 Yeah, so I will give a survey lecture about Surgery, Borders and Scalar Curvature. And then in the second part of our today's seminar, we will have a second talk by Johannes Ebert from Münster, who will then talk about more recent research results. To give a start, I would like to let's see. Okay, so let me just give a gentle start and a reminder of some examples of flexibility in geometry. And then when it goes on and during my talk, I will then more move into the direction of geometry topology. And then actually it will have a sort of different flavor. But the actual source of all these developments is in the flexibility in geometry. So we have the Nash-Kooper result, which says that if you have a short smooth embedding of a Riemannian manifold in Euclidean space of dimension at least one higher, then this can be C0 approximated by isometric embeddings, which are C1 and C2 or higher regularities, of course not possible. So these are some famous pictures which illustrate that. And then Amisha proved that for non-connected, for non-compact manifolds, which are connected, there are no global curvature restrictions. This is the famous H principle. It's also an example of a flexibility phenomenon. And together with Christian Bayer, I proved a flexibility result or existence result, which is of a very general nature of C11 regular metrics. So this is just a little below C2 of any kind on open dense subsets on smooth manifolds. So there are C11 regular metrics such that there are no curvature restrictions on open dense subsets of some given manifold V. So these are just general flexibility examples. And now let me turn to scalar curvature. And here on this slide, we have some examples in green and in red, which actually are some of them already occurred in some recent talks and some will occur in my talk or in later talks. Namely, we can ask if scalar curvature, this curvature notion, geometric notion, is it flexible or rigid? So do we have the possibility to deform things easily or are there obstructions to certain, to realizing certain kinds of restrictions on scalar curvature? And so I would like to start with a flexibility result by low-comp from 1995. Namely, this says that if you look at the space of all Romanian metrics with a certain upper scalar curvature bound C, then this subspace is actually C0 dense in all C infinity metrics. So every Romanian metric can C0 approximated by a metric with some given upper scalar curvature bound. And this is actually the reason why it's more interesting to look for restrictions for metrics with lower scalar curvature bounds. And so then we have a rigidity result, which actually was already mentioned in some previous talks, namely that if you look at all metrics which have a lower scalar curvature bound, then this subspace is actually C0 closed in all C infinity metrics. So if we have a sequence of C infinity metrics which converges C0 to another C infinity metric, then lower scalar curvature bounds are preserved. Then some existence and non-existence results in the second square. So we have a very general existence result of Gromov and Lawson for non-spin simply connected manifolds of dimension at least 5. We have the existence of positive scalar curvature metrics in dimension at least 5. And then on the contrary, there are interesting obstructions to positive scalar curvature metrics, for example on exotic spheres. This is a very interesting phenomenon was investigated by Nigel Hitchin in 1974 in his famous paper about harmonic spinners. And then we have another result which bears a lot of names, Hitchin, Gromov, Lawson, Kahr, then myself, Schick, Steinle, Crowley, Schick, Bodvinnik, Emil Randall, Williams, Perlmutter. And this gives a kind of conclusive result about non-triviality of higher homotopic groups, Pi K, of the space of positive scalar curvature metrics. And here I restrict on the n-sphere of dimension at least 5. And this tells us that in all dimensions, where we have a possible or potential index theoretic invariant, which may be non-trivial, and this can be realized on the space of positive scalar curvature metrics. So we have many non-trivial higher homotopic groups of spaces of positive scalar curvature metrics. And recently, together with Christian Bayer, I proved a flexibility result concerning boundary conditions for scalar curvature. And then on the contrary, we have a rigidity result telling us that if we are on the n-disc, we look at non-negative scalar curvature metrics, which restrict to the standard metric on the boundary and have mean curvature at least the one of a sphere, of an n-1 sphere, then this must be actually isometric to the standard metric. So this is a rigidity result, some of which we have already seen in previous talks. And let me just tell you what this boundary condition for scalar curvature is. This is quite recent result. And therefore I would like to mention it, namely, we look at a smooth manifold with compact boundary. Let's take a constant C. And then we look at the space of of metrics with scalar curvature bound below by C. And the mean curvature along the boundary is non-negative. So these are mean convex metrics of positive scalar curvature. And here we have a matrix of positive scalar curvature whose second fundamental form along the boundary is zero. So this means totally geodesic boundary. And then we have a theorem whose formulation will be similar to the formulation of other theorems that we will see on the next slides. Namely, we look at the space of metrics. This means R for Romanian. Look