 Punkeray's work on topology. Thank you very much. So it's really just a great honor to be able to come to Institute on Read Punkeray to speak about Punkeray's work on topology. And well, so these papers are Punkeray's, Annalise C2 papers. I mean, that's Annalise C2s. That's what they call topology in those days. And there's, well, there's several Comperondu announcements and five papers, but actually six, the original paper and five complements. But really the hardcore topology and the stuff I'll mainly talk about is in the first paper, the first and second and fifth complements. So John Morgan, in his lecture yesterday, said that most topologists think of Punkeray as the founder of topology. And part of the goal of this lecture is to make this argument, not that he's the first mathematician that did sort of impressive results in topology, rather Punkeray was really the first person. Well, he really created topology as sort of a separate field of mathematics, a field which has its own internal problems and issues. But Punkeray demonstrated that topology is a field that gets a lot of input and inspiration from other fields of mathematics and science, and in turn gives a lot of sort of contributions to other subjects. So John, yesterday sort of gave some hints about what topology is. And well, I think I'm just refrained from giving sort of a formal definition. But rather I'd like sort of Punkeray in his own words to tell us, rather not the technical definition of what topology is, but sort of what its structure is. So in the introduction to his first paper, he says, geometry, in fact, has a unique raison d'etre as the immediate description of the structures which underline our senses is above all the analytic study of a group. Consequently, there's nothing to prevent us from proceeding to study other groups which are analogous but more general. So I think what he's saying is that he was looking at, well, before he had the study of Fuxian groups, he was looking at these very special things. And Fuxian groups are the groups underlying hyperbolic geometry, the non-Euclidean geometry of the plane. And now he recognized that there's really other groups which describe sort of some more general objects, manifolds. So but then he makes this statement, which I really like, is, perhaps these reasons are not sufficient in themselves. It is not enough, in fact, for science to be legitimate. Its utility must be incontestable. So many objects demand our attention that only the most important have the right to be considered. And then he goes further and gives some examples of why it's important to study topology. But I'd like actually to quote, which actually John paraphrased yesterday, but just give the direct quote from this other paper, which he wrote in 1901, which was sort of a summary of the scientific works. And he says, it has been said that I've written it in the preface to Analyst of Cetus, which he did. That geometry is the art of reasoning well with badly made figures. Yes, without doubt, but with one condition. The proportions of the figures might be grossly altered, but their elements must not be interchanged, must conserve their relative situation. In other words, one does not worry about qualitative properties, but one must respect the qualitative properties. Sorry, one must not worry about quantitative properties. One must respect the qualitative properties. That is to say, precisely those which are the concern of Analyst Cetus. Then he mentions that there was an early stuff studied by Riemann and Betty, but no one really is followed up under works. And now here, he gives this long, just incredible reason which drove him to think about topologies. He says, well, as for me, all the diverse paths which I was successfully engaged in led me to Analyst Cetus. And he goes on and on. Finally, I glimpsed in Analyst Cetus a means of attacking an important problem in the theory of groups. The search for discrete or finite groups contained in a given continuous group is for all these reasons that I've devoted to this science a fairly long work. So that was the introduction. And before I go on and talk about what Ponqueray actually did, I want to just give you an idea of what the state of topology was before Ponqueray. So this is the first page of Euler's famous paper on the bridges of Konigsberg. And actually, this is sort of a picture in that paper where the problem is, well, here is like the city. And think of it being in four parts. And the question was, can you just take a walk starting at one spot and just go over all the seven bridges without going over a bridge twice? And Euler sort of solved that problem. And maybe that was the first serious result in topological type of argument. So Euler also discovered the Euler characteristic. And Gauss discovered this idea of linking numbers. So he had these two curves that somehow had to measure how much they go one around the other. Of course, he did a spectacular Gauss-Benet theorem, which found sort of a connection between geometry of the surface and its Euler characteristic. Riemann, well, he introduced the idea of a manifold. And in general, but in practice, he just looked at the two sphere and looked at really branched covers. But for surfaces, he had this measure of how complicated the surface was. He called it the order of connectivity, how