 Thank you. Thank you very much. Wherever you are in Australia or in any other place in the world. I wanted to share with you a kind of a story of, this is a number theory seminar, but basically we'll start with a number theory, but we'll talk about some applications outside number theory to combinatorics and hopefully we'll have time also to some topology and geometry. So let me start with what was in the some sense the origin of this and this is the story of Ramanujan graph. So what are Ramanujan graphs? Let X be a connected k-regular finite graph. We're talking about finite graph, although maybe sometimes it's better not to think about it, not only about finite graph, but we care mainly about finite graph and A will be the JCC matrix. So this is just an N by N matrix when N is the number of vertices and A VU is the number of edges between U and V, which is of course usually either 0 or 1. If you want to allow multiple edges, it's also fine. It's not difficult to see that for such a matrix the eigenvalues are always, and this is a symmetric matrix, so all eigenvalues are real, but it's not difficult to see that they lie between k and minus k where k is the degree of the graph. k always appears in an eigenvalue and minus k appears if and only if the graph is bipartite, that's easy exercises. These two are considered as the trivial eigenvalues and X, the graph X is called Ramanujan if every non-trivial eigenvalue is less than 2 square root of k minus 1. Now where this number 2 square root k minus 1 come from? Where is the magic? Let me just give you a little hint that for all that if you can think of A and you really should think of A not just as a matrix, but this is kind of the AJCC operator. You take functions on the graph and A on the vertices of the graph and the action of A on such a function is by summing up around that, by summing up around a vertex and as such operator you can think of A is also an operator on infinite graph. In particular you can think of it as an operator on the k-regular tree which is the universal cover of this k-regular graph and then you can take it as an operator on L2 of t of tk and then a classical theorem of Kasten tells us that the spectrum of that operator is the full interval between minus 2 square root k minus 1 and plus 2 square root of k minus 1. So what we are really saying here that all the non-trivial eigenvalues are in some sense coming from above, from the infinite tree. Before the talk we had a little chat and we mentioned Plato, the Greek philosophy and maybe it's an opportunity to be mentioning again. If you know in this philosophy every object in the real world is a kind of an ideal object in the world of ideas and in a way if you want to give the essence of this definition we are saying that the ideal k-regular graph is the infinite tree but okay this is in the infinite world, not in our world. Basically we want something which is like the ideal object except on the two trivial eigenvalues which are kind of enforced on us because we live in the finite world. Anyway besides the philosophy that these eigenvalues, it's always if you take an eigenvalues of a finite k-regular graph the eigenvalues control the rate of convergence of a random work to uniform distribution and the fastest possible you can opt for is the Ramanujan. That's a well-known theorem of Alon Mbappana which said you cannot get a better bound than this 2 square root of k minus 1. So ideally you want the bound eigenvalues to that interval. I don't have time to talk about that but these such graphs, Ramanujan graphs are best possible expanders and expand are fundamental important in computer science. I will not touch that at all because we need not just a lecture, we need a full course for that. Okay but it's not at all clear that Ramanujan graph exists but here is an explicit construction and you will see somehow you will start to smell number theory. So let P and Q be two different primes and for simplicity of the exposition we will assume that P is congruent and both of them are congruent to 1 more 4. The classical Jacobi theorem tells us that the number of 4 tuples in Z to the 4, the sum of those squares is 1, is exactly 8 times sigma over d where d runs over all the divisor of n's provided 4 does not divide d. This is a beautiful classical 19th century theorem which in a way was the best day of the whole area of automorphic form. Now if you apply this theorem for a prime P then you get, because a prime P does not have too many divisors, only 1 and P then you get that R4 of P is 8 times P plus 1. So every prime number can be expressed in 8 times P plus 1 ways as a sum of 4 squares. Now if R of P was assumed to be congruent to 1 more 4, so if you think about it for a second you see that this must implies that one of the X i's is odd and the three other are even. So let's just kind of reorganize them and assume that will take S to be the 4 tuples such that the first X 0 is odd n positive and the X 1, X 2 and X 3 are even and they can be