 Good morning, good afternoon, good evening, good night, wherever you are. I'm Steve Cantrell. I'm the director of the Institute of the Mathematical Sciences here of the Americans here at University of Miami. And so it's my pleasure to welcome you to the second of our first Distinguished Colloquium series for our consortium. And it's my pleasure to introduce Dr. Mina Tysher, a member of our Scientific Advisory Committee, who will be chairing this morning's session. Mina. Good morning. It's a great pleasure to introduce Professor Don Zagir. I mean, not only because of being one of the greatest number theorists of the 20th century and the 21st century, but due also to his influence on many other fields of mathematics. I mean, he is really unique in the way that he sees how number theory and his knowledge is connected to any other field of mathematics, that he can do mathematics in the middle of the night or in a boat or on a mountain. And he's concerted on mathematics everywhere. And thirdly, that he knows how to connect and talk to people. So at the end, we have a great mathematician with a great influence. And I will not describe everything that he contributed during his career. Phil Griffith, that introduces him for the first talk, did it beautifully. I just wanted to describe these three very big qualities on the image of a number, on concentration and on collaboration, that together with that, this gave this amazing influence. So Don, the floor is yours. Thank you very much, Mina. I don't remember when I've had such an exaggerated amount of praise. The only thing that is definitely true is that I sometimes do mathematics in the middle of the night. But I'm not so sure about all the rest, but it would be nice to think so. So I'm very happy to be here again. I'm going to start by reviewing a little bit, so maybe the first quarter of today's lecture in the somewhat different terms what I did in the first lecture. First of all, I've been told by a few people, actually by hundreds of people that I talk too fast. And so you probably missed a lot even if you were there and many people couldn't make the first lecture, didn't know about it. So I'm going to summarize a little bit the main directions, but presenting it differently from the way I did before. So as you know, the pair of lectures is called from knots to number theory. I'm more a number theorist, and today I want to start the other way with number theory. So there were two, I mentioned many things last time that are number theory like say to functions, but there were two that are especially important to me, and that are not so familiar to a wide group of people, even number theorists, not always and certainly people outside. So on the number theory side, I want to emphasize two main things, the block group, which I explained last time, I'll repeat again, but more briefly. So B is for block, F is for field. So this for us, it'll be a number field, it could in principle be any field. And the other is the Habira ring, which is a very, very beautiful object which somehow only to publishers usually know a few number theorists. And part of what came out of this work, in particular in joint work with Stavros and Peter Schultz, is that this Habira ring is actually just H sub q, and there are much more complicated Habira rings for every number field, which are no longer just one object, but they're graded. So there's an index, and the index is an element of this block group, which for q is trivial. So it has a new structure, but I won't be able to go into that. So here I'll say Sg, and here I'll say Sg and two, but I won't talk about those things today, but I certainly wanted to mention Peter Schultz, and another part that I probably won't get to is with Rina Khashayev. And I remind you from last time, the main collaborator in everything I'm telling is joint work with Stavros Gadrofalitis, who's now in Sus-Tex, Southern University of Science, technology in Shenzhen. And the paper, in some sense, we finished it today. Many people ask me when it's finally going to be ready, we'll only put it on the upload to the archive that may be a week, but we'll send out copies today to people in the field, including many who are at this lecture, and wait for a week to catch typos and make a few last minute improvements, but at least the... So what I'm talking about, there will be a written source soon, and also I mentioned last time I gave a course three months ago, and that's online on the Max Planck Institute, both the recorded lectures and the fair number of notes. So if you're interested in some part, you can look. So let me start with the number theory. So the block group is a certain set divided by, well, a group divided by another group. So A of F, I explained last time, these are formal integer combinations of elements of the field. So let's say some N, I, Z, I, it's finite, I goes from somewhere to somewhere, these are integers, these are elements of the field. So that would be, if I just said that, that would be Z of F, but I don't want them all, I want with the special property. And so I'll say first the way I did it last time, such that the sum over I of N, I times the determinant of the two by two matrix. If you take any two homomorphisms from the multiplicative group of the field to Z, so for instance, if it were Q, that might be the valuation at some prime, then you can make this two by two determinant. And if you add them all up with these weights, you should get zero. So there's an actually somewhat fancy way to say that, which is actually, once you get used to it, much easier, it means you take Z of F and you have a map to this exterior wedge of F cross and the map sends any generator Z to Z wedge one minus Z. And this is very confusing. If you don't know it, you can forget it, but it just means the wedge is a formal symbol, which satisfies that it's linear. So if you take X wedge Y one, Y two, it's linear in the sense of the multiplicative group because it's S cross. This would be the sum. And secondly, X wedge Y plus Y wedge X is zero. So it's anti-symmetric. So that's exactly the equivalent of that. If you think about it a bit, and C of F is the span of the so-called five-term relation. And that's a very, very beautiful thing, which goes back actually to the beginning of the 19th century. So it's now 200 years old. So let me tell you what the five-term relation is. So here's the theorem I like to call the easiest theorem of mathematics, meaning that there are even easier statements that are completely trivial, but they're not interesting. I wouldn't call them theorems. They're just an identity or an exercise, but this is really the deep theorem, the sense that it's trivial to prove, but has many, many ramifications in a huge number of fields of mathematics that I've won't list. And the theorem is this, if you have a collection of numbers in some field, so Z, ZN, let's say different from zero, for N is an integer. And if they satisfy the following recursion, that each ZN, if you take its one complement, one minus ZN, this will be the product of the two neighbors. So it's a doubly infinite sequence. And from this, you can work out the sequence in both directions. And the theorem is that then it automatically has period five. So then you get a collection of five numbers, Z1, well, any five successive ones, but for instance, Z1 plus Z5, that could be this element. I usually use XI generically for this, here it is, XI is such an element that satisfies this condition. And now you see that if I compute this D of XI, which sends each Z to Z wedge one minus Z, let's do it. Then you see that D of XI is the sum of N model of five because it's periodic by the easiest theorem. And then I'll have ZN wedge one minus ZN, but ZN is ZN minus one times ZN plus one because that's the defining relation. But because the wedge is additive, this is the sum ZN wedge ZN minus one, N mod five. And the other term simply is the sum ZN wedge ZN plus one. But now you see that since this is periodic, it doesn't matter if it's N and N plus one or N minus one and N, I can shift the index by one. And then I would have ZN minus one wedge ZN, which is the opposite of this and my symbol is anti-symmetric and so this is zero. So that's the very short proof that these special relations are in this kernel of D, so they're in A. And if you take arbitrary linear combination of them, we throw that away. It's exactly like homology. A homology group is cycles, multiple boundaries or homology, co-cycles, multiple boundaries. So this is the block group. It looks very abstract and it is a bit abstract, but it's absolutely comprehensible and calculatable in any given case. And I'll talk about that and you can write down elements easily. And the other thing is that you have the so-called regulator map if the field is C, but we'll be looking at number fields, but every number field can be embedded into C in as many ways as the degree. So the B of F will map to B of C and then I can take this regulator, it's