many cuts you need to chop a surface up into a disk. And then there was Listing, who discovered the Mobius band and actually coined the word topology. The topology is spelled in English. It was first used by Peter Tain as a bituary. Well, Mobius independently discovered the Mobius band. Cayley and Maxwell independently discovered this idea that if you have an island, then if you look at the high points and the low points and the saddles, they syncs minus the saddles plus a sum equals 1. And then there is this Jordan curve theorem, which said that if you have a closed curve in the plane, it separates into two pieces. There are various works in knot theory. And then there is a very influential work by Betty, which tried to generalize what Riemann did to higher dimensions. But all these works, and this really summarizes the whole extent of knowledge of topology before Pocherey's works. I mean, they're sort of very interesting isolated results, but not really part of a general theory. So now I'd like to just say some things about, well, what did Pocherey do in his foundational paper, which actually was preceded by two announcements? Well, to start with, well, he actually gives a little sharper definition of manifold, which was just along the lines that Betty and Riemann did. But another thing which actually John Morgan mentioned yesterday was the current, I mean, well, the standard definition now of a manifold is just being built up by charts, by pieces. I mean, that sort of is an innovation in this paper. He defines this notion of orientation. Well, actually, the idea of orientation existed earlier, but using sort of the termines, so between the transition maps, of course, the spectacular contribution of defining the fundamental group. So if you're given a manifold, then associated to that is a group, the fundamental group. And he defines it both sort of as a group of covering transformations in universal cover and generated in terms of loops. And this itself is quite revolutionary, because previous to this, to topological things, there is just sort of numbers. There is invariance. But here he's associating a group. Now he defines this idea of Betty number, which, well, in this form, it was 1 plus the maximum number of the kth Betty number is 1 plus the maximum number of k-dimensional orientable sub-manifolds, so that over the integers, these sub-manifolds are homologically independent. So in other words, if you just take some linear copies of this one and copies of that one and copies of that one, and if they all bound a manifold of one higher dimension, that's what a homology is. Well, more or less, in modern words, he recognized that homology, H1, is the abelianization of pi1. He defines this idea of algebraic intersection number and shows that if you have a manifold of dimension five and a manifold of dimension three living in a manifold of dimension eight, they'll intersect in points. But rather than just counting number of points, count number of points with signs. And you get a number, which if you just take the intersection of m and n versus n and m, they differ by a certain sign. And these intersection numbers are invariant under homology. And I should say this, for those of you who aren't mathematicians, these are just incredibly fundamental concepts that it's just amazing that just in one paper that they're introduced, he uses integration theory to distinguish homology classes, generalizing methods of Betty. He gives really interesting examples of manifolds. I mean, the examples that existed through Betty, through Klein, and students were not really too exciting. But I mean, it's amazing that he just, for example, looked at tourist bundles over the circle. And of course, he had a spectacular, where all this stuff is spectacular, but the punkerate duality theorem that says that if you have a manifold of dimension n, then the Betty numbers that are, in his words, that are sort of equidistant from n, the kth Betty number and the n minus kth Betty number, those are equal. And while he thought about this because he had some, well to him, he had this idea that a class was homologically trivial, if in all of you it's algebraic intersection number with classes of complementary dimension or trivial. And I should say he was very much inspired by Picard, who noted that the first and third Betty numbers were equal for complex surfaces. So, right, he defined this idea of cellulation, which, I mean, was very similar to, I mean, this idea of he just takes some space and just sort of decompose it into cells. I mean, that's, it was just a variation of an existing theme, but a very important sort of psychological way of looking at things. Of course, the famous Euler-Ponqueray characteristic relating to classical Euler characteristic is the alternating sum of Betty numbers. So, in this one paper, just incredible, truly incredible, just one after another, just absolutely fundamental seminal ideas. So, I'd like to mention a quotation of Darbell in his eulogy. He said, I saw a Ponqueray at the Sorbonne, the Bureau of De Launcetudes at the Academy. Whenever asked to solve a problem, his answer came with the speed of an arrow. When he wrote a memoir, he wrote by a single drift, making a few changes without returning to what he had written. I should say I made a serious attempt to read