Let's start lecture two with a small review of lecture one. So yesterday, we were studying the scattering of gluons. And we said that when you scatter n-gluons, you have to specify data. You specify momentum, polarization vectors, and a color index for each particle. And there was a clever way of separating the color structure so that you only had to worry about the kinematics. And that's called the color decomposition. And we wrote it in a form that looked like this, where w is some permutation of n elements, modulus cyclic transformations. These objects over here were called partial amplitudes. And we said that they satisfy some amazing properties. Well, at least they look amazing from their point of view, because they don't know anything about color. But once you know about Lagrangian, you know that they have to satisfy some properties. As yesterday, we mentioned some of them. The first one is cyclicity. Imagine we have the partial amplitude with the canonical ordering. This has to be equal to this one. So it has to be cyclic in the labels. The second one, and today I'm not going to try to spell it, I did it for you, the KK relations. So cyclicity tells us that from the n factorial, independent ones, we go down to a minus 1 factorial. But KK were smarter. And they said that you can actually bring any two labels, say one and n, to become adjacent by taking appropriate linear combinations. And I hope somebody tried to do it. So we had a formula like this. What's important is that they told us that any object of this form can be brought or can be written as a linear combination of amplitudes where one and n are adjacent to each other. And that brings any a minus 1 factorial down to a minus 2 factorial. And the third property was the BCJA identities, which tell us that you can even do the same kind of procedure but with a third particle. Here they chose one and two to be adjacent. After you use KK, you can bring one and two to be adjacent. And now you can use a further relation to bring three next to two with some function that depends on the permutation and only depends on Mandelstein variables. That was the magic that this only depended on Mandelstein variables. I told you that these functions were complicated, but that these identities could all be derived from something called the fundamental BCJA identity. And this is an identity of this form. So I want to get it right because we're going to use it later on. So let me get it from here. So just so that everybody's on the same page, let me remind you that throughout this talk, SAB is the same as KA plus KB squared. KA, KA mu is 0 because it's our massless particles. And therefore, this is always 2 times KA dot KB. So we have this identity. Exactly. Very good. So Jiu-Jitsu, a very good student. Actually, the physical origin of BCJA is not very clear. So this is something that in 2008 they discovered. Later on it was realized that it can be derived from a string theory by taking vertex operators and moving them around the boundary of a disk. So once you go around, you pick up a monodromy factor when you expand that, you would think that only the real part, or only the alpha prime, the leading order in alpha prime corrections, or sorry, the leading order in alpha prime, we have a field theory interpretation. But it turns out that the leading order is a real object. And the imaginary part also can be shown to be independent of alpha prime. Or not independent, but it doesn't receive any higher order corrections from the effective action. So there is yet another identity. And that's the BCJA identity. OK. The last thing we said, like yesterday, was that MHV amplitudes, the formula discovered by Park and Taylor, is Taylor made to satisfy these identities. So the formula has all gluons with plus elicity, except for two gluons, say i and j, that have minus elicity. And this is a formula that by now you're very familiar with. So this i and j is a little bit boring. The engine behind the triviality of all these identities, or how simple they look like, is actually this factor, which I would like to define as the part Taylor factor of 1, 2, up to n, just to simplify the notation. So I'm going to call this the part Taylor 1, 2, all the way to n. So a linear combination of these identities gives rise to the starting point that we had yesterday, which was called the U1 decoupling. And let me write it today in a slightly different form. It's exactly the same identity, but I'm going to start from the n. Yesterday I started with one position for the particle that I'm going to move around. So today I'm going to start from the n point. So I'm going to put particle n between 1 and 2. And you're going to see why. And no, I'm not going to forget it. Yes. So that's the U1 decoupling identity, the one that tells you that if you are scattering n minus 1 gluons and you want to know the probability of producing a photon that has to be 0. And we said that for MHB this was easy to prove. So we use this formula. Remember we can forget about this factor that's boring. So yesterday what we did was to say, well, let's factor out what everybody has, or what almost everybody has. And that was a part Taylor factor of particles 1, 2, up to n minus 1. So remember the definition. So I'm going to write it for you again here. So we have 1, 2, 2, 3, up to n minus 1, n minus 2, n minus 1, n minus 1, 1. And then we're going to put in here whatever I need to multiply this with to produce the corresponding one. So I have this formula, and I want to get this one. What do I have to multiply by to get it? Well, what I have to do is to multiply by, so in this formula 1 and 2 are not adjacent. But this formula has them adjacent. So that's bad. So I have to remove that factor. This formula has a 1n factor in the denominator. This one doesn't. So I have to put it in the denominator. And this one likewise. The next one, I do the same thing. And now you see why I reordered them. So yesterday I wasn't very smart. But today I am. So we go all the way to n minus 1, 1, n minus 1, n, and n, 1. And this thing has to be 0. I encourage you to prove this by a telescopic argument. But I'm going to tell you today about my favorite, actually my favorite way of proving it. And this was actually done by Hodges in 2008. And what I'm going to tell you is not the paper that he wrote down. It's just basically a footnote in his paper. So the paper is about something else. But the idea comes from there. So here is a deep identity we're going to use. Let me remind you before we do it that for any spinner, we can write it as an orbital factor. This is a two-component object. So we can write it as an orbital factor and 1 in the first component and ZA in the second component. And this identity now becomes an identity. See that the factors, the overall scale for factor 2 cancels out. The overall scale for the spinner 3 cancels out. And the only thing that appears here is the overall scale for n. But n is a common factor. So the overall scale factors out. And the identity becomes an identity between differences of complex numbers or among differences of complex numbers. Now I can tell you the trick. The trick is to realize that if you have ZA minus ZA plus 1 over ZA minus ZN ZM minus ZA plus 1, this is nothing but the contour integral in the complex plane Z. Imagine you have ZA here and ZA plus 1 here. Choose a path that goes from ZA to ZM plus 1 and integrate this function. Now it's a double pole. So the path doesn't matter. You can deform the path in any way you want, even pass it through the pole because the residue is going to be zero. So no problem. Now let's write down the identity. So the identity tells us that we have to start here. The first factor is the integral from Z1 to Z2. The second factor is the integral from Z2 to Z3, Z4, and so on. And guess what happens? We're going to close the contour. And when we close the contour, we deform it and we get zero. Isn't that nice? But this identity or this way of proving it also shows us something interesting, which is that if we start with any point here, say ZA, and we go to any point ZA plus M, the sum, say, of B from A to A plus M of these factors is equal to the same to the integral from here to here. But then I don't have to pass through all the points here. I can deform the contour and just go from A from ZA to ZA M plus 1, how appropriate. It ended up in the right place to be there, to be the upper limit of the integration, over Z minus ZN square.  