 Okay, so our next talk is by John Morgan, as you can see in the Poincaré Conjecture in 1905-2002. Thank you very much. It's a great honor and pleasure to be here to help examine the effect of Poincaré's work for the 100 years following his life. Like most, if not all, topologists, I consider Poincaré the founder of the subject of topology. There, of course, had been topological work before Poincaré, whether, for example, our Betty. But the first time topology became a subject in its own right was, as far as I'm concerned, in the work of Poincaré. Now, I'm not going to try to give a general discussion of all of Poincaré's topological work. I'm going to concentrate on one very famous but small piece of his work, the Poincaré Conjecture. And I believe, Dave, goodbye later in this conference. We'll talk more about Poincaré's topological work. So the story really begins with work in the latter part of the last half of the 19th century on surfaces. They were, by the time Poincaré started his work on higher dimensional spaces, very well understood. And in the 1890s, Poincaré began a long study of higher dimensional spaces. And, as I said, he introduced many of the techniques and ideas that still today form the basis of the subject of topology. And this work really culminated in 1905 in the San Quiem Complément de Analyses Citus, where he formulated a question, which later became known as the Poincaré Conjecture. He believed this was a central question in the theory of topology. And it concerned the first dimension in which things were not understood at that time. Everything was understood for curves and surfaces, but not for three-dimensional spaces. And it also concerned what Poincaré considered, and he considered the simplest of all three-dimensional spaces, the three-dimensional sphere. And he asked a topological question about that simplest case in the next dimension. And that's the Poincaré Conjecture. And through my talk, I will formulate the Conjecture for you if you don't know the statement. So, as I said, Poincaré thought he was asking the next question, the next one that you should look at in order to make progress in the subject. The first dimension that we didn't understand, so we move up by dimension, they'll just get more and more complicated as we go up. So we're in the next dimension, dimension three, and we'll look at the simplest of all three-dimensional spaces. Well, he was wrong. He was wrong on both accounts. It turned out that his question was generalized in many ways to other three-dimensional spaces and to higher-dimensional spaces. And this question, rather than being the next one to be resolved, was the final one to be resolved. So much was learned about higher-dimensional spaces before we understood Poincaré's question. And much was understood about other seemingly more complicated, larger three-dimensional spaces than about the simplest of all three-dimensional spaces, the sphere. So it took mathematicians almost 100 years, 1905, when the question was formulated, to 2002, before this particular question was solved. But in trying to understand this question and analogues in higher dimensions and for other three-dimensional spaces, 100 years of topology developed and grew out of these studies. And from my own somewhat biased perspective, I think of the 20th century as the century of topology, and it's due in great measure to trying to understand the question that Poincaré asked and its various generalizations using many of the techniques that he laid down. By the way, the answer to his question is yes. Poincaré posed the question, and I will show you the way he formulated it, in purely topological terms. He had surfaces and curves on surfaces and the way they intersect each other. And out of all of that, you can formulate the question. I think he clearly had in mind this was how you would solve the question. But in fact, once again, his intuition was not correct if this is indeed what his intuition was, because the solution came from a completely different part of mathematics. Maybe most surprising of all, that part of mathematics is closely related to other parts of Poincaré's work, namely his work on Fuchsian and more particularly on Kleinian groups. So let's go back and put ourselves at the turn of the century or maybe the 1890s and discuss briefly what was understood about surfaces. So first of all, maybe I should tell you what I mean by a surface. So these surfaces are going to be non-singular. It's possible to think about singular spaces. They're more complicated. We're going to concentrate today on manifolds for non-singular spaces. So the defining property of a surface is really the one I've shown you here, that every point on the surface has a neighborhood which looks topologically just like an open wall in the plane. And surface or two-dimensional space is reflected in the fact that our model is the plane, which is two-dimensional, meaning that locally we have two coordinates and you see I've drawn the coordinate lines there. So that's the defining property of a surface. For technical reasons, we're going to think of only compact surfaces. The topologists or the mathematicians in the audience will be very familiar with this. For those of you who are not, compact simply means the surface in question is a finite extent and there are no missing points or edges that you can fall off of. Well, there's a topological classification of these compact surfaces. I've drawn here some of the first few. These are all two-sided surfaces or the technical term is orientable. They're classified by one integer, the genus, which starts at zero, ones the torus we were just hearing about, genus two, three, four. And you can imagine how this series continues. You simply put in more and more poles. And the topological classification of these orientable surfaces is that every compact orientable surface is