Ponqueray's paper. And I believe what Darbell says is true. But this is actually, to me, a very interesting point that his writing style might have been just one go. But his working style, completely different. And I'll try to say just by quoting his own words that he constantly came back and rethought about central issues. And you'll see in the case of topology that each time he sort of rethought the subject, just incredible sort of the new breakthroughs that he was doing. So, well, du Donné was a little bit more direct. He said, it is often necessary to guess what he had in mind by interpreting the context. For many results, hardly a single argument does not raise doubts. The paper is really a blueprint for future developments of entirely new ideas, each of which demanded the creation of a new technique to put it on a sound basis. So, well, I mean, just have a list of things here. I mean, for example, this idea of just thinking of classes as sub-manifolds, well, that's actually a very delicate issue that took, I mean, there's really sophisticated algebraic topology and was finally sorted out by Rene Tom in 54. As far as using integration to understand classes, well, I can't do better than quoting sort of the anonymous author of the Samir in Volume 6 of Pancras' Collected Works, where he said, paragraph seven introduces cohomology. He posed problems that Elie Cartin explained and George Durom resolved. So the famous Durom theorem, which relates sort of the Durom cohomology based on forms with cohomology based on in real coefficients. And anyway, Cartin sort of inspired by what Pancras wrote, sort of conjectured these Durom conjectures, which Durom proved. Anyway, there's also other things, topological invariance homology. So this business with intersection numbers, well, to really put that in a firm foundation, you really have to understand sort of general position, which was done by Whitney. And of course, incidentally, Whitney proved this embedding theorem, which showed that actually all the manifolds that Hunkray considered are actually just takes here of all the manifolds. So Heiguard, in his 1898 thesis, well, he had some complaints, too. And in particular, he noted that RP3 was a counter-example to Punkray duality because the first betting number is one, the second betting number is two. Now, so in one due paper of 1899, well, Punkray responded to that by pointing out that his definition of betting number is actually different than Betty's definition of betting number. And well, the first approximation, well, definitely his betting number was just a rank of homology and Q coefficients, while Betty's number was approximately, well, to some extent, the first approximation, the rank of just in Z coefficients. So in particular, for RP3 rationally, it's just homology is just trivial in dimensions one and two, while in Z coefficients, of course, you know that betting number is one, and the second betting number is two. But going from Betty to Punkray, just looking at Betty numbers, Punkray had this huge breakthrough where he was allowing to add and subtract and divide homologies by integers, which, well, and today we think of that. I mean, that's just sort of in rational coefficients. But this is Hegard's objection. But I quote a gift because Hegard was really the first person to really look at what Punkray did. And I should say that Punkray, to his credit, he could have just blown off Betty by just this, sorry, by Hegard, just sort of dismissed Hegard by saying, Hegard, look, you misunderstood what my definition of betting number was. But Punkray took this as an opportunity just to sort of rethink the whole subject. And right. So then he just quickly followed up with the first and second complements. So here's a quotation from the introduction of the second one. It says, nevertheless, the question, that is, topology, is far from being exhausted. And I shall doubtless be forced to return to it several times. This time I shall confine myself to certain considerations that are likely to simplify, clarifying, complement the previous results. So in this one, he really just goes back and just really drops this idea of homology classes being represented by manifolds and introduces what we now have the standard thing of simplicial homology. And so in particular, if you have a space, it's built up by cells. Say there's CI cells and CI minus 1, I minus 1 cells. And he recognized how to compute the homology by just taking boundary because he sort of understood orientation, boundary orientation, from his first paper. And therefore, the boundary operator is this CI minus 1 times CI matrix. And then just by elementary linear algebra, then the ith betty number is just the number of I cells minus the rank of the ith boundary operator minus the rank of the I plus first matrix. So therefore, there's this nice, clean definition of homology, and therefore, he just gets really clean proof of the Euler-Punkray formula because when you're taking the alternating sum, well, you can see that the number of cells alternating some of the cells, which is just what Euler's definition is, well, in the low dimension case, is these bi's, they just cancel each other out. And in this paper, he recognized that there's the torsion part. And what I should just do a little side point is that it's interesting that