at those Romanian metrics with a scalar curvature bounded below by C. And then we have a mean convex boundary. And here we have a subspace with a stronger boundary condition, namely with totally geodesic boundary. And so we can look at the canonical inclusion here. And the statement is that this is a weak homotopy equivalence. So weak homotopy equivalence means that it uses isomorphisms in all homotopy groups. But it is for this talk enough to think of actual homotopy equivalences. So this means basically that whenever you have a metric in the larger space, or if you have a family of such metrics, then you can deform these metrics to metrics which lie in the smaller space. So you can preserve the lower scalar curvature bound and improve the boundary condition from mean convex to totally geodesic. So this is a typical kind of flexibility result in scalar curvature geometry. And here we are talking about boundary conditions. So similar results we have for other boundary conditions. And so there are previous results going in a similar direction. And now let me explain you what the surgery theorem tells us. And this is exactly the border between... Can I ask a question? Yes, you can ask. So on the previous slide, do you have to assume it's strictly greater or can it be greater or equal to? You mean here the scalar curvature? Yeah. Yeah. So during the deformation, the scalar curvature may decrease a little bit. But how much? It's up to you. But so for this result, we cannot... So in our result, we cannot say scalar larger than or equal. Because during our deformation, the scalar curvature will be decreased a little. I see. Yes. Thank you. So the surgery theorem is exactly at the border between the geometric intuition of flexibility. And then it opens a big world pointing into geometric topology. And because we have very powerful results in this area, this will lead to very powerful and amazing results in scalar curvature geometry. So the basic result is the following. We take an embedded sphere with trivialized normal bundle in our smooth manifold V. And then let's take the upper hemisphere and a standard sphere. It's just the upper hemisphere with the induced metric. It's a half sphere. And we call a Riemannian metric on V adapted to this embedded sphere. If in the normal direction, if the normal exponential map induces an isometric diffeomorphism of the product, SD cross this upper hemisphere together with isometric diffeomorphism to the normal bundle of radius pi half around SD. So here I should have written SD. Okay. So these are metrics in standard form around an embedded sphere. And then I can look at the space of positive scalar curvature metrics, which are adapted near SD or adapted to SD. So which are of this very special form. And then we can again look at the canonical inclusion. And this flexibility theorem here, the surgery theorem tells us that this is a weak homotopy equivalence, as long as the code dimension of the embedded sphere is at least three. Again, this means that if you have such a metric, then you can deform the metric through positive scalar curvature metrics into a very specific shape around SD without leaving the world of positive scalar curvature metrics. So it's again a flexibility result. And so you see that somehow here the names of Gromov Laws and Schoen-Jau are missing, but this will be done on the next slide. Actually, this formulation here is due to Gaia from 1993. And the full statement that this is a weak homotopy equivalence is a paper by Chernish in 2004. And we have another statement for other so-called surgery stable curved conditions, which I will not talk about right now. Okay. So let's keep that in mind. We can always achieve a very specific form near an embedded sphere. And now we can do a lot with that because this opens the way to apply surgery in this situation. And this is exactly what happens on the slide. So this is the first main application. Namely, we can compare spaces of positive scalar curvature metrics. So we are given a manifold V and again, and we assume that V hat is obtained from V by a surgery along an embedded sphere SD. Surgery means we have a trivialized normal bundle. We remove the disk bundle of the normal bundle around V. And then we glue in a new manifold with the same boundary as the one that we removed earlier. So the boundary is SD cross SN minus D minus 1. And here the boundary is SD cross SN minus D minus 1. This is the surgery step in geometric topology. And so now we can use the previous result to prove two of this very interesting statement here due to Gromov Lawson Schoen-Yau. Namely, if the coder measure is at least three and V admits a positive scalar curvature metric, then also V hat admits a positive scalar curvature metric. And if both surgeries, so we say that N minus D is the codimension of the surgery sphere, but if we have such a surgery going from V to V hat, then we can apply the inverse surgery going from V hat to V. And then the codimension of that surgery will be D plus 1. And if both surgeries going back and forth at least three, then actually the two spaces are homotopy equivalent. So this is a stronger statement. So it's not just an existence statement, but we have a stronger statement about the nature of these two spaces. And let me just give a one line proof of the first of the second statement. And the first statement is similar. So how does this work? Let's say we are interested in the space of positive scalar curvature metrics on V. Then up to homotopy, we can assume that all these