This video is the first in a series of videos on Wienberg's algorithm and Katz-Mudi algebras. It's an expanded version of the Wienberg lecture I gave a couple of days ago. This is the first video in the series and will be mostly on Wienberg's paper. I plan to have later videos covering topics like John Conway's reflection group in 25 dimensions and discuss some Katz-Mudi algebra as an automatic form related to it. I'll start by introducing Wienberg's paper. Wienberg's paper is one of the papers in this book here. I'll actually try to remember to put a link to this book since you can get it online these days. His paper is this one on Arithmetical Discrete Groups in Lobochevsky Space. Lobochevsky Space is another name for hyperbolic space. I came across this paper many years ago and I noticed in the back of it there are these really intriguing diagrams. So if you look at these diagrams, well, these are just the Coxtor diagrams of finite reflection groups that I'll discuss briefly later. But down here there are some new diagrams I hadn't come across before and I wonder what they are. There are some even stranger ones here. You see these sort of rather impressive mysterious circular diagrams. So what I'll try and do in this first lecture is explain what these diagrams in Wienberg's paper is. Somehow there's something very compelling about doing deep high level mathematics just by drawing these tiny little doodles on a piece of paper. So I'll start by just summarising or reviewing some basic material about reflection groups. So here's an example of a reflection group. What we do is we take three mirrors. So you think of these as being mirrors in the plane. Here the mirrors are just one-dimensional lines because I'm working in the plane rather than in the three-dimensional space. And if you take a point here, you can reflect it in these mirrors. You get a better colour that doesn't really show up. If you take this point and reflect it, you get another point here. And then if you reflect it, you get points here and here and here and here. So you get a sort of kaleidoscope effect. And this reflection group can be described as follows. What we do is we take a fundamental domain. So this region here is called a fundamental domain. And what this means is that every point in the plane is congruent under some series of reflections to a unique point in this fundamental domain. For instance, this point here is congruent to this point in the fundamental domain. And the mirrors that bound this fundamental domain are called the simple roots or rather they correspond to the simple roots. They're not quite the simple roots. Strictly speaking, the simple roots are little vectors orthogonal to these hyperplanes. So I really ought to call this the simple root, but someone called the hyperplane the simple root. And Coxeter found a very neat way of drawing pictures of reflection groups by just drawing what is called the Coxeter diagram. And the Coxeter diagram works like this. What you do is you draw a point for each simple root. So these points correspond to simple roots. So there are two simple roots and therefore there are two points in the Coxeter diagram. And you indicate the angle between these simple roots by drawing various lines between these two points. For instance, if there's an angle of pi over three or 60 degrees, you indicate this by drawing a single line between these points. So here's another reflection root. This time I'm just going to take two mirrors that are orthogonal to each other and here a fundamental domain looks like this. And the Coxeter diagram of this looks like this. There are two simple roots here and here. So there are two points in the Coxeter diagram and there are the angle between them is pi over two. And you indicate this by drawing no lines between these points. There's actually a slight difference between Coxeter diagrams and Dinken diagrams. So the differences as follows. The Dinken diagrams keep track of lengths of the root. So if we've got a mirror here, then the Coxeter diagram just keeps track of the mirror, but there's also a root perpendicular to it or orthogonal to it. And the Dinken diagram sort of has some extra information telling what the length of this vector is. And secondly, these roots must lie in a lattice. So an example of a reflection group whose roots don't lie in a lattice is if you take this reflection group. So here I've got five planes and you get a little reflection group with 10 elements. And if you draw the roots orthogonal to these, you find that the roots don't lie in a lattice, a lattice is some sort of regular array of pots in the plane like this. So if you take the linear combinations of these roots, they're dense in the plane. So there are three sorts of reflection groups that we're going to be concerned with because there are three different sorts of geometries. So there is spherical geometry and there is Euclidean geometry and there is hyperbolic geometry. And you can classify the reflection groups in spherical or Euclidean geometry. There are actually some rather nice pictures of spherical reflection groups. So here are some old pictures in the book by Klein and Fricker on elliptic modular functions. And this is the 19th century German, very famous German book. And recently it was translated to English by Arthur Dupre. So if you want to guess, I think the AMS is now selling the English translation. So here's an example of a spherical reflection group. There it's been sort of projected into a plane. And here's another example of the icosahedral spherical reflection group. You can even make three dimensional models of the icosahedral reflection group if you want to make it look like this. Anyway, the spherical reflection groups are classified as follows. They have this rather unimaginative notation of An, Bn or Cn, Dn, E6, E7, E8, F4, G2 and then there's also In, H3 and H4. And these ones I'm not going to be interested in because these don't correspond to Dinkin diagrams. They only correspond to coxster diagrams. And here there are two Dinkin diagrams, Bn and Cn, which correspond to the same coxster diagrams. So there are some slight technical differences. If you want to see what the diagrams look like, instead of writing them all out, I'm just going to show you the pictures of them in Binberg's paper. So here is a list of the coxster diagrams for spherical geometry. The classification of Euclidean reflection groups is very similar. So here's an example of Euclidean reflection group. What I can do is I can just take a lot of mirrors that look like this in the plane. And you see there's a fundamental domain here. And now if you look at one of these points of the fundamental domain, you notice what you're getting. If you just look at the six, three hyperplanes through this point, you're just getting the spherical reflection group that we had in the previous, that we had a couple of pages earlier. So there's quite a close correspondence between spherical and Euclidean reflection groups. It's not quite one-to-one because I could take a different vertex, for example. And in this case, all three vertices give us the same spherical reflection group, but sometimes it's a bit more complicated. So for each spherical reflection group, you tend to get one or two Euclidean reflection groups, or sometimes you get no Euclidean ones. If you want to see pictures of the Euclidean reflection groups, they were also given by Binberg, and here's a picture of the Euclidean reflection groups. You can see they look quite similar to the spherical ones. So here, for example, is a spherical reflection group called F4, and here is a corresponding Euclidean reflection group called F4 with a twiddle on it. And you see that this diagram is the same as that one, except you've added this extra node at the end, and all the others are kind of similar. So you get a Euclidean one by adding one extra node to a spherical reflection group. Klein and Fricker also have a few pictures of some of the Euclidean reflection groups, if I can find them. So here are Klein and Fricker's pictures of three of the Euclidean reflection groups. So I'm just going to give two examples of Euclidean, or no, sorry, of spherical reflection groups that I'm going to need later. First of all, we can take the lattice In, and the lattice In is just the same as z to the n, so it's all vectors m1, mn with mi, just an integer, and it has the obvious in a product. So the norm of a vector, that's the square of the length, is going to be m1 squared and so on, plus mn squared. So you just do the most obvious thing. And then if you look at the, you can ask what are the, what are the simple roots of this? Well, first of all, we have to figure out what the roots are. Well, a root is going to be a vector v, so a vector r, such that reflection in the orthogonal complement of r takes In to In. And what does reflection do? Well, it takes a vector v to v minus 2rv over vv times v. So it's quite easy to check. I mean, this is obvious for vectors that are orthogonal to r, and it's also obvious for r, so it's true for all vectors. And if you notice, we want this to be in the lattice we first started with, and this will always be true, provided this bit here is an integer. So we really want vv to be equal to 1 or 2. And for more complicated lattices, it's possible to have reflections where vv has more ratio than 2. But I'm mostly going to be doing what are called unimodular lattices, which means the volume of a fundamental domain is just 1, and these are the easiest ones to deal with. So we can now try and find the roots of In. So we just want vectors of norm 1 or 2. Well, there are some obvious vectors of norm 1. We could just take a lot of zeros and then have one vector that's plus or minus 1. So these are norm 1. Next, we could have vectors of norm 2. So we take two entries that are plus or minus 1, and these give norm 2. And now we can ask, what