usually called D, it's mapped to the real numbers. And it simply sends the sum of Zi satisfying this famous condition. I can put the answer, I'll just drop them. It sends them to the sum of D of Zi and D of Z. I wrote out the form last time. Maybe I won't bother. It's the Bloch-Wigner dialog rhythm. It's a completely explicit function programmed on many computers. It's like art tangent. It's just a fixed function called the Bloch-Wigner dialog rhythm. Very simple to compute. And then the point of this function, D of Z is that it was discovered in the early 19th century about by five different people independently, Spence and Abl and Lovachevsky and several other people later, they discovered that if you have five numbers related in this way, then the sum of the five values of Z of D is always zero. And that means that D vanishes on the group I'm dividing by for the complex numbers. And that means that this becomes a well-defined function on the Bloch group. Okay, so so much for the Bloch group. Now the Habiro ring. The Habiro ring H has as elements, things that I write as A of Q, where Q sometimes will be written as E to the two pi's times something else. Sometimes it won't. But these are not functions as I emphasized last time. It's not function on the complex numbers. A prior, if you put for Q some complex number like two, it won't make any sense. But what it is, is let me see how I wrote it. So the Habiro ring formally is this limit. Now I'm gonna use slightly fancier notation than last time. It's the inverse limit of the ring of the wrong polynomials in Q with coefficients in C, multilow QN. And I remind you that X semicolon QN, this is the Pochamer symbol, is one minus X times one minus QX all the way up to one minus Q to the N minus one X. And if I don't put two arguments, QN always means you just repeat the argument Q, QN. Okay, so it's one minus Q times one minus Q squared up to one minus Q to the N. So this again looks abstract, but what it means is for instance, if A is in H and if Zeta is a root of unity, then even though I said you can't evaluate A of points, you can evaluate it at that point and you'll get an actual number in Z of Zeta. And I explained that last time, this is a limit of polynomials, but after a while those polynomials differ by multiples of the Nth Pochamer symbol. But if Zeta is a fifth root of unity, let's say, then this Pochamer symbol one minus Q, one minus Q squared up to one minus Q to the N will vanish for all N equal or bigger, bigger than or equal to five. And so the thing will simply converge. So we have something given by sequence of polynomials, but that sequence will stabilize, it will just be constant from some point on for any root of unity. And therefore we get a map from H into the map that usually we write mu infinity for the set of roots of unity into actually Q bar because this is an algebraic number and that map is injective, that's not at all obvious. So even though I'm only telling you the value of A at all roots of unity, think of this on unit circle, you know, the rational points, but in fact, you can get the whole function back. But if you have A and H, then you also can take A infinitesimally, I always use E to the minus H near the origin. And then for the same reason, this will make sense as a power series in H with rational coefficients because you approximate this by sequence of polynomials. And after a while, two polynomials differ by this Pochamer symbol where a multiple of this where N is very big. But each of these terms is divisible by H because if Q is one, there's zero. And so that term is divisible by H to the N and therefore to any given order like H to the 10th, the series will simply stabilize. So each coefficient of these, this expansion will stabilize to a number. And in fact, you can do it even better if I don't use E to the minus H, but if I use one plus H, where H is still in, so of course it's still in Q of epsilon, that would be the same. But actually now it's even a power series with integer coefficients. And the reason is clear because the original polynomial that integer coefficients, so each polynomial does and the value stabilize, multiple any power of epsilon. So that means that we also have a map into Z of epsilon. So power series with integer coefficients. And this map is also injective. So you can recognize a Habira element either by its power series expansion around one point, one, or its values without the power series at all points. But actually you can combine it if Z is any root of unity, then you have a value and this will now be in Z of zeta, double bracket eight. So you have a bunch of power series, but already these special cases are enough. And maybe I won't say why it's enough unless there's a question at the end if somebody wants to ask in one line why you have this injectivity, how you go between these things, that's a very beautiful idea uses periodic congruences of these things. So just congruences mod p, just p is enough. I was shown by Ortsky quite a few years ago. And so that tells you that from the nature of these coefficient of p and the nature of this at roots of unity, p through its unity, you can go back and forth. Okay, so that's now the number theory. And now I assume that I've a knot and I want to remind you last time a bit. So now I've proposed it, so I've either three manifolds or in our case, it will mostly be, well, the first thing I say is for all three manifolds the decks will be knot complements, knots k, so m would be the complement. And so now I claim that for any m, it doesn't have to be a knot complement. I will get an element. Well, first of all, I'll get a field and then a number field. Actually, it's a product of several number fields I don't want, I've mentioned this last time a little I'm not going to repeat that. And an element, so this is FM if you want and an element, which also depends on m in the block group of this field. So that's one of the things that automatically is just given to you when you have an ultra three manifold you get some number field or sometimes a product of fields and elements of their block groups. And how does this work? So the basic fact is this, if you have let delta be a tetrahedron. So but it's a hyperbolic tetrahedron. So that means in one of the standard models you can think of the vertices as you know somewhere in the upper half plane and it looks, it doesn't look very Euclidean. But actually it's an ideal hyperbolic tetrahedron which means all of the vertices are at infinity. And infinity, the boundary of hyperbolic three space is the Riemann sphere. It's the complex line at the point of infinity. And then you can always essentially uniquely once you've numbered the vertices you can put one vertex at zero, one vertex at one, one at z, which is some complex number. Sorry, I don't want those sides and one at infinity. So I can always make the vertices. If I didn't do that, I would have four vertices P1, P2, P3, P4, they would have cross-race and that cross-race is this number z. So the whole thing only depends on z and the picture in hyperbolic space is the vertices are zero and one, zero and z, the geodesics between points on the boundary are vertical semicircles. The geodesics joining these points to infinity go like that and then there's one behind that you can't see. So you have such a tetrahedron, ideal tetrahedron parameterized by single complex numbers z, which to be a little more precise is an element of z different from zero and one. So delta of z. So now if you think of a usual Riemann surface you can triangulate it into triangles and we use that a lot to find homology. So in the same way M3, you can triangulate into a practically disjoint union of tetrahedron. They meet along their edges. You glue tetrahedron along a common face, a common triangle, but otherwise they're disjoint. And then each one, because of what I just told you each one has an associated number which is its cross-racial. And so you get a bunch of numbers. And because of most of rigidity these numbers are all algebraic and so they all lie in a certain number field and that's the definition essentially of F. It's the field that you get by adjoining to Q all of these so-called shape parameters. So these are the shape parameters because they describe the shape of the triangles infinity. So therefore what I didn't tell you here but we'll know is that because hyperbolic space is a metric with constant negative curvature constant curvature minus one but this thing is a volume and it's finite because even though it goes to infinity integral of the thing converges and it's exactly given that's the reason for the importance of that function it's exactly given by the block veganar dialog rhythm function. So therefore the volume of this thing is exactly the sum of the DMZI which is by the formula I already told you exactly D of psi M, psi M. Where psi M now I told you that there will be an element of the block group this psi will simply be the sum Zi but of course in order to