Okay, let's get started. Welcome to sadly the last of Tim's lectures on inverse Galois theory. Thank you, Tim. Yeah, thank you, Akshay. So to remind you where we stand, I started to talk a little bit about the group SL2 of three. The reason for that is, I mean, there are a couple of reasons it's the smallest group which is not semi-Abelian. And that makes it interesting because we haven't so far got any constructions out to get its families. And so this is an example of a group which is in fact an extension of C3 by Q8. But in this case, you see this group is non-Abelian here. And so our theorem doesn't apply. So what we're going to prove that in this case, actually the embedding problem is soluble and you can extend every C3 field, putting it inside into an SL2 of three field. So I finished last lecture by just writing down the list of subgroups of SL2 of three which are relevant. So it misses two subgroups that we don't care about but it has a whole tower for Q8 and then going up to SL2 of three. And by Galois correspondence, it gives us an indication of what we're going to try to do. We'll start with any cubic extension depending on what this field is, think of it as a Q. So cubic extension of Q or maybe Q of T. So then it's a family then. And then I would like to compute a quaternion extension on top of that, on which C3 acts like it acts on Q8 naturally as automorphisms by permuting i, j and k cyclically. So this is what we're going to try to do. And for that, we're not quite going, I mean, in theory, we know everything about Q8 extensions now we have a theorem due to Witt. But there is a slightly different characterization of quaternion extensions. A lot of people that were interested in quaternion extensions, which even goes earlier than Witt. It's a theorem of Bucht from 1910, which is essentially equivalent to what Witt has done in this case. And what he proves is the following, that if you have a bipoderative extension of some field, so let me call it capital K because that will be like in my diagram. And then I would like to compute a quaternion extension on top of it. So I start with a bi-quadratic and I ask, can I embed it into a quaternion extension? Bicoderatics are easy, take two elements and join the square roots, but quaternion extensions are harder. So, and here, because this problem is abstracted, if you remember, and in this case, what he showed is that such a bi-quadratic is embeddable in a quaternion extension, if and only if this U and V are up to squares in K, so squares in K obviously don't change which quadratic you get, but up to squares in K, U and V have the following sort of funny form. There exists alpha, beta and gamma in K, such that U is one plus alpha squared plus alpha squared beta squared times the same thing but shifted where you commute cyclically alpha, beta, gamma, similarly for V, when you shift this whole construction a bit further, and if you multiply these two together, you'll also see that one of these terms, maybe this one, becomes a square, so you can absorb it into square in K, so the expression for U, V, it's also very symmetric in this context. And now, so if you stare at this, well, first of all, I call this bi-quadratic square root of U and square root of V, but actually it has three quadratic subfields in it which are quite symmetric and they're given by square root U, square root V and square root U, V, so these are the three numbers whose square roots I'm joining, and you see his construction is very symmetric in U, V and W in a sense that if it respects the C3 symmetry, if I take this alpha, beta and gamma in such a way that they're invariant that C3 permutes them cyclically, it'll permute these three guys cyclically, and that's exactly what we want because we want not just any quaternary extension of the top of a C3, it'll give us a large risk product, but the one which respects the C3 action. And this, now with this, it's very easy both sort of in practice and in theory now to construct a sort of three extensions, so this is how we go about it. We take now, as I said, where essentially solving an embedded problem, so take any C3 extension, let's say of Q, and then you wait to put it in the family, so take any cubic Galois extension of the rational, which is going through the bottom of our diagram. So maybe let me go back for a second, so recall that again, in SL2 is three, quaternions are an index two subgroup, and then inside there are three subgroups of index two permuted by C3, and then there's a center and then this whole thing, and then identity, and the corresponding tower of fields is that we started with the cubic extension of little k, and then we are going to construct three quadratic extensions, which are permuted by C3. So what we are going to do, we take alpha to be, well, basically anything you want, so take it to be a general enough element of k, I'll comment to this in a second, sort of like when we constructed cyclic families and so on, when we said, well, take a general element of Q zeta n, and then apply its construction. So let's take alpha to be a general element in this cubic extension of Q, and now everything is fixed, because what I would like to do, I would like to have these three quadratics permuted by gamma, so I would like alpha beta at gamma to be permuted by gamma, so I'm going to let the gamma group of C3, so sorry, the gamma group of k over Q, the cyclic group of order three, let's pick a generator called G, I'm going to hit alpha with G, I get beta in k, hit it again by G, you get gamma in k, so alpha beta gamma would be three elements in this field, which are permuted by this C3 downstairs. Okay, this gives us alpha beta gamma, then we divide, define UV and W by this formula, and by his criterion now, this extension is first of all embedded into a quaternion extension, and secondly, everything is so symmetric that it's not hard at the quaternion extensions, if you recall from Witt's theorem, I essentially unique that it forces the whole thing to be Galois over Q with Galois group SL2 and 3. In fact, because Witt is very explicit, it tells you what is the extension that you get, you can unravel it, and then you find that the SL2 and 3 extension that we're after is given by this, very nice, very symmetric, explicit formula. It's K, I join root U, root V, so that's a bipodratic, and then after that, to get the quaternion thing on top, we're joined square root of this expression, which is, you see again, symmetrical UV and W. So the only thing is that if you do this theoretically, we haven't quite solved an embedding problem yet, because it could be, you know, that something goes terribly wrong and there are no such thing as general enough elements of K for this to work. So for example, for some reason, you always get smaller extensions, that all these elements, for example, for some reason, if you take them in less three extension, could be, I don't know, a square. So this style on top that we've constructed is going to be trivial. So you have to do a little bit of work to prove that this is not the case and these things exist, no matter what a little capital K or a little K is, but it's not terribly hard. And there is a theorem, which I think Fadeev was first to prove in 1945. So he proved that this, what I called picking in general enough element of Ks for this construction to work is always possible. So he showed that over Q or Q of T, or in fact, over any hill version field, this construction works and every C three extension of little K can be embedded into an SL two or three extension of little K. In particular, these things exist and there are regular families of SL two or three extensions. So again, this construction is kind of interesting is that it does solve an embedding problem in the case where the kernel is a non-ability group and it works well. And of course, you can ask yourself, well, like when does this work? Can you generalize it to other groups and semi-direct Q where N is non-ability? In other words, suppose I have a family which has a Galois group Q or some field Q or Q of T or some regression field, whatever, and I want to extend it to a family. I want to put it inside a larger group which has the following shape and semi-direct Q where N is an arbitrary group for which N and Q you can do this. If you could do this, at least soluble groups I think you could get very far with this process. This is not known. Well, in fact, this is not known for which class of groups it is possible but there's a big conjecture by the Bayer-Deschamps that say that it's always possible. So the conjecture says that every what we've now considered the obvious name split embedding problem. So lifting a Q extension to an N semi-direct Q extension for any N and any Q it's always possible over any hill version field. Now, of course, it's a difficult conjecture because it's much stronger than the inverse Galois problem. You can take N can be arbitrary and Q could be trivial. So for example, it says that you can embed any C1 extension into an extension with an arbitrary group, a Galois group. So this is a very strong statement but at least it gives you an indication of what we expect to be true and maybe hopeful to find some techniques which work and there are some very interesting groups that you could try to play with and see if you can generalize this construction in some way. So the difficulty here really is that what happens here secretly is that, remember again, the Quaternion group, it has an auto-automorphism of all the three and here we've got a family or construction whatever of Quaternion extensions which respects this auto-automorphism action. And this was what make, and this is now what makes it possible to push it further to the two or three extensions. And normally when you construct a family for some group G it is not going to be invariant on the auto-automorphism of G in any way. Well, unless maybe you try hard or I don't know. So it really raised an obvious question is how far you can push this. In chat, if there are any questions, if no. So I think that's all I want to say about this whole embedding problem and to finish by talking a little bit about two methods. So we've talked about semi-Abelian groups and split embedding problems. This is a very nice channel method and we know when it applies more or less except there are conjunctures that say that it applies much, much generally. But let us look at two other methods. One which is called descent to subgroups and one which called rigidity, which again, I think have a potential to be pushed further and they would be sort of very interesting I think to study as research problems. So one obvious question I talked a lot about with products, direct products, quotient groups, but well, except for an answer to one of the questions and one of the first lectures, I didn't talk much about subgroups and there is an obvious question. Suppose you constructed a family which has Galois group U, let's say, and then you have a subgroup G U and the question is can you find a subfamily which has this Galois group G? So for example, what Hilbert's construction of AN extension essentially does, it says let's start with an SN family such as X to the N plus AX plus B and then find some specialization. So try to find let's say A and B to be rational functions on some variable T that the discriminant of this thing becomes a square and the family drops from SN to AM. And you can ask yourself, when is this possible? And again, that's a big question. If this were always possible, you could solve the inverse Galois problem because every group is a subgroup of SN and we know how to construct SN families. So if we could always descend to a subgroup, then we could construct an arbitrary group G. We don't expect this really. As asked like this for an arbitrary family, this is certainly not true and you can not always do that. So let us look at example where you can construct C3 families starting from an S3 family like this by starting this descent problem. And already this case shows very nicely what sort of problems you encounter along the way. So again, what I would like to do, well, essentially this is Hilbert's case because this is A3 inside S3. I would like to construct families of extensions of cubic Galois extensions, let's say a cube starting from the fact that I know a lot of S3 extensions because just