are the simple roots of a fundamental domain? And this isn't terribly difficult to work out. So in two dimensions, the reflections look like this, and you can see for a fundamental domain, there are going to be two roots like that. And in general, the roots look like this. We can take norm 1 root, which looks like that, and then a norm 2 root that looks like this, and another norm 2 root that looks like this, and we go all the way down to this root here. And if you draw the corresponding Coxter diagram or Dinken diagram, rather it looks like this. We draw a point for each simple root, and all of these ones have inner product minus 1, which means we draw them by a single line, and these ones have... the angle between them is now pi over 4, which we indicate by drawing a double line, and a Dinken diagram, you draw a little inequality sign to point towards the smallest root. So the Dinken diagram of this is just a line with a line of points with something funny going on at one end of it. So the other example we're going to use is the famous E8 lattice. So the E8 lattice consists of the following vectors. It consists of all points m1 up to m8 in eight dimensions with either all the mi in z, or all the mi in z plus a half. So we have one of these two conditions, and we need an extra condition for some of the mi is even. So roughly speaking, this condition throws away half the vectors, and then this condition adds in the same number of vectors again. So this is sort of the same size as i to the n, except it's got this following extra property that the norm of all vectors is even. You notice this, if you take a vector where all the entries are a half, say a half up to a half, then the norm is a half squared times 8, which is 2, which is now even. So this is something funny that happens whenever the dimension is divisible by 8. You can do this sort of trick, and the norm of all vectors is even. If the dimension wasn't divisible by 8, then the norm of this vector would not be an even number. So we can ask how many roots does it have? Well, there are two sorts of roots. First of all, we can take two entries that are plus or minus one, and you can see there are 112 of these. Or we can take plus or minus a half, or up to plus or minus a half, and there are 128 of these. You might think there are 256, because there are eight choices for sign, but then we've got this condition that some of the vectors has to be even, which cuts down half of them. So all together there are 240 simple roots. And the Coxtor diagram or Dinker diagram looks like this. So you take 1, 2, 3, 4, 5, 6, 7 vectors like that, and then you have one extra vector there, and the vectors are like this. This one here could be the vector 1, minus 1, 0, minus 1, 1, 0. This one could be the vector 0, minus 1, 1, and this one has a lot of plus or minus a half, which I can't be bothered to work at, where all the signs go, and you can fill in the rest for yourself. We can also ask what is the corresponding Euclidean reflection group, which looks like the spherical E8 reflection group, except you add in translations of the lattice, and this is the effect that you add in one extra simple root onto the end, like this. So this is the Euclidean E8 lattice, and this is the spherical E8 lattice. And this diagram contains an awful lot of information. In fact, you can do some quite difficult calculations just by putting your finger over nodes. For instance, we can take the following question. Suppose you take the E8 Li algebra, whatever that is, something with corresponding to this, and ask what sub-algebras does it have? Well, you can find sub-algebras of the E8 Li algebra, or at least some of them, by taking this avineate Dinkin diagram and just putting your finger over various points. So if I put my finger over this point here, I see I've got a line of eight points, which is the An Dinkin diagram. And the An Dinkin diagram is just the Dinkin diagram with the Li algebra SL9 of R. So we've discovered Li algebra SL9 of R is contained in the Li algebra E8 just by this trivial operation of hiding a point. And similarly, you can find other sub-algebras of the E8 by putting your finger over other points. For instance, here we've got... that's actually the E6 Dinkin diagram. This is the A2 Dinkin diagram. That gives you another sub-algebra of the E8 Li algebra. So this diagram contains an enormous amount of high-level information, slightly hidden in it. Another example of a hyperbolic... Now we move on to hyperbolic reflection groups. So let's have some examples of this. Well, first of all, here's a rather nice picture in Klein and Fritter's book. So this is the fundamental... This is part of a hyperbolic reflection group in the hyperbolic plane. All these curved lines here are really straight lines in the hyperbolic space. They sort of look curved because if you embed hyperbolic space into Euclidean space, you don't really have enough room to do it properly. So the straight lines end up curved due to some sort of optical illusion. The fundamental domain for this reflection group consists of these little triangles here on the sides that are pi over 2, pi over 3, and pi over 7. The artist Escher also used hyperbolic space in some of his work. I think he discovered this by Thornton de Coxtor. So here's a typical example of one of Escher's pictures of hyperbolic space where he's sort of a tessellated hyperbolic space with a lot of fish. And again, if you look carefully, you can see there are some reflection hyperplanes.  And if you look very carefully, you can see that's a sort of... you can sort of reflect in that. The picture looks as if it's getting... as if the fish are getting very, very small near the boundary, but they're not really. That's just another optical illusion. You should think of all these fishers as really being the same size. And it looks as if it's getting really hot and crowded around here, but it's not at all. Again, it's just... it's just actually really no more crowded down here than it is up there. Another example of a hyperbolic reflection group which you've seen if you've covered modular forms is the group GL2 over the integers. So this is just the set of all points A, B, C, D. There's a 2 by 2 matrices. So A, B, C, D are integers. And A, D minus B, C is equal to plus or minus 1. And this acts on the upper half plane H which consists of all complex numbers tau x plus iy with y greater than 0. And it acts by A, B, C, D, tau equals A tau plus B over C tau plus D. Except you've got to be a little bit careful because tau might end up having negative imaginary parts. So what we need to do is identify tau with its complex conjugate. And then you get an action of GL2Z on the upper half plane. And you can ask what does the fundamental domain look like? And it looks like this. And what you do is you take... this is going to be the real line and this is going to be the complex line.  And the fundamental domain now looks like this. And if you want to know what its cox to diagram looks like, it again looks like this. So there are three bases of this fundamental domain. So we have three points. Look at the angles. The angles here are pi over 2, pi over 3 and there's something funny going on here. So we draw a single line between these because there's an angle of pi over 3. I should say which of these points corresponds to which point. Well, this wall here is going to correspond to this point and this wall here will correspond to this point and this wall will correspond to that point. And now we've got to do something about this angle here and there's really no good convention for it. Sometimes people indicate it by drawing a very thick line like this and other people do things like draw a double line or something. There was no fixed convention. So here the points meet at infinity. Sometimes you can get mirrors in hyperbolic space that don't meet even at infinity and then you need to think of something even more complicated to draw here and again there's really no good convention for this. This reflection, by the way, corresponds to taking tau goes to minus 1 over tau, a very famous Jacobi's imaginary transformation. By the way, if you've done modular forms you've probably seen a slightly different fundamental domain that looks a bit bigger. So if you add in this, the fundamental domain for SL2 looks like this. So SL2z just as AD minus BC equals plus 1. The problem is that the fundamental domain for SL2z is not the fundamental domain of a reflection group. These walls here are not reflections for SL2z. So if you want reflection groups, you have to use GL2z rather than SL2z in hyperbolic space. In high dimensions, there turn out to be absolutely massive numbers of examples, even in dimension 2. What I'm going to do is look at a particular series of examples studied by Wienberg, which are given by automorphism groups of Lorentzian lattices. So let me explain what these are. So Lorentzian lattice is a lattice, except instead of being contained in Euclidean space Rn, I'm going to have it contained in Lorentz space Rn comma 1. So this consists of all the points x1 up to xn, and then I put a vertical line for xn plus 1 to warn you there's something different going on. And the norm is going to be x1 squared all the way up to plus xn squared minus xn plus 1 squared. So you need to pay careful attention to this minus sign here, which makes everything a little bit weird. By the way, there are two conventions for Lorentzian space. You can also use R1 comma n, where you have one plus sign and n minus signs in the metric. And physicists are divided into two groups of people who won't speak to each other, one of whom uses this convention and the other of whom uses this convention, because this is, of course, the spacetime in special relativity. I'm going to use this convention because it's a little bit easier when you're talking about lattices because it will make various lattices Euclidean. In fact, if you go further into the theory, you find there are very strong reasons for using this convention instead. And unfortunately, this means I constantly get confused about which convention I'm using. So there are probably going to be a few sign errors later on in the talk. So how does this tie up with reflection, metabolic reflection groups? Well, let's draw a picture of Lorentzian space. Well, what you do is you draw a picture of the vectors v with v, v equals zero, and these form a famous double cone. So this is the light cone. You're doing special relativity. These are the vectors in momentum space that light travels along. And then we can have vectors r with r squared greater than zero, and these are space-like. So these sort of something to do with points of space. And then we have vectors of negative norm. So suppose I take the vectors of norm minus one. So the vectors of norm minus one are going to form a two-sheet of hyperboloids. There's one sheet up there and the other sheet down there. And what you do is you throw away one of these sheets and fix this sheet here. And it turns out to be a copy of hyperbolic space Hn. And the metric on hyperbolic space is very easy. You just take the metric of Euclidean space, except it's a kind of funny Lorentz metric where you have a minus sign here and just restrict it to this hyperboloid.  