metrics are of standard form near SD. So I have a very specific form. Namely, there are a product metric on SD cross a cap. And actually then we can remove this handle with a standard metric blue in the mirror handle for the surgery also with a standard metric. And this gives us exactly then a metric of positive scalar curvature on V hat, which is adapted to the dual surgery sphere. And then we can again use the homotopy equivalence given by the inclusion of the adaptive metric to the full space of metrics. Okay, so and the proof of the first statement is is actually the same. But of course, the second statement is a lot stronger because we are comparing spaces, and we are not just proving an existence result. Okay, so up to now, it's pretty, I would say intuitive what's going on because we know exactly what surgery means. And we can believe we can kind of think within our head what it means to deform a metric near a surgery sphere and then replace it by by the steel will handle in order to perform the surgery so far so good. But now, we have the borders and theorem and from this point onwards, the story becomes so actually it takes a different flavor. It becomes much less geometric at least for our intuition. And but on the other hand, we can use a very powerful and strong results from geometric topology. And I think it's one of the the wonders kind of in our field that during the times 80s and 90s of the last century, there were so powerful results available in geometric topology, which were developed previously in the 60s and 70s. And so these two branches then could be combined in a very efficient way, namely by this surgery theorem, due to Gormov and Lawson. So it's a very existent, it's a very general existence result for positive scalar curvature metrics. Namely, let's take a simply connected closed manifold of dimension at least five. So the whole story concerns manifolds of high dimension, at least five and lower dimensions, the world is a little different, because then the surgery theory doesn't work so well anymore. But in these dimensions, at least five, we know that if we is a spin manifold and spin boredant to a manifold, admitting a positive scalar curvature metric, then we itself admits a positive scalar curvature metric. Let me remind you what boredism means. This means that we have a manifold of one dimension higher, and its boundary is a disjoint union of V and V hat, right. And then we say these two manifolds are boredant. And if we require that the big manifold is a spin manifold, and we say it's so these two manifolds are spin boredant. Okay, so if V does not admit a spin structure and is oriented boredant to a closed manifold, admitting a positive scalar curvature metric, then the given manifold admits a positive scalar curvature metric. Okay, so this is a sort of reformulation of the previous result, but in a different language, what's going on? Right, so under each of these conditions, it turns out that V can be obtained from V hat by a sequence of surgeries of co-dimension at least three. And this uses the handle cancellation technique, which was used earlier in the proof of the H. Corbaudism theorem. And yeah, right, so it can be used very efficiently to prove such an existence result. But it's not very explicit, because if we have a bored, a boredism between two manifolds, it's in general not at all clear how a handle decomposition going from one manifold to the other looks like. It's just an abstract existence result. But we have, on the other hand, very powerful techniques to decide whether two manifolds are boredant or not. And this makes these results very interesting from geometric topological point of view. So let me just give, in order to give a complete picture, a general formulation of the Baudism theorem. Then you may wonder why you have to use, in one case, the spin Baudism and in the other case, the Oriented Baudism. So this is a little tricky, actually, in the story. And it's a little confusing, I would say. But there is a general formulation, and this uses the language of theta structure. So these are refinements of tangential structures. I don't want to go deeply into it, but I just would like to mention it so that you know what the state of the art really is. Namely, let's take the classifying space of vector bundles of dimension n plus 1. And let's look at a vibration over this classifying space. So for example, we could look at B spin n plus 1. So then a map into this B spin space would give spin structures on an Oriented vector bundle and so on. But this language or this notion is more general. We can also look at the product of B spin and also the fundamental group of some manifold W. And then just taking the projection in the first factor to Baudn plus 1, we again have a vibration over the space. So then we can take the total logical vector bundle of dimension n plus 1 over Baudn plus 1. And we call a theta structure on W. It's a kind of general kind of refinement of the given tangential structure of W is a bundle map, a vector bundle map. We have a map from W to B. And here each fiber of the tangent space is mapped isomorphically to the fiber of this pulled back vector bundle pulled back via the map theta. So here we can pull back the total logical bundle along theta. We get a bundle over B. And we require that we have a map of vector bundles here. And then we have a very general formulation of the Baudnism theorem due to Ebert and Frank. You can look at this paper on the archive. And it's also published already. Namely, we have a