positive or negative, right? You can see that we can see signs and we are still in the set. So this kind of get rid of the number 8 and it's easy to see that this S is a set of size P plus 1. And I would suggest you to think of such an alpha in S as a quaternion as an integral quaternion, namely alpha is equal X 0 plus X 1 i plus X 2 j plus X 3 k. And then you can see that if alpha is in S then the quaternion conjugate of alpha is also in S and P is just the norm of alpha as a quaternion. Okay, this was for P but we had another Q. Now Q was also assumed to be 1 mod 4 and therefore it's a minus 1 is quadratic mod Q is known to be quadratic residue. So let's choose some epsilon in the finite field of order Q, so that epsilon square is equal minus 1. Okay, now for every alpha in S, S was the set of P plus 1 element. For every alpha in S we associate the following matrix X 0 plus epsilon X 1 X 2 plus epsilon X 3 minus X 2 plus epsilon X 3 X 0 minus epsilon X 1. I guess many of you recognize that what I'm doing here is splitting the quaternion algebra, the quaternion algebra, the standard Hamiltonian quaternion algebra over FQ is isomorphic to the two by two matrices. And this is an explicit matrix. It's no it's determinant is equal to the norm of alpha namely it's equal to P if it's equal to P and P is different than Q. Then this is an invertible matrix and I prefer to think of it not as a matrix in GL to but in P GL to namely divide model of the center. Okay. And now the claim is that this is an old theorem for I look at it and I see 35 years ago my god. I'm so old. I am that if we look at the group generated, look at the subgroup of P GL to FQ generated by this P plus one element. And let X PQ be the K a graph of this subgroup with respect to this P plus one generator. Now these are a it's not difficult to see that with every generator you get also it's inverse in fact the quaternion conjugate of alpha gives you the inverse that's a little easy computation. And this is really a K a graph on P plus one element, namely for every element in the in the group H, you connect it with with the element is the let's say gamma you multiply gamma by this alpha alpha tilde. This is a P plus one regular. Now, in fact, the theorem is slightly more precise. If P itself is quadratic residue module q, then the group H is PSL to FQ, which is an index to subgroup. And if P is not quadratic residue, then the group H is the full group. And in this case, you get a bipartite Ramanujan Ramanujan graph of these these details are not so important. Now, to see that. So what what what does it mean in this case that this is Ramanujan graph you remember Ramanujan graph means that all the eigen that the non trivial eigen that all eigen values are either equal to K K in this case is P plus one. They are less than to square root of K minus one. In this case, it means that the absolute value is less than two square root of P plus one minus one, namely two square root P. And I guess that in such a seminar I don't have to remind you that that you smell the Riemann hypothesis over finite field as and in fact, the proof of the theorem eventually boils down to use this Riemann hypothesis over finite field via the work of the lean as I'm going to explain a minute. It's very frustrating someone to talk and not see any of you and not see. I hope just say something to let me know that I can continue and this is okay. We see you and everything is okay. Okay, it's kind of an army people now I see there are other under people and I see none of you and I talk to myself. Okay. All right. So just disturb me from time to time just that I feel that I'm talking. Okay. Why why why Ramanujan is here I don't really have time to explain let me just give a given in what Ramanujan has to do with this and why why we named it Ramanujan or why we call it Ramanujan. Well, we all know this Ramanujan tau function, the famous function which is given here and Ramanujan original conjecture was that for every prime P, the coefficients, the coefficients of this is less than two times P to the power 11 over two. And now if you write Q, Q is not the prime from before I'm sorry. It's just variable Q. If you write Q as e to the two pi i z then this delta of z is the cast form of white at 12 on the half plane with respect to the full modular group. And in fact, this is an eke Eigen form for all the eke operators and the eigenvalue of the pth eke operator acting on this cast form is exactly tau p. And I wanted for a minute to put it, I'm not going to explain I mean I guess that in such a seminar many of you know that or even better than me but let me just put the connection because that will help us to generalize it to higher dimensional. For every f eke, and now the Peterson generalized the Ramanujan conjecture and say let's look at all the s, s, k and w all the cast forms on the this eke congruent subgroup with the character w. And his conjecture was that for them, the eigenvalues are bounded by two p to the k minus one over