take an arbitrary cubic and then, well, you've got most likely it's going to be an S3 extension when it's going to be small. So let's start with the S3 family which is built in the package for no particular reason. It's just has small coefficients. It's an obvious family where you take a polynomial X cubed plus X plus A. Right, so again, what I would like to do, I would like to see if I think of this as a family of S3 extensions, so I take A to be one, two, three, any rational number, it'll give me an extension. And a question is, does it sometimes become dropped? Does the group sometimes drop to C3 and can you put these exceptions into a family? Well, how do you know whether it drops to C3? You have to compute the discriminant and see if the discriminant is a square. So you compute the discriminant of this cubic up to squares, it's minus 27 A squared minus four. And what I'd like to see is that if there are specific rational values for A or ideally a whole rational function, let's say A of T such that minus 27 A of T squared minus four is equal to a square. And if yes, then by plugging in instead of A, this sub family A of T will give me a family now over P1 with a variable T, a larger degree possibly, but now it will have the Galois group contained in C3. Now, unfortunately for this particular family, this is clearly impossible because you can't make this into a square. Minus 27 A squared minus four is very negative. So if you want this to be equal to B squared, this gives you a two to zero curve over the rationals which doesn't have any rational points because it doesn't have any real points. So there is a local obstruction in this case. And therefore, you can forget about this family. It doesn't have any C3 specializations. You might as well sort of throw it away and start over with you. So let's do that. Let's take another S3 family which is X cubed plus AX plus one. So now, I let this last coefficient to be one and the linear term to vary with A. The discriminant of this is minus four A cubed minus 27, oops, sorry about that. This B squared shouldn't be there because B is one. So it should be just minus four A cubed minus 27, like here. And again, you can ask yourself, can this expression be a square sometimes? So in other words, what happens if I equate this expression to B squared? What's the set of rational solutions to this equation look like? Well, if you look at the equation B squared equals minus four A cubed minus 27, this is what's called an elliptic curve. It's a curve of genus one. It may have rational points. So it is quite possible that you can find values for A and B such that this expression here on the right is a square. I didn't check this to be honest. But in any case, what we know for sure is that because it's a high genus curve, it certainly doesn't admit any maps of P1 into it. So you cannot map rationally P1 into an elliptic curve. And therefore there is no rational function like this A of t such that minus four A of t cubed minus 27 is always a square because this is a non-singular curve of genus one. So in other words, using this family here, you may be able to construct C three extensions over Q. So for example, to solve inverse go problem for C three over Q, by just saying, well, let's take A to be this number, then this thing just happens to draw to C three, the Galois group, but you cannot solve the inverse Galois or remove Q of t with it. And what you find out if you play with this in practice, a lot of families that are built in, for example, in the package, if you do this, so you take a family for a specific group and let you say, for instance, what happens if I try to go to a, try to descend to a quadratic subfield. So this quadratic subfield will be like here, given by an equation B squared or X squared, if you want equals some polynomial in A. And sometimes this will be a genus zero curve, which doesn't have rational points. Sometimes it will be a high genus curve like here. Sometimes you're lucky and it will be a genus zero curve with rational points, but that's not going to happen very often. So there will be very few situations where this descent is going to work for one-dimensional families. Yeah, so can you explain again why the discriminant being a square means that there is C3 extension? So generally, when you have an S3 extension, so remember what an S3 extension looks like. So S3 is a normal subgroup, which is C3, and that corresponds by Galois theory to a quadratic extension of Q contained inside the specific field. So when you have, for example, this S3 extension, the quadratic inside it, it's given by the square root of the discriminant of this polynomial. So inside this field here, there is a Q a joint square root of this. But of course, if this expression is a square, then Q a joint square root of this becomes a trivial extension. And in that case, your group, the Galois group is not S3 anymore. It has to be a smaller group. In fact, it's has to be contained in the alternating group A3. So in general, when you have a family, let's say given by a polynomial of degree N, its Galois group is some, let's say transitive subgroup of SN, transitive if the polynomial is useful. But if the discriminant of that polynomial is square, then it is in fact contained in the alternating group A. So if you can force a discriminant to be square, you can force your group to drop into the alternating group. And that's what I'm trying to do here because the alternating group here is just C3. So I'm really forcing the discriminant of the cubic to be a square. Okay, and now, but what happens here is that if you forget for the moment, like I spoke from most of my lectures so far, about one dimensional families, about families with just one parameter eight. We said we're interesting in an inverse Galois problem of Q of t or Q of a. If you allow families with larger number of variables, then suddenly you can push this method much further. And let me illustrate it in this case, let's take the generic S3 family, XQ plus AX plus B where A and B are two parameters. So this now in fact covers all S3 extensions of Q. And it's discriminant again, it's given by this formula and minus four AQ minus 27 B squared. And again, I'm asking myself the same question, can I make this expression of square? Can I find some substitutions for A and B, some rational functions, which satisfy the property that minus four times, this rational function Q minus this rational function squared is always a square. So in other words, I'm interested now in equation C squared equals minus four AQ minus 27 B squared. And this equation is not a curve anymore, it's a surface because it's now given by one equation, three variables. And because this equation has the largest degree, you see here is three is what's called the cubic surface, it's a surface of degree three. And such surfaces, if you're lucky and they have rational points at all, then there was called rational. So they are birational to the projective plane. So there is a map, which is defined as an open subset of P2 from a projective space with two variables, let's say capital A, capital B, or three variables you make homogeneous to our surface S here. And I left a similar exercise, I just put it up on the web for slightly more interesting groups that S3, but various computer algebra systems including magma have a machinery to do that. So you can give it a surface and you ask, is the surface rational? And if he asks, can you give me, can you give me a rational parameterization please? And in this case, what it says, it says yet surely the surface is rational and it can be parameterized as follows. If you take A to be this expression here minus a quarter of 27 A squared plus B squared and B equals this and C equals this, then you take this polynomials and capital A and capital B, you plug them in here, you find out that this equation is universally satisfied. So this gives a rational parameterization P2 to S. So this map is more or less an isomorphism in the sense that there are rational functions which give you that the other way expressing A and B in terms of these things which is kind of quite easy from these formulas. And yes, that's probably true. Yes, thank you, Bjorn. Yes, in this case, you could easily see that this surface is rational and just so for T. Yeah, so you don't need much for this case here. Yeah, I think you're right. So but in any case, either way, once you found a parameterization like this, then you see that you can substitute this A, B and C into the family of mine, into this family here. Yeah, so C I don't really care about. It's some sort of, it's what solves what makes, it's a square root of this expression. But the A and B I do care about. So if I plug A to be this and B to be this, then I find a family like this of cubic equations which is now, they have square discriminant and the discriminant is C squared where C is this expression here. So I find now a family of C3 extensions of QAB and in fact, this family is generic in a sense that it covers every single cubic extension, Galois cubic extension of Q, which is easy to see because every cubic extension of Q can be given by an equation like this. So it does correspond to a rational point on the surface and because we found that by rational isomorphism like this, we covered all rational points on the surface possibly with some exceptions, but I don't think in this case they matter. So this family here is what's called generic. It covers every single C3 extension of Q or in fact, over any extension of Q. So this is sort of a nice baby example. And it turns out that this method works in examples which are much more interesting. So let me give you one such example and look a little bit at the smallest groups for which the inverse Galois problem is unknown over Q of T because we've talked a lot about methods which allow you to go this far. So if you've looked at the list of semi-herbillian groups of order less than 64, you would have found that there are, I forgot I mentioned last time, I think six exceptions, one of them is A5 which we know how to do over Q of T by Hilbert and the other all relate to SL2F3. And it turns out that once you can construct SL2F3 by again fiddling with split extensions which you can now do for that family, you can construct the other groups there as well. So the smallest groups which are not known to be Galois groups with regularizations over Q of T are groups of what is 64. So again, if you've ever generated this list of non semi-herbillian groups, you'd find that there are 10 groups of what is 64 which are not semi-herbillian. So in a small group database, there's 250, I think groups of what is 64, it's a long list but there is 10 of them with numbers 8, 9, 10, 11, 12, 13, 14, 41, 42, 43 that turn out to be non semi-herbillian. To answer the question, is there any indication when a family will be generic, not as far as I know. So I mean, there are quite a few babies where people prove that certain families are generic, but they always like in my case essentially kind of generic by construction rather than you first construct a family and then you try to figure out whether it's generic or not generic. So I do not know, I don't know what answer to this. We're just looking at a family to decide that it covers every single extension of with that Galois group. I don't know if any such results to be honest. Okay, so there are 10 groups of what is 64 which are not semi-herbillian, in other words, which we don't know how to construct over QFT. And now groups, they all need to be generic. Groups, they all need potent and the new potent groups are classified by what's called the new potency class, the length of the central series. If you take the center of a group and divide by the center, you take the center of a group again and so on until you exhaust the whole group which you can always do in a new potent group. I mentioned very briefly there is a result by Thompson that every group from the potency class two is semi-herbillian and of these 10, seven are have new potency class three and three have new potency class four. And there's a paper by Leila Schnepz where she proves by extending this semi-herbillian, no, sorry, by looking, by studying the obstructions in this case of various lifting problems, she proves that the new potency class three groups of these seven, they in fact have