natural notion of theta Baudnism and theta manifolds, which are introduced here. And then we see if one inclusion of V1 into this classifying space is too connected, it means it induces bijections on pi 0 and pi 1 and a surjection on pi 2. Then each metric of positive scalar curvature on V0 induces a metric on the other boundary component, V1. And actually we have a map between spaces of positive scalar curvature metrics. And if these two structure maps are too connected, then again these two spaces are homotopic equivalent. So this is a general situation, general formulation. And this applies to all kinds of manifolds, spin, non-spin connected or not simply connected or not simply connected. And for example, if we take V0 and V1 simply connected and spin and spin bordant, which was on the previous slide, then for B, we just take B spin n plus 1. And this theorem then gives us the previous theorem on the previous slide. Blame me that in this case, these two spaces are homotopic equivalent, which can be deduced from Janusz's result. And okay, so from the Borderson theorem, we can do another rather large step using the classification of the computation of Borderson groups, which was achieved in the 60s. And it was really one of the enormous great achievements in geometric topology. Going back to Tom and then a lot of other people later. Namely, let's take a simply connected manifold of dimension at least five. And so as already decided on the second slide, if V does not emit a spin structure, then V admits a positive scalar curvature metric. And if V does emit a spin structure, then we have an obstruction, which is due to index theory. It's a refinement of the hatchiness. And this is called the alpha invariant of Hitchin. And if this is zero, then V actually admits a metric of positive scalar curvature. If alpha V is not zero, then Hitchin already proved that this is an obstruction to positive scalar curvature metrics by index theory and the Lichenhowitz formula. And so the proof uses the Borderson theorem together with a very interesting computation due to Stefan Stoltz from 1992. He proves that if V is a spin manifold with vanishing Hitchin alpha, then V is actually spin-bordant to the total space of an HP2 bundle. HP2 is the quaternionic projective space. It's an eight-dimensional manifold which admits a metric of positive scalar curvature. And if you have the total space of such a bundle, then by shrinking the fibers, the total space also admits a positive scalar curvature metric. And then using the Borderson theorem, we can conclude that V itself must admit a positive scalar curvature using the Borderson theorem. And for the non-spin case, we can either use an explicit computation or explicit list of generators of the oriented Borderson ring, or we can use another computation which is analogous, a little easier than the one of Stoltz. It's a former student of Mainz when fearing in 2008 in his diploma thesis, he proved that if you have a non-spin manifold, then V is oriented bored into the total space of a CP2 bundle. Actually, this is not quite, it's not stated correctly here. So it's independent whether V admits a spin structure or not. So every oriented manifold is oriented bored into the total space of such a bundle. Okay, so in the non-simply connected case, the situation is far less understood. And for example, I refer to the previous talk by Schmull Weinberger and Gulliang Hu, which showed us that there are powerful techniques using K theory of C star algebras. And then we enter the world of the Novikov conjecture, which is, I would say largely unexplored. I mean, we have very interesting results, but there are still very interesting questions that are unanswered so far. And now let me go back to the space of positive scalar curvature matrix. Again, it's not just existence, but we compare the spaces of such metrics. And here we have a quite interesting result you to young Bernard Cordas from the just two years ago, namely, he proved that the spaces of positive scalar curvature matrix on the four dimensional, four K dimensional sphere is homotopic equivalent to the space of positive scalar curvature metrics on HPK, the four K dimensional quaternionic projective space. This is quite amazing, since these two spaces are not spin bored. It's quite quite a surprise, I would say for me, it was a surprise to see such a result. And the trick is that we can, it's an observation due to Cordas, that actually the surgery theorem can be generalized to adapted metrics around sub manifolds, which are more general than just spheres, and who may also be non have non trivial, normal bundles. So we need not require that the normal bundles are trivial. So actually, so the surgery theorem can then be proved be proved in a pretty much the same way as before. But now we can derive such an interesting result by observing that in HPK, you can look at HPK minus one, it's a sub manifold in here. And if you look at the normal bundle, it's of course non trivial. But if you remove the disk bundle of the normal bundle, then the complement is just a four K disk. And hence you can remove this part and glue in a complimentary disk for K disk, and you get the four K sphere. So it's a generalized sort of surgery, I would say. And here this still gives us the possibility to compare these two spaces, but really on the level of surgery or this more generalized surgery, rather than on the level of borders. And then starting from this result, you can prove much more general result. Namely, if you have a simply connected