although Punkray introduced the fundamental group, the first time a group is associated as an invariant of a manifold, nevertheless, he and for the next 10 years, some very talented top level topologists thought about homology not as a group, which we do now, but in terms of just numbers and torsion numbers and betty numbers. And she just made this just a very simple observation, which psychologically changed the way people thought and just really a big advance, which was that you should just think of these things as groups. So right, and there's certain things that had to be cleaned up later, going from smooth, the PL, and invariance of these as topological invariance. And then Punkray goes on to just, again, completely fundamental ideas, which are the idea of barycentric subdivision, star of vertex, dual salulation. Anyway, well, if you're a topologist to recognize this picture where there is the black, which is, say, the original salulation, and there's the red, which is the dual salulation. And so these are two different salulations of the same manifold, but they both sort of subdivide to the barycentric subdivision. And anyway, staring at this picture and thinking about it, you get sort of a clean proof of at least in the simplicial world of Punkray duality. And well, in here, he sort of announces his first version of the Punkray conjecture, that he thought that if you have a manifold and all its bedding numbers, well, in the middle, except for the top and bottom dimension is zero, then the manifold's a sphere. And I should say, actually, this idea goes back to Riemann, the idea of if you have a manifold, try to find some invariance that just determined that sort of uniquely. And one of his big motivations was with these tourist bundles. Those were examples of manifolds which had the same bedding numbers, but were different manifolds. And so this is sort of, again, searching for the right sort of criterion. So now let me just go to the fifth compliment. So again, in the introduction, he says, I now return to the same question. Persuade that one can come to the end only by repeated efforts, and it's important and sufficient to justify further effort. This time, I can find myself to the study of certain three-dimensional manifolds, but the method used without doubt are of more general applicability. So of course, this is the famous paper where he entered, finally, after this long process, gives the right form of the Punkray conjecture. But ignoring that, this is just an incredible paper. And in this paper, he basically creates the Morse theory and handle-body theory. So in Morse theory, you have a manifold and you have a function from that manifold to the real numbers. And that's a way of slicing up the manifold using this function. Think of it like some type of height function. But there's certain singularities. And in the Morse theory, well, you show the function can be realized so that the singularities are all of certain types, and which you can then build up the manifold by just understanding these singularities. And what's incredible is all of that is in this paper. So here, you can't read it. But there's, well, it's really, it's right there. I mean, well, if I could actually make it bigger, it's just incredible. I mean, because, well, I don't know. Maybe it's the French education. Everyone knows this. But to me, I'm just totally shocked to see this. I mean, right here, it basically says that you have this height function on this manifold. First, you make a perturbation. Then you change the coordinates. And this function looks like a quadratic function. It's exactly Morse theory. And then sort of here's like the standard picture of sort of the index one singularity on a surface. Anyway, that just, that's, and so when he, in his introduction where he says this stuff, I have more equitable ability to higher dimensions. Well, this is exactly what sort of smells starting point for proving the high-dimensional Poincare conjecture. So, right. So anyway, so Morse and Morse didn't actually come out and say this isn't Morse theory. This is Poincare theory. But he does, in the introduction to his book, The Calculus of Variations, he gives sort of a clear acknowledgement to Poincare. So just, so he says at some point, such conceptions are not due to the author. It'll be sufficient to say that Henry Poincare was among the first to have a conscious theory of macroanalysis. And of all mathematicians, was doubtless the one who most effectively put such a theory into practice. So, that's more, so here's sort of other results from the, from this fifth compliment. He recognized that you could, there's a symplectic form and you use that to compute algebraic intersection number. He had a criterion for showing when a homology class is represented by a simple closed curve. Another criterion for when a homology, when a simple closed curve, sorry. Here's another criterion when a homotopy class is realized by a simple curve. And this to me, he just says this just casually in a couple lines, really incredible. So he just says when you take a curve, just homotopic to a geodesic, lift it to the universal cover. And if you just see that these, these are all just disjoint curves, just assuming it's not a power, then this is, it's gonna be, it's simple. On the other hand, if you homotopic to a geodesic