplace operator and it's known as area then the average of the first eigen first eigen values here extremely for the usual sphere so there are low eigen values how you prove that I give the proof and then it's forgotten all theorem forgotten who that Huber Huber you can map it to the usual sphere we can form a mapping but when you do it of course if you want to harmonic function eigen function go almost to eigen function but the problem is measures measures not being preserved but this exactly problem when it's trying to prove the big manifold must have small eigen values you map it to some a small map to the sphere they pull back something from the sphere but measure may be distorted tremendously and here measure distorted tremendously but my simple topology can balance in such a way that it will be actually partitioned for every sphere this measure will have zero integrals long directions and then when you pull them back they give you comparison function for eigen values and you give immediate two or three and therefore there are first two eigen values are small and so you apply this combine this with combine this with this formula for second variation and then you need to it's quite simple but quite remarkable and this for strange it doesn't work if topology different from the sphere of course when topology thinking some other surfaces because this theorem is not true anymore for when there are no spheres it depends on it depends on conformal structure with which degree it goes to the sphere if you have a surface of high genus and when you map it to the sphere with a low degree it's okay but if high degree it's not true and we know it is not true because there are expanders but here it still looks very plausible and there is no clear understanding what happens for high dimension of this sphere what happens if there is anything comparable to that or you can sweep it by two dimensional surfaces in such a nice homologically nice way but again as I said it will bring a little bit suspiciously close to a vertical connection it's not exactly like that wouldn't imply it etc but there may be some fundamental obstruction to the boss which may be hidden at all transparent here but again I just repeat the most challenging thing is to find formalism which would incorporate both methods simultaneously and the object would be not manifold anymore but subject much more general objects because here they are so soft very atypical for geometry when you have something like convexity it's object very very soft very weak links between them which however is efficient for many purposes how to bring it to the open so what is keeping the thing together so why I find this subject so different and interesting in Riemannian geometry the most profound structure which we know here incomparably if you look at other theorems Riemannian geometry they are much more primitive the logic they are much more primitive and another point amazingly that even when you have you replace condition by scalar curvature by section curvature it doesn't help in geometric theorems like I described very simple one when you have the square cube and then if you just say section curvature is great with glass one how to show the distance between some of them might be like over 1 square root of n there is absolutely no ways to see it by usually everybody in geometry and again funny enough is scalar curvature or section curvature yes assume many for the spin in the proof I don't know here you don't have to be spin I'm sorry you don't have to be spin but you have to fight with singularities or something in higher dimensions but you there is no direct operator proof of this which I can see so and this direct operator again that interesting point that you makes artificial argument and you go to some limit object there are apparently something related to direct operators which are not direct operators what are these objects all the time you have to go like that because you don't know you don't have the right concepts so this I think just more or less what I what I wanted to say and and of course there are kind of not so many geometric theorems I just I may bring force I formulate it more or less everything I just want to repeat another one quite nice theorem which I don't quite you'll follow the proof which certainly must have profound link with everything I said and this theorem by by this guy Shane Tom and it says the following if I take convex hypersurface in the Euclidean space and then I change it from this kind of side and I take another manifold corresponding to this hole with this convex maybe ball ball is good enough such that this has scalar curvature non-negative and mean curvature of this there are two statements if I say mean curvature is here point wise greater than mean curvature then it's easier to show it's impossible get glue in together smooth the hole but what's remarkable even if this integral of this one cannot be it must be necessarily smaller than the integral of mean curvature in this canonical picture and and the proof is as follows so that you if you have too much mean curvature integral even on mean curvature too much you can propagate this point of curvature to infinity by solving some parabolic equation and arrive at this point of mass theorem asymptotically will be wrong so you can redistribute which I must admit I haven't studied that you can redistribute in some situation mean curvature with in the integral form with scalar curvature and this very well corresponds to the fact that when you to the written proof of the positive mass theorem when you write this the formula is a boundary term and it's a boundary term when you write it because the formula itself is algebraic but if you remember it came from D squared integration by part it becomes kind of integration by part there is a boundary term and this exactly will give you the mass in the positive mass theorem and again so you see this again enter in dirac operator and here they enter in the kind of variation or differential equations but there is no universal way to speak about that it's kind of really piecemeal think linked a little bit and kind of common common ground for many things but there is no absolutely no no hint of general mathematics behind it there might be simple kind of algebra analysis that tie good theory here and it's absent and there might be kind of and this is another kind of probably now I formulated all kind of basic geometric theorem which are known there they have slightly more inequalities related in dimension 3 because there is Gauss-Bernet and I think related to relativity but there is altogether five, six, seven theorems and Penrose the only one which tells you about stability and there is no good stability in theorems and then one simple thing but kind of also essential everything for Scaly-Kovich has count apart from in-Kovich and this is usually kind of strange enough harder to prove there might be more elementary just an inclusion space but they are usually harder to prove in all non-trivial cases they were proved to reduce it to Scaly-Kovich use Dirac operators which are absurd which you don't see directly there right so they they appear as boundary of some many faults and smoothness boundary so they make Dirac operator method work tata tata so it gives you a bit stage so and so altogether I want to say the more you prove the less you understand that is very good of course because much, much to be understood ok so my time is over and so I have posted something and I will add more of course I wrote more and I add to today a little bit more to this lectures and so and if there are questions I will be happy to answer them what is the finger's must here what is the positive must theorem so the positive must theorem I explained the exact formulation all beyond myself but the meaning is as follows of course it's basic positive must theorem so there are cells you can formulate in a very stupid way but in a very very efficient