 to the first IMSA consortium distinguished lecture. IMSA, what we call the consortium, is one of the main ventures of IMSA, and it has become an even more important outreach mechanism in our new world. To say a few more words about the consortium and its importance, I'm very pleased to introduce the person who brought the idea to me originally, Professor Ernesto Lupercio of Simvastoff. Ernesto, thank you for your insight and vision, and welcome. Well, thank you very much for the words, but of course, IMSA, both has been so far very, very important to many, especially young mathematicians in the Americas, and in Latin America particularly, and also it could not exist at all without the infinite energy and considerable talents of professor control. So yes, IMSA is, well, an extraordinary venture that brings together something that should have been natural from a long time ago, and that shares a lot of the spirit with what ICTP has been doing now for the kids, that is to bring together North and South in the universal language of mathematics, but it was very natural in particular, and especially at the location of Miami, and considering also the long-term plans of the president of the University of Miami, Julio Frank, that there should be an institution, an established institution with long-term, ambitious programs to integrate the mathematics of the old continental mathematics from North to South, and indeed, IMSA has created collaborations going from all the way South, from Argentina, Chile, and certainly, especially Mexico, that is a country that in many ways they represent at the consortium, and although the pandemic, the timing of the opening of IMSA was almost in a step with the pandemic, even with these enormous challenges, IMSA has been very successful in creating collaborations and in changing the mathematical lives of young mathematicians in the Americas, that through the activities both in Miami and elsewhere and the postdoctoral programs and many other things, especially in this impossibly challenging times for mathematics in Latin America, where scientific budgets have evaporated, well, IMSA has been a crucial institution for many real people, real mathematicians in all these countries, and especially in Mexico. So it's an honor to be here sharing this floor with Professor Cantrell and now Professor Philip Griffiths on the board of IMSA, and it is a pleasure to have many people from Latin America connecting now to listen to the probably extremely interesting lectures of the great Don Saguer. Thank you very much. Now I'll ask Professor Griffiths to introduce the speaker for this morning. I will look at it on time. Thank you, Steve. It is a great pleasure to introduce Don Saguer, the first in the IMSA's distinguished lecture series. Don is one of the leading mathematicians of our class. His broad interest in the number theory low dimensional topology and algebraic geometry. In number theory, together with Dick Groves, they proved a special case of the famous Clinton first in the Clinton-Tandire trajectory, one of the millennium problems. In low dimensional topology, together with John Carr, they calculated the overhaul of one of the characteristics of the modular space on curve, table curve, beautiful formulas involving the data function. On a personal note, some years ago at the Aval Centennial, I was giving a talk in which I mentioned a special function. And I should mention that Don is an absolute expert in special functions. I mentioned a particular special function that arose out of Aval's work. Don is in the audience and I asked if I knew the functional equation and I said no. And if you don't, nobody does. So it's a great pleasure to introduce Don Saguer, who will speak on two of his laws. The title of his talk is From Nots to Number Theory. So should I start or no? So thank you very much. I'll hold three of you for the introduction. I'm very excited about this possibility, about the new institute. As Ernesto said, the aims of IMSA are somehow very, very similar to those of the ICTP that has existed now for I think 60 years that I've been connected with for seven, which is to both do science at a high level but also make communication between various countries, various continents and especially poorer countries or people who cannot travel as easily to main centers to encourage that, to do everything possible in both directions. And so it's very exciting at ICTP to be connected with that and of course I was really happy that there's such an initiative now for the Americas. So I will start as you know, this is one of two lectures. And as I prepared, I realized I'm going to split it a little. Today will be mostly earlier work. I wrote in the abstract that I will talk about ongoing work with Stavros Garufalidis, whose name I'll write on the board very soon because it's a little long. But in fact, today I'll talk mostly about earlier things and about where the whole story comes from. So I'll use the board and I hope everything will be visible. And then my handbook is there. The title is from Lotz, number three. And already this title is supposed to be surprising or at least it was very surprising to me in the course of the last 20 years and beginning to get used to it. There doesn't seem to be really any connection. One is pure topology and one is obviously number theory. I'm more of a number practical person than a number theorist. So for me, for instance, one of the knots which I'll draw in a second is called four one and it's going to lead to some actual numbers like one, 11, 697 over two or maybe another sequence one, zero minus one, zero, 47 over 24 if I remember correct. Maybe it's 12. My memory is abandoning me. So there are actual numbers but of course number theory is much more than just numbers. There's theory as well. So there are fancy things like algebraic number theory. So number fields and related things, number fields, related things might be Dedekind's data functions and such things. But there are also even fancier things like algebraic K theory. And finally there's the field which as Philip said is I love special functions and especially the ones that go back to the 19th century and are connected with number theory with dive functions. Excuse me, the screen is not seen as a big screen. It is only a small screen. I was told that it should be visible. Is it? Yes, it is visible but only as a small screen. It is probably your connection. I see it large. It's probably your bandwidth is small. Well, right now. I should try again. Right now I don't see it. Excuse me for your interview. You need to exit the gallery mode. It's the problem is on your side. Although right now I don't see it either. And now I see it. By the way, thank you for asking a question. I forgot to say, none of you have been muted. If you want to speak, just switch on your microphone and ask a question. Don't send me emails by chat because I can't read them. But feel free to interrupt. I mean, that's what a semi-live lecture is supposed to be. So can you hear now? Can you see? Yes, we can. Very well. Very easy. You couldn't hear. You wouldn't have heard that question. So it wasn't very logical. OK, so I was reminding that on the side of number theory we have both numbers, but we have fancy things like algebraic number theory, zeta functions, and the like, which I'll come to very soon. But we also have things like k-theory and above all, I would say my favorite, is modular forms, which will play some role today in the background and a lot of role on Friday. So in the continuation of the second lecture, I'll talk only about the recent work. Well, recent, it's over the last six or seven years. It's a big ongoing project with Garfaldides, only briefly today, but modern forms will become, or rather new kinds of modern forms will enter. But on the left, we have Knott's. And Knott's is pure topology. And by the way, I should say that one of the beautiful things in mathematics for everybody who loves mathematics is that different fields sometimes interconnect, but often they don't. And you have to be very specialized. But this interconnects with everything. So there will be analysis. Not too much in the lectures because of lack of time, but there's a lot of asymptotic analysis, some complex analysis. There will be implicitly some representation theory, although that will be much in the background. There's certainly quite a bit of pure algebra, and et cetera, et cetera. So it's not just these two fields, but for the moment, let's think of that. So I remind you what a Knott is. I think everyone knows what a Knott is. Even non-mathematicians, you draw a picture of a circle, well, except that normal human beings aren't mathematicians, but Knott has two ends. But for mathematicians, it should be a closed curve, which is somehow embedded in three space in such a way that you can't untie it. So the simplest ones, this one, what if I already did something wrong? This is supposed to work, but it didn't. And this is under, so here it's supposed to go over. There we go. And now this will go over, and then it goes under. It doesn't matter. I'll never use these pictures again. I just want to show that I can do it. But actually, I can't do it. There are many Knott's. And these strange names, 3, 1, 4, 1, 5, 2, minus 2, 3, 7, they come from various collections, atlases of Knott's. This means in the list of all Knott's that need at least four crossings, this is the first one, but there might be more. This is the second one. This is the first, actually, the only one. This is called the Trefoil Knott. This is called the Figure 8. This, I think, has a name, but I've forgotten. This is the minus 2, 3, 7 pretzel. And if I were a real topologist, I could draw pictures, but I'm not, and I can't. So these Knott's are specific things. And when I talk about topology, of course, this is a perfectly good manifold. But it's a one-dimensional manifold, and they're all kind of diffeomorphic. So what one means is the topology of the 3-manifold, the Knott sits in the 3-sphere, in S3, and you take the complement, and that's an open 3-manifold. So it's three-dimensional topology. But one of the beautiful things, actually very beautiful for me, since I'm not a good topologist, although I started as a topologist, but never a good one, is that the topology goes away almost immediately. What we'll actually be talking about is that these Knott's have various