So we got ourselves nice identities. So why are they important for us? Well, because it will allow us to rewrite this PCJA identity in a nicer way. And that's what I'm going to do now. So let's study the fundamental PCJA identity for part Taylor factors. Once again, I'm forgetting about the overall ij to the fourth. I don't care about that, because it's not going to do anything interesting. So again, I look at this sum and say, well, this sum looks pretty much like one of these. You want the coupling identities, but it's incomplete. You see that it doesn't go all the way around. It starts at B and it goes all the way to M minus 1. But can we use anything on the blackboard to simplify this problem? Well, exactly this identity here. Because the sum, so the fundamental PCJA identity becomes SMB, B1 to N. And now this sum over here becomes the sum from A equals B to M minus 1. So let me write it A, A plus 1. So I'm getting a little bit bored of writing Z and Z all the time. So let me introduce a convenient notation. ZA minus ZB is going to be just this.  Here, note that this A plus 1 is actually a number modulo M minus 1. So don't worry when we get to A equals to M minus 1 that we're going to be in trouble. Because when you put it in here, you get 1. You don't get N. So that's what the identity tells us. Or that that's what we want to prove. But now using our identity here, we can re-sum each term and get something that looks like this. That looks pretty simple. It's just a sum over B of these factors. Now this factor is trivial. It's a common factor. And we want to prove that this is 0. So we can cancel it out. Oops, I don't want to fall. All right, so we can cancel this out. Now we are ready to prove the identity. Let's go back to our spinors. We can multiply again by the scaling factors and bring this notation back to the angular bracket notation. And the identity we have to prove is that when we go from 1 to M minus 1, we multiply the Mandelstand variables. This spinner over BN, this has to be equal to 0. So let's see how many people are familiar with this formula. Well, I wrote it down yesterday, but did you see it in Simone's lecture? Well, if you haven't, know that this is the only Lorentz invariant possibility for this. There could be an overall factor, maybe a factor of 2 or something, but it turns out to just be 1. So you can try to work it out and show that indeed the factor is 1. So we're going to use that here and see why. Because when we write SNB as that product, this factor cancels. So we get MBB1 from 1 to M minus 1, and this has to be 0. Is this 0 now? Well, let me rewrite this in a spinor notation a little bit. So I'm going to write it like this. So this means that we have particle N, lambda tilde of particle N, alpha dot. This is the spinor index contracted with the spinor index of particle B. Sorry, these two are contracted. And this one is now contracted with the spinner of particle 1. But what is this object? This is a momentum of particle B. Now I can put the sum in and get, and now we use the observation that somebody made yesterday that I told you we have it in our imaginations, but we keep it always in mind, which is that we're always dealing with something that satisfies this momentum conservation, and therefore the sum of all momentum is equal to 0. And therefore the sum of all momentum from 1 to M minus 1 happens to be minus the momentum KN. But if we have the momentum KN contracted with the spinor lambda N, we get immediately 0, OK? So that's the proof of the identity. Now I want to rewrite the identity a little bit. Actually, I'm going to take this form, and I'm going to do the following. Once again, these brackets indicate differences of complex numbers. So I'm going to write the numerator, which is ZB minus Z1, in a slightly different form. Sorry, not even Georgie cut this because he's reading. So I'm rewriting the identity slightly. The idea is that this factor now cancels with a denominator and gives me a sum over SMB from B1 to M minus 1, plus the next factor, which is ZN minus 1, comes out of the sum. And we get SMB over ZB minus ZN. Now what is this? If you sum over all the particles from B to M minus 1, from 1 to M minus 1, know the definition of SAB, which was KN dot KB. We can put the sum inside. If you put the sum inside, you get, again, KN square, and this is 0. This is an overall factor, and we discover this new form for the BCJA identity. Note that this, we discover by moving particle M. But if we move any other particle, it should also be true. So it must be true that if we sum from B to N and B different from A, SAB over ZA minus ZB, this must be 0 for all A. Now we want to make a bold claim. And the claim is the following. That any rational function, some variables, x1, x2, up to xN. Now this could be independent variables, or they could be sets of variables. So x could represent many variables. But you have N of them that satisfies 1 and 2, cyclicity and KK. So we have a function 1, 2, up to N of these variables. And the function satisfies this. There is cyclic in the permutation of the labels and satisfies this identity. Then it must be the algebraic transform of Park and Taylor. We all know, of course, what algebraic transforms are, right? No? Good, because I made it up. So let me give you the definition of this. So what I mean by that is that you take Park and Taylor in terms of the complex variables Z. Now let me change the name, because it's going to become useful in the future. Because we're not going to be talking about spinors anymore. So I'm going to change Z for sigma, just to remind myself that I'm not talking about spinors. I'm talking about something more general than spinors. So for me, Park and Taylor, from now on, if I say Park and Taylor of 1, 2, up to N, these I take to mean the differences of sigmas, where the sigmas are complex numbers. So I'm introducing as many different notations as possible so that I confuse you maximally. So hopefully I won't succeed with that. So let's keep going. So we have this Park and Taylor. So what do I mean by the algebraic transform? The claim is that our function x1 and x2 up to xn is an integral over all these sigma variables of Park and Taylor of some product of constraints that relate the sigmas to the x variables using polynomials times possibly an integral that depends on the x's and the sigmas. And this integrand is permutation invariant in the labels. And these constraints are also permutation invariant. When you take them all at the same time, it's a list of constraints that are permutation invariant. These are delta functions. So many of you should be wondering what the heck am I doing, because I have integrals over complex variables and I have delta functions. And they don't go together very nicely, as you know. But by this, what I really mean is poles. So we have denominators. And we're doing a contour integral in a multi-dimensional space. But that's too long. So I prefer just to write delta functions and remember that I'm solving all these equations simultaneously. And I'm summing over all possible solutions. I don't care if they are real or complex. So this is a claim that if the function satisfies that, then it must be of this form. So the contour is actually a sum of contours. It's a sum over n-dimensional tori. And I cannot say that they enclose the singularity, because in more than one complex dimensions, you cannot enclose a singularity with the product of s1s. But each one of these guys defines a hyperplane. And then the Tn, in a sense, encloses the union of these hyperplanes. All of them. All of them. You might be worried, because you would say, well, if you pick all of them, you can use a residue theorem. But remember that there is a factor here. So this is not 0. You cannot use what is called the global residue theorem, because there is an extra piece that also has poles. So let me give you a more or a less complicated way of thinking about this. You solve all these equations, find all