 This talk will be about sporadic groups and number theory. So it was originally an introductory talk for a number theory discussion meeting. And the idea was to try and explain to number theorists why they might be interested in sporadic simple groups. So the original example of this was monstrous moonshine that I'll just quickly recall. So we have the monster group, which is the largest of the sporadic simple groups of order about 10 to 54. And we have the elliptic modular function j of tau equals q to minus one plus 744 plus 196884q plus 21493760q squared and so on. So coefficients are quite big. So the elliptic modular function is essentially what classifies elliptic curves. Two elliptic curves are isomorphic over the complex numbers if they have the same value of the elliptic modular function on them. And it satisfies j of tau equals j of tau plus one, which is obvious, q is equal to e to the two pi i tau and a much more subtle properties that j of minus one over tau is equal to j of tau. And the relation between the monster group was first observed by John Mackay, who noticed that this coefficient here is equal to 196883 plus one. And the point being that 196883 plus one is the dimension of the smallest representation of the monster group. And similarly, this coefficient is the sum of the first three representations of the monster group. So there's obviously some very weird coincidence going on. And a short afterwards notice that similar things happen for other sporadic groups. For example, the baby monster is related to a certain modular function of level two, which looks like q to the minus one plus 4372q and so on. And again, we find 4371, this is 4371 plus one, and 4371 is the dimension of a representation of the baby monster group. So John Thompson made the suggestion that these groups should have graded representations whose dimensions are given by the coefficients of these functions. For example, for the monster, these would have dimensions 10196884 and so on. And he also suggested that you should look at the trace of elements of the monster or to have a group you have on this. And this would give you another function. Conway Norton calculated these and found they were all various sorts of modular functions of high level, which are modular functions like J of tau, only more so. Well, most of these conjectures have now been proved. I think the final ones were cleaned up by Scott Carnahan a few years ago. And the problem is to ask what is actually going on? Because if all you've got is a graded representation of a group, that is frankly not all that exciting by itself. It's not very difficult to find graded representations of groups with degrees given by the coefficients of various modular forms. The point is for something to be a graded representation of a group, all you have to do is to check a lot of congruences and modular forms and functions are notorious for satisfying huge numbers of congruences. So it's not terribly difficult to check these by brute force. The real problem is to find some deeper structure going on underneath this. So we can put this as the problem, find algebraic structures underlying the object of these. So what we want is not just a graded representation of a group, but we want this graded representation to have some sort of structure. For instance, it might be a vertex algebra or a lee algebra. And I'm going to concentrate mainly on lee algebras because I don't have much time and trying to explain what a vertex algebra is takes more than the 20 minutes I've got. You can think of a vertex algebra as being very roughly something like a commutative ring except the product is not really defined everywhere. It behaves more like a rational function with singularities than a regular function. So what I want to talk about is lee algebras related to this that are acted on by sporadic groups. And the key point about lee algebras is that they satisfy a vile denominator formula or rather a vile-cat's denominator formula. And this looks like the following. It looks like e to the row times product of one minus e to the alpha is sum over w of minus one to the epsilon omega times e to the omega row here. Row is something called the vile vector and w is something called the vile group. And instead of explaining this, I will just give an example. So there's one example everyone has come across which is the Vandemonde determinant. So if you look at the determinant of this. So this as everyone who's done a graduate algebra course knows can be given as a product over x i minus x j by less than j. And what we see here is a product which with a little bit of fudging is looks like this product here. And we have a determinant which is a sum over the symmetric group SN. So we're going to take the vile group to be SN. And this turns out to be the vile denominator formula for the lee algebra GLN over C which is just n by n complex matrices. And there are similar product formula for all finite dimensional simple lee algebras. And things get more interesting when you look at some infinite dimensional algebras. For instance, let's look at the infinite dimensional lee algebra. Let's take SL2 with coefficients in the fields of Laurent polynomials. And the first thing to do is to draw a picture of this lee algebra. And I'm going to draw a picture of its, what are called its weight spaces. So here, this is just going to be the weight spaces of SL2. So these would correspond for matrices naught one, naught, naught, one minus one, naught, naught, and naught, naught, one, naught. And I actually have Laurent polynomial, which means everything sort of gets shifted by T if you go in this direction. So if we end up drawing a picture of this lee algebra, it kind of looks like this in some sense. So this represents a basis for the lee algebra. So each point represents a basis element. And this lee algebra is graded by Z squared in the way that I've indicated. And now what you can do is you can take all the positive roots, which are these things here and form a product over them. So the terms here are going to look like TZ, TZ to the minus one, T squared, T cubed Z, T cubed Z to the minus one and so on. So the product I get is one minus TZ times one minus TZ to the minus one times one minus T squared times one minus T cubed Z, one minus T cubed Z to the minus one and so on. So I get a term in the product for each of these points here. And if you work this out, it turns out to be one minus TZ minus TZ to the minus one plus T to the four Z squared plus T to the four Z to the minus two and so on, which is sum over all N of minus one to the N times T to the N squared Z to the N. So this is the famous Jacobi triple product identity. And if you're