invariance, and in particular, what are called quantum invariance. I won't explain why they're called quantum invariance, but their definitions involve, or the original definitions, things like quantum groups, quantum field theories, sometimes actual physical theories of some sort, otherwise things looking like physical quantum field theories. And they lead to invariance, but these are completely explicit functions, or objects, which are algebraic. And there are algorithms to compute them. And to me, this will just be a black box, certainly in these lectures. So I will just inform you, if the Knott is called 4-1, here is the quantum invariant, it's given by a formula that everyone can understand, and then you believe it comes from topology, but there will be no topology involved. Okay, so that's just to orient us in this universe. So why do Knott's lead to number theory at all? They shouldn't. The whole point of topology is that you can move things around, you can budge them and move them continuously and push and pull, and the whole point of number theory is that you can't, that it's discreet, that it's rigid. So there are only countably many objects, say elements of a number field, even number fields and so on. So the reason it works is the famous theorem called Moscow rigidity, which says that a certain class called hyperbolic manifolds cannot be deformed at all in dimension three and dimension two, they absolutely can, but in dimension three they're rigid, and this combined with more directly relevant for me is Thurston's work, so his famous notes I think were 1983, where he realized that all of three dimensional topology, in some sense, should be geometrized, and that was the famous problem that was one of the millennium problems and that was solved, not just the Poincare conjecture, but his vision that all three manifolds can be broken up roughly into pieces, each one of which belongs to one of eight geometries, and of those eight one is much more rich and has many more than everybody else, and that is called hyperbolic three manifolds. So I will often use the word hyperbolic, I'm not going to say in great detail what it means, but just for the record, the examples I'll use in the talk will always be these three, the four one knot, the figure eight, the five two knot, and the minus two, three, seven, so I won't necessarily always mention them, but those are where we've done most of our very extensive calculations, sorry, not, excuse me, perhaps it was that, hyperbolic, and this is kind of a baby knot, it's not hyperbolic, it's a so-called tourist knot. So the theory I'm talking about is rather degenerate here, it's much easier, it still exists, but you can work out everything. So you have this rigidity, and the way it works is this, if a three manifold is hyperbolic, it means that it's a quotient of hyperbolic three space. So most of you, certainly all of you have seen the hyperbolic two space, which is usually modeled in the upper half plane with some nice geodesics that are semicircles, but this has constant curvature, I mean there's a natural metric, and the curvature is minus one, so it's saddle shaped, it's curved inwards, and now you have a group of isometries. A discrete group of isometries, just like for the two plane, you might have SL2Z, active with the famous fundamental domain, that I'm sure you've all seen, and so if you divide H3 by group, which is sufficiently big, you will get a manifold which is either compact, or maybe slightly non-compact, but it's finite volume. And those are the ones we have, in our case it won't be compact, because in the cases I'll be talking, this will always be S3 minus K, so it's missing one circle at infinity, and that circle is the knot. So we have these things, and because of rigidity, because this is a fixed curvature, it's got an absolutely well-defined metric structure. It means if I put my fingers on two points, and ask three different mathematicians, how far apart are they, they will give them the same answer in centimeters. It's not topology, you don't deform, it's rigid, it's very precise. So the diameter of the thing, well it's infinite, if there's a cusp, but also the volume, which will be finite, is finite. And so that's our first connection to number theory, is that the volume of a hyperbolic three-band fold, and in particular of a knot, one usually just puts a vol of K, of course the knot is no volume, it's the complement, that means the volume of these things, you integrate the volume form dx dy dz over z cubed, over this thing and you get a number. And so first in the notes that I talked about in 1983, proved that these numbers have a wonderful property, the spectrum of volumes, so they're all positive real numbers. And there's the smallest one, there's the smallest volume, it's what's called the well-ordered set, so there's the smallest element, the second smallest, the third smallest, so below some point here, they're only finitely many, but then those have a first limit point. And this first limit point, if I draw pictures pretending it's two dimensional, these would be closed manifolds, and this would be the first manifold that is a cusp, so it wanders off to infinity, and the metric structure is such that the cusp is very thin, so even though it's infinitely long, the volume remains finite, and then you can close up this cusp, I would call the Dane surgery, and get countably many closed manifolds, and those are the ones you see to its left. And then, however, there's the next one, because it's well-ordered, after this there's the smallest and the second smallest, and another limit point, which again is one cusp, but then there's another limit point, and then there's the first limit point of limit points, that would be something with two cusps, et cetera, et cetera. And in the first paper that I wrote on this subject, which was with Walter Neumann in 1985, and I'll come back to it next time, because not the part I'm talking about now, but what we had to do about the combinatorics plays a big role later. With Walter Neumann we found, how is the asymptotics, as you approach one of these limit points, how many points are there, let's say below a given point, so how quickly do they approach in the neighborhood of each cusp? And that turned out to be a question, basically a pure analysis and number theory that the tribology somehow goes away. So these volumes have a structure, and I can make it very clear that there is a kind of a number theoretical structure by giving you the formula for the volume. So you have your three manifold, which for us later will be a non-compair, but now it doesn't matter, and I'm going to triangulate it, but since it's three dimensional, they aren't actually triangles with tetrahedron, so you have a bunch of tetrahedron, and then you glue them together just like if you had a triangulated surface, you'd have adjacent triangles, somehow going around your surface. So you write it as a union of triangles, but they're tetrahedrons, so they have four vertices, and we assume that they're ideal tetrahedron, you certainly, Thurston showed you can always do that. So each one has four vertices, and the vertices are in the boundary of this hyperbolic three space, which has been known since Slobachevsky with a complex projective line. And it's been known since I think before the 19th century that if you have four points in the projective line, there's only one invariant because you can move the projective line by Mobius transformations, which would exactly be the isometries of H3, and you'd note that there's only one invariant. It gives, I shouldn't have called them PI, I shouldn't have called them PI, and then P1 up to P4 gives Z, which is a formula I won't bother to write it, I'm sure you've seen it, which is the cross ratio, and this is a complex number, different from zero and one. So each of these tetrahedron comes with a number, and now the first remarkable thing, so my manifold has been triangulated, it's the union, not of course disjoint, they meet along their sides, but their interiors are disjoint, not let's say N of these tetrahedra, but each of these tetrahedra, each of these symptoms is ideal, it goes off to infinity, so I should really draw it like this, it's sort of the four corners go off to infinity, but it has a well-defined volume, and therefore the volume of M, since the interiors are disjoint, is just the sum of the volumes, but now there's a formula, this whole tetrahedron is determined by just one number, so each delta i corresponds to a number, which is the cross-race for its vertices, and the function is a famous function, d z is not Don Sagueres, you might think it's dialog rhythm, i from one to N, and so d, I'll write down the formula just so you see it's explicit, it essentially goes back to the 19th century, the Lobachevsky, although this version, which is a much better one, is called the Bloch-Wiegler function, it's very much a 20th century thing, so d of z, but it doesn't matter, I'll write the formula, but don't worry about it, if z happens to be less than one in absolute value, then you take the dialog rhythm function that Euler discovered, so the sum z to the N over N squared, and that function is many valued, it converts in the unit disk, but if you move off the unit disk, wander around and come back to live a different value, but what was found by Bloch, so Spencer Bloch and David Wiegler, is that if you take the following combination, you add to the imaginary part of this function the log of the absolute value times the argument of one minus z, so theta where you write this as r e to the i theta, then this thing is one valued, of course it's real, because I took real numbers, but it's one valued, so it makes sense, and this function gives the volume of a single ideal tetrahedron, so you might say at this point, I haven't given you any number theory because here we have a transcendental function, it's an infinite sum, and the z's are complex numbers, z is not number theory, z is analysis, but in fact, what's very beautiful and absolutely crucial for my whole story is that these numbers, these cross-races always belong to a certain field associated to the not, so that's a number field, well of course there's a field, you just generate a field with those numbers, I mean a number field, so find that extension, and I can tell you what it is here, I'll take red again so that it's more, or maybe