And that just makes this into hyperbolic space. The other nice thing is if we take a vector V of positive norm here and look at the hyperplane orthogonal to it. So we get some sort of hyperplane here. So this is going to be bar perp. Then this is a mirror. We can reflect in it and this gives us a reflection of hyperbolic space. And we want it to preserve a lattice so we usually want r squared to be one or two.  What happens if you take r squared equals minus one or minus two? Well, the formula for reflection still works but what it does is it takes this copy of hyperbolic space to this other copy of hyperbolic space. So it doesn't quite give you a reflection of hyperbolic space. What it gives you is a sort of funny sort of automorphism of hyperbolic space where you fix a point and kind of invert in that point and map everything to minus itself. Anyway, these sorts of involutions give you reflections and these ones don't although they're still quite interesting. And we can ask what is the automorphism group of L? Well, what is L? L might be, for example, the lattice. It might be the lattice i n comma one which consists of all points m1, mn, mn plus one with mi in the integers and the norm, of course, is m1 squared plus mn squared minus mn plus one squared. So this will, and the automorphism group is, it has a reflection group inside it and this reflection group has some sort of coxster diagram. All right, I should really call it a Dinkin diagram. And we can also look at automorphisms of the Dinkin diagram and, in fact, we get a semi-direct product of a reflection group and the automorphisms of the Dinkin diagram and this is almost the same as the automorphisms of i n comma one. It's not quite the same, for instance, there's a factor of two coming from the fact that the automorphism minus one also acts on i n, but the point is if you find the reflection group and the Dinkin diagram, you've pretty nearly found the automorphism group of i n comma one. And there's a second lattice that Wienberg examined, which is the lattice, this is two i's indicating that it is an even lattice and this is constructed in a similar way to what we did for the E8 lattice. So we're going to take m1 up to mn mn plus one with all the mi in z or all the mi in z plus a half and we also want some of the mi to be even. And this works if n is congruent to one modulo 8 and if n is congruent to one modulo 8, the norm are always even. So this is called an even lattice because the norm of every vector is even. So what Wienberg did in his paper that we're going to go through is work out the automorphism groups of these two lattices for certain small values of n. So to do this, we used Wienberg's algorithm. So what's Wienberg's algorithm? Well, Wienberg's algorithm is very simple in principle. What you do is you take hyperbolic space.  So here are three reflections and here's a fundamental domain. And what Wienberg's algorithm does is as follows. First of all, you pick a point P in the fundamental domain. And now, so step one is you pick P2. We find the walls of the fundamental domain in order of the distance from P. And that's it. That's more or less what Wienberg's algorithm is. Of course, there are some details to fill in, like how do you find the walls in order of their distance from P? And for this, you need the key point. The angle between walls is always at most pi over 2. And what this means is if you're finding the walls in terms of distance from P, suppose you found walls A1 up to An, then this gives you a constraint on An plus 1. So you have to kind of use these constraints to narrow down the possibilities for the next wall. The reason why the angle between the walls is at most pi over 2 can be seen if suppose you've got two walls that look like this and suppose you've got an angle here which is bigger than pi over 2. Well, if it's bigger than pi over 2, then there's going to be another wall like that, which means that this wasn't really an angle of the fundamental domain. So that's what Wienberg's algorithm is. We're just going to find all the walls in order of the distance remembering that the angle is at most pi over 2. Now let's translate this. Let's look at Wienberg's algorithm for lattices. So that was Wienberg's algorithm for hyperbolic space. We've now got to translate this into the language of lattices. So let's see what happens. Well, so here's our Lorentzian lattice and here's its light cone. And we've got to pick a point P. Well, Lorentzian space, sorry, hyperbolic space is going to be, say the norm minus one vectors. And what we're going to do is we want to pick a point P in here. And this is just going to be a vector, vector P with P squared is less than zero. So it's going, we have to pick some time like vector. And we then need to find the reflection hyperplanes. Well, as we saw in the previous slide, the reflection hyperplane is just going to correspond to a picking a point R with R squared greater than zero. And of course we want, in fact, we want R squared equals one or two because we're working in a union modular lattice. And now we want to pick them in order of the distance of P from R. So what's the distance from P to R perp? Well, I don't really care what the exact formula from the distance is. The key point is that it increases with this thing here. First of all, we normalize R so that it has norm one. And then you just take it in a product with the vector P. And as I said, there's a formula for the hyperbolic distance involving this, which I don't really care about. All we need to know is that the distance increases with this vector V. So what we're going to do is we're going to find all the reflection hyperplanes by picking points R in increasing order of this value here. Now, what's the condition that the hyperplanes have angle less than or equal to pi over two? Well, this turns out to be the condition that R1, R2 have inner product less than or equal to zero. So all the conditions of Wienberg's algorithm can be written very neatly in terms of this inner product of Lorentzian space. So let's see how this works for IN1. So step one is we need to pick our vector P. And we're just going to take P to be the most obvious possible vector of negative norm, which is zero, zero, zero, one. So P squared is minus one. And next we find the roots with Rp equals zero. And this is just going to be the reflection group of IN. And notice we actually have to make a sort of choice here because we need to pick one of the fundamental domains containing P, which is like picking one of the fundamental domains of the reflection group of the lattice IN. And remember, this is the thing with Dink and Diagon B, M. And we can just pick the simple roots as follows. We have minus one, naught, naught, one minus one, naught, all the way up to naught, naught, one minus one. And as you remember, this gives us a Dink and Diagon that looks something like this. So the step naught of Binberg's algorithm, we just get the B and Dink and Diagon. And then let's start with N equals two. Well, for N equals two, we only get one extra root, which is this one here. And this is norm one. And the Dink and Diagon we get looks like this. So here you notice that we've got the B and Dink and Diagram. And we've got one extra root. So this is the extra root here. So if we go back to Binberg's paper, you can see that this is the first of these mysterious diagrams he had. Now let's try N equals three. In this case, the root we found here actually dies off. And instead we get this root here, which now has norm two. And the Dink and Diagon we get now looks like this. First we get this bit here, which you notice, of course, is just the B three diagram. And then we get one new vector, which looks like this. There should probably be a crow on it. And this vector actually turns out to be stable in the following sense that as we increase N, we just keep it. We always get a simple root that looks like this. So sometimes simple roots kind of get killed off and sometimes they turn out to be stable and you're keeping for all N.  