the solutions, and sum over all the solutions, 1 over the determinant of the Jacobian of the matrix made out of this times Park and Taylor evaluated on the solutions. The solutions are values for sigmas that depend on the x's times your integral. Now the claim I want to make is that this object, even though these polynomials could be very complicated, this object, if the polynomials have coefficients that are rational functions of the x's, this answer, after you sum over everybody, it has to be a rational function of the original variables as well. What's the proof? Well, you can use a little bit of Galois theory to prove it. But it's actually true, OK? That's very good. So now, why am I saying this? I told you that we were able to prove these identities for Park and Taylor amplitudes. These are the MHB amplitudes. But how about the other ones? How about n squared MHB, these complicated objects that the more negative electricity particles you put, the more complicated they start to become? But they all satisfy this identity. So my claim is that they can all be written in this form, or at least as a claim. But how about number three, the BCJA identity? So the BCJA identity ends up being a voice down to these constraints. How about putting this together with BCJA in the following form? How many equations do we have here? We have n equations because we have n variables. And these, after you put them in Mathematica and you type together, they become polynomial constraints on the signals with coefficients in the Mandelstam variables. How nice is that? That looks exactly what I need for the definition of this integral transform or this algebraic transform. So what if we replace these polynomial constraints by exactly those equations? And using George's technique, I'm going to cross this and put my equations here. So it seems pretty good. But the moment you try that, you find that there is a technical disaster. You find a small problem. So let's compute this Jacobian. So what is a Jacobian? So if these are my equations, so let me now call EA the sum of our SAP over sigma A minus sigma B. My Jacobian is a following matrix. So I have to make a matrix of this form. And the matrix, let me call it phi, is a matrix that has the following structure. So everywhere else, everywhere outside the diagonal, the matrix has an entry like this. And on the diagonal, it has, I mean, I'm being very lazy here. So I should have written something a little nicer. So say here we have 1, 2 over sigma 1 minus sigma 2 square, 1, 3 over sigma 1 minus sigma 3 square, and so on. And in the diagonal, we just get minus the sum of all the elements here. And the same thing happens in all the rows. Now when you take the determinant of this matrix, what do you get? Zero. Damn it. So close. I mean, this whole thing, we have worked so beautifully. Now you say, well, maybe it's not such a big problem. Only one null vector.  But then you discover that you have three null vectors. The null vectors are 1, 1, 1, sigma 1, sigma 2, sigma n, and sigma 1 square, sigma 2 square, sigma n square. That's a bit disappointing. So there are three linear combinations. So we only have n minus 3 independent equations. Yes. Well, because having one problem is a little better than having three. Usually, yes. Well, the reason having one sounded at first better than having three is that people were used to defining in graph theory, you can define matrices associated with graphs that have precisely core rank 1. And there is a notion of the, when you have a matrix where the sum of all rows and the sum of all columns set up to 0. It's a theorem that you can remove one row and one column and compute the determinant of the matrix left over. And that determinant doesn't depend on the row or the column that you removed. That's something very common in graph theory. So if you knew that, of course, I didn't know that. But if you knew that, you would say, oh, having one null vector is not such a bad thing. In fact, this matrix satisfies us. The sum of all rows is 0, and the sum of all columns is 0. So indeed, if you remove one row, one row and one column, and you compute the determinant, the answer is independent of the choice of which row and column you chose. Of course, the answer is still 0. As it turns out, if you treat this as a bisonic redundancy and you use Fajepov-Pov procedure, you can show that there is a way to define an object, which is the following. You compute the determinant of the matrix where you remove three rows and three columns, say the row P, the row Q, and the row R. You remove them, and you remove the columns P, Q, and R. And you take from this object here, you take the determinant of the 3 by 3 matrix made out of the ones that you chose. So you take the determinant of 1, 1, sigma p, sigma p square, sigma q, sigma q square, sigma R, sigma R square, and you call this P, Q, R. Of course, you all know what this determinant is. It's just the product of the differences of the three labels, of the sigmas of the three labels. This is called the Van der Monde determinant of these three particles, or these three labels. And then if you have this and you divide by this, this is independent of the choice of P, Q, and R. Now you would say, well, but we started with n sigmas, m complex variables.  Sure enough, with this trick, you can define a Jacobian. But what do you do with the other three sigmas? And then you stop and think for a while and think maybe for a few days, and then keep thinking. And then you realize, well, but we saw in Ashok's lectures that if the sigmas were punctures on a Riemann sphere, and a Riemann sphere is a complex manifold, so we should remember to mod out by the SL2C acting on the sphere. And SL2C allows us to fix the value of three punctures. As Ashok said yesterday, the modular space of a three puncture sphere is trivial. There is only one such thing, so it's rigid. So that would mean that if we have n variables, we can fix three of them if we only had SL2C invariance. So BCJ or the BCJ identities are telling us that these transforms or these integrals are supposed to mean integrals over the n puncture sphere, not just any random n dimensional integrals. And that's a resolution of the puzzle. So once again, the resolution of the puzzle is that any amplitude should have a form or should be written as an integral over the product of n sigmas. But of course, we know we have this redundancy. So I'm going to write it like this. I'm going to divide by the volume of the redundancy, even though it doesn't make sense here. But times the part Taylor times some definition of what I mean by the constraints on the particles on the sigmas coming from these equations, times some integrand that depends on what? Well, it depends on the data, the momentum of the particles. So let me say kc, the polarization vectors of the particles and the sigmas. So these were the axes where I started. So what do I mean by this? Well, we have the definition. We know that if we remove three labels and multiply by this, we get something that is meaningful. Likewise, we know how to gauge fix this SL2C. You're very familiar with that in string theory, right? So when you're doing three-level computations on this here, you gauge fix by removing three of the sigma variables, fixing them to whatever values you like, and multiplying by the Van der Monde of those three variables. So what I mean by this is the integral from A different from ijk. So those are the three values I'm fixing, times the Van der Monde of ijk. Now here, I take this to mean precisely this, or in other words, I multiply by the Van der Monde of PQR and only impose the equations coming from all variables except PQR times some integrand, divided by my part Taylor factor, which I'm going to write explicitly here. So that's a clean. Now, one exercise for you is to show that the twisted string formula we wrote down yesterday indeed can be written in this form. Instead of doing that for you, I've decided to study a little bit these equations. So these equations, if that formula is true, and it actually works for any amplitudes, I think these equations that are linking the modular space of the n-puncher Riemann sphere to the space of kinematic invariance, they