and you see in the universal cover that these things are crossing, then it's not really simple. That is, it crosses itself downstairs. And I mean, to me, this is really incredible because the fact is for him to make this statement means that he must have known. I mean, he says these things if it's obvious, but he must, it means that he really understood this idea that if you take something, you do a homotopy. Well, that sort of look, might be look like a radical change in the surface, but the homotopy is going to just nevertheless, you start with this curve, you do some crazy homotopy, you're still gonna get some other curve which has the same endpoints at infinity. So this is probably the first time that the apology of infinity of hyperbolic space is used to do topological stuff. Anyway, he just showed how to compute fundamental group. Again, this is, I mean, there's really, I mean, this is just the primal form of the well-known Cyford-Van-Kampen theorem. Of course, then there's the, his famous thing of showing the punk ray homology sphere, this sphere which is counter-exampled to his, his earlier form of the punk ray conjecture, first the famous picture of the punk ray homology sphere, and the final statement of the punk ray conjecture. Actually, just an aside, you know, he states so the punk ray conjecture, so this one question remains to be dealt with, is it possible for the fundamental group of this manifold to reduce the identity without being simply connected, then he just gets some more explanation. And then the last statement of the paper is, however, this question will take us too far. Now, you know, I never understood what does that mean? So, but you know, I have my computer, I have volume six, 1,000 pages. So I said, okay, well, why not just give this a shot? And so I just, you know, just type in on the, sort of the find feature, I type in, you know, the, you know, entran, entranter-ray-tro-loin, and I get one hit, and it comes up with that expression is actually used in chapter seven of Nalicitis where he uses sort of that same expression except sort of he says these conditions are quite easy to construct. That would take me too far away from my subject. So if he really meant the same thing here as there, then, and I think what he meant was that, well, whatever more thoughts he had on the subject was really a little distracting from the content of this paper. So anyway, this again, just incredible, amazing things that did, and fundamental seminal ideas, just independent of not including what he did about the Poincaré conjecture. So that sort of summarized his analyst C2's papers, but what he did in topology is just outside of that, just amazing. I mean, there's, he had a, this is just thinking just purely in terms of functions, but I mean, the modern way of constructing the universal covering space of a topological space in terms of equivalence classes of paths, well, that appeared in some paper in 1883, the Poincaré-Hoff index formula for calculating, showing the Euler characteristic of a surface could be relayed in terms of vector fields. Well, I mean, he did that for surfaces and Hoff got his name somehow because 30 years later, he proved the same theorem in higher dimensions. Well, he had this theorem at the degree one maps. Well, the idea of degree one didn't exist then, but he had a map of a sphere which had, which, well, somehow mapping, anyway, so he had proved something, a version like that in this, he, of course, he computed, understood isometry's hyperbolic space and had his models, which are just completely fundamental, of course, to modern day hyperbolic geometry. Now, I'd just like to mention just his dimension theory where in this philosophy paper, he introduces, he asks the question of, well, why is three-dimensional space three-dimensional? And from the point of view of sort of, well, a coordinates to us, well, it seems obvious it's three-dimensional. From the point of view of just as sort of, as a smooth manifold, while you do a diffeomorphism in modern day, we have its dimensions because of the tangent space and a smooth manifold's rank preserving. So, but on the other hand, should it really be three-dimensions because Cantor had observed that if you look at three-dimensional space, I mean, it has really, has the same number of points as the circle, which is one-dimensions. So what intrinsic about three-dimensional space makes it three-dimensional instead of one-dimensional? And, well, then he makes this incredible statement. He says, I'll base the determination of the number of dimensions with the notion of cut. Now, I'm sure this one sentence just changed the lives of just a bunch of people and, you know, in his day because this idea of, well, I mean, if you basically, you know, you're saying you have a wise three-dimensional space, three-dimensional because you take a point, a little neighborhood, well, it's the boundary of that neighborhood's a sphere. Well, that's two-dimensional. Why is that? Because if you have a point on the sphere, you take a little neighborhood of that, boundary of that's a circle. That's, well, that's a circle is one-dimensional. And why is that? Because if you have a point on the circle, a little neighborhood of that's two discrete points and discrete points are zero-dimensional. And he just says that sort of casually, but