topologically equivalent to one of these. And in fact, depending on how the surface is presented to you, there are fairly effective ways of deciding where on the list your surface sits. Now, I haven't talked about non-orientable surfaces, but there's a similar classification for those. So, in fact, we have a complete classification of all compact surfaces. I'm not going to talk too much about non-orientable surfaces, but I can't resist showing you a couple of them. These, unlike the orientable surfaces, which we have nice representations of in two-dimensional space, they sit embedded in two-dimensional space. Non-orientable ones don't, so we have no direct picture of them. This is a picture of the smallest non-orientable surface, the projective plane. And you can see it's a three-dimensional representation, but there's a place where the surface crosses itself right here. This is not actually an embedded representation of the surface. Another one very famous is the Klein model. Once again, you see this surface cutting itself right here, and that's not part of the, in the definition of the surface. These two sheets are separate, so this is not an embedding. The surface crosses itself. And this is the beginning of, these are the first two in the list of non-orientable surfaces. The list goes on analogous to the way it does for orientable surfaces. All right. Well, one question I haven't addressed at all, and which is somewhat delicate to talk about, is when are we going to consider two spaces as equivalent? Well, the answer, of course, depends on which kind of mathematics you're doing, what kind of structures you're interested in concentrating on, and here we're concentrating on topology only. So two spaces are the same. If there's a continuous mapping from one to the other, that's a bijection and doesn't do any tearing or ripping of the topological structure, that has to be true of the mapping and its inverse. That's a topological equivalent. So in particular, in topology, there's no notion of links of curves, area, volume, shape, or size. There's only the form that you have. A round sphere, an ellipse, a very wobbly thing like the surface of the earth. Those are all topologically equivalent. They're the two sphere. We have the nice round torus that we think about, but we can have a very elongated one, a very long thin one. Those are all topologically equivalent. Often one thinks of topology as the science of cut and paste. You present a topological space by giving various pieces and saying how they fit together. The pieces can be quite simple, but if there are enough of them and the gluing pattern is complicated enough, you can get quite interesting shapes, topological spaces, out of fairly simple pieces. And I want to show you a couple of examples and surfaces where we can really see what's going on before we get to our higher dimensional spaces. So I'm going to give you a cut and paste description of the two sphere. There it is. There's the two sphere. I think of it as the join of the upper and lower hemisphere with the two boundary circles glued together becoming the equator. So just to separate the pieces a little bit, I'll put the two hemispheres. I'll separate them. Again, their boundaries are to be identified together. Now I'll flatten out each of the hemispheres. And I can think of the two sphere as two discs with their boundaries glued together. Going around here is identified as going around there. So that's a cut and paste description of the two sphere, just so you get a little more sense. I'll do a cut and paste description of the two torus, a very famous one. We start with the square. And we're going to identify the top with the bottom as indicated here and the right side with the left side. So let's just see what happens when we do that. Well, if I identify the top and bottom together, I get a cylinder and now on the cylinder I need to identify the right-hand circle and the left-hand circle together. And that's not too hard to do. I'll bend this tube around until its ends are close to each other and then glue them together and you see you get the torus. So this is a cut and paste description of the torus. Let me give you a cut and paste description of this projective plane. And I'll start with the description of the projective plane as the space of straight lines in three-dimensional space that pass through the origin. Well, any straight line that passes through the origin meets the sphere, the unit sphere, in two antipodal points. So the real projective plane is the topological space you obtain by taking the two-sphere and identifying antipodal points. Well, you can't actually see that space, but every antipode has one representative in the upper hemisphere and one representative in the lower hemisphere with some exceptions. The exceptions are points on the equator, where both the point and its antipodes are on the equator. So the space of lines in three-dimensional space can be identified with the upper hemisphere with a gluing identification or gluing rule that says you take every point on the boundary and glue it to its antipode. Well, here I've flattened out the upper hemisphere to a disk and now this antipodal identification identifies the lower half-circle with the upper half-circle. Now, again, you can't really see that space, but there's another nice description of it, a neighborhood of what's happening out here on the boundary, in fact, is the Mobius band, and then the inside is just a nice disk. So the projective plane is the union of the Mobius band with the disk where the boundaries are glued together. So that's some cut-and-paste topology in dimension two. Now we're ready to move to higher dimensions. And unusual to our eyes and ears, Planckere started his treatise on high-dimensional spaces by defending the right to do this study. He said that, paraphrasing from the French to the English, even