be in the block group it has to satisfy this mysterious condition some Zi wedge one minus Zi is zero. And so the topology has to be such that the gluing conditions that these different things glue together to form a closed manifold forces the Zi's not to be independent that's why the way also the reason that they're automatically algebraic because you have so many equations you have roughly n equations and unknowns and it forces it to be rigid they're exactly n solutions although that's not a complete proof. So why is that true? So last time when I talked about volumes I mentioned that Thurston I think around 1983 had discovered that the volumes of all three manifolds are well ordered. So there's the smallest volume second smallest to third and then they have the first limit point and then there's the next smallest and the next limit point and then a third limit point and first limit point of limit points and so on. And in the paper of Walter Neumann and myself in 1985 we studied we wanted to know how quickly these volumes tend to the limit and it's nice result involves the dialogue. I don't want to say that but to do that almost the entire paper was devoted to one problem and that was to describe the combinatorics of these numbers Zi once you have that under control everything works. And the answer is of course it's topology because we're talking about a three manifold who it didn't have to be not come to me yet for that we're talking about a three manifold but in the end it leads to some very surprising number theory including a simpler definition of the block group. So what we found there now is kind of university called the Norman Seguir equations we find that there are two matrices N by N matrices over Z they satisfy condition that I'll say in one minute and then what you have is that each of these say parameters you have that the product of Zi to the A sorry, Zj to the Aij it's matrix multiplication, j from one to N is equal but some of this is assigned it's a terrible nuisance but that's how it is. Here you have one minus Zj and since these are N by N matrices this thing goes from one to N. So you'll see that this is a polynomial equation in N unknowns but there are N of them so N equations and N knowns typically a zero dimension to get algebraic numbers and that's what we're getting but these are the equations but not every A and B will do it but the condition turns out to be these are N by N matrices there exists two more N by N matrices A, B, C, D such that this whole thing is an element of the symplectic group of two N by two N matrices with coefficients in Z I'm not going to define this that takes one minute but it's if you know it, you know it and if not it's completely pointless it's just a simple definition but the point is that these equations tell you that and now when you look at the sum ZI which was my age and you apply D of it so that's the sum ZI wedge one minus ZI and now you use these relations they give you stacks of relations between the Zs and the one minus Zs and you combine with the symplectic property then you find that it's exactly set up so that this is zero and in fact we've developed this now a bit more and it turns out that there exists a completely nice description for any number field now nothing to with the apology there's a description of B of F of the block group of the field in terms of these matrices and we only need A and B because only N, B play a role but the condition on A and B is that they're half of a symplectic matrix so we call it the half symplectic description and the solution Z, Z1 up to ZN of these equations so if you give yourself the top half of the symmetric symplectic matrix over Z and you give you then you can write down these equations and you can solve them and if you take a solution which will be some algebraic numbers then there's some ZI will be in the block group you get all elements of the block group and what's very pretty when you do this remember to understand the block group it's not enough to know that you're in the kernel of X, Y, H1 minus X that was the consequence of these and Z equations you also have to know that the five term relation won't hurt you so let's look at the five term relation and it's something very familiar in triangulation in the apology if this for a surface you could do the following trick you could take two adjacent triangles remove the common edge and replace them by these two but then you'd sort of be at the end of your rope because if you do it again you go back but when you do it for hyperbolic with the three dimensional things then you can take hyperbolic tetrahedron that's called the vertices ABC and X but then because it's they fit together into a closed manifold then this phase cannot be there because there's no boundary so after another tetrahedron that shares that same phase Y, I drew this poorly it's gonna be hard to read so we have Y and so here are two adjacent tetrahedrons now if I remove I just make the vertical line from X to Y and here I take the edge BC and join it all up I can't draw for beans so I can't figure out how to do this I made a mess but you can see top logically it's correct of course because everything is correct top logically so you take the tetrahedron it's four versus our X, B X, Y and BC or X, Y and AB or X, Y and AC and you have divided up this thing into three and therefore you can take any two tetrahedron and change them into three this is called the two three Pachner move and it's a beautiful theorem that any two triangulations are related by this procedure you always take two, do this break them into three and do it again hundreds of times up and down and you can go from any triangulation to any other well doing this has an effect on these Norman's Iger equations and it's on the nose although it takes some working out it's on the nose the five term relation and therefore if you go from one triangle triangulation to the other you don't change the class of Xi M so therefore this Xi M which once again is the sum of these shape parameters you don't add the numbers you add the formal things this thing is in the block group of F so we have elements in a very natural way in the block group that sort of arise for real life and I told you this is highly computable and the examples where if I took the four one not so downtown but not confidence F there was Q squared of minus three and the Xi was two times a six third of unity which is kind of trivially in the block group but if you took five two the field was Q of Xi or Xi cubed minus Xi squared plus one zero so it's the field to describe that minus 23 and now the element if I didn't get the numbers wrong is this one and you can equally check that that is in the block group so these are very very concrete things and this means that there is a triangulation with three triangles with these shape parameters and something about orientation to get a sign okay so that's very concrete and we get to this abstract looking block group from another from a three manifold now the other I also explained last time but I won't go into more detail will associate to any not now it's only not it's not three manifolds something called J well now I told you that there's something called the Kashyap invariant which is an element of Q of Z to N Z to N is the standard N through the unity I won't write it again for all integers N so if you have a knot you get this called the Kashyap invariant key to find a sequence of numbers but now Habero proves I think I mentioned it last time I'm not sure I wrote it out with year and so I forgot and I think 2001 I've forgotten that there exists but completely well by what I told you here it'll be unique because of the injectivity I mentioned there's an element of the Habero ring which is uniquely determined by the fact that this KN for every N is JK of Z to N where Z to N again is the standard N through the unity each the two pi I over N now you may say, hang on before I told you that I needed the value of the element of Habero ring at all roots of unity and this is all roots of unity of course dense on the circle they're all rational points but now I only have minus one minus one heat to two pi per third four I mean I just have a small subset but this thing has to be Galway invariant and the Galway group sends every root of unity of order N to another and it acts trans to the only primitive N roots of unity so if you know the value of an element at these standard roots of unity by Galway invariance you know them all so this completely fixes JK but the fact that it exists is very remarkable and so again this is completely explicit I gave the formula last time for four one I'll give it again writing it very slightly differently if K is four one then this function JK of Q is the infinite sum but remember it converges because in the sense of this inverse limit and I'll write it slightly differently from what I did last time you take QN that I just defined one minus Q up to one minus Q then the same with Q inverse and then