 So after hearing my talk, I was trying to think back what connections I've had with Marcel Bergeri. So actually, I met him a couple of times in the early 90s, but I think I had much more mathematical contact in the sense that I was inspired by his work in various ways. In particular, I think he was very interested in positive curvature, which is a topic which has a very strong intuitive appeal from the point of view of visual geometry. And maybe one of the most fascinating questions there is this conjecture of Hop, that S2 cross S2 does not have a metric with positive sectional curvature. I think there are a number of us who've spent time thinking about this, maybe not with much, making much progress. So early in my career, I made this observation with Wu Yi Xiang, which is simply if there is a counter example, it can't have much symmetry. It's symmetry group has to be finite, possibly trivial, right? So at the time, initially I remember being quite excited by this because it seemed to be some sort of indication in support of the Hop conjecture. But the more I thought about it, the more I began to think that it was more of a sign that there's a tension between the Hop conjecture and symmetry. And in fact, I'll go as far as to make the following statement. So let's say GS2 is just a standard round metric on the two-sphere. So then I'll conjecture the opposite of Hop, that in fact not only is their metric a positive curvature, but you can approximate the product metric with this. Now before you dismiss this out of hand, let me mention that in fact there are, I mean, this isn't completely random. If you take, look at metrics which have a lot of symmetry, like they're invariant under the product SO3 cross SO3, then they're basically product metrics. They have to have a lot of zero curvature, okay? On the other hand, if you reduce the symmetry assumption, Chigur produced metrics which have fewer flat, fewer two-planes of zero curvature, right? And the more you reduce the symmetry assumption, the less flatness is required, okay? And to date, there's no argument that would contradict this. So that's, yeah. So, okay, so let me move on to the main subject of the talk, which is completely different. So I'm going to be looking at smooth manifold, so X is a compact connected smooth manifold, and some reasons just put it inserting extra, I don't know what's doing this. So diff, X is going to be the set of smooth diff amorphisms, okay, with the C infinity topology. Diff zero is just the component of the identity, okay? So these are, these are diff amorphisms that are isotopic to the identity. And now, met X is going to be my notation for the space of smooth metrics, okay? Smooth Rani metrics, again, C infinity topology. And met with the subscript K identically C is going to be the subspace of metrics with constant sexual curvature C, okay, with the subspace topology. Okay, so the main, main question motivating the talk is this, so what can you say about the topology of diff, the space of diff amorphisms, okay? So the, well, the motivation, there's several motivations, so just from a purely naive point of view, after you can address classification questions, better not touch that, right? After you can classify manifolds up to diff amorphism, then the next question is, can you classify the mappings, okay? Which means, which boils down to class of understanding the structure of diff, okay? And in fact, people were studying this since going back to the 1920s, although it's for the homomorphism group rather than the diff amorphism group. So another reason for looking at this is it's a natural refinement of, of the mapping class group. The mapping class group is just the quotient diff modulo, the identity component, right? So diff amorphisms modulo isotope, or isotope classes of diff amorphisms. And that's something that shows up all over the place, and it's been studied for a very long time, okay? Another motivation is, well, if you're studying smooth fiber bundles where the fiber is x, but the structure group is the full diff amorphism group, you need to understand the structure of diff in order to classify those, okay? Okay, so the talk is gonna focus on, well, on dimensions less than or equal to three. Of course, there's a discussion in higher dimensions, but well, at least compared to the low-dimensional case, very little is known, and what is known is mainly asymptotic. But I'm gonna be focusing almost entirely on dimensions three after some initial comments about the lower-dimensional case. Okay, so let's start with dimension one, though we're looking at the circle, right? There's not much going on there. So you have O2, the orthogonal group, acting on the unit circle. So O2 embeds in diff S1, right? And this is a deformation retract, okay? You can do this in any number of ways. You can apply a heat flow to deform a diff amorphism to an isometry, or you can lift the universal cover and use a straight line isotope. It's really trivial, okay? So now let's move to dimension two. So starting with the two-sphere in 59, Smale showed that if you look at O3, orthogonal group, so that's a group sitting in the, well, acting on the two-sphere by isometries, that's a subgroup of diff, and the analogous statement is true. This is a homotopy equivalence. So in fact, this is not, I guess, you know, using modern technology, you can say this is an elementary result, at least depending upon your definition of elementary, but it's decidedly non-trivial, okay, compared to the one-dimensional case. So let me just mention the analog for the homomorphism group was studied by Knazer back in the 20s. So what I want to do first is to give a proof of this theorem of Smale. It's going to be different from Smale's proof, but I want to present this argument because it will motivate what I do later in the talk, okay, and it'll also give some idea of flavor of the arguments that people have used in this area. Okay, so the first step is to, is this reduction. So in this lemma, I want to show that this assertion that the orthogonal group, the inclusion here is a homotopy equivalence, is equivalent to showing that this space of curvature one metrics on the two-sphere is contractable, okay. This is my first step. So what's the proof of this? Well let's look at the diffeomorphism group of