deserve a name for them. Probably this is as good as any scattering equations we can start calling them. Now, I derive these equations from BCGA, Identities, and so on. But these equations were actually first found by, so remember yesterday, Jiu-Jitsu thinks that everything was done after 2000. This one is even more surprising. It's 1972. I'm published. Yeah, well, I mean, Peter Goddard sent me a copy. Of course, his body's with it. Don't force me to say anything. Well, they were trying at the time to create some new dual resonance models. But it didn't quite, I mean, I think they wanted to remove the tachyon or something. So they tried different things, and then they ended up with something that produced those equations. OK, so as I said, so if we have the space of kinematic invariance, the equations are connecting them. Well, it's not equal, so maybe, I don't know how to do it. So they are connecting with the modular space of the n-puncher sphere. This was supposed to be a calligraphic m, but something happened. OK, so let me give you an example in there. So we have 10 more minutes. See that these equations are really special, and they know a lot of physics. So in a sense, Riemann knows a lot of physics. And the example is consider four particles. So the space of kinematic invariance for four particles is the space of these Mandelstein variables subject to the constraint, well, this is what you usually call s. This is what you usually call t, and this is what you usually call u, and this is s plus t plus u equal to 0. Now let me draw this in the space s and t. So the space of kinematic invariance is basically R2. So R2, just the plane, as far as I know, doesn't know any physics. There is nothing that the plane knows. Especially it doesn't know that this line is special. This line is a line when s is equal to 0, and amplitudes are supposed to have a singularity there. But the plane doesn't know that. For the plane, this is just a regular line. This line here is a line t equals 0, where we are also supposed to have a singularity because amplitudes are supposed to factorize. And last but not least is a line u equal to 0. The physical regions for scattering are, of course, these ones. And we know that crossing allows us by complexifying this space. We can move from one place to the other and so on. But for the time being, all I want to say is that the plane doesn't know anything about why these lines are special. But let's look at the scattering equations. So we have four particles, but three of the scattering equations are linearly dependent. So we should be able to do everything with only one of them. So I'm going to choose scattering equation 1. And this is the equation that says that 1, 2 over sigma 1 minus sigma 2, 1, 3 over sigma 1 minus sigma 3, plus s14. Sigma 1 minus sigma 4 has to be 0. And now I want to make a good choice of sl2c. So I'm going to set sigma 1 to be, let me check what I want to do, 0, of course, sigma 2, 1, sigma 3, infinity. And sigma 4 is going to be moving around. So as we know, the modulus space is only one dimensional. So I only have sigma 4 as a variable. Now in this, the equation, what happens to the equation? Well, after gauge fixing in this form, this drops out. This is not here. This is not here. And this is 1. So the equation becomes a very difficult equation. Now you put it in Mathematica. Shift Enter, and you get that sigma 4 is equal to minus s over t in our variables there. Now what happens? Well, now we see that if, now Ashok also explained to us that the modular space has singularities. The modular space of the n-puncher sphere has singularities. Now the modular space of the 4-puncher sphere, how many singularities does it have? Well, it turns out to have only three singularities, which is when the puncher that we have free to move approaches one of the three that were fixed. So I'd like to draw the space. I mean, people say that the modular space is actually a three-puncher sphere. But let me draw it like this. I'm going to draw m0,3. This is my picture of what it looks like. Of course, this is a three-puncher sphere. At a genetic point, we have a sphere just like the one I drew there. The three punchers are happily living each one of them on their own. But then when we approach one of the singularities, sigma 4 approaches one of the punchers. So if sigma 4 goes to 0, that happens when s is equal to 0. If sigma 4 approaches infinity, that happens when t is equal to 0. And last but not least, if sigma 4 approaches 1, know the minus sign here. That's when u is equal to 0. So somehow, the three singularities have been mapped to the three lines here. So the modular space knows about all the physical singularities that the amplitude is supposed to have. Moreover, when we approach a singularity, the sphere can be thought of as splitting into two spheres, where we have, say, sigma 1 and sigma 4, sigma 2 and sigma 3. Here we have, I hope you're a little bit familiar with this. What I'm doing here is using a conformal transformation to blow up the sphere that seems to be getting a shrink to 0. There is a new puncher that is being generated. And of course, you're familiar with this because juji kept drawing these pictures all over and over again. And here we have another one, where we have sigma 2 and sigma 4. And here we have sigma 3 and sigma 4. Thank you. Yes. Very good. Now we have five minutes to determine how many solutions these equations have. I already know for four particles, we found one solution. You can sit down and write down the equations for five particles, and you will find two solutions. But only two data points are not enough. Any guesses on how many solutions could this thing have? So here is how we're going to study. Let n be the number of solutions. So how are we going to determine this number? We're going to use a trick that is going to be useful later today, I hope, and it's a following. The number of solutions, thinking about this as some intersection in some algebraic variety, some complicated story like that, the number of solutions should not depend or should not jump, even though it's an integer number, should not jump as you deform the parameters smoothly. And if it jumps, it will jump by an infinite number. So if we don't find infinity, we're all fine. So what are we going to do? We're going to take particle n to be soft. Well, what that means is that you take the momentum of the particle n, you multiply, you think about it as being some fixed vector times a parameter tau that you're going to take to 0. So let me write the equations, the first n minus 1 equations, and see how they look like. Well, they look like this. So I'm on purpose putting the nth guy at the n. So it's going to be nA over sigma n minus sigma a equal to 0. But this thing contains kn, and therefore, this thing is of order tau. And I'm assuming that tau is very, very, very small. So in fact, in the limit, I get the equations that we have for a system with n minus 1 particles. So starting with any solution for a system of n minus 1 particles, I should get something that is very close to a solution of this problem. So I have n, n minus 1 solutions, which I'm going to denote by the index i. So i runs from 1 up to n, n minus 1, a number that I still don't know. So for each of these n, n minus 1 solutions, I can study the last equation, which is the en equation. And this equation is the sum over sv, nA, n minus b, b from 1 to n minus 1. Now I can factor out the tau. So I'm going to have tau q dot kb. And this has to be equal to 0. But tau is a common factor. So no matter how small tau is, I can always remove it from here. And this is going to be a constraint on sigma n. Remember that I'm assuming that I have found already all the other sigmas. So this is a constraint for each one of these solutions. Now once again, you put this in Mathematica, you type together, and you find that Mathematica will return something that has a denominator of the form sigma n minus sigma bi, i from 1 to n minus 1, and some polynomial in the numerator. Our task is to find the degree of that polynomial. Well, clearly, we started with n minus 1 factors. This is something that should go to infinity as 1 over sigma n when sigma n goes to infinity. This has degree n minus 1. So this polynomial should start at sigma n to the n minus 2 power with some coefficient plus lower order. Is that clear? Now let's compute this coefficient. This coefficient is nothing but the sum of all these factors. But what is the sum of all these factors? I can put the sum inside or equivalently take q outside. By momentum conservation, this thing is q dot kn. But it's the same as tau q square. And q square is assumed to be null. So this is 0. So the leading term is 0. Now what I wrote as a lower piece is actually relevant, is actually the most important piece, is n minus 3 plus lower. So the conclusion is that for each i, for each of the capital n, n minus 1 solutions, we have n minus 3 values of sigma n. And therefore, we have counted how many solutions we have for n particles if we only knew how many solutions are there for n minus 1 particles. This implies that an, and we know, now you know how many solutions are there. And I think I'll stop here.