wondering why it's called a triple product when it's rather obviously an infinite product, not a triple product, the word triple comes from the fact that you've got these terms here and these terms here and these terms here. So that's three different sorts of factors in the product. And this is very exciting because this is more or less a theta function, which is a special sort of modular form, well, more or less. And it has large numbers of unexpected extra symmetries and turns up a lot in number theory when you're trying to do things like count the number of ways an integer can be written as a sum of squares. So this is typical of a denominator form of an affinely algebra. So more generally, you can replace SL2 by your favorite finite dimensional, the algebra. And you get generalizations of these called the McDonald's identities. McDonald actually discovered his identities before the relation with affinely algebras was known. He was trying to find an analog with a valid denominator formula for reflection groups. And at the time people hadn't quite figured out these were associated with the algebras and Victor Katz noticed very shortly afterwards that McDonald's identities were just the denominator formulas of these algebras. So the next example of a denominator formula is going to be related to the elliptic modular function. And I'm going to write J of tau minus 744 is sum of Cnq to the n. So this is q to minus one plus 196884q and so on. So I'm missing out the constant term. And what I'm going to do is I'm going to write down a Z squared graded algebra. And as usual, I'll draw a basis for it, except this time the root spaces of dimension greater than one in general. So the dimensions of the root spaces look like this. Everything else here is zero and all these things are zero. So mostly entries are zero except for the two quadrants. So in general, the entries in this quadrant are Cmn and the coefficient Cmn are getting very large, very rapidly. So this is a really huge Lie algebras. It's really enormous in this region and really enormous in this region and vanishes in this region and this region. So this Lie algebra is acted on by the monster group. So you remember the monster acted on a graded representation with these coefficients as the dimensions of various pieces. And from that vertex algebra, you can construct a Lie algebra, which turns out to look like this. And now we can ask, what does the denominator formula look like? Well, the denominator formula looks like this. What you do is you take a product over all the positive roots. And the positive roots are going to be these things here and I'm going to use m and n to indicate the coordinate axes. So what I now get is a product formula that looks like this. You take a product over m greater than zero and n in z, which is a product of all these entries here. We take one minus p to the m, q to the n and then we should raise this to the power of Cmn because we should have a factor for every basis element. And then we should multiply by p to the minus one. So this corresponds to the value e to the row in the vial denominator formula. And this is now equal to an alternating sum over the vial group, which is j of sigma minus j of tau. Because, well, this is actually a sum over the vial group. The vial group actually is order two, which is a bit surprising for such a huge Lie algebra or whatever. So this is really an alternating sum over a group of order two. And this is the elliptic modular function. And we notice just as for the affine Lie algebra, the denominator formula was a Jacobi theta function which satisfies some extra unexpected identities. This also satisfies some extra identities because it's invariant under sigma goes to minus one over sigma and tau goes to minus one over tau, which is not at all obvious if you just look at the Lie algebra. It's very difficult to understand this transformation in terms of the Lie algebra. So what this really is, you should think of this as being a modular function in two variables. If you've done Hilbert modular forms it's sort of like a Hilbert modular function, except it's a rather degenerate Hilbert modular function because you're working over q times q instead of an imaginary quadratic field. By the way, people sometimes get a bit worried because the right-hand side of this is anti-symmetric and the left-hand side isn't anti-symmetric. I should have said p is equal to e to the two pi i sigma and q is equal to e to the two pi i tau. Well, in fact, this side is anti-symmetric because one of the factors here is one minus p q to the minus one. And if you multiply this by the p to the minus one at the front, it becomes p to the minus one minus q to the minus one, which is indeed anti-symmetric and all the other terms are symmetric. So that's not really a problem. So you can also do similar things for all the other, the algebras I mentioned at the beginning of the talk. So let me just sort of summarize some known variations. First of all, we can have other the algebras associated to other groups. For example, if we take the baby monster, then this also acts on a Z2 graded Z2 graded Lie algebra whose the dimensions are given by the coefficients of q to the minus one plus four, three, seven, two, q plus nine, six, two, five, six, q squared, except the dimensions of the pieces of the Lie algebra are actually slightly more complicated than these coefficients. Some of them turn out to be Lie super algebras, which are likely algebras only they have sine errors in them, and this is useful because sometimes you get modular functions with a negative coefficients and it's rather hard to have a negative dimensional vector space, but you can easily have a negative dimensional super vector space because you count the super part of it as being negative dimensional. So you can get super algebras and vertex algebras actually on my various groups related to various other modular functions. And secondly, we can reduce mod p. So if you take one of these algebras v that is quite often defined over the integers and you might think you just reduce it mod p, well, you can, but that doesn't give you anything terribly exciting. A better way is to take the Tate co-homology. That's the same Tate co-homology that you get when you're doing class field theory. If you take the Tate co-homology of an element of say the monster with coefficients in say the monster, the algebra, and add first Tate co-homology, then this gives you a least super algebra acted on by the centralizer of v. So you get lots of natural super algebras over finite fields. There's a third thing you can do. You can just take restrictions. This says if you've got a group G contained in a group H and H acts