two. Okay. Let's just one question please from. Yeah, do you want to unmute and ask yourself. Well, I think the question was answered but the question was why are these graphs named after Ramanujan. Okay, so, so, so, so, so actually, most people think that we named them after Ramanujan because in the original conjecture we conjecture we use Ramanujan but that's not true. I will explain it in a second. Okay. The answer will come up right away. One more question because Ramanujan himself, as far as I know, did not have any interest in graph theory. But let me, let me, I think it will be explained soon. It will be understood soon why the property of the bond or the eigenvalue is the kind of in the spirit of the Ramanujan conjecture. In fact, I want to take you even farther away, and I want to take, I want to give you the modern reformulation of this Ramanujan Peterson conjecture in the language of representation theory. As you know, modern number theory was shifted in to large extent into representation theoretical formulation. Langlam programs, et cetera, et cetera. But I want to now to be specific about the Ramanujan Peterson conjecture. And the Ramanujan Peterson conjecture is equivalent, basically. Okay, I'm telling you a story so maybe I won't be all the time so careful about all the technical details otherwise I will not finish in 50 minutes or maybe even not in 50 hours. Let A be the ring of adels of the rationales, and let Pi be an irreducible caspital representation in L2 of, okay, we look at GL2 of the ring of adels, the ring of adels is a locally compact ring, GL2 is a locally compact group. GL2Q sits there as a discrete subgroup. Okay, I have to divide by the center, but as I said, I'm not careful here, too much about details. But anyway, it's like a lattice. It's a discrete subgroup of finite covolume. And assume that I take a representation which appears as a subrepresentation here, and every such representation can be decomposed into a tensor product of irreducible representations of the local factors, namely the GL2 of QP. Assume that the Archimedean factor is square integrable, then the conjecture is that the other local P component, the P component at the non-archimedean places are all temper. Temper means that there are weakly contains in the L2 of GL2QP. Let me, I mean, I guess that because of modern number theory deals a lot with representation theory that many of you understand that very well, but even if you're interested in number theory is in a different direction, don't worry about it. Just let me, what I want to say that we have here a reformulation of this number theoretical conjecture in a language of representation theory, and this reformulation is going back to Saqqe already like 50 years ago or something. And the point is that this conjecture was proven, this is a theorem of delin, who solved the Ramanujan conjecture. What this has to do with our graph? So that's what I want to explain. Why this, what, where this graph has to do with this representation theory or with this number theory? That's I want to reveal now. Let F be now the local non-archimedean field, the periodic numbers, or if you prefer, take a local non-archimedean field in characteristic P. Fp double brackets T. And let R be the ring of integers in that field, which is the periodic integers in the first case, or the ring of power series in T inside the Lorentz series, in T inside the Lorentz series, the ring of integers. Look at the group PGL2F. K, PGL2O is a maximal compact subgroup of G. So if we do what classically over the real number, we take G mod K to be the associated symmetric space. In this case, when we do it over an non-archimedean field, we get the so-called Brouh-Titz building for the case PGL2. G mod K is nothing more than P plus 1 regulatory. This is the Brouh-Titz 3 associated with the periodically group G, PGL2F. Now, let gamma be a discrete co-compact subgroup of G, the so-called lattice. And let's look at the double-coset, namely we divide G from the right by K, and then we come and divide it from the left by gamma. So if we divide it by K, we get the tree T. Now we divide the tree by gamma, we get a graph, right? If you divide a tree by action of a group, you get the P plus 1 regular graph. But this is a finite graph because G mod gamma is compact, and this is a discrete object because this is a tree. So compact and finite means finite. So this is just a finite P plus 1 regular graph. Now, we are not yet there. These are not yet our illusion graph, but we are getting closer because every finite P plus 1 regular graph can be obtained like that. In fact, I suggested some of my colleagues who teach combinatorics to start their course and define P plus 1 regular graph in this way, because every P plus 1 regular graph is obtained like that. It can be expanded, Ramanujan or not. What is a bat? Even before we go, now I want to say something, theorem. Look at such G mod K mod gamma, which is really T, the tree mod gamma. This is a Ramanujan graph, if and