Hey everybody, Cyrus Shepherd here. I want to introduce our next speaker because she is one of my favorite people in the SEO industry and also one of the most brilliant, Alexis Sanders. You may have heard of Alexis if you've ever played an online SEO game. She has two brilliant games that she's developed, Technical SEO Expert and Technical SEO Guru. So that's technicalseo.expert and technicalseo.guru. Check them out, they're a lot of fun. So even though Alexis is a brilliant Technical SEO, she's going to talk today about something we don't always consider when speaking about Technical SEO, which is the audience and how important finding that audience is to your SEO propositions. So we're going to kick it over to Alexis. I know you're going to enjoy this session. Hi, Mozcon. Customers are the center of quality content user experience and digital marketing. And have you ever thought to yourself, how do I learn about my customers with the resources that are already currently available at my company? And that's what we're going to be getting into today. If you are interested in a copy of this presentation before we get started, it's available at this bit.ly link, Seeking Science, because we'll be going into the science of seeking your customers today. As I mentioned, we're going to be going into the what, why, when, where, and how of finding your specific audience. So getting started with the what. Finding an audience is both an art and a science. We have both the intuition as well as the data that goes behind it. It's about getting to know people and quality data, but it's also about quantitative data as well. It is both micro and macro in that we're looking at large audiences, but we're also looking at individuals as well. It is multi-faceted because the people that we're looking at are not just one dimensional. They have lives, they have depth to them and that makes finding an audience multi-dimensional. And it's something that's constantly evolving. As people evolve, products also evolve and their audience evolve with them. So as a product gets more mature in the cycle, we're going to see shifts in the audience and that's something that is going to constantly change. And finally, it's something that can relate to everyone in your company and something that everyone is interested in. The user is generally considered the North Star for many products, for brands, for the media, for IT, as well as ultimately executives. So something that everyone really cares about. And it really comes down to understanding these questions. Who are they? Why do they buy? What do they ultimately need in the product or service that you're offering? What type of search queries did they use and how are they getting to you? What is that journey? What does that look like? And what matters to them as people? And so that comes down to a few different key questions to answer. So there's first off, in terms of categories, there's key information, there's things that are core to their personality, there's product information, as well as their relationship with technology. So in terms of key information, that often looks like things like demographics. So what is their age? What is their gender, their income, their location, their number of children, and that type of information is very general.  Another top question that you can ask in terms of key information is what are the top five adjectives that describe your audience in particular? Personal core relates to, instead of looking at it from who's the person on the outside, looking at the question from who's the person on the inside. So what do they want in life? What are some of their key motivators? What do they value and see as a self-concept? What general attitudes do they have? Like, for instance, I like to learn a lot. That's a general attitude that some people have and some people do not. And what are their cited interests? What are things that they've noted that they are interested in outside of your product and service? It's just all about getting to know them as a full person. Product related. So this gets into the nitty gritty of the information that you're going to want to look for. And at the core of this is, how do you win over this group? What do they really want from that product? What do they look for in the product? And one easy question to ask in terms of your top competition is I am most likely to buy my product or service from X, Y, or Z. And finally, for the last core question, what is their relationship with technology? What is their relationship with the web, with your website? What is their annual spend on that product or service? Where are they engaging online already? And how do we find them and engage with them on digital forms that are meaningful to them? And really the whole point of finding user data and analyzing it and finding insights from it is because it changes the conversation. Really, if you have no data, you have no context and you going into that conversation blind can be much more challenging, which affects many different streams of digital marketing. Context helps us to ask the right questions, to target what drives decisions, to identify where to find online and offline and determine how to ultimately engage. It also helps us connect in a meaningful capacity, identify what actually matters to people and understand ultimately their values and the ultimate goal is to create better experiences overall. So, when would you do this type of analysis? When would you go out and seek information specifically about your users? And really audience profiling and finding data extends the entire journey of a product from its initial development to its introduction to market, to huge growth, maturity, and decline because our audience at each stage is going to be very, very different. And it expands across of course branding as well as media strategy. So every channel within media cares about this and it's something that expands and it's really relevant at any point in time. It also affects deliverables across different VAUs. So, here's just like a simple visual of a very small sample of deliverables that use user information and audience profiling to inform that particular deliverable. And you can see just in this very small micro example that your user information affects your content channel, your information architecture, user experience, your design as well as all of your analytics deliverables as well as many others as well, which makes it something that's very relevant and useful. So getting into the meat of today's presentation, where? So where do we actually ultimately find this data? And generally comes from one of two places. There's really three out there, but for the most part, most times it's related to first party or third party data. So the third one is second party data. First party data of course is your own data that you collect and assimilate. Third party data is data that you get from someone else. And second party data is kind of that weird ground where maybe you have a partnership with someone and it's kind of an in-between mix of your data and their data and cross-sharing. But that's more of a rare opportunity, although it can be a very fruitful one. So getting it today, we're gonna start