realizations over QFT. So the three groups, which as far as I know are not known to be realizable over QFT are these three, the small group of 64, 41, 42, and 43. Let me just call them G1, G2, and G3. And it turns out that for at least one of them, this descent method works even though it's reasonably convoluted because you don't need to have quite a few steps in it. And you can construct a family for the first one of these three groups using this method. So if you look at G1, small group of 64, 41, then it's a transitive group of 60 points, number 156. It's a group of over 64. And it says as a maximum subgroup of some group of 128, which sits in a maximum subgroup, specific group of 256, 512, 1024, of which the last one, this particular group of 1024 still acting on 16 points, turns out to be a nice semi-direct product, which you can construct very easily. You can construct lots of families for this group using just resultant. It's essentially a with product kind of group. And then you can do this descent. So you can construct a family for this group and then you can ask yourself, can you find, can you descend to a subgroup of index two? Turns out that you can. Can you descend again? Turns out that you can. Can you descend again? Turns out that you can. And finally to the group G1. Along the way, you get some rational curves or rational surfaces. And in all cases, they have rational privatizations. There are non-local obstructions and you can construct a family for this group G1. G2 is very similar, except that I haven't been able yet to construct, to find a family for this group here, the top group on top, which I would be able to do this set all the way. I can only get up to the last step up to here, but not up to this specific group. And the third group G3 only acts on 32 points rather than 16, which makes the whole thing just a little bit more cumbersome. So in any case, what I just want to say about this descent method is that it does seem to be very powerful in practice. So it has been very helpful to construct a lot of groups and especially nice families for a lot of groups that haven't been known before. But I find it's very difficult to analyze theoretically. That is to say, if I start with a group and I start with a subgroup or even a maximum subgroup, somehow predicting in advance, that this descent is or is not going to be possible, whether you will end up running into local obstructions or not, I do not know both methods for doing that. So it's a little bit hard to, as far as I'm aware, to prove theoretical results with this method. That would be, I think, very interesting if one could do something like this rather than just do it group by group, family by family. Well, you can do this, but it's hard to do it on some sort of general basis. So to predict at least some situations when it works, I think that would be quite amazing. Okay, so that's, I think, all I want to say about this descent thing and except that, as I said, it does work very nicely in practice. And I gave, again, one exercise, how to construct, in fact, a generic family for fraternian groups using this method here, where you need to do a descent through a delpets of surface of degree four and parameterizing. It's a very nice example. Okay, and finally, the last method that I would like to talk about, it's called rigidity. It's a very powerful method which is on the opposite spectrum of what we've looked so far. We've mostly looked at soluble groups and this method applies mostly to simple groups or groups which are very close to simple. So let me, there are many variants of it. People did try to push it quite far, even though I think the work there is certainly not exhaustive, I'm sure you can do more with it. But let me at least state some of the basic versions of the rigidity method. So first of all, let us start with something that's called the Riemann existence theorem, which, as I, again, briefly mentioned before, it settles the inverse Galois problem over C of T and are analogs by Harbater and Pop. And there is other people who replace C here by a complete field, but over C, this is a very classical and very strong result due to Riemann. Since the following, suppose you have a finite group and you pick any number, any generators of G that you want, which multiply to one. So you pick elements G1 up to GL and G, completely arbitrary with the only condition here that G1 times G2 times and so on GLs just in this order is equal to one. And then you take arbitrary L, so it's the same L as this number of element, points on a projective line, so distinct points P1 up to PL. And then the Riemann existence theorem tells you that there exists a Galois cover of P1. So in other words, this is the compact Riemann surface which maps to P1, which has G, this group here, acting when it is an automorphism group, such that if you take the quotient of this group by of this Riemann surface by G, you get P1. So in other words, you get a map next to P1, such that G acts faithfully, so most points, they have exactly the order of G pre-images and such points are called already-fied, except, finally, many points where this is not true and this points, over these points, you say that the map phi is Riemann-fied. So, and the Riemann-ification locus is exactly these points P1 up to PL here, at least assuming the G is our identity. And so in other words, this Galois cover here is un-Riemann-fied outside this fine set of points S. And associated to every Riemann-fied point is what's called the Riemann-fied group. It's exactly the same as certification groups you see number theory. If you think of a cover of curves like this, in terms of their fields of rational functions, it gives you a Galois extension of fields of rational functions, where it goes the other way. So the field of rational functions here is just Q of t and you embed it into a larger field which has Galois group G over Q of t, sorry, C of t. We're in the complex numbers, I have to remember that. So C of t here and a fine extension is Galois group G here. And points of P1 you think of as places of the field of rational functions and the ramifications, the usual notion of ramification and the ramification group is the usual notion of our ramification group, except that here the residue field complex numbers algebraically closed. So this group here is always cyclic. And so what's in existence says is that you will discover to be un-ramified outside S and exactly have prescribed ramification groups. So ramification group over PI is generated by GI. But to be precise, ramification group is only defined up to conjugacy because if you take a different point over PI, you get a conjugate group. But one of these conjugates is exactly the group generated by GI. So it's an extremely powerful statement. And in particular, it tells you that the inverse Galois problem is very much true over C of t because so let's just prove this. In other words, every finite group has a regular realization over C of t. So you take any group G, you take any generators which multiply to one. So for example, you take any generators of G call them G1 up to GL minus one. And then you define GL to be the inverse of G1 GL minus one is supposed to be inverse. So the product is equal to one. And well, you apply Riemann existence here. So you take completely arbitrary points P1 up to PL. You can take any choice you wish. And then by Riemann existence, you find a cover like this phi with Galois group G which again by Galois theory corresponds to a field extension where the bottom field is C of t the field of rational functions of P1 and the top field, the field of rational functions of this curve X has Galois group G over it. And of course, it's automatically irregular because the field of constants here is algebraically closed. So there are no algebraic elements in this field over C except C itself. So that's it. So over C of t, you can construct covers like this and there's a lot of choice involved. And the problem is descending from C to Q because in general, you can write down some sort of variety which is called Hurwitz schema when it's one-dimensional Hurwitz curve which parameterizes things like this which parameterizes covers with a specific group and specific verification data. But this variety is difficult to compute, difficult to understand and so on. And they don't necessarily have rational points. So the problem really is to somehow get from here to the rational maps and to do the same sort of thing but over Q of t. And the trick is there is a general principle sort of a number theory and Galois theory that you can construct any sort of object over the complex numbers which is unique then just uniqueness forces it to be defined over the rationales. Basically because if it wasn't defined over rationales you could hit it by Galois and find another object like this. But now if it's unique, then it's not possible. Of course, this is a more philosophy than a theorem but it very often works and it works here in the same way. So generally the covers which are given by this theorem are not unique. There's a lot of choices involved and choices of these points PI, choices of these GIs and so on. And but you can what's called rigidified by putting enough conditions on this conjugacy classes. So on this GIs and selecting PI is sort of fixing them to be certain rational points in such a way that discover can be proved them to be unique and then it's can also be proved to be defined over Q. So that's what this rigidity method does. So this is how it works. So suppose G is a fine group and you pick L, it's the same L as before as I had this elements G1 up to GL. You pick L conjugacy classes. So let's say you fix them, let's call them C1 up to CL. So conjugacy classes of G and then you look at, well, basically what we're looking up there, the set of L tuples of elements G1 up to GL where GI each lies in the corresponding conjugacy class CI and their product is one. So G1, G2 up to GL is equal to one and they generate the whole group. So the group generated by G1 up to GL is equal to G. Now, if you just pick group at random and conjugacy class at random, this set may be empty because for example, you could just say, I want all my CI to be identity. Well, sure you can, but then they are never going to generate the whole group. So these conditions here may be sort of usually exclusive, but whatever this set is, let's say assume for the moment it's non-empty, G acts on it by conjugation because you can take an element little G and G and then conjugate all the GI by G and then you'll see that these conditions are going to be preserved. Conjugacy classes of course are preserved by conjugation. This product will still be one if you put here G and G in versus everywhere and they will still generate the whole group G because essentially they generate a conjugate of G inside G. So in terms of freedom of surfaces, what this conjugation does, it actually produces an isomorphic group of surface. So, okay, and normally this group G will act freely on it by conjugation. So in other words, all elements except identity are not going to have any fixed points. So let's look at this condition a little bit. So if G, little G has a fixed point on sigma, so in other words, suppose you have such a tuple G1 and 2GL that you conjugate by G and you find exactly the same tuple. So again, G acts on the tuples by conjugation. So it means that G conjugates G1 to G1, G2 to G2 and so on. So in other words, G commutes with all the GI's but because GI's generate the whole group, G commutes with the whole group and that means that G is in the center of G. So this condition is just so having fixed points equivalent to being in the center. So from now on, when we're going to talk about this rigidity method, we're going to assume that the center of G is trivial. As I mentioned before in one of the first lectures, this rigidity is a powerful method. It applies to a lot of interesting groups like the monster groups, but it certainly is restrictive. And for example, it never applies to groups such as milpotent groups because of this condition, the center has to be trivial. So then the action G on sigma is free. So it's just a union of orbits which are just regular orbits. Each one is sort of like a little copy of G. And we say that this tuples C1 up to CL is a rigid L tuple of conjugacy classes. If