that infinite sum will certainly in the Habira ring trivially but this does give I'd say to N the Kashyap invariant so now we have examples of the knots and the three manifolds give us elements in these rather sophisticated number of theoretical gadgets let me just look at the notes to see if I dropped anything and I did drop something in this case we can both do here's the order of Q so remember we're only talking about roots of unity and it's called the invariant so all I care is the order and I'm going to leave a little space if you're leaving notes you also could because there'll be one more line later if I don't forget and so here's for the four one knot and I actually gave you these values last time but as real numbers or as and I write them as polynomials in Q so the first three the first four were integers anyways and integers also a polynomial but the next one last time was 46 plus two squared of two and now it's 44 minus four Q squared minus four Q cubed so if that's a minus I'm not an equal sign so Q here is the fifth root of unity and so Q squared plus Q cubed belongs to the field Q squared of five and it's just the number I told you last time so you have this so remember the Habira ring one way of looking is you give its values at roots of unity but every word of unity has some order one two three four five dot the next one is 89 they're all well defined but we can also look at j four one of each the minus H and I gave you that one last time too it's a power series it happens to be even because this knot is its own mirror image it's 47 12th H the fourth plus etc so again we see numbers mysteriously miraculously coming out of the knot out of the same invariance the same cache I've been varying which is this a priori meaningless infinite sum it only makes sense when you can make it converge which is either if Q is a root of unity or either the minus H or a combination of root of unity times an infinitesimal powers series deformation around that root of unity so now we have certainly seen that knots are producing a lot we saw it last time and I reviewed it now that a lot of number theory is showing up out of these things so now I come to the next ingredient which I said last time but much more briefly and I want to say it a little better I was also going to say something about petolomy coordinates for this but I think I'll drop the petolomy of coordinates if somebody's heard of them wants to know they can say afterwards what did Mr. Petolomy do rather long ago you didn't actually give coordinates for hyperbole three manifolds so so the next story which I talked about last time is quantum modularity so this was the conjecture let me even call it QMC quantum modularity conjecture which I gave in 2010 based on the pictures quantum modularity conjecture so it said that for any knot but let me make it very explicit if I look at for instance J41 of 1 over N sorry I'm using two notations I'm using J of Q for a knot which is in the Habir ring that I'm also using an ordinary J knot script of X where X is a rational number or later slightly different and they're related by this way so for what I'm doing now I want the function of X so if I look at J at 1 over N this is exactly J41 of Z to N because that is e to the 2 pi and this by the definition is the original Kashyap invariant and that's where Kashyap had his volume conjecture which was N to the 3 hat well he did not that he just had the volume volume over 2 pi times N but actually then remember there was a whole power series and this power series this was a calculation of many people in particular Sergei Gukov and collaborators including me but also Starvers-Garuf Halidis in a different collection of calculations with other collaborators I wrote this several times last time it's a power series whose first two coefficients well the first one is one the next two are 11 over 72 squared of minus 3 and 697 over 2 times 72 squared of minus 3 squared in this particular case they're all in the field q squared of minus 3 in general they would all be in f of h except for this prefactor whose square is in f of h up to root of unity and that's what we think will always happen but this and where h so I didn't say what h was I should have but now I'll write it again it's 2 pi i over N so this whole thing I'll call the power series associated for 1 of h and here I evaluated the 2 pi i over N this was the volume conjecture of Kachai very much refined to put in these things but then as I told last time if you let N be replaced by x which might be a thousand plus a third so x is still large and it is a very small denominator but it's not an integer it's an integer plus some let's say some fixed amount you'll have exactly the same x3 halves e to the volume over 2 pi times x and the same power series on the nose to all orders at 2 pi i over x except that that whole form is true to all orders up to a factor and that factor is exactly the original j of x and so you have a multiple relation between j of x in this case I can put minus 1 over x because it's even it looks better and more generally if I take j for 1 and actually now I can put any not the K actually I put upstairs I should remember my notations if you take any element of the multiple group abcd are integers with determinant 1 you'll get the usual modularity factor that you'd expect and then you'll get I'll call this whole thing phi hat I'm not going to write it again phi hat of h which here is 2 pi i over x will be the phi series multiplied by an exponential in 1 over h so it's the completed power series so here it'll be some completed power series depending on the knot of not 2 pi i over x anymore but c times c x plus d but this power series is not always the original one this is the one when a over c is zero so for the element minus 1 over x but if you take some other element you'll have other power series so now I'm coming to the actual content of the you know this new not new anymore several years investigation so what we find is that this quantum modularity gives us but it's completely conjectural but it's true at all but it gives us a collection of power series phi alpha of h all of which will belong conjecturally to q bar double bracket h and there's a lot of number theory about this that I went into a little bit last time it's actually a power series with coefficients in so alpha is a rational number and then if I take the corresponding root of unity this will actually be some pre-factor just a constant times just power series in h with coefficients in that psychotomic field so there's a lot of arithmetic not just one power series but if the many but even the existence of these was conjectural and we didn't know what they were we just said there should be some series that you find on the computer to lots of terms but then there were two papers in I think 2013 and 2018 by de Mofti to Doug de Mofti and again Stoppers Garrafalidis found a formula for phi now that doesn't mean that they for all of the well first for phi zero and first paper then for all phi so there's it doesn't mean they prove the conjecture but they wrote down a collection indexed by rational numbers of power series which are completely well defined by using the triangulation the normal and secure equations on there it's not actually proof that they're top-large invariance but it's believed but in many cases it's you can check different triangulations but they gave an actual formula so the conjecture which said before for some unknown power series you have this asymptotic behavior which I hope I didn't remove the asymptotic behavior that j of gamma x is cx plus c to the three abs j of x I forgot this factor and then times this power series they tell you not just there's some power series but this power series does it and it always works but now something very nice happens because these power series have coefficients in the number field you can sorry that's not q it's not at all q in our example it was q of square root of minus three it's f which is the trace field of the knot the field that remember for the four one knot was q of square root of minus three and for another knot it was q of xi to square root of twenty three so you take that basic field you join the word of unity you're looking at and that's where the coefficients lie so they gave a bunch of series but now you don't just get phi alpha for the knot k but you get sigma one which is the original one up to phi alpha k sigma r where these are the different embeddings of the field and they actually correspond to the parabolic flat connections and sometimes as I've mentioned it's not a single field and it's embeddings it's an algebra it could be a product to find many fields but don't worry so you get a whole vector of size r so remember I gave the examples for my favorite knots for the paper it's the favorite knots two three seven pretzels the last one r was two here r was three here r was six so we had a quadratic field the cubic field and here two distinct cubic fields so because they