S2, okay, so this acts on this space of metrics by push-forward, right? If you're given a metric with curvature one, given a diffeomorphism, well you push-forward that metric, you get another metric with curvature one, all right, so that's my action. And now this is a transitive action, right, because if you have two metrics with curvature one and the two-sphere, they're isometric to one another, right. All that means then that we have a transitive action of a group, so we can think of this space of metrics as a coset space of the group that's acting, modulo the stabilizer of our favorite element, right, which we can take to be, let's say the standard metric on S2. So this is the isometric group of the standard two-sphere with respect to, I mean the two-sphere with respect to the standard metric, which is just O3, right, this is really just, O3 in disguise, okay. And okay, so now we have a coset space in a finite dimensional case. We would immediately say, well, now we have a fiber bundle here, right, we have the big group, the subgroup, and the quotient, the coset space, and we have a vibration. Okay, this is infinite dimensional, but it turns out it's not hard to justify, this is again a fiber bundle, it's not that hard to check. And now, just use the exact homotopy sequence of the vibration, right, so we know that, you know, this inclusion is a homotopy equivalence if and only if it induces an isomorphism on homotopy groups, and that's equivalent to saying that this guy has trivial homotopy groups, right, where it's contractable, okay, so that's all there's to it. So we're reduced to showing that this space of curvature one metric is contractable. So let's use Ricci flow, right, so Hamilton showed if you have, if you have a compact smooth manifold, you pick your favorite Ramanian metric H, well, then you can evolve that in some canonical way, right, so there's a unique solution to the Ricci flow equation, time derivative is minus twice Ricci, with the given initial condition. And when I say maximal, I mean defined on a maximal time interval, so this, this solution is defined on a maximal interval, this capital T is maximal, and he also showed that if this right endpoint is finite, well, then what's going on is that the curvature is blowing up as you approach this right endpoint, okay, so it's usually called the blow up time. All right, so let's come back to our situation, so Hamilton, well, and Chow have the following result, so Hamilton did this for metrics with positive curvature, Chow did the general case. So they showed that if you take a Ricci, any Ricci flow in S2, regardless of the initial condition, well, then it always blows up in finite time, okay, and in fact what happens as the time approaches capital T, as it approaches this blow up time, then the curvature is blowing up everywhere, and in fact, modulo rescaling, so modulo appropriate normalization, something very simple is happening, namely, it's converging to say a constant curvature metric, right, so it's blowing up, but it's blowing up just because it's sort of shrinking down, diameter is going to zero, but modulo rescaling, it's actually converging nicely to this constant curvature metric, all right, okay, so that gives us this metric G bar, well, then a corollary, I don't know if you can see that unfortunately, well, okay, corollary, well, this, this, the space of curvature one metrics is a deformation retract of the space of all metrics, right, so we take any metric, we've deformed it to a metric with constant curvature, okay, well, there's something that has to be checked, you need to make sure that this process here depends continuously on the initial condition, but that follows in a straightforward way from the proofs, I mean, you have uniform exponential decay of curvature and derivatives, okay, so that takes care of the two sphere, but let me just mention that, so this argument used the equivalence of this, this assertion with the second guy here, but in fact, using a similar, I mean, arguments of a similar flavor using these vibrations, you can prove it's equivalent to any number of other nice, nice statements, for example, you can look at the space of embedded circles in R2, right, the contractability of that space is equivalent to this, so for example, you can prove the contractability of the space of circles using either curve shortening flow or maybe the Riemann mapping theorem, you look at the disc bounded by the curve and then use that to do it on top of it, you can also, it's also equivalent to saying that the, if you look at the two disc, well then the space of diffeomorphisms of the two disc, which are the identity on the boundary, it'll fix the boundary point-wise that that's contractable, okay, so this is something, this is the thing that Smale used, he has a beautiful low-tech proof of this, this statement, and you can also use conformal geometry, which is arguable, the cleanest way to do it, okay, what about other surfaces, so let's look at, say, an oriental surface of genus G at least one, okay, so then, so diff zero is the identity component, so then there are two cases here, if the genus is at least two, this identity component is contractable, right, and if the genus is one, then the identity component is a deformation retracts to a torus, okay, namely, you just, genus is one, you're looking at torus, the isometric group of that torus will be a deformation retract of the identity component of diff, and, well, the rest of the diff morphism group, you can sort of think of, you can think of the analysis of the diff morphism group as being broken up into two pieces, this is cheating a little bit, but anyway, roughly speaking, you look at the identity component, and then you look at the quotient, the math and class group, right, and that math and class group has been understood for a long time, based on this level, okay, so let's go to dimension three, and so the first case to look at is the three sphere, right, so Smale conjectured this in 61, so this is the analog of what we were just discussing in dimension two, so this, a few years later, Jean-Serve showed that this is true on the level of path components or on level of pi zero, right, that this inclusion induces a bijection on path components, but then there was little progress made on this for a long