the size of this set, well, it's not empty, but otherwise in that it's as small as possible. So the size of this set sigma is exactly the order of G. Equivalently, you have only one orbit. The action of G on sigma is transitable. So every such tuple can be obtained from every other such tuple by conjugation by G. So this is the set up. So this is the most important definition here. So think about like this, you have a group, you have a bunch of conjugacy classes, you compute the order of this set, and if it turns out to be order of G, then you're like, okay, great, this is a tuple to which rigidity method, at least some form is probably going to apply. And a second condition, which you have to impose in some form or other to make the resulting family defined over Q, is what's called rationality for conjugacy classes. So it's also even easier to define. So if you've got a conjugacy class in G, it's called rational. If it has the following property, you take an element in that conjugacy class and you raise it to a power, which is co-prime to the order of G. So for example, you take the conjugacy class of elements of order three, then you take an element there and then you square it or you raise it to the power of four or power five anything co-prime to three because you don't want to get identity or something which is in a smaller conjugacy class but something co-prime to the order of G. And then the condition is conjugacy class is rational is that you always end up with the same class. So for any element G in the conjugacy class and any power which is co-prime to the order of G, this power here lies again in the same conjugacy class. So there are some examples of groups which are called rational groups. In fact, where every conjugacy class is rational and the most famous of them is the symmetrical SN. So if you take G to be SN, then it's conjugacy class to be correspond to cycle types. So for instance, in S4 you have a cycle type, four cycles, three cycles, double transpositions and so on. And our cycle type is unchanged under this operation. If you take, I don't know, a four cycle and you raise it to the power of three anything co-prime to four, then it stays a four cycle. So you end up in exactly the same conjugacy class. And that means that every conjugacy class in SN is rational. So SN is one of these groups to which this rigidity applies quite nicely, except we don't particularly care because constructing SN extensions is not very exciting. You just take an arbitrary polynomial, you will get an SN extension more or less. So this is not a very interesting group from our perspective, but nevertheless it illustrates at least its definition of rational conjugacy classes. And to give an example of a non-rational conjugacy class and what sort of more typically happens, let's take the smallest group which is not a SN really, cyclic group of what is three. So it has three elements, one G and G squared. And because the group is a billion, every element is its own conjugacy class. So this group has three conjugacy classes, identity, G and G squared. And of course you see already here that when you take an element and you square it, and that's an operation which is allowed, it's called prime to threes to the order of the element, then you end up in a different class. So you've got three classes, one G and G squared, and this operation of squaring an element, it has a property that this identity classes definitely always going to be preserved, but these classes G and G squared are going to be swapped. So this is an example of a group where not every conjugacy class is rationed. And the reason here I draw a character table for this group is that there is a very nice little lemma which says that a conjugacy class is rational, if and only if from point of character theory, every irreducible character takes a rational value on it. So you can always spot them in the character table. If you look at the character table of a specific group, you just look at the table where all entries are integers because algebraic integers, they're rational, they're automatically integers. And if you find such column, then you've got to solve the rational conjugacy class. And if not, then it's not. And like here, the operation of squaring or cubic or whatever an element will swap it with some other class and you can see which ones they are again by looking at the character table. So in this case, for example, if you look at this character table, complete conjugation swaps zeta three and zeta three squared and so it swaps these two columns and it tells you that these two classes are not rational. And more generally, this operation if you want by taking a conjugacy class associating to it the corresponding column in the character table and hitting it by Galois defines a Galois action on the set of conjugacy classes which factors to some a billion extension of q. It's a reasonably simple kind of action but it's not always trivial. Okay. So now with this notion of rational conjugacy classes we can state the first rigidity theorem which is usually referred to as basic rigidity theorem and combined work of the variety of people to Billy Fried, Matzatschik and Thompson. So following, suppose you've got a rigid l-tuple of rational conjugacy classes. So here are two conditions and I've got a group g trivial center, so three conditions. First condition is trivial center. Secondly, I take a tuple of conjugacy classes which are all rational. And thirdly, I want to be rigid in the sense that I described. So if you do this column elements, little g i in here which multiplied to one, the number you get is known zero but otherwise smallest possible equal to the order of g. Take p1, pl to be arbitrary but now rational points on p1. Then there exists and in fact, it exists unique which is its uniqueness which forces it to be a query. There exists a unique regular g covering from x to p1 over q which is like in the Riemann existence theorem except now this is the current defined over q. This covering here is defined over q. It's still unregified outside p1, pl and the inertia group at p i is generated by an element of this conjugacy class c i. So this is a nice theorem. It doesn't often apply because, well, the two conditions, rigidity and rationality. Rigidity is kind of, well, you have to have it otherwise there is no hope but rationality that we asked for is a little too strong for applications. There are just not many groups to which this applies. So there's a variant which I think is due to salary in this combination, in this formulation that says almost the same but it weakens rationality a little bit in an obvious way. It says, well, suppose you have a rigid tuple again of conjugacy classes c1 and cl but we don't want every class to be rational. We have an action of gawa, q bar over q on this set and we just want it to be gawa stable. So in other words, like in this example, if I took this conjugacy class and this conjugacy class, then although these two conjugacy classes are not rational, gawa permutes them sort of like with divisors, a rational divisor consists of points such that the whole thing is gawa stable but the points are not necessarily defined over q. It's exactly the same thing here. If you take these two classes, then they are not rational but they form a gawa stable set. And then you take the points pi in a projective line which are now not rational but which correspond to this action and that's always easy and possible to do. So you choose points p1 up to pl, now points in p1 q bar which are anti-isomorphic as a gawa set to this set c1 up to cl. And then there exists exactly as before a regular g covering which you can make unique once you sort of say carefully, state carefully what happens with inertia groups. Again, covering defined over q under a modified outside p1 pl and which has inertia at qi generated by an element of ci just a slightly restricting condition of rationality. And this now applies to a few interesting groups though not so many still and a lot of work by people like Batsat, Thompson who pushed it to sporadic groups and so on they looked at various variants and pushed this even further to write down specific variants or specific groups which work in their context and again, guarantee the existence of such a covering. But this method sort of look a bit at Hock which you can do if you're dealing with 25 sporadic groups because only 25 of them but I don't know of like a very clean general result which is more general. Okay, so in the last, what is it, seven minutes let me just finish with an example of an interesting group to which this rigidity applies. So the group here is a second simple group that there is we've already dealt with a five that's a smaller simple group non-Abelian simple group for the 60 the next non-Abelian group has order 168 and it's PSL2F7 or GS3F2 same group. And here is its conjugacy classes and a character table. So it has six conjugacy classes and six irreducible characters, elements of order one, two, three, four and two conjugacy classes of size seven, sorry, of elements of order seven and I wrote the sizes here of conjugacy classes and I wrote the six irreducible characters and the character table. And you see here immediately from this table that these four classes are rational and these two classes are not they're commuted by Galois as you can see by these numbers here which are commuted by Galois which are not rational numbers. Okay, so now what we can do we can find a rigid tuple of Quanget's classes and what, I mean, this is such a small group that you can just write a code, you know which goes through elements in these classes and just finds tuples which multiplies to one it's not very hard to do and the process will terminate very quickly in large groups it's harder to do but there's a very nice neat formula that allows to count in any group L tuples of elements such as GI lies in CI where CI given conjugacy classes and which multiplied to one exactly the thing that you want except the condition that GI generate the whole group but the formula is very, very nice it's given in terms of characters it says that in any group G and conjugacy classes C1 up to CL if you are interested in a number of elements so L tuples GI in that class CI whose product is equal to one then the number is equal to one of the order of G times the product of the size of these conjugacy classes and then there is a sum of irreducible characters of the values of that character like here are the values of irreducible characters of various classes on the class C1 up to CL multiplied together and then divided by the dimension of chi to the power L minus two I mean it looks maybe a little convoluted but because character tables are easy to compute normally for even large simple groups this gives a very efficient way to find these things so in this group PSL 27 there are three classes which I sort of try to mark with bold which is these guys element of order three and the two classes of order seven so again this is a collection of so C3, C5 and C6 that's going to be my triple of conjugacy classes and these two are not rational but the whole collection is defined over Q so the Gallo group over Q permutes these two so this is a rational in the sense of serves Gallo stable triple of conjugacy classes and what we're going to check we're going to check that this is a rigid triple so first let's apply this formula here and count the number of triples of elements element in this class element for three in this group element for seven in this group another element for seven in this group such which is in different conjugacy class which all multiply to one and this number is given by this formula here so you can just compute it you don't even need computer algebra system for this so it's one over the order of group which is 168 times the product of the size of conjugacy classes which are 56, 24 and 24 easy to see if you compute so centralizes of various elements sorry I lost my page pressing something, there it is again and then you need this sum over the characters the values so you go through all the characters and then you look at these values on this class multiply by the value on this class multiply by the value on this class and now of course there's lots of zeros at my table so for these characters so two rows three, four and five the