gave a recipe for the series but it's algebraic you can automatically get all of the perturbed ones you get a whole vector of power series but then you can add one more so among the trivial flat connections there's one called the trivial connection sigma zero and he realized that you should incorporate that one too so that was his another beautiful observation or realization so if I do this for the four one knot then remember this was the series one over the fourth root of three times one plus 11h over whatever it was 72 square root of minus three and the next one has to be the conjugate it's actually i over the fourth root of three but then here it's one minus 11h because you take the other embedding of q squared of minus three so it just happens in this case it's kind of boring it's simply five the original five of minus h up to root of unity but now we don't just have two we have three and the third one is exactly that was what he realized what you have to put here is the series we saw before it's the expansion of the kashaif invariant itself around one or more generally around alpha so this one would be exactly this kashaif invariant evaluated at zeta which remembers e to the two pi alpha times e to the minus h so if zeta is one it was this for that special knot so this was this example is when the knot is the simplest knot figure eight and when alpha is zero but you know stacks of these things for other knots and for other alphas so you get a whole vector and so now there's maybe I'll stop talking for 30 seconds so you can catch your breath sip of water now comes the kind of key point that led to this whole thing that I told you very briefly at the end of the last lecture so first of all oh I'll ask Marker to show us the slides again what I told you last time is that this function j of x is periodic so you can draw pictures maybe you show us again slide one and what you get is this weird thing kind of the cloud of points going off to infinity with a lot of internal structure but it's certainly not a function on the real line because this x is periodic so you see all of these points it goes from zero to one and then it starts again repeats periodically but if you take instead the ratio of j for instance j of minus one of x divided by j of x then roughly you're getting this except there's some little fuzzy so if we go to picture two which I showed last time just to remind you what happened then we got a function that at least became a function of a real variable but it wasn't a smooth function but kind of a function with jumps at all rational points so this modularity led to something very concrete but completely mysterious and I already told you the solution last time the very last minute that actually there's a whole vector of power series as I just wrote but actually this vector is going to be part of the whole matrix so I'll first tell you that so there's going to be a matrix phi alpha will always use underline for these all of these for power series and this matrix in general will be r plus one by r plus one it's this number point it's a real number 0054454041 and that's all the but it's not like there are of course these you could make as many digits as you want it's an exact number that's this side but here you say I would like to write down something else and say what this is but it doesn't make sense because this thing is e to the volume times 100 roughly normalized some huge number and then times this power series one plus 11 over 72 squared of minus 3 times 2 pi i over 100 but this series although the first coefficient is one and the next one is much smaller but after a while the coefficients blow up because the nth coefficient grows like n factorial so it doesn't make sense but now we can remove this we can replace this thing but the same number and we do we truncate so if you have such a power series a and h to the n and the a's are too big to make it converge so a and grows let's say like n factorial or some constant at the end then this series always diverges but if you stop n from zero to the optimal term here I think is so what is it it's roughly e to the it should be 1 over h times c I think let's say you stop it roughly of course you have to round up or round down but if you take the right number of terms notice the terms at the beginning the first one is one the second is much smaller the second is yet much smaller at some point they're exponentially small then they start going up so you just stop like you do in perturbative field theory when you calculate you know fine structure constants and things you just stop the series when the stop is good and we'll call that phi of h optimal truncation optimal so that's a well-defined number it's not completely well defined in the sense you have to get your algorithm exactly where you stop but but if you do this here it works and so I've written down that number two so you can do this and the amazing thing is if you just took the power series you could only get like eight coefficients and you say okay I get eight one line eight five one eight eight I'm pretty happy but now this thing is a well-defined number and so I can compute it and here it is as you see I'm doing pretty well so far so at this point you might well say ah it's simply equality as Samir asked just to but it can't really be quality because this thing I stopped at some point through where the next term so what actually happens is that but the negative what I'm going to write now you can't see the four one four one anyways I'll just remove it and put a few fewer digits this thing you can still make sense up to about 12 digits beyond the decimal because the optimal truncation is not exact but the thing you're throwing away is exponentially small but it's multiplied by something exponentially big but the difference is still very small here's slide 10 to the minus 12 so now it makes sense to talk about the difference and you immediately see that the difference isn't zero because this precision is 12 digits but it's already off in the in the ones place so it's not exactly equal but when you compute it you find that the difference is 1.004 zero one one one four one eight five roughly and you can't get more digits because the optimal truncation procedure is approximate but it does give you roughly 11 digits so if you look at this you say well that's not a random number it's very near to one so let's change 100 to 200 and then this goes to roughly 0.002 and if you interpolate sorry 001 in fact it's like one or friend squared what you find is that the difference if this is h you know is exactly we interpolated there's this standard algorithm and when you do this what you find is that this difference is a power series in h and it's exactly 1 minus h squared plus 4712 h to the fourth it's exactly the one that we already saw which we now have decided is going to be the top element of this matrix so now you have something really kind of surprising you have the devaldeo of j of one over n let me put it on the left minus this truncation 2 pi i over n but it doesn't have to be even an integer it can have a denominator so here I'll put j of x so you find that much that was the original conjecture but now when you do it again you find that you always get this j evaluated at zeta which is e to the 2 pi i a over c times e to the minus h but multiplied by something that something here is always 1 so to but it's still not exact because we've still subtracted something approximate but now we've got a second term so now it's beginning to look very different it's not a multivariate statement anymore it's an expansion of one term which is the product of one of our pi hat series remember that was the middle one of the three and then of coefficient j of x which is periodic then here we have this one which I told you is the top one and also a periodic coefficient which happens to be one however we should have a third one because we should have remember the third series in this case with simply minus 2 pi i over x we know that we know that we have three series but this one is exponentially big if this was 10 to the 20 this is 10 to the minus 20 it's too small however we were able to improve this process of optimal truncation to a new one called smooth optimal truncation it's quite amusing I could give a whole talk about that it's a fun thing completely well-defined numerical procedure you can't improve arbitrarily but you can improve the exponential degree of approximation by considerable factor and the result is when you do this with smooth instead of optimal you get like 30 40 digits that are reliable instead of only 12 and so now you can compare that with this but it's also divergent but you do it the same way with optimal and you find that there's a third difference when you subtract the big term and the middle term this is small term and that was indeed exactly the other series to all orders and it's multiplied by new numerical coefficient and so when you do this you find that you have a new function