with first party data. There are many different places that you can collect first party data. The first and probably generally the most accessible one is web performance analytics. So that's something like Google Analytics or Adobe Analytics, your organic search performance, which is also free through Google Search Console and Big Webmaster Tools. You also have ranking data tools so you can find out what information people are getting to your site from for organic. There's surveying tools that you can use that can be free like Google Surveys and Survey Monkey. And finally, if you're lucky enough to have a CRM team, which we'll get into a little bit later in terms of third party data, some of the cool things about CRM. These are really generally the types of tools. And as you can see, they range from free to very, very expensive. So a lot of the times finding user information and user data is about identifying what's available to you and what's most cost-effective and time-effective. And here's some resources. So I wanted to go into some of the different things, different reports that I really like about within a few of these tools so that if you're looking for user information, you can more easily access that. So the first is GA, mostly because it's very accessible to most brands and most small businesses. There's a report in GA called Audience and Interests. And if you look in there, there's really two groups that you're mainly primarily looking at, the in-market and the affinity. The in-market is going to relate to your user's interest in relation to your current product. So a lot of times this is going to end up being a little bit more relevant, whereas affinity is more about people's general interest and hobby. And that can be interesting because it can tell you maybe a little bit more about like the sighted interests of people and things that they're doing in their free time outside of your product. But generally the in-market, to me, it seems like it's a little bit more useful generally. Other stuff that just kind of didn't fit in other places. So if you really want to get into the weeds, GA has a user explorer report that you can go into. You can filter by organic search. You can also filter by conversions. And you can go in and see the different steps that a user and I guess it's kind of like a user's ID, and that particular journey that has been tagged by GA. So you can literally follow them through a journey. Now, this is of course very in the leads, but if you're trying to get very specific or maybe you're trying to look to do some illustrative user journey mappings, this can be really, really useful. Okay, the last thing about GA that I think is pretty cool is you can actually ask questions in the search bar. I'm sure a lot of the audience has already explored this, but there's a bunch of different basic questions that you can ask relating to content, and Google actually breaks these out in their documentation by understanding trends, content analysis, and understanding user behavior. And a lot of times the user behavior relates to user engagement metrics, like time on site, bounce rate, but having these questions allows you to just ask in the search bar questions like, what are my top landing pages in terms of sessions? So you can get a little bit more, a better understanding of your content analysis, which is pretty cool. All right, of course, GSC, we all know it, we all love it. Yeah, there's a lot of stories to tell, and a lot of stories to tell in click behavior. What are people clicking onto your site? Categorization is obviously a very easy mechanism to identify patterns. There's a bunch of different ways to do this that a lot of people have been over. I have some links in this presentation on ideas for how to do it efficiently in Google Sheets, Jupyter Notebooks, there's a co-lab as well. You could also do it using machine learning. There's been a lot of great suggestions by like bringing more in everyone on how to sort keywords by using machine learning and with Batista as well. So definitely check out some of the stuff that they have. But you can also do it manually. So, but there's a lot of stories to tell about what users are doing and how they're interacting with your site. Surveys. Surveys can be a lot of times the only way to get very specific information that you want from a specific subset of people. Many times in third-party data structures, when they're asking questions on psychographic, they're doing it through a survey, but to a larger audience and with professionals that of course have written surveys for natives. Now, surveys for you can be a great way to get quick information, to get qualitative information, to make sure that you are getting exactly what you need. And I've included some tips here. The primary ones are to make sure you test the survey, to stay consistent as much as possible, to use plain clear lingo, to reference an expert style, keep it short, keep it punchy, and as much as possible, avoid leading questions. And if you have someone who's an expert in the field, definitely have them read it over and check because there's always some weird insights with surveys where it's like, oh, okay, well, I didn't expect that, but it's cool. All right, another thing that you can do if you're really close to some of your top customers and top fans and you wanna get an understanding of their general interests, you can ask them questions of course, but you can also ask them what their Discover interests are in Google. So anybody can access their Discover interests on Google by going through the mobile experience, clicking on the three little dots and checking out to see what your interests are. And if you're close enough with your users and can ask them, it may be a good area to explore commonalities. And if you can get a bunch of different users to agree to do this and to help you out, you can then take that, copy all that information down and count frequencies and see if anything interesting appears. So it can be kinda cool. All right, so third-party tools. There are a lot of different third-party tools and a lot of different types of third-party tools. First, of course, there is keyword search data. Many people in here are already familiar with that. I only have a short, short tidbit on that. Social data, audience tools, publicly available data sets, like the census or really anything like research, which is pretty cool, research publications, social listening, competitive tools, as well as CRM, which kind of makes finding third-party data this endless stream of information. All right, going into the specifics here, search data. So as many of you know, and this is my only slide on search data here because I wanted to keep it short because a lot of Mulscon attendees are experts, catarising that data by intent, by search features, so search intent proxy, basically, by brand and by different questions can be really useful to extrapolating information about how people are clicking through to the site. As well as using location data where available. Sometimes I've found in the past, I guess this is a more of a competitive type thing, but that certain brands were performing better because they were concentrating in specific markets more, which is just kind of interesting to see when you layer in location onto different competitive data. Yeah, can provide some additional context. Google Find My Audience has a free audience tool that's available. So if your audience aligns really well with the audience is that Google has outlined, you can use these tools and basically go through and input general information. It's very, very high level, but at the same time, you can get these nice PDFs back and you can go through and maybe get some additional insights or places to start more research. A little bit more useful is Facebook audience insights. There are two forms of Facebook audience insights to flavors, if you will. The first one is using your own Facebook profile, basically, or Facebook company information, which you can then load into the tool and look at that specific information. So it's really like looking at your own data on Facebook audience insights. The second is using Facebook's audiences. Overall, they're