time, so something like 20 years went by, and finally Alan Hatcher proved this conjecture in 83, so his approach is to use another equivalent statement, so this, what I proved earlier in the case of the two sphere has an analog in dimension three, so the Smale conjecture here is equivalent to the contractability of the space of embedded two spheres in R3, okay, it's also equivalent to the contractability of the space of curvature one-metrics on S3, so his argument, I mean it's a long rather subtle argument, it's somehow a mix of combinatorial techniques and smooth techniques, and I think it's fair to say that this argument is not very well understood, okay, so a question that's been around for a long time for people who work in the area, is there some way of proving this using geometric analysis, right, for example you look at the space of two spheres in R3, you could ask is there some natural geometric process which takes a two sphere and isotope, it's a two around two sphere, okay, using mean curvature flow or some other natural flow, okay, so depending upon your interpretation of that question, this might, what I'm going to say in the rest of the talk, might be an affirmative answer or it might not be depending upon your definitions, okay, so what about other three manifolds, so there's a, what's usually called the generalized, generalized snail conjecture or the snail conjecture for X, it goes as follows, so suppose I have a metric GX on X with constant curvature C, okay, here I'm going to require C is plus or minus one, okay, so C is one, then I have a metric of constant sectional curvature one which means I'm looking at a spherical space form, right, C is minus one then this is hyperbolic manifold, so the assertion, the assertion is that if I look at the isometry group of this nice constant curvature metric then the inclusion of that into diff is a homotopy equivalence, right, so you can see all the complexity topological homotopical complexity of the diffeomorphism group just in the isometry group of this maximally symmetric metric, right, and this is a generalization of the snail conjecture because if we take this to be a three sphere then this is just the standard three sphere, the isometry group is so forth, right, so note just as we had these other equivalent statements in the in the case of the three sphere, this generalized snail conjecture is equivalent to the contractability of the space of constant curvature metrics, okay, something using a similar argument, so the main the main theorem that I'm going to be discussing today, so this is joint work with Richard Bemler, is this, so the same setup is here, so the metric of constant curvature C, C is plus or minus one, and I'm going to exclude the case when x is the three sphere or rp3, okay, so then then this space is contractable, yeah, just to be sure, when C is minus one, doesn't musto rigidity answer the question? No, that just, that just, that just tells you something about the, about the isometry group, right, you're trying to, right, think about you, the statement is comparing the isometry group with diff, so morally speaking you're saying well given any diffeomorphism you need to isotop that to an isometry by a process which varies continuously with the diffeomorphism you started with, right, so musto tells you what, tells you that, right, that if you have a, well, it tells you different things, I mean, it tells you that if you have two metric, two hyperbolic metrics then, then any, any homotopy equivalence, between the diffeomorphisms, right, okay, so, so the idea, well, as I, based on what I've said so far, is to somehow use Ricci flow and one nice feature of this is that you have, we end up with a proof which kind of works uniformly, whether it, whether we're written positive curvature or negative curvature, okay, so let me mention, so the argument that I'm going to discuss today excludes these two cases and I'll explain why in a little bit, there's another argument also based on Ricci flow, it's, it's more involved which works, which includes these cases, so it doesn't exclude them, we're in the process of writing that now, I'm not going to, not going to discuss that today, okay, so let me mention what was previously known here, so first of all I already mentioned the case of S3 of the general, I'm talking about the previously known cases of the generalized mill conjecture, all right, so Hatcher did the case when X is S3, the case when, when you have a hyperbolic manifold was also known, so when it's hyperbolic and Haken, meaning there's an embedded incompressible surface, this, this goes back to, to Hatcher and Ivanov in, in 76, the non-Haken case was much more difficult and it was proved by Gabay using previous work joint with several people, I've forgotten the co-authors using, call it non-coalescable, non-coalescable insulator technology, you know, this is a very long story and the proof is, is partly computer-assisted and then in the case of spherical space forms, so there, there are different techniques and so those were able to handle certain spherical space forms, so lens spaces other than RP3 and then prism and quaternionic manifolds, so basically on the level of fundamental group, lens spaces are the ones with cyclic fundamental group, these guys are dihedral cross cyclic and these are quaternionic, quaternionic cross cyclic, okay, so these are, these are three infinite families and then there are three other infinite families, the guys which are tetrahedral octahedral and do a dihedral cross, cross cyclic groups, okay, those are the ones, the new ones that are covered by the theorem, okay, so let me say a little bit before I start talking about the proof, let me talk about other three manifolds, so the same question, what does, what does defects look like, so for simplicity I'll assume that I'll look at orientable manifolds and assume that they're irreducible, meaning any embedded two-sphere bounds a three-ball, okay, so, so let's look at the case when the manifold is Hawken, right, this means it has a closed-embedded pyroinjective surface, non-positive orally characteristic, so what can we once say about diff, well it's pretty well understood, so kind of on the level of pi zero, oops, so if you look at the mapping class group, diff mod diff zero, then, then Waldhausen