but this function is j of x this one happened to be constant but this is a new function q of x and it turns out that this q of x or rather the q of x if I translated into e to the 2 pi ix then this element again is in the Habir ring and that you can check by these old-school conferences that I mentioned or in this case we could write down an actual formula so okay so this is where these matrices came from you see there's no kind of choice when you do this your computer tells you there's this term this but now you might think now it goes on forever but no now this is it stops and what's more now you can put these new series like q replace j by q and find the new modularity and you find that you get new series but the new series after a while during in the case of four one they're exactly well we already had three so it's only three new ones but you fill up the whole matrix and then the thing stops it closes up and now it has a very simple interpretation because this matrix now one column is one zero zero this is the Habero round you know the j this was this q this is the original phi and some other phi and some other phi and some other phi but if you multiply two such matrices then each entry when you multiply matrices you have a sum of three terms this times this times that and that is these three terms and so this thing can now be interpreted as the final version what we call the refined modularity conjecture it says that this matrix valued invariant I'm an underlying member for the matrix now if I put x plus b over cx plus d and divide by cx plus d to the three of us I'll slightly lie to keep the form simpler then this is simply the product I forget if it's on the right for the left of five x as a matrix times this j like thing which is periodic so it's sorry this is the matrix but this is evaluated at a over c times another phi which is at zeta e to the h where h is what it was before 2 pi i the details of the formulas don't matter don't even try to follow it but and zeta is e to the 2 pi i a over c so I messed this up slightly but the point is that the thing turns into a product of two matrices have a formula to all orders and so now let me come cut to the chase as they say and say what this has given us so you can do this for every element gamma of the sl2z and for every knot and we will get a single object we call w gamma which I should take out my notes so I don't get completely bollock stop with the notations but I do anyway we're going to get a single function w gamma of x and this will be a cosy it's a matrix r plus 1 by r plus 1 matrix depending on gamma and on x but it's a cosycle because this x maybe it's a rational number and then what you have is that w gamma gamma prime of x I should get the order right is w gamma of gamma prime x times w gamma prime of x so that's called the cosycle and the reason that it's a cosycle is that it's given so to speak to all orders you interpret all of these things as this new Habero like function but only the top things will be in the usual Habero ring the other elements are going to be in these new Habero rings of the other number fields the f1 up to fr but whatever it is we'll have a new j which is a matrix value thing and that makes sense for everybody and w gamma of x will just be w j of gamma x inverse times j of x so if you know even a little commons you'll say oh it's a very boring commology class because it's a cosycle but this says that it's a co-boundary but it's not because this co-boundary that's the one we saw on the slide the function that jumped all around it's a terrible mess so this here it's a co-boundary in the space of functions whose graph is just a mess but this is the surprise maybe we can look again at slide the last slide slide 4 this is the function which when k was 4 1 and gamma was 0 minus 1 1 0 this can I have slide 4 please Marco no you saw it last time but I want to show you again no slide 4 yeah there this w gamma still has for 4 1 it is 9 entries 3 of them are trivial but these three are not trivial they happen to be real or i times real taking the real part and now suddenly all of the functions become smooth so this now becomes a cosycle in c infinity of r well actually of r and then it's actually is determinant 1 so it's an r by r matrix r plus 1 by r plus 1 matrix whose coefficients suddenly have lifted from meaningless power series to or functions only defined at rational numbers but jumping all over the place suddenly it lifts to a function on r and this was a big very very big surprise and kind of the the main part that came out of this and you see it visually or you saw it a second go visually in that picture so now my time is nearly up I mean we started certainly a few minutes after 4 30 I think I can go on a little so I'll take a few minutes but very few to tell you the other side of the story so what happens with this function so there's all of these properties that I've already written so what we found well what we already found is that just by these numerical complex like even for 4 1 we started with this j 4 1 of x that we already knew that I gave you the formula q minus 1 n q n but then we had this power series phi and then the second one was as it happened 5 minus h for this case but then we had another 5 star of h that I didn't actually show you but very similar coefficients and here there was another I called the q of q of x and it turned out that you can find the formula for that q is e to the 2 pi i x always and this one turns out be the same q inverse n times q n but multiplied by one half of q to the n plus 1 minus q to the minus n minus 1 that's kind of surprising it's the same kind of thing and so this I'll say very briefly for every knot we found it's most almost everything is conjectural but by this point that you have the original how do you remember is a limit of polynomials so you can write it's an infinite sum of polynomials in q which where the nth one is vis by pochamber and this is a similar thing but you insert here a polynomial in q and q to the n Laurent polynomial but a fixed one independent of n and when you do that you can of course insert infinitely many polynomials but because of the recursions because of the property of the pochamber symbol if you take all polynomials in q and q to the n here then all of these functions span a finite dimensional module here it's got rank three and that rank three module has contains one it contains this it contains this and this is actually a basis and so this is called the q holonomic structure it was very surprising and then the same thing is true for these power series so this one here so there are infinitely many things if I put q to the n q to the 2n q to the 3n q to the minus n but there are any three of them are linearly dependent and so I can take linear combinations and there are only three independent ones but there's not really a canonical basis but then these experiments show there is a canonical base we don't know how to define it a priori which is given by the coefficients of this matrix so one of the things that came out of this analysis was new elements of the Habir ring also these other power series they're also you can describe a posteriori by the de Moff de Garafalidis algorithm but then another very nice things that there's a holomorphic structure these functions are not just in c infinity of r but away from at zero they have to be divergent because around zero they have a power series expansion and that's these series which we know are factorally divergent so they can't be analytic but it turns out that they are analytics to c omega on r star so these graphs that you saw a minute ago in graph four at the origin you can't see with the naked eye they see infinity but everywhere else is actually real analytic and that you can test because from these formulas we get closed formulas for the power series of this at every point and at zero it's divergent you look at the hundredth coefficient it's it's got a 50 digits but if you take some other point it's a convergent power series and the hundredth coefficient is point zero one or something I've all of the numbers there I can't don't have time to show you so you can test purely numerically but we by that time you had to be true which is that these series which were originally divergent power series in particular each phi of alpha like the original one phi of h is now the tailored series but it happens to be divergent because that point is not analytic the tailor series at h equals zero of an of a function which I'll still call five h because it's the same function so to speak but which is analytic away from zero so you have a function which is smooth everywhere but at zero it has a power series expansion that that's divergent because it's it's not analytic so the tailor series can diverge but at all other points the tailor series as all it's smooth but the tailor series would have to find that rate of convergence up to some near-singularity so we have this holomorphic behavior and so now I