This root kind of turns into this root in some sense as you increase N from two to three. So what this means is the Dink and Diagram is always going to look a bit like this. There'll be something going on then. We always get a piece that looks like this from BN and we always get this one extra root here. So, and for a while, that's all that happens. Again, if you look at Binberg's paper, you can see that for the next few values of it here is five less than or equal to N less than or equal to nine and nothing much new happens. However, when you get to N equals 10, something changes. This time you get a new root which looks like this. One, two, three, four, five, six, seven, eight, nine, ten. It's very easy to lose track of how many of these we've got. Three. Now you see this is norm one because we have 10 times one minus three squared, which is one. And this means the Dink and Diagram now looks like this. I hope I've got the right number of bits on them. So we now get an extra root here. So this was the root with three ones in it that we had before. And there's an interesting new phenomenon we have here is that the number of simple roots is greater than the dimension. And this means in particular that these roots are not linearly independent. So up until now, all the roots were linearly independent as happens for spherical reflection groups. But now we've got this new phenomenon. So for N less than 10, the fundamental domain is in some sense some sort of simplex. But from now on, the fundamental domain starts to be more complicated. And again, this root turns out to be not quite stable. But if we go to N equals 11, we get one, two, three, four, five, six, seven, eight, nine, ten, eleven, three. And this is now stable. We retain it in all bigger dimensions. So for N equals 12 and 13, nothing much new happens. We get a diagram that looks a bit like that. And for N equals 14, we get yet another new root, which looks like this. I'm going to put some dots here because I need to lose track of how many ones I've got. So now we have a root that sort of has a four here. And this is norm one. And if we look at its Dink and Diagon, which I'm just going to use Lindberg's paper for because I get it wrong otherwise. So here's the diagram for I-14-1. We notice that we've got this extra phenomenon that the Dink and Diagram now has a symmetry of order two, because we can just flip it like that. So let's say there's the Dink and Diagram has symmetry. And from now on, the symmetry tends to increase as you make N bigger and bigger. We usually get more and more symmetries of the Dink and Diagram. And this goes on up to N equals 17. So here's this diagram for N equals 16. And you see this time the symmetry group now has gone up to order four. And here, if you look rather carefully, you see the symmetry has gone up to order eight, I think. And at N equals 18, Binberg and Kaplan-Scheier gave up doing it by hand because the number of roots starts to increase rather rapidly and used a computer. So for N equals 18 and 19, there's the following interesting new phenomenon. At about this point, the Dink and Diagram is almost transitive on simple roots. It's not transitive on the simple roots. These are the roots of the Dink and Diagram for two reasons. First of all, some roots of norm one and some roots of norm two. This is the obvious reason why it can't be transitive on them. The roots may have different norms. There's a second more subtle reason. The roots may be parity vectors. And this phenomenon is going to turn up a little bit later in some of the later lectures. So I'll explain what this is. So a parity vector is a vector R such that Rv is current to v squared. So vv mod two for all vectors v. So an example of a parity vector is just a vector that's all ones. Or more generally, if all the entries are odd, then you can see that R has this condition here. Now we want R to be, have norm one or two. So I suppose we take R to be this vector here. So I'm going to have a lot of ones and then a three, sorry, and then a three here and then a five here. So this is norm one if there are 17 ones here. So R squared equals one. Now if we want all the things to be odd here and we want the norm to be one, this can only happen if n is congruent to two modulo eight. Because if a number is odd, then it squares always one mod eight. So there's a modulo eight condition occurring here. So if we want norm one roots that are parity vectors, then n has to be congruent to two mod eight. Similarly, if R squared equals two, then this corresponds to n being congruent to three modulo eight. And you notice 18 and 19 are congruent to two and three mod eight. So for these two values of dimension, we sort of unexpectedly get two sorts of simple roots of norm one or two sorts of simple roots of norm two. Now we get to n equals 20. And this is where there's a really big change in the behavior. And here it turns out there are an infinite number of simple roots. So Vinberg's algorithm kind of breaks down. Well, it doesn't break down. It just goes on forever. I mean, it will find all the roots, but it will just take an infinite amount of time to do so. Furthermore, the symmetry group of the Dinkin diagram is infinite. So I'm going to give several explanations of why things change so much at n equals 20. Let's first of all give Vinberg's explanation. So Vinberg's explanation for why this happens is there are lattices in dimension 19 with determinant equal to one, such that the root system that's called lattice L has rank less than the rank of L. And there are two such lattices. And the root systems have Dinkin diagrams of A11, D7 and E6 cubed. So these root systems have rank 18, but the lattices themselves have ranked 19. So they're unimodulated lattices whose vector space is not actually spanned by their roots. And Vinberg showed that whenever you get a lattice with this property, then it causes an infinite symmetry of the Dinkin diagram. Roughly speaking, if you take the lattice, modulo the root lattice, then this tends to act on the Dinkin diagram of a corresponding reflection group. And the point is if you've got a lattice L, then you find that L is actually equal to omega bar modulo omega, or some omega squared equal to some norm 0 vector in I 19, sorry, I 21. So this is going to be a 19-dimensional lattice. And by sort of staring at this, you can see that the lattice modulo, the root lattice, actually acts as symmetries fixing this vector w. So if this group is infinite, then the Dinkin diagram turns out to be infinite. So that's as far as Vinberg went in for the odd unimodulated lattices. He worked out the Dinkin diagrams up to 19 and showed that in dimension above 19, the Dinkin diagram actually becomes infinite. He also did the case of even lattices. So we have I 9 1 and I 17 1, and I guess I 25 1. So let's look at these three lattices. Well, for this lattice, what we do is we can write this as E 8 plus this little two-dimensional hyperbolic lattice with inner products like that. And that means if we start Vinberg's algorithm, we start off with the Dinkin diagram of E 8. Just as for the odd lattice, we start off with the Dinkin diagram of B n. And in fact, we get not only E 8, but if you think about it, we actually get an E 9. So this is the E 9 Dinkin diagram of the Euclidean reflection group. And then if we apply Vinberg's algorithm, we find there's just one extra vector, which is this point here and here. This is for obvious reasons. This is usually called the E 10 Dinkin diagram. And then Vinberg showed that that's the only extra root you get. So this is Vinberg's Dinkin diagram for this lattice. Now for the 18-dimensional lattice, something rather similar happens, except we start with 1, 2, 3, 4, 5, 6, 7. We start with two copies of E 8 because we can write this lattice as E 8 squared plus 1, 1, minus 1. So as before, we get affine E 8 squared. And then there's one extra point here. And now if we look at this diagram here, we can see several rather interesting things in it. In particular, we can see all the even unimodal lattices of dimension 16. So one even unimodal lattice of dimension 16 is E 8 squared. And we see there's the affine Dinkin diagram of E 8 squared if I put my finger over that. But there's a second even unimodal lattice of dimension 16. And we can see it if I put my finger over these two vertices here. And if you look at this, you will see it's actually the affine D 16 Dinkin diagram. You remember the affine D 16, affine D and Dinkin diagram looks like this. So in fact, the affine Dinkin diagram of any even unimodal lattice of the right dimension is actually contained in the Dinkin diagram of this Lorentzian lattice. By the way, the same thing works for odd lattices. You may think, well, you can now classify unimodal lattices just by writing down these Dinkin diagrams and spotting affine Dinkin diagrams inside this. Unfortunately, this doesn't really work. The trouble is, except in a few cases like this, the Dinkin diagram just turns out to be too complicated. I mean, as we saw before, it's actually infinite. So good luck writing it down and trying to find all the affine Dinkin diagrams inside it. And in this case here, Wienberg showed that the Dinkin diagram is infinite. Well, for odd lattices, Wienberg showed the diagram was infinite by finding a unimodal lattice whose root system is a smaller rank than that of the lattice. And in 24 dimensions, there's a very famous lattice with this probably called the Leach lattice. It's unimodular and it's just no roots at all. So in particular, the rank of the root system is less than the rank of the lattice in a rather drastic way. The rank of the root system is actually zero. So for this root system, we can carry out Wienberg's algorithm just before. And in fact, we get the same thing as before. We get three copies of E8 and an extra point here, but we get lots of other points as well. In fact, an infinite number of them. So that's the summary of Wienberg's paper. We calculated all the Dinkin diagrams of the reflection groups of unimodular rents in lattices in the cases when they were all finite. So the next lecture, I'm going to talk about Conway's work on this lattice here. So Wienberg showed the reflection group was infinite and Conway came up with this absolutely spectacular calculation because he managed to work out what the Dinkin diagram of this is and it turns out to be more or less the Leach lattice.