only if, every infinite dimensional irreducible spherical sub-representation of L2 of G mod gamma, you look at L2 of G mod gamma as a georepresentation, and then you can ask whether you can decompose L2 of G mod gamma as a direct sum of irreducible representation. And we want that every irreducible spherical sub-representation is tampered. Okay, again, if you know what it means, fine, but even if you don't know, again, what you see here, that these are down to earth, the definition of a graph being Ramanujan is equivalent to a representation theoretical statement on some infinite dimensional representation of G, you know, on a specific unit representation on the infinite dimensional in-built space. Okay, spherical means that the maximal compact sub-representation as a fixed point and tampered means that matrix coefficients are in L2 plus epsilon, or if you want, as we said before, weakly contained in L2 of G. So what is important for us is that the combina, I am repeating this, that the combinatorial property for the finite graph of being Ramanujan is equivalent to a representation theoretical statement. And the latter, now, you see, we met this world, you see, we met this world, we wanted that some representation will be tampered, right? So let me tell you what the LIN theorem really gives you. It's not equivalent to the LIN, but it follows from the LIN. So just a moment, let me finish here, that the combinatorial property of being Ramanujan is equivalent to representation theoretical statement. And this representation theoretical statement is really a number theoretical in some sense by Satake, which is related to the Ramanujan Peterson conjecture. By the way, this last statement, this theorem about being Ramanujan and the representation being tampered has nothing to do with number theory. So you can formulate this also for the group G of the full automorphism group of the K regulatory and the theorem is still valid exactly as it is stated here. So that's not a number theoretical theorem. The number theoretical theorem is to find the situation in which this assumption is valid, that representations are tampered. And that's exactly what the LIN theorem, so I don't have time to go on the old details, but the theorem of the LIN, when he solved the Ramanujan conjecture, gave along the way that if gamma is an arithmetic lattice in PGL2QP and gamma IR is a congruent subgroup, be careful. There are also non-congruent subgroups in gamma, such gamma is the virtually free group, it has a lot of subgroups of finite index, most of them are not congruent subgroups. But if it is a congruent subgroup, then every reducible infinite dimension and spherical representation of L2 of G mod gamma i is tampered exactly the condition that we need. And therefore we can deduce that these specific graphs are Ramanujan. And so the G mod gamma i in the abstract way are Ramanujan. And the explicit construction that I gave you at the very beginning of the talk with choosing P, choosing Q, taking the quaternion, taking epsilon to be square root of minus one is simply writing down an explicit presentation of one, or well it's infinitely many, but of these examples and you can get many more by the same from the same theorem of the same technique. And later on this was generalized by a student of mine, Morgenstern, who used Drindfeld solution to the Ramanujan conjecture to get for other case, etc, etc, this has many applications and combinatorics but as this is a 50 minutes talk, I will rush. Alex. Yes. Excuse me just one question from Patrick. If you want to ask your question, then maybe I should ask it, can we use non congruent subgroups to make Ramanujan graphs. Probably some non congruent subgroups will also give you Ramanujan graph, but definitely not all of them. It's not difficult to show, as I said, that if you take arbitrary finite index subgroup, you can get an arbitrary P plus one regular graph. And most of them in some sense that can be made precise are not Ramanujan. There is an open problem about random ones, whether it's Ramanujan or not, but I am sure that some non congruent subgroup will give you Ramanujan, but we don't have any technique how to do it. So some of we are really using very heavy mathematics to construct such a Ramanujan graph, and probably there should be other methods. Recently Spilman and some other people have some non explicit methods or some explicit methods to do more Ramanujan graphs, but that's a different story. What I want to go ahead and to introduce you to this old mathematics already 30 years ago to a more recent one, which is about Ramanujan complexes. But now after this introduction, you can guess somehow what I'm going to do. So we are going to replace PGL2 by PGLD, let D be now any positive integral. And then instead of PGL2, we look at PGLDF. So this will be the periodic league group. And we'll take the maximal complex subgroup PGLDO. So this is the maximal complex subgroup. The quotient gmod k is exactly the so-called Brouh-Athit's