showed that this, so you have a natural map from this to the set of homotopy equivalences up to homotopy, okay, self-homotopy equivalences up to homotopy and, and Waldhausen showed that this map is an isomorphism, right, so in words it says, right, given a self-homotopy equivalence of X, you can realize that by a diffomorphism up to homotopy and given to diffomorphisms if they're homotopic they're isotopic, okay, so then you're in some sense reduced to understanding the identity component and this was handled by Hatcher and Ivanov, so this is actually predated Hatcher's theorem for S3 and so it was a conditional result at the time they proved this, it assumed the, the smell conjecture, in any case they showed diff zero is homotopy equivalent to a k-torus where k is the rank of the center, so for example you take a three-torus, right, the identity component is homotopy equivalent to a three-torus or if you take, say, a surface of genus two across a circle then the identity component just comes from a circle, right, okay, now so what does this tell you, so of course this wasn't known at the time of this work but after Perlman proved the geometrization conjecture and, and since this takes care of the Haken cases, the, what was left to be done is the non-Haken case, right, and those by geometrization are geometrisable manifolds, so those, those admit geometric structures modeled on one of these geometries here, okay, and so these, these, these last three are cypherd-fibered where the base, these two have hyperbolic base and these have Euclidean base, this one has Euclidean base, and so it was shown in 2013 that, that in these two, the case where the base is hyperbolic, so these two cases here, you have a statement similar to this, okay, so one understands diff in those cases, so that leaves only S3, H3 and nil, so H3 was already handled as I mentioned and so we've, we've just finished off the spherical case and actually the Ricci flow does give an approach to the, to the final case as well, so this will finish the story completely, okay, so I guess, yeah, so let's come back to the proof of the main theorem, so we're looking at, let's say we'll look at a spherical space form, I mean the proof in the hyperbolic cases is completely parallel, so we want to show that the space of curvature one matrix is contractable, right, so the idea is to use Ricci flow, the way we did for the two sphere, right, unfortunately Ricci flow in dimension three is vastly more complicated than Ricci flow in, in 2D, okay, so in particular in the 2D case when the thing blows up, that's blowing, blowing up because you're seeing an asymptotically round metric, you just rescale and, and you have a very nice picture and in dimension three it's much more complicated, so what, the idea is to actually implement this we need to, to use a notion of Ricci flow through singularities, so this is developed by John Lott, Richard Dumbler and myself in the last few years and putting this all together what you get is some canonical evolution that's defined for any initial metric and it depends in an appropriate sense, depends continuously on the initial condition, okay, so what I'm going to do next is spend a little time kind of recapping this, this theory, all right, so and then I'll come back to the proof at the end, okay, so in dimension three well if you start with a standard metric on the three sphere then the Ricci flow at time t is the standard, the original metric multiplied by this linearly decreasing function, right, so this blows up as t goes to one quarter, things ranks down to a point, on the other hand if you, if you have a kind of barbell metric on S3 then then what you see is a neck pinch singularity, so let me, so I'm gonna start, I'm gonna first give some heuristics and draw some nice pictures and do some storytelling and then I'll converge to some mathematics after a little while, okay, so so you should think of some sort of initial condition like this it has two roughly spherical pieces with some neck size sizes scales are one are three and in the middle are two, so if these are, if the neck is sufficiently close, neck size is sufficiently close to the size of these other two spherical regions then things kind of round off as you go forward in time it eventually has positive curvature and then you'll see asymptotically you'll see the same simple nice behavior that we saw before, okay, it comes asymptotically round, on the other hand if the neck size is a lot smaller than the sizes of these two guys then it will shrink faster, there's an instability here and you'll see a neck forming and then it will kind of want to pinch off, okay, and so what what will occur is that so if you ran this forward in time at the blow up time you would see that there's an open subset of the manifold on which the metric has a smooth limit locally as you approach the the singular time, okay, and elsewhere the metric is blowing up, all right, so this is very different from what we saw before, what we saw before was that at the blow up time it blows up everywhere, okay, and modulo just a global rescaling its convergent, okay, very different now, so so how does one deal with this, well this led to the notion of Ricci flow of surgery, so this was introduced by Hamilton and developed initially by him and then by Perlman and the cartoon is as follows, so you start with some initial metric, you run the Ricci flow until a neck starts to form or in Perlman's case you run it until you have a singularity but you can stop a little bit earlier and then inside this neck region you try to identify say here and here regions where you see neck leg geometry, so something that looks modulo rescaling like around two sphere for us are to small up to small error, okay, and then in those neck regions you cut the manifold and you throw away the part which has high curvature and then you cap off the two two sphere boundary components with three discs, okay, with appropriately chosen geometry, so after surgery you see something that looks like this, now you have a compact manifold, compact remodeling manifold and you can restart the Ricci flow, okay, so this kind of alternating process is the Ricci flow of surgery, okay, and even though this may seem a little bit ad hoc it has spectacular applications, right, so by using