though we cannot directly see these higher-dimensional spaces, they are susceptible to mathematical definition and thus subject to rigorous mathematical study. And furthermore, his studies in other areas of mathematics and physics always led him back to these questions about the topology of higher-dimensional spaces. So for those reasons, he proposed to make a study of higher-dimensional spaces, which he started with these treatises. Okay, well I want to concentrate, as I said, on three-dimensional spaces. So their defining property is the analogous property to surfaces, but now using three coordinates instead of two. So spaces that are locally modeled on ordinary three-dimensional Euclidean space. In other words, near every point we have three local coordinates, which look like the coordinates in Euclidean space. And again, I'm going to only consider the compact case. No missing points, no edges, and a finite extent. As Planckere said, we cannot directly see any of these spaces. Let's think about the simplest one, the one that the Planckere conjecture concerns. And that's the sphere, the three-dimensional sphere. So the three-dimensional sphere is given inside four-dimensional Euclidean space, the Euclidean space of four coordinates by a single equation, the analog of the equation for the unit sphere in three space. So this is a three-dimensional sphere. How are we going to understand the three-dimensional sphere? I'm going to give you two descriptions, and both of them are by analogy with what we've already, well, with things that are easy to see about the two sphere. So the first is, think of the three-dimensional sphere as a completion of ordinary three-dimensional space where we add one more point. And again, if we drop down a dimension to dimension two, this is something that you can see through what's known as stereographic projection. Imagine a light at the north pole of the sphere, and we have a plane which is tangent to the opposite south pole. The light rays coming out of here, straight lines from this point, will puncture the sphere in one point and continue down onto the plane. So this gives us a correspondence between all the points in the sphere and all the points on the plane, with one exception. There's no point in the plane that corresponds to the north pole. So if you take the north pole out of the two sphere, this straight line identification between points in the sphere and points in the plane identifies all of the two sphere except the north pole with the plane. I think of this as, I guess you have these in France, these balls of cheese and at Christmas time they're wrapped up in paper and then tied up with a nice little bow. Well this time our paper is not a finite extent, it's an infinite extent, and there's nothing left over when we wrap it up. As you go out to infinity in any direction on the paper, you're headed, you go out by a straight line, you're headed along a geodesic toward the north pole. So that's one description of the two sphere. There's a completely analogous description of the three-dimensional sphere. It is the completion of ordinary three-dimensional space by adding one more point at infinity. And if you start here and you go out in any direction toward infinity, you're headed toward that last point. That's one way to think about the three sphere. There's a cut and paste description of the three sphere, which is analogous to the cut and paste description I gave you of the two sphere. Namely the two sphere was the union of two discs glued together along their boundary. So the three sphere is the union of two solid three-dimensional balls glued together along their boundary. You can sort of almost imagine how you're going to do this. You start gluing the boundaries together, but it doesn't quite work at the end. We can't see this three-dimensional sphere. Nevertheless, we have a nice mathematical description of it. Now, it turns out that every three-dimensional manifold has a description similar to the one that I just gave of the sphere where you have to replace the ball by what's called a solid-handle body. So here I have two solid-handle bodies of genus three. So this time the boundary, which is what you see here, is one of these surfaces, the surface of genus three, but I'm now considering the region of three-dimensional space enclosed by this surface. So just like the balls were the regions enclosed by the spheres, here I'm talking about what are called solid-handle bodies, or the regions enclosed by these surfaces. So if I take these two solid-handle bodies, the boundaries are topologically equivalent. In fact, there are lots of topological equivalences between them, some very interesting ones. If you take any topological equivalence from one boundary to the other and you glue them together, you'll get a three-dimensional manifold and every orientable three-dimensional manifold can be presented that way for some genus-handle body. The sphere was made by using genus zero-handle bodies. Those with boundary is sphere, the two-sphere. The three-sphere is the only three-manifold that can be presented that way, but every three-manifold can be presented using some handle body of higher genus. Well, the trouble with this description is that there are many different ways, in equivalent ways of presenting the same manifold. So you can have lots and lots of different handle body decompositions. A presentation of a manifold is a union of two solid-handle bodies by some gluing, where the genus of the handle bodies are different, the gluings are completely different, yet the manifolds are the same. There is a sequence of elementary moves that allows you to go from any one presentation to the other, but one of the moves allows increasing the genus of the handle body. In fact, there's no effective algorithm for deciding when to handle body presentations give the same manifold. So