green, so the field here, so this is k, and here I'll put fk, so here there isn't really one in the sense of what I'm about to do because it's not hyperbolic, but here it's the field q of the square root of minus three, which is the very simplest non-real field, non-real number field, there is smallest degrees, smallest discriminant, this is the field q of xi, where xi cubed, I hope I get the signs right, if I don't I'll check it in a second, this is the field, if your number here is, you know that the simplest cubic field is imaginary, and it has discriminant minus 23, that's this one, and here the field is the same, so that's in fact why we chose that particular example because certain phenomena we want to compare when you have so-called sister knots that share certain data, and in particular these are sisters, and they share in particular their associated number field, it's more or less the same, it's what's called the trace field, which is the trace you would get by adjoining to this gamma, sits in SL2C, but it's only up to conjugation, but the elements in it have a well-defined trace because it's up to conjugation, and if you adjoin all those traces to q, you will get a field called the number field, and this field is often the same and sometimes a little different, but it's essentially, that it's a number field, and I showed you what it is in these things, so by miracle, but it's because of this rigidity, by miracle from something completely flabby, the original knot that you can move things around, you suddenly have something, it couldn't be more rigid than a number field with a given discriminant and a given, and I just checked it was a plus sign. Okay, so that's already a beginning, so we already have a number field, but now this d of z is a transcendental function, but we know that sometimes some combination like this sum d of zi, it looks completely transcendental because this is a transcendental function, even if I'd z to the one, z to the n over n to the one, though to the ordinary logarithm, the logarithm of an algebraic number is not algebraic, so this is not algebraic at all, we don't expect it to be, we don't want it to be, but it turns out that sometimes the volume is the zeta function up to a factor which doesn't matter some powers of pi and some rational numbers and so on, but it's essentially the value at s equals two of what's called the Dedekind zeta function, I'll just mention it many of you know, and if you don't it'll never come back, but every number field has a zeta function, if the number field is q, of course it's the so-called Riemann zeta function due to Euler and that we all know the sum one over n to the s, and this is the one that tells you how primes split in this number field, it's the key, object of key interest in all of higher number theory, algebraic number theory, and it's values at s equals two are always actually that's true for any field, the value at s equals two for any field is given in terms essentially of this dialog rhythm function. So there is already very serious number theory even at this level, and I can make it a little more serious and I'd like to do that, so I'm going to introduce another frightening word in that I don't expect you to know these words, and I think I was told I can go as far as the left as I want and the camera will somehow move, I hope it's true and you'll be able to read what I write. Yeah, so yeah, there's nothing more wonderful than giving a lecture of this sort that's already technical and not live and being in really good hands, it is just an absolute miracle. So I have it here interest and I have it in one and I'm very grateful. So let me go on, but I want to say that associated to any field but in particular to any number field, there's something absolutely frightening, which is called the block group, which I'll say a few words about, but you don't really have to know what it is, only that it's out there, but there's something that is known to be essentially isomorphic to it, let's say after you tense with Q, these are the alien groups and so to find over Z, if you wish that you can tense with Q, get a Q vector space and essentially this is the same as K3 and if you think that's frightening, believe me, this is much worse. Algebraic K theory is a definition that it takes months to absorb and I'm not sure anybody really has a deep in their heart about algebraic K groups, higher ones are really telling you, but whatever it is, this is something quite explicit, but an element of the block group of any field, there's an element, we'll have the following form, it's a formal, it's an integer, formal integer combination, integer combination of elements of F. But not every combination is allowed, it has to satisfy something very special and so it has the form Z1 up to Zn, where each Zi is in the number field F. Well, but if F, if K was my knot and this was K, gave this number field that I told you about FK, which in the three examples was Q of square root of minus three and then twice it was Q, I shouldn't have used the same Xi. Well, the heck with it, I'm old and I can't change my habits, elements of the block group I always call Xi, so it's not that Xi. So associated to K is this field, associated the field is this group, but associated to the knot is a particular element and that element is simply the formal integer linear combination whose entries are exactly these cross ratios. And I told you that not every combination of numbers in the field belongs to the block group, I'll tell you the exact condition and then you can forget it, I won't really use it, but at least then you've seen the definition of the block group. So Xi, now we're not talking about knots, just any old sum Zi belongs to BF, but I should warn you BF is the element some about to say, but modern and equivalence relations. So the representation may not be unique. There may be other Zi's that you can use and that's natural because of course, if you have a manifold, you can triangulate it in many different ways. And so there are many different formal elements I could write, but there's a relation in the block group and it's set up exactly in such a way that the moves that you go from one triangulation to the other in the triangulation are exactly the relations that define the block group. But I won't give the relation, they come from the functional equation of this dialogue but I'll say what when it belongs to the thing even though on the F, for all homomorphisms Phi and Psi from the field from the multiplicative group of the field, so I ran out of space, so I'll go, this is illegal. So you take a map from F cross to Z, for instance, if F was Q, you might take the two attic valuations, the power of two and your rational number. So pulse over negative depending whether two is the numerator denominator. So these are homomorphisms and the condition is that every time that you pick two of these things, if you take Phi of Zi and you take Psi of Zi, but you also take Phi of one minus Zi. So I didn't tell you, but if you permute these four points, it's the same tetrahedron but the Z will turn into something else which might be, for instance, one minus Z. So they both are kind of equally important. And so I take this two by two matrix and I take its determinant and the condition is that that should be zero. So it's a somewhat strange condition, it takes a lot of getting used to it and if you haven't seen it, if you don't try to get used to it, it won't come back. But just that you've seen that this block group is a fancy gadget, not nearly as fancy as algebraic K3, but fancy enough, but it comes down to something quite explicit. So I'll give an example and then you'll see. So example, which does come up for a certain three manifold, it's not a compliment. So some M3 is going to give me the field which will be Q of squared of minus seven and then the xi will be two times, I hope I remember it correctly, by heart, it's two times, so there would be three tetrahedra, two of them have cross ratio, one plus squared of minus seven over two and the other one, I think it's a plus sign, is minus one plus squared of minus seven over four. And if you look at the, if you compute the norms of these norms and one minus these norms, you find there are only twos and in this field there are only two primes that divide the prime two, P2 and P2 prime and so you can only take the two added, the P2 added evaluation, the P2 prime added evaluation. So that's the only fine psi you can take, so there's only one condition and you check that it works. So this is in the block group, whatever it is, of Q of squared of minus seven and now if you take D of it, which means two D of, I won't write it again of that number plus D of this number, so two D of one plus root minus seven over two, this is up to some easy constant, the value of the zeta function at s equals two of the field Q of squared of minus seven. So I think, I hope this is the first section of the talk, I hope that that convinces you that we've gone from something that looked very flabby to something rigid, hyperbolic geometry because of rigidity and then to number, really numbers, number fields but much more than number fields elements of those number fields that combine to give elements of the algebraic K theory or the so-called block group and that sometimes for instance, the volume will be something like a zeta function. So these volumes are absolutely numbers of arithmetic interest, in general it's the so-called Borel regulator of K theory. So that was the first of my examples. So now I've told you that, now I want to talk, I told you that the real, the actual things I'll talk about are quantum invariants. So there are several, there's one called the WRT invariant, Wittenbresch-T-Tureyev, I won't write it out because we won't use it. That's of a three manifold and it's much more general than the one I'm talking about. I studied it many years ago in a paper with Ruth Lawrence which was to me the first introduction to these things and the connection with number theory but that was a very, very easy three manifold kind of a very far from being hyperbolic and there are many, many people now, in particular Sergei Kukov and the whole team of people around them and connected with my mother's then there may be 15 names doing for several years, very exciting work and slowly developing a vision of how this thing should look and it's very