product value here times value here times value here is going to be zero so in actually it's only two characters that are relevant which is this one and that one where the values are known zero so what you find is 168 times 56, 24, 24 and then one times one times one with a trivial character divided by one dimension of chi and then for this last character here minus one times one times one divided by the dimension of chi to the power my L is three so to the power one so over eight so this is seven eighths I guess one minus one eighths so you multiply one over three times seven times eight by seven by eight by three by eight by three by eight and by seven eighths and if you do that you see that one of the sevens cancels one of the eights cancels and one of the threes cancels and you find that you get 168 which is the order of the group so in other words the number of triples which satisfy this condition there is exactly 168 of them and it turns out that every such trip will automatically generates chi I don't know a very quick way to see this but one way to see this for example is just to look at this group just find one triple of elements like this which does generate chi just find one example by hand and then once you know that there is one example we know that there is 168 of them because remember again the set of elements which generate chi they form a union of regular orbits and therefore you know that all of these 162 triples have this property and therefore this triple is rigid you could probably do something a little bit better by proving that two of these elements already generated group of order 21 at least element 43 and element 47 if you could just prove that this group has larger order then it will be enough to prove that it's whole group but I'm not sure I'm not entirely sure how to finish off this argument so in any case it's easy to find an example it's a small group and then you find that this triple is therefore rigid and by the rigidity theorem therefore this is a Galois group of q of t so this example is I mean by itself it's maybe not super interesting because I left as an exercise to prove that for some reason that I don't exactly know it's very easy to realize it's global q of t anyway you look at LMFDB you look at a bunch of fields there which have PSL2F7 group there's lots of them and by just basically interpolating them as I did there in an exercise you find lots and lots of one parameter families and you can in fact push it to two, three or even four parameter families that's the paper of Malle which does this so this group is actually quite easy to construct over q of t but rigidity applies to a lot of other groups where this is certainly not the case and generally just to give a last comment on PSL2F7 so Schich has proved a theorem using a different method which involves module forms and module curves but PSL2FP is a Galois group over q whenever two or three or seven is a quadratic non residue on UP and this particular in particular it covers PSL2F7 so what we just did is basically we proved one simple case of his result but there are many other methods and especially the other groups especially sporadic simple groups where it's only using rigidity that we know how to construct how to construct these groups so generally it's a very powerful method and it complements a lot of things that we've done so far which mostly applies to solute groups okay I think I exactly run out of time so this is a really good place to stop for me thank you very much okay let's thank Tim both for that lecture and for all his lectures we're a little past 10 but I think there's a break before the next problem session so I think we have time for questions are there modifications to the rigidity results if Z of G is known trivial that's a very good question so I think I've seen several so I've seen one paper where I think it's a book in Malematsat who say that it's you don't need a center to be trivial you need it to have a complement to the group so a subgroup which together with the center generates the whole group and which have a trivial intersection but as somebody else remarks in different paper that means that your group is a direct product by the center of something else a smaller group and in a smaller group you still have rigidity so in fact it doesn't add anything interesting so to answer your question I do not know personally of any modification which is interesting in a sense that it would allow you to do more groups than you could do otherwise with this rigidity method so it seems somehow a bit of a fundamental issue is this method because of the action of the group by conjugation and the fact that the center is just invisible Other questions? Could I ask a question? Please Yes, I heard there was a question Oh yeah so other questions so part of the problem is this variety which has a map to P1 there needs to be defined over Q so sort of for the most but is there any connection with Galatasent in this complex? Well certainly in a sense that what we are trying to do here is you know Galatasent we are trying to we start with a curve over C or more generally as I mentioned there is a theory of who with schemes and who with curves where you try to understand all colors which have this property and then to descend the corresponding scheme to Q or find and then after that find its rational points so this is I think a special case of Galatasent and in fact I think the proof in Serge's book Topics and Galatas theory of this basic rigidity theorem it's you know it's very much Galatasent like it uses arithmetic fundamental group and it's very much in that flavor and I think he even mentions connections to Galatasent theorems in there Okay, thank you and there was a question what is the status of PSO2FP? Yes, by now it's known for all primes so Sheikh proved it under these conditions of two, three and seven being quadratic non-residue and then it was extended I think by Matzat two and five as a quadratic non-residue and by now it is completely settled but I think it is apart from the alternating groups it's possibly the only family like complete family of simple groups for which it is known for which you know that these groups are known to be realized over Q of t There's also a work of Zhiyue Yun that did some exceptional groups I think G2, G2FP and E8FP but I guess those are yeah I think they're Okay simple Yeah, absolutely Thank you Okay, if there are no other questions let's thank Tim again for all of his nice lectures


We thought it'd be fun to talk a little bit about music as the show We don't normally talk about music although Johnny my co-host is a musician I'm more the business side So it was interesting to unpack your book from both of our lenses And I thought it'd be fun to just talk a little bit about what your first music memory is that shaped your life And I'll go first because I don't normally share this on the show But growing up with a single dad My first real music experience was the summer in michigan and my dad rolling down all the windows in our car and our family sedan and turning up the volume on eric claptons layla And singing every single word of that song just as gleeful as could be and I just remember him being so happy And starting to realize just how music was so powerful for him to evoke that emotion And that memory will forever stay with me So I was curious what your first musical memories are and how this shaped your lives I was thinking that there's a my first musical experience and then like my first Pop music experience. They're both significant to me. So you know, I I grew up in going to church with my family my both my parents Play piano and sing and so If my mom wasn't playing piano at the church, you know, my dad was singing in the choir and and I I learned music that way. So to speak I learned about harmonies and um orchestration arrangements and all but what moved me was um the first record I ever bought Which was Joan Jett in a black heart. So I love rock and roll And I just remember the first time I hear in those those chords like Da na da na da na It's like it just it made something inside me shake and I had to have it. It was so raw Um, I still love that song to this day. I think it's an amazing song I just did a a bit where I was discussing how she was my first crush Which is probably why everything went sideways Wonderful I can I can relate panos what she got for us I grew up in Cyprus, which actually is where I am right now by coincidence and I remember turning on the tv It was probably around 1978 or 79. So it was, you know, six years old and There was a black and white movie called king creole with Elvis Presley and walter map out Oh, yeah, and Elvis started as this awesome bass line And the eponymous song king creole that Elvis sings is still to this day one of my favorite songs And that trio, you know, the Elvis Presley trio I think is one of the most underappreciated musical trios in history because every one of those guys DJ Fontana and scottie more They're just amazing amazing musicians and so is Elvis and I just don't think you often get The credit that you should be as a musician. So that that is my first Childhood sort of musical experience that frankly set me down to a path of wanting to move to America And wanting to be music It's interesting that you say that Elvis had been responsible for a lot of people wanting to move to America and playing music We can certainly See his influence on the Beatles and and everyone else who had followed I I loved hearing that and scottie more. Oh, yes, the backup musicians Ridiculous Elvis doesn't get enough credit as you mentioned. I think his hips get more credit than him But he was just as Comparable as a musician as the rest of those guys For myself, it was it's difficult for me to pinpoint this because I grew up in a musical family Music was a big part of my household. My dad played in bands They played and rehearsed at my house. They played at the the club on the weekends and because of that There there was always a radio on or there was rehearsals. There was always musical instruments in the house However, when I heard AJ ask this question I can I can definitely pinpoint the first time music moved me in a way that was not Was feelings that I've never had felt before and We used in the summertime As kids would do we play pickup games with the other neighbor kids and I remember playing outside the neighbors playing football and The local radio rock radio station was on and they were playing purple haze by jimmy hendrix And for whatever reason When the guitar riff started and I remember Having feelings and being moved in such a way That I had never experienced before and I wanted a lot more of that and that radio couldn't go Any louder and of course following that because I that had led me to wanting to play the guitar And wanting to recreate those sounds and the feelings that I had Inside and and the feelings There was some that I didn't know what they were because they were more of sexual tones that were sort of brought out in those moments But I also felt invincible. I felt immortal I felt myself becoming larger from enveloping those colors and tones That Basically the rest of my life was was written in that in that moment What I love about all of these anecdotes and the anecdotes in the the book as well is just the powerful impact of listening and Johnny often in our growing the business over the last 15 years Uses a lot of music metaphors around you're not hearing it. You're not listening properly And I'm really curious to unpack what you both have learned from musicians and listening Especially when it comes to being innovative in business because that's such a powerful skill that I think is so underappreciated And underdeveloped in a lot of members in our audience we are in an era right now where I think there's a lot of broadcasting and not enough receiving And It tends to sort of mask in many ways the critical importance of just paying attention and As an entrepreneur I think I always thought as a musician because I am a musician by training and When you're playing music and especially if you play jazz When you get lost in the core changes as they say, what do you do? You just sort of like hang back and just stop You pay attention You know, you almost recalibrate where you are And in my entrepreneurial journey there were many