this sort of, you know, right away, Browler just tried to get sort of a more sort of mathematical definition. Well, that is mathematical actually what he said, but which is more ultimately formulated by maybe Manger in your zone, maybe 10 years after that. But this, you know, it's huge, well, subfield of mathematics based on this dimension theory. So let me just mention that it's just incredible what Punkeray did in topology. And yet, well, as I mentioned in the very beginning, I quoted from the introduction, that was about, took up, well, like a full page of Punkeray's study, well, view of his own view of his own scientific works, which was 100 pages, about 100 pages. And so of those 100 pages, about three and a half pages were devoted to topology, those three and a half pages felt the need to spend like a full page just explaining why he's, he'd wanna do this stuff. And so you might say, well, that doesn't seem like very much. And, but, you know, Punkeray actually, you know, offers a disclaimer for why he shouldn't be sort of overwhelmed by these numbers. And so there's an introduction, well, just a little two page O lecture paper written by Mittag Luffler, who sort of solicited this paper from Punkeray. He says, well, in the letter I wrote, which he agreed to my request, he wrote characteristically of his powerful capacity for work and activity. Punkeray wrote, I would point out that there is the disadvantage that too much time will last between when the proofs are returned and the publication is given. Making the record seem more complete than it actually is. So, right. So, the next thing is, so I wanna say a few words about Punkeray's sort of influence in topology. Well, of course the influence was just overwhelming, just in the Punkeray conjecture, the stuff that the mathematics that it led to in high dimensions, low dimensions, work in hierabolic geometry. But independent of that, there's just incredible amount of work that these papers sort of stimulated. And of course, there's the famous Punkeray question, Punkeray conjecture question, but here's sort of three questions that he asked from his analysis, C2's paper. It says, it would be interesting to treat the following questions. One, given a group defined by number of, well, basically it says given a finally presented group, does it correspond to a manifold of n dimensions? And if so, how would you create that manifold? And three is, if you have two manifolds of the same dimension, which have the same fundamental group, are they always homeomorphic? These questions will require difficult study and long development. I will not say more here. But so these were in his 1895 paper. Now, I should say that a lot of English version. Now, if you just take these questions literally, the answers are relatively simple. In fact, Cameron Gordon noted that Punkeray himself could have answered question three because these S2 cross tests, two and S4 are different manifolds of the same dimension four, but they both have trivial fundamental group. And there's also sort of relatively simple, I mean, clever, but simple examples, in particular for question one, given any group you could find and any dimension bigger than equal to four, you could find a closed oriented manifold of that dimension which has that group. So every group is realized by all manifolds of dimensions bigger than equal to four. But I'd like to take this point of view is that Punkeray, he did stuff and he sort of rethought it, rethought it, we find it understood it better. So I don't think that we should be frozen in 1895 with sort of the statement of these questions, rather that with the sort of the increased sophistication that these questions really should have evolved appropriately. And if you just modify them just a tiny bit to put it in sort of a more modern sort of understanding, then these questions are just, well, they're only at best sort of partially understood. But in particular, this question three, does the group determine the manifold? Well, if you restrict to manifolds of which have the property that all the higher homotopy groups are trivial, that's also known as a spherical, and you allow yourself sort of a weaker condition of what's known as homotopy equivalents, then Whitehead in the 40s noted that manifolds which are a spherical same fundamental group in fact are homotopy equivalent. And so these as spherical manifolds include sort of a wide class. But if you take the question three of, does the group determine the manifold and then restrict yourself to as spherical manifolds, well, this is what's now known as the Borel conjecture, that is closed as spherical manifolds with the same fundamental group or homeomorphic or sort of in different sort of just form, it's what's also known as topological rigidity of as spherical manifolds. So just said another way, I mean, what does the Poincare conjecture say? It says that if you have, say a three sphere and you have another manifold, which, well, is homotopic to that, so if you have a three sphere and another manifold homotopy equivalent to that, then the Poincare conjecture says that this manifold really is the three sphere. So you could just say the same exact thing if you have a manifold which is as spherical, another one the same homotopy type or has the same group, then the Borel conjecture is