I have to start with an apology. I don't know quite as much about gynomethematics as some of the other speakers at this conference. As a matter of fact, I'm not even going to speak about any particular gynomethematical texts at all. I'm going to talk about a bramanical mathematical text but as I will argue later, this particular text engages with another mathematician who is a gyn. And I will try to analyze the approach of the author I'm going to present to you. Now, this author is called Bhaskara. He is the first to write a commentary on the famous text by Arya Bhatta that has been mentioned earlier today. Arya Bhatta wrote his work probably in 510. There's some disagreement about that. Others think it was 499, but it was around the year 500. And Bhaskara, his commentator, he finished his work, his Bhashya, his commentary, the Arya Bhattiyya Bhashya in 629 of the Common Era. So that is some 120, perhaps 130 years later. Now, both Arya Bhatta's text and Bhaskara's commentary on it deal in part with geometry. So unlike some of the other speakers, I will not talk about the knowledge of numbers and the manipulation of numbers. I will talk about the parts in these texts that deal with geometry. And both these texts, they present a number of geometrical theorems. Now, Bhaskara's Bhashya, he also provides diagrams and exercises. In one sense, it's the earliest really surviving, really mathematical treatise, and it addresses a public of people who have to learn mathematics. There are exercises, there are diagrams, it's quite amusing to read. However, neither of these two texts of Arya Bhatta and of Bhaskara, they ever provide proofs for their theorems. This has given rise to quite a discussion among modern scholars. Why are there geometrical theorems which without any proofs being given, you cannot imagine in Euclid's geometry any theorem to be given without a proof. So people have often supposed that the proofs were there but they were simply not provided because they were orally transmitted. This assumption is demonstrably incorrect. At least for this case, I don't talk about other texts that I don't know half as well as this particular one. Both the texts by Arya Bhatta and the one by Bhaskara, they contain some theorems that are false, they are not correct. They were false in the form they were given by Arya Bhatta around the year 500, as we saw, and they were still false in Bhaskara's text in 629. They had apparently been handed down in this erroneous form for some 130 years and no one during these 130 years had ever noticed that they were mistaken. This would have been impossible if generations of students had bent over the proofs of these theorems. Discovering the mistakes would have been a matter of routine and Bhaskara's Baasje would not have repeated incorrect theorems. If you are interested, the two theorems, one of them is about the volume of a pyramid. Bhaskara says it is the surface of the bottom times half the height. And of course you know from school geometry it is a third of that. Bhaskara didn't know, Arya Bhatta didn't know and no one in between found out the mistake. And there is another one, I won't enter into details. It is perhaps not surprising that Bhaskara was not critical with regard to the text he had received from Arya Bhatta. In his comments on the first chapter he states, for example, that all knowledge derives from Brahma, Arya Bhatta pleased Brahma on account of his great aesthetic practices and could then compose for the wellbeing of the world his texts. That means he had not reasoned it out, he had received it from Brahma and Bhaskara, 130 years later, would not disagree with Arya Bhatta because the origin of this knowledge was from higher up, so to say. Elsewhere Bhaskara attributes superhuman qualities to Arya Bhatta, he calls him attindriyarte darshin, seeing things that are beyond the reach of the senses. Somewhere else, he quotes verse from earlier non-identified astronomical text and calls it smrti, that means sacred tradition. All this confirms that Bhaskara handed down a tradition that he looked upon as authoritative and more or less sacred and not a body of doctrines that could be improved upon by critical reflection. Clearly, independent critical reasoning with the possible result that a respected authority is shown to be wrong could have no place in his work. Now, we can contrast this with what we know about the history of Indian philosophy. Indian philosophers were not indifferent to proofs. They used proofs in their treatises and theorized about it. Clear ideas about what constitutes valid proof were well in place at a time when Bhaskara wrote his Arya Bhatiya Bhashya. Don't forget that this is in the seventh century. Dignaga, Dharmakirti, all the Nayayikas who had thought about the nature of proofs they had lived before him. The question is therefore, why was Bhaskara not interested? Why does he not give proofs? To find an answer, we must stay a little longer with the philosophers. Their interest in proofs and in logic in general has to be understood in the specific circumstances in which Indian philosophy developed. The history of Indian philosophy can at least to some extent be described as the history of different schools of thought that confront each other. The opposition between Buddhists and Brahmins is particularly important here, but it is not the only one. Philosophers try to defend their own positions and show that the positions of their opponents were incoherent or worse. In these confrontations, which may sometimes have taken the shape of debates, sometimes public debates, logic, and proofs were vital. Indian mathematicians at the time of Bhaskara did not normally find themselves in a situation of confrontation. They did not normally have to prove that others were wrong and only they themselves right. Debates about mathematics may have been rare or even nonexistent. Recall in this context that Buddhists did not practice mathematics. It's one field of knowledge where Buddhists had withdrawn from mathematics, astronomy, and astrology. And they had left this task essentially to Brahmins. I've explained that in a book that has come out a few years ago. Mathematicians as a result did not feel threatened and they did not have to prove that their theorems were correct. The result of this situation we have seen. Some of Bhaskara's theorems were in actual fact incorrect, but no one over a period of one and a half centuries and perhaps longer ever noticed. There is one important exception to what I have just said. In one passage, Bhaskara discusses an opinion with which he disagrees. A central point of disagreement is the value of pi and Mr. Shah earlier has pointed out that most giants and also, in fact, also many others believed that the value of pi is the square root of 10. Bhaskara did not agree. So Bhaskara's opponent, whom he quotes, holds that it is the square root of 10. Pi is the square root of 10. Bhaskara disagrees and he criticizes that but not without quoting his opponent and he does that in the line in Prakrit. Now, since Brahmins would never use Prakrit to my knowledge for in their writings, the opponent was not a Brahmin. He was not a Buddhist either and I conclude that the opponent of Bhaskara in this case was a giant and the fact that it was about the value of pi being the square root of 10, that makes it all the more plausible because as Mr. Shah pointed out earlier, the square root of 10 is the giant value of pi, not only giant, to be sure. There are, in fact, many examples of texts that use this value and I repeat, not only giant text but also others but I will not enter into these details. Now, Bhaskara doesn't accept it because he thinks that this value has not been demonstrated. He himself has a wonderful value for pi. I don't remember by heart but it is exactly but it is very close to what in modern mathematics we consider the value of pi. The interesting part follows next. So Bhaskara opposes the view that the value of pi is the square root of 10. And then in order to criticize this, he quotes a theorem of his opponent and this theorem concerns the length of an arch. Now I'll try to show the picture. Is it a PC that I should press, press? Ah, there it is. There it is. As you know, it concerns this part of the circle and it is like a bowl, this is the cord and this part is the arrow, the sagitta. And this opponent of Bhaskara has a formula to calculate the length of the arch. Once you know the value of the sagitta, of the arrow and of the cord. Now Bhaskara rightly points out that in a particular case where the arrow has a certain value, the cord has a certain value, if you apply that formula, you get the length of the arch which is shorter than the cord, which is totally bizarre, the arch than the cord. And so the formula which leads to a result in which the arch is shorter than the cord, of course, is totally incorrect. But interestingly, Bhaskara formulates that in a way as if that is the result of the incorrect value of pi. In fact, it isn't. You can take another value of i, pi, take the same formula and you still get a wrong result. You can even take the value of pi that Bhaskara himself presents, put it in the formula and still be shorter than all that in a criticism of an opponent and why is this opponent criticized because he thinks pi is the square root of 10. So here for once, we have a situation where Bhaskara does disagree with an opponent. But and he attacks the words of this opponent, but he also shows that he has no idea of logic. He mixes up the thing. He criticizes, he says, the wrong result is a result of pi being a square root of 10, which is wrong. But in fact, it is the result of a formula which would give wrong results wherever and whenever you would apply it. So I will not give you all the details and the exact formulation he chooses. You have to take that for the time being on trust. And of course, it's easy to verify what I say because I want to draw some conclusions from what I've said so far. What does it teach us? I think it shows that the same Bhaskara who could not discover mistakes made in his own tradition has no difficulty finding mistakes in a tradition different from his own. And this tradition different from his own in this case appears to be a giant tradition of geometry. However, it looks as if Bhaskara's criticism of the giant tradition contains a logical error. The sutra, that means the theorem he criticizes is no doubt incorrect. But this fact has nothing to do with the value that the giant attributed to Pi. Whatever value one attributes to Pi, the sutra will always lead to an absurdity. It is unlikely that the contemporary philosopher of his time, trained in logic, could have made such a mistake, really an elementary mistake. Now having made a comparison with the history of Indian philosophy, it is almost a pity that there were apparently so few occasions for confrontation between Bhaskara's and other traditions of geometry in classical India, at least at that period. If there had been, each of these traditions might have developed a deeper critical sense first with regard to others, ultimately also with regard to themselves. The outcome might have been positive for all. For one thing, more self-criticism might have induced Bhaskara and no doubt others to critically evaluate the theorems that had been handed down to them and ask themselves why they should accept them. Incorrect theorems might in this way have been weeded out. In reality, no such thing happened, not at least at the time of Bhaskara. We do not know whether mathematicians in the giant tradition cared about Bhaskara's criticism. We have seen that they continued to assign the value square root of 10 to pi, they did it for a thousand years to come after Bhaskara until many centuries after him. The problem that Bhaskara found with this value, as we have seen, was linked to a rule about the length of the arc of a circle. Interestingly, many giant mathematicians in works that have survived, they use an altogether different rule to calculate the length. I cannot project it, but in fact, Mr. Shah predicted one of these rules right here an hour ago. And that is a rule in which, as Mr. Shah pointed out, the outcome is very close to the correct value of the arc. But that is not the formula that Bhaskara is criticizing. He criticizes the formula that is quite different. And this other form that we find in so many giant texts is not open to Bhaskara's criticism. For here, if you apply it, the arc turns out to be certainly longer than the chord. And well, this better formula had already been known by Umaswati. And of course, it was known later to mathematicians like Mahavira, Arya Bhattattu, and there was a second one of the same name, Shri Patti, and a number of others. I don't need to go into the details. I conclude, it seems clear that geometry at a time of Bhaskara, and that is Bhaskara one, the one I've been talking about, was practiced in different schools. To some extent, these schools were independent of each other and concentrated on their own traditional teachings. If these traditional teachings were incorrect, they were preserved without anyone being aware of it. Criticism only came into play where representatives of different schools confronted each other. Only in such cases was there place for criticism. Unfortunately, such confrontations were, at a time of Bhaskara, too infrequent to allow a more general atmosphere of criticism to become part of the traditions. Thank you for your attention.