times When I did find myself down the path where I did get lost And I think it was that instinct of okay Let me just pause for a minute Pay attention to what's happening around me and I mean really pay attention And and then start jamming again And in many ways, I think we're We're taught the other way around And many of us tend to talk our way out of deals We tend to talk our way out of jobs so knowing When to talk when to lead But also when you just hang back and pay attention Either to the conversation you're having to the person that's talking Or to the situation that's at hand Is of critical nature and in the book We talk about Miles Davis and how He literally did play notes as if they cost a whole lot of money And he believed that the silence is even more important than the actual notes themselves So for me it's always been What are you not saying as much as what Are you saying or what's not being stated as much as what is stated that that counts Pan is perhaps you know this story with Miles It's with herbie hand cock when he had first joined the group And herbie was just very excited to be performing At a level with these people that he had looked up to and thought were the pinnacle of performing And I guess there was a maybe you know the story better But there was a live performance and herbie had played some wrong notes and he freaks out And He was so worried about what miles was going to say afterwards and what would the repercussions of this because of course miles had Was known as a hardcore Musician and guy when it came to those performances and herbie didn't really notice anything that was going on And except that the some of the tones that miles was playing had shifted and changed and after the gig Herbie was very nervous waiting to get chewed out and nothing ever came of it and I Herbie wanted to know perhaps maybe miles didn't catch it and said something just to see and Miles basically just said well, you didn't hear me change the whole Key of that section because of the notes that you played He said and it was basically you need to start listening The thing about listening for me is I come out of a design background first, I mean I play music too, but um When we were starting to Work on the ideas from this book like I I just sort of see the parallels between the creative acts of design and and music and um You know in in design we talk about observation as like the beginning of inspiration And and listening is the same you know like in in the book we talk about There's a uh antidote about Bjork sitting in in Reykjavik growing up in Reykjavik and the the sounds from the harbor You know to to most people who charge noise pollution right like The seagulls the boats the horns the whole thing for her that's music right and I mean she she I she literally actually took those sounds and Composed the beginning of a song um with those sounds right sounds like a random noise until it turns into harmonies But there's so much around us to be inspired by if you if you can do what pano said and and stop You start to you just naturally start to notice opportunities Right um or notice things that are different that cause you to think something new And I think the key to most Creativity new new ideas innovation happens with that simple act Which to upon us this point we live in a society that doesn't encourage it You know, it's a constant feat of information to us. I mean When you hear people say they're behind on their netflix, you know, there's a problem, you know Right From our work that we've done here at the art of charm over the years One of the biggest complaints that most of the guys have is I run out of things to say or I'm always so quiet in a conversation and it drives me nuts I want to be vocal well all the answers And all everything that you need to say is in the other person If you just take a moment to listen to what they have to say You always have something to say you'll always be able to contribute and what to what we're all saying here is We don't spend enough time Stopping and and listening to what's going on and and Bjork is a great example She was able to use those sounds to create some of the Most beautiful songs. Well in another way to think about it is sort of we're all biased towards action And even as an entrepreneur you are conditioned to think that It's all about what you do But it's often what you decide not to do That actually determines your fate So it's another way to look at this idea of of playing and silence of listening and talking Of of doing or not doing it's The space that's many times determines Where the opportunity is more so Than the other areas that you're so busy filling If you will and I had to learn this Lesson time and again as an entrepreneur whether it was The earlier thing that I said to you before of you know, yes, you can talk your way out of a deal you already had and it happened to me But And then also Sometimes you're so hell bent On what you think your company is doing That you don't pay attention To what your customers are saying you should be doing so that's how we've been Sort of using this concept of listening as Perhaps the most integral part to innovation to entrepreneurialism and ultimately For me to all all creation I really enjoyed that point because Yes, we're thinking about innovation from the teams that we're on and the company that we're leading But that innovation is also spurred by our customers our clients our audience and the feedback that they're giving us Hey, this is working. This isn't working. This is what I wanted. This is what I need And if we are to your point exactly hell bent on proving our point winning that argument having the most facts data accolades We're missing that feedback loop that generates the innovation that we're all after again We're conditioned to look at customers As sort of this passively receptive audience to our creation But every good musician knows that in every show that you go That audience is a unique audience in that unique moment of time and they're ultimately Co-creating with you that experience that you're both sharing It's not about you creating an experience and they're passively just receiving it That moment in time is generated by both audience and creator And I've always believed that business is the same way your customers, especially in a post industrial era in a digital era Your customer is every bit As much of a co-creator of the product that you're putting out there As as you are as the inventor Um, so again for us it goes back to these musical concepts that we're articulating in two beats ahead about This musician instinct that they have for listening for collaborating for co-creating That we believe is so applicable to to innovation and and certainly these are concepts that we've experienced firsthand in our journeys I'd love to unpack the collaboration Angle a bit more. I feel that Uh listening is certainly a component of that Much of the music that I think of that is timeless that is You know transcending of genres is when You have these completely different perspectives and musicians and styles coming together and collaborating to make something so unique That as we spoke about earlier just like reaches inside of us and like makes us stop Business is the same way like you need that level of collaboration at a team to really Make something impactful And you guys working together on this book. We're collaborating. So I'm sure there's a lot of lessons to impact there as well How do you approach collaboration? Apart from just the listening angle. I think one of the most interesting things we learned um as we talked to more musicians and entrepreneurs and writing the book is that collaboration was uh a very active choice by And part of the musicians to pursue other people that they felt were more talented or Um at least equally talented and they felt like they could have respect and trust in You know, and I felt like that was such an important idea because I I see um In education for example, there's a lot of push for project-based learning Which I think is a really great idea. I mean I work in a business. It's all about project-based. I mean, you're working with different people all the time, right? So I think it's a great idea But but what I what I've realized over time is like the emphasis on project-based learning in schools isn't enough because There's a missing element there of this desire to work together This pursuit of each other's talent this belief of one another's creativity It's just like random people just thrown together and especially if you're random teenagers throwing together You know how difficult it is to even get them to talk to each other let alone work together um, but I would I would but it carries on into adulthood. I was I was speaking once um Had a university and I you know, I said something about collaboration and in the q&a at the end Someone said uh, just raise their hand. It's like but what if you don't like your teammates, you know, I was like Like Yeah, it's a very fair question. So um, I think that's for for um I guess the biggest surprise for me then was like thinking about something like Beyonce who in her own right, you know, we know is a giant of talent Her her creative pursuit is like to choose people that are amazing at the thing that they do like she chose like She wanted to do a song with jack white because he's an amazing guitarist, right? or or um she wanted to work with um It was a Milo x would remix her music Put it up on Spotify and YouTube the record label kept taking it down because breaking copyright But she loved what he was doing so much that she asked him to go on tour with him um For her was all about just finding that great creative partner and so It really changed my even my own thinking about how to How to work with other people like at least, you know, even if you do get in that situation when you're randomly put together Trying to figure out what it is about them that you find inspiring or different Or exciting and then letting that be the thing that draws you together, right? it's it's a very um To me an exciting way to reframe what collaboration is Because it makes it a pursuit of other and a pursuit of something bigger Um versus I think what seems to be scary in this if you're raised in that high school model Just random putting people together to try stuff today at lunch I was talking with my twin daughters who are seven and as you know Seven-year-olds have a habit of asking questions about everything And sometimes the simplest things you just never stop to think about but We were having a discussion and we started talking about The new Pixar movie soul, which is all about jazz. Yeah, and um We started talking about Pixar and I used the word. Oh, they're they're a company And they said what's a company? And I had to stop and think about it for a minute How do you explain to seven seven-year-old? But then you say well A company is a group of people that come together just like we are here around the table to create something. So if me and mom Collaborate to create dinner. Well, I guess you can call us a company of sorts that produces the dinner um But What struck me as we were both writing the book But also reflecting on the journey that michael and I have had as collaborators in multiple projects Is that We're so conditioned to think of groups in business or companies And the way that you go about curating if you will who you hire Based on skills And you spend a lot of time evaluating people's skills Maybe they're the personality, but at the end of the day The magic sauce of anything is just this intuitive chemistry you have with somebody where actually It's almost as if the skill is less important to that intuitive chemistry And what we found about the way that most musicians collaborate is that first and foremost It starts with that instinctive connection that you have with somebody and Had michael and I when we first met We said well, we have an intention of writing a book together or designing a class or a major Or starting an initiative about identifying rights owners in the music business or any of the stuff that we've done together We would have probably been completely turned off from each other and never Had anything to do with each other after that initial meeting But it was this intuitive connection that we have that led us And if neither of us have written the book before so clearly we did not select each other Around our skill set, right? We just had an innate trust that the two of us would produce something interesting. We didn't know what it was but I think as a musician I thought I can rip with this guy. I don't know where it's going to take us. I don't know what's going to happen but We're approaching life and the world and business and everything else Both simultaneously from a very similar point of view, but also from a very different point of view. So That intuitively I think told us we can create something cool together And there's a quote in the book from t-bone brunette who's a very famous music producer And is this something along the line?  