building. What is the Brouh-Athit's building? This is a combinatorial object which can be identified with gmod k. It's a D minus one dimensional, contractible, infinite, simple complex. The vertices of the buildings. Now I have to define a co-operator. Now this is a little bit technical. I will rush quickly over that if you got it fine, if you don't get it also fine. It's not so crucial, but I will show you what is the kind of a generalization of the object that we had for PGL2. The vertices of the buildings comes naturally with colors. The colors are D colors. The nicest ways to think about them is element of the cyclic group of order D. That's the way you define them. And now we define colored adjacency operator. We are going to talk about the adjacency operator on functions. We had before a adjacency operator on the infinite tree. And we got, if you remember, the Ramanujan graph as a quotient of this infinite contractible graph. Contractible graph means a tree. Now we start with the contractible, simple complex. We look at the L2 functions on the vertices of this building. And we are going to define D minus 1 a co-operator, ai of f on a vertex x. You sum up over all the neighbors f y was such that the color of y minus the color of x is equal i. If you think about it, the adjacency operator is the sum of this D minus 1 operator. Now for the case D equal 2, we have only one operator. That's the previous game. But now we have to look at all these D minus 1 together. Now we have a bit of a problem here that the ai are normal operator but not self-adjoint. So the eigenvalues are not real. But they are normal and ai star is the equal to ad minus y. And they all commute with each other so we can diagonalize them simultaneously. And now let me tell you what we do. I mean, it looks maybe a bit strange if you never came across things like that. But I hope that after my preparation you can see what I'm doing. Now I will look at the spectrum of this operator on the infinite building, on the L2 of the infinite building. You remember the story about the Greek philosopher Plato. That's the ideal object. The spectrum of all of them, the simultaneous spectrum of them is going to be a subset of the complex of a vector space of dimension D minus 1 over the complex number. So what replaces a single eigenvalue is now a vector of eigenvalues. And I'm going to define, so now you remember what was the definition of a Ramanujan graph. The definition of Ramanujan graph was that every eigenvalue of the finite quotient is really an eigenvalue. Every non-trivial eigenvalue of the finite quotient is an eigenvalue coming from above, coming from the universal cover. So this will be my definition now. A finite quotient building mod gamma, gamma, a co-compact discrete subgroup of G of PGLDF will be called Ramanujan complex. This is a finite simplicial complex. If every non-trivial simultaneous eigenvalue lambda 1 up to lambda d minus 1 of a 1 up to a d minus 1 acting on this is inside the spectrum, sigma d, the spectrum of the ideal object, the spectrum of what it's building. So this generalized the definition of Ramanujan graph to what we call now Ramanujan complex. Okay, there is a settled issue. What do we mean by non-trivial eigenvalues here? So you have to really understand it and it took us a while to do that. In the case of d equal to, there are two trivial eigenvalues. You remember k and minus k. Here, there are d trivial eigenvalues that we should eliminate from the picture. Anyway, a nice theorem of Winnelly says, which is the analog of Alonbo Pan, I says that if you have an infinite family of such quotient, you cannot do better than this definition. Namely, take the spectrum of these operators on the infinite family, you get a subset of the d minus 1 space over the complex number, take the unit of them, take the closure. Then this is at least as large as this set sigma d, which means that you can never do better than Ramanujan. So again, it's a good sign that Ramanujan are optimal. Okay, now do they exist? Okay, so first of all, indeed we proved. Now what did we prove? We proved this is a joint work also a long ago with Uzi Bishna and Bet Samo. Bet Samo at that time was a student of PN. Unfortunately, she really passed away like two years after the cancer. This was a big tragedy. So we proved that the building mod gamma is a Ramanujan complex. Again, you see it's a purely group. It's a purely down to earth statement about a finite object, combinatorial object, if and only if every infinite dimension and irreducible spherical sub-representation of L2 or PGL2 of mod gamma is tampered exactly the same statement as before. Well, of course, we proved this theorem. We made the definition of Ramanujan complex to make sure that this theorem will be true, right? We look for is the combinatorial meaning of such a theorem. Why did we do it? Because we did it after the work of LaForgue, of Lorent LaForgue, for which he got the fields where Dal, you know, he proved this Ramanujan, well, he