parallel to what we're finding on our side. I haven't yet, I said the name but I didn't yet write it. Actually it hasn't yet occurred but I'll put it anyway. Stavros Garofalidis, Stavros, because he's my co-worker for everything although, as I said, the joint work will come later. So we have these quantum invariants and the work with him, this is the more important invariant, the WRT invariant because it's general for all three manifolds but there's a very beautiful invariant called the Kashyv invariant and the Kashyv invariant is something associated to a knot and so in the original way that it lives you have the knot and the notation. I think you the Kashyv, this is maybe 25, 30 years ago is for each number n, which is one, two, three, I'll always write as a to n for the standard nth root of unity. So on the unit circle, you take the nearest root of unity going counterclockwise from one. So you've got one nth of the circle. So this will be a number and already you'll see now that we're talking number theory because this is a number not just in the field q of z but even in the ring of integers which were six atomic fields is just z of z to n. So now we're full blast in number theory but it's a collection, remember? So we have infinitely many numbers. It's a very strange invariant but as I said before, it's completely computable. So even if you don't know any topology which puts more or less in the same boat as me and you don't even care about all these knots and you wouldn't know how to draw the figure eight or the four two even if you were asked but there's a black box. There are algorithms that are in the literature. People give you the formula for the Kashyap invariant for any given knot and once you have it, you can just study it and then you can actually forget the topology except you know that you're doing something of interest for topology because the rest of what you do is algebraic. So let me actually write down what it is. I could write down for four, five, two as well. So let me take this. If k is the four one, well, I don't have to write it called the k. I can just call it four one. That's the point of having a notation. This is given by a very beautiful formula. So I'm going to use the following notation very often. If q is a number and x is another number, then what's called the q factorial or shifted factorial or q pochamber symbol that has many names. It's the finite product is n factors, one minus x times one minus qx up to one minus q to the n minus one x. So in particular q, qn, which is what I'll mostly be using, then I'll just put, I think I'll just call it qn because why, well, that's the product one minus q up to one minus q to the n. You see it's like factorial because if q is kind of one plus epsilon, then the ith factor is roughly i times epsilon. So the whole thing is epsilon to the n times n factorial. So this is the infinitesimal version of the quantum version of factorials. It's called the quantum factorial. And so here the form is very beautiful. You take this n pochamber symbol or shifted factorial and then it's a complex number. So I can take, it's an algebraic number, but in particular complex number, I take its absolute value squared and that's also an algebraic number. And of course it belongs to z of z to n, each one individually because complex number, the absolute value squared is alpha times alpha bar and they both belong to this because z to n bars, z to n inverse. But now where do you stop? Well, you don't stop, you go to infinity. But actually, as you can see, you only go to n minus one because as soon as little n equals capital N or is bigger than in this product of n terms, let's say capital N is five, little n is seven, then this is one minus q to the seventh but on the way there was a one minus q to the five which vanishes and once one factor is zero, the product is zero. So as soon as n exceeds capital N or is equal, then this thing is zero. And so it's actually a finite sum and that's why not only the individual terms but this whole infinite sum is in that. So this would be the Kashayev invariant. Now, okay, now the Kashayev invariant is absolutely wonderful properties and actually essentially everything I'll be telling you, especially next time will be various, it'll all start with that and more general verse of the Kashayev invariant and their properties. But I want to tell you the first, the two most spectacular. So properties of the Kashayev invariant. So property one, but we don't know if it's a property, it's a conjecture. This is, it's a very famous conjecture. We set a conference in Japan about five years ago called 25 Years of the Volumes of the Volume Conjecture but it's still open. It's been set, checked for a handful of knots maybe 20 by an hour or 30 but a couple of years ago it was only two. But it's, so it's a property that we know is true but we don't know how to prove it. And the property is this, that if you take Kn, that it will grow exponentially like a certain constant in n and in the exponent little o of n. So it is exactly exponential growth. So it's nth root will have a limit and the wonderful conjecture about the name tells you all is that up to a factor of two pi this is the volume of the knot by which it's already set. I mean the volume of S3 minus the knot if it's hyperbolic and I should be honest, the volume as Thurston realized is a real number but it has an imaginary part which is the so-called Chern-Simons invariant and that's not quite a real number. It's a real number multiple of four pi squared and so this number is slightly ambiguous but if you change it by two pi squared it's actually i times the volume then you'll get e to the two pi and it's okay. So anyway, believe me, this makes sense but for the four one knot the volume is just 1.03 or whatever it is and so you get a number, 2.03. No sorry the volume, yeah that's right the volume of the two three knot is about 2.03 and then you divide by two pi. So let me give this to you so that to make it much more immediate and intuitive let me give you a little table here. So if I take the values for one, two, three, four then they're one, 13, no one, five, 13 and 27 and when I get to six it's 89 but when I get to five it's, if I remember correct I'll check in a second for, I better check right away because I don't remember it's, in fact, I had it written down 46 plus two squared of five. I didn't remember the multiple of squared of five. So in general as I already said these numbers are algebraic numbers but you see they're growing and if I get to a hundred then it's about 10 to the 16 it's growing exponential in Kashyap's conjecture. It's an amazing conjecture not because we care so much how these numbers grow. It's the answer like so often in mathematics the problem becomes good because the answer is good. The answer is it's not just some stupid exponential like it's always two to the n for every knot. No it involves the volume. So the volume which is a topological invariant by most of rigidity but now you can read it off from the growth of the Kashyap invariant. So there's another property which I'll come to in a second which in fact that's my next theme is that the Kashyap invariant belongs to what I'll just use in H. It's a wonderful thing that should be much better known in number theory. It's a purely number theory logic but many more typologists know it because it's been used a lot in knot theory and in three manifold theory. It hasn't yet really, I mean I have some number theories to know but it's not well known. It's a gadget called the Habiro ring. And so each of the for every knot this Kashyap invariant will lie in that Habiro ring. So this is something of again a piece of pure number theory and I'll show you how that works. So an element of the Habiro ring is I'll call it a function A of Q. But it's not a function in the usual sense, a function in the usual sense. You take Q to be a number as an argument and then you get a value. You cannot in general evaluate this function. So for instance if you take A of pi it will make no sense at all. However, a typical case would be this one. So for the for one knot this invariant I'll use a script J. For the for one knot this invariant is going to be the following. N goes from zero to infinity. And then it's just what I wrote here but I don't like to use complex conjugation because that's not now to break the thing. So instead I'll write out what that is if you compute it. It's very trivial calculation. So this formula that I just wrote here this thing is simply J for one evaluated at the nth root of unity. And that's true for every knot. So we're going to have an element of the Habiro ring. I haven't yet told you what the Habiro ring is. But in such a way that if you evaluate it at an nth root of unity the standard nth root of unity you'll get exactly the original Kashyap invariant. So actually that tells you what it is uniquely because if you know this thing then you know the values of this at roots of unity but it's Galois invariant. It's defined completely with polynomials with rational coefficients even integer coefficients. So you can just Galois conjugate but every root of unity if z is e to the 2 pi i alpha every root of unity is e to the 2 pi alpha with alpha a rational number. And so there's a numerator over n and then this zeta is Galois Kashyap to the standard zeta and my talk. And so you can automatically get the value at every q just by taking that formula and changing zeta to for instance zeta to the 17th the 17th is prime to capital N. So there's a recipe to go back and forth and so if you have an element of the Habiro ring and I can tell you very easily how it will look to any element of the Habiro ring will look like this. It's a sum n from zero to infinity and then it's some polynomial in z of q and q inverse. So like in my example it was minus one to the n q to the minus n times n plus one over two times qn and then times qn. So it's always got an expansion not at all unique but it can be expanded as an infinite sum of these q factorials, these qn's. I shouldn't put a line through that, it's unintentional. So it's got a sum but the nth term is missed by qn. So that means it will converge for the same reason I told you in the special case. If n is a fifth root of unity, sorry if q is a fifth root of unity then you can just stop the sum at four because you'll have a polynomial times zero. So these sums are