I care who's playing it And I love that quote because it's not so much. Well, I need a bass player I need a guitarist. I need a drummer. I need this. I need that is more What are you bringing to the set that's going to make it interesting? So first and foremost, I care about who you are Rather than what you are And and I think this is a discussion that we're not having enough in the business world about how you bring teams together and certainly anybody who started company knows that your co-founders are usually not people that you sort of consciously select as Well, you are this and I'm that so let's make a company together It's often just this intuitive connection that you have with somebody who you go about hey Maybe we can do something fun together How do you create that intuition in team members who maybe haven't worked together before or are placed there in a Circumstance just based on their skills. I ran the company that I founded sonic bids We were always especially in the early days as companies grow you do lose that Um, and I used to have a mentor who started a company called ricodesk Who said to me, you know, the company's kind of stopped being fun after we we we could no longer share two pizzas So I don't know what's that like depending on how much you eat is like eight or ten people, right? But those initial teams are so critical to the success of any business and I think in the in the History of any company those early teams tend to be Long forgotten like most people they'll tell you who the founder is but they have no idea who The second and third than fifth person actually are and the second person is often the most critical person in in the creation of something um, but they're they're they're often confined to uh, you know some asterisks somewhere if if that but For us it was all about this shared interest and because I guess it was easy to find that because sonic bids was in the music space so music acted as this binding agent That no matter what walk of life you were Uh from no matter what passport you carried No matter what your skill set quote unquote was whether you were an engineer or You know salesperson or whatever I think that commonality Had a reassured everybody that You know we vibe to use a music term And in the early days that made all the difference that because vibe for me is another way of saying I trust you Because there's something that connects us there that's beyond What's ever on a piece of paper or a resume or a linkedin profile? It's something that cannot actually be captured in any of these things.  I I love it. It's and for AJ and I we've been doing the art of charm for the last 15 years And though we're quite different in many aspects of our life. We're always able to figure out a way To continue working together because the mission that we sent out on in the very beginning had never changed Now if either of us Diverted from what this was about Then we wouldn't be able to work together and much like a lot of musicians You have an idea of what you want to do and what you want to create together and everyone starts to go for it Then if anyone deviates, let's say they choose. I'm going to chase the money now rather than the creative vision that we had for each other Well, you have deviated from that path chasing the money comparatively to chasing the innovation or the the mission that you sent out to do are two completely different trajectories that operate completely different and so For certain that to have that vision aligned And agreed upon and then the steps that we're going to take to get there Needs to be set up so those collaborations can maintain In fact, they should those collaborations should feed each other to where somebody's not feeling it The other person is able to carry it and pick it up till the other person finds their way back And I know for AJ and our In our lives and our business with the art of charm There certainly wasn't always days that we could share two pizzas either But it was the the vision that allowed us to to continue on and pick each other up when Either of us had faltered or felt defeated or or just a bit burned out And that's what made Our podcast successful and and allowed us within the that construct to be able to grow as well And and you know john as you're talking. I mean We've never met you guys before But when you just said, okay, we've been in business together for such a long time I'm not surprised because I think the four of us are having this conversation and I think You two know that michael and I have an intuitive connection And I know just seeing you guys interact with each other That you have that intuitive connection and another thing that Somebody that I I met early on in my business career told me Is don't confuse invention with intention And um there's a guy called bill warner who founded a company called avid and I never forgot that because I think we often confuse sort of The product that we're putting out with the reason and the intention and the purpose that we have with what we're trying to create And um, you you get too caught up sometimes in in that Whereas I think with michael and and and and may Again, we never sought out to do anything together. It's just one thing led to another led to another and it was that innate trust that we had um, and the curiosity that we have Uh in who each other is that that has kind of led us to where where we are Yeah, I think that's such a Um difficult thing to learn, you know, um Another person in a book is spencer tweedy. He's he's the son of jeff tweedy the um founder of wilco and um Spencers um A really cool guy I have to say I mean he he's He's a great drummer. He's played with norah jones and mavis staples, you know He's he's on all his prom dad's promotions right now because wilco can't get together so He's playing on his dad's solo album, but but he um He said something to us that I mean he's written a book now. He's got some records out. He's started a microphone cable business And we're trying to figure out like what is the what is the red thread across all these things and he said well I kind of learned I learned from my dad that the important thing about a songwriting process is Literally the process it's like if if you If you're willing to embrace the process enjoy and enjoy it for what it is You'll continue to be motivated and won't be distracted by whatever artifact comes from it you know and um It sounds kind of intuitive. I think for like especially for someone who's making a living making records and making books but it really is about that idea of of enjoying What you're learning through your curiosity is or enjoying the practice of your craft You know and then good things come from it I mean ponies is right like if we had met and like hey, let's let's set five goals together for the next five years I mean no way No way would that ever worked You know and think about that for anyone you've ever fell in love with I mean that like The first thing you don't like you like finds one really attractive. You don't start making life plans for your children's college You know in your retirement So, you know, it's it's um, but I think in in business we often do get confused by those things, you know and we and we we put our eyes on the wrong prize in fact when it really is this idea of shared purpose leads to great things versus trying to focus on the artifact that um Can be a distraction actually Johnny and I in 50 years of business. We've been on a lot of unsuccessful collaborations, we've had seven business partners over the years and part of their inability to collaborate is why they're no longer here And I feel that in collaboration. There's always a part of you that gets sacrificed for the collaboration And Johnny likes to say, you know, don't bring that tired riff back into the meeting We've heard it. We said no, it's not working rework it But you know, many of us who are struggling in in collaborations, especially in a school setting As you said where we don't have that intuition or that shared purpose yet We get so hung up on Us personally showcasing our talents or getting our ideas to come to fruition or being that leader in the group And we're unwilling to make that sacrifice for the collaboration What have you guys learned from musicians and how they approach collaborations when they know that riffs are going to be cut Lyrics are going to not make it in there that they thought when they were writing or when they were playing Were that thing were that spark? Yeah, that ego can gets in a way so easily big time if you believe if you believe enough in the pursuit together Then you can you can actually go overcome these things like I had to have a friend once he's actually in the book um as well I said a friend once as if he's no longer around he is around We used to work together. We no longer we he lives in a different city now, but um He he said in his collaborations, he never compromised And I was like that sounds impossible like what do you mean? And when he impacted for me, I thought it was very interesting. He's like well It was in my mind if either party has to compromise Then neither of you are going to be happy. So if you ever get to a point where either of you are Unhappy with what the other one is doing and you feel like you have to give in The best thing to do is both set your ego aside and try for a new idea Because eventually you'll both get to an idea that neither of you are compromising about and you both feel great about You know, it was one of those the discussions that aj and I were having before we got on with you guys And what we wanted to discuss and how we wanted to frame everything today and The that the section in the book about collaborating I was wanting to see and I now I have already known what what I had felt my musical background has I have brought into the company with me and I know what those aspects are and that I That you guys illustrated the collaborative part in a way and I was like, oh, I never I never saw that aspect of it and For myself and aj was talking about the riff idea I mean so many rehearsal rooms where everyone's bringing in these different ideas and what's getting cut and what's getting added and for myself The the riff never mattered Or the lyric never mattered If the end result was was was better So if everyone's riffs go in everyone's lyrics go in and we're looking for the best the best bits I had never gotten And I think this was just through that process where I had never gotten beat up or never felt I'm so mad at these guys. That was it. I can't believe they didn't they didn't use this bit Because for me, I was always happy that At the end we've all agreed that the song works The the song is strong and it is strong because of all of our efforts and aj knows Coming into and to meetings. I have viewed meetings in that the same setup For the last 15 years because that's what I'm so accustomed to and comfortable working in that And as aj mentioned, we certainly have guys who have come through and it's like well, listen. It's my way or the highway Well, you see there's only four of us on this call right now Yeah, and this you know this concept of trusting That your collaborator is bringing something to the partnership The the the collaboration that is additive Not because of what they do necessarily but because of who they are And the way that that's expressed Is again something that we've seen in talking with musicians time and again like um You know one of the examples we talk about in the book is is ferrell and actually how he successfully collaborated with so many people including brands like adidas and You