that's a way to present all three-dimensional manifolds and in some theoretical sense at least element, well, in a result that tells you element removes that will carry you from any one presentation to any other of the three-dimensional manifold. But we need to work on the other side as well. How can we tell when two spaces really are different? And, of course, the answer to that is invariance, tautological. Two manifolds have different invariance. Well, if two manifolds have different invariance, they're different. There are lists of invariance for manifolds, many of them introduced by Poincare, and we'll concentrate on one, but there's no general theorem when you have enough invariance. And, in fact, the Poincare conjecture is a question about whether or not a particular invariant is enough to distinguish the three-spheres from all others. So, as I said, Poincare introduced many invariance, which I won't talk about. I'll just list them, the homology. Well, parts of homology were known before Poincare, but Poincare focused on homology. He introduced a group that will be very important for us, that's called the Fundamental Group, or the group de Poincare. And that group is made from loops in the space. And in the case of three-dimensional manifolds, that group can actually be computed from the presentation of the manifold as a union of two-handle bodies. Let's just talk for a few minutes about loops in the space, and let's go back to the case of surfaces. Here I've drawn three surfaces, surfaces are indicated in red, and on each surface, well, on the sphere I have a loop, on the torus I have a loop, and on the sphere of genus two I have three loops. Now, it's not too hard to convince yourself that this loop on the two-sphere actually can be continuously deformed to a point. In fact, as you see it here, it bounds a disk on the surface of the sphere, a pathological disk, and you can use that disk to shrink the curve down. It's probably also fairly intuitively obvious that this loop on the torus cannot be shrunk to a point continuously. No matter how I try to move this loop around, keeping it on the surface of the torus, it's going to go around this hole so I can never shrink it to a point. And here I've drawn three loops on this surface. There are many, many more. I've chosen to draw these three, to be shrunk to points. Now, I've given, I gave you a brief argument that this loop on the surface of the sphere can be shrunk to a point. In fact, any loop on the surface of the sphere can be shrunk to a point. If you think of the sphere as two-dimensional space with one more point, as long as the loop avoids the extra point, it's a loop in the plane, any loop in the plane, we can just take a straight line so that works to show that any loop on the two-sphere shrinks to a point, but on every other surface, as I've indicated the next two here, there are loops that don't shrink to point. So the two-sphere is characterized by a topological property among all surfaces. It's the only surface with the property that every loop on it shrinks to a point. Well, surfaces and more generally spaces with this property are called simply connected. So the two-sphere is the only simply connected surface. So Poincare said, what about dimension three? Is the three-sphere the only three-dimensional manifold with the property that every loop deforms continuously to a trivial or point loop? That's the Poincare conjecture. And it's in complete analogy to what happens in dimension two. So I've said another way. If a three-manifold M is simply connected, then is M topologically equivalent to the sphere or topologically equivalent to the union of three balls with their boundaries identified. That's the Poincare conjecture. Poincare presented three-dimensional manifolds in exactly the way I have talked about them in terms of the unions of two solid handled bodies. And he gave some very beautiful examples of surfaces that were presented this way and explicit equivalences of boundaries of handled bodies. There was an earlier version of his question in which he asked whether or not the homology was enough to determine these three to characterize the three-sphere. Was the three-sphere the only three-dimensional manifold with its homology? And then he gave an example of a union of two genus-two handled bodies, an explicit example. And he computed the first homology and showed it was zero and computed the fundamental group or Poincare's group and showed that it was not trivial showing that this manifold was in fact not topologically equivalent to the three-sphere even though it had the same homology. That manifold now goes into the name of Poincare's homology sphere. So his first attempt at this question turned out not to be correct and he gave the counter example and then he formulated it this way. So as I said earlier doesn't stretch the imagination to believe he thought the question could be answered by studying these handled bodies and the way they're glued together in particular looking at various closed curves on the boundary surface that describes the two, the handled bodies on the two sides. I think he, reading now, it seems like he thought it should be possible to study those presentations and the various moves and prove his, answer his question. Well, not only did he believe that but generation after generation of leading topologists over the next almost 100 years believed that enough to attempt to answer the question in these terms and it all came to naught at least as far as the Poincare conjecture was concerned. Now, it didn't lead to naught it just led to naught in resolving this conjecture. It's through these attempts that much was learned about other three-dimensional manifolds more complicated three-dimensional manifolds in the 1950s and 1960s and also in higher dimensions. There's an analog of Poincare's conjecture about spheres in higher