this Perlman was able to implement Hamilton's original vision for proving the geometricization conjecture in particular three-dimensional conjecture, so actually even though, you know, so all the singularities look like this, like S2 cross R something, well, okay, so these are cartoons, right, so I'm suppressing a lot of complications but something I'll come to a little bit later may speak a little bit to your question but the idea is that when you look at any part of the Ricci flow of surgery where the curvature is large then the geometry looks like a model geometry, okay, so there's a classification of the models and what you see whenever the curvature is large is something that's geometrically close to one of these models, okay, so I'll say a little bit more about this in a few minutes, in any case, so this Ricci flow of surgery has a spectacular application but Perlman himself, so was not entirely satisfied with this, I guess that's kind of characteristic of Perlman, right, so actually he has the comments in both of his Ricci flow preprints, so he says it's likely that by passing to the limit in this construction one would get a canonically defined Ricci flow through singularities but at the moment I don't have proof of that, so what does he mean here, so imagine think about this Ricci flow of surgery process that I had in the previous cartoon, right, so you fix an initial metric, you run this Ricci flow through Ricci flow of surgery, okay, now you start again, you run the Ricci flow of surgery but you do the surgery at a finer scale, okay, so when you cut along necks, you do it along necks which have a smaller scale, okay, and now do a sequence of such Ricci flow as the surgery where the surgery scale goes to zero, all right, so what he's saying is that this whole sequence of processes is somehow converging to some canonical evolution, all right, so that's the intuition here and I think the intuition at the time is simply that well when you're doing the surgery at a fine scale and somehow your intervention is kind of ad hoc, you're definitely doing something, you're disturbing the flow but your intervention is so small it should be negligible, okay, that's somehow the intuition, and in the second preprint where you actually prove geometrization, this is in the first paragraph, he says our approach is aimed at eventually constructing a canonical Ricci flow, okay, so that was what motivated John and I to start thinking about this and then that was followed by the work with Richard, so the cartoon that you should have in mind for Ricci flow through singularities is something like this, so again think of time is moving upward, these dotted pictures here correspond to time slices, and in contrast to what we saw before, as you run forward in time this neck region here starts to pinch down, well and then at some moment you hit a singularity and then it just keeps going and it resolves itself, maybe at the singular time this thing would be maybe non-compact, some sort of cusp-like geometry and similarly on the other side, but after the singular time that will round off and resolve itself, okay, so this is what you should have in mind and what Perriman is suggesting is that not only can you make sense of such an object, but it's actually completely canonical and canonically determined by the initial metric, okay, all right, so now I'll sort of move toward the precise statement of this and the first ingredient is to start thinking about Ricci flow space times or a spacetime version of Ricci flow, so let's start by thinking about an ordinary Ricci flow defined say on some manifold M maybe compact, so it's defined on some time interval zero up to capital T, well from this data we can cook up a spacetime structure in kind of a completely obvious way, so let's let calligraphic M be the product, M cross the time interval, so on this product we have the time function, right, just the map to the second factor, then you have the time vector field, right, that's the vector field tangent to the second factor, right, and a Ramanian metric which isn't Ramanian metric on this spacetime, but it's defined along the time slices, right, defined along the foliation given by the time slices and the Ricci flow equation in the spacetime picture reads as follows, so the lead derivative of the metric which is defined just along the leaves, that's minus twice Ricci, okay, so all right, so this motivates should motivate the following definition, so Ricci flow spacetime is this is an object which locally looks like what I just talked about, okay, so so it's going to be since we're interested in 3D Ricci flow, it's going to be a smooth format of all the boundary, if this makes sense in any dimension, but who cares for now, we have a time function taking values and it's non-negative reals, and a time vector field when you hit the time function with a time vector field you get one, right, so if you take an integral curve with a time vector field well then the time function increases at rate one, right, and then we have a Ramanian metric defined on the foliation given by the level sets of time, right, so you have the time slices and a Ramanian metric defined along that foliation and then you have the Ricci flow equation as before, okay, so this is just, so locally, locally I can just take, locally I can look at, choose a point, look at the time slice and take a flow box for the time vector field and then up to diffeomorphism this is just the same as what I was talking about a second ago, all right, so the point is that globally this could look very different, right, it doesn't have to look like a product, things could be incomplete and it opens the door to a much more pathological structure or at least more general structure, so before I get to that some notation, so m sub t this will be the time t slice, m less than or equal to capital T, this is going to be the time slab ranging between zero and capital T and rather than using this big tuple I'll use capital M to denote the whole thing, the way you denote a Ramanian manifold just by m, right, okay, so this notion of a Ricci flow space-time is much too weak to actually do anything, we need to impose some additional conditions, so there are two conditions, there's a completeness type condition and another condition called a