way saying that you can so first I think again what is the idea have many of all this somehow goes to infinity and very naïve way different way to go into infinity may go by round course it may be cone which is sharper than flat cone or more open and even if it is sharper it's kind of infinite positive must if open it's negative infinity negative this negative must at infinity right energy infinity so what in fact the picture is you can see the flat situation perturbation which is smaller than conical the radii K is smaller and the model picture is a Schwarzfeld metric I don't remember what it is but it is unique spherical symmetric metric of this shape which invariant under rotations and essentially mass corresponds to this to this gap here right and so this is a model example of positive of positive mass behavior then there is the following theorem that if you have this many forms in the first order of magnitude approaching Euclidean if it has negative mass so it means it was it was diverging faster than Euclidean so the idea is if it has negative mass diverge faster it's bigger than Euclidean therefore you don't expect positive Euclidean to diverge however if you have it then you can extend it to a flat one with this positive curve so if somebody was explaining too fast with positive you can turn it and still become convex and going slightly slower right if it convex and diverges faster than flat you can turn it may flat and still keeping still keeping positive and this is a theorem you will all come so positive mass even only if I mean therefore if you have negative mass manifold of positive square equation of negative mass you have manifold which is flat and somewhere positive therefore you would have metric on the torus with this property and we know it's impossible right so it's quite easy and this argument is quite easy just exactly you solve some differential equations and here's a lead term theorem it's more powerful of the same type so very many manipulation this you you solve suitable differential equations and you can pump a little bit scary curvature from here to here and then to say it's impossible which is a little bit absurd all argument are like that you do many construction organizing and then say after all you know it's impossible it's a little bit a little bit bizarre but some of the results which come quite quite quite I mean using the sharp they're not they really kind of are the sharp theorem so that's that's about positive mass there is big activity also for which is for hyperbolic manifolds so again it was proven by Minou that hyperbolic metric you cannot be deformed in compact region and make scary curvature more positive and moreover then it was improved by various people using but he uses tricky Dirac operator so I must admit I don't quite understand them we take Dirac operator and add some scalar term and the moment you find the scalar term this square this Dirac operator become this plus something exactly this enters so so operate on this manifold on the hyperbolic space become non-negative that square that square plus I think it becomes some something with plus n n minus 1 plus scalar curvature before so this cancels this term but however you can't use induced theorem if I understand correctly which I am not certain so what happens it becomes it's still elliptic but it doesn't split of course you can't have anti-computation it disappears however the written argument works you don't need it right you still can construct and then there was variation of which term you add here and in different people you add some more different terms and have different results and different precision of this result on the other hand once you believe and this is was known theorem here whatever poison you must tell it reduces to that and this again can be proven by minimal surfaces by kind of kind of compactification partial compactification so but still I mean for negative curvature our understanding much more limited of what happens there are again two methods there are actually three methods one coming from Minou and its variation there are minimal surfaces and thirdly what I said people I apologize for what name of people when you use spectrum spectrum of the Yerevko period with many many many sections so this is what happens in negative and then people say quite a few papers I don't exactly understand the significance what happens for different perturbation or hyperbolic geometry they have really hold to be knowledge and about maybe a hundred papers and then it's unclear to me what they tell they're all very in a way special the concern of geometry but they still give you sometimes sharp inequalities it's unknown what happens for complex hyperbolic space probably you can deform it locally in larger poly-scaling curvature but it's unclear you cannot do in careless geometry it's not but but again it's lots of little kind of directions but the right is principle issues and so one of them overall shape stability and the most fundamental what are the correct objects where the theory should be applied to it's not there is definitely I was saying this before I'm pretty convinced everything which doesn't involve directly spin must be applicable to polyhedral spaces where singularity have positive sectional curvature which is called lexando spaces so if because if this is very simple singularity polyhedral spaces with sharp curves and already there become a rough method makes no sense the poly is already locally wrong and with minimal surfaces they should work but then singularity becomes really kind of rather unpleasant and if you pretend they are not there in some time you can do it it's okay but in general it's unknown so the whole geometric measure theory seems to be truthful in this case but but then well something is clear what to do and something you start making construction and then it's unclear if it keeps you in the right category but yeah up to dimension yeah up to dimension 7 maybe you can do it but again it has not been done or you see it fits very well if you blow up a tension cone whatever in minimal surface 30 in the cylinder of 30 they perfectly go along but this has not been done I think it's quite feasible but it has not been done but I don't think it's the issue that the main kind of it's kind of definite precisely formulated class of questions but not the most kind of exciting ones okay so unless you have questions again you talked about some kind of rigidity for Penrose inequality yes it is Penrose inequality is stability so the Penrose inequality is the one relating the Hawking mass and the ADM mass this for me I admit I don't know this terminology I keep forgetting who is who but what it says as follows that if I have space which at infinity looks like this Schwarz's metric with this kind of sphere and now it goes anyway with Poznan's but it may have these bubbles and finish it like that and you take each bubble the first bubble from each side these bubbles smaller than the sphere and this is inequality this kind of very nice inequality and where can can we find this rigidity statement? no it says we perturb if you perturb a little bit this metric this Schwarz's metric that all this stuff which appears can be cut away by small this whole how we interpret the stability this how we interpret the stability stability statement right but this this sharp result unavailable in any other situation and the proof is very two-dimensional but where I think in this final version by Bray but it was before proven by some other people under certain conditions there's only one bubble it was proven by somebody else before yes exactly but they approved for one bubble I haven't followed the proof yes I it's going by minimal surfaces and just it's very kind of convincing it's not not terrible not terrible difficult there is a heuristic argument by Penrose which is probably easier to make rigorous which actually these people done but I haven't followed in detail happy? okay