dimensions and in 1961 Smale resolved it and showed that the analog of Poincare's question was in fact true in all dimensions five and higher and in the early 80s Mike Friedman resolved the analogous question in dimension four leaving only the original question in dimension three about spheres and also as I said for three-dimensional spaces three-dimensional manifolds much more complicated manifolds were fairly well understood by this time as well. So we have the two axes dimension going up and complication going out we have the three sphere down here in the corner and we have all this information going up and this question left standing as the last last man standing. Well it was finally solved Poincare's question was resolved in 2002 by Grigory Perlman he showed that the Poincare conjecture is true, is true that is to say the answer to Poincare's question is yes if you have a simply connected three-dimensional manifold it is topologically equivalent to the three sphere but the method of solution unlike the previous hundred years of 97 years of attack was not a purely topological attack the solution came from very different areas of mathematics geometry and partial differential equations so rather than a direct topological attack we use this deep result out of geometry and partial differential equations and the beginning of this connection is the theorem from the 19th century mid-19th century that all these topological manifolds have on them what we call a Romanian metric so these manifolds have tangent planes so I've drawn a little bit like a surface a point on the surface there's a linear space associated with a point the tangent plane of the manifold a Romanian metric is simply a way to assign links and angles like in Euclidean space to the tangent planes to every tangent plane that very smoothly as you move around the manifold so called a Romanian metric well there's not just one Romanian metric there's an infinite dimensional family of Romanian metrics on any manifold each Romanian metric has associated with it curvature which is really the part of the metric that's independent of the coordinates used to write it down when you write a metric down you need local coordinates to write it you choose different local coordinates you'll get different formulas but the curvature remains as you change coordinates so this is the part of the metric that's independent of the coordinates and I've tried to indicate a little bit about curvature in dimension two which is the simplest the first dimension where there is curvature and I've drawn three pieces of surface this one has positive curvature this one is flat and this one has negative curvature in two dimensions curvature is as Gauss described it a measure of the discrepancy of area from the area in the flat plane so you look on your surface take a point take the ball of radius r on the surface and you compare the area of that ball with the area of the ball of radius r in the plane you take an appropriate limit as r goes to zero and you get a curvature it's actually got a sign reversal in it if your area is smaller than the planar area as it is in this case that's positive curvature here the area is the same as the plane so it's flat and here the area is bigger you know this has less area because anytime you take top of an orange peel and you flatten it out it rips because it doesn't have enough area to cover the area on the plane if you were to project this sort of saddle under the plane it would overlap on itself it has more area this one you just slit it open and rattle it and it flattens out under the plane so it has the same area so that's curvature in dimension two Gaussian curvature in higher dimensions at every point and in every two plane direction there's a curvature called the sectional curvature which is analogous to the curvatures we see here and all these two dimensional curvatures fit together into something called the Riemann curvature tensor and as I say that's the part of the metric that's invariant under coordinate changes for example Riemann proved that your if you have a romanian metric you can find coordinates where it looks like a Riemann metric if locally at least if and only if the Riemann curvature tensor is zero now the nicest metrics are those of constant curvature where all points all two plane directions at all points have the same curvature these are called metrics of constant curvature and manifolds of constant curvature are well understood through the theory of Lie groups and so on and in particular the three sphere we simply connected three dimensional manifold with constant positive curvature so this gives us a way to approach the Poincare conjecture start with a simply connected manifold and try to find on it a metric of constant positive curvature if you do you will have produced you will approve that the manifold is the three sphere and that's exactly what Perlman does he starts with an arbitrary romanian metric on our simply connected three manifold and he uses a partial differential equation to evolve that metric and shows that eventually it becomes round now the equation that Perlman uses is an equation involving the curvature of the manifold so you flow the metric it's an evolution it's a partial differential equation for an evolution of the metric it was first introduced in mathematics by Richard Hamilton and in physics by Dan Friedan where in fact it's a renormalization group flow it's called the Ricci flow equation because it deforms the metric using the Ricci curvature tensor and structurally as a differential equation as a partial differential equation it looks a lot like a nonlinear version of the heat equation but not for scalar functions but rather for these tensors the romanian metric and I've written the equation down the metric here is gij in local coordinates this is the Ricci tensor which is a curvature tensor derived