far-right propagandist Tucker Carlson released the multi-part series on the January 6th insurrection titled Patriot Purge and This is an infill wars like documentary that is Unapologetically unhinged so I'm not gonna talk that much about this documentary But I think you can put two into together It's Tucker Carlson and he's trying to do revisionist history about a piece of history that we all just witnessed Just this last year now political facts bill McCarthy does a write-up on the documentary and he fact-checks some of the most absurd claims made by Tucker Carlson and here are some of the claims that Tucker Carlson made so he claims that the government constructed a narrative about the event specifically to strip Trump supporters of their constitutional rights he downplays the role of far-right militia groups he blames the left because of course and He also speculates about whether or not it was a false flag I mean it's blatant propaganda But apparently it was a bridge too far for some individuals that work at Fox News because apparently the months of Misinformation about COVID-19 vaccines and the great replacement theory that wasn't too much But this is where they're finally drawing the line All right better late than never but you know for me I would have been really upset years ago with Tucker Carlson But again credit where it's due so to Fox News contributors actually quit over this with one saying that This was the last straw now the two contributors that quit Steven Hayes and Jonah Goldberg. These were regulars on Brett Baer show Now what's interesting is that? Other folks at Fox News like Brett Baer They too were mad that Tucker Carlson released this and they actually complained to management at Fox News So as Colby Hall of Mediaite explains Fox News hosts Brian Kilmeade on his radio show questioned his colleague Brett Baer on Whether he has an issue with Tucker Carlson's controversial series Patriot Purge The question from Kilmeade followed the resignations of two Fox News contributors Jonah Goldberg and Steven Hayes We've long been regular contributors to Baer's show special report They both said Carlson's Patriot Purge series about January 6th was the last straw a subsequent report said that Baer and Chris Wallace had objected to Fox News executives about the conspiracy laden series from the prime time host and PR's David Falkenflick Reported the terrific scoop veterans figures on Fox News's side Including political anchors Brett Baer and Chris Wallace shared their objections with Fox News media CEO Suzanne Scott and its president of news J Wallace those objections rose to Lachlan Murdoch the chairman and CEO of the network's parent company Fox Corporation through a senior spokeswoman Scott and Wallace declined comment Murdoch did not return a request for comment through a spokesman Now I just want to pause before we get to Baer's response to Kilmeade's question I mean let's just put this into perspective He was so mad and concerned with this so-called documentary that Tucker Carlson released That he spoke with the higher ups at Fox News You don't do that if you're just irritated by something that someone is putting out You do that if you have grave concerns about the content that is being produced not just because it's misinformation But because it reflects poorly On you as a host on a network that is putting out this type of misinformation that Gives info wars a run for its money. So of course when Baer was asked about this by Kilmeade He's going to explain how Look, this is bad and I unequivocally condemn Tucker Carlson This should not be aired on Fox News Fox News needs to do better and as an employee of Fox News I want to speak up and I want us to do news and I want us to push facts Not misinformation except he's not going to do that He is going to back down like the little beta milk cuck That he is Baer commented on his friends Hayes and Goldberg and their decision to depart Fox News in protest Quote, I think it was a tough choice But one that they made on principle he offered before adding, you know And I'm going to let them speak for themselves Baer then reaffirmed his interest in hearing all kinds of voices left right Trump Whoever supporters before admitting the departures of Hayes and Goldberg were sad for special report in Fox News But they made their choice some principles So let their statements stand Kilmeade noted that he had watched the fox nation series patriot purge Which he found to offer an interesting perspective about january 6th That he hadn't seen before I didn't get hurt by it I didn't get damaged by it were you bothered by it because that's the reporting A pregnant silence followed that very pointed question that pushed Baer to address an alliance to Hayes and Goldberg Or the host of the show at 8 p.m. Tucker Carlson quote There's a Brian I don't want to go down this road Baer answered perhaps with the slightest sense of annoyance towards Kilmeade You know, I mean there were concerns about it.  Tucker Carlson is also a coward because as much as he lambasts vaccine mandates He refuses to actually address the mandate that a fox news Has instituted so understand that all of these people on fox news every single pundit. These are not news people These are actors for the most part willing to bite their tongue to continue getting a paycheck it's just I mean on one hand, I don't want to be too horn on baer because It's good that he complained to management About Tucker Carlson, but they're not going to I don't know punish him or penalize him in any way because he brings in the ratings So if you truly want To stop Tucker Carlson's damage, then there needs to be pushback use what little Uh influence you have on fox news to push back Respond to him on your program But everyone at fox news is a coward because I mean the goal isn't to Provide viewers with information the goal is to peddle a right wing misinformation narrative overall And even if people like Brett Baer aren't as bad as Tucker Carlson He still Does bad like it's still all right wing propagandas.  You want to see these schools fight each other You want to see them tear each other down? But overall none of them are going to do anything that will impact their bottom line So Brett Baer will only speak out to the extent that it doesn't actually affect His ability to peddle his own propaganda on fox news So, uh, I mean either way Tucker Carlson is bad, but I just can't help but feel A little bit irritated with everyone who's only now speaking out about Tucker Carlson because he's given you So many examples as to why he's not just A bad propagandist not bad in terms of like he's not effective I think he is effective, but what he produces is genuinely damaging to american democracy And he's only gotten worse So i'm glad that there is a line that some people are drawing But the line Was crossed months ago and to only now think that He's you know problematic. I mean great better late than never. I'll put that out there right better late than ever But for fuck's sake, I mean he was peddling the great replacement theory the fact that there weren't mass Walkouts of fox news at that point goes to show you that many people who work here are cowards and they were refusing to speak up because They prioritized their own paychecks over, you know harming the country and that's unfortunate But that's the reality of america


Okay, so welcome to the Bookmat platform details webinar. It's going to be a short week here, just Monday, Tuesday, and Wednesday, and then that'll be it for for a bit. So, and then next week, we will not have webinars as well. Alright, so it's the summer vacation and and we'll resume in I think after Labor Day in September for the US session here. Okay, so let's get into it. Risk disclaimer, trading equities in futures involves substantial risk of loss is not suitable for all investors past performance is not indicative of future results. For more information, go to bookmap.com. Once you are in trial and become a member, you'll have access to a lot of free resources, including the bookmap advanced analysis webinar, which is following this webinar in about 28 minutes. So if you're interested in that, then, well, it's only for those that are in trial or current customers is we had a lot of complaints about the customers not being able to, you know, have more understanding of using bookmap understanding the order flow, when there were a lot of questions regarding the platform. So we separated the two out. And so the first one is understand what you're looking at here in the platform, answer any questions, and then we do get into some of the analysis of the live order flow. But then we'll just move directly into that in the following webinar. Okay, so that's the idea. Let me show you where you can find bookmap. So go to bookmap.com here. Let's click on explore. There's some intro videos here. If you're brand new, I would recommend watching a few of these to understand what you're looking at. And then let's click on connectivity. Okay, these are the methods how you will connect bookmap to the live markets. We are not a data provider. You will need to have a data provider or broker that we support. Okay, so these are the data feeds and APIs that in some cases like Ninja or TT or interactive brokers, we go through the API of these programs or platforms. Bookmap is a platform in itself though, so you can connect it directly to CQG, Rhythmic, Gain Capital, IQ Feed, Transact, etc. Okay, let's go down a little bit further and this is where you can give bookmap a try. So you get a 14 day trial period. There's a basic and the advanced version. 