always convergent so though it's not a function in the usual sense you cannot substitute for q an arbitrary complex number but you can substitute roots of unity. Okay so this is the next piece of really exciting number theory. So we've already have a knot lead us to a number field sometimes to Dedek and Zephanes but I'll admit it to the block group and algebraic k theory and now to this Habira ring. But now I can already start putting it together a little bit and showing you a little more. So let's take the simplest element. Let me look at my notes if I'm forgetting something and also look at the time. My time is of course moving along that I've been told that always happens. Whatever you do. Oh I see I did leave out something that I absolutely wanted to say which is on the volume conjecture. So because that's absolutely crucial to our story. If I go back to the volume conjecture then remember that we have this invariant so I gave the explicit formula and it was supposed to be e to a certain constant which is the volume over two pi times n but then in the original volume conjecture it's just something sub exponential but there's a much more precise version due originally to Gukov in this form. I think he was the first I'm not sure maybe also Garofa lead us various people and then various calculations including a paper typed with Gukov, Dimovt and Lennel's gave a more precise version and the precise version is that it has an asterotic expansion to all orders as the exponential we saw times n to the three halves times a power series which of course depends on the knot. Well so this would be for any knot. So there's a power series in two pi i over n and this power series is supposed to have algebraic coefficients. That's why I put the two pi i just to get the algebraicity. So let me give an example if I take the four one knot then I don't have to rewrite those factors they're the same five four one of h is the following power series there's a pre-factor which is one over the fourth root of three. But if you remember this the field here was q squared of minus three and you see that the square of this pre-factor is up to power of i in that field and that will always happen the pre-factor will always be an eighth root of unity times the square root of something that but who cares but now comes the actual series and these are the numbers I showed you before one plus 11 times h over 72 squared of minus three plus these are the numbers I wrote before one 11697 halves. So in this case it happens to be a power series with rational coefficients in h but in general it's not just q bar of h it's actually up to a pre-factor which here was the fourth root of three it's actually coefficients in the same field f so here the field remembers q of squared of minus three and you see that this whole power series is does a coefficient q of squared of minus three so now we're really honestly producing numbers the serious numbers it's much more than just this limiting value that gave the volume which is anyway transcendental and symmetric invariant differential geometry but this is now truly number theory we have a pre-factor which turns out very interesting and then we have these further factors and the same conjecture happens for every knot and there are numerical methods that you can check this to very, very high precision very far find dozens of coefficients we know 280 for this knot 200 for the five two knot and so on so this can be done to high level so that is important but now comes in a sense the really beautiful fact which is to me most beautiful because as I said I love multiple forms so this is well I won't start with the modularity I'll come back to that in the middle let me talk about this thing here so we just had to each knot we had this function jk of q where q is a formal letter but it will make sense if q is at a root of unity so in particular I can define another function I'll use an ordinary j not a script j of alpha which is simply jk of e to the 2 pi i alpha so now it becomes a function on q now you could say it's a function on q mod z because it's periodic but I don't want to do that it's a function on all of q and the reason that I want to do this is that we have my favorite group the modular group which consists of 2 by 2 matrices A, B, C, D, 4 rational integers with determinant 1 and that acts on the upper half plane as we all know by fractional linear transformations but it also acts on any complex number and also any real number the same way and so I can took if gamma is this element then gamma of alpha is a alpha plus b over c alpha plus d and so we have an action and so that means that we have an action of sl2z on this domain space so jk is now a function on q and so now what I wrote before becomes the following statement that jk of 1 over n actually it's minus 1 over n in general but for this not it happens to be an even function will be e to a constant times n times n to the 3 halves times this power series that I told you evaluated at 2 pi i over n but actually even here there's already a little multinarity because this is if n is, you know, a thousand n1 so n is large but it's going to infinity if this is asymptotic so n is big but it's an integer but let's say that n is going to infinity I mean I found this on the computer many years ago 10 years ago and published a paper called quantum multireforms and so now let's say that I let n take the values a thousand and one-third a thousand and one and a third a thousand and two and a third so I fixed the imaginary part then when you do it on the computer at a very high precision you find that for these shifted ends you have the same exponential the same n to the 3 halves the same power series to all orders I did it to 50 orders, 50 powers but then you have a constant and the constant in this particular case is 13 and the reason it's 13 is because that is this number by what I've defined is exactly this invariant evaluated at a third because remember if alpha is a third then q is e to the 2 pi over 3 which is the cube root of unity it's a to 3 so this 13 is that and so the full statement maybe it's confusing to call it n because it looks like an integer so I'll call it x so as x goes to infinity with say a fixed denominator a fixed rational part then this function is exactly true to all orders as it was before nothing has changed but up to a constant and that constant is exactly the value of the same function at x but remember it's periodic so that's why it's the same if I go to infinity through integers plus a third or integers plus any other fixed shift so this absolutely looks like multilayerity because if you know ordinary multilayer forms an ordinary multilayer form on SL2z is a function that transforms nice under the multilayer group but the multilayer is generated by only the translations the matrix 1101 and the inversion the matrix 0-110 which is f of minus 1 over tau and if it's a multilayer function then it's equal and if it's a multilayer form it's more or less equal so there's a transformation equation that tells you that you only have to know what happens for tau plus 1 minus 1 over tau well our j of x already is invariant under x plus 1 because it is a function of e to the 2 by i x x here could be alpha but here we're saying that it has some behavior but only asymptotically as x goes to infinity so it's very hard to interpret this in the sense you have to really get used to it it took us a long time to be understood how these things work because we're actually taking a kind of a double limit double scaling limit the physicists might call it in a double topology x is going to infinity so maybe I would make a graph if I've graphed this thing as a function of x it's very big because of the exponential function let's say I've graphed its log it would have some local behavior it would wobble around but then after a while this is a thousand and this is a thousand and one well it goes up a bit because it's growing that's the exponential factor but it would be more or less the load behavior would be the same so it's a combination of two factors one is a smooth function where x is in R it's just there's no arithmetic it's just a real number going to infinity this is some smooth function that would have some graph and then you multiply it by solving this one x which was also in Q but therefore it's not just in R it's also in Q mod Z and then j of x is the number which if x is a third would be 13 if it's a quarter of b27 27 and so on so you have this mixture of a purely arithmetic behavior depending on the fractional part of x and an asymptotic behavior depending on x as a large real number going to infinity well that's all well and good but as I just told you the reason that numbers here it's like to talk about functions that are invariant or nearly invariant to some factor under tau plus one and minus one for tau it's not just a random love of inverting and translating but because these two special matrices 1 1 0 1 and 0 minus 1 1 0 generate the group so here people really want to know what happens when you take a tau plus b over c tau plus d for anything and then it will be roughly up to a factor f of tau and of course as I'm sure all of you are certainly most of you have seen if it's a so-called multiform of weight k to factor c tau plus d to the k but it's not important it's something easy you can relate f of tau with f of gamma tau so when I did these experiments in 2010 I was looking now at j for my number field of my not ax plus b over cx plus d so this is gamma of x and gamma is some fixed element in sl2z so abc and dr fixed for instance 0 minus 1 1 0s the one I just did and x is going to infinity exactly as before that it both is going to infinity as a real number but also multital 1 it's kind of stabilizing maybe it's just constant it's always you know 1 3rd, multital 1 well then the same thing happens everywhere there's the product of a power series which now depends it's not always the same one I had before it'll depend on the number a over c so on the value of infinity and so there'll still be some power series and they'll again be an exponential factor that I won't write in a power of x that I won't write so that part's the same here it's not 1 over x it's just like usually in multiliarities well I will put some of this because remember we had n to the 3 halves now it's cx plus d to the 3 halves but then there'll be an exponential