spin manifold of dimension at least five, with vanishing alpha genus so that we have at least one positive scalar curvature metric, then the space of positive scalar curvature metrics on V is homotopic equivalent to the space of positive scalar curvature metrics on the end sphere. And this is quite interesting because before we had such a result, there was a possibility that by studying spaces of positive scalar curvature metrics or manifolds, we may derive interesting invariance about the underlying manifolds themselves. But with such a result, this says this is impossible. Because at least in the simply connected case, different manifolds cannot be distinguished by looking at their spaces of positive scalar curvature metrics up to homotopy. So the only space which is of interest which is of interest to us is the space of positive scalar curvature metrics on the end sphere, if we are interested in the homotopy type of such spaces. So this is the universal space in this respect. And okay, so the key idea is that you combine this observation with the Stoltz technology. You look at the HP2 bundles, which give all kinds of borders and classes with vanishing alpha genus. And then in these fibers, you can apply this construction, right? You can more or less replace each fiber by an eight sphere by applying this construction fiber wise. And then we have such a quite amazing result. Okay, so let me then come to the space of positive scalar curvature metrics on the end sphere. On the previous slide, we saw that this is the case which is of the main interest to us. And then we have a non-triviality result for the higher homotopy groups of spaces of positive scalar curvature metrics on the end sphere. Namely, there is a general construction which is due to Hitchin. It's called the index difference map. It maps from the space of positive scalar curvature metrics to so the index difference then gives an induced k theory class, real k theory. This is what we are using here. And this gives us then a map from the kth homotopy group of the space to the coefficients of real k theory in dimension k plus n plus one. Right, if k is zero, then we are in dimension n plus one. And this is a case which was already started by Hitchin. Okay, so these coefficients here, they are equal to z in dimensions divisible by four. They are equal to z modulo two in dimensions for k plus n plus one equal one or two modulo eight and zero otherwise. And so now we can ask the question if this index difference map is surjective or not. So can the invariance which appear here, either z or z two or zero, can they be realized by certain interesting homotopy classes within that space? And this is actually possible. This is what this result tells us. And as you could see on the previous slide, there were partial results going in the same direction by very many people. I do not want to repeat all the names, but this is kind of the most general formulation of such a result. And it's also the strongest result because it says that we really have a surjectivity of this map. Okay, but Winnig-Ebert-Reiner Williams proved this for n at least six for n equals five. It's a u2-pal motor. And I would like to present a construction of these homotopy classes using a technique from a paper of myself, Schick and Steinberg, from 2014. And so that's quite explicit. You can see it on the next slide, but it only applies to the case if n is large compared to k, right? So this is what Thomas and Wolfgang myself did in 2014. We proved a similar result in this case, in the case where k plus n plus one is divisible by four, but we had to assume in addition that n is large compared to k. And so this full result on this slide uses a completely different proof technique from ours. And it uses quite interesting developments in more recent geometric topology. So the keyword is co-bordism categories and the classification of different morphism type of manifolds using this very powerful technique, which was developed here in the last 15 years or so in geometric topology. But, okay, so let me construct these non-zero classes in higher homotopic groups of spaces of positive scalar curvature matrix on the n sphere. The key ingredient is the following. And if n is large compared to k, we have an explicit bound, actually. And if we are in the case that d plus n plus one is divisible by four, there are five bundles over the k plus one sphere whose total space has non-trivial a-head genus. And the total space is also spin. This is quite amazing because the a-head genus is not multiplicative in such five fiber bundles contrary to the signature. So the signature of such a total space would be zero because it's the product of the signature of base and fiber. But in the a-head genus, this is not the case. But on the other hand, it was before we wrote this paper actually unknown if there are examples of this kind where the total space has a non-trivial a-head genus. And we can also realize it in such a way that the fiber is a spin boundary. And now, as we learned in elementary classes in bundle theory, we write the total space in terms of a clutching construction. Namely, we divide the base into two halves, the upper and low hemisphere over which the bundle is trivial. And then we glue together these two parts using a clutching map, phi, which is defined on the equator of sk plus one is sk, and it goes to the diffeomorphisms of f. And also since f is a spin boundary, we have a metric of positive scalar curvature, this general existence result of a previous slide. Using