from the romanian tensor by tracing on two of the four indices and this is a parabolic evolution equation so this is the time variation of the metric given by the curvature the interesting minus sign so because it looks a little bit like the heat equation for tensors the Ricci flow has a dispersive quality to it and I think the intuition should be that we have curvature all over the manifold and this dispersive equation is going to equally distribute the curvature around the manifold and in the case of a simply connected manifold going to flow to curvature constant positive curvature and that's how we'll find this metric that we need in order to see our manifold is the three sphere well this is an oversimplification of what actually happens this is a nonlinear equation and as in all nonlinear equations one has to worry about the nonlinear terms forcing singularities driving you to singularities and in fact that happens this equation has finite time singularities doesn't always just flow nicely to the smooth round metric and in fact the most delicate part of Perlman's argument is dealing with these finite time singularities where the curvature is blowing up he was able to give a qualitative description of those regions and using that to show how to continue the topology and the geometry the topology and this flow equation through the singularities and keep the flow going until you converge to the round metrics at some later time this is exactly what he did his argument works as well not only for the three sphere but for all three dimensional manifolds and produces a description of every three dimensional manifold in terms of geometric pieces it does distribute the curvature evenly around the manifold but different pieces will have different kinds of curvature and it completely solves the question of what all three dimensional spaces look like most of them turn out to be the most interesting class turn out to be hyperbolic manifolds of constant negative curvature and these are exactly the three dimensional manifolds that come up in Kleinian group theory circling back to Poincare's original work on Kleinian groups so most three manifolds the most interesting three manifolds are accounted for by these constant negative curvature manifolds the Kleinian groups here we have the positive version of that in the three sphere and this geometric differential equation shows us how to find all these geometric structures starting with any Romanian metric and therefore understand all three dimensional manifolds this is nothing at all like Poincare's original presentation of the manifold I haven't mentioned Hagar decompositions or loops on surfaces or anything else I've used geometry to solve this problem there is now a very interesting new development in three dimensional topology which does go back to this point of view of the three manifold as a Hagar decomposition and studying the loops that bound discs on the two sides and their patterns of intersection we now have thanks again to some partial differential equations we now have ways to study those intersection patterns that are stronger than just purely topological methods that were available to Poincare and the next 100 years of topologists and it's not inconceivable that we might in the end be able to use these techniques to give an argument more like the one Poincare originally had in mind but that's what people thought for 100 years so maybe I'm just being overly optimistic and with that I'll stop thank you, time for questions yes no the differential equation exists in all dimensions but it's much more powerful in dimension 3 than it is so far in any higher dimensions and I think the fundamental reason is that dimension 3 is a dimension where Riemannian curvature and Ricci curvature are equivalent information in all higher dimensions the Riemannian metric is a much more complicated higher dimensional object and the Ricci curvature is just a shadow of Riemannian metric the flow equation is about the Ricci curvature what you need to know about is Riemannian curvature and this discrepancy makes the equation much less powerful in higher dimensions than it is in dimension 3 there are higher dimensional results now using this equation but they're in the complex scalar case where the metric is constrained quite a bit more than the general Riemannian metric and there are some positive results coming out of Ricci flow in that context more questions yes, Claire the handle body decomposition the question is how did he know about the handle body decomposition he got splitting this came out of his study of Poincare duality his approach to manifolds was to triangulate them and then he had the skeleton and the dual skeleton and if you do this in dimension 3 you have a neighborhood of the one skeleton which is a solid handle body and the dual which is a neighborhood of the dual one skeleton so there are your two solid handle bodies and what's in between is just a product so it was this picture of Poincare duality I mean now we would say look at a Morse function but that's not how he thought about it he thought about it in terms of a triangulation questions? I have the following question you mentioned that you believe that Poincare would have been surprised by Perlman's approach are you really convinced by that? I just want to say that one of the many proofs of Poincare or the uniformization theorem is really using the somehow regression of the heat equation so he was very well aware the fact that PDs are fundamental in topology and this is at least my point of view oh well let me let me by analogy say yes you may well be right when I talked to Bill Thurston about this he said I said what do you think of this Bill he said yeah well that's sort of how I thought it would go but but I wasn't the right person to try to do it he went on to say okay so if there is no more question thank you again