factor with x that I'll drop so it has the form it looks so this is again as a function function of a large number x but it's a real number going to infinity that is a real number so this is a smooth function this is some power series and then again there's a part that depends on x multital 1 and it's not just some function it's always the very same function we start with and that's why it's called multital because it's just like from multital form when you apply f of a tau plus b over c tau plus d you get c tau plus d to a power called the weight so here will be 3 halves times f of tau here j of x plus b over cx plus d is j of x times cx plus d the 3 halves except it isn't there's this huge factor which actually dominates this exponentially big factor but it splits into two parts the analytic part and the other and that was a big mystery which will become very clear in the second lecture when I talk about matrix invariance so here we have this and now I already gave you if I take the 4,1 node when a over c was 0 the original one there was a pre-factor which remember was the fourth root of 3 and then it started 1 plus 11 times h plus etc well h over 72 squared minus 3 if I don't lie but if I take for instance phi at a 5th you can do all these things numerically then you again get a pre-factor times some a0 plus a1h with algebraic numbers and so here all the coefficients lay in the same it's a power series in h but all the coefficients are not just algebraic but they all lie in that famous number field which for this particular node this was the 4,1 node happened to be q of the square root of minus 3 but then there was a pre-factor which didn't look like that well here it's much more beautiful these a n's all lie in fk but you have to join that n through divinity that's sort of natural so if n is 5th in this case you join the 5th through divinity in general you would join e to the 2 pi i a over c so it's the c through divinity where c is the c of the matrix so these numbers all line as an example a0 and they're quite complicated a0 is a number of norm 29 in this example and nobody knows you know it's just some random numbers so you get very complicated coefficients we know dozens of them by numerical calculations which are all done by numerical approximation because none of this is actually approved but now comes the interesting part there's still there's essentially the same pre-factor I may be lying maybe a factor squared of 3 but then there's a new factor and that factor is absolutely fascinating I found on the computer and I had no idea what it was it's not in that number field this epsilon is an element of f which here is q of square root of minus 3 where you would join z to n so here that would be actually q of square root of minus 3 and z to 5 but in general you would join fk and the root of divinity epsilon is in that field but you take the c through it so here at 5th we take the 5th root and it's not just that it's in the units of this so this thing has now produced something that is kind of the holy grail in mathematics except this doesn't solve the problem so a big problem, not in mathematics in algebraic number theory in algebraic number theory if you have a number field it has associated to it a group of units so for q the units are just plus or minus 1 but for most fields it's more of z plus z plus a finite group something like that so you have this group and it's impossible except for a very special class of fields like cytotomic fields or quadratic fields it's essentially impossible to write down units but here we're getting units actually coming out of topology here we've actually constructed a unit and so the final discovery so this was something that was kind of guessed in discussions of me with Frank Caligari and then he saw how to do it he's a very very strong number theorist against Stauffer and Scarif Alidis and myself so it's a paper I think it's on the archive in 2018 and it'll appear this year and it's accepted in the and also the IHES over the... I call it the Supergeur I mean so it turns out that there's always a unit like this and this unit does not depend so here we have an alpha which in my case was a fifth this alpha depends on of course alpha because it has to it's in a field which is you would join alpha but it does not depend on the not it turns out that was our conjecture and we know now it's true it actually only depends on this element I talked about which was in this mysterious thing called the block group so whatever it is I don't want to go into details but what we found is that you can associate to any element of the block group of any number field and to any rational number a unit which if alpha is has to nominate her in a C then this thing it's going to be the C-th root of a unit of f where you join e to the 2 pi i alpha so for any rational number and any number field any rational number alpha any number field f and any element of this block group you can have this so that doesn't depend on knots at all and that is what we did in this paper and then what we have not proved is that this one is that one and of course it should be because that's what motivated it but we don't know how to prove that these things even exist so for the knots it's not true that we know it's this thing but we know how to construct this thing for any element for any number field any rational number and any element of the block group and when you do it for the number field coming from a knot like q squared minus 3 and the element coming from the block group I didn't tell you what they were in our fields in our examples let me just do two of them so this is k and this is xi here it's just two times the sixth third of unity which is in the field q squared minus 3 and here if I remember correctly remember it was there was a xi a different xi it doesn't matter but just to say it's explicit these are the shape parameters of the tetrahedron that if you add up d of this plus d of that you'll get the volume so these are the things that the block group sits completely explicit and once you have that there's a computer recipe give them this thing to write down this unit and that is the unit that we find but that theorem of Caligari Garfaldides and myself is completely number theoretical there's no reference to knots except for motivation it's a general construction but I don't think anybody could have guessed it there was no reason to be looking for units associated to number fields in elements of the block groups but the topology simply informed us that they exist and then once we knew to look for it well you have to be quite brilliant and that was Frank Caligari we wouldn't have known where to look but he saw in principle how to construct it and after a couple of years' work everything worked so this was really an application of topology to number theory so I don't know first of all I don't know how long these lectures meant to be I was told there should be time for questions and answers but I wasn't told that there should be time also for the lecturer so what I want to do next is at least show you a couple of pictures and I had two small themes left to end today but at least one of them I want to say I've almost said it so I told you already that j of x let's say for the four not and I'll just always stick to the figure eight not because it's the easiest I told you that j of minus one of x is roughly j of x times something so that means if you this is exponentially big but if you divide through maybe it'll be a much better behaved function so that's what I want to show you so now I have to ask Marco who's the eminence please doing things if you could show us the first slide so he has to share the screen so what you see there is a picture so we have this function j of x and remember that j of x is periodic so whatever I show you it repeats when you get here so for each number it's only defined at rational numbers a half a third and so on but when you graph it you can't actually graph j of x because it's exponentially big so the graph is actually log j of x otherwise it would go up to the moon so what you see is a picture of this function and you see it's not devoid of structure here you can see rational numbers like here's a half it's shooting up the actual value of the half is down near the bottom of the screen but nearby points near half are going off to infinity in some special way and similar near one and then it repeats periodically so it's a very mysterious function but now you're seeing a picture of a typical element of the Habiru ring which here is the Kashayf invariant of the figure eight knot but it's completely mysterious nobody can see what's going on but that was the function j of x but now modularity tells us that at least it should become easier if you take the quotient j of x over j of minus one of x and again I take the log for the same reason so now if we go to slide two which Marko will click immediately Marko did you hear a slide two? Yeah, there we go here the graph is completely different it's still not a smooth function it jumps but it only jumps and it has little breaks but it's a function that function if you take that graph and take the closure you get a well-defined one-dimensional thing in R2 whereas the one before if you took its closure you actually got the same set because it's a discrete set getting denser and denser as you go to infinity so here you see the modularity in a visual way this is not the new discovery that these lectures are about it's the one of ten years ago this is a paper of mind quantum holomorphic holomorphic modern forms and it led to the thing that I'll talk next time where this will become suddenly enlightening now it's still meant to be mysterious so here's the picture and just for fun if we look now at slide three slide three is a blow up of a small neighborhood at the point three eighths so three eighths is kind of a typical point rational but with a very small denominator so it's a simple rational number and now if you look at a small neighborhood blowing up very much what you see is that the function jumps so the point in the middle is the value of three eighths but the points above are the values just to the left and below is just to the right and as you see what you see is that the function indeed jumps don't worry about the value at three eighths because later you actually emit it and you do something else so what you see is a function