canonical neighborhood assumption, so let me start with the completeness assumption, so let's look at this cartoon, so in fact the Ricci flow space-times are going to be typically incomplete in two senses, so look at what happens when this neck forms and pinches, right, so look at this singular time slice here and the cartoon, you should imagine that this time slice is actually a non-compact manifold which has two ends, one corresponding to this, one corresponding to that, the ends are dipymorphic to S2 cross the non-negative reels, okay, and so for a generic neck pinch these will be incomplete manifolds, okay, so you can find with respect to the Ramanian distance you can find a Cauchy sequence going out this end which has no limit, okay, so the time slice is an incomplete Ramanian manifold, okay, now if you look at the vector field, the time vector field, imagine starting on the point on the initial manifold which has sort of the misfortune of going into the singularity here, right, then if you try to follow that trajectory it's going to be an incomplete trajectory, right, so this is kind of built into the setup, right, we have this, we need to accommodate that, and the I, it's not part of M, okay, that's a very important point, so our whole philosophy here, so for people who work in other geometric flows and are used to thinking about kind of singular objects like, you know, verifold, Brachy flows and whatever, we're taking completely the opposite approach, everything is smooth, and part of the reason for this is just simply, if you did try to try to put in, if you tried to re-complete, you'd end up with some sort of singular object, and you'd have to spend a lot of time worrying, you know, what does Ricci curvature mean there, and so on and so forth, right, so it turns out that you get a completely tractable approach by working with smooth objects, okay, okay, so what is the notion of completeness, so the idea before you read this, the idea is just simply, okay, you can have incompleteness of time slices, incompleteness of the time vector field, but you'll only encounter that if you move through regions with unbounded curvature, okay, so if you stay in a region with uniformly bounded curvature, then you don't experience incompleteness, okay, so the definition, I'll say it's zero complete if whenever you have a curved gamma, so an integral curve of plus or minus the time vector field, or a unit speed curve in a time slice, then, well, if the curvature, the norm of the curvature tensor remains bounded along this curved gamma, then the limit exists, okay, so that's the notion of completeness, and now I want to talk about this other notion, the canonical neighborhood assumption, so M satisfies the epsilon r canonical neighborhood assumption if the following holds, so you take a point in space time, let's say it's time t slice, and then let's suppose the norm of the curvature tensor at that particular point exceeds r to the minus two times t, so think of this r as being like a scale, so that has scales like distance, so then the corresponding curvature threshold would be r to the minus two, so that's what this is doing on the right hand side, okay, so if you look at a point in the time t slice where the curvature exceeds the threshold, this threshold, well then in fact the time slice with base point at that x is epsilon close modulo rescaling by this curvature to a point of time slice in a cap of solution, okay, so this comes back to the question that was posed earlier, a cap of solution is, or cap of solutions are these model solutions in dimension, the model Ricci flow solutions in dimension three, I'm not going to define them, but I'll give you some examples, right, so these are an example would be a shrinking round spherical space form, or a shrinking round cylinder, or a Bryant soliton, right, which is a model for, it's a steady stall soliton, which is a model, which lives on r3, and it's going to have the model for the evolution near kind of a fingertip region, you can think of after the neck pinch occurs, then you have these, I mean intuitively you have these receding tip regions here, and these are modeled, you can think of these being modeled on these Bryant solitons, so currently the, I'm not going to give you the definition of cap of solution, there are a few more known examples, which I don't want to go into, but these have been sort of, I mean Perlman classified them in a qualitative sense, which was enough to implement his surgery method and it's enough for us to implement our construction, but you should think intuitively that these are the models, so what we're saying is that if you're at any point in the space time where the curvature exceeds its threshold, then you pick your point while you rescale it to normalize the curvature to be one, and after you rescale, what you see looks close to the model, so after you rescale a big ball looks almost isometric in CK topology, but large K to one of these models, so that's the canonical neighbor assumption, and then our definition is singular Ricci flow is a Ricci flow space time where the initial time slice is compact and zero complete, and it satisfies this canonical neighbor assumption, so John and I proved that if you have any compact Ramanian manifold, this is a three dimension three of course, then there exists such a singular Ricci flow, with this is the initial condition, the disometry, and this satisfies the, this particular singular Ricci flow will satisfy, given an Epsilon, it satisfies the Epsilon R canonical neighbor assumption, where this R depends on Epsilon and the bounds on the geometry of the initial data, and then Richard and I proved uniqueness, so there's some threshold value for Epsilon, so this is the quality of approximation by these model solutions, cap of solutions, so that if you have two singular Ricci flows, and they satisfy the Epsilon can or canonical neighbor assumption, for some R, well then an isometry between their initial conditions extends to an isometry between space times, okay, so as long as the quality of this canonical neighbor assumption is better than some threshold, you get uniqueness, okay, and the same methods tell you that in an appropriate sense the space time depends continuously on the initial time slice, okay, if you think for a minute you