this clutching map, we can then fabricate a homotopic class. Namely, we can just map each point of sk to the metric g0 on f pulled back along the diffeomorphism phi evaluated on psi. So phi is a map from sk to diff, phi of psi is a diffeomorphism and we pull back g0 along this diffeomorphism. This gives a map from sk to the space. And I claim that the induced homotopic class is different from zero, and then you can use Baudism invariance of the space or the general result by Ebert and B Miller to compare these two spaces. Okay, let's assume that this is homotopic to a constant map. Then we look at the parametrized mapping cylinder of this family of diffeomorphisms. Why parametrized mapping cylinder? Okay, for each psi, you get a mapping cylinder by gluing 0, 1 cross f along the two boundaries. So 0, f will be identified with 1 phi of psi f. But we do this for any parametric psi, and therefore we have an additional factor sk. There's a parametrized mapping cylinder construction. So we glue together the two ends. And this admits a metric, a fiber-wise metric of positive scalar curvature, actually, if we assume that this family is homotopic to the constant family. Because then we can, before gluing this thing together, adapt the metrics in each fiber so that the two metrics match at the gluing region. And so we can make sure that in this non-trivial fiber bundle, each fiber carries a metric of positive scalar curvature, assuming that this family of pulled-back metrics is homotopic to a constant family. And hence, by shrinking the fibers, we see that the space of positive scalar curvature metrics on n must be non-trivial, non-empty. So we have at least one metric. But these two manifolds are spin-bordant. That's quite easy to see. This parametrized mapping cylinder is spin-bordant to p. So the dimension of the base manifolds is more or less equal. And you just have to play a little bit around with these two constructions. But then we get a contradiction here. And you see that on this slide, we are not so much using any surgery theory or anything like that. But it's the old-fashioned classical A-head obstruction, which is used here. But in order to compare two spaces, like the space for f and the space for sn, we use the surgery in the borders and techniques. Okay, so this means that we have higher homotopic classes, higher non-trivial homotopic classes in this space. And now let me say a few words about modularized basis of positive scalar curvature metrics. Namely, we can look at the space of positive scalar curvature metrics and observe that the diffeomorphism group acts by pull-back on that space. The quotient is called the modularized space of positive scalar curvature metrics on v. And I would say for technical reasons, it's not a very good excuse, but this is what we can do here, is we see the problem is that this diffeomorphism group is not acting freely on the space of Romanian metrics. But there is a smaller group, a certain subgroup, namely the group of diffeomorphisms, fixing a point and acting as the identity on the tangent space. This is what we call the observer modular, the observer diffeomorphism group, fixing an observer. And this group acts freely on the space of Romanian metrics as long as v is connected. And therefore, we look at this quotient here, so we quotient out a slightly smaller group. This is called the observer modularized space of positive scalar curvature metrics. Let me now just cite two results going in the direction that kind of marking the state of the art, what we know here. Namely, on the previous slide, I constructed non-trivial, higher homotopic classes of the space of positive scalar curvature metrics on Sn. And now, if you pass to the quotient in the observer modularized space, and these classes survive this process. And this is quite interesting. It looks almost contradictory because when we go back to the previous slide, the construction of these classes was by a pullback construction along a family of diffeomorphisms. And this means that the homotopic class, which we see here, for trivial reasons, is killed by passing to the observer modularized space. But once we apply Bordersman variance, which says, okay, these two spaces are equal, but the actual of the diffeomorphism group of Sn is different here. And here we can pass to the observer modularized space without killing these classes. This is quite interesting. And we also have a result concerning the full modularized space, namely for any queue, the resist examples of closed manifolds such that the full modularized space dividing out the full diffeomorphism group with all its non-trivial stabilizers and so on has a non-trivial homotopic group in degree 4q, even with rational coefficients. And for example, what we do not know is if these classes, which I constructed on the previous slide, observe to the homotopic groups of the full modularized space. So there are lots of questions which are still open in this direction. But what you could see during my talk is that, at least for me, it's an amazing combination of very strong techniques in geometric topology. And in the end, we get geometric results because these are existence or non-existence and classification results in positive scalar curvature geometry. But I have to admit, the flavor of these results is much less geometry than what we saw in some of the previous talks. But I think it's just a different perspective on our subject and it shows that it's extremely rich. Okay, thank you for your attention and I would like to finish here.