49 per month, 99 per month, they're billed quarterly. And the difference between the two. One is the ability to trade from the chart, which is really nice because we have the liquidity in front of us. And we understand where we're going to be positioning front running high liquidity to get our entries, higher probability of our entries on in the market, or hide behind our stops behind high liquidity as well. So we have less of a chance of getting hit. And then the other difference here, majority of the differences are these add-on indicators that are more geared for order flow and the high frequency environment. So not your typical type of add-ons here. And then you can see there's the basic and advanced over here as well, but it's packaged with DX feed. Now the DX feed, it's only for U.S. equities. It's not for futures. It's an excellent data feed. It gives you full depth of market. And we can take a look at some equities if you have any questions about that. It's really interesting to see the equities compared to the futures. Anyway, just, you know, these sessions are for you guys. Answer any questions that you have, but let's jump in here. And well, let me show you first the Twitter account here. You can follow us on Twitter and get updated information. You can also go to the YouTube page and you can subscribe here. If you're new, I would recommend going through the features and components here. Okay, just get an overview, understanding what Bookmap is. Then you might want to watch a few of these order flow video snippets. They're very short. And these videos cover a lot of the content that we see in the live order flow advanced analysis webinar. So that said, right down below here, the selected webinars that we have, there's a few of the advanced ones that are in here. So a few are these platform webinars and then a few are also from the advanced webinars. So this was actually from Fridays, I believe. And great stuff, great to see. So yeah, I recommend watching some of those if you have a moment. And then here's a professional trader that we like very much. He's excellent and he uses Bookmap so you can see how he integrates it within his trading. Okay, let's jump in. All right, so let's take a look at Bookmap and we're going to go through this process like we do every day. So we're going to make it very, very simple and we'll get more advanced very quickly here. Okay, so what is it you're looking at in Bookmap? And let's turn off a lot of the data here. And we're just going to look at some candlestick charts and zoom out a little bit. Okay, so this is a five minute candlestick chart. Each vertical dotted line here is 15 minutes of data. So you have five, 10, 15. All right, so you can see we're looking at the NASDAQ. You can see the move down here, a lot of red. Now the candlestick chart is showing you open, high, low, close of five minute period. And that's it. Only four data points. There is a lot being left out. And we're so accustomed to looking at candlestick charts for the most part. And we're making decisions based on candlestick charts, but there's just so little here to go on. And that's a problem. We want a lot more transparency and insight. For example, where did the volume trade? Where are traders committed? Now, most of us also have a volume subchart as we do here. You can see where the volume is spiking. That's good. However, we want to know exactly when the volume spiked, where on the candle, and how much, and what type as well. Was it aggressive selling or aggressive buying? It gives you tremendous insight to what's going on. Because if you know that there are a lot of sellers down here on this candle here, down at the lows, and we see buyers come back in and they lift the offer. Well, these guys are going to cover. They're going to be buyers, and they're going to be squeezed out of the market probably up at this area here. So that's information we want to see, and we're not getting it here in the candlestick. So that is a problem, and bookmap solves that issue. So let me first just turn on the best bid and offer. Already, we get an understanding here of microstructure within that five-minute period, which you couldn't see here at all. You just see the big down bar here. Well, what occurred in that five-minute period was, you can see we get a quick move down, a retest back, and then another move to the downside, and then it consolidated here. So there's a lot of volume that actually traded down here. And actually, that scenario of it being trapped volume is looking pretty good, because we actually see this pattern again and again. And let me show you what I mean. And I'm just looking at historical best bid and offer, and I'm already getting that insight. Okay, without even looking at the volume, but I know that there's going to be a lot of volume here. So these guys are going to be trapped. They'll cover actually looking at the structure. I originally looked up here. Looking at the structure now, though, I think they'll be covering here. So if we can spike up above here and maybe ignite some of those stops, we can have a nice quick move back up into maybe 5780. That's just by reading the order flow here. Well, not even the order flow, we're just looking at the structure. So we already have tremendous insight by looking at that. But let's turn the volume dots on, and here we go. So now we can see all of this volume that traded down in this area here. And let's zoom in a little bit. Let me take this off here. Here comes the initiated bind that we're looking for already. As you can see, look at the green dots here. That's aggressive bind. So it's looking pretty good. This scenario that we just outlined is looking pretty good. It would be really hard pressed to see that in the candlestick pattern here. So anyway, let's zoom in and look at some of this traded volume here. So here's our five minute period, and now we've zoomed in a little bit, and we can see each vertical dotted line is a minute. Look at the structure that is broken here. Look at the breakdown, and you see the aggressive selling. A red dot is an aggressive market sell order. A green dot is an aggressive market buy order. Let me zoom into this level here, and I'll show you in more detail. You can see that here's a green dot. It occurred on the historical best offer, which is the red line. The best bid is the green line. So this is a market buy. So someone crossed the spread, and they paid up for it, and then they got filled here on the best offer. In fact, it looks like they may have gotten slipped a tick, thinking that maybe I'll get filled at 5760. Well, the offer came up to 5760 in a quarter, and they got filled there. So that's what we're showing here now, the red dot. Let's zoom in to here, and we can see the red dot here is on the historical best bid. So this is an aggressive market sell order. Someone hit the market sell button. They took liquidity off of the best bid, and the transaction occurs here with the dot. So that's what you're looking at, and you can continue to zoom into these levels. Now I know that this is a small trade, so we're not going to get too much out of it, but as you can see here in the timeline, we're down at microsecond level. So we're talking millions of seconds. So notice how as I zoom out, there's actually a few different trade events in here. And as I zoom back out, though, we consolidate visually and aggregate just visually this data because it happens so quickly that it doesn't really have meaning for us. So as I zoom back out, it consolidates into one red dot, but this red dot will continue to consolidate and it will get bigger. And it gets bigger and bigger and bigger to a certain point where we're giving you the overall understanding of what occurred here, what traded, what type of volume, and you can see in this case, it's consolidated into a pie display that gives us both buying and selling, so the delta between the two, or the delta between the buying and the selling. So we have an understanding of what occurred at some of these levels in these areas. Now, if you want the exact number and a lot of traders do, you just click on the data tip tool and you can hover over this dot and it gives you the date, the time, what was on the ask at this price level and the volume at this price level. So this dot here signifies 176 contracts that traded. So that's the volume. And there's, let's zoom out here a little bit as I was talking. Here's the scenario that unfolded just like we were talking about. We came up into above this area here and you can see the aggressive buyers here. All right, hold on just a moment. Okay, and buyers are still engaged here, it looks like. So maybe we'll climb up into this 57, 81, 82 area now. Let's see if we have enough buying here, aggressive buying to get there. They need to get above the first stop it would be this 57, 73. All right, so anyway, that's the volume and now we have tremendous insight here to what really unfolded and that was lacking within this candlestick chart. Now, we also, another part of order flow and understanding where price might go is to understand the limit order book. So let's zoom in here a little bit. And I've got it set up here. Let's hide a couple of these columns, make it simple. Okay, so the dome, most of us are very familiar looking at a dome. So this is the dome here in book map, the COB column, current order book. Now I have two of them. Okay, I have one that's here, I'll just right click and format and you can see it's bars and numbers. Actually, let's just