on some nice algebraic structure that obviously G does as well. And this quite often gets rediscovered by people. For example, any symmetric group acts on a natural module whose dimensions are given by coefficients of modular forms. And you can embed any group into a symmetric group. And sometimes people get rather excited and say, that means any finite group has moonshine associated to it. Well, no, it doesn't really. As I said, graded representations of coefficients that are modular forms are not by themselves all that interesting because they're actually quite easy to construct using variations of this construction. So a lot of these, the algebras actually give rise to automorphic forms. So an example of something that's almost an automorphic form is this, this isn't quite an automorphic form because it's not actually holomorphic at CUSPs, but it's pretty close to being an automorphic form. So I'll give just one example of this. And this really is the simplest example. It's kind of funny that the simplest thing lives in 26 dimensions. And there's a le algebra graded by this lattice and its denominator formula looks like this. So this is a similar example of a le algebra. And this is a similar example of an automorphic form. So I'll give just one example of what the automorphic forms associated with these le algebras look like. And its denominator formula looks like this. Mine's e to the alpha to the p24 of one plus alpha squared over two. So this is number of partitions of a number into parts of 24 different colors. So it's sort of like the partition function only somewhat more exciting. And this is again equal to a sum over the vial group and sum over all integers of minus one to the epsilon of omega as in the vial denominator formula. And now you sum over tau n e to the omega n row where this is Ramanogen's famous tau function defined by q times product over one minus q to the n 24. We've got the sum of tau n q to the n. And what's interesting about this is that this is actually an automorphic form for the orthogonal group two comma 26 over the reals. That's not a misprint. You change one 25 to 26. What this means is there's a lot of unexpected extra symmetry. So this automorphic form is trivially invariant under the symmetry group of this lattice here but it turns out to be invariant under the symmetry group of a bigger lattice I 226 which makes it into an automorphic form for this orthogonal group. In general, what happens is the automorphic forms you get for the knownly algebras tend to be automorphic forms for orthogonal groups of the form two comma n plus two. So there are various special cases of it. Two comma one is locally isomorphic SL two. O two comma two is locally isomorphic SL two times SL two which gives you Hilbert modular forms. And we saw an example of this earlier when we had J sigma minus J tau which is a sort of modular function for this group here. O two comma three is isomorphic to SP four. So we get Ziegler modular forms of genus two. And in general, when if you've got an infinite product that is an automorphic form, it's an indication there's probably an algebra lurking around there somewhere in the background. For example, if we take the classical infinite product for the Dedekind Ata function and so on, this is sort of related to the Virasura algebra in physics. So if you see an automorphic form that's an infinite product you should at least think about the possibility there's a knee algebra somewhere in the background. So finally, I want to talk about unknown moonshine. So so far what I've been talking about moonshine for the monster and so on is reasonably well understood and that there aren't that many open questions we don't know how to answer. What I'm going to talk about now is umbral moonshine which is still unexplained as far as I know. So this was first observed by Iguchi and Iqami and generalized by Cheng, Duncan and Harvey. So I'll just say very briefly what they observed. So Iguchi and Iqami were looking at some modular functions that turned up in physics and they discovered this one. So these were its coefficients and so on. And these are the coefficients of a certain mock theta function. So mock theta functions were discovered by Ramanujan in the early 20th century. And for a long time there was no good definition of them. Mock theta functions were defined as the functions that Ramanujan said were mock theta functions and that was about it. About 10 or 20 years ago, Zavegas actually managed to figure out what these were. They're essentially harmonic mass waveforms. And this sort of revolutionized the theory of mock theta functions because you could finally study them systematically and actually know what they are. Anyway, this is a particular example of a mock theta function. And the surprising thing about it is that these numbers here are all dimensions of irreducible representations of the material group M24, which is one of the sporadic simple groups. This number isn't but it's equal to 3520 plus 10395. And these are dimensions of irreducible representations of M24. So it sort of suggests there should be a representation of M24 as a graded representation with correspondence of these coefficients. Well, there's one slight problem with that. This coefficient has a minus sign in it. Well, this doesn't completely rule things out. I mean, maybe you don't involve this coefficient for some reason or maybe there's a super algebra, but even if it's a super algebra, it seems very funny just having a one dimensional super piece and all the rest ordinary. So there's something a lot understood going on here. So the open question is, is there some algebraic structure? I mean, maybe a vertex algebra, the algebra, something more general that someone hasn't thought of underlying this. Well, of course, M24 isn't the only group you can play this with. So as I said, Cheng, Duncan and Harvey generalizes quite a lot and found variations of this corresponding to all neomyel lattices. So one of the neomyel lattices is more or less actual by M24 and corresponds to this case. And there are similar examples going on for other neomyel lattices. So there's another neomyel lattices associated M12 and there's a mock theta function associated M12 or rather lots of mock theta functions because you can take traces of elements of M12 on this. So we've got this completely open problem. We've got this, these mock theta functions that seem to be related to sporadic simple groups and nobody really knows why. There are also some examples related to some other groups like the O'nan group and the Rude-Vallis group which are even more mysterious. So this is just a completely open problem for research. I think I'll leave it at that.