basically their base audiences and you can identify different themes and different targets and topics and pull out different demographics from that. So that can be another way to get more general, a general foothold into what you are interested in. So for example, if you know that you need to do research into cloud software, you can see what does the cloud software audience on Facebook look like? What are some of their job titles? What are some of their top categories and favorite things on Facebook? What location are they primarily in and what device usages do they have? So it's a great way if you are unfamiliar or maybe don't have a super high budget, can go through and pick out the information and identify some things about user audiences that Facebook has. Or of course, if you have your Facebook audience data, you can use that too. So you can get insights about your own audience that way, which is pretty cool. Of course, I would be a failure to not mention SparkToro, of course, which I'm sure many of you guys have already seen. So I just had some quick little things that I really loved about SparkToro here. And I think it's really useful if you're approaching audience insights from a content or influencer perspective, which of course was their target. It allows you to break data down in a variety of ways. It is pretty straightforward overview of the social accounts and the top accounts that are out there. If you're looking for the high influencers in the field, you'll be able to find those. It shows a lot of online consumption data as well as top-act network. So where's your audience actually engaging online, which of course is a core question. If you're doing work primarily on your website and everyone else is on Reddit, then they're not gonna find you. But yeah. It also includes some useful social listening to bits which can be really fun and if you love keyword research, then you probably also really like this part as well because it just kind of tries to create a broader person which is super interesting. Okay. Next, one of the things, I'm not sure how many other people in the industry do this, but I love looking at research publications. Some of my favorite include Pew Research for just general information. They have a lot of really good stuff. If you have a health client, the CDC has a lot of really great stuff. The US Census is great. There's also a plethora of other data sets that are published by companies like PWC as well as researchers. And this can be a great place to find general high-level audience information that allows you to make assumptions about things, things that are going on in the world. I also highly suggest the Mary Meeker Report every year, the Internet Trends Report. It's very, very good. And it kind of shows the idea of how do you take all this high-level data, compile it into something that's useful and pick out the trends that are going to affect businesses. I think she does a really good job of that, but that's the idea here is basically there's endless rooms of research out there that you can use and you can tap into to help understand what's going on in terms of your audience. All right. The last section I have here is a little bit expanded. And of course CRM fits into both the first party and third-party data world, but I wanted to talk a little bit more about that because I think that CRM isn't something that typically people talk about in SEO. And I think it's something that's super fascinating and a lot of times teams already have a CRM team that's built in another part of the organization and it's a matter of knowing what to ask, what data to ask them for. So one of the things that's really interesting about CRM is generally media of teams, think of CRM as something that's more useful on digital for targeted campaigns. And in truth, it kind of is in that what you're looking for with targeted campaigns is less discerning than what you're looking for if you're looking for a general audience like content. Now there are really two primary metrics that you're going to see in CRM data. The first is frequency. So the percentage of respondents who met a specific criteria and the index, which is the comparison to the base audience. So regardless of whether you're doing work for content or targeted campaigns, you always want to have a high index because that's what makes your audience unique. For example, if you are in Canada and you're Disney or something like that, they may over index on French speakers and that's what in Quebec and that's what makes them in Quebec unique, which is what you're ultimately looking for. The difference really between targeted campaigns and general content and publishing type initiatives is that when you're looking for content, you don't really want to have a low frequency. You really want to have high frequency that you look for. And that's because you don't want to just target those niche populations that maybe you over index on. You want to target the majority of people and you want to target what makes them unique. So, yeah. Okay, and the other thing that I wanted to introduce is this idea that there is a hierarchy of data and that a lot of times what we do in SEO sits somewhere in the middle of the bottom to the bottom of the hierarchy. And that's in my opinion, okay, because we really don't generally do one-to-one personal targeting, but it's something that can be done. So, panel data at the bottom is things like surveys, personas, sample customer data, segment level analysis, audiences, and that's kind of a lot of the comfort area for SEO. I think ultimately, probably in the future, as websites evolve, maybe, I don't know, five, 10 years or something like that, we'll get into something that can address more personal level computer experiences for non-logged-in users or non-authenticated users. As of now, I think a lot of that split is between new users and existing users. If you're, for instance, cookie does one of those. There's cookies, which basically is what that is. So, if you're cookie does an existing user, then you may get an existing user experience. And if you're not cookieed as an existing user, maybe you'll get a new user, or maybe if you, for instance, cleared your cookies or something, you'll get the new user experience. But there is this hierarchy that at the top of it sits personal level data. And it's not really, you know, Ted to Bob or something like that or specific people. And it's more about your IDs that they assign you on one, which is a very, I guess, interesting concept to that it's not personally identifiable information if it's not you, kind of interesting. And this data expands very far. There are many companies out there and pretty much, I guess, every company that you could name probably collects data about you and all this interconnects and you can associate all that data back with your ID if you're lucky enough. And there's something that comes into CRM, which is called the closed loop. And what that means, and that's what they really strive to do to get personal one-to-one data. And what that means is that if on platform one, we have cute bow tie guy, and on platform three, you have cute bow tie guy.  