know, this is a situation where there's a successful partnership there because There's an inherent trust between company and artist That even though the artist is not a classically trained designer Apparel uh designer shoe manufacturer. That's not what they bring He is not a supplier to adidas, right? That's not what you're that you don't bring ferrell in for a collaboration Because of his amazing ability to manufacture foam soles, right? That's not what you're hiring this person for So if all you see shoe manufacturing As it's basically a bunch of components that come together to create something then You sure miss the whole idea of bringing in a ferrell and I think when you hire people in companies Or when companies acquire smaller companies, they often make that mistake they They don't see The people for who they are and what their talents and how they can be additive They're just trying to fit them within a very particular sort of frame Um, and it doesn't work It just doesn't work because you're leaving out the very uniqueness of that person um And what they bring to it and it's something that I think you see throughout the book this idea that successful creative collaborations are not Because of the very specific very well defined skill um, it's about What the other person brings to the collaboration that That's michael was saying actually is uncompromising because everybody is individual as an individual And it's that individuality That they bring that merges and becomes part of a braider hole That is what makes a hole so cool That is not a bunch of people who compromise their own individuality is a bunch of people who brought their individuality and one plus one equals 10 precisely because Of how they come together and their respect that they have about each other's difference rather than each other's commonality and in your experience of assembling teams And then joining teams that you necessarily didn't pick From the start right because in entrepreneurship when you're starting your own thing Your your hand picking the team and then when you move to a corporate setting you didn't get to pick the talent But you're still leading the team Is there a difference to how you approach collaboration in that or identify How to bring that individuality out in those team members to get the best out of that collaboration well, i'll tell you my experience and i'm i'm sure michael can share his but um After I I I sold My company and I went to berkeley I went from a company of about a hundred people to an organization of 2000 plus people and The areas that I lead at berkeley are actually all over the All over the world A campus in abu dhabi a campus in valencia spain in new york city and I also inherited a number of of of um of people that Were kind of cobbled together to create a new area of berkeley that we call global global strategy and innovation and I have spent and spend still a lot of time trying to understand people's individuality rather than How they fit into my grand plan? So it's almost like you're reversing the process and my plan is developed Out of their individualities Rather than me trying to shoehorn them into a grand plan that I prefabbed and brought to and brought to the table because then you're missing the Very strength of all the people that you have working with you um so Honestly, it's been a process and it still is I there are people that are just Not that easily knowable or sometimes they don't know their own strengths and I think your job Is to help them figure out Where they can be the strongest and how they can express themselves in a different plateau so Or a different platform, I should say um So that's been sort of my approach and it's been an adjustment for me going from a company that I created To an organization that is several times bigger That's a 75 year old institution and working with people that Didn't know me at all Up until I became their their manager there's a A guy in our book named steve stout who he wrote a book called the tanning of america he's basically This genius businessman that connected hip-hop culture with lifestyle culture and has been one of the major deal makers for a lot of these artists to get um outside of their music into other kinds of businesses, you know But in his consultancy He talks about this idea of majors and minors. He's like your major is your role Like your job title But your minor is your passion And he's an incident in a business. I want to know everybody's major and minor because when we build teams Your role may not matter You know um What could matter would be your passion so He's primarily um You know a marketing organization. So he's like His business is like technologists storytellers and creatives Because but in those three main categories of employees, they could have passion around music. They could have passion around sports. They could have passion around uh Food or luxury or whatever it is and so You know what even though they're in the technology team, they might be a software coder If they have a deep love for luxury brands, I want them on the project You know whether it has anything to do with software or not Because they're going to bring that energy that knowledge um to whatever we're doing and I think it's a lot like what you're talking about panos.  I mean job titles don't really even mean anything if you like try to actually deconstruct them um We're motivated by the things that Make us enjoy life, right? And so if you can figure out how to unlock that It on teams for your job. I mean That's great and that feels great. You know now that we have the the collaboration We got we got the pressure cooker of a bunch of guys putting something together We gotta we gotta eke something out and in the book you guys talked about putting your putting a demo together And for I want to make this quick connection Because when facebook came out or sonic bids or any of the technology that we're using We got a beta version. We we saw the first version and we all liked it and so they decided hey Let's develop it. So what could you tell us? About that section and what you guys had gotten out of that It's the bravery of doing even when you don't know what you're doing and The truth is You just don't know what you're doing even when you think you do and I feel The the the most brave thing that you can do is just dare to take that first step And almost trusting that number one it's going to turn out different than you had envisioned it in your head But that is to michael's point earlier It is the most important part of beginning a process That will lead you somewhere interesting, but you have no idea what that interesting is And We I have another great quote from an interview that we have with Justin Timberlake and he says, you know Learn how to dare to suck And it it it takes Bravery and courage To not be afraid to suck, but it's in that Moment If you will Where the sucking can turn into something truly amazing and different Um, so for me, that's The whole concept behind demoing if you're so focused on the polished product And if you so over legislate and over design what that Polish product ought to be I think you miss All the opportunity that may exist by coming up with something totally different than you intended to begin with and many a great song and a great product a great company Heck a great marriage are often byproducts of of just Being comfortable with the fact that you're going to end up somewhere that you had absolutely no idea you would be when you started out um, and I think it gives you confidence actually that Even when you are making what you would think are mistakes. They're just parts of that process, right and for anybody who plays an instrument you know I don't know about you AJ, but Johnny you do and Michael you do, uh I remember the first time that I picked up the guitar I went to the store and I bought a An imitation Fender Stratocaster that looked like Clapton's Blackie, you know And I plugged it in and The edge also played a similar guitar and I had seen you two love it red rocks And I thought I was gonna pick up this concert. Yeah, and I thought I was gonna pick up the guitar and sound Like the edge or or or clap or Clapton. I mean, I guess we're starting with Leila and we're finishing with Leila And I played that note and it was so Different that what was in my spirit in my heart in my head in And it was both exhilarating and utterly crushing At the same time because it sucked And my finger hurt and it it it didn't it didn't feel or sound like anything Then what I thought the experience would be watching The edge play Gloria or Clapton play Leila Uh, I don't know any other guitar player playing any other song named after a woman And but it was that first note that sucked That ended up being the first step that I took to the life that I have today, right? Um and to me that's that's the bravery around demoing. It's just Doing and not being afraid to get it wrong. One of my favorite stories that An examples of that is Is the Ramones and you have these four guys who really didn't have much of a shot at really doing anything else And it was the new new york in the 70s the places just Decomposing as they sat there and Had this idea that we're going to put a band together and we're going to try to make something happen and You see Film of them at cbgb's like their third show where they all started the wrong song and then they get on a fist fight on stage and but yet fast forward We have now have a band Who took their limited skills? Crafted songs that fit within the skills that they had And not only made a name for themselves, but have inspired There's probably not a kid who picks up a guitar Who doesn't go through the Ramones path as they start to learn how these things come together And realize that oh the three chords I have learned that hurt my fingers That are are killing me and are bumming me out because I only know three Well, that's all it takes for me to write an album and show three other guys how to play these these three chords In a rudimentary drum beat and we are off to the races And then to take what you've put together in that rehearsal room That painstakingly get on stage and then you're going to go all start the wrong song and get a fist fight on stage Knowing we're on a path to success Pretty brave And one more question before before we wrap and we love asking Everyone of our guests what their x factor is And that's either a skill set or a mindset that has allowed you to reach your full potential in life Michael anything come to mind for your x factor, you know one one thing I've a mentor taught me In the last couple years was this idea of anytime that I feel defensive about something to switch that feeling of defensiveness to curiosity And uh, man, that that has been life changing for me It's opened me up to so many things that I think normally I would have shied away from or conversations. I wouldn't have Um or even heated moments. So that would be mine beautiful realization You know, I'm gonna give an answer that probably would have not given even a year ago, but I would say it's sort of this combination of both confidence and humility um Confidence because it's about what we were talking earlier, right like the Daring to act even when you just don't know But then the humility to learn when What you didn't know ends up being something that maybe I don't know you should have known or Or or uh, or you're learning from right Because if you don't have that humility, you're just not going to learn from the inevitable Um wrong notes that you're playing um, so for me, it's been this this duality that I'm not afraid to try Which in and of itself takes an act of courage But then if it doesn't it's not accompanied by The humility that it takes to learn Then I just don't feel you make forward progress you remain stagnant either in just playing those three chords if you will And you're just writing the same song over and over again um Or in never playing or learning how to play one chord And therefore there is no second or third. So there is no song and there is no album Uh, it's really no rock star. Uh, so that yeah, that's that's been on on my end Thank you both so much for joining us and writing such an enjoyable and insightful book on innovation and music two beats ahead And have a wonderful day. Thanks for having us. Thank you guys