realize there's something to be said here about what continuity means, because I'm saying you have continuity even in the presence of these singularities, okay, but I'm not going to say what that is at the moment, and also the same methods prove the conjecture that Perlman had made about the convergence of Ricci flow with surgery to a canonical process, okay, which is the singular Ricci flow, all right, so what properties do these guys have, so suppose we have a spherical space form other than s3rp3 and take some initial metric g and calligraphic m will be the singular Ricci flow with that as initial data, okay, so what are the properties, well, so this is kind of upgrading the things that I've said so far, so for every t, at most one connected component of the time slice is dip hemorphic to x with finitely many punctures, I mean finitely many points removed, okay, so we start with some initial manifold x, some singularities may form, you may have, a time slice might have infinitely many connected components, okay, but at most one of them is topologically non-trivial, could look like x with some points removed, the rest, the remaining components are topologically trivial, they're basically s3 with finitely many points removed, okay, and there's some finite time capital T such that all the time slices are empty beyond that time, so this is using the fact that we have a spherical space form to get an extinction result, it's based on the work of Coding and Minna-Causie or Perlman, they have the either extinction argument can be used, and now, all right, let's let omega g be the supreme time where the time t-slice has a topologically non-trivial component, right, as in this, right, so then as you approach this time omega g, this time slice, mt has a unique component, dip hemorphic to x, so now there are no punctures, like as we approach this time, we see a copy of x, and this becomes asymptotically round as you approach this time, right, so in other words, this family of Riemannian manifolds converges modular rescaling to a x with some constant keratometric, okay, so let me just recap this in terms of the picture, okay, so we start with our initial manifold, this should be x, we run forward, maybe we see a neck pinch here, so this little symbol here is supposed to mean interesting topology, right, Richard said this is the international symbol for topology, non-trivial topology, so wait a minute, we've got tourists, spent two days trying to figure out how to represent non-trivial topology in a diagram, anyway, so, right, so the neck pinch occurs here, this is our three sphere, so it's an uninteresting going forward, maybe it does something, but we don't care, we keep track of the one with not a component with non-trivial topology, maybe another neck pinch occurs, but eventually to the statement I just made, we will see this topologically interesting component will eventually become round and go extinct, okay, so up here we're going to see this asymptotically shrinking round metric, okay, so this is much more complicated than when we saw in dimension two, but nonetheless there's some remnant of the two-dimensional story, okay, so the next ingredient was also in the paper with John, if you have a singular Ricci flow and you take some component Z of a time slice, well, then you can find some finite subset of that time slice so that every point in the complement lies on an integral curve of the time vector field starting at time zero, okay, right, so we have this Ricci flow spacetime, if you pick a point, then you can try to follow the time, the integral curve of the time vector field either forward or backward, right, so what this is saying is if you try to go backward to time zero, you'll succeed at all but finitely many points in the time slice, okay, this is very different from trying to go forward, when you go forward lots of points can be destroyed, okay, so this we call this finite, call these these guys in S at the points with bad world lines and there are finitely many of those, so here's the picture, so let's think about what happens to this component that's asymptotically around, now suppose you try to go backward in time, well if you're in a point here you'll just kind of go backward to time zero but if you start over here now maybe you aren't a part which is as you go backward at time it emerges from this thing which was sort of a hospital singularity, okay, so there's turns out the argument actually shows that what's happening there'll be at most one point on that fingertip, there'll be exactly one point in that fingertip which goes singular as you go backward in time and every other point manages to avoid that singularity, okay, but the upshot is that I can take this thing here, I can puncture it at finitely many points, okay, and everything else can be flowed all the way back to the time zero slice, okay, so this gives me a dipmorphism and embedding from that punctured time slice to an open subset of the time zero slice, right, it's all called W, and so then we can use that flow, we can push forward this family of asymptotically round metrics, right, so I'm going to take, think about the metric on these time slices as we go forward and they become asymptotically round, I'll rescale them to normalized curvature and then push them all forward, they're going to live on, that's going to give me a family of metrics here which are going to limit to a metric with constant curvature one, okay, so what we get is a canonical partially defined metric, curvature one metric, defined on an open set that depends on G and let's call it metric G check, okay, and by construction this is going to be isometric to X with some finite set deleted with a metric with curvature one, okay, this is my my favorite curvature one metric on X, okay, okay, so now, yeah, so using a continuity property that I mentioned before, you can show that this assignment, you start with the metric G, you go to this canonical partially defined curvature one metric, this is continuous in an appropriate sense, okay, so now let's go back to the proof of the main theorem, so remember, so we have a spherical space form different from Sphere RP3, we pick, yeah, this is our curvature one metric and we want to show that this space is contractable, right, so we want to show it's homotopy groups