that in fact they're the same. And this has led to just so tremendous amount of mathematical activity. Chris Punker I knew this in dimension two. The famous Moss-Dahl rigidity theorem says that if both those manifolds were hyperbolic, then in fact you could homotopic to an isometry and that's a, well, theorem in all dimensions bigger than or equal to three. And in high dimensions there's sort of a sharper form of this Moss-Dahl rigidity due to Farrell and Jones and very recently Bartels and Luke sort of generalize the Moss-Dahl and then Farrell, Jones theorem to grow of hyperbolic groups and cat zero groups. So, but actually just as an aside, let me mention that you could sharpen this question a little bit further is that if you have two homotopy equivalent manifolds and you have a homeomorphism, how unique is that homeomorphism? And well certainly you need at least unique up to isotope and in low dimensions, at least dimension three it's true that two homotopic, homeomorphisms are isotopic, but so this is interesting in high dimensions, it's actually false that you could have you could have a homeomorphism of a hyperbolic manifold to itself but the isometry of homotopic to it could very well not be isotopic and that could happen in dimensions 10 and higher now. So that was, so question three is very open for many contexts and so okay, that's just a situation there and now here's a formulation of question one which our question one is given a group is it the group of a manifold? Well, if you restrict your attention to manifolds which again have this asphiricity property then again this is something that just stimulates this tremendous amount of work and well it's maybe it's known as the wall conjecture that basically these punk or a duality groups these if you have a group that when now you look at from the point of view of well a complex which has somehow given a group you could find some topological space which has that as the fundamental group and if that space has well certain properties then that group is called a punk or a duality group and those are the groups that are candidates for being groups of asphirical manifolds so this is I call this sort of the modern form of punk or a sort of first question and so again this is now known in dimension for punk or in dimension two and it's ready for dimension three it's still very much not known and there's been a huge amount of work in this progress in this direction but it's been a consequence of if you have a group which satisfies certain properties that you can do various things well basically it's all been reduced now to two interesting problems which are the weak hydrobalization conjecture which is whether these PD three groups are hyperbolic in the sense of Gromov and well, seemingly they have sort of some what's known as atroidal and then this other thing which is the cannon conjecture which is a very interesting conjecture which you have a group which in some sense again you look at its boundary like punk or a was doing in the fifth complement and if that boundary is a two sphere then this group is actually the group of a, well, well if it's torsion free that's the group of a hyperbolic three manifold so this is extremely interesting question which is basically saying that you have a group of such and such type can you actually sort of uniformize it and there's been various partial results and maybe just point out this result of Kleiner and Bonk which is that if you have a space well in certain circumstances the answer is yes if the certain space satisfies what's known as like a certain punk array inequality so right, so I just want to just briefly just to indicate just a direct consequence of punk rays initial paver and topology already there's these very interesting questions that are just very open and so anyway this, if you want to learn more about punk rays work on topology there's really a lot of really great resources I mean and I should thank, I just had a huge amount of help in preparing this lecture from Michelle Brawlow and John Stillwell and most of the translations were direct from Stillwell's papers on, well it's not his papers but it's, he translated the analyst Cetus into five supplements to English and gave some annotation and where I, and there's Sarkarya sort of did one of that I mentioned power graphs in the introduction and Google Translate helped out a little bit too. So anyway thank you for your attention. We have time for some questions now. Is the question here? No. Yeah, in your introduction you show briefly the notice of punk array on this motivation for analysis Cetus and in that there is, there was one big motivation which was to understand more the differential equation in the three body problem and also the disturbing function in the planetary problem and these were very, he was very much motivated for the expansion of this perturbing function which involved very complicated algebraic equation. So he had a very concrete problem to solve and my question, and a difficult problem. So my question is whether, how much discount in the success of his development? The fact that he has this very difficult problem to solve. That I have no idea. I mean clearly he was, I don't know. You certainly covered a lot of territory there. Last minute chance. Okay, well let's thank David. Goodbye again. Thank you.