proved the Ramanujan conjecture for PGLD over a positive characteristic field. So we really, we were after understanding the combinatorial meaning of his work because he generalized Trinfeld from GL2 to GLD. So we wanted to understand what is the combinatorial meaning of the work of LaForgue. And that we were able to say, okay, now only positive characteristic if gamma is an arithmetic subgroup of PGLDF and gamma are a congruent subgroup. And then under some technical restriction that I want, every blah blah infinite dimension, irreducible sub-representation is tampered. And therefore this enable us to deduce that these objects are Ramanujan complexes. And we went ahead and we did, and we constructed Ramanujan complexes, finite complexes, which are Ramanujan. Some work is done along these lines, more or less, parallely by Winnie Lee and by Silver. But to be honest, when we did this work, we had no idea why we are doing it. In some sense, there was no real goal, there was no combinatorial application that we had in mind. When 35 years ago, when in the work I did with Sonic and Philips, people were looking for better and better expander. So we knew at that time what we were after. Here, it was just kind of a curiosity, what is the combinatorial meaning of the work of Lafour. There was a feeling that something should come up of it. But we didn't have, and indeed nobody paid attention to our work, except of the government of India, which called a place, our paper called Ramanujan complex, and they gave a name to a place in India, Ramanujan complex, after recognizing the importance of our paper, what is even more impressive that they did it 15 years before we wrote our paper. So they kind of knew that there would be such an important paper about Ramanujan complex. One question from Joel Rosenberg. Joel, would you like to unmute? What is the content? Yes, please. What is the content of the definition of Ramanujan complex? Are you saying that in general there are not eigenvectors on the quotient that lift, otherwise it seems like... No, no, the other way around. I mean, all eigenvalues on the quotients are somehow, except on the trivial one, those which appears also on the infinite building. What the meaning of that? I don't know. But you'll see in a minute that there is some highly non-trivial applications. That's the rest of my, the pure last minute of my talk. But I admit that when we did it, we didn't know really what we were after. We wanted to understand the... We had the feeling, you know, I didn't give you the full story, but Ramanujan grass was such a successful story in the center. There are so many applications in combinatorics of the Ramanujan grass. There was a feeling that maybe the Ramanujan complexes, some higher dimensional version will be also of some importance to combinatorics. But I have to be honest with you that the work started because Beds Samuels, I have a part-time position at Yale. I go there every year for a few weeks. And she came to me and she was a little bit stuck in her work. She learned Potomofic Rome. And she asked me, maybe you can suggest me something. And I told her, you know, Lafort, prove this. Why don't you check the combinatorial meaning of his work? It turns out to be much, much, much more difficult than I assume. And so eventually I joined in and we worked very hard on that and we got these complexes. But as I said, when we wrote the paper, we have a beautiful object, which I have no application. So let me, I have only five minutes. So let me go very quickly on some dramatic change in the story. And this came up now, even if you missed me at the rest of the talk, I'm starting a five minutes completely different talk. The talk also, the story of this talk also starts like 35 years ago with a beautiful harem of Bosch and Furedi, who were undergraduate students at that time, and answered a problem of Erdos. They proved that given a subset P of the plane of size N, there exists a point in the plane which is covered by two ninths of the N choose three triangles determined by P, right? For every three points, you can, you can draw a triangle which is determined by that point. And then you have a point which is, which is covered by two ninths of them. But this two ninth is optimal, et cetera. Shortly after that, Barani proved an higher dimensional version of this theorem. He proved that for every D, there exists some constant, depending only on the, only on the dimension, such that for every subset P of R to the D with N points, there exists a point Z which is covered by at least CD, by at least these constant times N choose D plus one. You have N choose D plus one, these simplices determined by this set P, and they have a non-trivial, some positive proportion of them as a non-trivial intersection. It's still a big open problem in this convexity theory to know what is the CD and what even the rate of decay, et cetera, et cetera, et cetera. Gromov completely changed the point of view on this theorem, but let me bring his definition