are trivial, so for every m, pi m of this guy is trivial, so let's pick an m sphere in map, map of an n-sphere into this space, continuous map, and right, so we want to extend that to a map of the m plus one disk in here, right, that's saying that it's an homotopic, well let's first extend it to a map into the space of all metrics, okay, space of all metrics is contractable, all right, it's just a convex subset of an affine space, so we can do that, then, so then for every point in this m plus one disk, right, we get this extension which is some some metric on on X, and then we can apply this Regiflow construction I had before and we get a partially defined metric, that's W sub GP G check of P, okay, so as P varies we see the subsets that are varying, but on each subset we see a constant curvature metric, okay, so the idea, so to complete the proof, we extend this metric to all of X, okay, so let me just give you a quick sketch of how that extension works, so so let's look at, right, we have this last minute, this metric here, this is in some late time slice with finding the many punctures, we have a metric here which is modular scaling converging to something with curvature one, okay, so we can think of having a metric here which has curvature identically one, and then we flow this back using the trajectories of the time vector field to a subset of X, so it's going to be, its complement is going to be some possibly horrible closed set, okay, but we push forward this curvature one metric, and so what we see here is a metric with curvature one on the complement of this closed set, okay, so now let's come back here, so we have finding the many punctures, each of them, so we can cover them by disjoint three discs here, so here we have a three disc, okay, but the part that lies away from the puncture is going to be an end, which is an S2 cross non-negative reels, okay, so mapping those collars around these points under the dipymorphism, what we get is some sort of collar around each of these guys here, okay, so I'm talking about this region here and similarly for the others, okay, so we have, so this is the picture I have for a single point in this M plus one disc, right, so the task is to extend this metric over this region, right, it has curvature one, but I want to extend it over this region, well just from topology turns out you can see that this entire region here, all these guys, these are all three discs, so let's call these, these are three discs, so we have the task, we have a three disc here, we have a curvature one metric defined near the boundary of the three disc and we want to extend it over the three disc, okay, so from the construction it's not hard to see that, well okay, so let's think about this, conceptually we have a three disc, we have some neighborhood of the boundary two sphere and here we have a curvature one metric which is defined, okay, so for such metrics you have a developing map dev which goes to the three sphere with the standard metric, so this developing map is an isometric immersion, right, and it's canonical up to post-composition with an isometry, okay, now in this case you can show this is actually an embedding, okay, so you have the embedding of this region here into S3, so something like this, okay, but since we're in the three sphere this region bounds a three disc, okay, so you can extend this maybe after restricting it to a slightly smaller region, you can extend it to a diffeomorphism of this three disc to this three disc here, okay, so when you pull back the metric you get an extension which has curvature one, okay, so that's how you extend the metric for a single guy, now the point is this extension process is basically it's unique up to making a different choice of the parameterization of this three disc, right, but it's a different choice which agrees near the boundary, okay, so the ambiguity corresponds to pre-composing with the diffeomorphism of the three disc which is the identity on the boundary, okay, and Hatcher's theorem for the three sphere tells you that that's contractable ambiguity, okay, so the idea is now if you have a parameterized family it's a little bit like obstruction theory where you have a trivial obstruction group coming from the contractability of these guys, okay, so I'll stop there. So the very last step you mentioned looks like you're using Hatcher's theorem for general theorem, yes, and now you're in the process of proving Hatcher's theorem with similar ideas. Using Ricci flow, it's a different, the last part of this, yeah, uses Ricci flow, uses the space-time geometry, but it is in a different way, yeah. You mentioned not simple manifolds, isn't it something interesting story, I think you made some projections years ago about B.D. for non-simple manifolds in particular, one can make a guess for candle body, say, yeah. Oh, right, okay, and some complicated processes next and so on. Yes, so I said, I assumed you're reducible, so there is some work on this, Hatcher has work, I don't know the literature that well, but the picture is, as you said, it's more complicated. I suspect that the analysis won't really tell you anything. Yeah, for example, candle body, one can guess, it's a big heap of candle body for genus B, it's the following one, it's really more for more space, and on the boundary it's the pulse subset when you get genus zero curves connected by things, you will enable for it to intercept the interior of more of a space, so it will be some complicated, not keep a long space. Do you know, I think it's Hatcher has some work on this, oh, someone's indicating, talking to the wrong person, do you know the literature on this? There's some work, but I mean it's, I'm not the right person to be talking about it, yeah. Well, maybe we can get you two together afterwards. If you have a compact legal action on a space form, this also prove it's conjugate to a subtle ASL of people? Yes, but I mean, this was already known by the Warbeful theorem, in fact, yeah. Is there a smooth policy? Yeah, I mean, in the case, I think the only interesting, the only case where you have to do work is when the group is zero-dimensional, because otherwise it boils down to something two-dimensional, I mean, where you have to work, you have to use serious technology, and that's the question. Thank you.