come to the very last part of the talk the time is running out it's actually a second paper but a much simpler one that is the companion to this a much simpler one one's understood the first one and that's so this the last topic is from knots from knots to q series so by q series that just means some sum you know a and q to the n typically for me always in fact with coefficients of q but now convergent for q less than one so if I think of this as a function of tau f of tau where q is always in the theory of multiple forms is e to the 2 pi i tau tells in the upper half plane so I get functions that are now defining the upper half plane which is completely different from what I had before I had functions at the rational numbers which are kind of disjoint from the upper half plane so subset of the reals and so we have no reason to think that there would be q series associated to knots and I'll tell very briefly I've told it often lecture some of you may have heard the story of how they we found this it's kind of very amusing because it shows that you can discover things exactly when you don't know what you're doing so in work of Garoufalides on what's called the stability of the coefficients of the evaluation of the regular quantum spin network is nothing to do with topology or knots he found a certain series which I'll write down G of q which you can write in many ways but it's the Pochamer symbol we had q infinity times the sum from 0 to infinity minus 1 to the n q to the power of 3 n squared plus n over 2 divided by q n cubed okay so it's just some series so this is what I mean by q series and this particular kind is what's called the hyper q hyper geometric series and maybe I should mention that this kind of series where you have q with or without a sign q to quadratic form divided by something Pochamer symbols that's a very common class and a specific class of those are called Naam sums after Werner Naam and Naam had a beautiful conjecture for many years that said that those things are occasionally multidare the world's Ramanujan example is the most famous example but they can only be multidare if some element of the block group that he defined was a torsion element and nobody ever thought of relating these things to the block group it is a very esoteric thing and five years ago because of the paper with Garafalidis and Caligara on these units that I talked last time we actually proved his conjecture about modularity so roughly the conjecture says if something is supposed to be true multidare form which it might be that can only happen if the element of the block group is zero but that can only happen if the volume is zero so for hyperbolic manifolds this will never be multidare so this you don't want it to be multidare it can't be multidare it's impossible so this is not a multidare function but that doesn't mean sorry I wanted this it's a function of tau or again q no this was called g of q let's call it okay let's think of it as a function of tau so now we looked at this and so he won't know the asymptotics when q went to one and so he asked me if I could do that numerically because we have good numerical methods and after a lot of work we found that g of e to the 2 pi i tau when tau tends to zero so here's zero in the upper half things you come let's say vertically it was very hard to do because it oscillates and it's very hard to recognize oscillatory things if you don't realize a certain trick that we only realize a year later and it's very easy but what you find after a lot of numerical work is exactly that to all orders this function is the difference of the same power series I've been talking the whole time with the 1, 11, 6, 97, number 2 and it's so to speak the odd part of that and we found this completely numerically we didn't know we just started this numerically it oscillated but the oscillatory thing there's a rotation by i so it was an exponential term turned out to be exactly the number is 0.326 and Stavros recognized he said wait a second that's the volume of the 4, 1, 0 there's nothing to do with knot theory and then we found the coefficients and the the second coefficient was the 6, 97 and I remembered that we had seen the 6, 97 somewhere and it was exactly the 6, 97 you've been seeing so in other words these things that come from the 4, 1, 0 were related to this tree which came from somewhere else but it means that somehow the 4, 1, 0 has to know about this new theory as he gives it but there should be two and so we look for a long time to find two and it turns out that there's a second one I won't write the form the bit similar it's completely explicit so we have two functions so the 4, 1, 0 the story is for every knot but let's think 4, 1 gives me two knots two series we call them simply little g and big g both of which are given by this kind of Q hyper geometric series so by very explicit formula and I gave the first coefficients of little g and maybe I can give you a couple of capital G if I can see where they're written down so here I told you this starts one as you are remembering the time right sorry you remember the time oh sorry then I'll just tell you in one minute okay less than a minute 30 seconds I thought I still had five minutes because I'm very optimistic about time so we found after some work there are two power series and now they have an amazing relation with each other which is that each of these is completely divergent outside of unit circle so if you think of them as a function of tau on the upper half plane they have the whole line so let me put g of tau instead of it's the same function but if you take minus one of her tau and then there's a factor I think tau or something like that and here you take minus one of her tau g of tau this function and this function and this function this function are defined in the upper half plane and they're singular at every point they have the line just same similarities but this one is analytic in the cut plane you have the upper half plane and the lower half plane you have to cut from zero to infinity but it extends holomorphically across the real axis and so just to say the final sentence it turns out there's a huge class of such functions we now call them holomorphic quantum multiforms in general it's matrix value this is matrix one coefficient of a matrix but the final statement is that these invariants that I'm defining can extend to holomorphic quantum multiforms which are functions that are suddenly not defined at roots of unity are just as power series perturbational but they're actual holomorphic functions in the upper half plane and they have a quantum modularity property but it's completely different from ordinary multiforms you don't have that the function at tau and the function at minus one over tau are related but what you have is that the discrepancy the difference is the function that suddenly is holomorphic in a much bigger region so I'm sorry I went on the amount of material I told it's a 96 page paper and even leaving out everything there's still too much left so thank you very much thank you very much thank you very much this very beautiful lecture combining as we said before we have here geometry topology number two or even analysis everything so 10 minutes more for combining all these beautiful things together I think it was a very good it's very nice that you open this series of consortium lectures and I must say that even though Zoom usually is not our preferred media but the way that you talked on a on a blackboard with an audience and just getting photographed that is really it almost felt that I'm in the in the hall in the hall where you are where you are talking so this is so this is very nice and I open the floor for questions if anybody has a question then you can put it in the chat or just wave your hand or something it's not so easy well I think nobody is muted if they want they can just ask I was told they can turn on their microphone and just speak okay okay fine so unmute yourself and and ask a question no we are usually if two people unmute themselves we'll manage it so any any question well you have this way to put to take difficult things and make them look simple I have I'm afraid I did the opposite today because I was to to rush the little so I'm afraid some things must have been quite incomprehensible but I hope there some questions maybe something that I could elaborate a little more what's what's the what's the next space I mean if anybody else that's okay I have a I have a question yes go ahead how does this invariance behave with respect to finite covers they say you take a feed you take a figure eight you take a finite cover you have another hyperbolic manifold are there any interesting relation between the invariance of the or is it trivial or what Donald again last time you asked a very interesting question that I'd never thought of and now you asked another very interesting question that I never thought of I will ask Stavros if he can see anything we have experimental data in