make it bars only and that's fine. I like it like that. And then I have another COB column here and it is formatted as numbers only and I've split out the data between the buyers and sellers or bidders and sellers and I inverse that split so it kind of mimics regular dome here. Okay, so we can see the depth of market here. Traders lining up to provide liquidity here on the bid side and traders on the sell side here lining up and providing liquidity up here as well. Now, these numbers change all day long if we zoom in here. And we can gain a lot of information by reading the dome here. We understand areas where there's liquidity where the market can trade like right here there's pretty high liquidity. Let's see if they come up and test that area. Okay, because the market can trade up here at this 70 and three quarters area up here. And just to get an understanding of the auction, what's really going on in the auction here and it gives us a lot of insight. The problem with the dome is that these numbers are constantly changing. When they change, we have no reference of what now occurred there historically. And that's a big problem because this is rather tedious to read and to remember there was high liquidity down here. They started to bid up in front of it. They were front running it. They were really interested and then that changes as the say price goes away. Now you don't remember when and maybe even that price level. And what if it comes back in the next 15 or 20 minutes? You'll have to record and remember all of that which is rather tedious. So the way book map solves that issue is through visualization here in the heat map. So here's your dome and here's your price ladder. And then this window here is also the current market window. It's the best bid and offer right now currently with the dashed lines. And then this is the last traded volume, this number here. So now I'm going to turn on the heat map and I'm going to show you and let me adjust this heat map a little bit. Now we have an understanding in this window here. We'll just start with this window. You can see these very bright areas. That's high liquidity. And we know there's 55 contracts there. It's pretty high for this particular time. And that's on the bid and then on the offer we see 51 up here. So we know in the current market window here where they're bidding and offering where this really gets interesting is we take that data here and we record it. And then it's transposed on the chart historically. So now we have an understanding of these areas here. You see these striations in the heat map. It got kind of gray. It got a brighter shade of gray and then it got darker and then it got brighter. That is the adding and pulling of liquidity. It's a little bit better example here at 63. So I know that they're interested down here but they keep on adding and pulling. So they're not that interested. So now I'm starting to gauge and understand within the context of the chart and the dome how this auction is going. And it gives a lot of value and insight that is lacking here in the dome. That's the problem with the dome and we solve that issue by recording it transposing and onto the chart. So we get the current market as well as the historical market. Now as I zoom out this is where it gets really interesting because we have a much bigger picture of what's going on. Now I'm going to have to... You can see here we can adjust for that heat map and get some more insight. So look at our area up here that we were looking at previously. We were looking for stops being hit up in this area. Look at all the liquidity that was provided up here. Very high liquidity. Larger traders, they know what they're looking for. They were looking to grab those stops and they did providing their limit orders up in that area. And we can see that we rotated lower. Came back down, rotated once more a little bit lower and then we saw here now we'd have to zoom back in there to see that and maybe adjust for the heat map at this point. But we saw them starting to get interested here. And we can see now we've rotated back up and we've hit high liquidity here up at 71 and it looks like we want to come up and test into 72 and 73. So now we have a complete picture and in fact let's take the candlesticks off. Now we have a complete picture of the volume that traded, the microstructure with the historical best bid and offer, the volume that traded on that in those areas where, how much, when and what type. And then we also have the understanding of the auction where they're bidding and offering. And if they mean business in that or if they're just going to pull that liquidity. And now we have a much clearer picture of what's going on at some of these levels. So any questions? Let me know. That's just the basics of book map and there are so many other features here that we haven't even touched yet. There's all sorts of volume columns over here to take a look at. You can also right click here and we can look at volume columns. We can look at trades counter columns. Not the number of contracts that traded, but the number of trade events that took place. We can also look at quotes that were refreshed in those areas. And we can look at the delta of the quotes. We can add custom notes. We have a time and sales column and lots of details and settings for all those various columns. There's other features here to go through. So we have automated trading strategies. There's three that we have here. There's a chase, escape and execute. Now you can trade right from the book map chart as well. And let's enable it. And you will need the book map advanced version to be able to do this. And let's just choose one here. And let me enable automated trading strategy. Whoops. I'm going to actually engage two at the same time. The chase is like a trailing stop, but it's a trailing limit order for entry. So you'll chase after the market and then it will stay at a specific level. And then you're looking for a pullback and getting filled on the pullback. You can chase by a number of ticks here, but that you can input yourself. The escape is a little different. Basically, the escape is like a currency price coming up into an area. And let's say I want to be a seller up in this area here. And let's say I don't have high liquidity to support my decision. I want to trade along with the bigger players. So if a certain condition is not met here that I don't have liquidity to support my decision to go short, then it engages in action to move my order away by a number of ticks here. So in fact, let me make this condition rather hard to be fulfilled on purpose. Let me engage these strategies and I'll show you. So I'm basically going to have a market buy limit here and then market sell up here. And you can see that I'm chasing it here on the bid. And I just got filled. So all right. And I just got filled there. All right. Well, let me, I'll have to make these a little more. Let's see. Oh, okay. That's why I want 500 here. I thought I put that in. Okay, let's try that again. Okay, so I'm going to be a seller here. I'm going to be a buyer down here. And you can see I'm already chasing and then and I just got filled again. All right, so let me flatten this and let's see what I have here. Okay, so it went back to 62. That's not what I'm looking for. Okay, and I just want one price level here. Okay. This ought to do the trick. Okay, so now I'm trying not to get filled here, basically. And because I want to show you how the algos work. Okay, so you can see that a price moves up and it's moving my order away. Okay, so this is the escape here on the way up. Okay, now it's engaging chase as if the market comes back down. Okay, the bid here is obviously chasing. It has not escaped yet. There's an escape. Okay, so you can see I'm straddling just like a market-making algo in this sense here. Just staying a slightly away from the market. Okay, but still staying in line here and when I get to a certain point here, what I can do is I'll just turn these off. All right, I'll just say, okay, I want to be engaged. I want to be in the market. There I am. And let's see if it can come up and just come a couple ticks here and hit me at, I need to also dis-unengage from that one on the chase. Okay, here we go. Okay, and I just got filled here. All right, so you can see how it works. And that's just another feature. That's just one more feature. There are so many others. Ask away, take a look, give it a trial, free trial period and start to explore. There's iceberg detectors that we have here looking for iceberg orders or looking for maybe larger players as well. Now these are some of the book map advanced features. We also have an indicator panel on the bottom, the sub chart here and the delta as you can see here. And there's many different settings that you can have also for this indicator panel. In fact, if you're a programmer written in Java, you can write your own programs within book map. Yeah, Tony, will it help you get in and out of your trades? Yeah, I mean, if you're looking for a specific setup for the orders we're working here. Right? So that's exactly what you can do. Okay. Anyway, there are quite a few other things to take a look at. We don't have time. We'll have to jump over to the other webinar. So we'll catch up with you tomorrow. All right? So yeah, if you have any questions, just reach out to me support at bookmap.com and happy to answer any of your questions or if you guys are looking for a walkthrough of the product, just let me know. Okay? All right. All right, thanks. Take care. Bye-bye.