And the same for platform two and platform four, you know that each of these are the same. But for whatever reason, we find out that platform one guy is also the same as platform four guy. We can actually close this loop and determine that this person is the same person that we're all talking about. Even though we don't know this person's name, we don't know any of their personal information, obviously, we know that when we're talking about him on platform one, we're also talking about him on platform two as well as three and four. And that's what closing the loop is. Now, you can't always have a perfect data set and any database cleanser will tell you that. But there's just something known as match rate, which is basically like how many of these were we able to match and determine that we have data on this person and that matches. And that's pretty much how that number calculates out. And at Merkle, we have an internal system called M1, which is something that mostly are media teams use to for remarketing purposes and to get more specific data on audiences and they have their own thing going on. But if our client has purchased it, it's something that we can also use for SEO. And I just wanted to share with you some of the things that I thought were interesting here. So, and the importance, of course, of having an analyst. So data, of course, is not just, as I mentioned before, a science. It's also ended up in art and there's a lot of intuition that goes behind it. So I just wanna share with you some examples from that particular system. So, all right, minute made. According to data, 35 through 49 year olds crush minute made. And as you can see here, the gray, of course, is the base audience. The blue is, of course, index. So we're looking for how much the data overlaps here in this visual. So you can see for the 35 through 49 year olds, the data just explodes. Like people who are 35 through 49 love minute made. And of course, you look at like their net worth and like, whoa, these people are worth 150,000 to $375,000. Like what is in that choose? Pretty cool. But if you're, of course, an analyst and digging through the data, you may also discover, okay, well, maybe it's not that these guys love minute made. Maybe they're not super pumped about getting some OJ in the morning, but more about the fact that they all have children. So it's kind of one of the more fun things. When people have kids, it kind of messes up the data set, but it's use it in a way that makes it kind of interesting. You know, data looking at this type of stuff and serum data can also make you introspective in such that things like, why do 76 year olds index so low on eating and going out to Chipotle? Even more interesting, why is it that people don't get Chipotle as much when there's a grandparent in the house? I don't know. Maybe make that 2020 initiative, get your grandparent to a Chipotle. Anyways, okay. So diving into a little bit of the how, how do we format this data? How do we compile it? Generally, this ends up being a lot of information. So you have to have a way of processing it and thinking about it and how do we present this so that we can develop something that we can simulate to send out to the rest of the teams. My suggestion is two things. The first, brainwriting. The second being mini shreds. So brainwriting is exactly like brainstorming, but instead of having everybody get in the same room together and talk out ideas initially, you have people go and write their thoughts out and process the information. And oftentimes what that allows people to do is come back to their roots to idea before talking and process information. It also allows for if you have a few extroverted voices or people that are of influence that are more extroverted and more authoritative, it allows people who are maybe more introverted or take a little bit more time to process to think about their thoughts, go through and process them and have them all written out and prepared ahead of time. So you can have a productive meeting. Where the charade comes in is basically a mini meeting where everybody comes together and talks about their ideas and you compile it into something that's useful and kind of pulls out the top insights from all different angles here. And of course, I included some screenshots down here for you guys to see. Some examples where we used Microsoft whiteboard and you can see in the brainwriting side on the blue side is one person, the red overlay is another, the yellow overlay is another. And you can see these can get kind of exciting and it's kind of has like a dynamic feel to it.  How do you present this information? So I have three different ideas of how I personally think are different ways to present this information. But I mean, really it's endless. Like there's many different things you could use data for and really doesn't necessarily need to be something that's presented, it could be bullets in an email, but here's some ideas. So the first is an illustrative user journey for content. And what that looks like is basically picking out either an audience or a specific user, identifying different stages along their journey. And of course you would fill in all these little boxes here, identifying and aligning what website touchpoints exist. And then finally identifying the primary focal point of the user. Are they looking for something like convenience at this stage? Are they looking for information, trying to figure out what is the one specific user audience? And sometimes this can help because instead of trying to think of it as this big problem, you can reduce it into a problem that's more accessible and more relatable. Then later you can take it and expand this out to be something a little bit bigger. But it's something that can, as I mentioned, reduce the problem, which is key in solving complex issues. The next is, I wanted to introduce before I got started into the audience journeys and briefs, the means and model. And there's a ton of different psychological models that you can use. The means and model is just one. But what it relates to is this idea that there are product attributes and product benefits that are things that are on the surface that people see and that are tangible, but then underline our feelings and a sense of self that come with every single thing that we as human beings do. We seek to align. And a simple sentence to kind of describe how this goes is what are the tangible benefits of the product that lead to functional benefits for the person? Basically, what are they getting out of it on the surface? Which affect how that user or customer feels and reflects on their personal values. And that's something that I think fits really well into briefs because you can basically identify those underlying components that go into each person. But that of course requires a lot of data, a lot of analysis, a lot of chunking, a lot of different ideas coming into the room. So yeah, this is one example of what an audience brief may look like. Of course, there's a lot of flexibility there. And finally, there's this idea of mapping user journey. For content, I think there are a few different rows of information that are really important. So making sure that you identify what is the primary goal of the user at that particular stage, the types of content that they would need, the common traits, I think is something that's very useful. Content location, you can keep it or leave it. The next stage is also really important. Like how are we pushing them down the particular funnel? And of course, starting with the different columns, acceptance, I'm sorry, awareness, acceptance, consideration, decision, it makes sense. Sometimes you need to expand and contract that model depending on the different audience. And you can see here we modified it a little bit. And journey mapping is something that can also expand to all channels. It's not something that necessarily is just about content. It also can be about every single channel and every single touch point and how all those points engage. And here's just one small screenshot example of what that looks like at a larger strategic level. So our TLDR for today, our summary, is that in my opinion, for doing user research, use what data you have available. And there is data available to you out there. Communicate to others on teams to get a fuller picture of what's going on with users because other teams may have different data than you and you can use that. The third is that user research is something that's constantly evolving and something that you can consistently do but is something that makes more sense to do at specific intervals that makes sense for you and is efficient for your team and something that you need to almost constantly be doing as you go and constantly learning, which is exciting. And finally, that personal one-to-one campaigns are our thing. And to close out today, I just wanted to mention this kind of closing thought that it's about constant change and that is the new norm. It's not about guessing right and having a project and going back to some limo. It's about learning fast and trying different things. I have a few special things to you, Abby, Chris, Max, Caitlin and Adam. Thank you so much.  Once again, virtual. Yay. Thanks everyone.