that's actually C infinity in the limit both to the left and to the right as you approach any rational points here it's the rational point three eighths and that will lead to a whole new class of functions that we need for the full paper called asymptotic functions near Q so here near Q is near rational points near three eighths this function is a well-defined asymptotic behavior which means a well-defined Taylor series so it's smooth to the right and another one to the left this curve doesn't get quite there because I graphed a thousand points but at some point if you get too close to three eighths the denominator is too big, the computer stopped but the curve of course would continue smoothly all the way to that point from above and below so this was one big surprise and so now, well I kind of asked a question but nobody answered maybe nobody has the right theoretically I started at 430 but of course we didn't because I was introduced by several people I think I've talked for an hour am I meant to stop or can I take another five minutes three minutes, ten minutes I have no idea what the organizers want assuming at least one of them is still there eight last minutes sorry, three, five minutes yeah, I think it's time to take five minutes okay, so I just want to round this up by the way I don't need the slides anymore Markov at least not these I'm going to want one more very very soon so what you see is that from this whole story something really to me amazing has happened that from nots we already made the list somewhere but I've removed it we already from a not oh here it is we got a number field we got the block group we even have zeta functions although I didn't really talk about them the block group, the Habira ring but at the last part we got units a construction of units related elements of the block group so a connection between two purely number theoretical things that now we know is very clearly connected and I should say that in the continuation what I'll talk about next time all of this number theory goes much further but this Habira ring will eventually get replaced but we don't yet have a complete definition or at least not as completely satisfactory one there will be a new Habira ring for every number field and in particular the ones I'm talking about here and the classical one is the one for Q but these are much more subtle and an element of this Habira ring has a weight in the case of Q the weight was always zero so it's a graded ring and the weight will be an element of its block group Q doesn't have a block group but most fields have a non-trivial block group and so all of this will be part of a new structure and this will be its joint work we've been discussing a lot we're far from writing anything up with stills, Stavros Garufalidis and Peter Schultz so there's a very beautiful number theoretical story that's slowly emerging which generalizes some of the number theory I've told but I'll say even next time quite little and for today this is just a hint but I wanted to end very briefly with the last topic which is the last thing I talked about here was my paper of 2010 so it's still a decade in back and in the last 10 years together with mostly Stavros but also Indiumoft and Caligari and other people we've been looking and also they've been working with groups so there's kind of a group working on this whole complex ideas but in particular with Garufalidis what we now found is a new picture so the final picture that I'll talk about next time suddenly makes this completely mysterious thing equally mysterious because you don't know why it's true but at least you know what is supposed to be true so this is once again with Stavros and myself and somehow let's say over the last five years roughly the papers now by the way for those of you who want literature I gave a course about this partly in this room a few months ago and it's most of the I think the recording is not always good quality there were some problems and lecture notes on many of the lectures are online on the MPI website and also the big papers Stavros will be ready well for our friends within a few days and then on the archive I hope within two weeks or something we've finished a bit it's an over 90 page paper it's a long story so if you want more details then I'm telling here you might have to wait a few days or look at the MPI website but the final thing is this to each note we had already this Kashayev invariant which had two versions remember it was J of X well alpha let's call it X again this was a rational number which was also script J of E to the 2 pi I X so that's a root of unity and script J is in the simplest case in the Habiro ring but in fact they won't be they'll be in these more general Habiro rings so the original picture was this but the new picture is that actually so this was the original and absolutely beautiful it's one of the most wonderful invariants certainly in topology but maybe anywhere the Kashayev invariant but actually we have a whole matrix and the size of the matrix is r plus one by r plus one where r is usually the degree of that number field but I actually lied to you a little that field is a field and remember for my three favorite examples 237 I told you that it was Q of square root of minus three so the degree was so r here is two here it was Q of xi which was a cubic field and here I told you that the original field is also Q of xi but actually when you do the full calculation there's a second field which in this case is Q of cosine two pi over seven which is a cubic field so it's actually three plus three so r is actually six in this case and then actually the story is even slightly bigger there's always another factor these are labeled by the so-called connections and there's the trivial connection and that always gives you field Q so if you take the three one not reach remember it wasn't hyperbolic r will be zero but r plus one is non-zero you're still of Q so the total size of this algebra here is one plus three plus three it's seven so it's always r plus one we'll have an r plus one matrix and that matrix I'll just write down how it looks for four one and then I'll stop and I'll be very approximate so this matrix we still call it j but we underline it but then take its bold face it's now a matrix value thing and again it's labeled by k and by alpha but I'll just do it for alpha so I'll do it just for the four one not and zero so then this matrix is going to be a function of let's not even worry too much what it's a function of it's going to be a three by three matrix but the first column for every matrix is one zero zero zero so this thing will actually have blocked triangular form for every not it will be one and then zeros then something here and then something here so here r is just two so here this corresponds to Q and here this corresponds to Q this corresponds to Q squared of minus three and this corresponds if you'll let me say that to Q squared of minus three it's the same abstract number field but differently embedded in the complex numbers and the same this way but the matrix is not at all symmetric but the labeling is symmetric it's given by these so-called flat parabolic connections so minus the square root of minus three whatever that means so here I'm going to have this Habero element so remember I told you that for the four one not if I took the Habero element and didn't compute it at root subunity but instead near one maybe I didn't tell you that if you computed it e to the h where h is infinitesimal then for the same reason that the nth coefficient has that nth polymer symbol so it's n terms one minus x one minus one minus Q one minus Q squared it'll be divisible by h to the n so if the nth term is divisible by h to the n the whole thing makes sense it's a power series and I think I did write it down at least I wrote down the numbers it happened to be this particular power series for this particular not that it's even as a fluke it's because it's its own mirror image if it weren't it wouldn't be okay so here I'll have this one minus h squared plus etc and here I'll have something similar so this one is in the Habero ring but it's the element of the Habero ring evaluated the element of the Habero ring you can evaluate it at root subunit but you can also evaluate infinitesimally near roots of unity for instance one or any other root of unity so then you get this so this is also something else in the Habero ring with an equally explicit formula that I won't bother to write down here we'll have the series that I started with the phi zero of h maybe you remember the phi zero of h was the one that had the 11 h over 72 squared of minus 3 plus 697 halves times h over 72 squared of minus 3 squared and so on so three of the the function we saw already both occur here but there are different positions of the matrix before they played wildly different roles and it happens that the third one usually be something else but this notice I said it's its own mirror image and so when you go to the complex conjugate here you just change h to minus h and similarly there's a similar that's called the size zero of h with similar coefficients it starts one this I think is 37 I've forgotten some other number over 2 it's exactly the same sort so what you get is a whole matrix and you might say well so what everybody can generalize we had one invariant but actually we had two we had this power series the Habero ring one and this infinite power series which were mysteriously linked but now I can show you and then comes the clue in the fast final slide and I'll stop remember before what we did is we took j of gamma x and we divided by j of x so for instance I took j of minus 1 of x and I divided by j of x and what had been a whole cloud of functions suddenly became but it wasn't a very nice function at least it was a function of a real number now I do the same but now there's a big difference when you have a matrix you can divide one matrix by the other so for instance I could take this whole matrix this 3 by 3 matrix and take it at minus 1 of x and divide it you can't put slash because the matrix you have to decide on the order you would divide j of x on the left let's say by j of minus 1 of x and so now that was the picture before when you only took one single component of the thing and then it was all messed up this was the original j but now could I have slide 4 and then I'll finish slide 