first, and then we'll see his theorem, and then I will end up by the connection with these romanism complexes. Gromov defined a simplicity complex dimension D to be epsilon geometric, respectively epsilon topological, to have the property, sorry, it has epsilon geometric on topological overlapping property, if for every function from the vertices of X, I'm sorry, I had to say, yeah, I don't know, I said, this is dimension D. We take a similar common dimension D. We map the vertices to R D, and then once you map the vertices to R D, you can make an affine extension of this map from the simplicity complex to R to the D. And then you want to say there exists a point Z in R to the D, which is covered by epsilon fractions of the images of the D cells. Okay, let me jump right here. What's the meaning now of the two theorems which I quoted before. They say that if you take the complete D dimensional simplification complexes on N vertices, then they are geometric expander, which means they have this property for some constant. That's exactly what they prove. But it gave a much stronger definition of being topological overlapping property, or what we call it now topological expander, which means that for every map from the vertices to R to the D, and for every continuous extension, the affine extension is essentially unique. The continuous extensions that are plenty, you can do it in a million different ways to make the extension. And he wants to go more, that still there will be a point covered by all of them. And he proved the following remarkable theorem, really, if you look at that, if you think, I don't have time to explain, but this is a remarkable theorem of Gromov, that the complete K regular graph, the complete, sorry, did they measure a simplification complexes, are topological expander. Try even to draw to yourself a picture of the K is D equal to, to see that it's really counterintuitive. I tell you the truth when I look first time at the paper of Gromov, I thought that it's, it's cannot be true. And he probably does not really express, does not really say what he really means, but he tells not to be correct. It's amazing. And he made the fundamental problem. I didn't talk about the analogy with the expander's, but somehow in the expander theory, to prove that the complete graph is expander is triviality. The whole point is to prove that there exists bounded degree expander. And he asked, are there simplification complexes of bounded degree which are geometric or topological expander. And shortly after that, few years after that, suddenly in a joint paper with Jacob Fox and LaForge, this is not the same LaForge, this is, which was mentioned before, this is his brother, Vincent LaForge, Asaf Naor and Janusz Pach, they prove that the Ramanujan complexes of dimension D that we talked about before, at least when the prime is large enough, they are geometric expander. So suddenly they brought to life our Ramanujan complexes, proving that they have the weaker property of being geometric expander, namely that they have the property of the overlapping property with respect to affine extension. But they left open the more difficult question. In fact, at that time, Gromov told me that he's not sure what's the right answer. Are there topological expander or bounded degree of not? And more recently, this was resolved in a very strange way. The first work was by Tali Kaufman, David Kasdanen and me. We resolved it in dimension two. And then my student, Shayevra with Tali Kaufman, showed it for everybody in a strange way. You want to guess that the Ramanujan complexes are also topological expander. They satisfy also that the topological, the overlapping property. We don't know that. That's an open, that's still an open problem. What we prove, and this is because of the structure of the proof, which uses a comological methods over the phylocardistic two, which I don't have time to explain. We prove the following, the following strange, strange results. Again, we have to take a Q, I mean P, very, very large. Then the D scale term of the D plus one, dimension Ramanujan complexes. You see, the Ramanujan complex of dimension D plus one is a D plus one, similar complex. So ignore the top phases. These are still bounded degree complexes. When we fix the prime P, we fix the prime P. We have to be very loud, but we fix it. We get infinitely many Ramanujan complexes of dimension D after we throw out the top cells. And they are the dimensional topological expander. So this really gave a highly non-trivial application of these Ramanujan complexes. The last statement is that these Ramanujan complexes were suddenly also discovered by the computer scientists in the last two, three years, and there is a lot of work and application of them back to also to computer science. This was my org 15 years ago. It took time, but now it seems like it's really going ahead even much faster than I have seen. Okay, my time is over. I'm sorry, I was a little bit over time. Thank you very much.