great detail but it takes months to get it for each not for the three notes I mentioned I don't even know how to write down you could I guess get triangulations just by repeating the take the inverse of the given triangulation so if it's a double cover you can take quite twice the number of triangles and glue together it may be that there's a simple answer to your question but to be honest I'm a little ashamed I never thought of it I have no idea I will certainly tell Stavros I don't know if he's listening but he'll certainly talk to him tomorrow whether he sees whether there is such relation it's an extremely reasonable question and I have no idea I'm sorry yeah I see yes yes I have a question yeah don't you have this series small g and capital G yeah with integer coefficients but if you will have actually is a kind of like integers depending on two on two entries of your like two elements from r plus one element sets like enter your matrix and some integer oh thank you that's I was in principle going to say that so this is what we found originally the little g from the experiment I told you and then big g but if you look at this you see little g big g little g big g it looks like a bilinear expression but actually it's just a straight product but it's a product of matrices because let me call it again G tar of bar of tau now that's only the two by two part the top part we don't know how to do the extra row but here you put g of tau or g of q here you put g of q and here there are two others that we found later that belong to a q homonomic system just like I explained before so for instance if you remember how I went from the first element you don't have to remember I think it's still written from the one element the Habir ring to the next you just insert a q to the n plus one then if I remember correctly this is exactly the same I simply insert here one half of q to the n plus one minus q to the minus n minus one it's essentially that might not be exactly so you get actually in many functions g g one g two but again they satisfy recursion they form a q homonomic system it's a finite rank here of rank two and there's a canonical basis or fairly canonical it's not quite this so we actually have four series and now once you have this then you see that the previous statement is again if I take this function at tau which means that q and it minus one over tau and multiply when you multiply two matrices the upper left hand entry is this one times that one these two here I don't want to get it try to get exactly right this is just one entry of the product but this whole thing suddenly goes into it's a two by two matrix of determinant one in holomorphic functions in C prime which is C minus the negative axis so this quantum modularity is not about two functions it's about r plus one by r plus one times the whole matrix and the statement is much simpler and what's more it's not just this it's for any gamma except this this will change very slightly it's a different C prime so you get this and each of these functions each gene is only defined in the upper half plane actually upper and lower they do not extend at all but now you can make this and it's the same stories before so you make a co-cycle this co-cycle is of course a co-cycle because in the upper half plane it's a co-boundary it's just a function of tau minus or divided by but this function is the one that's holomorphic here this function isn't so this becomes a non-triple co-cycling this and now this is the holomorphic one it's exactly analogous to the other but now comes the wonderful thing that's the culmination of the whole thing it's not exactly analogous to the other it's equal to the one I gave before coming from the Habiro invariance the perturbative invariance is the same co-cycle at the end as you get so it's like in motives you can realize the same motive by very different objects but at the end it has the same underlying properties and different avatars different manifestations of it it's the same here we have one co-cycle and that's the basic invariant of the knot is this co-cycle gamma goes to w gamma so it's a co-cycle all together with coefficients of sl2z in coefficients slr plus 1 of holomorphic functions so we get a single co-cycle that's the final invariant it's a unique object but it becomes a co-boundary in two related worlds the world of holomorphic functions in the upper half plane and the way the world of functions on the rational numbers are with asymptotic expansions near all rational numbers and then somehow they fit together in this rather surprising way so it's the same w at the end and this is just one term of that expansion so indeed you were right that it's a whole r plus 1 story not just two functions at random so thanks for your distance to say that because that's the end of the story that the same function comes from completely different worlds from this world perturbed to things around infinity or around rational numbers and this thing from the series you did your coefficients any other question can I have a question yeah go ahead so if you have partner move moves in principle in any dimensions for example in four dimensions okay I can hardly hear in four dimensions we have three three move partner move is there is some natural generalization of block group which involves something like six six term relation or some something like that I can see you but I can't hear you very well okay but I don't know what generalization is this with the six term relation I'm not aware so you should tell me privately because I certainly can't answer since I've never heard of this that's my question is it is there some natural no natural generalization is there a known is there some known generalization of the block group yes there's something called higher block groups which I know very well because I invented them for 35 years ago also experimentally completely conjecturally but you can write down block groups depending on the number m and the original one the b1 is kind of triple b2 is the original and these higher block groups have a regulator and remember from my last lecture this is related to zeta f of 2 and these are related to zeta f of m conjecturally and some part of that was proved for m equals 3 a few years later by Sasha Goncharov and about three or four years ago the case m equals 4 have been done there's a huge number of huge activity trying to prove those old conjectures and the conjectures that these groups are to the fairly explicit description should be isomorphic to higher k groups but there's so if m is 2 you get k3 and that's connected with three-dimensional hyperbolic geometry but they are not directly connected in the same way with higher dimensional hyperbolic geometry and also you don't have the rigidity theorems and you don't the arithmetic doesn't work the same way so I think the passage from number theory to publish it in number theory fails but on the number theory side there is indeed a series of higher ones by the way you said maybe six term I thought you meant you knew one the next one's a to three Goncharov found the what we believe is the unique defining relations of 22 term relation so it's much much more complicated but that's not how you do it you do it in a different way and then you can work out the relations after so there are indeed higher block groups which are also defined by combination of numbers satisfying some condition and multiple sum relations and they're conjectually equal to higher k groups and then by Borel's famous theorem that the Borel regulator of k theory gives you the value of the zeta function that it can say to function at m we believe here this will be with the poly not the dialogue with the mth polylogithm and that's you can check on the computer and I wrote a paper many years go check it up to m equals 17 so the 35th k group for a field to pretend with Anthony Cohen and Luan and so it's been checked we know it's true but but except for m equals three and four we don't know anything in those case are extremely difficult already so yes there is a beautiful story but it's not really connected with geometry yet as far as I know okay thanks thank you wonderful questions the questions are better than the lecture okay if there are no more questions I before I'm thinking again the speaker I want to thank the staff at ICTP and in UM for doing a wonderful job to put this together with a live audience and and virtual audience and it was a very good experience so thank you very much everybody and thank you very much dawn for this beautiful lecture and time invested etc thank you very much thank you don't thank you very much so we have another lecture on Monday at 11am Miami time thank you on Monday and Tuesday 9am no 11am 11am 11am Miami time yeah at 11am it was changed to 11am because I thought it was changed to 9am so Monday and Tuesday both at 11 o'clock yeah Adi Parazito will give the second consortium series of lectures so you are all welcome thank you okay bye bye everybody bye bye