4 slide 4 there we go these are the six components well a 3 by 3 matrix you'll tell me should have 9 but if you divide 2 matrix in block form it's still in block form so 3 of the components are 1 and 0 I didn't graph them because we know what they look like but the other six stoppers graph and as you can see suddenly they've all become smooth functions and in fact we actually know that they're real analytic functions with the well-defined holomorphic continuation and so that will be the story next time these functions actually become holomorphic functions they leak from the real first from the rational numbers which is where they were defined into the reals and then from the reals into the upper and lower half planes and they acquire some sort of modularity properties and that will be the main theme well that is the theme of the second lecture so this at least gives you a foretaste of things to come and I'm sorry that I went on a little bit but I usually go on even more okay so thank you for now and I hope there will be questions I've got any questions for the speaker I have one question Professor Saggy so in your lecture you see the unity of mathematics can you hear? here but I can't see can you put on your camera just for fun I don't know who's I have it I'm from Mexico coming from Mexico okay I can see you thank you I couldn't I didn't know who it was and I couldn't see you thank you so I mean in your beautiful lecture you see the unity of mathematics not theory algebraic topology k-theory et cetera one ingredient that you can measure is dynamical systems because the volume of the hyperbolic manifest also given by theorem of use is given by ergodic theory by the horocyclic and geodesic flows a theorem of mine I know what theorem you mean but that's for SL2Z that's two-dimensional this is three-dimensional but actually it can be generalized to PSL2 can be applied for the case of the figure eight knots also so you could add my only comment is that you could have added dynamic assistance in the least that you gave that's all I have to say I hope that at some point you will send me an email and say a bit more obviously I didn't talk about it because I didn't know and it's not part of this complex of ideas it's meant to be connected and not just lose things but that's absolutely fascinating it sounds very convincing that there will be a really interesting connection and I hope I certainly won't work on it because I already don't know the policy but I know dynamical systems much less but there are other people who do and I hope somebody will be inspired so thank you very much that was a beautiful comment and addition thank you now no one else will dare ask a question because you've set the bar too high any other questions for Professor Sagia so don't be shy Don I have a question on the this is Sameer hello can you see me recognize your voice anyway but I can even see you on the on the blackboard to the extreme right there's a modular type equation for J is that supposed to be an equality or an asymptotic symbol oh sorry this of course is complete nonsense it's of course asymptotic equality what I mean is and it's I mean in the paper we write out this same mantra 500 times so I think we had to tech macro as x tends to infinity through rational numbers so it's not on the reels through rational numbers and now you can put the last part in two different ways and I said them both but very quickly everything I say is very quick with bounded denominator or you can say with fixed fractional part because fixed fractional part would mean as I said you might take a thousand and a third a thousand and one and a third and so on so you just go up in steps of one and then you'll get this asymptotically to all orders of course it's not an equality these aren't exact funds that couldn't possibly be they're all in different number fields and so on but it's all orders but whether I say that I fixed the fractional part exactly and say it's a third or say it's bounded number let's say it's denominator at most five well then there are only five denominators and if it's denominator three there are only two numbers a third and two thirds and so I define it then it's the same so there's this double topology that I mentioned x is going to infinity as a real number and so half of the formula the cx plus d to the three halves the e to the constant times x and the power series in one over roughly cx plus d that part is a completely smooth function that you could draw a graph and if you drew a graph of that it would be exponentially big it would be completely smooth and then you would multiply that by a periodic function so it's not that it oscillates but whatever it does it's the same in every group of size one except it's not a smooth function it's the cloud that you saw before so if I actually draw a graph you would see the cloud but when I divide one by the other then that turns into the graph that had this behavior so thank you very much that was simply a mistake of course it's not equality it's of course asymptotic but it's only not as x goes to infinity in some rational way if you take a thousand times pi you will get nonsense it has to be either an integer or an integer by a fixed rational shift or a bounded denominator rational shift so now people have two choices they can have a brand new idea or they can point out a mistake Any other questions? Well of course we'll have a second chance to ask you in the second lecture on Friday and we would like to thank you very much all of you for attending in person and also connecting and let's thank Professor Sagir again for his very nice lecture This is a chance Thank you very much Ah, hello Ah, hello Don I wasn't really connected my mic wasn't working I had a question actually Another question, yeah I can see who it is and I can hear who it is Yeah, it's me Yeah, about maybe seven, eight years ago with Herbert Gungel we're also using from elements of local roots extract some roots in some number fields I don't remember that we had the units Are you aware of this? Actually we never wrote it Maybe we don't This little paper that you wrote with Herbert he's told me about it several times right now I think it's very related this is to do with the so-called cyclic dialogue rhythm Yes, yes If I had taken five minutes I've often given a lecture on this story with units the whole lectures about those units then I certainly would speak about the cyclic dialogue and I would have mentioned your joint I think never published paper with him and two but today I simply skipped it I said there is a beautiful story with the unit and all I said is you can associate your development of the block group but indeed that's true and it's absolutely connected Your story didn't go I think nearly as far in that you had the cyclic dialogue rhythm in its properties but what I think you didn't have is that if you take certain combinations of those cyclic diagrams that are associated onto the block groups that a miracle happens and you get units in a psychotomic extension of your original number field Yes, that's a problem we don't have So that was the discovery and I don't see how anybody could have discovered that by pure thought I don't see how you would ever expect it We didn't do it by pure thought It came out of the numerical calculations you had these numbers I mean I found that in my paper of 2010 that number that I told you the fifth root I had to calculate 300 digits of it by numerical interpolation and then identified as an algebraic number and actually it wasn't even the fifth root it was in the tenth root because I was in a smaller field I finally identified it as the product of a private norm 29 with the tenth root of a unit in the real part of the 15th roots of unity and it was like completely miraculous I had no idea why that was happening and then we worked for several years to understand where are those units coming from So that's the really subtle part of the story but indeed the main ingredient on the way is the cyclic dialog which I could write down, I won't and it's essential I can say in one word for the audience we have the Pochhammer symbol one minus q, one minus q squared and so on or one minus x, one minus qx and if you put instead one minus q, one minus q squared one minus q cubed, cubed and so on then you get this corresponding thing that's the analog, not of the log but of the dialog and this is the fact that actually several people have found in various guises and in particular Herbert Gangel and Maxime Konsevich So indeed that's true I think it was Voit Kovac maybe Sorry? Voit Kovac I think it was before Voit Kovac, I think he discovered this earlier I still didn't catch the name Voit Kovac I don't remember that I didn't even know in fact I know many of these things I didn't know that but anyway we're getting now two technologists were talking about the thing that wasn't actually in the talk I didn't mention the cyclic dialog in the middle I mean I told way too much but I sure you I skipped a lot more than I told I mean it's as I told you it's a 90 page paper I'm just giving little glimpses of little pieces So Maybe I could ask you whether this lecture was recorded So unfortunately I came in late In fact I only saw the last two minutes of your talk Yes, to my Somewhere one can see Yes, to my delight It's recorded, yes At the inter-site I'm sure I see, excellent It will be of you How many people have asked me that and I'm very happy Usually I don't like lectures being recorded because it reminds me what a mess I made but I mean one's responsible for the mess one makes but the fact is people can't always come it might be three in the morning where they are So I'm very happy this lecture was recorded and the second one, of course, also will be Driving right now So, yes So it would be great to have a link to that Yeah, it's nice to see you Hi, nice to see you You don't often see Vikings drive Nice to see you too, Don Okay So yeah, but okay I will look forward to finding the link and then I'll join for the talk on next Friday So thanks Okay Anyone else people seem to be thawing as it's turned off then they suddenly realise they have a question Are there some questions? No, seems not Well, again, thank you very much for such a nice lecture and we'll see everyone on Friday for the second part All again on Friday Well, I don't know all those who survived See you on Friday See you on Friday